Born-Oppenheimer approximation in an effective field theory language
Brambilla, Nora; Krein, Gastão; Tarrús Castellà, Jaume; Vairo, Antonio
2018-01-01
The Born-Oppenheimer approximation is the standard tool for the study of molecular systems. It is founded on the observation that the energy scale of the electron dynamics in a molecule is larger than that of the nuclei. A very similar physical picture can be used to describe QCD states containing heavy quarks as well as light-quarks or gluonic excitations. In this work, we derive the Born-Oppenheimer approximation for QED molecular systems in an effective field theory framework by sequentially integrating out degrees of freedom living at energies above the typical energy scale where the dynamics of the heavy degrees of freedom occurs. In particular, we compute the matching coefficients of the effective field theory for the case of the H2+ diatomic molecule that are relevant to compute its spectrum up to O (m α5). Ultrasoft photon loops contribute at this order, being ultimately responsible for the molecular Lamb shift. In the effective field theory the scaling of all the operators is homogeneous, which facilitates the determination of all the relevant contributions, an observation that may become useful for high-precision calculations. Using the above case as a guidance, we construct under some conditions an effective field theory for QCD states formed by a color-octet heavy quark-antiquark pair bound with a color-octet light-quark pair or excited gluonic state, highlighting the similarities and differences between the QED and QCD systems. Assuming that the multipole expansion is applicable, we construct the heavy-quark potential up to next-to-leading order in the multipole expansion in terms of nonperturbative matching coefficients to be obtained from lattice QCD.
DEFF Research Database (Denmark)
Tolstikhin, Oleg I.; Madsen, Lars Bojer
2013-01-01
We show that retardation in adjusting an electronic state to an instantaneous internuclear configuration caused by the finiteness of the electron’s velocity breaks the validity of the Born-Oppenheimer (BO) approximation at large electron-nuclei distances. This applies even to the ground state. As...... approximation and the asymptotic expansion are compared with numerical results for a one-dimensional three-body system modeling a diatomic molecule, with both electronic and nuclear motions treated exactly....
Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations
International Nuclear Information System (INIS)
Stottmeister, Alexander; Thiemann, Thomas
2016-01-01
This article, as the first of three, aims at establishing the (time-dependent) Born-Oppenheimer approximation, in the sense of space adiabatic perturbation theory, for quantum systems constructed by techniques of the loop quantum gravity framework, especially the canonical formulation of the latter. The analysis presented here fits into a rather general framework and offers a solution to the problem of applying the usual Born-Oppenheimer ansatz for molecular (or structurally analogous) systems to more general quantum systems (e.g., spin-orbit models) by means of space adiabatic perturbation theory. The proposed solution is applied to a simple, finite dimensional model of interacting spin systems, which serves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is explained how the content of this article and its companion affect the possible extraction of quantum field theory on curved spacetime from loop quantum gravity (including matter fields).
On the Mass of Atoms in Molecules: Beyond the Born-Oppenheimer Approximation
Directory of Open Access Journals (Sweden)
Arne Scherrer
2017-08-01
Full Text Available Describing the dynamics of nuclei in molecules requires a potential energy surface, which is traditionally provided by the Born-Oppenheimer or adiabatic approximation. However, we also need to assign masses to the nuclei. There, the Born-Oppenheimer picture does not account for the inertia of the electrons, and only bare nuclear masses are considered. Nowadays, experimental accuracy challenges the theoretical predictions of rotational and vibrational spectra and requires the participation of electrons in the internal motion of the molecule. More than 80 years after the original work of Born and Oppenheimer, this issue has still not been solved, in general. Here, we present a theoretical and numerical framework to address this problem in a general and rigorous way. Starting from the exact factorization of the electron-nuclear wave function, we include electronic effects beyond the Born-Oppenheimer regime in a perturbative way via position-dependent corrections to the bare nuclear masses. This maintains an adiabaticlike point of view: The nuclear degrees of freedom feel the presence of the electrons via a single potential energy surface, whereas the inertia of electrons is accounted for and the total mass of the system is recovered. This constitutes a general framework for describing the mass acquired by slow degrees of freedom due to the inertia of light, bounded particles; thus, it is applicable not only in electron-nuclear systems but in light-heavy nuclei or ions as well. We illustrate this idea with a model of proton transfer, where the light particle is the proton and the heavy particles are the oxygen atoms to which the proton is bounded. Inclusion of the light-particle inertia allows us to gain orders of magnitude in accuracy. The electron-nuclear perspective is adopted, instead, to calculate position-dependent mass corrections using density functional theory for a few polyatomic molecules at their equilibrium geometry. These data can
Cavity Born-Oppenheimer Approximation for Correlated Electron-Nuclear-Photon Systems.
Flick, Johannes; Appel, Heiko; Ruggenthaler, Michael; Rubio, Angel
2017-04-11
In this work, we illustrate the recently introduced concept of the cavity Born-Oppenheimer approximation [ Flick et al. PNAS 2017 , 10.1073/pnas.1615509114 ] for correlated electron-nuclear-photon problems in detail. We demonstrate how an expansion in terms of conditional electronic and photon-nuclear wave functions accurately describes eigenstates of strongly correlated light-matter systems. For a GaAs quantum ring model in resonance with a photon mode we highlight how the ground-state electronic potential-energy surface changes the usual harmonic potential of the free photon mode to a dressed mode with a double-well structure. This change is accompanied by a splitting of the electronic ground-state density. For a model where the photon mode is in resonance with a vibrational transition, we observe in the excited-state electronic potential-energy surface a splitting from a single minimum to a double minimum. Furthermore, for a time-dependent setup, we show how the dynamics in correlated light-matter systems can be understood in terms of population transfer between potential energy surfaces. This work at the interface of quantum chemistry and quantum optics paves the way for the full ab initio description of matter-photon systems.
Stanke, Monika; Adamowicz, Ludwik
2013-10-03
In this work, we describe how the energies obtained in molecular calculations performed without assuming the Born-Oppenheimer (BO) approximation can be augmented with corrections accounting for the leading relativistic effects. Unlike the conventional BO approach, where these effects only concern the relativistic interactions between the electrons, the non-BO approach also accounts for the relativistic effects due to the nuclei and due to the coupling of the coupled electron-nucleus motion. In the numerical sections, the results obtained with the two approaches are compared. The first comparison concerns the dissociation energies of the two-electron isotopologues of the H2 molecule, H2, HD, D2, T2, and the HeH(+) ion. The comparison shows that, as expected, the differences in the relativistic contributions obtained with the two approaches increase as the nuclei become lighter. The second comparison concerns the relativistic corrections to all 23 pure vibrational states of the HD(+) ion. An interesting charge asymmetry caused by the nonadiabatic electron-nucleus interaction appears in this system, and this effect significantly increases with the vibration excitation. The comparison of the non-BO results with the results obtained with the conventional BO approach, which in the lowest order does not describe the charge-asymmetry effect, reveals how this effect affects the values of the relativistic corrections.
International Nuclear Information System (INIS)
Bardo, R.D.; Wolfsberg, M.
1977-01-01
The wave equation for a nonlinear polyatomic molecule is formulated in molecule-fixed coordinates by a method originally due to Hirschfelder and Wigner. Application is made to a triatomic molecule, and the wave equation is explicitly presented in a useful molecule-fixed coordinate system. The formula for the adiabatic correction to the Born--Oppenheimer approximation for a triatomic molecule is obtained. The extension of the present formulation to larger polyatomic molecules is pointed out. Some terms in the triatomic molecule wave equation are discussed in detail
Li, Chen; Requist, Ryan; Gross, E K U
2018-02-28
We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length R = R c , where an avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the R-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical R c by 0.5 bohr. This effect is very accurately captured by a simple and rigorously derived correction, with an M -1 prefactor, to the exchange-correlation potential in density functional theory, M = reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wave function and conditional electronic density, ∇ R χ(R) and ∇ R n(r, R), it couples the Kohn-Sham equations at neighboring R points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation-an approximation that reduces the search for nonadiabatic density functionals to the search for a single function y(n).
Mass distribution of fission fragments within the Born-Oppenheimer approximation
Pomorski, K.; Ivanyuk, F. A.; Nerlo-Pomorska, B.
2017-03-01
The fission fragments mass-yield of 236 U is obtained by an approximate solution of the eigenvalue problem of the collective Hamiltonian that describes the dynamics of the fission process whose degrees of freedom are: the fission (elongation), the neck and mass-asymmetry modes. The macroscopic-microscopic method is used to evaluate the potential energy surface. The macroscopic energy part is calculated using the liquid drop model and the microscopic corrections are obtained using a Woods-Saxon single-particle levels. The four-dimensional modified Cassini ovals shape parametrization is used to describe the shape of the fissioning nucleus. The mass tensor is taken within a cranking-type approximation. The final fragment mass distribution is obtained by weighting the adiabatic density distribution in the collective space with the neck-dependent fission probability. The neck degree of freedom is found to play a significant role in determining the final fragment mass distribution.
International Nuclear Information System (INIS)
Zheng Renhui; Jing Yuanyuan; Chen Liping; Shi Qiang
2011-01-01
Graphical abstract: An analytically solvable model was employed to study proton coupled electron transfer reactions. Approximated theories are assessed, and vibrational coherence is observed in case of small reorganization energy. Research highlights: → The Duschinsky rotation effect in PCET reactions. → Assessment of the BO approx. for proton motion using an analytically solvable model. → Vibrational coherence in PCET in the case of small reorganization energy. - Abstract: By employing an analytically solvable model including the Duschinsky rotation effect, we investigated the applicability of the commonly used Born-Oppenheimer (BO) approximation for separating the proton and proton donor-acceptor motions in theories of proton coupled electron transfer (PCET) reactions. Comparison with theories based on the BO approximation shows that, the BO approximation for the proton coordinate is generally valid while some further approximations may become inaccurate in certain range of parameters. We have also investigated the effect of vibrationally coherent tunneling in the case of small reorganization energy, and shown that it plays an important role on the rate constant and kinetic isotope effect.
Energy conserving, linear scaling Born-Oppenheimer molecular dynamics.
Cawkwell, M J; Niklasson, Anders M N
2012-10-07
Born-Oppenheimer molecular dynamics simulations with long-term conservation of the total energy and a computational cost that scales linearly with system size have been obtained simultaneously. Linear scaling with a low pre-factor is achieved using density matrix purification with sparse matrix algebra and a numerical threshold on matrix elements. The extended Lagrangian Born-Oppenheimer molecular dynamics formalism [A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008)] yields microcanonical trajectories with the approximate forces obtained from the linear scaling method that exhibit no systematic drift over hundreds of picoseconds and which are indistinguishable from trajectories computed using exact forces.
Analysis of the Time Reversible Born-Oppenheimer Molecular Dynamics
Lin, Lin; Lu, Jianfeng; Shao, Sihong
2013-01-01
We analyze the time reversible Born-Oppenheimer molecular dynamics (TRBOMD) scheme, which preserves the time reversibility of the Born-Oppenheimer molecular dynamics even with non-convergent self-consistent field iteration. In the linear response regime, we derive the stability condition as well as the accuracy of TRBOMD for computing physical properties such as the phonon frequency obtained from the molecular dynamic simulation. We connect and compare TRBOMD with the Car-Parrinello molecular...
Sirjoosingh, Andrew Rajendra
Nuclear quantum effects such as zero-point energy and hydrogen tunneling play an important role in a wide variety of chemical reactions. Moreover, non-Born-Oppenheimer effects are important in reactions such as proton-coupled electron transfer (PCET), which are integral to various electrocatalytic applications and bioenzymatic processes. The breakdown of the Born-Oppenheimer approximation between electronic and nuclear motions engenders the need for accurate characterization of the degree of nonadiabaticity. Furthermore, in regimes where the inclusion of these effects is vital, as it is for PCET systems, the development of non-Born-Oppenheimer quantum chemical methods is increasingly important. In this dissertation, we present diagnostics of electron-proton nonadiabaticity that can be obtained from standard electronic structure calculations and describe their application to representative systems, highlighting the mechanistic differences between two subclasses of PCET. In addition, we describe the development of new electronic structure methods within the nuclear-electronic orbital (NEO) framework, which is an orbital-based approach that inherently includes electron-proton nonadiabaticity by treating electrons and select protons quantum mechanically on equal footing. Previous studies using NEO involved applying mean-field-based approaches, which lacked sufficient electron-proton dynamical correlation, leading to overlocalized nuclear densities. Subsequent efforts focused on the development of explicitly correlated NEO approaches which, although accurate, were too computationally intractable to be practical for the study of PCET systems. In this dissertation, we describe two approaches to develop tractable NEO methods. Firstly, we describe the formulation of a multi-component density functional theory approach within the NEO framework, which involves the derivation of several electron-proton correlation functionals to accurately account for electron
Born-Oppenheimer potential energy for interaction of antihydrogen with molecular hydrogen
International Nuclear Information System (INIS)
Strasburger, Krzysztof
2005-01-01
Inelastic collisions with hydrogen molecules are claimed to be an important channel of antihydrogen Hbar losses (Armour and Zeman 1999 Int. J. Quantum Chem. 74 645). In the present work, interaction energies for the H 2 -Hbar system in the ground state have been calculated within the Born-Oppenheimer approximation. The leptonic problem was solved variationally with the basis of explicitly correlated Gaussian functions. The geometry of H 2 was fixed at equilibrium geometry and the Hbar atom approached the molecule from two directions-along or perpendicularly to the bond axis. Purely attractive potential energy curve has been obtained for the first nuclear configuration, while a local maximum (lower than the energy at infinite separation) has been found for the second one
Born Oppenheimer potential energy for interaction of antihydrogen with molecular hydrogen
Strasburger, Krzysztof
2005-09-01
Inelastic collisions with hydrogen molecules are claimed to be an important channel of antihydrogen (\\overlineH) losses (Armour and Zeman 1999 Int. J. Quantum Chem. 74 645). In the present work, interaction energies for the H_{2}\\--\\overlineH system in the ground state have been calculated within the Born-Oppenheimer approximation. The leptonic problem was solved variationally with the basis of explicitly correlated Gaussian functions. The geometry of H2 was fixed at equilibrium geometry and the \\overlineH atom approached the molecule from two directions—along or perpendicularly to the bond axis. Purely attractive potential energy curve has been obtained for the first nuclear configuration, while a local maximum (lower than the energy at infinite separation) has been found for the second one.
Wave function continuity and the diagonal Born-Oppenheimer correction at conical intersections
Energy Technology Data Exchange (ETDEWEB)
Meek, Garrett A.; Levine, Benjamin G., E-mail: levine@chemistry.msu.edu [Department of Chemistry, Michigan State University, East Lansing, Michigan 48824 (United States)
2016-05-14
We demonstrate that though exact in principle, the expansion of the total molecular wave function as a sum over adiabatic Born-Oppenheimer (BO) vibronic states makes inclusion of the second-derivative nonadiabatic energy term near conical intersections practically problematic. In order to construct a well-behaved molecular wave function that has density at a conical intersection, the individual BO vibronic states in the summation must be discontinuous. When the second-derivative nonadiabatic terms are added to the Hamiltonian, singularities in the diagonal BO corrections (DBOCs) of the individual BO states arise from these discontinuities. In contrast to the well-known singularities in the first-derivative couplings at conical intersections, these singularities are non-integrable, resulting in undefined DBOC matrix elements. Though these singularities suggest that the exact molecular wave function may not have density at the conical intersection point, there is no physical basis for this constraint. Instead, the singularities are artifacts of the chosen basis of discontinuous functions. We also demonstrate that continuity of the total molecular wave function does not require continuity of the individual adiabatic nuclear wave functions. We classify nonadiabatic molecular dynamics methods according to the constraints placed on wave function continuity and analyze their formal properties. Based on our analysis, it is recommended that the DBOC be neglected when employing mixed quantum-classical methods and certain approximate quantum dynamical methods in the adiabatic representation.
International Nuclear Information System (INIS)
Huo, W.M.
1977-01-01
The convergence and range of applicability of the Bonham series representation of the Born--Oppenheimer exchange amplitude is investigated. Numerical calculations on 1 1 S→2 3 S and 1 1 S→2 3 P of He by electron impact demonstrate that the first three terms of the Bonham series can provide an adequate representation of the Born--Oppenheimer amplitude from high energies down to near threshold. The three-term Bonham series is then used to represent the Hartree--Fock exchange kernel in momentum space, which has been shown by Lassettre to be proportional to the Born--Oppenheimer amplitude. Inverse Fourier transform, plus an additional approximation of replacing the momentum which appears as an expansion parameter in the Bonham series by its averge value, gives us a local exchange potential. If the averge momentum in the Thomas--Fermi model is used, the first term of the local exchange potential is just the electron gas exchange potential. A second term corresponding to a correction for the inhomogeneity in the electron gas density, is also obtained. No adjustable parameter is involved. Exchange energies of He, Be, and Ne calculated using the local exchange potential agree much better with the Hartree--Fock results than the electron gas model
The structure of the second-order non-Born-Oppenheimer density matriz D2:
Ludena, Eduardo; Iza, Peter; Aray, Yosslen; Cornejo, Mauricio; Zambrano, Dik
Properties of the non-Born-Oppenheimer 2-matrix are examined. Using a coordinate system formed by internal translationally invariant plus the total center-of-mass coordinates it is shown that regardless of the point of reference selected, the operator for the reduced second order density matrix, 2-RDM, solely depends upon the translationally invariant internal coordinates. We apply this result to examine the nature of the 2-RDM extracted from the exact analytical solutions for model non-Born-Oppenheimer four-particle systems of the Coulomb-Hooke and Moshinsky types. We obtain for both these models explicit closed-form analytic expressions for the electron and nuclear 2-RDM. An explicit expression is also obtained for the electron-nuclear 2-RDM in the Moshinsky case, which shows coupling between the electron and nuclear coordinates. EVL and YA acknowledge support of SENESCYT's Prometheus Program.
Borkowski, Mateusz; Buchachenko, Alexei A.; Ciuryło, Roman; Julienne, Paul S.; Yamada, Hirotaka; Kikuchi, Yuu; Takahashi, Kakeru; Takasu, Yosuke; Takahashi, Yoshiro
2017-12-01
We present high-resolution two-color photoassociation spectroscopy of Bose-Einstein condensates of ytterbium atoms. The use of narrow Raman resonances and careful examination of systematic shifts enabled us to measure 13 bound-state energies for three isotopologues of the ground-state ytterbium molecule with standard uncertainties of the order of 500 Hz. The atomic interactions are modeled using an ab initio based mass-scaled Born-Oppenheimer potential whose long-range van der Waals parameters and total WKB phase are fitted to experimental data. We find that the quality of the fit of this model, of about 112.9 kHz (rms) can be significantly improved by adding the recently calculated beyond-Born-Oppenheimer (BBO) adiabatic corrections [J. J. Lutz and J. M. Hutson, J. Mol. Spectrosc. 330, 43 (2016), 10.1016/j.jms.2016.08.007] and by partially treating the nonadiabatic effects using distance-dependent reduced masses. Our BBO interaction model represents the experimental data to within about 30.2 kHz on average, which is 3.7 times better than the "reference" Born-Oppenheimer model. We calculate the s -wave scattering lengths for bosonic isotopic pairs of ytterbium atoms with error bars over two orders of magnitude smaller than previous determinations. For example, the s -wave scattering length for 174Yb is +5.55812 (50 ) nm.
McKemmish, Laura K; McKenzie, Ross H; Hush, Noel S; Reimers, Jeffrey R
2015-10-14
Entanglement is sometimes regarded as the quintessential measure of the quantum nature of a system and its significance for the understanding of coupled electronic and vibrational motions in molecules has been conjectured. Previously, we considered the entanglement developed in a spatially localized diabatic basis representation of the electronic states, considering design rules for qubits in a low-temperature chemical quantum computer. We extend this to consider the entanglement developed during high-energy processes. We also consider the entanglement developed using adiabatic electronic basis, providing a novel way for interpreting effects of the breakdown of the Born-Oppenheimer (BO) approximation. We consider: (i) BO entanglement in the ground-state wavefunction relevant to equilibrium thermodynamics, (ii) BO entanglement associated with low-energy wavefunctions relevant to infrared and tunneling spectroscopies, (iii) BO entanglement in high-energy eigenfunctions relevant to chemical reaction processes, and (iv) BO entanglement developed during reactive wavepacket dynamics. A two-state single-mode diabatic model descriptive of a wide range of chemical phenomena is used for this purpose. The entanglement developed by BO breakdown correlates simply with the diameter of the cusp introduced by the BO approximation, and a hierarchy appears between the various BO-breakdown correction terms, with the first-derivative correction being more important than the second-derivative correction which is more important than the diagonal correction. This simplicity is in contrast to the complexity of BO-breakdown effects on thermodynamic, spectroscopic, and kinetic properties. Further, processes poorly treated at the BO level that appear adequately treated using the Born-Huang adiabatic approximation are found to have properties that can only be described using a non-adiabatic description. For the entanglement developed between diabatic electronic states and the nuclear motion
Cawkwell, M J; Niklasson, Anders M N; Dattelbaum, Dana M
2015-02-14
The initial chemical events that occur during the shock compression of liquid phenylacetylene have been investigated using self-consistent tight binding molecular dynamics simulations. The extended Lagrangian Born-Oppenheimer molecular dynamics formalism enabled us to compute microcanonical trajectories with precise conservation of the total energy. Our simulations revealed that the first density-increasing step under shock compression arises from the polymerization of phenylacetylene molecules at the acetylene moiety. The application of electronic structure-based molecular dynamics with long-term conservation of the total energy enabled us to identify electronic signatures of reactivity via monitoring changes in the HOMO-LUMO gap, and to capture directly adiabatic shock heating, transient non-equilibrium states, and changes in temperature arising from exothermic chemistry in classical molecular dynamics trajectories.
On the inclusion of the diagonal Born-Oppenheimer correction in surface hopping methods
Energy Technology Data Exchange (ETDEWEB)
Gherib, Rami; Ryabinkin, Ilya G.; Izmaylov, Artur F. [Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4 (Canada); Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6 (Canada); Ye, Liyuan [Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4 (Canada)
2016-04-21
The diagonal Born-Oppenheimer correction (DBOC) stems from the diagonal second derivative coupling term in the adiabatic representation, and it can have an arbitrary large magnitude when a gap between neighbouring Born-Oppenheimer (BO) potential energy surfaces (PESs) is closing. Nevertheless, DBOC is typically neglected in mixed quantum-classical methods of simulating nonadiabatic dynamics (e.g., fewest-switch surface hopping (FSSH) method). A straightforward addition of DBOC to BO PESs in the FSSH method, FSSH+D, has been shown to lead to numerically much inferior results for models containing conical intersections. More sophisticated variation of the DBOC inclusion, phase-space surface-hopping (PSSH) was more successful than FSSH+D but on model problems without conical intersections. This work comprehensively assesses the role of DBOC in nonadiabatic dynamics of two electronic state problems and the performance of FSSH, FSSH+D, and PSSH methods in variety of one- and two-dimensional models. Our results show that the inclusion of DBOC can enhance the accuracy of surface hopping simulations when two conditions are simultaneously satisfied: (1) nuclei have kinetic energy lower than DBOC and (2) PESs are not strongly nonadiabatically coupled. The inclusion of DBOC is detrimental in situations where its energy scale becomes very high or even diverges, because in these regions PESs are also very strongly coupled. In this case, the true quantum formalism heavily relies on an interplay between diagonal and off-diagonal nonadiabatic couplings while surface hopping approaches treat diagonal terms as PESs and off-diagonal ones stochastically.
Curl condition for a four-state Born-Oppenheimer system employing the Mathieu equation.
Sarkar, Biplab; Adhikari, Satrajit
2008-10-09
When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.
Reimers, Jeffrey R; McKemmish, Laura K; McKenzie, Ross H; Hush, Noel S
2015-10-14
Using a simple model Hamiltonian, the three correction terms for Born-Oppenheimer (BO) breakdown, the adiabatic diagonal correction (DC), the first-derivative momentum non-adiabatic correction (FD), and the second-derivative kinetic-energy non-adiabatic correction (SD), are shown to all contribute to thermodynamic and spectroscopic properties as well as to thermal non-diabatic chemical reaction rates. While DC often accounts for >80% of thermodynamic and spectroscopic property changes, the commonly used practice of including only the FD correction in kinetics calculations is rarely found to be adequate. For electron-transfer reactions not in the inverted region, the common physical picture that diabatic processes occur because of surface hopping at the transition state is proven inadequate as the DC acts first to block access, increasing the transition state energy by (ℏω)(2)λ/16J(2) (where λ is the reorganization energy, J the electronic coupling and ω the vibration frequency). However, the rate constant in the weakly-coupled Golden-Rule limit is identified as being only inversely proportional to this change rather than exponentially damped, owing to the effects of tunneling and surface hopping. Such weakly-coupled long-range electron-transfer processes should therefore not be described as "non-adiabatic" processes as they are easily described by Born-Huang ground-state adiabatic surfaces made by adding the DC to the BO surfaces; instead, they should be called just "non-Born-Oppenheimer" processes. The model system studied consists of two diabatic harmonic potential-energy surfaces coupled linearly through a single vibration, the "two-site Holstein model". Analytical expressions are derived for the BO breakdown terms, and the model is solved over a large parameter space focusing on both the lowest-energy spectroscopic transitions and the quantum dynamics of coherent-state wavepackets. BO breakdown is investigated pertinent to: ammonia inversion, aromaticity
Born-Oppenheimer molecular dynamics simulation of pentanoic acid adsorption on α-Al2O3
Martinotto, André L.; Zorzi, Janete E.; Perottoni, Cláudio A.
2017-11-01
Adsorption of a single pentanoic acid (C5H10O2) molecule on (0001) α-Al2O3 in a vacuum was explored with the aid of Born-Oppenheimer molecular dynamics simulations. Computer simulations were carried out considering two different situations, namely a clean Al/O-terminated surface and, also, a (0001) α-Al2O3 surface saturated with doubly-coordinated, isolated hydroxyls. In the first case, pentanoic acid adsorbs dissociatively, with the creation of an isolated surface hydroxyl, while the oxygen from the molecule's former carbonyl makes a bond to a nearby surface Al. On the other hand, pentanoic acid adsorbs on hydroxylated alumina by making a strong hydrogen bond to a surface oxygen, with the molecule aligning itself nearly parallel to the surface after full relaxation. For each case (i.e., pentanoic acid adsorption on Al/O-terminated or hydroxylated corundum surface), the different adsorption mechanism has a marked impact on the respective calculated infrared absorption spectrum, which can be of further use as an analytical tool to determine the underlying adsorption mechanism in actual experiments.
Heshmat, Mojgan; Privalov, Timofei
2017-09-01
Using Born-Oppenheimer molecular dynamics (BOMD), we explore the nature of interactions between H2 and the activated carbonyl carbon, C(carbonyl), of the acetone-B(C6F5)3 adduct surrounded by an explicit solvent (1,4-dioxane). BOMD simulations at finite (non-zero) temperature with an explicit solvent produced long-lasting instances of significant vibrational perturbation of the H—H bond and H2-polarization at C(carbonyl). As far as the characteristics of H2 are concerned, the dynamical transient state approximates the transition-state of the heterolytic H2-cleavage. The culprit is the concerted interactions of H2 with C(carbonyl) and a number of Lewis basic solvent molecules—i.e., the concerted C(carbonyl)⋯H2⋯solvent interactions. On one hand, the results presented herein complement the mechanistic insight gained from our recent transition-state calculations, reported separately from this article. But on the other hand, we now indicate that an idea of the sufficiency of just one simple reaction coordinate in solution-phase reactions can be too simplistic and misleading. This article goes in the footsteps of the rapidly strengthening approach of investigating molecular interactions in large molecular systems via "computational experimentation" employing, primarily, ab initio molecular dynamics describing reactants-interaction without constraints of the preordained reaction coordinate and/or foreknowledge of the sampling order parameters.
Ludeña, E V; Echevarría, L; Lopez, X; Ugalde, J M
2012-02-28
We consider the calculation of non-Born-Oppenheimer, nBO, one-particle densities for both electrons and nuclei. We show that the nBO one-particle densities evaluated in terms of translationally invariant coordinates are independent of the wavefunction describing the motion of center of mass of the whole system. We show that they depend, however, on an arbitrary reference point from which the positions of the vectors labeling the particles are determined. We examine the effect that this arbitrary choice has on the topology of the one-particle density by selecting the Hooke-Calogero model of a three-body system for which expressions for the one-particle densities can be readily obtained in analytic form. We extend this analysis to the one-particle densities obtained from full Coulomb interaction wavefunctions for three-body systems. We conclude, in view of the fact that there is a close link between the choice of the reference point and the topology of one-particle densities that the molecular structure inferred from the topology of these densities is not unique. We analyze the behavior of one-particle densities for the Hooke-Calogero Born-Oppenheimer, BO, wavefunction and show that topological transitions are also present in this case for a particular mass value of the light particles even though in the BO regime the nuclear masses are infinite. In this vein, we argue that the change in topology caused by variation of the mass ratio between light and heavy particles does not constitute a true indication in the nBO regime of the emergence of molecular structure.
Andermatt, Samuel; Cha, Jinwoong; Schiffmann, Florian; VandeVondele, Joost
2016-07-12
In this work, methods for the efficient simulation of large systems embedded in a molecular environment are presented. These methods combine linear-scaling (LS) Kohn-Sham (KS) density functional theory (DFT) with subsystem (SS) DFT. LS DFT is efficient for large subsystems, while SS DFT is linear scaling with a smaller prefactor for large sets of small molecules. The combination of SS and LS, which is an embedding approach, can result in a 10-fold speedup over a pure LS simulation for large systems in aqueous solution. In addition to a ground-state Born-Oppenheimer SS+LS implementation, a time-dependent density functional theory-based Ehrenfest molecular dynamics (EMD) using density matrix propagation is presented that allows for performing nonadiabatic dynamics. Density matrix-based EMD in the SS framework is naturally linear scaling and appears suitable to study the electronic dynamics of molecules in solution. In the LS framework, linear scaling results as long as the density matrix remains sparse during time propagation. However, we generally find a less than exponential decay of the density matrix after a sufficiently long EMD run, preventing LS EMD simulations with arbitrary accuracy. The methods are tested on various systems, including spectroscopy on dyes, the electronic structure of TiO2 nanoparticles, electronic transport in carbon nanotubes, and the satellite tobacco mosaic virus in explicit solution.
Coutinho, Nayara D; Aquilanti, Vincenzo; Silva, Valter H C; Camargo, Ademir J; Mundim, Kleber C; de Oliveira, Heibbe C B
2016-07-14
Among four-atom processes, the reaction OH + HBr → H2O + Br is one of the most studied experimentally: its kinetics has manifested an unusual anti-Arrhenius behavior, namely, a marked decrease of the rate constant as the temperature increases, which has intrigued theoreticians for a long time. Recently, salient features of the potential energy surface have been characterized and most kinetic aspects can be considered as satisfactorily reproduced by classical trajectory simulations. Motivation of the work reported in this paper is the investigation of the stereodirectional dynamics of this reaction as the prominent reason for the peculiar kinetics: we started in a previous Letter ( J. Phys. Chem. Lett. 2015 , 6 , 1553 - 1558 ) a first-principles Born-Oppenheimer "canonical" molecular dynamics approach. Trajectories are step-by-step generated on a potential energy surface quantum mechanically calculated on-the-fly and are thermostatically equilibrated to correspond to a specific temperature. Here, refinements of the method permitted a major increase of the number of trajectories and the consideration of four temperatures -50, +200, +350, and +500 K, for which the sampling of initial conditions allowed us to characterize the stereodynamical effect. The role is documented of the adjustment of the reactants' mutual orientation to encounter the entrance into the "cone of acceptance" for reactivity. The aperture angle of this cone is dictated by a range of directions of approach compatible with the formation of the specific HOH angle of the product water molecule; and consistently the adjustment is progressively less effective the higher the kinetic energy. Qualitatively, this emerging picture corroborates experiments on this reaction, involving collisions of aligned and oriented molecular beams, and covering a range of energies higher than the thermal ones. The extraction of thermal rate constants from this molecular dynamics approach is discussed and the systematic
Liévin, J; Demaison, J; Herman, M; Fayt, A; Puzzarini, C
2011-02-14
The equilibrium structure of acetylene (also named ethyne) has been reinvestigated to resolve the small discrepancies noted between different determinations. The size of the system as well as the large amount of available experimental data provides the quite unique opportunity to check the magnitude and relevance of various contributions to equilibrium structure as well as to verify the accuracy of experimental results. With respect to pure theoretical investigation, quantum-chemical calculations at the coupled-cluster level have been employed together with extrapolation to the basis set limit, consideration of higher excitations in the cluster operator, inclusion of core correlation effects as well as relativistic and diagonal Born-Oppenheimer corrections. In particular, it is found that the extrapolation to the complete basis set limit, the inclusion of higher excitations in the electronic-correlation treatment and the relativistic corrections are of the same order of magnitude. It also appears that a basis set as large as a core-valence quintuple-zeta set is required for accurately accounting for the inner-shell correlation contribution. From a pure experimental point of view, the equilibrium structure has been determined using very accurate rotational constants recently obtained by a "global analysis" (that is to say that all non-negligible interactions are explicitly included in the Hamiltonian matrix) of rovibrational spectra. Finally, a semi-experimental equilibrium structure (where the equilibrium rotational constants are obtained from the experimental ground state rotational constants and computed rovibrational corrections) has been obtained from the available experimental ground-state rotational constants for ten isotopic species corrected for computed vibrational corrections. Such a determination led to the revision of the ground-state rotational constants of two isotopologues, thus showing that structural determination is a good method to identify
Czech Academy of Sciences Publication Activity Database
Veis, Libor; Višňák, Jakub; Nishizawa, H.; Nakai, H.; Pittner, Jiří
2016-01-01
Roč. 116, č. 18 (2016), s. 1328-1336 ISSN 0020-7608 R&D Projects: GA ČR GA203/08/0626 Institutional support: RVO:61388955 Keywords : Born-Oppenheimer approximation * nuclear orbital plus molecular orbital method * phase estimation Subject RIV: CF - Physical ; Theoretical Chemistry Impact factor: 2.920, year: 2016
Pu, Maoping; Heshmat, Mojgan; Privalov, Timofei
2017-07-01
Using Born-Oppenheimer molecular dynamics (BOMD) with density functional theory, transition-state (TS) calculations, and the quantitative energy decomposition analysis (EDA), we examined the mechanism of H2-liberation from LB—H(+) + (-)H—LA ion-pair, 1, in which the Lewis base (LB) is (o-C6H4Me)3P and the Lewis acid (LA) is B(p-C6F4H)3. BOMD simulations indicate that the path of H2 liberation from the ion-pair 1 goes via the short-lived transient species, LB⋯H2⋯LA, which are structurally reminiscent of the TS-structure in the minimum-energy-path describing the reversible reaction between H2 and (o-C6H4Me)3P/B(p-C6F4H)3 frustrated Lewis pair (FLP). With electronic structure calculations performed on graphics processing units, our BOMD data-set covers more than 1 ns of evolution of the ion-pair 1 at temperature T ≈ 400 K. BOMD simulations produced H2-recombination events with various durations of H2 remaining fully recombined as a molecule within a LB/LA attractive "pocket"—from very short vibrational-time scale to time scales in the range of a few hundred femtoseconds. With the help of perturbational approach to trajectory-propagation over a saddle-area, we directly examined dynamics of H2-liberation. Using EDA, we elucidated interactions between the cationic and anionic fragments in the ion-pair 1 and between the molecular fragments in the TS-structure. We have also considered a model that qualitatively takes into account the potential energy characteristics of H—H recombination and H2-release plus inertia of molecular motion of the (o-C6H4Me)3P/B(p-C6F4H)3 FLP.
Bond selective chemistry beyond the adiabatic approximation
Energy Technology Data Exchange (ETDEWEB)
Butler, L.J. [Univ. of Chicago, IL (United States)
1993-12-01
One of the most important challenges in chemistry is to develop predictive ability for the branching between energetically allowed chemical reaction pathways. Such predictive capability, coupled with a fundamental understanding of the important molecular interactions, is essential to the development and utilization of new fuels and the design of efficient combustion processes. Existing transition state and exact quantum theories successfully predict the branching between available product channels for systems in which each reaction coordinate can be adequately described by different paths along a single adiabatic potential energy surface. In particular, unimolecular dissociation following thermal, infrared multiphoton, or overtone excitation in the ground state yields a branching between energetically allowed product channels which can be successfully predicted by the application of statistical theories, i.e. the weakest bond breaks. (The predictions are particularly good for competing reactions in which when there is no saddle point along the reaction coordinates, as in simple bond fission reactions.) The predicted lack of bond selectivity results from the assumption of rapid internal vibrational energy redistribution and the implicit use of a single adiabatic Born-Oppenheimer potential energy surface for the reaction. However, the adiabatic approximation is not valid for the reaction of a wide variety of energetic materials and organic fuels; coupling between the electronic states of the reacting species play a a key role in determining the selectivity of the chemical reactions induced. The work described below investigated the central role played by coupling between electronic states in polyatomic molecules in determining the selective branching between energetically allowed fragmentation pathways in two key systems.
Energy Technology Data Exchange (ETDEWEB)
Ramakrishnan, Raghunathan [Institute of Physical Chemistry and National Center for Computational Design and Discovery of Novel Materials (MARVEL), Department of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel (Switzerland); Rauhut, Guntram, E-mail: rauhut@theochem.uni-stuttgart.de [Institute for Theoretical Chemistry, University of Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart (Germany)
2015-04-21
Semi-quartic force fields (QFF) rely on a Taylor-expansion of the multi-dimensional Born-Oppenheimer potential energy surface (PES) and are frequently used within the calculation of anharmonic vibrational frequencies based on 2nd order vibrational perturbation theory (VPT2). As such they are usually determined by differentiation of the electronic energy with respect to the nuclear coordinates. Alternatively, potential energy surfaces can be expanded in terms of multi-mode expansions, which typically do not require any derivative techniques. The computational effort to retrieve QFF from size-reduced multi-mode expansions has been studied and has been compared with standard Taylor-expansions. As multi-mode expansions allow for the convenient introduction of subtle approximations, these will be discussed in some detail. In addition, a preliminary study about the applicability of a generalized Duschinsky transformation to QFFs is provided. This transformation allows for the efficient evaluation of VPT2 frequencies of isotopologues from the PES of the parent compound and thus avoids the recalculation of PESs in different axes systems.
International Nuclear Information System (INIS)
Ramakrishnan, Raghunathan; Rauhut, Guntram
2015-01-01
Semi-quartic force fields (QFF) rely on a Taylor-expansion of the multi-dimensional Born-Oppenheimer potential energy surface (PES) and are frequently used within the calculation of anharmonic vibrational frequencies based on 2nd order vibrational perturbation theory (VPT2). As such they are usually determined by differentiation of the electronic energy with respect to the nuclear coordinates. Alternatively, potential energy surfaces can be expanded in terms of multi-mode expansions, which typically do not require any derivative techniques. The computational effort to retrieve QFF from size-reduced multi-mode expansions has been studied and has been compared with standard Taylor-expansions. As multi-mode expansions allow for the convenient introduction of subtle approximations, these will be discussed in some detail. In addition, a preliminary study about the applicability of a generalized Duschinsky transformation to QFFs is provided. This transformation allows for the efficient evaluation of VPT2 frequencies of isotopologues from the PES of the parent compound and thus avoids the recalculation of PESs in different axes systems
Energy Technology Data Exchange (ETDEWEB)
Bjorgaard, J. A., E-mail: jbjorgaard@lanl.gov [Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Velizhanin, K. A. [Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Tretiak, S., E-mail: serg@lanl.gov [Center for Integrated Nanotechnologies, Center for Nonlinear Studies, and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2016-04-21
The effects of solvent on molecular processes such as excited state relaxation and photochemical reaction often occurs in a nonequilibrium regime. Dynamic processes such as these can be simulated using excited state molecular dynamics. In this work, we describe methods of simulating nonequilibrium solvent effects in excited state molecular dynamics using linear-response time-dependent density functional theory and apparent surface charge methods. These developments include a propagation method for solvent degrees of freedom and analytical energy gradients for the calculation of forces. Molecular dynamics of acetaldehyde in water or acetonitrile are demonstrated where the solute-solvent system is out of equilibrium due to photoexcitation and emission.
Indian Academy of Sciences (India)
IAS Admin
V S Borkar is the Institute. Chair Professor of. Electrical Engineering at. IIT Bombay. His research interests are stochastic optimization, theory, algorithms and applica- tions. 1 'Markov Chain Monte Carlo' is another one (see [1]), not to mention schemes that combine both. Stochastic approximation is one of the unsung.
Mukherjee, Soumya; Mukherjee, Bijit; Adhikari, Satrajit
2017-08-24
We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of non-adiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wave functions. ADT matrices eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrödinger equation to its diabatic form. Two real molecular systems NO 3 and C 6 H 6 + (Bz + ) are selectively chosen for the demonstration of workability of those equations. We examine the NACTs among the lowest five electronic states of the NO 3 radical [X̃ 2 A 2 ' (1 2 B 2 ), Ã 2 E″ (1 2 A 2 and 1 2 B 1 ) and B̃ 2 E' (1 2 A 1 and 2 2 B 2 )], in which all types of non-adiabatic interactions, that is, Jahn-Teller (JT) interactions, Pseudo Jahn-Teller (PJT) interactions, and accidental conical intersections (CIs) are present. On the other hand, lowest five electronic states of Bz + [X̃ 2 E 1g (1 2 B 3g and 1 2 B 2g ), B̃ 2 E 2g (1 2 A g and 1 2 B 1g ), and C̃ 2 A 2u (1 2 B 1u )] depict similar kind of complex feature of non-adiabatic effects. For NO 3 radical, the two components of degenerate in-plane asymmetric stretching mode are taken as a plane of nuclear configuration space (CS), whereas in case of Bz + , two pairs are chosen: One is the pair of components of degenerate in-plane asymmetric stretching mode, and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asymmetric stretching modes. We calculate ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chemistry package. Subsequently, the ADT is performed using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, symmetric, and continuous diabatic PESs for both the molecular systems.
CERN. Geneva
2015-01-01
Most physics results at the LHC end in a likelihood ratio test. This includes discovery and exclusion for searches as well as mass, cross-section, and coupling measurements. The use of Machine Learning (multivariate) algorithms in HEP is mainly restricted to searches, which can be reduced to classification between two fixed distributions: signal vs. background. I will show how we can extend the use of ML classifiers to distributions parameterized by physical quantities like masses and couplings as well as nuisance parameters associated to systematic uncertainties. This allows for one to approximate the likelihood ratio while still using a high dimensional feature vector for the data. Both the MEM and ABC approaches mentioned above aim to provide inference on model parameters (like cross-sections, masses, couplings, etc.). ABC is fundamentally tied Bayesian inference and focuses on the “likelihood free” setting where only a simulator is available and one cannot directly compute the likelihood for the dat...
The off-resonant aspects of decoherence and a critique of the two-level approximation
International Nuclear Information System (INIS)
Savran, Kerim; Hakioglu, T; Mese, E; Sevincli, Haldun
2006-01-01
exponential independently from the number of levels. The latter and the coupling strengths play the major role in the quantitative values of the rates and the rates are independent of the other spectral parameters. The validity of the presented results is restricted only by their reliance on the Born-Oppenheimer approximation. This approximation is strongly dependent on the external observational time and its reliability depends on an additional timescale. In the rest of the work, the crossover between the short and the long time behaviour of the density matrix of the multilevelled system is examined using an intuitive argument. It is shown that the Born approximation weakens as the resonant couplings become more effective at long times. This implies that, in calculations made with this approximation in the long time regime, a need for a justification arises for the reliability of the results. This justification is made for the present work
Diophantine approximation and badly approximable sets
DEFF Research Database (Denmark)
Kristensen, S.; Thorn, R.; Velani, S.
2006-01-01
Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X. The clas......Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega which `stay clear' of some given set of points in X....... The classical set Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Omega have full Hausdorff dimension...
International Nuclear Information System (INIS)
Ginsburg, C.A.
1980-01-01
In many problems, a desired property A of a function f(x) is determined by the behaviour of f(x) approximately equal to g(x,A) as x→xsup(*). In this letter, a method for resuming the power series in x of f(x) and approximating A (modulated Pade approximant) is presented. This new approximant is an extension of a resumation method for f(x) in terms of rational functions. (author)
Approximate iterative algorithms
Almudevar, Anthony Louis
2014-01-01
Iterative algorithms often rely on approximate evaluation techniques, which may include statistical estimation, computer simulation or functional approximation. This volume presents methods for the study of approximate iterative algorithms, providing tools for the derivation of error bounds and convergence rates, and for the optimal design of such algorithms. Techniques of functional analysis are used to derive analytical relationships between approximation methods and convergence properties for general classes of algorithms. This work provides the necessary background in functional analysis a
Sparse approximation with bases
2015-01-01
This book systematically presents recent fundamental results on greedy approximation with respect to bases. Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications. The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and do...
Topological approximations of multisets
Directory of Open Access Journals (Sweden)
El-Sayed A. Abo-Tabl
2013-07-01
Full Text Available Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concept of rough set theory are information systems and approximation operators of approximation spaces. In this paper, we define and investigate three types of lower and upper multiset approximations of any multiset. These types based on the multiset base of multiset topology induced by a multiset relation. Moreover, the relationships between generalized rough msets and mset topologies are given. In addition, an illustrative example is given to illustrate the relationships between different types of generalized definitions of rough multiset approximations.
Expectation Consistent Approximate Inference
DEFF Research Database (Denmark)
Opper, Manfred; Winther, Ole
2005-01-01
We propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation. The approximation can be understood from replacing an average over the original intractable distribution with a tractable one. It requires two tractable probability...
Approximations of Fuzzy Systems
Directory of Open Access Journals (Sweden)
Vinai K. Singh
2013-03-01
Full Text Available A fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy. Such results can be viewed as an existence of optimal fuzzy systems. Li-Xin Wang discussed a similar problem using Gaussian membership function and Stone-Weierstrass Theorem. He established that fuzzy systems, with product inference, centroid defuzzification and Gaussian functions are capable of approximating any real continuous function on a compact set to arbitrary accuracy. In this paper we study a similar approximation problem by using exponential membership functions
Indian Academy of Sciences (India)
First page Back Continue Last page Overview Graphics. Loose-cluster approximation. Continuous curve Our Theory. Dashed curve Our Simulation. Loose cluster approx. not only. captures -the anomalous. qualitative features but is also,. quantitatively, quite accurate. Notes:
Bosma, Wieb
1990-01-01
The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given.
International Nuclear Information System (INIS)
Knobloch, A.F.
1980-01-01
A simplified cost approximation for INTOR parameter sets in a narrow parameter range is shown. Plausible constraints permit the evaluation of the consequences of parameter variations on overall cost. (orig.) [de
Gautschi, Walter; Rassias, Themistocles M
2011-01-01
Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences. This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovia, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational alg
Approximation Behooves Calibration
DEFF Research Database (Denmark)
da Silva Ribeiro, André Manuel; Poulsen, Rolf
2013-01-01
Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009.......Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009....
On Convex Quadratic Approximation
den Hertog, D.; de Klerk, E.; Roos, J.
2000-01-01
In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of
Improved Approximation Algorithm for
Byrka, Jaroslaw; Li, S.; Rybicki, Bartosz
2014-01-01
We study the k-level uncapacitated facility location problem (k-level UFL) in which clients need to be connected with paths crossing open facilities of k types (levels). In this paper we first propose an approximation algorithm that for any constant k, in polynomial time, delivers solutions of
Prestack wavefield approximations
Alkhalifah, Tariq
2013-09-01
The double-square-root (DSR) relation offers a platform to perform prestack imaging using an extended single wavefield that honors the geometrical configuration between sources, receivers, and the image point, or in other words, prestack wavefields. Extrapolating such wavefields, nevertheless, suffers from limitations. Chief among them is the singularity associated with horizontally propagating waves. I have devised highly accurate approximations free of such singularities which are highly accurate. Specifically, I use Padé expansions with denominators given by a power series that is an order lower than that of the numerator, and thus, introduce a free variable to balance the series order and normalize the singularity. For the higher-order Padé approximation, the errors are negligible. Additional simplifications, like recasting the DSR formula as a function of scattering angle, allow for a singularity free form that is useful for constant-angle-gather imaging. A dynamic form of this DSR formula can be supported by kinematic evaluations of the scattering angle to provide efficient prestack wavefield construction. Applying a similar approximation to the dip angle yields an efficient 1D wave equation with the scattering and dip angles extracted from, for example, DSR ray tracing. Application to the complex Marmousi data set demonstrates that these approximations, although they may provide less than optimal results, allow for efficient and flexible implementations. © 2013 Society of Exploration Geophysicists.
Generalized Approximate Message Passing
DEFF Research Database (Denmark)
Oxvig, Christian Schou; Arildsen, Thomas; Larsen, Torben
2017-01-01
This tech report details a collection of results related to the Generalised Approximate Message Passing (GAMP) algorithm. It is a summary of the results that the authors have found critical in understanding the GAMP algorithm. In particular, emphasis is on the details that are crucial in implemen...
Wolff, Hans
This paper deals with a stochastic process for the approximation of the root of a regression equation. This process was first suggested by Robbins and Monro. The main result here is a necessary and sufficient condition on the iteration coefficients for convergence of the process (convergence with probability one and convergence in the quadratic…
DEFF Research Database (Denmark)
Madsen, Rasmus Elsborg
2005-01-01
The Dirichlet compound multinomial (DCM), which has recently been shown to be well suited for modeling for word burstiness in documents, is here investigated. A number of conceptual explanations that account for these recent results, are provided. An exponential family approximation of the DCM...
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2011-01-01
Most prestack traveltime relations we tend work with are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multi-focusing or double square-root (DSR) and the common reflection stack (CRS) equations. Using the DSR equation, I analyze the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I derive expansion based solutions of this eikonal based on polynomial expansions in terms of the reflection and dip angles in a generally inhomogenous background medium. These approximate solutions are free of singularities and can be used to estimate travetimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. A Marmousi example demonstrates the usefulness of the approach. © 2011 Society of Exploration Geophysicists.
Topology, calculus and approximation
Komornik, Vilmos
2017-01-01
Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdős, Fejér, Stieltjes, and Turán. The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdős and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations. Students of both pure an...
Fragments of approximate counting
Czech Academy of Sciences Publication Activity Database
Buss, S.R.; Kolodziejczyk, L. A.; Thapen, Neil
2014-01-01
Roč. 79, č. 2 (2014), s. 496-525 ISSN 0022-4812 R&D Projects: GA AV ČR IAA100190902 Institutional support: RVO:67985840 Keywords : approximate counting * bounded arithmetic * ordering principle Subject RIV: BA - General Mathematics Impact factor: 0.541, year: 2014 http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9287274&fileId=S0022481213000376
Optimization and approximation
Pedregal, Pablo
2017-01-01
This book provides a basic, initial resource, introducing science and engineering students to the field of optimization. It covers three main areas: mathematical programming, calculus of variations and optimal control, highlighting the ideas and concepts and offering insights into the importance of optimality conditions in each area. It also systematically presents affordable approximation methods. Exercises at various levels have been included to support the learning process.
Approximate Bayesian recursive estimation
Czech Academy of Sciences Publication Activity Database
Kárný, Miroslav
2014-01-01
Roč. 285, č. 1 (2014), s. 100-111 ISSN 0020-0255 R&D Projects: GA ČR GA13-13502S Institutional support: RVO:67985556 Keywords : Approximate parameter estimation * Bayesian recursive estimation * Kullback–Leibler divergence * Forgetting Subject RIV: BB - Applied Statistics, Operational Research Impact factor: 4.038, year: 2014 http://library.utia.cas.cz/separaty/2014/AS/karny-0425539.pdf
Topics in Metric Approximation
Leeb, William Edward
This thesis develops effective approximations of certain metrics that occur frequently in pure and applied mathematics. We show that distances that often arise in applications, such as the Earth Mover's Distance between two probability measures, can be approximated by easily computed formulas for a wide variety of ground distances. We develop simple and easily computed characterizations both of norms measuring a function's regularity -- such as the Lipschitz norm -- and of their duals. We are particularly concerned with the tensor product of metric spaces, where the natural notion of regularity is not the Lipschitz condition but the mixed Lipschitz condition. A theme that runs throughout this thesis is that snowflake metrics (metrics raised to a power less than 1) are often better-behaved than ordinary metrics. For example, we show that snowflake metrics on finite spaces can be approximated by the average of tree metrics with a distortion bounded by intrinsic geometric characteristics of the space and not the number of points. Many of the metrics for which we characterize the Lipschitz space and its dual are snowflake metrics. We also present applications of the characterization of certain regularity norms to the problem of recovering a matrix that has been corrupted by noise. We are able to achieve an optimal rate of recovery for certain families of matrices by exploiting the relationship between mixed-variable regularity conditions and the decay of a function's coefficients in a certain orthonormal basis.
Finite elements and approximation
Zienkiewicz, O C
2006-01-01
A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises.Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher o
International Nuclear Information System (INIS)
El Sawi, M.
1983-07-01
A simple approach employing properties of solutions of differential equations is adopted to derive an appropriate extension of the WKBJ method. Some of the earlier techniques that are commonly in use are unified, whereby the general approximate solution to a second-order homogeneous linear differential equation is presented in a standard form that is valid for all orders. In comparison to other methods, the present one is shown to be leading in the order of iteration, and thus possibly has the ability of accelerating the convergence of the solution. The method is also extended for the solution of inhomogeneous equations. (author)
Cyclic approximation to stasis
Directory of Open Access Journals (Sweden)
Stewart D. Johnson
2009-06-01
Full Text Available Neighborhoods of points in $mathbb{R}^n$ where a positive linear combination of $C^1$ vector fields sum to zero contain, generically, cyclic trajectories that switch between the vector fields. Such points are called stasis points, and the approximating switching cycle can be chosen so that the timing of the switches exactly matches the positive linear weighting. In the case of two vector fields, the stasis points form one-dimensional $C^1$ manifolds containing nearby families of two-cycles. The generic case of two flows in $mathbb{R}^3$ can be diffeomorphed to a standard form with cubic curves as trajectories.
Approximate Euclidean Ramsey theorems
Directory of Open Access Journals (Sweden)
Adrian Dumitrescu
2011-04-01
Full Text Available According to a classical result of Szemerédi, every dense subset of 1,2,…,N contains an arbitrary long arithmetic progression, if N is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of {1,2,…,N}d contains an arbitrary large grid, if N is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i every dense separated set of points in some interval [0,L] on the line contains an arbitrary long approximate arithmetic progression, if L is large enough. (ii every dense separated set of points in the d-dimensional cube [0,L]d in Rd contains an arbitrary large approximate grid, if L is large enough. A further generalization for any finite pattern in Rd is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in Rd contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.
Approximate Bayesian computation.
Directory of Open Access Journals (Sweden)
Mikael Sunnåker
Full Text Available Approximate Bayesian computation (ABC constitutes a class of computational methods rooted in Bayesian statistics. In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood function might be computationally very costly to evaluate. ABC methods bypass the evaluation of the likelihood function. In this way, ABC methods widen the realm of models for which statistical inference can be considered. ABC methods are mathematically well-founded, but they inevitably make assumptions and approximations whose impact needs to be carefully assessed. Furthermore, the wider application domain of ABC exacerbates the challenges of parameter estimation and model selection. ABC has rapidly gained popularity over the last years and in particular for the analysis of complex problems arising in biological sciences (e.g., in population genetics, ecology, epidemiology, and systems biology.
The quasilocalized charge approximation
International Nuclear Information System (INIS)
Kalman, G J; Golden, K I; Donko, Z; Hartmann, P
2005-01-01
The quasilocalized charge approximation (QLCA) has been used for some time as a formalism for the calculation of the dielectric response and for determining the collective mode dispersion in strongly coupled Coulomb and Yukawa liquids. The approach is based on a microscopic model in which the charges are quasilocalized on a short-time scale in local potential fluctuations. We review the conceptual basis and theoretical structure of the QLC approach and together with recent results from molecular dynamics simulations that corroborate and quantify the theoretical concepts. We also summarize the major applications of the QLCA to various physical systems, combined with the corresponding results of the molecular dynamics simulations and point out the general agreement and instances of disagreement between the two
Approximate quantum Markov chains
Sutter, David
2018-01-01
This book is an introduction to quantum Markov chains and explains how this concept is connected to the question of how well a lost quantum mechanical system can be recovered from a correlated subsystem. To achieve this goal, we strengthen the data-processing inequality such that it reveals a statement about the reconstruction of lost information. The main difficulty in order to understand the behavior of quantum Markov chains arises from the fact that quantum mechanical operators do not commute in general. As a result we start by explaining two techniques of how to deal with non-commuting matrices: the spectral pinching method and complex interpolation theory. Once the reader is familiar with these techniques a novel inequality is presented that extends the celebrated Golden-Thompson inequality to arbitrarily many matrices. This inequality is the key ingredient in understanding approximate quantum Markov chains and it answers a question from matrix analysis that was open since 1973, i.e., if Lieb's triple ma...
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2012-05-01
Many of the explicit prestack traveltime relations used in practice are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multifocusing, based on the double square-root (DSR) equation, and the common reflection stack (CRS) approaches. Using the DSR equation, I constructed the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I recasted the eikonal in terms of the reflection angle, and thus, derived expansion based solutions of this eikonal in terms of the difference between the source and receiver velocities in a generally inhomogenous background medium. The zero-order term solution, corresponding to ignoring the lateral velocity variation in estimating the prestack part, is free of singularities and can be used to estimate traveltimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. The higher-order terms include limitations for horizontally traveling waves, however, we can readily enforce stability constraints to avoid such singularities. In fact, another expansion over reflection angle can help us avoid these singularities by requiring the source and receiver velocities to be different. On the other hand, expansions in terms of reflection angles result in singularity free equations. For a homogenous background medium, as a test, the solutions are reasonably accurate to large reflection and dip angles. A Marmousi example demonstrated the usefulness and versatility of the formulation. © 2012 Society of Exploration Geophysicists.
Pramana – Journal of Physics | Indian Academy of Sciences
Indian Academy of Sciences (India)
Home; Journals; Pramana – Journal of Physics; Volume 66; Issue 2. Volume 66, Issue 2. February 2006, pages 313-478. pp 313-324 Research Articles. A new algorithm for anisotropic solutions · M Chaisi S D .... Coulomb-Born-Oppenheimer approximation in Ps-H scattering · Hasi Ray · More Details Abstract Fulltext PDF.
Mass-imbalanced Three-Body Systems in Two Dimensions
DEFF Research Database (Denmark)
F. Bellotti, F.; Frederico, T.; T. Yamashita, M.
2013-01-01
We consider three-body systems in two dimensions with zero-range interactions for general masses and interaction strengths. The momentum-space Schr\\"odinger equation is solved numerically and in the Born-Oppenheimer (BO) approximation. The BO expression is derived using separable potentials and y...
Indian Academy of Sciences (India)
coriolis coupling between the different electronic states. In fact, the Renner-Teller effect arises due to breakdown of the Born-Oppenheimer approximation. Several theo- retical developments have been reported on this effect for linear triatomic and tetraatomic molecules. For details, see references8,9 and references therein.
Electron-nuclear coupling in time-dependent multicomponent density functional theory
Butriy, Olena O.
2008-01-01
In this thesis we developed the time-dependent version of the multicomponent density functional approach to treat time-dependent electron-nuclear systems. The method enables to describe the electron-nuclear coupling fully quantum mechanically. No Born-Oppenheimer approximation is involved in the
A pion and gluon exchange model of the non-annihilation part of the nucleon-antinucleon interaction
International Nuclear Information System (INIS)
Fernandez, F.
1988-01-01
The effect of quark antisymmetrization on the one-pion and one-gluon exchange between nucleons and antinucleons is explicitly calculated within the framework of the Born-Oppenheimer approximation. We found an effective short range part of the N-anti N interaction which is repulsive in all the spin-isospin channels. (Author)
The local quantum-mechanical momentum balance and its consequencies
International Nuclear Information System (INIS)
Zieche, P.; Diener, G.; Graafenstejn, J.; Kaschner, R.; Nielsen, O.H.
1988-01-01
Many-electron systems are considered within Born-Oppenheimer approximation. The balance equation is derived; its first application concerns force sum rules, involving alternative possibilities for the calculation of relaxation forces. An extended crystal with force-free bulk atoms but under a non-vanishing stress is considered
Efimov States of Heavy Impurities in a Bose-Einstein Condensate
DEFF Research Database (Denmark)
Zinner, Nikolaj Thomas
2013-01-01
We consider the problem of two heavy impurity particles embedded in a gas of weakly-interacting light mass bosonic particles in the condensed state. Using the Bogoliubov approach to describe the bosonic gas and the Born-Oppenheimer approximation for the three-body dynamics, we calculate the modif...
International Conference Approximation Theory XV
Schumaker, Larry
2017-01-01
These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22–25, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type. The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, a...
Hierarchical low-rank approximation for high dimensional approximation
Nouy, Anthony
2016-01-07
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. Such high-dimensional approximation problems naturally arise in stochastic analysis and uncertainty quantification. In many practical situations, the approximation of high-dimensional functions is made computationally tractable by using rank-structured approximations. In this talk, we present algorithms for the approximation in hierarchical tensor format using statistical methods. Sparse representations in a given tensor format are obtained with adaptive or convex relaxation methods, with a selection of parameters using crossvalidation methods.
Comparative Study of Approximate Multipliers
Masadeh, Mahmoud; Hasan, Osman; Tahar, Sofiene
2018-01-01
Approximate multipliers are widely being advocated for energy-efficient computing in applications that exhibit an inherent tolerance to inaccuracy. However, the inclusion of accuracy as a key design parameter, besides the performance, area and power, makes the identification of the most suitable approximate multiplier quite challenging. In this paper, we identify three major decision making factors for the selection of an approximate multipliers circuit: (1) the type of approximate full adder...
Forms of Approximate Radiation Transport
Brunner, G
2002-01-01
Photon radiation transport is described by the Boltzmann equation. Because this equation is difficult to solve, many different approximate forms have been implemented in computer codes. Several of the most common approximations are reviewed, and test problems illustrate the characteristics of each of the approximations. This document is designed as a tutorial so that code users can make an educated choice about which form of approximate radiation transport to use for their particular simulation.
International Conference Approximation Theory XIV
Schumaker, Larry
2014-01-01
This volume developed from papers presented at the international conference Approximation Theory XIV, held April 7–10, 2013 in San Antonio, Texas. The proceedings contains surveys by invited speakers, covering topics such as splines on non-tensor-product meshes, Wachspress and mean value coordinates, curvelets and shearlets, barycentric interpolation, and polynomial approximation on spheres and balls. Other contributed papers address a variety of current topics in approximation theory, including eigenvalue sequences of positive integral operators, image registration, and support vector machines. This book will be of interest to mathematicians, engineers, and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis, and related approximation areas.
Exact constants in approximation theory
Korneichuk, N
1991-01-01
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are base
Approximation by planar elastic curves
DEFF Research Database (Denmark)
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2016-01-01
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven...
Anytime classification by ontology approximation
Schlobach, S.; Blaauw, E.; El Kebir, M.; Ten Teije, A.; Van Harmelen, F.; Bortoli, S.; Hobbelman, M.C.; Millian, K.; Ren, Y.; Stam, S.; Thomassen, P.; Van Het Schip, R.; Van Willigem, W.
2007-01-01
Reasoning with large or complex ontologies is one of the bottle-necks of the Semantic Web. In this paper we present an anytime algorithm for classification based on approximate subsumption. We give the formal definitions for approximate subsumption, and show its monotonicity and soundness; we show
Some results in Diophantine approximation
DEFF Research Database (Denmark)
Pedersen, Steffen Højris
This thesis consists of three papers in Diophantine approximation, a subbranch of number theory. Preceding these papers is an introduction to various aspects of Diophantine approximation and formal Laurent series over Fq and a summary of each of the three papers. The introduction introduces...... the basic concepts on which the papers build. Among other it introduces metric Diophantine approximation, Mahler’s approach on algebraic approximation, the Hausdorff measure, and properties of the formal Laurent series over Fq. The introduction ends with a discussion on Mahler’s problem when considered...... in the formal Laurent series over F3. The first paper is on intrinsic Diophantine approximation in the Cantor set in the formal Laurent series over F3. The summary contains a short motivation, the results of the paper and sketches of the proofs, mainly focusing on the ideas involved. The details of the proofs...
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-08-18
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Approximate Implicitization Using Linear Algebra
Directory of Open Access Journals (Sweden)
Oliver J. D. Barrowclough
2012-01-01
Full Text Available We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.
Rollout sampling approximate policy iteration
Dimitrakakis, C.; Lagoudakis, M.G.
2008-01-01
Several researchers have recently investigated the connection between reinforcement learning and classification. We are motivated by proposals of approximate policy iteration schemes without value functions, which focus on policy representation using classifiers and address policy learning as a
Shearlets and Optimally Sparse Approximations
DEFF Research Database (Denmark)
Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q
2012-01-01
of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide...... optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction...... to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field....
Mathematical algorithms for approximate reasoning
Murphy, John H.; Chay, Seung C.; Downs, Mary M.
1988-01-01
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away
Approximate number sense theory or approximate theory of magnitude?
Content, Alain; Velde, Michael Vande; Adriano, Andrea
2017-01-01
Leibovich et al. argue that the evidence in favor of a perceptual mechanism devoted to the extraction of numerosity from visual collections is unsatisfactory and propose to replace it with an unspecific mechanism capturing approximate magnitudes from continuous dimensions. We argue that their representation of the evidence is incomplete and that their theoretical proposal is too vague to be useful.
Approximate Matching of Hierarchial Data
DEFF Research Database (Denmark)
Augsten, Nikolaus
The goal of this thesis is to design, develop, and evaluate new methods for the approximate matching of hierarchical data represented as labeled trees. In approximate matching scenarios two items should be matched if they are similar. Computing the similarity between labeled trees is hard...... formally proof that the pq-gram index can be incrementally updated based on the log of edit operations without reconstructing intermediate tree versions. The incremental update is independent of the data size and scales to a large number of changes in the data. We introduce windowed pq...... as in addition to the data values also the structure must be considered. A well-known measure for comparing trees is the tree edit distance. It is computationally expensive and leads to a prohibitively high run time. Our solution for the approximate matching of hierarchical data are pq-grams. The pq...
Approximations to camera sensor noise
Jin, Xiaodan; Hirakawa, Keigo
2013-02-01
Noise is present in all image sensor data. Poisson distribution is said to model the stochastic nature of the photon arrival process, while it is common to approximate readout/thermal noise by additive white Gaussian noise (AWGN). Other sources of signal-dependent noise such as Fano and quantization also contribute to the overall noise profile. Question remains, however, about how best to model the combined sensor noise. Though additive Gaussian noise with signal-dependent noise variance (SD-AWGN) and Poisson corruption are two widely used models to approximate the actual sensor noise distribution, the justification given to these types of models are based on limited evidence. The goal of this paper is to provide a more comprehensive characterization of random noise. We concluded by presenting concrete evidence that Poisson model is a better approximation to real camera model than SD-AWGN. We suggest further modification to Poisson that may improve the noise model.
Face Recognition using Approximate Arithmetic
DEFF Research Database (Denmark)
Marso, Karol
Face recognition is image processing technique which aims to identify human faces and found its use in various diﬀerent ﬁelds for example in security. Throughout the years this ﬁeld evolved and there are many approaches and many diﬀerent algorithms which aim to make the face recognition as eﬀective...... as possible. The use of diﬀerent approaches such as neural networks and machine learning can lead to fast and eﬃcient solutions however, these solutions are expensive in terms of hardware resources and power consumption. A possible solution to this problem can be use of approximate arithmetic. In many image...... processing applications the results do not need to be completely precise and use of the approximate arithmetic can lead to reduction in terms of delay, space and power consumption. In this paper we examine possible use of approximate arithmetic in face recognition using Eigenfaces algorithm....
Diophantine approximation and Dirichlet series
Queffélec, Hervé
2013-01-01
This self-contained book will benefit beginners as well as researchers. It is devoted to Diophantine approximation, the analytic theory of Dirichlet series, and some connections between these two domains, which often occur through the Kronecker approximation theorem. Accordingly, the book is divided into seven chapters, the first three of which present tools from commutative harmonic analysis, including a sharp form of the uncertainty principle, ergodic theory and Diophantine approximation to be used in the sequel. A presentation of continued fraction expansions, including the mixing property of the Gauss map, is given. Chapters four and five present the general theory of Dirichlet series, with classes of examples connected to continued fractions, the famous Bohr point of view, and then the use of random Dirichlet series to produce non-trivial extremal examples, including sharp forms of the Bohnenblust-Hille theorem. Chapter six deals with Hardy-Dirichlet spaces, which are new and useful Banach spaces of anal...
Approximate reasoning in physical systems
International Nuclear Information System (INIS)
Mutihac, R.
1991-01-01
The theory of fuzzy sets provides excellent ground to deal with fuzzy observations (uncertain or imprecise signals, wavelengths, temperatures,etc.) fuzzy functions (spectra and depth profiles) and fuzzy logic and approximate reasoning. First, the basic ideas of fuzzy set theory are briefly presented. Secondly, stress is put on application of simple fuzzy set operations for matching candidate reference spectra of a spectral library to an unknown sample spectrum (e.g. IR spectroscopy). Thirdly, approximate reasoning is applied to infer an unknown property from information available in a database (e.g. crystal systems). Finally, multi-dimensional fuzzy reasoning techniques are suggested. (Author)
Approximations to the Newton potential
International Nuclear Information System (INIS)
Warburton, A.E.A.; Hatfield, R.W.
1977-01-01
Explicit expressions are obtained for Newton's (Newton, R.G., J. Math. Phys., 3:75-82 (1962)) solution to the inverse scattering problem in the approximations where up to two phase shifts are treated exactly and the rest to first order. (author)
Approximation properties of haplotype tagging
Directory of Open Access Journals (Sweden)
Dreiseitl Stephan
2006-01-01
Full Text Available Abstract Background Single nucleotide polymorphisms (SNPs are locations at which the genomic sequences of population members differ. Since these differences are known to follow patterns, disease association studies are facilitated by identifying SNPs that allow the unique identification of such patterns. This process, known as haplotype tagging, is formulated as a combinatorial optimization problem and analyzed in terms of complexity and approximation properties. Results It is shown that the tagging problem is NP-hard but approximable within 1 + ln((n2 - n/2 for n haplotypes but not approximable within (1 - ε ln(n/2 for any ε > 0 unless NP ⊂ DTIME(nlog log n. A simple, very easily implementable algorithm that exhibits the above upper bound on solution quality is presented. This algorithm has running time O((2m - p + 1 ≤ O(m(n2 - n/2 where p ≤ min(n, m for n haplotypes of size m. As we show that the approximation bound is asymptotically tight, the algorithm presented is optimal with respect to this asymptotic bound. Conclusion The haplotype tagging problem is hard, but approachable with a fast, practical, and surprisingly simple algorithm that cannot be significantly improved upon on a single processor machine. Hence, significant improvement in computatational efforts expended can only be expected if the computational effort is distributed and done in parallel.
Approximate Reanalysis in Topology Optimization
DEFF Research Database (Denmark)
Amir, Oded; Bendsøe, Martin P.; Sigmund, Ole
2009-01-01
In the nested approach to structural optimization, most of the computational effort is invested in the solution of the finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures...
Ultrafast Approximation for Phylogenetic Bootstrap
Bui Quang Minh, [No Value; Nguyen, Thi; von Haeseler, Arndt
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and
APPROXIMATE MODELS FOR FLOOD ROUTING
African Journals Online (AJOL)
kinematic model and a nonlinear convection-diffusion model are extracted from a normalized form of the St. Venant equations, and applied to ... normal ﬂow condition is moderate. Keywords: approximate models, nonlinear kinematic ... The concern here is with the movement of an abnormal amount of water along a river or ...
On badly approximable complex numbers
DEFF Research Database (Denmark)
Esdahl-Schou, Rune; Kristensen, S.
We show that the set of complex numbers which are badly approximable by ratios of elements of , where has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably...
Rational approximation of vertical segments
Salazar Celis, Oliver; Cuyt, Annie; Verdonk, Brigitte
2007-08-01
In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.
All-Norm Approximation Algorithms
Azar, Yossi; Epstein, Leah; Richter, Yossi; Woeginger, Gerhard J.; Penttonen, Martti; Meineche Schmidt, Erik
2002-01-01
A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ p norms. We address this problem by introducing the concept of an All-norm ρ-approximation
Approximate Reasoning with Fuzzy Booleans
van den Broek, P.M.; Noppen, J.A.R.
This paper introduces, in analogy to the concept of fuzzy numbers, the concept of fuzzy booleans, and examines approximate reasoning with the compositional rule of inference using fuzzy booleans. It is shown that each set of fuzzy rules is equivalent to a set of fuzzy rules with singleton crisp
Hydrogen: Beyond the Classic Approximation
International Nuclear Information System (INIS)
Scivetti, Ivan
2003-01-01
The classical nucleus approximation is the most frequently used approach for the resolution of problems in condensed matter physics.However, there are systems in nature where it is necessary to introduce the nuclear degrees of freedom to obtain a correct description of the properties.Examples of this, are the systems with containing hydrogen.In this work, we have studied the resolution of the quantum nuclear problem for the particular case of the water molecule.The Hartree approximation has been used, i.e. we have considered that the nuclei are distinguishable particles.In addition, we have proposed a model to solve the tunneling process, which involves the resolution of the nuclear problem for configurations of the system away from its equilibrium position
Hydrogen Beyond the Classic Approximation
Scivetti, I
2003-01-01
The classical nucleus approximation is the most frequently used approach for the resolution of problems in condensed matter physics.However, there are systems in nature where it is necessary to introduce the nuclear degrees of freedom to obtain a correct description of the properties.Examples of this, are the systems with containing hydrogen.In this work, we have studied the resolution of the quantum nuclear problem for the particular case of the water molecule.The Hartree approximation has been used, i.e. we have considered that the nuclei are distinguishable particles.In addition, we have proposed a model to solve the tunneling process, which involves the resolution of the nuclear problem for configurations of the system away from its equilibrium position
Good points for diophantine approximation
Indian Academy of Sciences (India)
n=1 of real numbers in the interval [0, 1) and a sequence. (δn)∞ n=1 of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be 'well approximated' by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − xn| < δn holds for infinitely many positive integers n ...
Dimensionality Reduction with Adaptive Approximation
Kokiopoulou, Effrosyni; Frossard, Pascal
2007-01-01
In this paper, we propose the use of (adaptive) nonlinear approximation for dimensionality reduction. In particular, we propose a dimensionality reduction method for learning a parts based representation of signals using redundant dictionaries. A redundant dictionary is an overcomplete set of basis vectors that spans the signal space. The signals are jointly represented in a common subspace extracted from the redundant dictionary, using greedy pursuit algorithms for simultaneous sparse approx...
Ultrafast approximation for phylogenetic bootstrap.
Minh, Bui Quang; Nguyen, Minh Anh Thi; von Haeseler, Arndt
2013-05-01
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and the Shimodaira-Hasegawa-like approximate likelihood ratio test have been introduced to speed up the bootstrap. Here, we suggest an ultrafast bootstrap approximation approach (UFBoot) to compute the support of phylogenetic groups in maximum likelihood (ML) based trees. To achieve this, we combine the resampling estimated log-likelihood method with a simple but effective collection scheme of candidate trees. We also propose a stopping rule that assesses the convergence of branch support values to automatically determine when to stop collecting candidate trees. UFBoot achieves a median speed up of 3.1 (range: 0.66-33.3) to 10.2 (range: 1.32-41.4) compared with RAxML RBS for real DNA and amino acid alignments, respectively. Moreover, our extensive simulations show that UFBoot is robust against moderate model violations and the support values obtained appear to be relatively unbiased compared with the conservative standard bootstrap. This provides a more direct interpretation of the bootstrap support. We offer an efficient and easy-to-use software (available at http://www.cibiv.at/software/iqtree) to perform the UFBoot analysis with ML tree inference.
Initially Approximated Quasi Equilibrium Manifold
International Nuclear Information System (INIS)
Shahzad, M.; Arif, H.; Gulistan, M.; Sajid, M.
2015-01-01
Most commonly, kinetics model reduction techniques are based on exploiting time scale separation into fast and slow reaction processes. Then, a researcher approximates the system dynamically with dimension reduction for slow ones eliminating the fast modes. The main idea behind the construction of the lower dimension manifold is based on finding its initial approximation using Quasi Equilibrium Manifold (QEM). Here, we provide an efficient numerical method, which allow us to calculate low dimensional manifolds of chemical reaction systems. This computation technique is not restricted to our specific complex problem, but it can also be applied to other reacting flows or dynamic systems provided with the condition that a large number of extra (decaying) components can be eliminated from the system. Through computational approach, we approximate low dimensional manifold for a mechanism of six chemical species to simplify complex chemical kinetics. A reduced descriptive form of slow invariant manifold is obtained from dissipative system. This method is applicable for higher dimensions and is applied over an oxidation of CO/Pt. (author)
Approximate Inference for Wireless Communications
DEFF Research Database (Denmark)
Hansen, Morten
to the optimal one, which usually requires an unacceptable high complexity. Some of the treated approximate methods are based on QL-factorization of the channel matrix. In the work presented in this thesis it is proven how the QL-factorization of frequency-selective channels asymptotically provides the minimum......-phase and all-pass filters. This enables us to view Sphere Detection (SD) as an adaptive variant of minimum-phase prefiltered reduced-state sequence estimation. Thus, a novel way of computing the minimum-phase filter and its associated all-pass filter using the numerically stable QL-factorization is suggested...
Generalized Gradient Approximation Made Simple
International Nuclear Information System (INIS)
Perdew, J.P.; Burke, K.; Ernzerhof, M.
1996-01-01
Generalized gradient approximations (GGA close-quote s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. copyright 1996 The American Physical Society
Wavelet Approximation in Data Assimilation
Tangborn, Andrew; Atlas, Robert (Technical Monitor)
2002-01-01
Estimation of the state of the atmosphere with the Kalman filter remains a distant goal because of high computational cost of evolving the error covariance for both linear and nonlinear systems. Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics on the the error correlation field from a global two-dimensional chemical constituent assimilation, and implement an adaptive wavelet approximation scheme on the assimilation of the one-dimensional Burger's equation. In the former problem, we show that 99%, of the error correlation can be represented by just 3% of the wavelet coefficients, with good representation of localized features. In the Burger's equation assimilation, the discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6%, of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.
Plasma Physics Approximations in Ares
International Nuclear Information System (INIS)
Managan, R. A.
2015-01-01
Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, Fn( μ/θ ), the chemical potential, μ or ζ = ln(1+e μ/θ ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A α (ζ ),A β (ζ ), ζ, f(ζ ) = (1 + e -μ/θ )F 1/2 (μ/θ), F 1/2 '/F 1/2 , F c α , and F c β . In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.
Approximation by double Walsh polynomials
Directory of Open Access Journals (Sweden)
Ferenc Móricz
1992-01-01
Full Text Available We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh-Fourier series of a function in a homogeneous Banach space X. In particular, X may be Lp(I2, where 1≦p<∞ and I2=[0,1×[0,1, or CW(I2, the latter being the collection of uniformly W-continuous functions on I2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by-products, we deduce sufficient conditions for convergence in Lp(I2-norm and uniform convergence on I2 as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.
Approximating the minimum cycle mean
Directory of Open Access Journals (Sweden)
Krishnendu Chatterjee
2013-07-01
Full Text Available We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1 First we show that the algorithmic question is reducible in O(n^2 time to the problem of a logarithmic number of min-plus matrix multiplications of n-by-n matrices, where n is the number of vertices of the graph. (2 Second, when the weights are nonnegative, we present the first (1 + ε-approximation algorithm for the problem and the running time of our algorithm is ilde(O(n^ω log^3(nW/ε / ε, where O(n^ω is the time required for the classic n-by-n matrix multiplication and W is the maximum value of the weights.
Nonlinear approximation with dictionaries I. Direct estimates
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
2004-01-01
We study various approximation classes associated with m-term approximation by elements from a (possibly) redundant dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation w...
The exact molecular wavefunction as a product of an electronic and a nuclear wavefunction
International Nuclear Information System (INIS)
Cederbaum, Lorenz S.
2013-01-01
The Born-Oppenheimer approximation is a basic approximation in molecular science. In this approximation, the total molecular wavefunction is written as a product of an electronic and a nuclear wavefunction. Hunter [Int. J. Quantum Chem. 9, 237 (1975)] has argued that the exact total wavefunction can also be factorized as such a product. In the present work, a variational principle is introduced which shows explicitly that the total wavefunction can be exactly written as such a product. To this end, a different electronic Hamiltonian has to be defined. The Schrödinger equation for the electronic wavefunction follows from the variational ansatz and is presented. As in the Born-Oppenheimer approximation, the nuclear motion is shown to proceed in a potential which is the electronic energy. In contrast to the Born-Oppenheimer approximation, the separation of the center of mass can be carried out exactly. The electronic Hamiltonian and the equation of motion of the nuclei resulting after the exact separation of the center of mass motion are explicitly given. A simple exactly solvable model is used to illustrate some aspects of the theory.
Energy Technology Data Exchange (ETDEWEB)
Albert, Julian; Falge, Mirjam; Hildenbrand, Heiko; Engel, Volker [Universität Würzburg, Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, Campus Nord, Am Hubland, 97074 Würzburg (Germany); Gomez, Sandra; Sola, Ignacio R. [Departamento de Quimica Fisica, Universidad Complutense, 28040 Madrid (Spain)
2015-07-28
We theoretically investigate the photon-echo spectroscopy of coupled electron-nuclear quantum dynamics. Two situations are treated. In the first case, the Born-Oppenheimer (adiabatic) approximation holds. It is then possible to interpret the two-dimensional (2D) spectra in terms of vibrational motion taking place in different electronic states. In particular, pure vibrational coherences which are related to oscillations in the time-dependent third-order polarization can be identified. This concept fails in the second case, where strong non-adiabatic coupling leads to the breakdown of the Born-Oppenheimer-approximation. Then, the 2D-spectra reveal a complicated vibronic structure and vibrational coherences cannot be disentangled from the electronic motion.
How large are nonadiabatic effects in atomic and diatomic systems?
Yang, Yubo; Kylänpää, Ilkka; Tubman, Norm M; Krogel, Jaron T; Hammes-Schiffer, Sharon; Ceperley, David M
2015-09-28
With recent developments in simulating nonadiabatic systems to high accuracy, it has become possible to determine how much energy is attributed to nuclear quantum effects beyond zero-point energy. In this work, we calculate the non-relativistic ground-state energies of atomic and molecular systems without the Born-Oppenheimer approximation. For this purpose, we utilize the fixed-node diffusion Monte Carlo method, in which the nodes depend on both the electronic and ionic positions. We report ground-state energies for all systems studied, ionization energies for the first-row atoms and atomization energies for the first-row hydrides. We find the ionization energies of the atoms to be nearly independent of the Born-Oppenheimer approximation, within the accuracy of our results. The atomization energies of molecular systems, however, show small effects of the nonadiabatic coupling between electrons and nuclei.
Low Temperature Hydrogen Antihydrogen Interactions
International Nuclear Information System (INIS)
Armour, E. A. G.; Chamberlain, C. W.
2001-01-01
In view of current interest in the trapping of antihydrogen (H-bar) atoms at low temperatures, we have carried out a full four-body variational calculation to determine s-wave elastic phase shifts for hydrogen antihydrogen scattering, using the Kohn Variational Principle. Terms outside the Born-Oppenheimer approximation have been taken into account using the formalism of Kolos and Wolniewicz. As far as we are aware, this is the first time that these terms have been included in an H H-bar scattering calculation. This is a continuation of earlier work on H-H-bar interactions. Preliminary results differ substantially from those calculated using the Born-Oppenheimer approximation. A method is outlined for reducing this discrepancy and taking the rearrangement channel into account.
Neutron Resonances in Systems of Few Nuclei and Their Possible Role in Radiation of Overdense Stars
International Nuclear Information System (INIS)
Takibayev, N. Zh.
2011-01-01
Exact analytical solutions of three- and four-body systems made of one light particle and other heavy particles have been obtained in the model of Born-Oppenheimer approximation with two-body separable interactions. In the case of neutron scattering on a subsystem of few fixed nuclei the appearance of new resonance quantum states has been shown as well as their dependence on distances between heavy nuclei. The applications of new phenomena to overdense stars radiation are considered. (author)
International Nuclear Information System (INIS)
Pen'kov, F.M.
1998-01-01
The Born-Oppenheimer approximation is used to obtain an equation for the effective interaction in three atoms bound by a single electron. For low binding energies in an 'electron + atom' pair, long-range forces arise between the atoms, leading to bound states when the size of the three-atom cluster is a few tens of angstrom. A system made of alkali-metal atoms is considered as an example
Approximation properties of fine hyperbolic graphs
Indian Academy of Sciences (India)
is called the metric invariant translation approximation property for a countable dis- crete metric space. Moreover ... Uniform Roe algebras; fine hyperbolic graph; metric invariant translation approximation property. ..... ate Studies in Mathematics, Volume 88 (2008) (Rhode Island: American Mathematical. Society Providence).
Approximate Uniqueness Estimates for Singular Correlation Matrices.
Finkbeiner, C. T.; Tucker, L. R.
1982-01-01
The residual variance is often used as an approximation to the uniqueness in factor analysis. An upper bound approximation to the residual variance is presented for the case when the correlation matrix is singular. (Author/JKS)
Reduction of Linear Programming to Linear Approximation
Vaserstein, Leonid N.
2006-01-01
It is well known that every Chebyshev linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to a Chebyshev linear approximation problem.
HE11 radiation patterns and gaussian approximations
International Nuclear Information System (INIS)
Rebuffi, L.; Crenn, J.P.
1986-12-01
The possibility of approximating the HE11 radiation pattern with a Gaussian distribution is presented. A numerical comparison between HE11 far-field theoretical patterns and Abrams and Crenn approximations permits an evaluation of the validity of these two approximations. A new numerically optimized HE11 Gaussian approximation for the far-field, extended to great part of the near field, has been found. In particular, the value given for the beam radius at the waist, has been demonstrated to give the best HE11 Gaussian approximation in the far-field. The Crenn approximation is found to be very close to this optimal approximation, while the Abrams approximation is shown to be less precise. Universal curves for intensity, amplitude and power distribution are given for the HE11 radiated mode. These results are of interest for laser waveguide applications and for plasma ECRH transmission systems
Analytical approximations of Chandrasekhar's H-Function
International Nuclear Information System (INIS)
Simovic, R.; Vukanic, J.
1995-01-01
Analytical approximations of Chandrasekhar's H-function are derived in this paper by using ordinary and modified DPN methods. The accuracy of the approximations is discussed and the energy dependent albedo problem is treated. (author)
Axiomatic Characterizations of IVF Rough Approximation Operators
Directory of Open Access Journals (Sweden)
Guangji Yu
2014-01-01
Full Text Available This paper is devoted to the study of axiomatic characterizations of IVF rough approximation operators. IVF approximation spaces are investigated. The fact that different IVF operators satisfy some axioms to guarantee the existence of different types of IVF relations which produce the same operators is proved and then IVF rough approximation operators are characterized by axioms.
Truth Approximation, Social Epistemology, and Opinion Dynamics
Douven, Igor; Kelp, Christoph
This paper highlights some connections between work on truth approximation and work in social epistemology, in particular work on peer disagreement. In some of the literature on truth approximation, questions have been addressed concerning the efficiency of research strategies for approximating the
Operator approximant problems arising from quantum theory
Maher, Philip J
2017-01-01
This book offers an account of a number of aspects of operator theory, mainly developed since the 1980s, whose problems have their roots in quantum theory. The research presented is in non-commutative operator approximation theory or, to use Halmos' terminology, in operator approximants. Focusing on the concept of approximants, this self-contained book is suitable for graduate courses.
Approximate Nearest Neighbor Queries among Parallel Segments
DEFF Research Database (Denmark)
Emiris, Ioannis Z.; Malamatos, Theocharis; Tsigaridas, Elias
2010-01-01
We develop a data structure for answering efficiently approximate nearest neighbor queries over a set of parallel segments in three dimensions. We connect this problem to approximate nearest neighbor searching under weight constraints and approximate nearest neighbor searching on historical data...
Nonlinear approximation with dictionaries I. Direct estimates
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
2004-01-01
We study various approximation classes associated with m-term approximation by elements from a (possibly) redundant dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation...... with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove...... that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in L^p spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss som new estimates they provide....
Nonlinear approximation with dictionaries, I: Direct estimates
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
We study various approximation classes associated with $m$-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best $m$-term approximation is compared to new classes defined by considering $m......$-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space......, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in $L^p$ spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss som new estimates...
Bounded-Degree Approximations of Stochastic Networks
Energy Technology Data Exchange (ETDEWEB)
Quinn, Christopher J.; Pinar, Ali; Kiyavash, Negar
2017-06-01
We propose algorithms to approximate directed information graphs. Directed information graphs are probabilistic graphical models that depict causal dependencies between stochastic processes in a network. The proposed algorithms identify optimal and near-optimal approximations in terms of Kullback-Leibler divergence. The user-chosen sparsity trades off the quality of the approximation against visual conciseness and computational tractability. One class of approximations contains graphs with speci ed in-degrees. Another class additionally requires that the graph is connected. For both classes, we propose algorithms to identify the optimal approximations and also near-optimal approximations, using a novel relaxation of submodularity. We also propose algorithms to identify the r-best approximations among these classes, enabling robust decision making.
Mapping moveout approximations in TI media
Stovas, Alexey
2013-11-21
Moveout approximations play a very important role in seismic modeling, inversion, and scanning for parameters in complex media. We developed a scheme to map one-way moveout approximations for transversely isotropic media with a vertical axis of symmetry (VTI), which is widely available, to the tilted case (TTI) by introducing the effective tilt angle. As a result, we obtained highly accurate TTI moveout equations analogous with their VTI counterparts. Our analysis showed that the most accurate approximation is obtained from the mapping of generalized approximation. The new moveout approximations allow for, as the examples demonstrate, accurate description of moveout in the TTI case even for vertical heterogeneity. The proposed moveout approximations can be easily used for inversion in a layered TTI medium because the parameters of these approximations explicitly depend on corresponding effective parameters in a layered VTI medium.
Multilevel Monte Carlo in Approximate Bayesian Computation
Jasra, Ajay
2017-02-13
In the following article we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given.
Approximate unitary equivalence of normaloid type operators
Zhu, Sen
2015-01-01
In this paper, we explore approximate unitary equivalence of normaloid operators and classify several normaloid type operators including transaloid operators, polynomial-normaloid operators and von Neumann operators up to approximate unitary equivalence. As an application, we explore approximation of transaloid operators with closed numerical ranges. Among other things, it is proved that those transaloid operators with closed numerical ranges are norm dense in the class of transaloid operators.
Uniform analytic approximation of Wigner rotation matrices
Hoffmann, Scott E.
2018-02-01
We derive the leading asymptotic approximation, for low angle θ, of the Wigner rotation matrix elements, dm1m2 j(θ ) , uniform in j, m1, and m2. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.
A Note on Generalized Approximation Property
Directory of Open Access Journals (Sweden)
Antara Bhar
2013-01-01
Full Text Available We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for ( has also been characterized.
Local density approximations for relativistic exchange energies
International Nuclear Information System (INIS)
MacDonald, A.H.
1986-01-01
The use of local density approximations to approximate exchange interactions in relativistic electron systems is reviewed. Particular attention is paid to the physical content of these exchange energies by discussing results for the uniform relativistic electron gas from a new point of view. Work on applying these local density approximations in atoms and solids is reviewed and it is concluded that good accuracy is usually possible provided self-interaction corrections are applied. The local density approximations necessary for spin-polarized relativistic systems are discussed and some new results are presented
Approximate maximum parsimony and ancestral maximum likelihood.
Alon, Noga; Chor, Benny; Pardi, Fabio; Rapoport, Anat
2010-01-01
We explore the maximum parsimony (MP) and ancestral maximum likelihood (AML) criteria in phylogenetic tree reconstruction. Both problems are NP-hard, so we seek approximate solutions. We formulate the two problems as Steiner tree problems under appropriate distances. The gist of our approach is the succinct characterization of Steiner trees for a small number of leaves for the two distances. This enables the use of known Steiner tree approximation algorithms. The approach leads to a 16/9 approximation ratio for AML and asymptotically to a 1.55 approximation ratio for MP.
Golibrzuch, Kai; Bartels, Nils; Auerbach, Daniel J; Wodtke, Alec M
2015-04-01
We review studies of molecular interactions and chemical reactions at metal surfaces, emphasizing progress toward a predictive theory of surface chemistry and catalysis. For chemistry at metal surfaces, a small number of central approximations are typically made: (a) the Born-Oppenheimer approximation of electronic adiabaticity, (b) the use of density functional theory at the generalized gradient approximation level, (c) the classical approximation for nuclear motion, and (d) various reduced-dimensionality approximations. Together, these approximations constitute a provisional model for surface chemical reactivity. We review work on some carefully studied examples of molecules interacting at metal surfaces that probe the validity of various aspects of the provisional model.
Non-Linear Approximation of Bayesian Update
Litvinenko, Alexander
2016-06-23
We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.
Diagonal Pade approximations for initial value problems
International Nuclear Information System (INIS)
Reusch, M.F.; Ratzan, L.; Pomphrey, N.; Park, W.
1987-06-01
Diagonal Pade approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained. 20 refs., 3 figs., 1 tab
Simultaneous approximation in scales of Banach spaces
International Nuclear Information System (INIS)
Bramble, J.H.; Scott, R.
1978-01-01
The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods
Approximation properties of fine hyperbolic graphs
Indian Academy of Sciences (India)
2016-08-26
Aug 26, 2016 ... In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use ... Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, People's Republic of China ...
Nonlinear approximation with general wave packets
DEFF Research Database (Denmark)
Borup, Lasse; Nielsen, Morten
2005-01-01
We study nonlinear approximation in the Triebel-Lizorkin spaces with dictionaries formed by dilating and translating one single function g. A general Jackson inequality is derived for best m-term approximation with such dictionaries. In some special cases where g has a special structure, a complete...
Quirks of Stirling's Approximation
Macrae, Roderick M.; Allgeier, Benjamin M.
2013-01-01
Stirling's approximation to ln "n"! is typically introduced to physical chemistry students as a step in the derivation of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar "toy…
On approximating multi-criteria TSP
Manthey, Bodo; Albers, S.; Marion, J.-Y.
2009-01-01
We present approximation algorithms for almost all variants of the multi-criteria traveling salesman problem (TSP), whose performances are independent of the number $k$ of criteria and come close to the approximation ratios obtained for TSP with a single objective function. We present randomized
On approximating multi-criteria TSP
Manthey, Bodo
We present approximation algorithms for almost all variants of the multicriteria traveling salesman problem (TSP). First, we devise randomized approximation algorithms for multicriteria maximum traveling salesman problems (Max-TSP). For multicriteria Max-STSP where the edge weights have to be
Boundary Value Problems and Approximate Solutions ...
African Journals Online (AJOL)
In this paper, we discuss about some basic things of boundary value problems. Secondly, we study boundary conditions involving derivatives and obtain finite difference approximations of partial derivatives of boundary value problems. The last section is devoted to determine an approximate solution for boundary value ...
Polynomial approximation approach to transient heat conduction ...
African Journals Online (AJOL)
This work reports polynomial approximation approach to transient heat conduction in a long slab, long cylinder and sphere with linear internal heat generation. It has been shown that the polynomial approximation method is able to calculate average temperature as a function of time for higher value of Biot numbers.
Approximation algorithms for guarding holey polygons ...
African Journals Online (AJOL)
Guarding edges of polygons is a version of art gallery problem.The goal is finding the minimum number of guards to cover the edges of a polygon. This problem is NP-hard, and to our knowledge there are approximation algorithms just for simple polygons. In this paper we present two approximation algorithms for guarding ...
Similarity based approximate reasoning: fuzzy control
Raha, S.; Hossain, A.; Ghosh, S.
2008-01-01
This paper presents an approach to similarity based approximate reasoning that elucidates the connection between similarity and existing approaches to inference in approximate reasoning methodology. A set of axioms is proposed to get a reasonable measure of similarity between two fuzzy sets. The
Improved Dutch Roll Approximation for Hypersonic Vehicle
Directory of Open Access Journals (Sweden)
Liang-Liang Yin
2014-06-01
Full Text Available An improved dutch roll approximation for hypersonic vehicle is presented. From the new approximations, the dutch roll frequency is shown to be a function of the stability axis yaw stability and the dutch roll damping is mainly effected by the roll damping ratio. In additional, an important parameter called roll-to-yaw ratio is obtained to describe the dutch roll mode. Solution shows that large-roll-to-yaw ratio is the generate character of hypersonic vehicle, which results the large error for the practical approximation. Predictions from the literal approximations derived in this paper are compared with actual numerical values for s example hypersonic vehicle, results show the approximations work well and the error is below 10 %.
Approximate error conjugation gradient minimization methods
Kallman, Jeffrey S
2013-05-21
In one embodiment, a method includes selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, calculating an approximate error using the subset of rays, and calculating a minimum in a conjugate gradient direction based on the approximate error. In another embodiment, a system includes a processor for executing logic, logic for selecting a subset of rays from a set of all rays to use in an error calculation for a constrained conjugate gradient minimization problem, logic for calculating an approximate error using the subset of rays, and logic for calculating a minimum in a conjugate gradient direction based on the approximate error. In other embodiments, computer program products, methods, and systems are described capable of using approximate error in constrained conjugate gradient minimization problems.
The tendon approximator device in traumatic injuries.
Forootan, Kamal S; Karimi, Hamid; Forootan, Nazilla-Sadat S
2015-01-01
Precise and tension-free approximation of two tendon endings is the key predictor of outcomes following tendon lacerations and repairs. We evaluate the efficacy of a new tendon approximator device in tendon laceration repairs. In a comparative study, we used our new tendon approximator device in 99 consecutive patients with laceration of 266 tendons who attend a university hospital and evaluated the operative time to repair the tendons, surgeons' satisfaction as well as patient's outcomes in a long-term follow-up. Data were compared with the data of control patients undergoing tendon repair by conventional method. Totally 266 tendons were repaired by approximator device and 199 tendons by conventional technique. 78.7% of patients in first group were male and 21.2% were female. In approximator group 38% of patients had secondary repair of cut tendons and 62% had primary repair. Patients were followed for a mean period of 3years (14-60 months). Time required for repair of each tendon was significantly reduced with the approximator device (2 min vs. 5.5 min, ptendon repair were identical in the two groups and were not significantly different. 1% of tendons in group A and 1.2% in group B had rupture that was not significantly different. The new nerve approximator device is cheap, feasible to use and reduces the time of tendon repair with sustained outcomes comparable to the conventional methods.
Hardness and Approximation for Network Flow Interdiction
Chestnut, Stephen R.; Zenklusen, Rico
2015-01-01
In the Network Flow Interdiction problem an adversary attacks a network in order to minimize the maximum s-t-flow. Very little is known about the approximatibility of this problem despite decades of interest in it. We present the first approximation hardness, showing that Network Flow Interdiction and several of its variants cannot be much easier to approximate than Densest k-Subgraph. In particular, any $n^{o(1)}$-approximation algorithm for Network Flow Interdiction would imply an $n^{o(1)}...
Regression with Sparse Approximations of Data
DEFF Research Database (Denmark)
Noorzad, Pardis; Sturm, Bob L.
2012-01-01
We propose sparse approximation weighted regression (SPARROW), a method for local estimation of the regression function that uses sparse approximation with a dictionary of measurements. SPARROW estimates the regression function at a point with a linear combination of a few regressands selected...... by a sparse approximation of the point in terms of the regressors. We show SPARROW can be considered a variant of \\(k\\)-nearest neighbors regression (\\(k\\)-NNR), and more generally, local polynomial kernel regression. Unlike \\(k\\)-NNR, however, SPARROW can adapt the number of regressors to use based...
Approximately Liner Phase IIR Digital Filter Banks
Directory of Open Access Journals (Sweden)
J. D. Ćertić
2013-11-01
Full Text Available In this paper, uniform and nonuniform digital filter banks based on approximately linear phase IIR filters and frequency response masking technique (FRM are presented. Both filter banks are realized as a connection of an interpolated half-band approximately linear phase IIR filter as a first stage of the FRM design and an appropriate number of masking filters. The masking filters are half-band IIR filters with an approximately linear phase. The resulting IIR filter banks are compared with linear-phase FIR filter banks exhibiting similar magnitude responses. The effects of coefficient quantization are analyzed.
Mathematical analysis, approximation theory and their applications
Gupta, Vijay
2016-01-01
Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.
Approximation of the semi-infinite interval
Directory of Open Access Journals (Sweden)
A. McD. Mercer
1980-01-01
Full Text Available The approximation of a function f∈C[a,b] by Bernstein polynomials is well-known. It is based on the binomial distribution. O. Szasz has shown that there are analogous approximations on the interval [0,∞ based on the Poisson distribution. Recently R. Mohapatra has generalized Szasz' result to the case in which the approximating function is αe−ux∑k=N∞(uxkα+β−1Γ(kα+βf(kαuThe present note shows that these results are special cases of a Tauberian theorem for certain infinite series having positive coefficients.
Pion-nucleus cross sections approximation
International Nuclear Information System (INIS)
Barashenkov, V.S.; Polanski, A.; Sosnin, A.N.
1990-01-01
Analytical approximation of pion-nucleus elastic and inelastic interaction cross-section is suggested, with could be applied in the energy range exceeding several dozens of MeV for nuclei heavier than beryllium. 3 refs.; 4 tabs
Square well approximation to the optical potential
International Nuclear Information System (INIS)
Jain, A.K.; Gupta, M.C.; Marwadi, P.R.
1976-01-01
Approximations for obtaining T-matrix elements for a sum of several potentials in terms of T-matrices for individual potentials are studied. Based on model calculations for S-wave for a sum of two separable non-local potentials of Yukawa type form factors and a sum of two delta function potentials, it is shown that the T-matrix for a sum of several potentials can be approximated satisfactorily over all the energy regions by the sum of T-matrices for individual potentials. Based on this, an approximate method for finding T-matrix for any local potential by approximating it by a sum of suitable number of square wells is presented. This provides an interesting way to calculate the T-matrix for any arbitary potential in terms of Bessel functions to a good degree of accuracy. The method is applied to the Saxon-Wood potentials and good agreement with exact results is found. (author)
Seismic wave extrapolation using lowrank symbol approximation
Fomel, Sergey
2012-04-30
We consider the problem of constructing a wave extrapolation operator in a variable and possibly anisotropic medium. Our construction involves Fourier transforms in space combined with the help of a lowrank approximation of the space-wavenumber wave-propagator matrix. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. We present a mathematical derivation of this method, a description of the lowrank approximation algorithm and numerical examples that confirm the validity of the proposed approach. Wave extrapolation using lowrank approximation can be applied to seismic imaging by reverse-time migration in 3D heterogeneous isotropic or anisotropic media. © 2012 European Association of Geoscientists & Engineers.
Steepest descent approximations for accretive operator equations
International Nuclear Information System (INIS)
Chidume, C.E.
1993-03-01
A necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi-accretive operators defined on a uniformly smooth Banach space. (author). 49 refs
Degree of Approximation and Green Potential
Directory of Open Access Journals (Sweden)
M. Simkani
2009-03-01
Full Text Available We will relate the degree of rational approximation of a meromorphic function f to the minimum value, on the natural boundary of f, of Green potential of the weak∗ limit of the normalized pole-counting measures
Low Rank Approximation Algorithms, Implementation, Applications
Markovsky, Ivan
2012-01-01
Matrix low-rank approximation is intimately related to data modelling; a problem that arises frequently in many different fields. Low Rank Approximation: Algorithms, Implementation, Applications is a comprehensive exposition of the theory, algorithms, and applications of structured low-rank approximation. Local optimization methods and effective suboptimal convex relaxations for Toeplitz, Hankel, and Sylvester structured problems are presented. A major part of the text is devoted to application of the theory. Applications described include: system and control theory: approximate realization, model reduction, output error, and errors-in-variables identification; signal processing: harmonic retrieval, sum-of-damped exponentials, finite impulse response modeling, and array processing; machine learning: multidimensional scaling and recommender system; computer vision: algebraic curve fitting and fundamental matrix estimation; bioinformatics for microarray data analysis; chemometrics for multivariate calibration; ...
Methods of Fourier analysis and approximation theory
Tikhonov, Sergey
2016-01-01
Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.
APPROXIMATE DEVELOPMENTS FOR SURFACES OF REVOLUTION
Directory of Open Access Journals (Sweden)
Mădălina Roxana Buneci
2016-12-01
Full Text Available The purpose of this paper is provide a set of Maple procedures to construct approximate developments of a general surface of revolution generalizing the well-known gore method for sphere
An overview on Approximate Bayesian computation*
Directory of Open Access Journals (Sweden)
Baragatti Meïli
2014-01-01
Full Text Available Approximate Bayesian computation techniques, also called likelihood-free methods, are one of the most satisfactory approach to intractable likelihood problems. This overview presents recent results since its introduction about ten years ago in population genetics.
Approximation for the adjoint neutron spectrum
International Nuclear Information System (INIS)
Suster, Luis Carlos; Martinez, Aquilino Senra; Silva, Fernando Carvalho da
2002-01-01
The proposal of this work is the determination of an analytical approximation which is capable to reproduce the adjoint neutron flux for the energy range of the narrow resonances (NR). In a previous work we developed a method for the calculation of the adjoint spectrum which was calculated from the adjoint neutron balance equations, that were obtained by the collision probabilities method, this method involved a considerable quantity of numerical calculation. In the analytical method some approximations were done, like the multiplication of the escape probability in the fuel by the adjoint flux in the moderator, and after these approximations, taking into account the case of the narrow resonances, were substituted in the adjoint neutron balance equation for the fuel, resulting in an analytical approximation for the adjoint flux. The results obtained in this work were compared to the results generated with the reference method, which demonstrated a good and precise results for the adjoint neutron flux for the narrow resonances. (author)
TMB: Automatic differentiation and laplace approximation
DEFF Research Database (Denmark)
Kristensen, Kasper; Nielsen, Anders; Berg, Casper Willestofte
2016-01-01
are automatically integrated out. This approximation, and its derivatives, are obtained using automatic differentiation (up to order three) of the joint likelihood. The computations are designed to be fast for problems with many random effects (approximate to 10(6)) and parameters (approximate to 10...... computations. The user defines the joint likelihood for the data and the random effects as a C++ template function, while all the other operations are done in R; e.g., reading in the data. The package evaluates and maximizes the Laplace approximation of the marginal likelihood where the random effects......(3)). Computation times using ADMB and TMB are compared on a suite of examples ranging from simple models to large spatial models where the random effects are a Gaussian random field. Speedups ranging from 1.5 to about 100 are obtained with increasing gains for large problems...
Approximations of Stochastic Partial Differential Equations
Di Nunno, Giulia; Zhang, Tusheng
2014-01-01
In this paper we show that solutions of stochastic partial differ- ential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.
Saddlepoint approximation methods in financial engineering
Kwok, Yue Kuen
2018-01-01
This book summarizes recent advances in applying saddlepoint approximation methods to financial engineering. It addresses pricing exotic financial derivatives and calculating risk contributions to Value-at-Risk and Expected Shortfall in credit portfolios under various default correlation models. These standard problems involve the computation of tail probabilities and tail expectations of the corresponding underlying state variables. The text offers in a single source most of the saddlepoint approximation results in financial engineering, with different sets of ready-to-use approximation formulas. Much of this material may otherwise only be found in original research publications. The exposition and style are made rigorous by providing formal proofs of most of the results. Starting with a presentation of the derivation of a variety of saddlepoint approximation formulas in different contexts, this book will help new researchers to learn the fine technicalities of the topic. It will also be valuable to quanti...
Approximative solutions of stochastic optimization problem
Czech Academy of Sciences Publication Activity Database
Lachout, Petr
2010-01-01
Roč. 46, č. 3 (2010), s. 513-523 ISSN 0023-5954 R&D Projects: GA ČR GA201/08/0539 Institutional research plan: CEZ:AV0Z10750506 Keywords : Stochastic optimization problem * sensitivity * approximative solution Subject RIV: BA - General Mathematics Impact factor: 0.461, year: 2010 http://library.utia.cas.cz/separaty/2010/SI/lachout-approximative solutions of stochastic optimization problem.pdf
Nonlinear approximation with nonstationary Gabor frames
DEFF Research Database (Denmark)
Ottosen, Emil Solsbæk; Nielsen, Morten
2018-01-01
We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions...... resolution. Based on this characterization we prove an upper bound on the approximation error occurring when thresholding the coefficients of the corresponding frame expansions. We complement the theoretical results with numerical experiments, estimating the rate of approximation obtained from thresholding...
Lattice quantum chromodynamics with approximately chiral fermions
Energy Technology Data Exchange (ETDEWEB)
Hierl, Dieter
2008-05-15
In this work we present Lattice QCD results obtained by approximately chiral fermions. We use the CI fermions in the quenched approximation to investigate the excited baryon spectrum and to search for the {theta}{sup +} pentaquark on the lattice. Furthermore we developed an algorithm for dynamical simulations using the FP action. Using FP fermions we calculate some LECs of chiral perturbation theory applying the epsilon expansion. (orig.)
On surface approximation using developable surfaces
DEFF Research Database (Denmark)
Chen, H. Y.; Lee, I. K.; Leopoldseder, s.
1999-01-01
We introduce a method for approximating a given surface by a developable surface. It will be either a G(1) surface consisting of pieces of cones or cylinders of revolution or a G(r) NURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produ...... robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding. (C) 1999 Academic Press....
On surface approximation using developable surfaces
DEFF Research Database (Denmark)
Chen, H. Y.; Lee, I. K.; Leopoldseder, S.
1998-01-01
We introduce a method for approximating a given surface by a developable surface. It will be either a G_1 surface consisting of pieces of cones or cylinders of revolution or a G_r NURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produce...... robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding....
Lattice quantum chromodynamics with approximately chiral fermions
International Nuclear Information System (INIS)
Hierl, Dieter
2008-05-01
In this work we present Lattice QCD results obtained by approximately chiral fermions. We use the CI fermions in the quenched approximation to investigate the excited baryon spectrum and to search for the Θ + pentaquark on the lattice. Furthermore we developed an algorithm for dynamical simulations using the FP action. Using FP fermions we calculate some LECs of chiral perturbation theory applying the epsilon expansion. (orig.)
Nonlinear Stochastic PDEs: Analysis and Approximations
2016-05-23
Approximation to Nonlinear SPDEs with Discrete Random Variables , SIAM J Scientific Computing, (08 2015): 1872. doi: R. Mikulevicius, B. Rozovskii. On...multiplicative discrete random variables , ( ) S. Lototsky, B. Rozovsky. Stochastic Partial Differential Equations, (09 2015) B. Rozovsky, R...B. Rozovsky and G.E. Karniadakis, "Adaptive Wick-Malliavin approximation to nonlinear SPDEs with discrete random variables ," SIAM J. Sci. Comput., 37
Polynomial approximation of functions in Sobolev spaces
International Nuclear Information System (INIS)
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomical plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces
Approximation properties ofλ-Bernstein operators.
Cai, Qing-Bo; Lian, Bo-Yong; Zhou, Guorong
2018-01-01
In this paper, we introduce a new type λ -Bernstein operators with parameter [Formula: see text], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of [Formula: see text] to [Formula: see text], and we see that in some cases the errors are smaller than [Formula: see text] to f .
Rough Sets Approximations for Learning Outcomes
Encheva, Sylvia; Tumin, Sharil
Discovering dependencies between students' responses and their level of mastering of a particular skill is very important in the process of developing intelligent tutoring systems. This work is an approach to attain a higher level of certainty while following students' learning progress. Rough sets approximations are applied for assessing students understanding of a concept. Consecutive responses from each individual learner to automated tests are placed in corresponding rough sets approximations. The resulting path provides strong indication about the current level of learning outcomes.
Approximations for the Erlang Loss Function
DEFF Research Database (Denmark)
Mejlbro, Leif
1998-01-01
Theoretically, at least three formulae are needed for arbitrarily good approximates of the Erlang Loss Function. In the paper, for convenience five formulae are presented guaranteeing a relative error <1E-2, and methods are indicated for improving this bound.......Theoretically, at least three formulae are needed for arbitrarily good approximates of the Erlang Loss Function. In the paper, for convenience five formulae are presented guaranteeing a relative error
Approximating centrality in evolving graphs: toward sublinearity
Priest, Benjamin W.; Cybenko, George
2017-05-01
The identification of important nodes is a ubiquitous problem in the analysis of social networks. Centrality indices (such as degree centrality, closeness centrality, betweenness centrality, PageRank, and others) are used across many domains to accomplish this task. However, the computation of such indices is expensive on large graphs. Moreover, evolving graphs are becoming increasingly important in many applications. It is therefore desirable to develop on-line algorithms that can approximate centrality measures using memory sublinear in the size of the graph. We discuss the challenges facing the semi-streaming computation of many centrality indices. In particular, we apply recent advances in the streaming and sketching literature to provide a preliminary streaming approximation algorithm for degree centrality utilizing CountSketch and a multi-pass semi-streaming approximation algorithm for closeness centrality leveraging a spanner obtained through iteratively sketching the vertex-edge adjacency matrix. We also discuss possible ways forward for approximating betweenness centrality, as well as spectral measures of centrality. We provide a preliminary result using sketched low-rank approximations to approximate the output of the HITS algorithm.
The Grammar of Approximating Number Pairs
Eriksson, Kimmo; Bailey, Drew H.; Geary, David C.
2009-01-01
We studied approximating pairs of numbers (a, b) used to estimate quantity in a single phrase (“two, three years ago”). Pollmann and Jansen (1996) found that only a few of the many possible pairs are actually used, suggesting an interaction between the ways in which people estimate quantity and their use of quantitative phrases in colloquial speech. They proposed a set of rules that describe which approximating pairs are used in Dutch phrases. We revisit this issue in an analysis of Swedish and American language corpora and in a series of three experiments in which Swedish and American adults rated the acceptability of various approximating pairs, and created approximating pairs of their own in response to various estimation tasks. We find evidence for Pollmann’s and Jansen’s rules in both Swedish and English phrases, but also identify additional rules and substantial individual and cross-language variation. We discuss implications for the origin of this loose “grammar” of approximating pairs. PMID:20234023
'LTE-diffusion approximation' for arc calculations
International Nuclear Information System (INIS)
Lowke, J J; Tanaka, M
2006-01-01
This paper proposes the use of the 'LTE-diffusion approximation' for predicting the properties of electric arcs. Under this approximation, local thermodynamic equilibrium (LTE) is assumed, with a particular mesh size near the electrodes chosen to be equal to the 'diffusion length', based on D e /W, where D e is the electron diffusion coefficient and W is the electron drift velocity. This approximation overcomes the problem that the equilibrium electrical conductivity in the arc near the electrodes is almost zero, which makes accurate calculations using LTE impossible in the limit of small mesh size, as then voltages would tend towards infinity. Use of the LTE-diffusion approximation for a 200 A arc with a thermionic cathode gives predictions of total arc voltage, electrode temperatures, arc temperatures and radial profiles of heat flux density and current density at the anode that are in approximate agreement with more accurate calculations which include an account of the diffusion of electric charges to the electrodes, and also with experimental results. Calculations, which include diffusion of charges, agree with experimental results of current and heat flux density as a function of radius if the Milne boundary condition is used at the anode surface rather than imposing zero charge density at the anode
Approximate Bayesian evaluations of measurement uncertainty
Possolo, Antonio; Bodnar, Olha
2018-04-01
The Guide to the Expression of Uncertainty in Measurement (GUM) includes formulas that produce an estimate of a scalar output quantity that is a function of several input quantities, and an approximate evaluation of the associated standard uncertainty. This contribution presents approximate, Bayesian counterparts of those formulas for the case where the output quantity is a parameter of the joint probability distribution of the input quantities, also taking into account any information about the value of the output quantity available prior to measurement expressed in the form of a probability distribution on the set of possible values for the measurand. The approximate Bayesian estimates and uncertainty evaluations that we present have a long history and illustrious pedigree, and provide sufficiently accurate approximations in many applications, yet are very easy to implement in practice. Differently from exact Bayesian estimates, which involve either (analytical or numerical) integrations, or Markov Chain Monte Carlo sampling, the approximations that we describe involve only numerical optimization and simple algebra. Therefore, they make Bayesian methods widely accessible to metrologists. We illustrate the application of the proposed techniques in several instances of measurement: isotopic ratio of silver in a commercial silver nitrate; odds of cryptosporidiosis in AIDS patients; height of a manometer column; mass fraction of chromium in a reference material; and potential-difference in a Zener voltage standard.
Semiclassical initial value approximation for Green's function.
Kay, Kenneth G
2010-06-28
A semiclassical initial value approximation is obtained for the energy-dependent Green's function. For a system with f degrees of freedom the Green's function expression has the form of a (2f-1)-dimensional integral over points on the energy surface and an integral over time along classical trajectories initiated from these points. This approximation is derived by requiring an integral ansatz for Green's function to reduce to Gutzwiller's semiclassical formula when the integrations are performed by the stationary phase method. A simpler approximation is also derived involving only an (f-1)-dimensional integral over momentum variables on a Poincare surface and an integral over time. The relationship between the present expressions and an earlier initial value approximation for energy eigenfunctions is explored. Numerical tests for two-dimensional systems indicate that good accuracy can be obtained from the initial value Green's function for calculations of autocorrelation spectra and time-independent wave functions. The relative advantages of initial value approximations for the energy-dependent Green's function and the time-dependent propagator are discussed.
Trajectory averaging for stochastic approximation MCMC algorithms
Liang, Faming
2010-10-01
The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400-407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and it has also served as a prototype for the development of adaptive algorithms for on-line estimation and control of stochastic systems. Recently, it has been used in statistics with Markov chain Monte Carlo for solving maximum likelihood estimation problems and for general simulation and optimizations. In this paper, we first show that the trajectory averaging estimator is asymptotically efficient for the stochastic approximation MCMC (SAMCMC) algorithm under mild conditions, and then apply this result to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305-320]. The application of the trajectory averaging estimator to other stochastic approximationMCMC algorithms, for example, a stochastic approximation MLE algorithm for missing data problems, is also considered in the paper. © Institute of Mathematical Statistics, 2010.
The grammar of approximating number pairs.
Eriksson, Kimmo; Bailey, Drew H; Geary, David C
2010-04-01
In the present article, we studied approximating pairs of numbers (a, b) that were used to estimate quantity in a single phrase ("two, three years ago"). Pollmann and Jansen (1996) found that only a few of the many possible pairs are actually used, suggesting an interaction between the ways in which people estimate quantity and their use of quantitative phrases in colloquial speech. They proposed a set of rules that describe which approximating pairs are used in Dutch phrases. We revisited this issue in an analysis of Swedish and American language corpora and in a series of three experiments in which Swedish and American adults rated the acceptability of various approximating pairs and created approximating pairs of their own in response to various estimation tasks. We found evidence for Pollmann and Jansen's rules in both Swedish and English phrases, but we also identified additional rules and substantial individual and cross-language variation. We will discuss implications for the origin of this loose "grammar" of approximating pairs.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef
2017-06-30
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
Numerical approximation of partial differential equations
Bartels, Sören
2016-01-01
Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular ...
Fast wavelet based sparse approximate inverse preconditioner
Energy Technology Data Exchange (ETDEWEB)
Wan, W.L. [Univ. of California, Los Angeles, CA (United States)
1996-12-31
Incomplete LU factorization is a robust preconditioner for both general and PDE problems but unfortunately not easy to parallelize. Recent study of Huckle and Grote and Chow and Saad showed that sparse approximate inverse could be a potential alternative while readily parallelizable. However, for special class of matrix A that comes from elliptic PDE problems, their preconditioners are not optimal in the sense that independent of mesh size. A reason may be that no good sparse approximate inverse exists for the dense inverse matrix. Our observation is that for this kind of matrices, its inverse entries typically have piecewise smooth changes. We can take advantage of this fact and use wavelet compression techniques to construct a better sparse approximate inverse preconditioner. We shall show numerically that our approach is effective for this kind of matrices.
Piecewise-Cubic Approximation in Autotracking Mode
Dikoussar, N D
2004-01-01
A method for piecewise-cubic approximation within the frame of four-point transforms is proposed. The knots of the segments are detected in autotracking mode using a digitized curve. A three-point cubic parametric spline (TPS) is used as a model of a local approximant. A free parameter $\\theta$ (a coefficient at $x^{3}$) is found in a line following mode, using step-by-step averaging. A formula for expression of the free parameter via a length of the segment and values of a function and derivatives in joining points is received. The $C^{1}$-smoothness depends on the accuracy of the $\\theta$-estimate. The stability of the method w.r.t. input errors is shown as well. The key parameters of the approximation are: the parameters of the basic functions, the variance of the input errors, and a sampling step. The efficiency of the method is shown by numerical calculations on test examples.
On transparent potentials: a Born approximation study
International Nuclear Information System (INIS)
Coudray, C.
1980-01-01
In the frame of the scattering inverse problem at fixed energy, a class of potentials transparent in Born approximation is obtained. All these potentials are spherically symmetric and are oscillating functions of the reduced radial variable. Amongst them, the Born approximation of the transparent potential of the Newton-Sabatier method is found. In the same class, quasi-transparent potentials are exhibited. Very general features of potentials transparent in Born approximation are then stated. And bounds are given for the exact scattering amplitudes corresponding to most of the potentials previously exhibited. These bounds, obtained at fixed energy, and for large values of the angular momentum, are found to be independent on the energy
The adiabatic approximation in multichannel scattering
International Nuclear Information System (INIS)
Schulte, A.M.
1978-01-01
Using two-dimensional models, an attempt has been made to get an impression of the conditions of validity of the adiabatic approximation. For a nucleon bound to a rotating nucleus the Coriolis coupling is neglected and the relation between this nuclear Coriolis coupling and the classical Coriolis force has been examined. The approximation for particle scattering from an axially symmetric rotating nucleus based on a short duration of the collision, has been combined with an approximation based on the limitation of angular momentum transfer between particle and nucleus. Numerical calculations demonstrate the validity of the new combined method. The concept of time duration for quantum mechanical collisions has also been studied, as has the collective description of permanently deformed nuclei. (C.F.)
The binary collision approximation: Background and introduction
International Nuclear Information System (INIS)
Robinson, M.T.
1992-08-01
The binary collision approximation (BCA) has long been used in computer simulations of the interactions of energetic atoms with solid targets, as well as being the basis of most analytical theory in this area. While mainly a high-energy approximation, the BCA retains qualitative significance at low energies and, with proper formulation, gives useful quantitative information as well. Moreover, computer simulations based on the BCA can achieve good statistics in many situations where those based on full classical dynamical models require the most advanced computer hardware or are even impracticable. The foundations of the BCA in classical scattering are reviewed, including methods of evaluating the scattering integrals, interaction potentials, and electron excitation effects. The explicit evaluation of time at significant points on particle trajectories is discussed, as are scheduling algorithms for ordering the collisions in a developing cascade. An approximate treatment of nearly simultaneous collisions is outlined and the searching algorithms used in MARLOWE are presented
Minimal entropy approximation for cellular automata
International Nuclear Information System (INIS)
Fukś, Henryk
2014-01-01
We present a method for the construction of approximate orbits of measures under the action of cellular automata which is complementary to the local structure theory. The local structure theory is based on the idea of Bayesian extension, that is, construction of a probability measure consistent with given block probabilities and maximizing entropy. If instead of maximizing entropy one minimizes it, one can develop another method for the construction of approximate orbits, at the heart of which is the iteration of finite-dimensional maps, called minimal entropy maps. We present numerical evidence that the minimal entropy approximation sometimes outperforms the local structure theory in characterizing the properties of cellular automata. The density response curve for elementary CA rule 26 is used to illustrate this claim. (paper)
APPROXIMATE INTEGRATION OF HIGHLY OSCILLATING FUNCTIONS
Directory of Open Access Journals (Sweden)
I. N. Melashko
2017-01-01
Full Text Available Elementary approximate formulae for numerical integration of functions containing oscillating factors of a special form with a parameter have been proposed in the paper. In this case general quadrature formulae can be used only at sufficiently small values of the parameter. Therefore, it is necessary to consider in advance presence of strongly oscillating factors in order to obtain formulae for numerical integration which are suitable in the case when the parameter is changing within wide limits. This can be done by taking into account such factors as weighting functions. Moreover, since the parameter can take values which cannot always be predicted in advance, approximate formulae for calculation of such integrals should be constructed in such a way that they contain this parameter in a letter format and they are suitable for calculation at any and particularly large values of the parameter. Computational rules with such properties are generally obtained by dividing an interval of integration into elementary while making successive approximation of the integral density at each elementary interval with polynomials of the first, second and third degrees and taking the oscillating factors as weighting functions. The paper considers the variant when density of the integrals at each elementary interval is approximated by a polynomial of zero degree that is a constant which is equal to the value of density in the middle of the interval. At the same time one approximate formula for calculation of an improper integral with infinite interval of the function with oscillating factor of a special type has been constructed in the paper. In this case it has been assumed that density of the improper integral rather quickly goes to zero when an argument module is increasing indefinitely. In other words it is considered as small to negligible outside some finite interval. Uniforms in parameter used for evaluation of errors in approximate formulae have been
Stopping Rules for Linear Stochastic Approximation
Wada, Takayuki; Itani, Takamitsu; Fujisaki, Yasumasa
Stopping rules are developed for stochastic approximation which is an iterative method for solving an unknown equation based on its consecutive residuals corrupted by additive random noise. It is assumed that the equation is linear and the noise is independent and identically distributed random vectors with zero mean and a bounded covariance. Then, the number of iterations for achieving a given probabilistic accuracy of the resultant solution is derived, which gives a rigorous stopping rule for the stochastic approximation. This number is polynomial of the problem size.
Computational topology for approximations of knots
Directory of Open Access Journals (Sweden)
Ji Li
2014-10-01
• a sum of total curvature and derivative. High degree Bézier curves are often used as smooth representations, where computational efficiency is a practical concern. Subdivision can produce PL approximations for a given B\\'ezier curve, fulfilling the above two conditions. The primary contributions are: (i a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original B\\'ezier curve, and (ii improved iteration bounds over those previously established.
On the dipole approximation with error estimates
Boßmann, Lea; Grummt, Robert; Kolb, Martin
2018-01-01
The dipole approximation is employed to describe interactions between atoms and radiation. It essentially consists of neglecting the spatial variation of the external field over the atom. Heuristically, this is justified by arguing that the wavelength is considerably larger than the atomic length scale, which holds under usual experimental conditions. We prove the dipole approximation in the limit of infinite wavelengths compared to the atomic length scale and estimate the rate of convergence. Our results include N-body Coulomb potentials and experimentally relevant electromagnetic fields such as plane waves and laser pulses.
Static correlation beyond the random phase approximation
DEFF Research Database (Denmark)
Olsen, Thomas; Thygesen, Kristian Sommer
2014-01-01
derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly...... for intermediate binding distances. A Hubbard model for the dimer allows us to obtain exact analytical results for the various approximations, which is readily compared with the exact diagonalization of the model. Moreover, the model is shown to reproduce all the qualitative results from the ab initio calculations...
Faster and Simpler Approximation of Stable Matchings
Directory of Open Access Journals (Sweden)
Katarzyna Paluch
2014-04-01
Full Text Available We give a 3 2 -approximation algorithm for finding stable matchings that runs in O(m time. The previous most well-known algorithm, by McDermid, has the same approximation ratio but runs in O(n3/2m time, where n denotes the number of people andm is the total length of the preference lists in a given instance. In addition, the algorithm and the analysis are much simpler. We also give the extension of the algorithm for computing stable many-to-many matchings.
An approximate analytical approach to resampling averages
DEFF Research Database (Denmark)
Malzahn, Dorthe; Opper, M.
2004-01-01
Using a novel reformulation, we develop a framework to compute approximate resampling data averages analytically. The method avoids multiple retraining of statistical models on the samples. Our approach uses a combination of the replica "trick" of statistical physics and the TAP approach for appr......Using a novel reformulation, we develop a framework to compute approximate resampling data averages analytically. The method avoids multiple retraining of statistical models on the samples. Our approach uses a combination of the replica "trick" of statistical physics and the TAP approach...
APPROXIMATION OF PROBABILITY DISTRIBUTIONS IN QUEUEING MODELS
Directory of Open Access Journals (Sweden)
T. I. Aliev
2013-03-01
Full Text Available For probability distributions with variation coefficient, not equal to unity, mathematical dependences for approximating distributions on the basis of first two moments are derived by making use of multi exponential distributions. It is proposed to approximate distributions with coefficient of variation less than unity by using hypoexponential distribution, which makes it possible to generate random variables with coefficient of variation, taking any value in a range (0; 1, as opposed to Erlang distribution, having only discrete values of coefficient of variation.
Approximate Controllability of Fractional Integrodifferential Evolution Equations
Directory of Open Access Journals (Sweden)
R. Ganesh
2013-01-01
Full Text Available This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory, p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.
Approximated Fractional Order Chebyshev Lowpass Filters
Directory of Open Access Journals (Sweden)
Todd Freeborn
2015-01-01
Full Text Available We propose the use of nonlinear least squares optimization to approximate the passband ripple characteristics of traditional Chebyshev lowpass filters with fractional order steps in the stopband. MATLAB simulations of (1+α, (2+α, and (3+α order lowpass filters with fractional steps from α = 0.1 to α = 0.9 are given as examples. SPICE simulations of 1.2, 1.5, and 1.8 order lowpass filters using approximated fractional order capacitors in a Tow-Thomas biquad circuit validate the implementation of these filter circuits.
Approximate Inference and Deep Generative Models
CERN. Geneva
2018-01-01
Advances in deep generative models are at the forefront of deep learning research because of the promise they offer for allowing data-efficient learning, and for model-based reinforcement learning. In this talk I'll review a few standard methods for approximate inference and introduce modern approximations which allow for efficient large-scale training of a wide variety of generative models. Finally, I'll demonstrate several important application of these models to density estimation, missing data imputation, data compression and planning.
Approximation result toward nearest neighbor heuristic
Directory of Open Access Journals (Sweden)
Monnot Jér"me
2002-01-01
Full Text Available In this paper, we revisit the famous heuristic called nearest neighbor (N N for the traveling salesman problem under maximization and minimization goal. We deal with variants where the edge costs belong to interval Ša;taĆ for a>0 and t>1, which certainly corresponds to practical cases of these problems. We prove that NN is a (t+1/2t-approximation for maxTSPŠa;taĆ and a 2/(t+1-approximation for minTSPŠa;taĆ under the standard performance ratio. Moreover, we show that these ratios are tight for some instances.
Radiation forces in the discrete dipole approximation
Hoekstra, A.G.; Frijlink, M.O.; Waters, L.B.F.M.; Sloot, P.M.A.
2001-01-01
The theory of the discrete-dipole approximation (DDA) for light scattering is extended to allow for the calculation of radiation forces on each dipole in the DDA model. Starting with the theory of Draine and Weingartner [Astrophys. J. 470, 551 (1996)] we derive an expression for the radiation force
Perturbation of operators and approximation of spectrum
Indian Academy of Sciences (India)
The pure linear algebraic approach is the main advantage of the results here. Keywords. Operator .... The paper is organized as follows. In §2, the approximation results are extended to the case of a one-parameter norm continuous family of operators. In §3, the spectral gap prediction results are proved with some examples.
Isotopic Approximation of Implicit Curves and Surfaces
Plantinga, Simon; Vegter, Gert
2004-01-01
Implicit surfaces are defined as the zero set of a function F: R3 → R. Although several algorithms exist for generating piecewise linear approximations, most of them are based on a user-defined stepsize or bounds to indicate the precision, and therefore cannot guarantee topological correctness.
A rational approximation of the effectiveness factor
DEFF Research Database (Denmark)
Wedel, Stig; Luss, Dan
1980-01-01
A fast, approximate method of calculating the effectiveness factor for arbitrary rate expressions is presented. The method does not require any iterative or interpolative calculations. It utilizes the well known asymptotic behavior for small and large Thiele moduli to derive a rational function w...
An approximate analytical approach to resampling averages
DEFF Research Database (Denmark)
Malzahn, Dorthe; Opper, M.
2004-01-01
Using a novel reformulation, we develop a framework to compute approximate resampling data averages analytically. The method avoids multiple retraining of statistical models on the samples. Our approach uses a combination of the replica "trick" of statistical physics and the TAP approach for appr...
Approximate solutions to variational inequalities and applications
Directory of Open Access Journals (Sweden)
M. Beatrice Lignola
1994-11-01
Full Text Available The aim of this paper is to investigate two concepts of approximate solutions to parametric variational inequalities in topological vector spaces for which the corresponding solution map is closed graph and/or lower semicontinuous and to apply the results to the stability of optimization problems with variational inequality constrains.
Nanostructures: Scattering beyond the Born approximation
Grigoriev, S.V.; Syromyatnikov, A. V.; Chumakov, A. P.; Grigoryeva, N.A.; Napolskii, K.S.; Roslyakov, I. V.; Eliseev, A.A.; Petukhov, A.V.; Eckerlebe, H.
2010-01-01
The neutron scattering on a two-dimensional ordered nanostructure with the third nonperiodic dimension can go beyond the Born approximation. In our model supported by the exact theoretical solution a well-correlated hexagonal porous structure of anodic aluminum oxide films acts as a peculiar
Large hierarchies from approximate R symmetries
International Nuclear Information System (INIS)
Kappl, Rolf; Ratz, Michael; Vaudrevange, Patrick K.S.
2008-12-01
We show that hierarchically small vacuum expectation values of the superpotential in supersymmetric theories can be a consequence of an approximate R symmetry. We briefly discuss the role of such small constants in moduli stabilization and understanding the huge hierarchy between the Planck and electroweak scales. (orig.)
Approximability and Parameterized Complexity of Minmax Values
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Hansen, Thomas Dueholm; Miltersen, Peter Bro
2008-01-01
We consider approximating the minmax value of a multi player game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of ε log n digits (for any constant ε > 0) is NP-hard, where n is the size of the game. On the other hand......, approximating the value with a precision of c log log n digits (for any constant c ≥ 1) can be done in quasi-polynomial time. We consider the parameterized complexity of the problem, with the parameter being the number of pure strategies k of the player for which the minmax value is computed. We show...... that if there are three players, k = 2 and there are only two possible rational payoffs, the minmax value is a rational number and can be computed exactly in linear time. In the general case, we show that the value can be approximated wigh any polynomial number of digits of accuracy in time n^O(k) . On the other hand, we...
An Approximate Bayesian Fundamental Frequency Estimator
DEFF Research Database (Denmark)
Nielsen, Jesper Kjær; Christensen, Mads Græsbøll; Jensen, Søren Holdt
2012-01-01
Joint fundamental frequency and model order estimation is an important problem in several applications such as speech and music processing. In this paper, we develop an approximate estimation algorithm of these quantities using Bayesian inference. The inference about the fundamental frequency...
Pade approximant calculations for neutron escape probability
International Nuclear Information System (INIS)
El Wakil, S.A.; Saad, E.A.; Hendi, A.A.
1984-07-01
The neutron escape probability from a non-multiplying slab containing internal source is defined in terms of a functional relation for the scattering function for the diffuse reflection problem. The Pade approximant technique is used to get numerical results which compare with exact results. (author)
Uniform semiclassical approximation for absorptive scattering systems
International Nuclear Information System (INIS)
Hussein, M.S.; Pato, M.P.
1987-07-01
The uniform semiclassical approximation of the elastic scattering amplitude is generalized to absorptive systems. An integral equation is derived which connects the absorption modified amplitude to the absorption free one. Division of the amplitude into a diffractive and refractive components is then made possible. (Author) [pt
Hierarchical matrix approximation of large covariance matrices
Litvinenko, Alexander
2015-11-30
We approximate large non-structured Matérn covariance matrices of size n×n in the H-matrix format with a log-linear computational cost and storage O(kn log n), where rank k ≪ n is a small integer. Applications are: spatial statistics, machine learning and image analysis, kriging and optimal design.
Nonlinear approximation with dictionaries. II. Inverse Estimates
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
2006-01-01
In this paper, which is the sequel to [16], we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block...
Nonlinear approximation with dictionaries,.. II: Inverse estimates
DEFF Research Database (Denmark)
Gribonval, Rémi; Nielsen, Morten
In this paper we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for separated decomposable dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually...
Generalized Lower and Upper Approximations in Quantales
Directory of Open Access Journals (Sweden)
Qimei Xiao
2012-01-01
Full Text Available We introduce the concepts of set-valued homomorphism and strong set-valued homomorphism of a quantale which are the extended notions of congruence and complete congruence, respectively. The properties of generalized lower and upper approximations, constructed by a set-valued mapping, are discussed.
Uncertainty relations for approximation and estimation
Energy Technology Data Exchange (ETDEWEB)
Lee, Jaeha, E-mail: jlee@post.kek.jp [Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 (Japan); Tsutsui, Izumi, E-mail: izumi.tsutsui@kek.jp [Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 (Japan); Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801 (Japan)
2016-05-27
We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cramér–Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is available as well, our inequality can treat both the position–momentum and the time–energy relations in one framework albeit handled differently. - Highlights: • Several inequalities interpreted as uncertainty relations for approximation/estimation are derived from a single ‘versatile inequality’. • The ‘versatile inequality’ sets a limit on the approximation of an observable and/or the estimation of a parameter by another observable. • The ‘versatile inequality’ turns into an elaboration of the Robertson–Kennard (Schrödinger) inequality and the Cramér–Rao inequality. • Both the position–momentum and the time–energy relation are treated in one framework. • In every case, Aharonov's weak value arises as a key geometrical ingredient, deciding the optimal choice for the proxy functions.
An approximate classical unimolecular reaction rate theory
Zhao, Meishan; Rice, Stuart A.
1992-05-01
We describe a classical theory of unimolecular reaction rate which is derived from the analysis of Davis and Gray by use of simplifying approximations. These approximations concern the calculation of the locations of, and the fluxes of phase points across, the bottlenecks to fragmentation and to intramolecular energy transfer. The bottleneck to fragment separation is represented as a vibration-rotation state dependent separatrix, which approximation is similar to but extends and improves the approximations for the separatrix introduced by Gray, Rice, and Davis and by Zhao and Rice. The novel feature in our analysis is the representation of the bottlenecks to intramolecular energy transfer as dividing surfaces in phase space; the locations of these dividing surfaces are determined by the same conditions as locate the remnants of robust tori with frequency ratios related to the golden mean (in a two degree of freedom system these are the cantori). The flux of phase points across each dividing surface is calculated with an analytic representation instead of a stroboscopic mapping. The rate of unimolecular reaction is identified with the net rate at which phase points escape from the region of quasiperiodic bounded motion to the region of free fragment motion by consecutively crossing the dividing surfaces for intramolecular energy exchange and the separatrix. This new theory generates predictions of the rates of predissociation of the van der Waals molecules HeI2, NeI2 and ArI2 which are in very good agreement with available experimental data.
Markov operators, positive semigroups and approximation processes
Altomare, Francesco; Leonessa, Vita; Rasa, Ioan
2015-01-01
In recent years several investigations have been devoted to the study of large classes of (mainly degenerate) initial-boundary value evolution problems in connection with the possibility to obtain a constructive approximation of the associated positive C_0-semigroups. In this research monograph we present the main lines of a theory which finds its root in the above-mentioned research field.
Orthorhombic rational approximants for decagonal quasicrystals
Indian Academy of Sciences (India)
Unknown
An important exercise in the study of rational approximants is to derive their metric, especially in relation to the corresponding quasicrystal or the underlying clusters. Kuo's model has ..... the smaller diagonal of the fat rhombus in the Penrose tiling. This length scale is obtained by a section along a1 in the Penrose tiling and ...
Uncertainty relations for approximation and estimation
International Nuclear Information System (INIS)
Lee, Jaeha; Tsutsui, Izumi
2016-01-01
We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cramér–Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is available as well, our inequality can treat both the position–momentum and the time–energy relations in one framework albeit handled differently. - Highlights: • Several inequalities interpreted as uncertainty relations for approximation/estimation are derived from a single ‘versatile inequality’. • The ‘versatile inequality’ sets a limit on the approximation of an observable and/or the estimation of a parameter by another observable. • The ‘versatile inequality’ turns into an elaboration of the Robertson–Kennard (Schrödinger) inequality and the Cramér–Rao inequality. • Both the position–momentum and the time–energy relation are treated in one framework. • In every case, Aharonov's weak value arises as a key geometrical ingredient, deciding the optimal choice for the proxy functions.
Approximation properties of fine hyperbolic graphs
Indian Academy of Sciences (India)
groups have invariant translation approximation property (ITAP, see Definition 2.2). He also pointed out that there was no serious difficulty in extending the main theorem to fine hyperbolic graphs, but he did not outline the proof. So in this paper, we first give a proof for this extension, see Theorem 1.1 below. Then we define ...
Approximate Dynamic Programming by Practical Examples
Mes, Martijn R.K.; Perez Rivera, Arturo Eduardo; Boucherie, Richard; van Dijk, Nico M.
2017-01-01
Computing the exact solution of an MDP model is generally difficult and possibly intractable for realistically sized problem instances. A powerful technique to solve the large scale discrete time multistage stochastic control processes is Approximate Dynamic Programming (ADP). Although ADP is used
Approximation Algorithms for Model-Based Diagnosis
Feldman, A.B.
2010-01-01
Model-based diagnosis is an area of abductive inference that uses a system model, together with observations about system behavior, to isolate sets of faulty components (diagnoses) that explain the observed behavior, according to some minimality criterion. This thesis presents greedy approximation
Quasilinear theory without the random phase approximation
International Nuclear Information System (INIS)
Weibel, E.S.; Vaclavik, J.
1980-08-01
The system of quasilinear equations is derived without making use of the random phase approximation. The fluctuating quantities are described by the autocorrelation function of the electric field using the techniques of Fourier analysis. The resulting equations posses the necessary conservation properties, but comprise new terms which hitherto have been lost in the conventional derivations
Statistical model semiquantitatively approximates arabinoxylooligosaccharides' structural diversity
DEFF Research Database (Denmark)
Dotsenko, Gleb; Nielsen, Michael Krogsgaard; Lange, Lene
2016-01-01
(wheat flour arabinoxylan (arabinose/xylose, A/X = 0.47); grass arabinoxylan (A/X = 0.24); wheat straw arabinoxylan (A/X = 0.15); and hydrothermally pretreated wheat straw arabinoxylan (A/X = 0.05)), is semiquantitatively approximated using the proposed model. The suggested approach can be applied...
Upper Bounds on Numerical Approximation Errors
DEFF Research Database (Denmark)
Raahauge, Peter
2004-01-01
This paper suggests a method for determining rigorous upper bounds on approximationerrors of numerical solutions to infinite horizon dynamic programming models.Bounds are provided for approximations of the value function and the policyfunction as well as the derivatives of the value function...
Multi-Interpretation Operators and Approximate Classification
Engelfriet, J.; Treur, J.
2003-01-01
In this paper non-classical logical techniques are introduced to formalize the analysis of multi-interpretable observation information, in particular in approximate classification processes where information on attributes of an object is to be inferred on the basis of observable properties of the
Approximate Furthest Neighbor in High Dimensions
DEFF Research Database (Denmark)
Pagh, Rasmus; Silvestri, Francesco; Sivertsen, Johan von Tangen
2015-01-01
-dimensional Euclidean space. We build on the technique of Indyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk’s approximation factor and reducing the running time by a logarithmic factor. We also present a variation...
Padé approximations and diophantine geometry.
Chudnovsky, D V; Chudnovsky, G V
1985-04-01
Using methods of Padé approximations we prove a converse to Eisenstein's theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function, for classes of functions, parametrized by meromorphic functions. This result is applied to the Tate conjecture on the effective description of isogenies for elliptic curves.
tt in the soft-gluon approximation
Indian Academy of Sciences (India)
April 2002 physics pp. 575–590. QCD corrections to decay-lepton polar and azimuthal angular distributions in ee+ee- t t in the soft-gluon approximation. SAURABH D RINDANI ... accurate determination of its couplings will have to await the construction of a linear e ·e collider. ..... is the azimuthal angle of the l· momentum.
Stability of approximate factorization with $ heta $-methods
W. Hundsdorfer (Willem)
1997-01-01
textabstractApproximate factorization seems for certain problems a viable alternative to time splitting. Since a splitting error is avoided, accuracy will in general be favourable compared to time splitting methods. However, it is not clear to what extent stability is affected by factorization.
Decision-theoretic troubleshooting: Hardness of approximation
Czech Academy of Sciences Publication Activity Database
Lín, Václav
2014-01-01
Roč. 55, č. 4 (2014), s. 977-988 ISSN 0888-613X R&D Projects: GA ČR GA13-20012S Institutional support: RVO:67985556 Keywords : Decision-theoretic troubleshooting * Hardness of approximation * NP-completeness Subject RIV: BB - Applied Statistics, Operational Research Impact factor: 2.451, year: 2014
Comparison of Two Approaches to Approximated Reasoning
van den Broek, P.M.; Wagenknecht, Michael; Hampel, Rainer
A comparison is made of two approaches to approximate reasoning: Mamdani's interpolation method and the implication method. Both approaches are variants of Zadeh's compositional rule of inference. It is shown that the approaches are not equivalent. A correspondence between the approaches is
Lognormal Approximations of Fault Tree Uncertainty Distributions.
El-Shanawany, Ashraf Ben; Ardron, Keith H; Walker, Simon P
2018-01-26
Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed-form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling-based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed-form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks's method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models. © 2018 Society for Risk Analysis.
Counting independent sets using the Bethe approximation
Energy Technology Data Exchange (ETDEWEB)
Chertkov, Michael [Los Alamos National Laboratory; Chandrasekaran, V [MIT; Gamarmik, D [MIT; Shah, D [MIT; Sin, J [MIT
2009-01-01
The authors consider the problem of counting the number of independent sets or the partition function of a hard-core model in a graph. The problem in general is computationally hard (P hard). They study the quality of the approximation provided by the Bethe free energy. Belief propagation (BP) is a message-passing algorithm can be used to compute fixed points of the Bethe approximation; however, BP is not always guarantee to converge. As the first result, they propose a simple message-passing algorithm that converges to a BP fixed pont for any grapy. They find that their algorithm converges within a multiplicative error 1 + {var_epsilon} of a fixed point in {Omicron}(n{sup 2}E{sup -4} log{sup 3}(nE{sup -1})) iterations for any bounded degree graph of n nodes. In a nutshell, the algorithm can be thought of as a modification of BP with 'time-varying' message-passing. Next, they analyze the resulting error to the number of independent sets provided by such a fixed point of the Bethe approximation. Using the recently developed loop calculus approach by Vhertkov and Chernyak, they establish that for any bounded graph with large enough girth, the error is {Omicron}(n{sup -{gamma}}) for some {gamma} > 0. As an application, they find that for random 3-regular graph, Bethe approximation of log-partition function (log of the number of independent sets) is within o(1) of corret log-partition - this is quite surprising as previous physics-based predictions were expecting an error of o(n). In sum, their results provide a systematic way to find Bethe fixed points for any graph quickly and allow for estimating error in Bethe approximation using novel combinatorial techniques.
Electron-hydrogen atom collisions in the presence of a laser field
International Nuclear Information System (INIS)
Brandi, H.S.; Koiller, B.; Barros, H.G.P.L. de
1978-01-01
The collision of an electron and a hydrogen atom in the presence of a laser field is studied within a previously proposed approximation (based on the space translation approximation) for the bound states of the hydrogen atom. Fhe Green's function formalism is applied to derive an expression for the scattering amplitude associated to multiphoton processes. The Born-Oppenheimer approximation is obtained and numerical calculations are performed for the ls→2s inelastic excitation. It is shown as expected that exchange effects are important only for scattering processes involving low energy electrons [pt
Low-energy Scattering of Positronium by Atoms
Ray, Hasi
2007-01-01
The survey reports theoretical studies involving positronium (Ps) - atom scattering. Investigations carried out in last few decades have been briefly reviewed in this article. A brief description of close-coupling approximation (CCA), the first-Born approximation (FBA) and the Born-Oppenheimer approximation (BOA) for Ps-Atom systems are made. The CCA codes of Ray et a1 [1-6] are reinvestigated using very fine mesh-points to search for resonances. The article advocates the need for an extended basis set & a systematic study using CCAs.
Analytical Ballistic Trajectories with Approximately Linear Drag
Directory of Open Access Journals (Sweden)
Giliam J. P. de Carpentier
2014-01-01
Full Text Available This paper introduces a practical analytical approximation of projectile trajectories in 2D and 3D roughly based on a linear drag model and explores a variety of different planning algorithms for these trajectories. Although the trajectories are only approximate, they still capture many of the characteristics of a real projectile in free fall under the influence of an invariant wind, gravitational pull, and terminal velocity, while the required math for these trajectories and planners is still simple enough to efficiently run on almost all modern hardware devices. Together, these properties make the proposed approach particularly useful for real-time applications where accuracy and performance need to be carefully balanced, such as in computer games.
Analysing organic transistors based on interface approximation
International Nuclear Information System (INIS)
Akiyama, Yuto; Mori, Takehiko
2014-01-01
Temperature-dependent characteristics of organic transistors are analysed thoroughly using interface approximation. In contrast to amorphous silicon transistors, it is characteristic of organic transistors that the accumulation layer is concentrated on the first monolayer, and it is appropriate to consider interface charge rather than band bending. On the basis of this model, observed characteristics of hexamethylenetetrathiafulvalene (HMTTF) and dibenzotetrathiafulvalene (DBTTF) transistors with various surface treatments are analysed, and the trap distribution is extracted. In turn, starting from a simple exponential distribution, we can reproduce the temperature-dependent transistor characteristics as well as the gate voltage dependence of the activation energy, so we can investigate various aspects of organic transistors self-consistently under the interface approximation. Small deviation from such an ideal transistor operation is discussed assuming the presence of an energetically discrete trap level, which leads to a hump in the transfer characteristics. The contact resistance is estimated by measuring the transfer characteristics up to the linear region
Nonlinear analysis approximation theory, optimization and applications
2014-01-01
Many of our daily-life problems can be written in the form of an optimization problem. Therefore, solution methods are needed to solve such problems. Due to the complexity of the problems, it is not always easy to find the exact solution. However, approximate solutions can be found. The theory of the best approximation is applicable in a variety of problems arising in nonlinear functional analysis and optimization. This book highlights interesting aspects of nonlinear analysis and optimization together with many applications in the areas of physical and social sciences including engineering. It is immensely helpful for young graduates and researchers who are pursuing research in this field, as it provides abundant research resources for researchers and post-doctoral fellows. This will be a valuable addition to the library of anyone who works in the field of applied mathematics, economics and engineering.
Simple Lie groups without the approximation property
DEFF Research Database (Denmark)
Haagerup, Uffe; de Laat, Tim
2013-01-01
For a locally compact group G, let A(G) denote its Fourier algebra, and let M0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-∗ topology...... on the space M0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely, SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center...... and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP....
The optimal XFEM approximation for fracture analysis
International Nuclear Information System (INIS)
Jiang Shouyan; Du Chengbin; Ying Zongquan
2010-01-01
The extended finite element method (XFEM) provides an effective tool for analyzing fracture mechanics problems. A XFEM approximation consists of standard finite elements which are used in the major part of the domain and enriched elements in the enriched sub-domain for capturing special solution properties such as discontinuities and singularities. However, two issues in the standard XFEM should specially be concerned: efficient numerical integration methods and an appropriate construction of the blending elements. In the paper, an optimal XFEM approximation is proposed to overcome the disadvantage mentioned above in the standard XFEM. The modified enrichment functions are presented that can reproduced exactly everywhere in the domain. The corresponding FORTRAN program is developed for fracture analysis. A classic problem of fracture mechanics is used to benchmark the program. The results indicate that the optimal XFEM can alleviate the errors and improve numerical precision.
Approximated solutions to Born-Infeld dynamics
Energy Technology Data Exchange (ETDEWEB)
Ferraro, Rafael [Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA),Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires (Argentina); Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Pabellón I, 1428 Buenos Aires (Argentina); Nigro, Mauro [Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Pabellón I, 1428 Buenos Aires (Argentina)
2016-02-01
The Born-Infeld equation in the plane is usefully captured in complex language. The general exact solution can be written as a combination of holomorphic and anti-holomorphic functions. However, this solution only expresses the potential in an implicit way. We rework the formulation to obtain the complex potential in an explicit way, by means of a perturbative procedure. We take care of the secular behavior common to this kind of approach, by resorting to a symmetry the equation has at the considered order of approximation. We apply the method to build approximated solutions to Born-Infeld electrodynamics. We solve for BI electromagnetic waves traveling in opposite directions. We study the propagation at interfaces, with the aim of searching for effects susceptible to experimental detection. In particular, we show that a reflected wave is produced when a wave is incident on a semi-space containing a magnetostatic field.
Incomplete Sparse Approximate Inverses for Parallel Preconditioning
International Nuclear Information System (INIS)
Anzt, Hartwig; University of Tennessee, Knoxville, TN; Huckle, Thomas K.; Bräckle, Jürgen; Dongarra, Jack
2017-01-01
In this study, we propose a new preconditioning method that can be seen as a generalization of block-Jacobi methods, or as a simplification of the sparse approximate inverse (SAI) preconditioners. The “Incomplete Sparse Approximate Inverses” (ISAI) is in particular efficient in the solution of sparse triangular linear systems of equations. Those arise, for example, in the context of incomplete factorization preconditioning. ISAI preconditioners can be generated via an algorithm providing fine-grained parallelism, which makes them attractive for hardware with a high concurrency level. Finally, in a study covering a large number of matrices, we identify the ISAI preconditioner as an attractive alternative to exact triangular solves in the context of incomplete factorization preconditioning.
Approximate Solutions in Planted 3-SAT
Hsu, Benjamin; Laumann, Christopher; Moessner, Roderich; Sondhi, Shivaji
2013-03-01
In many computational settings, there exists many instances where finding a solution requires a computing time that grows exponentially in the number of variables. Concrete examples occur in combinatorial optimization problems and cryptography in computer science or glassy systems in physics. However, while exact solutions are often known to require exponential time, a related and important question is the running time required to find approximate solutions. Treating this problem as a problem in statistical physics at finite temperature, we examine the computational running time in finding approximate solutions in 3-satisfiability for randomly generated 3-SAT instances which are guaranteed to have a solution. Analytic predictions are corroborated by numerical evidence using stochastic local search algorithms. A first order transition is found in the running time of these algorithms.
Time Stamps for Fixed-Point Approximation
DEFF Research Database (Denmark)
Damian, Daniela
2001-01-01
Time stamps were introduced in Shivers's PhD thesis for approximating the result of a control-flow analysis. We show them to be suitable for computing program analyses where the space of results (e.g., control-flow graphs) is large. We formalize time-stamping as a top-down, fixed-point approximat......Time stamps were introduced in Shivers's PhD thesis for approximating the result of a control-flow analysis. We show them to be suitable for computing program analyses where the space of results (e.g., control-flow graphs) is large. We formalize time-stamping as a top-down, fixed...
Traveltime approximations for inhomogeneous HTI media
Alkhalifah, Tariq Ali
2011-01-01
Traveltimes information is convenient for parameter estimation especially if the medium is described by an anisotropic set of parameters. This is especially true if we could relate traveltimes analytically to these medium parameters, which is generally hard to do in inhomogeneous media. As a result, I develop traveltimes approximations for horizontaly transversely isotropic (HTI) media as simplified and even linear functions of the anisotropic parameters. This is accomplished by perturbing the solution of the HTI eikonal equation with respect to η and the azimuthal symmetry direction (usually used to describe the fracture direction) from a generally inhomogeneous elliptically anisotropic background medium. The resulting approximations can provide accurate analytical description of the traveltime in a homogenous background compared to other published moveout equations out there. These equations will allow us to readily extend the inhomogenous background elliptical anisotropic model to an HTI with a variable, but smoothly varying, η and horizontal symmetry direction values. © 2011 Society of Exploration Geophysicists.
The approximability of the String Barcoding problem
Directory of Open Access Journals (Sweden)
Rizzi Romeo
2006-08-01
Full Text Available Abstract The String Barcoding (SBC problem, introduced by Rash and Gusfield (RECOMB, 2002, consists in finding a minimum set of substrings that can be used to distinguish between all members of a set of given strings. In a computational biology context, the given strings represent a set of known viruses, while the substrings can be used as probes for an hybridization experiment via microarray. Eventually, one aims at the classification of new strings (unknown viruses through the result of the hybridization experiment. In this paper we show that SBC is as hard to approximate as Set Cover. Furthermore, we show that the constrained version of SBC (with probes of bounded length is also hard to approximate. These negative results are tight.
Approximations in the PE-method
DEFF Research Database (Denmark)
Arranz, Marta Galindo
1996-01-01
Two differenct sources of errors may occur in the implementation of the PE methods; a phase error introduced in the approximation of a pseudo-differential operator and an amplitude error generated from the starting field. First, the inherent phase errors introduced in the solution are analyzed...... for a case where the normal mode solution to the wave equation is valid, when the sound is propagated in a downward refracting atmosphere. The angular limitations for the different parabolic approximations are deduced, and calculations showing shifts in the starter as the second source of error...... is investigated. Numerical and analytical starters are compared for source locations close to the ground. The spectral properties of several starters are presented....
A Varifold Approach to Surface Approximation
Buet, Blanche; Leonardi, Gian Paolo; Masnou, Simon
2017-11-01
We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold, in particular, for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of d-surfaces in the Euclidean n-space, without restrictions on dimension and codimension. The variational nature of the approach also allows us to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method.
Approximations and Solution Estimates in Optimization
2016-04-06
for applications in machine learning and stochastic optimization . In this paper, we quantify the error in optimal values, optimal solutions, near...problem from that of another rather different problem is especially important in stochastic optimization , optimal control, and semi-infinite...approximate solutions to convex stochastic programs. SIAM J. Optimization , 18(3):961–979, 2007. [26] J. O. Royset and R. J-B Wets. From data to
Finite element approximation of the Isaacs equation
Salgado, Abner J.; Zhang, Wujun
2015-01-01
We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size $h$ whereas the coarse scale $\\varepsilon$ is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical s...
Mean-field approximation minimizes relative entropy
International Nuclear Information System (INIS)
Bilbro, G.L.; Snyder, W.E.; Mann, R.C.
1991-01-01
The authors derive the mean-field approximation from the information-theoretic principle of minimum relative entropy instead of by minimizing Peierls's inequality for the Weiss free energy of statistical physics theory. They show that information theory leads to the statistical mechanics procedure. As an example, they consider a problem in binary image restoration. They find that mean-field annealing compares favorably with the stochastic approach
Fast Approximate Joint Diagonalization Incorporating Weight Matrices
Czech Academy of Sciences Publication Activity Database
Tichavský, Petr; Yeredor, A.
2009-01-01
Roč. 57, č. 3 (2009), s. 878-891 ISSN 1053-587X R&D Projects: GA MŠk 1M0572 Institutional research plan: CEZ:AV0Z10750506 Keywords : autoregressive processes * blind source separation * nonstationary random processes Subject RIV: BB - Applied Statistics, Operational Research Impact factor: 2.212, year: 2009 http://library.utia.cas.cz/separaty/2009/SI/tichavsky-fast approximate joint diagonalization incorporating weight matrices.pdf
Fast approximate convex decomposition using relative concavity
Ghosh, Mukulika
2013-02-01
Approximate convex decomposition (ACD) is a technique that partitions an input object into approximately convex components. Decomposition into approximately convex pieces is both more efficient to compute than exact convex decomposition and can also generate a more manageable number of components. It can be used as a basis of divide-and-conquer algorithms for applications such as collision detection, skeleton extraction and mesh generation. In this paper, we propose a new method called Fast Approximate Convex Decomposition (FACD) that improves the quality of the decomposition and reduces the cost of computing it for both 2D and 3D models. In particular, we propose a new strategy for evaluating potential cuts that aims to reduce the relative concavity, rather than absolute concavity. As shown in our results, this leads to more natural and smaller decompositions that include components for small but important features such as toes or fingers while not decomposing larger components, such as the torso, that may have concavities due to surface texture. Second, instead of decomposing a component into two pieces at each step, as in the original ACD, we propose a new strategy that uses a dynamic programming approach to select a set of n c non-crossing (independent) cuts that can be simultaneously applied to decompose the component into n c+1 components. This reduces the depth of recursion and, together with a more efficient method for computing the concavity measure, leads to significant gains in efficiency. We provide comparative results for 2D and 3D models illustrating the improvements obtained by FACD over ACD and we compare with the segmentation methods in the Princeton Shape Benchmark by Chen et al. (2009) [31]. © 2012 Elsevier Ltd. All rights reserved.
Approximations of Two-Attribute Utility Functions
1976-09-01
06320 i Technology Cambridge, Massachusetts 02139 Dr. Jack R. Borsting, Chairman Dept. of Operations Research and Professor Oskar Morgenstern ... Morgenstern utility functions u defined on two attributes from the viewpoint of mathematical --roxi-n’tion theory.r It focuses on approximations v of u that...von Neumann and Morgenstern (1947) or an equivalent system (Herstein and Milnor, 1953; Fishburn, 1970) so that there exists u: T + Re such that P - Q
Approximate Inverse Preconditioners with Adaptive Dropping
Czech Academy of Sciences Publication Activity Database
Kopal, J.; Rozložník, Miroslav; Tůma, Miroslav
2015-01-01
Roč. 84, June (2015), s. 13-20 ISSN 0965-9978 R&D Projects: GA ČR(CZ) GAP108/11/0853; GA ČR GA13-06684S Institutional support: RVO:67985807 Keywords : approximate inverse * Gram-Schmidt orthogonalization * incomplete decomposition * preconditioned conjugate gradient method * algebraic preconditioning * pivoting Subject RIV: BA - General Mathematics Impact factor: 1.673, year: 2015
Solving Math Problems Approximately: A Developmental Perspective.
Directory of Open Access Journals (Sweden)
Dana Ganor-Stern
Full Text Available Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults' ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger than the exact answer and when it was far (vs. close from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner.
Optical pulse propagation with minimal approximations
Kinsler, Paul
2008-01-01
Propagation equations for optical pulses are needed to assist in describing applications in ever more extreme situations -- including those in metamaterials with linear and nonlinear magnetic responses. Here I show how to derive a single first order propagation equation using a minimum of approximations and a straightforward "factorization" mathematical scheme. The approach generates exact coupled bi-directional equations, after which it is clear that the description can be reduced to a singl...
Hierarchical matrix approximation of large covariance matrices
Litvinenko, Alexander
2015-01-07
We approximate large non-structured covariance matrices in the H-matrix format with a log-linear computational cost and storage O(n log n). We compute inverse, Cholesky decomposition and determinant in H-format. As an example we consider the class of Matern covariance functions, which are very popular in spatial statistics, geostatistics, machine learning and image analysis. Applications are: kriging and optimal design
Hierarchical matrix approximation of large covariance matrices
Litvinenko, Alexander
2015-01-05
We approximate large non-structured covariance matrices in the H-matrix format with a log-linear computational cost and storage O(nlogn). We compute inverse, Cholesky decomposition and determinant in H-format. As an example we consider the class of Matern covariance functions, which are very popular in spatial statistics, geostatistics, machine learning and image analysis. Applications are: kriging and op- timal design.
Solving Math Problems Approximately: A Developmental Perspective.
Ganor-Stern, Dana
2016-01-01
Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults' ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner.
Markdown Optimization via Approximate Dynamic Programming
Directory of Open Access Journals (Sweden)
Cos?gun
2013-02-01
Full Text Available We consider the markdown optimization problem faced by the leading apparel retail chain. Because of substitution among products the markdown policy of one product affects the sales of other products. Therefore, markdown policies for product groups having a significant crossprice elasticity among each other should be jointly determined. Since the state space of the problem is very huge, we use Approximate Dynamic Programming. Finally, we provide insights on the behavior of how each product price affects the markdown policy.
Factorized Approximate Inverses With Adaptive Dropping
Czech Academy of Sciences Publication Activity Database
Kopal, Jiří; Rozložník, Miroslav; Tůma, Miroslav
2016-01-01
Roč. 38, č. 3 (2016), A1807-A1820 ISSN 1064-8275 R&D Projects: GA ČR GA13-06684S Grant - others:GA MŠk(CZ) LL1202 Institutional support: RVO:67985807 Keywords : approximate inverses * incomplete factorization * Gram–Schmidt orthogonalization * preconditioned iterative methods Subject RIV: BA - General Mathematics Impact factor: 2.195, year: 2016
An analytical approximation for resonance integral
International Nuclear Information System (INIS)
Magalhaes, C.G. de; Martinez, A.S.
1985-01-01
It is developed a method which allows to obtain an analytical solution for the resonance integral. The problem formulation is completely theoretical and based in concepts of physics of general character. The analytical expression for integral does not involve any empiric correlation or parameter. Results of approximation are compared with pattern values for each individual resonance and for sum of all resonances. (M.C.K.) [pt
Fast algorithms for approximate circular string matching.
Barton, Carl; Iliopoulos, Costas S; Pissis, Solon P
2014-03-22
Circular string matching is a problem which naturally arises in many biological contexts. It consists in finding all occurrences of the rotations of a pattern of length m in a text of length n. There exist optimal average-case algorithms for exact circular string matching. Approximate circular string matching is a rather undeveloped area. In this article, we present a suboptimal average-case algorithm for exact circular string matching requiring time O(n). Based on our solution for the exact case, we present two fast average-case algorithms for approximate circular string matching with k-mismatches, under the Hamming distance model, requiring time O(n) for moderate values of k, that is k=O(m/logm). We show how the same results can be easily obtained under the edit distance model. The presented algorithms are also implemented as library functions. Experimental results demonstrate that the functions provided in this library accelerate the computations by more than three orders of magnitude compared to a naïve approach. We present two fast average-case algorithms for approximate circular string matching with k-mismatches; and show that they also perform very well in practice. The importance of our contribution is underlined by the fact that the provided functions may be seamlessly integrated into any biological pipeline. The source code of the library is freely available at http://www.inf.kcl.ac.uk/research/projects/asmf/.
Approximate particle number projection in hot nuclei
International Nuclear Information System (INIS)
Kosov, D.S.; Vdovin, A.I.
1995-01-01
Heated finite systems like, e.g., hot atomic nuclei have to be described by the canonical partition function. But this is a quite difficult technical problem and, as a rule, the grand canonical partition function is used in the studies. As a result, some shortcomings of the theoretical description appear because of the thermal fluctuations of the number of particles. Moreover, in nuclei with pairing correlations the quantum number fluctuations are introduced by some approximate methods (e.g., by the standard BCS method). The exact particle number projection is very cumbersome and an approximate number projection method for T ≠ 0 basing on the formalism of thermo field dynamics is proposed. The idea of the Lipkin-Nogami method to perform any operator as a series in the number operator powers is used. The system of equations for the coefficients of this expansion is written and the solution of the system in the next approximation after the BCS one is obtained. The method which is of the 'projection after variation' type is applied to a degenerate single j-shell model. 14 refs., 1 tab
Feedforward Approximations to Dynamic Recurrent Network Architectures.
Muir, Dylan R
2018-02-01
Recurrent neural network architectures can have useful computational properties, with complex temporal dynamics and input-sensitive attractor states. However, evaluation of recurrent dynamic architectures requires solving systems of differential equations, and the number of evaluations required to determine their response to a given input can vary with the input or can be indeterminate altogether in the case of oscillations or instability. In feedforward networks, by contrast, only a single pass through the network is needed to determine the response to a given input. Modern machine learning systems are designed to operate efficiently on feedforward architectures. We hypothesized that two-layer feedforward architectures with simple, deterministic dynamics could approximate the responses of single-layer recurrent network architectures. By identifying the fixed-point responses of a given recurrent network, we trained two-layer networks to directly approximate the fixed-point response to a given input. These feedforward networks then embodied useful computations, including competitive interactions, information transformations, and noise rejection. Our approach was able to find useful approximations to recurrent networks, which can then be evaluated in linear and deterministic time complexity.
Impulse approximation versus elementary particle method
International Nuclear Information System (INIS)
Klieb, L.
1982-01-01
Calculations are made for radiative muon capture in 3 He, both in impulse approximation and with the elementary particle method, and results are compared. It is argued that a diagrammatic method which takes a selected set of Feynman diagrams into account only provides insufficient warrant that effects not included are small. Therefore low-energy theorems are employed, as first given by Adler and Dothan, to determine the amplitude up to and including all terms linear in photon momentum and momentum transfer at the weak vertex. This amplitude is applied to radiative muon capture with the elementary particle method (EPM). The various form factors needed are discussed. It is shown that the results are particularly sensitive to the π- 3 He- 3 H coupling constant of which many contradictory determinations have been described in the literature. The classification of the nuclear wave function employed in the impulse approximation (IA) is summarized. The ν-decay of 3 H and (radiative muon capture in 3 He is treated and numerical results are given. Next, pion photoproduction and radiative pion capture are considered. IA and EPM for radiative muon capture are compared more closely. It is concluded that two-step processes are inherently difficult; the elementary particle method has convergence problems, and unknown parameters are present. In the impulse approximation, which is perhaps conceptually more difficult, the two-step interaction for the nucleon is considered as effectively point-like with small non-local corrections. (Auth.)
Ranking Support Vector Machine with Kernel Approximation
Directory of Open Access Journals (Sweden)
Kai Chen
2017-01-01
Full Text Available Learning to rank algorithm has become important in recent years due to its successful application in information retrieval, recommender system, and computational biology, and so forth. Ranking support vector machine (RankSVM is one of the state-of-art ranking models and has been favorably used. Nonlinear RankSVM (RankSVM with nonlinear kernels can give higher accuracy than linear RankSVM (RankSVM with a linear kernel for complex nonlinear ranking problem. However, the learning methods for nonlinear RankSVM are still time-consuming because of the calculation of kernel matrix. In this paper, we propose a fast ranking algorithm based on kernel approximation to avoid computing the kernel matrix. We explore two types of kernel approximation methods, namely, the Nyström method and random Fourier features. Primal truncated Newton method is used to optimize the pairwise L2-loss (squared Hinge-loss objective function of the ranking model after the nonlinear kernel approximation. Experimental results demonstrate that our proposed method gets a much faster training speed than kernel RankSVM and achieves comparable or better performance over state-of-the-art ranking algorithms.
CMB-lensing beyond the Born approximation
International Nuclear Information System (INIS)
Marozzi, Giovanni; Fanizza, Giuseppe; Durrer, Ruth; Dio, Enea Di
2016-01-01
We investigate the weak lensing corrections to the cosmic microwave background temperature anisotropies considering effects beyond the Born approximation. To this aim, we use the small deflection angle approximation, to connect the lensed and unlensed power spectra, via expressions for the deflection angles up to third order in the gravitational potential. While the small deflection angle approximation has the drawback to be reliable only for multipoles ℓ ∼< 2500, it allows us to consistently take into account the non-Gaussian nature of cosmological perturbation theory beyond the linear level. The contribution to the lensed temperature power spectrum coming from the non-Gaussian nature of the deflection angle at higher order is a new effect which has not been taken into account in the literature so far. It turns out to be the leading contribution among the post-Born lensing corrections. On the other hand, the effect is smaller than corrections coming from non-linearities in the matter power spectrum, and its imprint on CMB lensing is too small to be seen in present experiments.
Function approximation of tasks by neural networks
International Nuclear Information System (INIS)
Gougam, L.A.; Chikhi, A.; Mekideche-Chafa, F.
2008-01-01
For several years now, neural network models have enjoyed wide popularity, being applied to problems of regression, classification and time series analysis. Neural networks have been recently seen as attractive tools for developing efficient solutions for many real world problems in function approximation. The latter is a very important task in environments where computation has to be based on extracting information from data samples in real world processes. In a previous contribution, we have used a well known simplified architecture to show that it provides a reasonably efficient, practical and robust, multi-frequency analysis. We have investigated the universal approximation theory of neural networks whose transfer functions are: sigmoid (because of biological relevance), Gaussian and two specified families of wavelets. The latter have been found to be more appropriate to use. The aim of the present contribution is therefore to use a m exican hat wavelet a s transfer function to approximate different tasks relevant and inherent to various applications in physics. The results complement and provide new insights into previously published results on this problem
Simultaneous perturbation stochastic approximation for tidal models
Altaf, M.U.
2011-05-12
The Dutch continental shelf model (DCSM) is a shallow sea model of entire continental shelf which is used operationally in the Netherlands to forecast the storm surges in the North Sea. The forecasts are necessary to support the decision of the timely closure of the moveable storm surge barriers to protect the land. In this study, an automated model calibration method, simultaneous perturbation stochastic approximation (SPSA) is implemented for tidal calibration of the DCSM. The method uses objective function evaluations to obtain the gradient approximations. The gradient approximation for the central difference method uses only two objective function evaluation independent of the number of parameters being optimized. The calibration parameter in this study is the model bathymetry. A number of calibration experiments is performed. The effectiveness of the algorithm is evaluated in terms of the accuracy of the final results as well as the computational costs required to produce these results. In doing so, comparison is made with a traditional steepest descent method and also with a newly developed proper orthogonal decompositionbased calibration method. The main findings are: (1) The SPSA method gives comparable results to steepest descent method with little computational cost. (2) The SPSA method with little computational cost can be used to estimate large number of parameters.
Odic, Darko; Lisboa, Juan Valle; Eisinger, Robert; Olivera, Magdalena Gonzalez; Maiche, Alejandro; Halberda, Justin
2016-01-01
What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience has suggested that our representations of approximate number, time, and spatial extent are fundamentally linked and constitute a "generalized magnitude system". But, the shared behavioral and neural signatures between number, time, and space may alternatively be due to similar encoding and decision-making processes, rather than due to shared domain-general representations. In this study, we investigate the relationship between approximate number and time in a large sample of 6-8 year-old children in Uruguay by examining how individual differences in the precision of number and time estimation correlate with school mathematics performance. Over four testing days, each child completed an approximate number discrimination task, an approximate time discrimination task, a digit span task, and a large battery of symbolic math tests. We replicate previous reports showing that symbolic math abilities correlate with approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate number correlates more strongly with formal math compared to time and continues to correlate with math even when precision in time and individual differences in working memory are controlled for. These results suggest that there are important differences in the mental representations of approximate number and approximate time and further clarify the relationship between quantity representations and mathematics. Copyright
Photoelectron spectroscopy and the dipole approximation
Energy Technology Data Exchange (ETDEWEB)
Hemmers, O.; Hansen, D.L.; Wang, H. [Univ. of Nevada, Las Vegas, NV (United States)] [and others
1997-04-01
Photoelectron spectroscopy is a powerful technique because it directly probes, via the measurement of photoelectron kinetic energies, orbital and band structure in valence and core levels in a wide variety of samples. The technique becomes even more powerful when it is performed in an angle-resolved mode, where photoelectrons are distinguished not only by their kinetic energy, but by their direction of emission as well. Determining the probability of electron ejection as a function of angle probes the different quantum-mechanical channels available to a photoemission process, because it is sensitive to phase differences among the channels. As a result, angle-resolved photoemission has been used successfully for many years to provide stringent tests of the understanding of basic physical processes underlying gas-phase and solid-state interactions with radiation. One mainstay in the application of angle-resolved photoelectron spectroscopy is the well-known electric-dipole approximation for photon interactions. In this simplification, all higher-order terms, such as those due to electric-quadrupole and magnetic-dipole interactions, are neglected. As the photon energy increases, however, effects beyond the dipole approximation become important. To best determine the range of validity of the dipole approximation, photoemission measurements on a simple atomic system, neon, where extra-atomic effects cannot play a role, were performed at BL 8.0. The measurements show that deviations from {open_quotes}dipole{close_quotes} expectations in angle-resolved valence photoemission are observable for photon energies down to at least 0.25 keV, and are quite significant at energies around 1 keV. From these results, it is clear that non-dipole angular-distribution effects may need to be considered in any application of angle-resolved photoelectron spectroscopy that uses x-ray photons of energies as low as a few hundred eV.
Product-State Approximations to Quantum States
Brandão, Fernando G. S. L.; Harrow, Aram W.
2016-02-01
We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.
Dynamic system evolution and markov chain approximation
Directory of Open Access Journals (Sweden)
Roderick V. Nicholas Melnik
1998-01-01
Full Text Available In this paper computational aspects of the mathematical modelling of dynamic system evolution have been considered as a problem in information theory. The construction of mathematical models is treated as a decision making process with limited available information.The solution of the problem is associated with a computational model based on heuristics of a Markov Chain in a discrete space–time of events. A stable approximation of the chain has been derived and the limiting cases are discussed. An intrinsic interconnection of constructive, sequential, and evolutionary approaches in related optimization problems provides new challenges for future work.
The EH Interpolation Spline and Its Approximation
Directory of Open Access Journals (Sweden)
Jin Xie
2014-01-01
Full Text Available A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.
On one approximation in quantum chromodynamics
International Nuclear Information System (INIS)
Alekseev, A.I.; Bajkov, V.A.; Boos, Eh.Eh.
1982-01-01
Form of a complete fermion propagator near the mass shell is investigated. Considered is a nodel of quantum chromodynamics (MQC) where in the fermion section the Block-Nordsic approximation has been made, i. e. u-numbers are substituted for ν matrices. The model was investigated by means of the Schwinger-Dyson equation for a quark propagator in the infrared region. The Schwinger-Dyson equation was managed to reduce to a differential equation which is easily solved. At that, the Green function is suitable to represent as integral transformation
Topics in multivariate approximation and interpolation
Jetter, Kurt
2005-01-01
This book is a collection of eleven articles, written by leading experts and dealing with special topics in Multivariate Approximation and Interpolation. The material discussed here has far-reaching applications in many areas of Applied Mathematics, such as in Computer Aided Geometric Design, in Mathematical Modelling, in Signal and Image Processing and in Machine Learning, to mention a few. The book aims at giving a comprehensive information leading the reader from the fundamental notions and results of each field to the forefront of research. It is an ideal and up-to-date introduction for gr
Shape theory categorical methods of approximation
Cordier, J M
2008-01-01
This in-depth treatment uses shape theory as a ""case study"" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. It offers students a unified and consolidated presentation of extensive research from category theory, shape theory, and the study of topological algebras.A short introduction to geometric shape explains specifics of the construction of the shape category and relates it to an abstract definition of shape theory. Upon returning to the geometric base, the text considers simplical complexes and
Mean field approximation to QCD, 1
International Nuclear Information System (INIS)
Tezuka, Hirokazu.
1987-09-01
We apply mean field approximation to the gluon field in the equations of motion derived from the QCD lagrangian. The gluon mean field is restricted to the time-independent 0th component, and color exchange components are ignored. The equation of motion for the gluon mean field turns into a Poisson equation, and that for the quark field into a Dirac equation with a potential term. For example, assuming spherical symmetric box-like distribution and Gauss-like distribution to quarks, we try to solve these two equations simultaneously. (author)
Nonlinear higher quasiparticle random phase approximation
Smetana, Adam; Šimkovic, Fedor; Štefánik, Dušan; Krivoruchenko, Mikhail
2017-10-01
We develop a new approach to describe nuclear states of multiphonon origin, motivated by the necessity for a more accurate description of matrix elements of neutrinoless double-beta decay. Our approach is an extension of the Quasiparticle Random Phase Approximation (QRPA), in which nonlinear phonon operators play an essential role. Before applying the nonlinear higher QRPA (nhQRPA) to realistic problems, we test its efficiency with exactly solvable models. The first considered model is equivalent to a harmonic oscillator. The nhQRPA solutions follow from the standard QRPA equation, but for nonlinear phonon operators defined for each individual excited state separately. The second exactly solvable model is the proton-neutron Lipkin model that describes successfully not only energy spectrum of nuclei, but also beta-decay transitions. Again, we reproduce exactly the numerical solutions in the nhQRPA framework. We show in particular that truncation of the nonlinear phonon operators leads to an approximation similar to the self-consistent second QRPA, given the phonon operators are defined with a constant term. The test results demonstrate that the proposed nhQRPA is a promising tool for a realistic calculation of energy spectra and nuclear transitions.
Fast approximate hierarchical clustering using similarity heuristics
Directory of Open Access Journals (Sweden)
Kull Meelis
2008-09-01
Full Text Available Abstract Background Agglomerative hierarchical clustering (AHC is a common unsupervised data analysis technique used in several biological applications. Standard AHC methods require that all pairwise distances between data objects must be known. With ever-increasing data sizes this quadratic complexity poses problems that cannot be overcome by simply waiting for faster computers. Results We propose an approximate AHC algorithm HappieClust which can output a biologically meaningful clustering of a large dataset more than an order of magnitude faster than full AHC algorithms. The key to the algorithm is to limit the number of calculated pairwise distances to a carefully chosen subset of all possible distances. We choose distances using a similarity heuristic based on a small set of pivot objects. The heuristic efficiently finds pairs of similar objects and these help to mimic the greedy choices of full AHC. Quality of approximate AHC as compared to full AHC is studied with three measures. The first measure evaluates the global quality of the achieved clustering, while the second compares biological relevance using enrichment of biological functions in every subtree of the clusterings. The third measure studies how well the contents of subtrees are conserved between the clusterings. Conclusion The HappieClust algorithm is well suited for large-scale gene expression visualization and analysis both on personal computers as well as public online web applications. The software is available from the URL http://www.quretec.com/HappieClust
Traveling cluster approximation for uncorrelated amorphous systems
International Nuclear Information System (INIS)
Kaplan, T.; Sen, A.K.; Gray, L.J.; Mills, R.
1985-01-01
In this paper, the authors apply the TCA concepts to spatially disordered, uncorrelated systems (e.g., fluids or amorphous metals without short-range order). This is the first approximation scheme for amorphous systems that takes cluster effects into account while preserving the Herglotz property for any amount of disorder. They have performed some computer calculations for the pair TCA, for the model case of delta-function potentials on a one-dimensional random chain. These results are compared with exact calculations (which, in principle, taken into account all cluster effects) and with the CPA, which is the single-site TCA. The density of states for the pair TCA clearly shows some improvement over the CPA, and yet, apparently, the pair approximation distorts some of the features of the exact results. They conclude that the effects of large clusters are much more important in an uncorrelated liquid metal than in a substitutional alloy. As a result, the pair TCA, which does quite a nice job for alloys, is not adequate for the liquid. Larger clusters must be treated exactly, and therefore an n-TCA with n > 2 must be used
APPROXIMATING INNOVATION POTENTIAL WITH NEUROFUZZY ROBUST MODEL
Directory of Open Access Journals (Sweden)
Kasa, Richard
2015-01-01
Full Text Available In a remarkably short time, economic globalisation has changed the world’s economic order, bringing new challenges and opportunities to SMEs. These processes pushed the need to measure innovation capability, which has become a crucial issue for today’s economic and political decision makers. Companies cannot compete in this new environment unless they become more innovative and respond more effectively to consumers’ needs and preferences – as mentioned in the EU’s innovation strategy. Decision makers cannot make accurate and efficient decisions without knowing the capability for innovation of companies in a sector or a region. This need is forcing economists to develop an integrated, unified and complete method of measuring, approximating and even forecasting the innovation performance not only on a macro but also a micro level. In this recent article a critical analysis of the literature on innovation potential approximation and prediction is given, showing their weaknesses and a possible alternative that eliminates the limitations and disadvantages of classical measuring and predictive methods.
Analytic approximate radiation effects due to Bremsstrahlung
Energy Technology Data Exchange (ETDEWEB)
Ben-Zvi I.
2012-02-01
The purpose of this note is to provide analytic approximate expressions that can provide quick estimates of the various effects of the Bremsstrahlung radiation produced relatively low energy electrons, such as the dumping of the beam into the beam stop at the ERL or field emission in superconducting cavities. The purpose of this work is not to replace a dependable calculation or, better yet, a measurement under real conditions, but to provide a quick but approximate estimate for guidance purposes only. These effects include dose to personnel, ozone generation in the air volume exposed to the radiation, hydrogen generation in the beam dump water cooling system and radiation damage to near-by magnets. These expressions can be used for other purposes, but one should note that the electron beam energy range is limited. In these calculations the good range is from about 0.5 MeV to 10 MeV. To help in the application of this note, calculations are presented as a worked out example for the beam dump of the R&D Energy Recovery Linac.
Approximate reversal of quantum Gaussian dynamics
Lami, Ludovico; Das, Siddhartha; Wilde, Mark M.
2018-03-01
Recently, there has been focus on determining the conditions under which the data processing inequality for quantum relative entropy is satisfied with approximate equality. The solution of the exact equality case is due to Petz, who showed that the quantum relative entropy between two quantum states stays the same after the action of a quantum channel if and only if there is a reversal channel that recovers the original states after the channel acts. Furthermore, this reversal channel can be constructed explicitly and is now called the Petz recovery map. Recent developments have shown that a variation of the Petz recovery map works well for recovery in the case of approximate equality of the data processing inequality. Our main contribution here is a proof that bosonic Gaussian states and channels possess a particular closure property, namely, that the Petz recovery map associated to a bosonic Gaussian state σ and a bosonic Gaussian channel N is itself a bosonic Gaussian channel. We furthermore give an explicit construction of the Petz recovery map in this case, in terms of the mean vector and covariance matrix of the state σ and the Gaussian specification of the channel N .
Approximating Markov Chains: What and why
International Nuclear Information System (INIS)
Pincus, S.
1996-01-01
Much of the current study of dynamical systems is focused on geometry (e.g., chaos and bifurcations) and ergodic theory. Yet dynamical systems were originally motivated by an attempt to open-quote open-quote solve,close-quote close-quote or at least understand, a discrete-time analogue of differential equations. As such, numerical, analytical solution techniques for dynamical systems would seem desirable. We discuss an approach that provides such techniques, the approximation of dynamical systems by suitable finite state Markov Chains. Steady state distributions for these Markov Chains, a straightforward calculation, will converge to the true dynamical system steady state distribution, with appropriate limit theorems indicated. Thus (i) approximation by a computable, linear map holds the promise of vastly faster steady state solutions for nonlinear, multidimensional differential equations; (ii) the solution procedure is unaffected by the presence or absence of a probability density function for the attractor, entirely skirting singularity, fractal/multifractal, and renormalization considerations. The theoretical machinery underpinning this development also implies that under very general conditions, steady state measures are weakly continuous with control parameter evolution. This means that even though a system may change periodicity, or become chaotic in its limiting behavior, such statistical parameters as the mean, standard deviation, and tail probabilities change continuously, not abruptly with system evolution. copyright 1996 American Institute of Physics
Analytic approximate radiation effects due to Bremsstrahlung
International Nuclear Information System (INIS)
Ben-Zvi, I.
2012-01-01
The purpose of this note is to provide analytic approximate expressions that can provide quick estimates of the various effects of the Bremsstrahlung radiation produced relatively low energy electrons, such as the dumping of the beam into the beam stop at the ERL or field emission in superconducting cavities. The purpose of this work is not to replace a dependable calculation or, better yet, a measurement under real conditions, but to provide a quick but approximate estimate for guidance purposes only. These effects include dose to personnel, ozone generation in the air volume exposed to the radiation, hydrogen generation in the beam dump water cooling system and radiation damage to near-by magnets. These expressions can be used for other purposes, but one should note that the electron beam energy range is limited. In these calculations the good range is from about 0.5 MeV to 10 MeV. To help in the application of this note, calculations are presented as a worked out example for the beam dump of the R and D Energy Recovery Linac.
On some applications of diophantine approximations.
Chudnovsky, G V
1984-03-01
Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E-functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E-functions at rational points. Our results show that values of E-functions at rational points have measures of diophantine approximations typical to "almost all" numbers. In particular, any such number has the "2 + epsilon" exponent of irrationality: Theta - p/q > q(-2-epsilon) for relatively prime rational integers p,q, with q >/= q(0) (Theta, epsilon). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. & Chudnovsky, G. V. (1983) Proc. Natl. Acad. Sci. USA 80, 5158-5162].
Regularity and approximability of electronic wave functions
Yserentant, Harry
2010-01-01
The electronic Schrödinger equation describes the motion of N-electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, with three spatial dimensions for each electron. Approximating these solutions is thus inordinately challenging, and it is generally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to show readers that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as...
Adaptive and Approximate Orthogonal Range Counting
DEFF Research Database (Denmark)
Chan, Timothy M.; Wilkinson, Bryan Thomas
2013-01-01
-case optimal query time O(log_w n). We give an O(n loglog n)-space adaptive data structure that improves the query time to O(loglog n + log_w k), where k is the output count. When k=O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan, Larsen, and Pătraşcu, SoCG 2011......]. •We give an O(n loglog n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1+δ)-factor approximation to the count in O(loglog n) time for any fixed constant δ>0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem. •Lastly......Close Abstract We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model. •It is well known that there are linear-space data structures for 2-D orthogonal range counting with worst...
DEFF Research Database (Denmark)
Sadegh, Payman
1997-01-01
This paper deals with a projection algorithm for stochastic approximation using simultaneous perturbation gradient approximation for optimization under inequality constraints where no direct gradient of the loss function is available and the inequality constraints are given as explicit functions...... of the optimization parameters. It is shown that, under application of the projection algorithm, the parameter iterate converges almost surely to a Kuhn-Tucker point, The procedure is illustrated by a numerical example, (C) 1997 Elsevier Science Ltd....
Hydromagnetic turbulence in the direct interaction approximation
International Nuclear Information System (INIS)
Nagarajan, S.
1975-01-01
The dissertation is concerned with the nature of turbulence in a medium with large electrical conductivity. Three distinct though inter-related questions are asked. Firstly, the evolution of a weak, random initial magnetic field in a highly conducting, isotropically turbulent fluid is discussed. This was first discussed in the paper 'Growth of Turbulent Magnetic Fields' by Kraichnan and Nagargian. The Physics of Fluids, volume 10, number 4, 1967. Secondly, the direct interaction approximation for hydromagnetic turbulence maintained by stationary, isotropic, random stirring forces is formulated in the wave-number-frequency domain. Thirdly, the dynamical evolution of a weak, random, magnetic excitation in a turbulent electrically conducting fluid is examined under varying kinematic conditions. (G.T.H.)
Discrete Spectrum Reconstruction Using Integral Approximation Algorithm.
Sizikov, Valery; Sidorov, Denis
2017-07-01
An inverse problem in spectroscopy is considered. The objective is to restore the discrete spectrum from observed spectrum data, taking into account the spectrometer's line spread function. The problem is reduced to solution of a system of linear-nonlinear equations (SLNE) with respect to intensities and frequencies of the discrete spectral lines. The SLNE is linear with respect to lines' intensities and nonlinear with respect to the lines' frequencies. The integral approximation algorithm is proposed for the solution of this SLNE. The algorithm combines solution of linear integral equations with solution of a system of linear algebraic equations and avoids nonlinear equations. Numerical examples of the application of the technique, both to synthetic and experimental spectra, demonstrate the efficacy of the proposed approach in enabling an effective enhancement of the spectrometer's resolution.
Nanostructures: Scattering beyond the Born approximation
Grigoriev, S. V.; Syromyatnikov, A. V.; Chumakov, A. P.; Grigoryeva, N. A.; Napolskii, K. S.; Roslyakov, I. V.; Eliseev, A. A.; Petukhov, A. V.; Eckerlebe, H.
2010-03-01
The neutron scattering on a two-dimensional ordered nanostructure with the third nonperiodic dimension can go beyond the Born approximation. In our model supported by the exact theoretical solution a well-correlated hexagonal porous structure of anodic aluminum oxide films acts as a peculiar two-dimensional grating for the coherent neutron wave. The thickness of the film L (length of pores) plays important role in the transition from the weak to the strong scattering regimes. It is shown that the coherency of the standard small-angle neutron scattering setups suits to the geometry of the studied objects and often affects the intensity of scattering. The proposed theoretical solution can be applied in the small-angle neutron diffraction experiments with flux lines in superconductors, periodic arrays of magnetic or superconducting nanowires, as well as in small-angle diffraction experiments on synchrotron radiation.
UFBoot2: Improving the Ultrafast Bootstrap Approximation.
Hoang, Diep Thi; Chernomor, Olga; von Haeseler, Arndt; Minh, Bui Quang; Vinh, Le Sy
2018-02-01
The standard bootstrap (SBS), despite being computationally intensive, is widely used in maximum likelihood phylogenetic analyses. We recently proposed the ultrafast bootstrap approximation (UFBoot) to reduce computing time while achieving more unbiased branch supports than SBS under mild model violations. UFBoot has been steadily adopted as an efficient alternative to SBS and other bootstrap approaches. Here, we present UFBoot2, which substantially accelerates UFBoot and reduces the risk of overestimating branch supports due to polytomies or severe model violations. Additionally, UFBoot2 provides suitable bootstrap resampling strategies for phylogenomic data. UFBoot2 is 778 times (median) faster than SBS and 8.4 times (median) faster than RAxML rapid bootstrap on tested data sets. UFBoot2 is implemented in the IQ-TREE software package version 1.6 and freely available at http://www.iqtree.org. © The Author 2017. Published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution.
Approximation by max-product type operators
Bede, Barnabás; Gal, Sorin G
2016-01-01
This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product operators as those which allow one to obtain, in many cases, more valuable estimates than those obtained by classical approaches. The text considers a wide variety of operators which are studied for a number of interesting problems such as quantitative estimates, convergence, saturation results, localization, to name several. Additionally, the book discusses the perfect analogies between the probabilistic approaches of the classical Bernstein type operators and of the classical convolution operators (non-periodic and periodic cases), and the possibilistic approaches of the max-product variants of these operators. These approaches allow for two natural interpretations of the max-product Bernstein type operators and convolution type operators: firstly, as possibilistic expectations of some fuzzy variables, and secondly,...
PROX: Approximated Summarization of Data Provenance.
Ainy, Eleanor; Bourhis, Pierre; Davidson, Susan B; Deutch, Daniel; Milo, Tova
2016-03-01
Many modern applications involve collecting large amounts of data from multiple sources, and then aggregating and manipulating it in intricate ways. The complexity of such applications, combined with the size of the collected data, makes it difficult to understand the application logic and how information was derived. Data provenance has been proven helpful in this respect in different contexts; however, maintaining and presenting the full and exact provenance may be infeasible, due to its size and complex structure. For that reason, we introduce the notion of approximated summarized provenance, where we seek a compact representation of the provenance at the possible cost of information loss. Based on this notion, we have developed PROX, a system for the management, presentation and use of data provenance for complex applications. We propose to demonstrate PROX in the context of a movies rating crowd-sourcing system, letting participants view provenance summarization and use it to gain insights on the application and its underlying data.
Exact and Approximate Probabilistic Symbolic Execution
Luckow, Kasper; Pasareanu, Corina S.; Dwyer, Matthew B.; Filieri, Antonio; Visser, Willem
2014-01-01
Probabilistic software analysis seeks to quantify the likelihood of reaching a target event under uncertain environments. Recent approaches compute probabilities of execution paths using symbolic execution, but do not support nondeterminism. Nondeterminism arises naturally when no suitable probabilistic model can capture a program behavior, e.g., for multithreading or distributed systems. In this work, we propose a technique, based on symbolic execution, to synthesize schedulers that resolve nondeterminism to maximize the probability of reaching a target event. To scale to large systems, we also introduce approximate algorithms to search for good schedulers, speeding up established random sampling and reinforcement learning results through the quantification of path probabilities based on symbolic execution. We implemented the techniques in Symbolic PathFinder and evaluated them on nondeterministic Java programs. We show that our algorithms significantly improve upon a state-of- the-art statistical model checking algorithm, originally developed for Markov Decision Processes.
Approximate analytical modeling of leptospirosis infection
Ismail, Nur Atikah; Azmi, Amirah; Yusof, Fauzi Mohamed; Ismail, Ahmad Izani
2017-11-01
Leptospirosis is an infectious disease carried by rodents which can cause death in humans. The disease spreads directly through contact with feces, urine or through bites of infected rodents and indirectly via water contaminated with urine and droppings from them. Significant increase in the number of leptospirosis cases in Malaysia caused by the recent severe floods were recorded during heavy rainfall season. Therefore, to understand the dynamics of leptospirosis infection, a mathematical model based on fractional differential equations have been developed and analyzed. In this paper an approximate analytical method, the multi-step Laplace Adomian decomposition method, has been used to conduct numerical simulations so as to gain insight on the spread of leptospirosis infection.
Analytical approximations for wide and narrow resonances
International Nuclear Information System (INIS)
Suster, Luis Carlos; Martinez, Aquilino Senra; Silva, Fernando Carvalho da
2005-01-01
This paper aims at developing analytical expressions for the adjoint neutron spectrum in the resonance energy region, taking into account both narrow and wide resonance approximations, in order to reduce the numerical computations involved. These analytical expressions, besides reducing computing time, are very simple from a mathematical point of view. The results obtained with this analytical formulation were compared to a reference solution obtained with a numerical method previously developed to solve the neutron balance adjoint equations. Narrow and wide resonances of U 238 were treated and the analytical procedure gave satisfactory results as compared with the reference solution, for the resonance energy range. The adjoint neutron spectrum is useful to determine the neutron resonance absorption, so that multigroup adjoint cross sections used by the adjoint diffusion equation can be obtained. (author)
Analytical approximations for wide and narrow resonances
Energy Technology Data Exchange (ETDEWEB)
Suster, Luis Carlos; Martinez, Aquilino Senra; Silva, Fernando Carvalho da [Universidade Federal, Rio de Janeiro, RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia. Programa de Engenharia Nuclear]. E-mail: aquilino@lmp.ufrj.br
2005-07-01
This paper aims at developing analytical expressions for the adjoint neutron spectrum in the resonance energy region, taking into account both narrow and wide resonance approximations, in order to reduce the numerical computations involved. These analytical expressions, besides reducing computing time, are very simple from a mathematical point of view. The results obtained with this analytical formulation were compared to a reference solution obtained with a numerical method previously developed to solve the neutron balance adjoint equations. Narrow and wide resonances of U{sup 238} were treated and the analytical procedure gave satisfactory results as compared with the reference solution, for the resonance energy range. The adjoint neutron spectrum is useful to determine the neutron resonance absorption, so that multigroup adjoint cross sections used by the adjoint diffusion equation can be obtained. (author)
Evaluating methods for approximating stochastic differential equations.
Brown, Scott D; Ratcliff, Roger; Smith, Philip L
2006-08-01
Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler's method for solving ordinary differential equations. The accuracy of Euler's method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matrix method proposed by Diederich and Busemeyer (2003) yielded significant improvements. The accuracy of all methods depended critically on the size of the approximating time step. The large (∼10 ms) step sizes often used by psychological researchers resulted in large and systematic errors in evaluating RT distributions.
Efficient Approximate OLAP Querying Over Time Series
DEFF Research Database (Denmark)
Perera, Kasun Baruhupolage Don Kasun Sanjeewa; Hahmann, Martin; Lehner, Wolfgang
2016-01-01
The ongoing trend for data gathering not only produces larger volumes of data, but also increases the variety of recorded data types. Out of these, especially time series, e.g. various sensor readings, have attracted attention in the domains of business intelligence and decision making. As OLAP...... queries play a major role in these domains, it is desirable to also execute them on time series data. While this is not a problem on the conceptual level, it can become a bottleneck with regards to query run-time. In general, processing OLAP queries gets more computationally intensive as the volume...... are either costly or require continuous maintenance. In this paper we propose an approach for approximate OLAP querying of time series that offers constant latency and is maintenance-free. To achieve this, we identify similarities between aggregation cuboids and propose algorithms that eliminate...
Approximate truncation robust computed tomography—ATRACT
International Nuclear Information System (INIS)
Dennerlein, Frank; Maier, Andreas
2013-01-01
We present an approximate truncation robust algorithm to compute tomographic images (ATRACT). This algorithm targets at reconstructing volumetric images from cone-beam projections in scenarios where these projections are highly truncated in each dimension. It thus facilitates reconstructions of small subvolumes of interest, without involving prior knowledge about the object. Our method is readily applicable to medical C-arm imaging, where it may contribute to new clinical workflows together with a considerable reduction of x-ray dose. We give a detailed derivation of ATRACT that starts from the conventional Feldkamp filtered-backprojection algorithm and that involves, as one component, a novel original formula for the inversion of the two-dimensional Radon transform. Discretization and numerical implementation are discussed and reconstruction results from both, simulated projections and first clinical data sets are presented. (paper)
Approximate Sensory Data Collection: A Survey
Directory of Open Access Journals (Sweden)
Siyao Cheng
2017-03-01
Full Text Available With the rapid development of the Internet of Things (IoTs, wireless sensor networks (WSNs and related techniques, the amount of sensory data manifests an explosive growth. In some applications of IoTs and WSNs, the size of sensory data has already exceeded several petabytes annually, which brings too many troubles and challenges for the data collection, which is a primary operation in IoTs and WSNs. Since the exact data collection is not affordable for many WSN and IoT systems due to the limitations on bandwidth and energy, many approximate data collection algorithms have been proposed in the last decade. This survey reviews the state of the art of approximatedatacollectionalgorithms. Weclassifythemintothreecategories: themodel-basedones, the compressive sensing based ones, and the query-driven ones. For each category of algorithms, the advantages and disadvantages are elaborated, some challenges and unsolved problems are pointed out, and the research prospects are forecasted.
Approximate Sensory Data Collection: A Survey.
Cheng, Siyao; Cai, Zhipeng; Li, Jianzhong
2017-03-10
With the rapid development of the Internet of Things (IoTs), wireless sensor networks (WSNs) and related techniques, the amount of sensory data manifests an explosive growth. In some applications of IoTs and WSNs, the size of sensory data has already exceeded several petabytes annually, which brings too many troubles and challenges for the data collection, which is a primary operation in IoTs and WSNs. Since the exact data collection is not affordable for many WSN and IoT systems due to the limitations on bandwidth and energy, many approximate data collection algorithms have been proposed in the last decade. This survey reviews the state of the art of approximatedatacollectionalgorithms. Weclassifythemintothreecategories: themodel-basedones, the compressive sensing based ones, and the query-driven ones. For each category of algorithms, the advantages and disadvantages are elaborated, some challenges and unsolved problems are pointed out, and the research prospects are forecasted.
The Bloch Approximation in Periodically Perforated Media
International Nuclear Information System (INIS)
Conca, C.; Gomez, D.; Lobo, M.; Perez, E.
2005-01-01
We consider a periodically heterogeneous and perforated medium filling an open domain Ω of R N . Assuming that the size of the periodicity of the structure and of the holes is O(ε),we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ω ε (Ω ε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis R N and then localize the problem for abounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω
Coated sphere scattering by geometric optics approximation.
Mengran, Zhai; Qieni, Lü; Hongxia, Zhang; Yinxin, Zhang
2014-10-01
A new geometric optics model has been developed for the calculation of light scattering by a coated sphere, and the analytic expression for scattering is presented according to whether rays hit the core or not. The ray of various geometric optics approximation (GOA) terms is parameterized by the number of reflections in the coating/core interface, the coating/medium interface, and the number of chords in the core, with the degeneracy path and repeated path terms considered for the rays striking the core, which simplifies the calculation. For the ray missing the core, the various GOA terms are dealt with by a homogeneous sphere. The scattering intensity of coated particles are calculated and then compared with those of Debye series and Aden-Kerker theory. The consistency of the results proves the validity of the method proposed in this work.
DEFF Research Database (Denmark)
Sadegh, Payman; Spall, J. C.
1998-01-01
The simultaneous perturbation stochastic approximation (SPSA) algorithm has attracted considerable attention for challenging optimization problems where it is difficult or impossible to obtain a direct gradient of the objective (say, loss) function. The approach is based on a highly efficient...... simultaneous perturbation approximation to the gradient based on loss function measurements. SPSA is based on picking a simultaneous perturbation (random) vector in a Monte Carlo fashion as part of generating the approximation to the gradient. This paper derives the optimal distribution for the Monte Carlo...... process. The objective is to minimize the mean square error of the estimate. The authors also consider maximization of the likelihood that the estimate be confined within a bounded symmetric region of the true parameter. The optimal distribution for the components of the simultaneous perturbation vector...
DEFF Research Database (Denmark)
Sadegh, Payman; Spall, J. C.
1997-01-01
The simultaneous perturbation stochastic approximation (SPSA) algorithm has recently attracted considerable attention for optimization problems where it is difficult or impossible to obtain a direct gradient of the objective (say, loss) function. The approach is based on a highly efficient...... simultaneous perturbation approximation to the gradient based on loss function measurements. SPSA is based on picking a simultaneous perturbation (random) vector in a Monte Carlo fashion as part of generating the approximation to the gradient. This paper derives the optimal distribution for the Monte Carlo...... process. The objective is to minimize the mean square error of the estimate. We also consider maximization of the likelihood that the estimate be confined within a bounded symmetric region of the true parameter. The optimal distribution for the components of the simultaneous perturbation vector is found...
Some properties of dual and approximate dual of fusion frames
Arefijamaal, Ali Akbar; Neyshaburi, Fahimeh Arabyani
2016-01-01
In this paper we extend the notion of approximate dual to fusion frames and present some approaches to obtain dual and approximate alternate dual fusion frames. Also, we study the stability of dual and approximate alternate dual fusion frames.
Approximate Model for Turbulent Stagnation Point Flow.
Energy Technology Data Exchange (ETDEWEB)
Dechant, Lawrence [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2016-01-01
Here we derive an approximate turbulent self-similar model for a class of favorable pressure gradient wedge-like flows, focusing on the stagnation point limit. While the self-similar model provides a useful gross flow field estimate this approach must be combined with a near wall model is to determine skin friction and by Reynolds analogy the heat transfer coefficient. The combined approach is developed in detail for the stagnation point flow problem where turbulent skin friction and Nusselt number results are obtained. Comparison to the classical Van Driest (1958) result suggests overall reasonable agreement. Though the model is only valid near the stagnation region of cylinders and spheres it nonetheless provides a reasonable model for overall cylinder and sphere heat transfer. The enhancement effect of free stream turbulence upon the laminar flow is used to derive a similar expression which is valid for turbulent flow. Examination of free stream enhanced laminar flow suggests that the rather than enhancement of a laminar flow behavior free stream disturbance results in early transition to turbulent stagnation point behavior. Excellent agreement is shown between enhanced laminar flow and turbulent flow behavior for high levels, e.g. 5% of free stream turbulence. Finally the blunt body turbulent stagnation results are shown to provide realistic heat transfer results for turbulent jet impingement problems.
Configuring Airspace Sectors with Approximate Dynamic Programming
Bloem, Michael; Gupta, Pramod
2010-01-01
In response to changing traffic and staffing conditions, supervisors dynamically configure airspace sectors by assigning them to control positions. A finite horizon airspace sector configuration problem models this supervisor decision. The problem is to select an airspace configuration at each time step while considering a workload cost, a reconfiguration cost, and a constraint on the number of control positions at each time step. Three algorithms for this problem are proposed and evaluated: a myopic heuristic, an exact dynamic programming algorithm, and a rollouts approximate dynamic programming algorithm. On problem instances from current operations with only dozens of possible configurations, an exact dynamic programming solution gives the optimal cost value. The rollouts algorithm achieves costs within 2% of optimal for these instances, on average. For larger problem instances that are representative of future operations and have thousands of possible configurations, excessive computation time prohibits the use of exact dynamic programming. On such problem instances, the rollouts algorithm reduces the cost achieved by the heuristic by more than 15% on average with an acceptable computation time.
Adaptive approximation of higher order posterior statistics
Lee, Wonjung
2014-02-01
Filtering is an approach for incorporating observed data into time-evolving systems. Instead of a family of Dirac delta masses that is widely used in Monte Carlo methods, we here use the Wiener chaos expansion for the parametrization of the conditioned probability distribution to solve the nonlinear filtering problem. The Wiener chaos expansion is not the best method for uncertainty propagation without observations. Nevertheless, the projection of the system variables in a fixed polynomial basis spanning the probability space might be a competitive representation in the presence of relatively frequent observations because the Wiener chaos approach not only leads to an accurate and efficient prediction for short time uncertainty quantification, but it also allows to apply several data assimilation methods that can be used to yield a better approximate filtering solution. The aim of the present paper is to investigate this hypothesis. We answer in the affirmative for the (stochastic) Lorenz-63 system based on numerical simulations in which the uncertainty quantification method and the data assimilation method are adaptively selected by whether the dynamics is driven by Brownian motion and the near-Gaussianity of the measure to be updated, respectively. © 2013 Elsevier Inc.
Rainbows: Mie computations and the Airy approximation.
Wang, R T; van de Hulst, H C
1991-01-01
Efficient and accurate computation of the scattered intensity pattern by the Mie formulas is now feasible for size parameters up to x = 50,000 at least, which in visual light means spherical drops with diameters up to 6 mm. We present a method for evaluating the Mie coefficients from the ratios between Riccati-Bessel and Neumann functions of successive order. We probe the applicability of the Airy approximation, which we generalize to rainbows of arbitrary p (number of internal reflections = p - 1), by comparing the Mie and Airy intensity patterns. Millimeter size water drops show a match in all details, including the position and intensity of the supernumerary maxima and the polarization. A fairly good match is still seen for drops of 0.1 mm. A small spread in sizes helps to smooth out irrelevant detail. The dark band between the rainbows is used to test more subtle features. We conclude that this band contains not only externally reflected light (p = 0) but also a sizable contribution f rom the p = 6 and p = 7 rainbows, which shift rapidly with wavelength. The higher the refractive index, the closer both theories agree on the first primary rainbow (p = 2) peak for drop diameters as small as 0.02 mm. This may be useful in supporting experimental work.
Approximate von Neumann entropy for directed graphs.
Ye, Cheng; Wilson, Richard C; Comin, César H; Costa, Luciano da F; Hancock, Edwin R
2014-05-01
In this paper, we develop an entropy measure for assessing the structural complexity of directed graphs. Although there are many existing alternative measures for quantifying the structural properties of undirected graphs, there are relatively few corresponding measures for directed graphs. To fill this gap in the literature, we explore an alternative technique that is applicable to directed graphs. We commence by using Chung's generalization of the Laplacian of a directed graph to extend the computation of von Neumann entropy from undirected to directed graphs. We provide a simplified form of the entropy which can be expressed in terms of simple node in-degree and out-degree statistics. Moreover, we find approximate forms of the von Neumann entropy that apply to both weakly and strongly directed graphs, and that can be used to characterize network structure. We illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real-world data sets, including structures from protein databases and high energy physics theory citation networks.
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
Coronal Loops: Evolving Beyond the Isothermal Approximation
Schmelz, J. T.; Cirtain, J. W.; Allen, J. D.
2002-05-01
Are coronal loops isothermal? A controversy over this question has arisen recently because different investigators using different techniques have obtained very different answers. Analysis of SOHO-EIT and TRACE data using narrowband filter ratios to obtain temperature maps has produced several key publications that suggest that coronal loops may be isothermal. We have constructed a multi-thermal distribution for several pixels along a relatively isolated coronal loop on the southwest limb of the solar disk using spectral line data from SOHO-CDS taken on 1998 Apr 20. These distributions are clearly inconsistent with isothermal plasma along either the line of sight or the length of the loop, and suggested rather that the temperature increases from the footpoints to the loop top. We speculated originally that these differences could be attributed to pixel size -- CDS pixels are larger, and more `contaminating' material would be expected along the line of sight. To test this idea, we used CDS iron line ratios from our data set to mimic the isothermal results from the narrowband filter instruments. These ratios indicated that the temperature gradient along the loop was flat, despite the fact that a more complete analysis of the same data showed this result to be false! The CDS pixel size was not the cause of the discrepancy; rather, the problem lies with the isothermal approximation used in EIT and TRACE analysis. These results should serve as a strong warning to anyone using this simplistic method to obtain temperature. This warning is echoed on the EIT web page: ``Danger! Enter at your own risk!'' In other words, values for temperature may be found, but they may have nothing to do with physical reality. Solar physics research at the University of Memphis is supported by NASA grant NAG5-9783. This research was funded in part by the NASA/TRACE MODA grant for Montana State University.
Kim, SungKun; Lee, Hunpyo
2017-06-01
Via a dynamical cluster approximation with N c = 4 in combination with a semiclassical approximation (DCA+SCA), we study the doped two-dimensional Hubbard model. We obtain a plaquette antiferromagnetic (AF) Mott insulator, a plaquette AF ordered metal, a pseudogap (or d-wave superconductor) and a paramagnetic metal by tuning the doping concentration. These features are similar to the behaviors observed in copper-oxide superconductors and are in qualitative agreement with the results calculated by the cluster dynamical mean field theory with the continuous-time quantum Monte Carlo (CDMFT+CTQMC) approach. The results of our DCA+SCA differ from those of the CDMFT+CTQMC approach in that the d-wave superconducting order parameters are shown even in the high doped region, unlike the results of the CDMFT+CTQMC approach. We think that the strong plaquette AF orderings in the dynamical cluster approximation (DCA) with N c = 4 suppress superconducting states with increasing doping up to strongly doped region, because frozen dynamical fluctuations in a semiclassical approximation (SCA) approach are unable to destroy those orderings. Our calculation with short-range spatial fluctuations is initial research, because the SCA can manage long-range spatial fluctuations in feasible computational times beyond the CDMFT+CTQMC tool. We believe that our future DCA+SCA calculations should supply information on the fully momentum-resolved physical properties, which could be compared with the results measured by angle-resolved photoemission spectroscopy experiments.
The (e,2e) reaction in molecules
International Nuclear Information System (INIS)
Dey, S.; Dixon, A.; Teubner, P.J.O.; Weigold, E.
1975-01-01
The aplication of the (e,2e) technique is discussed in the framework of (e,2e) on molecular hydrogen. It is shown that the technique is sufficiently sensitive to distinguish between simple wavefunctions and those containing configuration interactions. By comparing the data on H 2 and D 2 is shown that the Born-Oppenheimer approximation is confirmed to an accuracy of about 3 per cent. The data is also used to contrast other methods of determining electron momentum distributions in molecules. Data on methane, carbon monoxide and molecular nitrogen is also presented. (author)
Effect of nucleon and hadron structure changes in-medium and its impact on observables
Energy Technology Data Exchange (ETDEWEB)
K. Saito; K. Tsushima; A.W. Thomas
2005-07-05
We study the effect of hadron structure changes in a nuclear medium using the quark-meson coupling (QMC) model. The QMC model is based on a mean field description of non-overlapping nucleon (or baryon) bags bound by the self-consistent exchange of scalar and vector mesons in the isoscalar and isovector channels. The model is extended to investigate the properties of finite nuclei, in which, using the Born-Oppenheimer approximation to describe the interacting quark-meson system, one can derive the effective equation of motion for the nucleon (or baryon), as well as the self-consistent equations for the meson mean fields.
Ab initio vibration rotation spectra of triterated isotopologues of H3+
Amin, Amit; Nazareth, Jude; Tennyson, Jonathan
2008-11-01
Vibration-rotation energy levels for T3+, H 2T +, T 2H +, D 2T + and T 2D + are reported. These were calculated using the high accuracy model of Polyansky and Tennyson (J. Chem. Phys. 110 (1999) 5056) which explicitly allows for both adiabatic and non-adiabatic corrections to the Born-Oppenheimer approximation. These levels should be reliable to better than 0.05 cm -1 and can thus be used to make reliable predictions of infrared spectra for the triterated H3+ species.
Wouters, Sebastian; Limacher, Peter A; Van Neck, Dimitri; Ayers, Paul W
2012-04-07
We have implemented the sweep algorithm for the variational optimization of SU(2) U(1) (spin and particle number) invariant matrix product states (MPS) for general spin and particle number invariant fermionic Hamiltonians. This class includes non-relativistic quantum chemical systems within the Born-Oppenheimer approximation. High-accuracy ab initio finite field results of the longitudinal static polarizabilities and second hyperpolarizabilities of one-dimensional hydrogen chains are presented. This allows to assess the performance of other quantum chemical methods. For small basis sets, MPS calculations in the saturation regime of the optical response properties can be performed. These results are extrapolated to the thermodynamic limit.
Herrmann, Thomas; Ren, Qinghua; Balint-Kurti, Gabriel G.; Manby, Frederick R.
2007-06-01
Optimal control of rovibrational excitations of the CO molecule using picosecond infrared laser pulses is described in the framework of the electric-nuclear Born-Oppenheimer approximation [G. G. Balint-Kurti et al., J. Chem. Phys. 122, 084110 (2005)]. The potential energy surface of the CO molecule in the presence of an electric field is calculated using coupled cluster theory with a large orbital basis set. The quantum dynamics of the process is treated using a full three dimensional treatment of the molecule in the laser field. The detailed mechanisms leading to efficient control of the selected excitation processes are discussed.
The role of meson dynamics in nuclear matter saturation
International Nuclear Information System (INIS)
Goncalves, E.
1988-01-01
The problem of the saturation of nuclea matter in the non-relativistic limit of the model proposed by J.D. Walecka is studied. In the original context nuclear matter saturation is obtained as a direct consequence of relativistic effects and both scalar and vector mesons are treated statically. In the present work we investigate the effect of the meson dynamics for the saturation using a Born-Oppenheimer approximation for the ground state. An upper limit for the saturation curve of nuclear matter and are able to decide now essential is the relativistic treatment of the nucleons for this problem, is obtained. (author) [pt
Directory of Open Access Journals (Sweden)
H. Krienke
2013-01-01
Full Text Available Theoretical calculations of the conductivity of sodium nitrate in water are presented and compared with experimental measurements. The method of direct correlation force in the framework of the interionic theory is used for the calculation of transport properties in connection with the associative mean spherical approximation (AMSA. The effective interactions between ions in solutions are derived with the help of Monte Carlo and Molecular Dynamics calculations on the Born-Oppenheimer level. This work is based on earlier theoretical and experimental studies of the structure of concentrated aqueous sodium nitrate solutions.
Stability of two-electron diatomic molecules
International Nuclear Information System (INIS)
Ferron, Alejandro; Serra, Pablo
2007-01-01
We present a detailed study of the ground state behaviour of two-electron diatomic molecules. The ground state stability diagram for diatomic molecules in the Born-Oppenheimer approximation is obtained and the behaviour of the ground state near the stability line is studied. Two different cases are analysed: the homonuclear two-centre two-electron molecule with the internuclear distance as a free parameter and the diatomic two-electron molecule (in this case, the internuclear distance is determined by equilibrium conditions). Analytical and numerical results for these systems are presented
Vibrational Excitation of Diatomic Molecular Ions in Strong Field Ionization of Diatomic Molecules
International Nuclear Information System (INIS)
Kjeldsen, Thomas K.; Madsen, Lars Bojer
2005-01-01
A model based on the strong-field and Born-Oppenheimer approximations qualitatively describes the distribution over vibrational states formed in a diatomic molecular ion following ionization of the neutral molecule by intense laser pulses. Good agreement is found with a recent experiment [X. Urbain et al., Phys. Rev. Lett. 92, 163004 (2004)]. In particular, the observed deviation from a Franck-Condon-like distribution is reproduced. Additionally, we demonstrate control of the vibrational distribution by a variation of the peak intensity or a change of frequency of the laser pulse
Tunneling Ionization of Diatomic Molecules
DEFF Research Database (Denmark)
Svensmark, Jens Søren Sieg
2016-01-01
When a molecule is subject to a strong laser field, there is a probability that an electron can escape, even though the electrons are bound by a large potential barrier. This is possible because electrons are quantum mechanical in nature, and they are therefore able to tunnel through potential...... of tunneling ionizaion of molecules is presented and the results of numerical calculations are shown. One perhaps surprising result is, that the frequently used Born-Oppenheimer approximation breaks down for weak fields when describing tunneling ionization. An analytic theory applicable in the weak-field limit...
Approximate solutions of the Wei Hua oscillator using the Pekeris ...
Indian Academy of Sciences (India)
The analytical exact solutions of the wave equation with some exponential-type potentials are impossible for l = 0 states. So, the next best thing to do is to find approximate analytical solutions of a given potential by appropriate approximation techniques. Therefore, approximate schemes like the Pekeris approximation [6–8] ...
Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines
Directory of Open Access Journals (Sweden)
Xiaolei Zhang
2009-01-01
Full Text Available This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a global G2 continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.
Cheon, Sooyoung
2013-02-16
Importance sampling and Markov chain Monte Carlo methods have been used in exact inference for contingency tables for a long time, however, their performances are not always very satisfactory. In this paper, we propose a stochastic approximation Monte Carlo importance sampling (SAMCIS) method for tackling this problem. SAMCIS is a combination of adaptive Markov chain Monte Carlo and importance sampling, which employs the stochastic approximation Monte Carlo algorithm (Liang et al., J. Am. Stat. Assoc., 102(477):305-320, 2007) to draw samples from an enlarged reference set with a known Markov basis. Compared to the existing importance sampling and Markov chain Monte Carlo methods, SAMCIS has a few advantages, such as fast convergence, ergodicity, and the ability to achieve a desired proportion of valid tables. The numerical results indicate that SAMCIS can outperform the existing importance sampling and Markov chain Monte Carlo methods: It can produce much more accurate estimates in much shorter CPU time than the existing methods, especially for the tables with high degrees of freedom. © 2013 Springer Science+Business Media New York.
Peng, Degao; Yang, Yang; Zhang, Peng; Yang, Weitao
2014-12-07
In this article, we develop systematically second random phase approximations (RPA) and Tamm-Dancoff approximations (TDA) of particle-hole and particle-particle channels for calculating molecular excitation energies. The second particle-hole RPA/TDA can capture double excitations missed by the particle-hole RPA/TDA and time-dependent density-functional theory (TDDFT), while the second particle-particle RPA/TDA recovers non-highest-occupied-molecular-orbital excitations missed by the particle-particle RPA/TDA. With proper orbital restrictions, these restricted second RPAs and TDAs have a formal scaling of only O(N(4)). The restricted versions of second RPAs and TDAs are tested with various small molecules to show some positive results. Data suggest that the restricted second particle-hole TDA (r2ph-TDA) has the best overall performance with a correlation coefficient similar to TDDFT, but with a larger negative bias. The negative bias of the r2ph-TDA may be induced by the unaccounted ground state correlation energy to be investigated further. Overall, the r2ph-TDA is recommended to study systems with both single and some low-lying double excitations with a moderate accuracy. Some expressions on excited state property evaluations, such as ⟨Ŝ(2)⟩ are also developed and tested.
Coefficients Calculation in Pascal Approximation for Passive Filter Design
Directory of Open Access Journals (Sweden)
George B. Kasapoglu
2018-02-01
Full Text Available The recently modified Pascal function is further exploited in this paper in the design of passive analog filters. The Pascal approximation has non-equiripple magnitude, in contrast of the most well-known approximations, such as the Chebyshev approximation. A novelty of this work is the introduction of a precise method that calculates the coefficients of the Pascal function. Two examples are presented for the passive design to illustrate the advantages and the disadvantages of the Pascal approximation. Moreover, the values of the passive elements can be taken from tables, which are created to define the normalized values of these elements for the Pascal approximation, as Zverev had done for the Chebyshev, Elliptic, and other approximations. Although Pascal approximation can be implemented to both passive and active filter designs, a passive filter design is addressed in this paper, and the benefits and shortcomings of Pascal approximation are presented and discussed.
Comparison of four support-vector based function approximators
de Kruif, B.J.; de Vries, Theodorus J.A.
2004-01-01
One of the uses of the support vector machine (SVM), as introduced in V.N. Vapnik (2000), is as a function approximator. The SVM and approximators based on it, approximate a relation in data by applying interpolation between so-called support vectors, being a limited number of samples that have been
Meta-Regression Approximations to Reduce Publication Selection Bias
Stanley, T. D.; Doucouliagos, Hristos
2014-01-01
Publication selection bias is a serious challenge to the integrity of all empirical sciences. We derive meta-regression approximations to reduce this bias. Our approach employs Taylor polynomial approximations to the conditional mean of a truncated distribution. A quadratic approximation without a linear term, precision-effect estimate with…
Three-loop Phi-derivable approximation in QED
Andersen, J.O.; Strickland, M.
2005-01-01
In this paper we examine 9Φ-derivable approximations in QED, General theorems tell us that the gauge dependence of the n-loop Φ-derivable approximation shows up at order g2n where g is the coupling constant. We consider the gauge dependence of the two-loop Φ-derivable approximation to the Debye mass
On Love's approximation for fluid-filled elastic tubes
International Nuclear Information System (INIS)
Caroli, E.; Mainardi, F.
1980-01-01
A simple procedure is set up to introduce Love's approximation for wave propagation in thin-walled fluid-filled elastic tubes. The dispersion relation for linear waves and the radial profile for fluid pressure are determined in this approximation. It is shown that the Love approximation is valid in the low-frequency regime. (author)
Approximation Properties of Certain Summation Integral Type Operators
Directory of Open Access Journals (Sweden)
Patel P.
2015-03-01
Full Text Available In the present paper, we study approximation properties of a family of linear positive operators and establish direct results, asymptotic formula, rate of convergence, weighted approximation theorem, inverse theorem and better approximation for this family of linear positive operators.
Iterative Approximate Solutions of Kinetic Equations for Reversible Enzyme Reactions
Khoshnaw, S.
2012-01-01
We study kinetic models of reversible enzyme reactions and compare two techniques for analytic approximate solutions of the model. Analytic approximate solutions of non-linear reaction equations for reversible enzyme reactions are calculated using the Homotopy Perturbation Method (HPM) and the Simple Iteration Method (SIM). The results of the approximations are similar. The Matlab programs are included in appendices.
Approximate Noether symmetries and collineations for regular perturbative Lagrangians
Paliathanasis, Andronikos; Jamal, Sameerah
2018-01-01
Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approximate symmetries. Finally, some applications are presented.
Perturbative corrections for approximate inference in gaussian latent variable models
DEFF Research Database (Denmark)
Opper, Manfred; Paquet, Ulrich; Winther, Ole
2013-01-01
Expectation Propagation (EP) provides a framework for approximate inference. When the model under consideration is over a latent Gaussian field, with the approximation being Gaussian, we show how these approximations can systematically be corrected. A perturbative expansion is made of the exact b...... illustrate on tree-structured Ising model approximations. Furthermore, they provide a polynomial-time assessment of the approximation error. We also provide both theoretical and practical insights on the exactness of the EP solution. © 2013 Manfred Opper, Ulrich Paquet and Ole Winther....
Methods of Approximation Theory in Complex Analysis and Mathematical Physics
Saff, Edward
1993-01-01
The book incorporates research papers and surveys written by participants ofan International Scientific Programme on Approximation Theory jointly supervised by Institute for Constructive Mathematics of University of South Florida at Tampa, USA and the Euler International Mathematical Instituteat St. Petersburg, Russia. The aim of the Programme was to present new developments in Constructive Approximation Theory. The topics of the papers are: asymptotic behaviour of orthogonal polynomials, rational approximation of classical functions, quadrature formulas, theory of n-widths, nonlinear approximation in Hardy algebras,numerical results on best polynomial approximations, wavelet analysis. FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics for orthogonal polynomials associated with exponential weights on R.- A.L. Levin, E.B. Saff: Exact Convergence Rates for Best Lp Rational Approximation to the Signum Function and for Optimal Quadrature in Hp.- H. Stahl: Uniform Rational Approximation of x .- M. Rahman, S.K. ...
Discrete extrinsic curvatures and approximation of surfaces by polar polyhedra
Garanzha, V. A.
2010-01-01
Duality principle for approximation of geometrical objects (also known as Eu-doxus exhaustion method) was extended and perfected by Archimedes in his famous tractate “Measurement of circle”. The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to spherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies. The Suggested approach can be applied to nonconvex surfaces and in the case of multiple dimensions.
Approximate solutions for certain bidomain problems in electrocardiography
Johnston, Peter R.
2008-10-01
The simulation of problems in electrocardiography using the bidomain model for cardiac tissue often creates issues with satisfaction of the boundary conditions required to obtain a solution. Recent studies have proposed approximate methods for solving such problems by satisfying the boundary conditions only approximately. This paper presents an analysis of their approximations using a similar method, but one which ensures that the boundary conditions are satisfied during the whole solution process. Also considered are additional functional forms, used in the approximate solutions, which are more appropriate to specific boundary conditions. The analysis shows that the approximations introduced by Patel and Roth [Phys. Rev. E 72, 051931 (2005)] generally give accurate results. However, there are certain situations where functional forms based on the geometry of the problem under consideration can give improved approximations. It is also demonstrated that the recent methods are equivalent to different approaches to solving the same problems introduced 20years earlier.
On approximation of flat Banach modules by free modules
International Nuclear Information System (INIS)
Aristov, O Yu
2005-01-01
The local structure of flat Banach modules is considered; in particular, it is shown that if a flat module has the approximation property, then it is freely approximable, that is, the identity operator on it is approximated by operators each of which admits factorization through a free Banach module satisfying a natural finiteness condition. Among the maps involved in the factorization, the first is approximately multiplicative up to ε on compact sets, and the second is exactly a morphism of modules. The properties of freely approximable and approximately projective modules are studied. It is proved that the standard complex for calculating the derived functor Ext is locally asymptotically exact in the first term for an arbitrary second argument if and only if its first argument is a flat Banach module.
Precise analytic approximations for the Bessel function J1 (x)
Maass, Fernando; Martin, Pablo
2018-03-01
Precise and straightforward analytic approximations for the Bessel function J1 (x) have been found. Power series and asymptotic expansions have been used to determine the parameters of the approximation, which is as a bridge between both expansions, and it is a combination of rational and trigonometric functions multiplied with fractional powers of x. Here, several improvements with respect to the so called Multipoint Quasirational Approximation technique have been performed. Two procedures have been used to determine the parameters of the approximations. The maximum absolute errors are in both cases smaller than 0.01. The zeros of the approximation are also very precise with less than 0.04 per cent for the first one. A second approximation has been also determined using two more parameters, and in this way the accuracy has been increased to less than 0.001.
Space-efficient path-reporting approximate distance oracles
DEFF Research Database (Denmark)
Elkin, Michael; Neiman, Ofer; Wulff-Nilsen, Christian
2016-01-01
We consider approximate path-reporting distance oracles, distance labeling and labeled routing with extremely low space requirements, for general undirected graphs. For distance oracles, we show how to break the nlogn space bound of Thorup and Zwick if approximate paths rather than distances need...... to be reported. For approximate distance labeling and labeled routing, we break the previously best known space bound of O(logn) words per vertex. The cost for such space efficiency is an increased stretch....
Approximation for a large-angle simple pendulum period
International Nuclear Information System (INIS)
Belendez, A; Rodes, J J; Belendez, T; Hernandez, A
2009-01-01
An approximation scheme to obtain the period for large amplitude oscillations of a simple pendulum is analysed and discussed. The analytical approximate formula for the period is the same as that suggested by Hite (2005 Phys. Teach. 43 290), but it is now obtained analytically by means of a term-by-term comparison of the power-series expansion for the approximate period with the corresponding series for the exact period. (letters and comments)
Multijet final states: exact results and the leading pole approximation
International Nuclear Information System (INIS)
Ellis, R.K.; Owens, J.F.
1984-09-01
Exact results for the process gg → ggg are compared with those obtained using the leading pole approximation. Regions of phase space where the approximation breaks down are discussed. A specific example relevant for background estimates to W boson production is presented. It is concluded that in this instance the leading pole approximation may underestimate the standard QCD background by more than a factor of two in certain kinematic regions of physical interest
Fractal image coding by an approximation of the collage error
Salih, Ismail; Smith, Stanley H.
1998-12-01
In fractal image compression an image is coded as a set of contractive transformations, and is guaranteed to generate an approximation to the original image when iteratively applied to any initial image. In this paper we present a method for mapping similar regions within an image by an approximation of the collage error; that is, range blocks can be approximated by a linear combination of domain blocks.
Upper bounds on minimum cardinality of exact and approximate reducts
Chikalov, Igor
2010-01-01
In the paper, we consider the notions of exact and approximate decision reducts for binary decision tables. We present upper bounds on minimum cardinality of exact and approximate reducts depending on the number of rows (objects) in the decision table. We show that the bound for exact reducts is unimprovable in the general case, and the bound for approximate reducts is almost unimprovable in the general case. © 2010 Springer-Verlag Berlin Heidelberg.
LCAO approximation for scaling properties of the Menger sponge fractal.
Sakoda, Kazuaki
2006-11-13
The electromagnetic eigenmodes of a three-dimensional fractal called the Menger sponge were analyzed by the LCAO (linear combination of atomic orbitals) approximation and a first-principle calculation based on the FDTD (finite-difference time-domain) method. Due to the localized nature of the eigenmodes, the LCAO approximation gives a good guiding principle to find scaled eigenfunctions and to observe the approximate self-similarity in the spectrum of the localized eigenmodes.
Analysis of the dynamical cluster approximation for the Hubbard model
Aryanpour, K.; Hettler, M. H.; Jarrell, M.
2002-01-01
We examine a central approximation of the recently introduced Dynamical Cluster Approximation (DCA) by example of the Hubbard model. By both analytical and numerical means we study non-compact and compact contributions to the thermodynamic potential. We show that approximating non-compact diagrams by their cluster analogs results in a larger systematic error as compared to the compact diagrams. Consequently, only the compact contributions should be taken from the cluster, whereas non-compact ...
Approximation theorems by Meyer-Koenig and Zeller type operators
International Nuclear Information System (INIS)
Ali Ozarslan, M.; Duman, Oktay
2009-01-01
This paper is mainly connected with the approximation properties of Meyer-Koenig and Zeller (MKZ) type operators. We first introduce a general sequence of MKZ operators based on q-integers and then obtain a Korovkin-type approximation theorem for these operators. We also compute their rates of convergence by means of modulus of continuity and the elements of Lipschitz class functionals. Furthermore, we give an rth order generalization of our operators in order to get some explicit approximation results.
Co-Occuring Directions Sketching for Approximate Matrix Multiply
Mroueh, Youssef; Marcheret, Etienne; Goel, Vaibhava
2016-01-01
We introduce co-occurring directions sketching, a deterministic algorithm for approximate matrix product (AMM), in the streaming model. We show that co-occuring directions achieves a better error bound for AMM than other randomized and deterministic approaches for AMM. Co-occurring directions gives a $1 + \\epsilon$ -approximation of the optimal low rank approximation of a matrix product. Empirically our algorithm outperforms competing methods for AMM, for a small sketch size. We validate empi...
Exchange energy in the local Airy gas approximation
DEFF Research Database (Denmark)
Vitos, Levente; Johansson, B.; Kollár, J.
2000-01-01
The Airy gas model of the edge electron gas is used to construct an exchange-energy functional that is an alternative to those obtained in the local-density and generalized-gradient approximations. Test calculations for rare-gas atoms, molecules, solids, and surfaces show that the Airy gas...... functional performs better than the local-density approximation in all cases and better than the generalized-gradient approximation for solids and surfaces....
A simple approximation for eigenvalues in quantum theory
International Nuclear Information System (INIS)
Mitter, H.; Yamazaki, K.
1983-01-01
An approximation method for the determination of energy levels in quantum theory is discussed, which starts from a scaled set of eigenstates of a solvable model. It is shown, that the lowest approximation fulfills certain relations, which hold also exactly. The approximation is tested by comparison with numerically computed eigenvalues in several cases. The errors turn out to be moderate in most of these and depend very little on typical coupling constants. (Author)
Perturbative corrections for approximate inference in gaussian latent variable models
DEFF Research Database (Denmark)
Opper, Manfred; Paquet, Ulrich; Winther, Ole
2013-01-01
orders, corrections of increasing polynomial complexity can be applied to the approximation. The second order provides a correction in quadratic time, which we apply to an array of Gaussian process and Ising models. The corrections generalize to arbitrarily complex approximating families, which we...... illustrate on tree-structured Ising model approximations. Furthermore, they provide a polynomial-time assessment of the approximation error. We also provide both theoretical and practical insights on the exactness of the EP solution. © 2013 Manfred Opper, Ulrich Paquet and Ole Winther....
Algorithms For Approximation IV. Proceedings of the 2001 International Symposium
National Research Council Canada - National Science Library
Levesley, Jeremy
2002-01-01
...: Algorithms- Approximation of Functions, Data Fitting, Geometric and Surface Modelling, Splines, Wavelets, Radial Basis Functions, Support Vector Machines, Norms and Metrics, Errors in Data, Uncertainty Estimation...
Recursive B-spline approximation using the Kalman filter
Directory of Open Access Journals (Sweden)
Jens Jauch
2017-02-01
Full Text Available This paper proposes a novel recursive B-spline approximation (RBA algorithm which approximates an unbounded number of data points with a B-spline function and achieves lower computational effort compared with previous algorithms. Conventional recursive algorithms based on the Kalman filter (KF restrict the approximation to a bounded and predefined interval. Conversely RBA includes a novel shift operation that enables to shift estimated B-spline coefficients in the state vector of a KF. This allows to adapt the interval in which the B-spline function can approximate data points during run-time.
Scattering theory and effective medium approximations to heterogeneous materials
International Nuclear Information System (INIS)
Gubernatis, J.E.
1977-01-01
The formal analogy existing between problems studied in the microscopic theory of disordered alloys and problems concerned with the effective (macroscopic) behavior of heterogeneous materials is discussed. Attention is focused on (1) analogous approximations (effective medium approximations) developed for the microscopic problems by scattering theory concepts and techniques, but for the macroscopic problems principally by intuitive means, (2) the link, provided by scattering theory, of the intuitively developed approximations to a well-defined perturbative analysis, (3) the possible presence of conditionally convergent integrals in effective medium approximations
Approximation of quadrilaterals by rational quadrilaterals in the plane
Indian Academy of Sciences (India)
Keywords. Rational triangles and quadrilaterals; rational approximability of polygons; rational points on quartic curves; elliptic curves; torsion points; rational points on varieties and their density.
Embedding impedance approximations in the analysis of SIS mixers
Kerr, A. R.; Pan, S.-K.; Withington, S.
1992-01-01
Future millimeter-wave radio astronomy instruments will use arrays of many SIS receivers, either as focal plane arrays on individual radio telescopes, or as individual receivers on the many antennas of radio interferometers. Such applications will require broadband integrated mixers without mechanical tuners. To produce such mixers, it will be necessary to improve present mixer design techniques, most of which use the three-frequency approximation to Tucker's quantum mixer theory. This paper examines the adequacy of three approximations to Tucker's theory: (1) the usual three-frequency approximation which assumes a sinusoidal LO voltage at the junction, and a short-circuit at all frequencies above the upper sideband; (2) a five-frequency approximation which allows two LO voltage harmonics and five small-signal sidebands; and (3) a quasi five-frequency approximation in which five small-signal sidebands are allowed, but the LO voltage is assumed sinusoidal. These are compared with a full harmonic-Newton solution of Tucker's equations, including eight LO harmonics and their corresponding sidebands, for realistic SIS mixer circuits. It is shown that the accuracy of the three approximations depends strongly on the value of omega R(sub N)C for the SIS junctions used. For large omega R(sub N)C, all three approximations approach the eight-harmonic solution. For omega R(sub N)C values in the range 0.5 to 10, the range of most practical interest, the quasi five-frequency approximation is a considerable improvement over the three-frequency approximation, and should be suitable for much design work. For the realistic SIS mixers considered here, the five-frequency approximation gives results very close to those of the eight-harmonic solution. Use of these approximations, where appropriate, considerably reduces the computational effort needed to analyze an SIS mixer, and allows the design and optimization of mixers using a personal computer.
Nonlinear optics principles and applications
Rottwitt, Karsten
2014-01-01
IntroductionReview of linear opticsInduced polarizationHarmonic oscillator modelLocal field correctionsEstimated nonlinear responseSummaryTime-domain material responseThe polarization time-response functionThe Born-Oppenheimer approximationRaman scattering response function of silicaSummaryMaterial response in the frequency domain, susceptibility tensorsThe susceptibility tensorThe induced polarization in the frequency domainSum of monochromatic fieldsThe prefactor to the induced polarizationThird-order polarization in the Born-Oppenheimer approximation in the frequency domainKramers-Kronig relationsSummarySymmetries in nonlinear opticsSpatial symmetriesSecond-order materialsThird-order nonlinear materialsCyclic coordinate-systemContracted notation for second-order susceptibility tensorsSummaryThe nonlinear wave equationMono and quasi-monochromatic beamsPlane waves - the transverse problemWaveguidesVectorial approachNonlinear birefringenceSummarySecond-order nonlinear effectsGeneral theoryCoupled wave theoryP...
Force-Field Functor Theory: Classical Force-Fields which Reproduce Equilibrium Quantum Distributions
Directory of Open Access Journals (Sweden)
Ryan eBabbush
2013-10-01
Full Text Available Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory.
Babbush, Ryan; Parkhill, John; Aspuru-Guzik, Alán
2013-01-01
Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory.
Efficient approximation of black-box functions and Pareto sets
Rennen, G.
2009-01-01
In the case of time-consuming simulation models or other so-called black-box functions, we determine a metamodel which approximates the relation between the input- and output-variables of the simulation model. To solve multi-objective optimization problems, we approximate the Pareto set, i.e. the
Approximation of the inverse G-frame operator
Indian Academy of Sciences (India)
In this paper, we introduce the concept of (strong) projection method for -frames which works for all conditional -Riesz frames. We also derive a method for approximation of the inverse -frame operator which is efficient for all -frames. We show how the inverse of -frame operator can be approximated as close as we ...
On Approximations of Compact Sets of Functions by Algebraic Surfaces
Kudryavtsev, S. N.
1986-06-01
This article deals with approximations of certain compact sets of smooth and analytic functions by families of functions depending in a polynomial fashion on parameters. The connection between the accuracy of the approximations of the compact sets by such families and the number of parameters and their degree is studied. Bibliography: 3 titles.
The modified signed likelihood statistic and saddlepoint approximations
DEFF Research Database (Denmark)
Jensen, Jens Ledet
1992-01-01
SUMMARY: For a number of tests in exponential families we show that the use of a normal approximation to the modified signed likelihood ratio statistic r * is equivalent to the use of a saddlepoint approximation. This is also true in a large deviation region where the signed likelihood ratio...... statistic r is of order √ n. © 1992 Biometrika Trust....
Approximation of the inverse G-frame operator
Indian Academy of Sciences (India)
... projection method for -frames which works for all conditional -Riesz frames. We also derive a method for approximation of the inverse -frame operator which is efficient for all -frames. We show how the inverse of -frame operator can be approximated as close as we like using finite-dimensional linear algebra.
Performance approximation of pick-to-belt orderpicking systems
M.B.M. de Koster (René)
1994-01-01
textabstractIn this paper, an approximation method is discussed for the analysis of pick-to-belt orderpicking systems. The aim of the approximation method is to provide an instrument for obtaining rapid insight in the performance of designs of pick-to-belt orderpicking systems. It can be used to
An approximate confidence interval for recombination fraction in ...
African Journals Online (AJOL)
user
2011-02-14
Feb 14, 2011 ... proposed a two stage Markov Chain Monte Carlo (MCMC) method to calculate an approximate confidence interval (ACI) ... Key words: Markov Chain Monte Carlo (MCMC), Gibbs sampler, approximate confidence interval, simulation size. ... from local conditional distributions at parameter valuesθ , given the ...
Approximation properties of the neuro-fuzzy minimum function
Gottschling, Andreas; Kreuter, Christof
1999-01-01
The integration of fuzzy logic systems and neural networks in data driven nonlinear modeling applications has generally been limited to functions based upon the multiplicative fuzzy implication rule for theoretical and computational reasons. We derive a universal approximation result for the minimum fuzzy implication rule as well as a differentiable substitute function that allows fast optimization and function approximation with neuro-fuzzy networks.
Finite approximate controllability for semilinear heat equations in noncylindrical domains
Directory of Open Access Journals (Sweden)
Menezes Silvano B. de
2004-01-01
Full Text Available We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.
A simple approximation method for dilute Ising systems
International Nuclear Information System (INIS)
Saber, M.
1996-10-01
We describe a simple approximate method to analyze dilute Ising systems. The method takes into consideration the fluctuations of the effective field, and is based on a probability distribution of random variables which correctly accounts for all the single site kinematic relations. It is shown that the simplest approximation gives satisfactory results when compared with other methods. (author). 12 refs, 2 tabs
Topology optimization of dynamics problems with Padé approximants
DEFF Research Database (Denmark)
Jensen, Jakob Søndergaard
2007-01-01
An efficient procedure for topology optimization of dynamics problems is proposed. The method is based on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using the adjoint method. This gives an accurate approximation of the frequency response over ...
The generalized Mayer theorem in the approximating hamiltonian method
International Nuclear Information System (INIS)
Bakulev, A.P.; Bogoliubov, N.N. Jr.; Kurbatov, A.M.
1982-07-01
With the help of the generalized Mayer theorem we obtain the improved inequality for free energies of model and approximating systems, where only ''connected parts'' over the approximating hamiltonian are taken into account. For the concrete system we discuss the problem of convergency of appropriate series of ''connected parts''. (author)
On a saddlepoint approximation to the Markov binomial distribution
DEFF Research Database (Denmark)
Jensen, Jens Ledet
A nonstandard saddlepoint approximation to the distribution of a sum of Markov dependent trials is introduced. The relative error of the approximation is studied, not only for the number of summands tending to infinity, but also for the parameter approaching the boundary of its definition range. ...
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
On approximation of Lie groups by discrete subgroups
Indian Academy of Sciences (India)
2016-08-26
Aug 26, 2016 ... The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete ...
Approximate solutions of the Wei Hua oscillator using the Pekeris ...
Indian Academy of Sciences (India)
The approximate analytical bound-state solutions of the Schrödinger equation for the. Wei Hua oscillator are carried out in N-dimensional space by taking Pekeris approximation scheme to the orbital centrifugal term. Solutions of the corresponding hyper-radial equation are obtained using the conventional Nikiforov–Uvarov ...
Reply to Steele & Ferrer: Modeling Oscillation, Approximately or Exactly?
Oud, Johan H. L.; Folmer, Henk
2011-01-01
This article addresses modeling oscillation in continuous time. It criticizes Steele and Ferrer's article "Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes" (2011), particularly the approximate estimation procedure applied. This procedure is the latent version of the local linear approximation procedure…
Merging Belief Propagation and the Mean Field Approximation
DEFF Research Database (Denmark)
Riegler, Erwin; Kirkelund, Gunvor Elisabeth; Manchón, Carles Navarro
2010-01-01
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al., which allows to use the same objective function (Kullback-Leibler divergence...
OPTICAL QUANTITATION AND RADIOGRAPHIC DIAGNOSIS OF INCIPIENT APPROXIMAL CARIES LESIONS
VERDONSCHOT, EH; VANDERIJKE, JW; BROUWER, W; TENBOSCH, JJ; TRUIN, GJ
1991-01-01
The objectives of this study were to test the applicability of photocell measurements in approximal caries diagnosis and to evaluate the use of radiographs as validating criterion. Forty extracted premolars were selected, and the progression of the approximal lesions was graded clinically and
Function approximation using combined unsupervised and supervised learning.
Andras, Peter
2014-03-01
Function approximation is one of the core tasks that are solved using neural networks in the context of many engineering problems. However, good approximation results need good sampling of the data space, which usually requires exponentially increasing volume of data as the dimensionality of the data increases. At the same time, often the high-dimensional data is arranged around a much lower dimensional manifold. Here we propose the breaking of the function approximation task for high-dimensional data into two steps: (1) the mapping of the high-dimensional data onto a lower dimensional space corresponding to the manifold on which the data resides and (2) the approximation of the function using the mapped lower dimensional data. We use over-complete self-organizing maps (SOMs) for the mapping through unsupervised learning, and single hidden layer neural networks for the function approximation through supervised learning. We also extend the two-step procedure by considering support vector machines and Bayesian SOMs for the determination of the best parameters for the nonlinear neurons in the hidden layer of the neural networks used for the function approximation. We compare the approximation performance of the proposed neural networks using a set of functions and show that indeed the neural networks using combined unsupervised and supervised learning outperform in most cases the neural networks that learn the function approximation using the original high-dimensional data.
The high intensity approximation applied to multiphoton ionization
International Nuclear Information System (INIS)
Brandi, H.S.; Davidovich, L.; Zagury, N.
1980-08-01
It is shown that the most commonly used high intensity approximations as applied to ionization by strong electromagnetic fields are related. The applicability of the steepest descent method in these approximations, and the relation between them and first-order perturbation theory, are also discussed. (Author) [pt
A Multi-wavelet type limiter for discontinuous Galerkin approximations
Cheruvu, V.; Ryan, J.K.
2010-01-01
In this report, we present a multi-wavelet type limiter for the discontinuous Galerkin method for limiting the solution when spurious oscillations develop near a shock. This limiting leads to a loss of information in the approximation that can be detrimental to a higher order approximation (k > 2).
The second Born approximation of electron–argon elastic scattering ...
Indian Academy of Sciences (India)
We study the elastic scattering of atomic argon by electron in the presence of a bichromatic laser ﬁeld in the second Born approximation. The target atom is approximated by a simple screening potential and the continuum states of the impinging and emitting electrons are described as Volkov states. We evaluate the S-matrix ...
The second Born approximation of electron–argon elastic scattering ...
Indian Academy of Sciences (India)
Abstract. We study the elastic scattering of atomic argon by electron in the presence of a bichro- matic laser field in the second Born approximation. The target atom is approximated by a simple screening potential and the continuum states of the impinging and emitting electrons are described as Volkov states. We evaluate ...
Practical error analysis of the quasi-steady-state approximation ...
African Journals Online (AJOL)
The Quasi-Steady-State Approximation (QSSA) is a method of getting approximate solutions to differential equations, developed heuristically in biochemistry early this century. It can produce acceptable and important results even when formal analytic and numerical procedures fail. It has become associated with singular ...
A-Statistical extension of the Korovkin type approximation theorem
Indian Academy of Sciences (India)
imation, in various areas of functional analysis [1,3,5,12,15]. The study of the Korovkin type approximation theory is a well-established area of research, which deals with the problem of approximating a function f by means of a sequence {Lnf } of positive lin- ear operators. Statistical convergence, which was introduced nearly ...
A periodic basis system of the smooth approximation space
Czech Academy of Sciences Publication Activity Database
Segeth, Karel
2015-01-01
Roč. 267, 15 September (2015), s. 436-444 ISSN 0096-3003 R&D Projects: GA ČR GA14-02067S Institutional support: RVO:67985840 Keywords : smooth approximation * data approximation * data interpolation * Fourier transform Subject RIV: BA - General Mathematics Impact factor: 1.345, year: 2015 http://www.sciencedirect.com/science/article/pii/S0096300315001526
Fifth International Conference on "Approximation and Optimization in the Caribbean"
Approximation, Optimization and Mathematical Economic
2001-01-01
The articles in this proceedings volume reflect the current trends in the theory of approximation, optimization and mathematical economics, and include numerous applications. The book will be of interest to researchers and graduate students involved in functional analysis, approximation theory, mathematical programming and optimization, game theory, mathematical finance and economics.
On root mean square approximation by exponential functions
Sharipov, Ruslan
2014-01-01
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is solved. Then the nonlinear problem is studied in some particular example.
Approximate first integrals of a chaotic Hamiltonian system | Unal ...
African Journals Online (AJOL)
Approximate first integrals (conserved quantities) of a Hamiltonian dynamical system with two-degrees of freedom which arises in the modeling of galaxy have been obtained based on the approximate Noether symmetries for the resonance ω1 = ω2. Furthermore, Kolmogorov-Arnold-Moser (KAM) curves have been ...
The Log-Linear Return Approximation, Bubbles, and Predictability
DEFF Research Database (Denmark)
Engsted, Tom; Pedersen, Thomas Quistgaard; Tanggaard, Carsten
2012-01-01
We study in detail the log-linear return approximation introduced by Campbell and Shiller (1988a). First, we derive an upper bound for the mean approximation error, given stationarity of the log dividend-price ratio. Next, we simulate various rational bubbles which have explosive conditional expe...
Approximate furthest neighbor with application to annulus query
DEFF Research Database (Denmark)
Pagh, Rasmus; Silvestri, Francesco; Sivertsen, Johan von Tangen
2016-01-01
Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries and present a simple, fast, and highly practical data structure for answering AFN queries in high-dimensional Euclid......Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries and present a simple, fast, and highly practical data structure for answering AFN queries in high...... a variation based on a query-independent ordering of the database points; while this does not have the provable approximation factor of the query-dependent data structure, it offers significant improvement in time and space complexity. We give a theoretical analysis and experimental results. As an application...
The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes
Blas, Diego; Tram, Thomas
2011-01-01
Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10'000 to 100'000 Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic LambdaCDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.
Ordering, symbols and finite-dimensional approximations of path integrals
International Nuclear Information System (INIS)
Kashiwa, Taro; Sakoda, Seiji; Zenkin, S.V.
1994-01-01
We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of O(ε), where ε of is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of pq- and qp-ordering. As an explicit example the fermionic oscillator is considered in detail. (author)
Approximate models for broken clouds in stochastic radiative transfer theory
International Nuclear Information System (INIS)
Doicu, Adrian; Efremenko, Dmitry S.; Loyola, Diego; Trautmann, Thomas
2014-01-01
This paper presents approximate models in stochastic radiative transfer theory. The independent column approximation and its modified version with a solar source computed in a full three-dimensional atmosphere are formulated in a stochastic framework and for arbitrary cloud statistics. The nth-order stochastic models describing the independent column approximations are equivalent to the nth-order stochastic models for the original radiance fields in which the gradient vectors are neglected. Fast approximate models are further derived on the basis of zeroth-order stochastic models and the independent column approximation. The so-called “internal mixing” models assume a combination of the optical properties of the cloud and the clear sky, while the “external mixing” models assume a combination of the radiances corresponding to completely overcast and clear skies. A consistent treatment of internal and external mixing models is provided, and a new parameterization of the closure coefficient in the effective thickness approximation is given. An efficient computation of the closure coefficient for internal mixing models, using a previously derived vector stochastic model as a reference, is also presented. Equipped with appropriate look-up tables for the closure coefficient, these models can easily be integrated into operational trace gas retrieval systems that exploit absorption features in the near-IR solar spectrum. - Highlights: • Independent column approximation in a stochastic setting. • Fast internal and external mixing models for total and diffuse radiances. • Efficient optimization of internal mixing models to match reference models
Dynamic flow-driven erosion - An improved approximate solution
Yu, Bofu; Guo, Dawei; Rose, Calvin W.
2017-09-01
Rose et al. (2007) published an approximate solution of dynamic sediment concentration for steady and uniform flows, and this approximate solution shows a peak sediment concentration at the early stage of a runoff event, which can be used to describe and explain the first flush effect, a commonly observed phenomenon, especially in the urban environment. However the approximate solution does not converge to the steady state solution that is known exactly. The purpose of the note is to improve the approximate solution of Rose et al. (2007) by maintaining its functional form while forcing its steady state behaviour for sediment concentration to converge to the known steady state solution. The quality of the new approximate solution was assessed by comparing the new approximate solution with an exact solution for the single size class case, and with the numerical solution for the multiple size classes. It was found that 1) the relative error, or discrepancy, decreases as the stream power increases for all three soils considered; 2) the largest discrepancy occurs for the peak sediment concentration, and the average discrepancy in the peak concentration is less than 10% for the three soils considered; 3) for the majority of the 27 slope-flow combinations and for the three soils considered, the new approximate solution modestly underestimates the peak sediment concentration.
Confidence Intervals for Asbestos Fiber Counts: Approximate Negative Binomial Distribution.
Bartley, David; Slaven, James; Harper, Martin
2017-03-01
The negative binomial distribution is adopted for analyzing asbestos fiber counts so as to account for both the sampling errors in capturing only a finite number of fibers and the inevitable human variation in identifying and counting sampled fibers. A simple approximation to this distribution is developed for the derivation of quantiles and approximate confidence limits. The success of the approximation depends critically on the use of Stirling's expansion to sufficient order, on exact normalization of the approximating distribution, on reasonable perturbation of quantities from the normal distribution, and on accurately approximating sums by inverse-trapezoidal integration. Accuracy of the approximation developed is checked through simulation and also by comparison to traditional approximate confidence intervals in the specific case that the negative binomial distribution approaches the Poisson distribution. The resulting statistics are shown to relate directly to early research into the accuracy of asbestos sampling and analysis. Uncertainty in estimating mean asbestos fiber concentrations given only a single count is derived. Decision limits (limits of detection) and detection limits are considered for controlling false-positive and false-negative detection assertions and are compared to traditional limits computed assuming normal distributions. Published by Oxford University Press on behalf of the British Occupational Hygiene Society 2017.
Weak Approximation of SDEs by Discrete-Time Processes
Directory of Open Access Journals (Sweden)
Henryk Zähle
2008-01-01
Full Text Available We consider the martingale problem related to the solution of an SDE on the line. It is shown that the solution of this martingale problem can be approximated by solutions of the corresponding time-discrete martingale problems under some conditions. This criterion is especially expedient for establishing the convergence of population processes to SDEs. We also show that the criterion yields a weak Euler scheme approximation of SDEs under fairly weak assumptions on the driving force of the approximating processes.
Approximation for a Coulomb-Volkov solution in strong fields
Reiss, H. R.; Krainov, V. P.
1994-08-01
A simple analytical approximation is found for the wave function of an electron simultaneously exposed to a strong, circularly polarized plane-wave field and an atomic Coulomb potential. The approximation is valid when α0>>1, where α0 is the classical radius of motion of a free electron in the plane-wave field. This constraint is sufficiently mild at low frequencies that it makes possible a major extension of the lower bound of laser intensities for which Volkov-solution-based approximations are useful.
Pade approximants and efficient analytic continuation of a power series
International Nuclear Information System (INIS)
Suetin, S P
2002-01-01
This survey reflects the current state of the theory of Pade approximants, that is, best rational approximations of power series. The main focus is on the so-called inverse problems of this theory, in which one must make deductions about analytic continuation of a given power series on the basis of the known asymptotic behaviour of the poles of some sequence of Pade approximants of this series. Row and diagonal sequences are studied from this point of view. Gonchar's and Rakhmanov's fundamental results of inverse nature are presented along with results of the author
Good and Bad Neighborhood Approximations for Outlier Detection Ensembles
DEFF Research Database (Denmark)
Kirner, Evelyn; Schubert, Erich; Zimek, Arthur
2017-01-01
Outlier detection methods have used approximate neighborhoods in filter-refinement approaches. Outlier detection ensembles have used artificially obfuscated neighborhoods to achieve diverse ensemble members. Here we argue that outlier detection models could be based on approximate neighborhoods...... in the first place, thus gaining in both efficiency and effectiveness. It depends, however, on the type of approximation, as only some seem beneficial for the task of outlier detection, while no (large) benefit can be seen for others. In particular, we argue that space-filling curves are beneficial...
Approximate Quantum Adders with Genetic Algorithms: An IBM Quantum Experience
Directory of Open Access Journals (Sweden)
Li Rui
2017-07-01
Full Text Available It has been proven that quantum adders are forbidden by the laws of quantum mechanics. We analyze theoretical proposals for the implementation of approximate quantum adders and optimize them by means of genetic algorithms, improving previous protocols in terms of efficiency and fidelity. Furthermore, we experimentally realize a suitable approximate quantum adder with the cloud quantum computing facilities provided by IBM Quantum Experience. The development of approximate quantum adders enhances the toolbox of quantum information protocols, paving the way for novel applications in quantum technologies.
Error bounds for approximations with deep ReLU networks.
Yarotsky, Dmitry
2017-10-01
We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In particular, we prove that deep ReLU networks more efficiently approximate smooth functions than shallow networks. In the case of approximations of 1D Lipschitz functions we describe adaptive depth-6 network architectures more efficient than the standard shallow architecture. Copyright © 2017 Elsevier Ltd. All rights reserved.
An approximation to the interference term using Frobenius Method
Energy Technology Data Exchange (ETDEWEB)
Palma, Daniel A.P.; Martinez, Aquilino S.; Silva, Fernando C. da [Universidade Federal, Rio de Janeiro, RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia. Programa de Engenharia Nuclear; E-mail: aquilino@lmp.ufrj.br
2007-07-01
An analytical approximation of the interference term {chi}(x,{xi}) is proposed. The approximation is based on the differential equation to {chi}(x,{xi}) using the Frobenius method and the parameter variation. The analytical expression of the {chi}(x,{xi}) obtained in terms of the elementary functions is very simple and precise. In this work one applies the approximations to the Doppler broadening functions and to the interference term in determining the neutron cross sections. Results were validated for the resonances of the U{sup 238} isotope for different energies and temperature ranges. (author)
An approximation to the interference term using Frobenius Method
International Nuclear Information System (INIS)
Palma, Daniel A.P.; Martinez, Aquilino S.; Silva, Fernando C. da
2007-01-01
An analytical approximation of the interference term χ(x,ξ) is proposed. The approximation is based on the differential equation to χ(x,ξ) using the Frobenius method and the parameter variation. The analytical expression of the χ(x,ξ) obtained in terms of the elementary functions is very simple and precise. In this work one applies the approximations to the Doppler broadening functions and to the interference term in determining the neutron cross sections. Results were validated for the resonances of the U 238 isotope for different energies and temperature ranges. (author)
Approximating the ground state of gapped quantum spin systems
Energy Technology Data Exchange (ETDEWEB)
Michalakis, Spyridon [Los Alamos National Laboratory; Hamza, Eman [NON LANL; Nachtergaele, Bruno [NON LANL; Sims, Robert [NON LANL
2009-01-01
We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Z{sup d}. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset {chi} {contained_in} V the ground state projector can be approximated by the product of two projections, one supported on {chi} and one supported on {chi}{sup c}, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. Such an approximation was useful in proving an area law in one dimension, and this result corresponds to a multi-dimensional analogue.
Kullback-Leibler divergence and the Pareto-Exponential approximation.
Weinberg, G V
2016-01-01
Recent radar research interests in the Pareto distribution as a model for X-band maritime surveillance radar clutter returns have resulted in analysis of the asymptotic behaviour of this clutter model. In particular, it is of interest to understand when the Pareto distribution is well approximated by an Exponential distribution. The justification for this is that under the latter clutter model assumption, simpler radar detection schemes can be applied. An information theory approach is introduced to investigate the Pareto-Exponential approximation. By analysing the Kullback-Leibler divergence between the two distributions it is possible to not only assess when the approximation is valid, but to determine, for a given Pareto model, the optimal Exponential approximation.
The degenerate-internal-states approximation for cold collisions
Maan, A.C.; Tiesinga, E.; Stoof, H.T.C.; Verhaar, B.J.
1990-01-01
The Degenerate-Internal-States approximation as well as its first-order correction are shown to provide a convenient method for calculating elastic and inelastic collision amplitudes for low temperature atomic scattering.
High-order finite element approximations of the Maxwell equations
Sarmany, D.
2010-01-01
This thesis discusses numerical approximations of electromagnetic wave propagation, which is mathematically described by the Maxwell equations. These equations are typically either formulated as integral equations or as (partial) differential equations. Throughout this thesis, the numerical
Comparison of different caries detectors for approximal caries detection
Directory of Open Access Journals (Sweden)
Esin Bozdemir
2016-09-01
Conclusion: The ability of bitewing radiography to identify sound surfaces was better than that of the other methods. The LF device was the most sensitive tool for detecting approximal surfaces with caries, followed by the LED device.
Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems
Energy Technology Data Exchange (ETDEWEB)
Benzi, M. [Universita di Bologna (Italy); Tuma, M. [Inst. of Computer Sciences, Prague (Czech Republic)
1996-12-31
A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.
Embedding relations connected with strong approximation of Fourier ...
Indian Academy of Sciences (India)
. © Indian Academy of Sciences. Embedding relations connected with strong approximation of Fourier series. BOGDAN SZAL. Faculty of Mathematics, Computer Science and Econometrics,. University of Zielona Góra, 65-516 Zielona Góra, ul.
Approximating one class of finitely differentiable functions by another
Kudryavtsev, S. N.
1997-04-01
Necessary and sufficient conditions are found for approximating (with preassigned accuracy in an integral metric) a class of functions with a given majorant for the moduli of continuity of the highest derivatives by another class.
Rotational and Helical Surface Approximation for Reverse Engineering
DEFF Research Database (Denmark)
Randrup, Thomas; Pottmann, Helmut
1997-01-01
to basic shapes used in computer aided design. The algorithms apply methods of line geometry to the set of surface normals in combination with techniques of numerical approximation. The presented results possess applications in reverse engineering and computer aided manufacturing....
Interpolation function for approximating knee joint behavior in human gait
Toth-Taşcǎu, Mirela; Pater, Flavius; Stoia, Dan Ioan
2013-10-01
Starting from the importance of analyzing the kinematic data of the lower limb in gait movement, especially the angular variation of the knee joint, the paper propose an approximation function that can be used for processing the correlation among a multitude of knee cycles. The approximation of the raw knee data was done by Lagrange polynomial interpolation on a signal acquired using Zebris Gait Analysis System. The signal used in approximation belongs to a typical subject extracted from a lot of ten investigated subjects, but the function domain of definition belongs to the entire group. The study of the knee joint kinematics plays an important role in understanding the kinematics of the gait, this articulation having the largest range of motion in whole joints, in gait. The study does not propose to find an approximation function for the adduction-abduction movement of the knee, this being considered a residual movement comparing to the flexion-extension.
A weighted random walk approximation to fractional Brownian motion
Lindstrøm, Tom
2007-01-01
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.
A random walk approximation to fractional Brownian motion
Lindstrøm, Tom
2007-01-01
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.
Technical notes. Spherical harmonics approximations of neutron transport
Energy Technology Data Exchange (ETDEWEB)
Demeny, A.; Dede, K.M.; Erdei, K.
1976-12-01
A double-range spherical harmonics approximation obtained by expanding the angular flux separately in the two regions combined with the conventional single-range spherical harmonics is found to give superior description of neutron transport.
Approximate equations at breaking for nearshore wave transformation coefficients
Digital Repository Service at National Institute of Oceanography (India)
Chandramohan, P.; Nayak, B.U.; SanilKumar, V.
Based on small amplitude wave theory approximate equations are evaluated for determining the coefficients of shoaling, refraction, bottom friction, bottom percolation and viscous dissipation at breaking. The results obtainEd. by these equations...
Real-time creased approximate subdivision surfaces with displacements.
Kovacs, Denis; Mitchell, Jason; Drone, Shanon; Zorin, Denis
2010-01-01
We present an extension of Loop and Schaefer's approximation of Catmull-Clark surfaces (ACC) for surfaces with creases and corners. We discuss the integration of ACC into Valve's Source game engine and analyze performance of our implementation.
APPECT: An Approximate Backbone-Based Clustering Algorithm for Tags
DEFF Research Database (Denmark)
Zong, Yu; Xu, Guandong; Jin, Pin
2011-01-01
resulting from the severe difficulty of ambiguity, redundancy and less semantic nature of tags. Clustering method is a useful tool to address the aforementioned difficulties. Most of the researches on tag clustering are directly using traditional clustering algorithms such as K-means or Hierarchical...... algorithm for Tags (APPECT). The main steps of APPECT are: (1) we execute the K-means algorithm on a tag similarity matrix for M times and collect a set of tag clustering results Z={C1,C2,…,Cm}; (2) we form the approximate backbone of Z by executing a greedy search; (3) we fix the approximate backbone...... Agglomerative Clustering on tagging data, which possess the inherent drawbacks, such as the sensitivity of initialization. In this paper, we instead make use of the approximate backbone of tag clustering results to find out better tag clusters. In particular, we propose an APProximate backbonE-based Clustering...
Approximation with positive linear operators and linear combinations
Gupta, Vijay
2017-01-01
This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian–Ivanov for linear combinations of Bernstein and Bernstein–Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduate and postgraduate courses as we...
Galerkin approximations of nonlinear optimal control problems in Hilbert spaces
Directory of Open Access Journals (Sweden)
Mickael D. Chekroun
2017-07-01
Full Text Available Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere $\\mathbb{S}^2$.
Approximate eigensolutions of Dirac equation for the superposition ...
Indian Academy of Sciences (India)
2014-07-02
Uvarov (NU) method is used to examine the approximate analytical energy eigenvalues and two-component wave function of the Dirac equation with the Hellmann potential for arbitrary spin-orbit quantum number κ in the ...
Numerical approximation of random periodic solutions of stochastic differential equations
Feng, Chunrong; Liu, Yu; Zhao, Huaizhong
2017-10-01
In this paper, we discuss the numerical approximation of random periodic solutions of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to -∞ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and modified Milstein scheme. Subsequently, we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of √{Δ t} in the mean square sense in Euler-Maruyama method and Δ t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.
Approximate relationship between frequency-dependent skin depth ...
Indian Academy of Sciences (India)
Home; Journals; Journal of Earth System Science; Volume 125; Issue 7. Approximate relationship between frequency-dependent skin depth resolved from geoelectromagnetic pedotransfer function and depth of investigation resolved from geoelectrical measurements: A case study of coastal formation, southern Nigeria.
The Space Complexity of 2-Dimensional Approximate Range Counting
DEFF Research Database (Denmark)
Wei, Zhewei; Yi, Ke
2013-01-01
with respect to orthogonal ranges, and the best lower bound is . The ε-approximation is a rather restricted data structure, as we are not allowed to store any information other than the coordinates of a subset of points in P. In this paper, we explore what can be achieved without any restriction on the data...... with additive error εn. A well-known solution for this problem is the ε-approximation. Informally speaking, an ε-approximation of P is a subset A ⊆ P that allows us to estimate the number of points in P ∩ R by counting the number of points in A ∩ R. It is known that an ε-approximation of size exists for any P...
Approximation generation for correlations in thermal-hydraulic analysis codes
International Nuclear Information System (INIS)
Pereira, Luiz C.M.; Carmo, Eduardo G.D. do
1997-01-01
A fast and precise evaluation of fluid thermodynamic and transport properties is needed for the efficient mass, energy and momentum transport phenomena simulation related to nuclear plant power generation. A fully automatic code capable to generate suitable approximation for correlations with one or two independent variables is presented. Comparison in terms of access speed and precision with original correlations currently used shows the adequacy of the approximation obtained. (author). 4 refs., 8 figs., 1 tab
Compound Poisson Approximations for Sums of Random Variables
Serfozo, Richard F.
1986-01-01
We show that a sum of dependent random variables is approximately compound Poisson when the variables are rarely nonzero and, given they are nonzero, their conditional distributions are nearly identical. We give several upper bounds on the total-variation distance between the distribution of such a sum and a compound Poisson distribution. Included is an example for Markovian occurrences of a rare event. Our bounds are consistent with those that are known for Poisson approximations for sums of...
Approximate quantitative relationships for rotating magnetic field current drive
International Nuclear Information System (INIS)
Hugrass, W.N.; Ohnishi, M.
1999-01-01
A simplified model for the rotating magnetic field (RMF) current drive in an infinitely long cylindrical plasma is used to obtain approximate relationships between the fluid flow velocities, collisionality and degree of nonlinearity. These approximate relationships provide simple quantitative estimates for the basic conditions required for the RMF current drive technique to be applied successfully. In particular, the condition required for the motion of the ion fluid not to be flux-preserving, is evaluated quantitatively for the first time. (author)
Comparison of interpolation and approximation methods for optical freeform synthesis
Voznesenskaya, Anna; Krizskiy, Pavel
2017-06-01
Interpolation and approximation methods for freeform surface synthesis are analyzed using the developed software tool. Special computer tool is developed and results of freeform surface modeling with piecewise linear interpolation, piecewise quadratic interpolation, cubic spline interpolation, Lagrange polynomial interpolation are considered. The most accurate interpolation method is recommended. Surface profiles are approximated with the square least method. The freeform systems are generated in optical design software.
Approximate Dynamic Programming Based on High Dimensional Model Representation
Czech Academy of Sciences Publication Activity Database
Pištěk, Miroslav
2013-01-01
Roč. 49, č. 5 (2013), s. 720-737 ISSN 0023-5954 R&D Projects: GA ČR(CZ) GAP102/11/0437 Institutional support: RVO:67985556 Keywords : approximate dynamic programming * Bellman equation * approximate HDMR minimization * trust region problem Subject RIV: BC - Control Systems Theory Impact factor: 0.563, year: 2013 http:// library .utia.cas.cz/separaty/2013/AS/pistek-0399560.pdf
Approximate Compilation of Constraints into Multivalued Decision Diagrams
DEFF Research Database (Denmark)
Hadzic, Tarik; Hooker, John N.; O’Sullivan, Barry
2008-01-01
We present an incremental refinement algorithm for approximate compilation of constraint satisfaction models into multivalued decision diagrams (MDDs). The algorithm uses a vertex splitting operation that relies on the detection of equivalent paths in the MDD. Although the algorithm is quite gene...... by replacing the equivalence test with a constraint-specific measure of distance. We demonstrate the value of the approach for approximate and exact MDD compilation and evaluate its benefits in one of the main MDD application domains, interactive configuration....
Approximate Method for Solving the Linear Fuzzy Delay Differential Equations
Directory of Open Access Journals (Sweden)
S. Narayanamoorthy
2015-01-01
Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.
Approximation of a Class of Incompressible Third Grade Fluids Equations
Directory of Open Access Journals (Sweden)
Zeqi Zhu
2015-01-01
Full Text Available This paper discusses the approximation of weak solutions for a class of incompressible third grade fluids equations. We first introduce a family of perturbed slightly compressible third grade fluids equations (depending on a positive parameter ϵ which approximate the incompressible equations as ϵ→0+. Then we prove the existence and uniqueness of weak solutions for the slightly compressible equations and establish that the solutions of the slightly compressible equations converge to the solutions of the incompressible equations.
Complexity of Gaussian-Radial-Basis Networks Approximating Smooth Functions
Czech Academy of Sciences Publication Activity Database
Kainen, P.C.; Kůrková, Věra; Sanguineti, M.
2009-01-01
Roč. 25, č. 1 (2009), s. 63-74 ISSN 0885-064X R&D Projects: GA ČR GA201/08/1744 Institutional research plan: CEZ:AV0Z10300504 Keywords : Gaussian-radial-basis-function networks * rates of approximation * model complexity * variation norms * Bessel and Sobolev norms * tractability of approximation Subject RIV: IN - Informatics, Computer Science Impact factor: 1.227, year: 2009
Discussion of CoSA: Clustering of Sparse Approximations
Energy Technology Data Exchange (ETDEWEB)
Armstrong, Derek Elswick [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-03-07
The purpose of this talk is to discuss the possible applications of CoSA (Clustering of Sparse Approximations) to the exploitation of HSI (HyperSpectral Imagery) data. CoSA is presented by Moody et al. in the Journal of Applied Remote Sensing (“Land cover classification in multispectral imagery using clustering of sparse approximations over learned feature dictionaries”, Vol. 8, 2014) and is based on machine learning techniques.
Approximate MAP Decoding on Tail-Biting Trellises
Madhu, A. S.; Shankar, Priti
2005-01-01
We propose two approximate algorithms for MAP decoding on tail-biting trellises. The algorithms work on a subset of nodes of the tail-biting trellis, judiciously selected. We report the results of simulations on an AWGN channel using the approximate algorithms on tail-biting trellises for the $(24,12)$ Extended Golay Code and a rate 1/2 convolutional code with memory 6.
Averaging approximation to singularly perturbed nonlinear stochastic wave equations
Lv, Yan; Roberts, A. J.
2012-06-01
An averaging method is applied to derive effective approximation to a singularly perturbed nonlinear stochastic damped wave equation. Small parameter ν > 0 characterizes the singular perturbation, and να, 0 ⩽ α ⩽ 1/2, parametrizes the strength of the noise. Some scaling transformations and the martingale representation theorem yield the effective approximation, a stochastic nonlinear heat equation, for small ν in the sense of distribution.
Solving microwave heating model using Hermite-Pade approximation technique
International Nuclear Information System (INIS)
Makinde, O.D.
2005-11-01
We employ the Hermite-Pade approximation method to explicitly construct the approximate solution of steady state reaction- diffusion equations with source term that arises in modeling microwave heating in an infinite slab with isothermal walls. In particular, we consider the case where the source term decreases spatially and increases with temperature. The important properties of the temperature fields including bifurcations and thermal criticality are discussed. (author)
A simple approximation of productivity scores of fuzzy production plans
DEFF Research Database (Denmark)
Hougaard, Jens Leth
2005-01-01
This paper suggests a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production plans. The procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making it highly operational......This paper suggests a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production plans. The procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making it highly operational...
Efficient approximation of random fields for numerical applications
Harbrecht, Helmut
2015-01-07
We consider the rapid computation of separable expansions for the approximation of random fields. We compare approaches based on techniques from the approximation of non-local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. We provide an a-posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples validate and quantify the considered methods.
Computing gap free Pareto front approximations with stochastic search algorithms.
Schütze, Oliver; Laumanns, Marco; Tantar, Emilia; Coello, Carlos A Coello; Talbi, El-Ghazali
2010-01-01
Recently, a convergence proof of stochastic search algorithms toward finite size Pareto set approximations of continuous multi-objective optimization problems has been given. The focus was on obtaining a finite approximation that captures the entire solution set in some suitable sense, which was defined by the concept of epsilon-dominance. Though bounds on the quality of the limit approximation-which are entirely determined by the archiving strategy and the value of epsilon-have been obtained, the strategies do not guarantee to obtain a gap free approximation of the Pareto front. That is, such approximations A can reveal gaps in the sense that points f in the Pareto front can exist such that the distance of f to any image point F(a), a epsilon A, is "large." Since such gap free approximations are desirable in certain applications, and the related archiving strategies can be advantageous when memetic strategies are included in the search process, we are aiming in this work for such methods. We present two novel strategies that accomplish this task in the probabilistic sense and under mild assumptions on the stochastic search algorithm. In addition to the convergence proofs, we give some numerical results to visualize the behavior of the different archiving strategies. Finally, we demonstrate the potential for a possible hybridization of a given stochastic search algorithm with a particular local search strategy-multi-objective continuation methods-by showing that the concept of epsilon-dominance can be integrated into this approach in a suitable way.
Spectral flow of trimer states of two heavy impurities and one light condensed boson
DEFF Research Database (Denmark)
Zinner, Nikolaj Thomas
2014-01-01
The spectral flow of three-body (trimer) states consisting of two heavy (impurity) particles sitting in a condensate of light bosons is considered. Assuming that the condensate is weakly interaction and that an impurity and a boson have a zero-range two-body interaction, we use the Born-Oppenheim......The spectral flow of three-body (trimer) states consisting of two heavy (impurity) particles sitting in a condensate of light bosons is considered. Assuming that the condensate is weakly interaction and that an impurity and a boson have a zero-range two-body interaction, we use the Born......-Oppenheimer approximation to determine the effective three-body potential. We solve the resulting Schr\\"odinger equation numerically and determine the trimer binding energies as a function of the coherence length of the light bosonic condensate particles. The binding energy is found to be suppressed by the presence...... of the condensate when the energy scale corresponding to the coherence length becomes of order the trimer binding energy in the absence of the condensate. We find that the Efimov scaling property is reflected in the critical values of the condensate coherence length at which the trimers are pushed...
Alonso, J L; Castro, A; Clemente-Gallardo, J; Echenique, P; Mazo, J J; Polo, V; Rubio, A; Zueco, D
2012-12-14
At non-zero temperature and when a system has low-lying excited electronic states, the ground-state Born-Oppenheimer approximation breaks down and the low-lying electronic states are involved in any chemical process. In this work, we use a temperature-dependent effective potential for the nuclei which can accommodate the influence of an arbitrary number of electronic states in a simple way, while at the same time producing the correct Boltzmann equilibrium distribution for the electronic part. With the help of this effective potential, we show that thermally activated low-lying electronic states can have a significant effect in molecular properties for which electronic excitations are oftentimes ignored. We study the thermal expansion of the Manganese dimer, Mn(2), where we find that the average bond length experiences a change larger than the present experimental accuracy upon the inclusion of the excited states into the picture. We also show that, when these states are taken into account, reaction-rate constants are modified. In particular, we study the opening of the ozone molecule, O(3), and show that in this case the rate is modified as much as a 20% with respect to the ground-state Born-Oppenheimer prediction.
Validity of the Aluminum Equivalent Approximation in Space Radiation Shielding
Badavi, Francis F.; Adams, Daniel O.; Wilson, John W.
2009-01-01
The origin of the aluminum equivalent shield approximation in space radiation analysis can be traced back to its roots in the early years of the NASA space programs (Mercury, Gemini and Apollo) wherein the primary radiobiological concern was the intense sources of ionizing radiation causing short term effects which was thought to jeopardize the safety of the crew and hence the mission. Herein, it is shown that the aluminum equivalent shield approximation, although reasonably well suited for that time period and to the application for which it was developed, is of questionable usefulness to the radiobiological concerns of routine space operations of the 21 st century which will include long stays onboard the International Space Station (ISS) and perhaps the moon. This is especially true for a risk based protection system, as appears imminent for deep space exploration where the long-term effects of Galactic Cosmic Ray (GCR) exposure is of primary concern. The present analysis demonstrates that sufficiently large errors in the interior particle environment of a spacecraft result from the use of the aluminum equivalent approximation, and such approximations should be avoided in future astronaut risk estimates. In this study, the aluminum equivalent approximation is evaluated as a means for estimating the particle environment within a spacecraft structure induced by the GCR radiation field. For comparison, the two extremes of the GCR environment, the 1977 solar minimum and the 2001 solar maximum, are considered. These environments are coupled to the Langley Research Center (LaRC) deterministic ionized particle transport code High charge (Z) and Energy TRaNsport (HZETRN), which propagates the GCR spectra for elements with charges (Z) in the range I aluminum equivalent approximation for a good polymeric shield material such as genetic polyethylene (PE). The shield thickness is represented by a 25 g/cm spherical shell. Although one could imagine the progression to greater
Dissociation between exact and approximate addition in developmental dyslexia.
Yang, Xiujie; Meng, Xiangzhi
2016-09-01
Previous research has suggested that number sense and language are involved in number representation and calculation, in which number sense supports approximate arithmetic, and language permits exact enumeration and calculation. Meanwhile, individuals with dyslexia have a core deficit in phonological processing. Based on these findings, we thus hypothesized that children with dyslexia may exhibit exact calculation impairment while doing mental arithmetic. The reaction time and accuracy while doing exact and approximate addition with symbolic Arabic digits and non-symbolic visual arrays of dots were compared between typically developing children and children with dyslexia. Reaction time analyses did not reveal any differences across two groups of children, the accuracies, interestingly, revealed a distinction of approximation and exact addition across two groups of children. Specifically, two groups of children had no differences in approximation. Children with dyslexia, however, had significantly lower accuracy in exact addition in both symbolic and non-symbolic tasks than that of typically developing children. Moreover, linguistic performances were selectively associated with exact calculation across individuals. These results suggested that children with dyslexia have a mental arithmetic deficit specifically in the realm of exact calculation, while their approximation ability is relatively intact. Copyright © 2016 Elsevier Ltd. All rights reserved.
An approximate fractional Gaussian noise model with computational cost
Sørbye, Sigrunn H.
2017-09-18
Fractional Gaussian noise (fGn) is a stationary time series model with long memory properties applied in various fields like econometrics, hydrology and climatology. The computational cost in fitting an fGn model of length $n$ using a likelihood-based approach is ${\\\\mathcal O}(n^{2})$, exploiting the Toeplitz structure of the covariance matrix. In most realistic cases, we do not observe the fGn process directly but only through indirect Gaussian observations, so the Toeplitz structure is easily lost and the computational cost increases to ${\\\\mathcal O}(n^{3})$. This paper presents an approximate fGn model of ${\\\\mathcal O}(n)$ computational cost, both with direct or indirect Gaussian observations, with or without conditioning. This is achieved by approximating fGn with a weighted sum of independent first-order autoregressive processes, fitting the parameters of the approximation to match the autocorrelation function of the fGn model. The resulting approximation is stationary despite being Markov and gives a remarkably accurate fit using only four components. The performance of the approximate fGn model is demonstrated in simulations and two real data examples.
Optimal Approximation of Biquartic Polynomials by Bicubic Splines
Kačala, Viliam; Török, Csaba
2018-02-01
Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline's first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of class C2 and also the best approximant to the polynomial. Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of class C2 if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question. The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.
NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES
Energy Technology Data Exchange (ETDEWEB)
Christensen, Max La Cour [Technical Univ. of Denmark, Lyngby (Denmark); Villa, Umberto E. [Univ. of Texas, Austin, TX (United States); Engsig-Karup, Allan P. [Technical Univ. of Denmark, Lyngby (Denmark); Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-01-22
The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.
An overview on polynomial approximation of NP-hard problems
Directory of Open Access Journals (Sweden)
Paschos Vangelis Th.
2009-01-01
Full Text Available The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
Analysis of 2D NMR relaxation data using Chisholm approximations.
Huber, S; Haase, A; Gleich, B
2017-08-01
To analyze 2D NMR relaxation data based on a discrete delta-like relaxation map we extended the Padé-Laplace method to two dimensions. We approximate the forward Laplace image of the time domain signal by a Chisholm approximation, i.e. a rational polynomial in two dimensions. The poles and residues of this approximation correspond to the relaxation rates and weighting factors of the underlying relaxation map. In this work we explain the principle ideas of our algorithm and demonstrate its applicability. Therefore we compare the inversion results of the Chisholm approximation and Tikhonov regularization method as a function of SNR when the investigated signal is based on a given discrete relaxation map. Our algorithm proved to be reliable for SNRs larger than 50 and is able to compete with the Tikhonov regularization method. Furthermore we show that our method is also able to detect the simulated relaxation compartments of narrow Gaussian distributions with widths less or equal than 0.05s -1 . Finally we investigate the resolution limit with experimental data. For a SNR of 750 the Chisholm approximation method was able to resolve two relaxation compartments in 8 of 10 cases when both compartments differ by a factor of 1.7. Copyright © 2017 Elsevier Inc. All rights reserved.
A quantum relaxation-time approximation for finite fermion systems
Energy Technology Data Exchange (ETDEWEB)
Reinhard, P.-G., E-mail: paul-gerhard.reinhard@fau.de [Institut für Theoretische Physik, Universität Erlangen, D-91058 Erlangen (Germany); Suraud, E. [Université de Toulouse, UPS, Laboratoire de Physique Théorique, IRSAMC, F-31062 Toulouse Cedex (France); Laboratoire de Physique Théorique, Université Paul Sabatier, CNRS, F-31062 Toulouse Cédex (France); Physics Department, University at Buffalo, The State University New York, Buffalo, NY 14260 (United States)
2015-03-15
We propose a relaxation time approximation for the description of the dynamics of strongly excited fermion systems. Our approach is based on time-dependent density functional theory at the level of the local density approximation. This mean-field picture is augmented by collisional correlations handled in relaxation time approximation which is inspired from the corresponding semi-classical picture. The method involves the estimate of microscopic relaxation rates/times which is presently taken from the well established semi-classical experience. The relaxation time approximation implies evaluation of the instantaneous equilibrium state towards which the dynamical state is progressively driven at the pace of the microscopic relaxation time. As test case, we consider Na clusters of various sizes excited either by a swift ion projectile or by a short and intense laser pulse, driven in various dynamical regimes ranging from linear to strongly non-linear reactions. We observe a strong effect of dissipation on sensitive observables such as net ionization and angular distributions of emitted electrons. The effect is especially large for moderate excitations where typical relaxation/dissipation time scales efficiently compete with ionization for dissipating the available excitation energy. Technical details on the actual procedure to implement a working recipe of such a quantum relaxation approximation are given in appendices for completeness.
Analytical models approximating individual processes: a validation method.
Favier, C; Degallier, N; Menkès, C E
2010-12-01
Upscaling population models from fine to coarse resolutions, in space, time and/or level of description, allows the derivation of fast and tractable models based on a thorough knowledge of individual processes. The validity of such approximations is generally tested only on a limited range of parameter sets. A more general validation test, over a range of parameters, is proposed; this would estimate the error induced by the approximation, using the original model's stochastic variability as a reference. This method is illustrated by three examples taken from the field of epidemics transmitted by vectors that bite in a temporally cyclical pattern, that illustrate the use of the method: to estimate if an approximation over- or under-fits the original model; to invalidate an approximation; to rank possible approximations for their qualities. As a result, the application of the validation method to this field emphasizes the need to account for the vectors' biology in epidemic prediction models and to validate these against finer scale models. Copyright © 2010 Elsevier Inc. All rights reserved.
Sequential function approximation on arbitrarily distributed point sets
Wu, Kailiang; Xiu, Dongbin
2018-02-01
We present a randomized iterative method for approximating unknown function sequentially on arbitrary point set. The method is based on a recently developed sequential approximation (SA) method, which approximates a target function using one data point at each step and avoids matrix operations. The focus of this paper is on data sets with highly irregular distribution of the points. We present a nearest neighbor replacement (NNR) algorithm, which allows one to sample the irregular data sets in a near optimal manner. We provide mathematical justification and error estimates for the NNR algorithm. Extensive numerical examples are also presented to demonstrate that the NNR algorithm can deliver satisfactory convergence for the SA method on data sets with high irregularity in their point distributions.
Integral approximants for functions of higher monodromic dimension
Energy Technology Data Exchange (ETDEWEB)
Baker, G.A. Jr.
1987-01-01
In addition to the description of multiform, locally analytic functions as covering a many sheeted version of the complex plane, Riemann also introduced the notion of considering them as describing a space whose ''monodromic'' dimension is the number of linearly independent coverings by the monogenic analytic function at each point of the complex plane. I suggest that this latter concept is natural for integral approximants (sub-class of Hermite-Pade approximants) and discuss results for both ''horizontal'' and ''diagonal'' sequences of approximants. Some theorems are now available in both cases and make clear the natural domain of convergence of the horizontal sequences is a disk centered on the origin and that of the diagonal sequences is a suitably cut complex-plane together with its identically cut pendant Riemann sheets. Some numerical examples have also been computed.
Finite element approximation to the even-parity transport equation
International Nuclear Information System (INIS)
Lewis, E.E.
1981-01-01
This paper studies the finite element method, a procedure for reducing partial differential equations to sets of algebraic equations suitable for solution on a digital computer. The differential equation is cast into the form of a variational principle, the resulting domain then subdivided into finite elements. The dependent variable is then approximated by a simple polynomial, and these are linked across inter-element boundaries by continuity conditions. The finite element method is tailored to a variety of transport problems. Angular approximations are formulated, and the extent of ray effect mitigation is examined. Complex trial functions are introduced to enable the inclusion of buckling approximations. The ubiquitous curved interfaces of cell calculations, and coarse mesh methods are also treated. A concluding section discusses limitations of the work to date and suggests possible future directions
Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties
DEFF Research Database (Denmark)
Christensen, Max la Cour; Vassilevski, Panayot S.; Villa, Umberto
2017-01-01
This paper introduces a nonlinear multigrid solver for mixed finite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The AMGe coarse spaces with approximation properties used in this work enable us to overcome the difficulties...... discretizations on general unstructured grids for a large class of nonlinear partial differential equations, including saddle point problems. The approximation properties of the coarse spaces ensure that our FAS approach for general unstructured meshes leads to optimal mesh-independent convergence rates similar...... to those achieved by geometric FAS on a nested hierarchy of refined meshes. In the numerical results, Newton’s method and Picard iterations with state-of-the-art inner linear solvers are compared to our FAS algorithm for the solution of a nonlinear saddle point problem arising from porous media flow...
Nonlinear ordinary differential equations analytical approximation and numerical methods
Hermann, Martin
2016-01-01
The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...
Space-angle approximations in the variational nodal method
International Nuclear Information System (INIS)
Lewis, E. E.; Palmiotti, G.; Taiwo, T.
1999-01-01
The variational nodal method is formulated such that the angular and spatial approximations maybe examined separately. Spherical harmonic, simplified spherical harmonic, and discrete ordinate approximations are coupled to the primal hybrid finite element treatment of the spatial variables. Within this framework, two classes of spatial trial functions are presented: (1) orthogonal polynomials for the treatment of homogeneous nodes and (2) bilinear finite subelement trial functions for the treatment of fuel assembly sized nodes in which fuel-pin cell cross sections are represented explicitly. Polynomial and subelement trial functions are applied to benchmark water-reactor problems containing MOX fuel using spherical harmonic and simplified spherical harmonic approximations. The resulting accuracy and computing costs are compared
On the boundary conditions in cylindrical cell approximation
International Nuclear Information System (INIS)
Altiparmakov, D.V.
1980-01-01
A solution of the integral transport equation for an arbitrary boundary condition is obtained by solving the integral transport equation for homogeneous (vacuum) boundary condition and using the neutron balance condition. An effective boundary condition satisfying the zero gradient of the neutron flux on the cell boundary is assumed. The numerical solution is obtained by using a pointwise approximation based on a polynomial flux approximation. Disadvantage factor calculations of the Thie lattice cells are carried out. Comparisons are performed with the results obtained for the actual cells by two-dimensional methods as well as their cylindrical approximations applying various boundary conditions. It is obvious from the results shown here that the proposed boundary condition has advantages in respect to others. The errors introduced by the proposed boundary condition are of the lower order in respect to the inaccuracy of the existing transport methods. Thus, the applications of the two-dimensional methods for regular lattice calculations is unnecessary. (author)
Bilinear reduced order approximate model of parabolic distributed solar collectors
Elmetennani, Shahrazed
2015-07-01
This paper proposes a novel, low dimensional and accurate approximate model for the distributed parabolic solar collector, by means of a modified gaussian interpolation along the spatial domain. The proposed reduced model, taking the form of a low dimensional bilinear state representation, enables the reproduction of the heat transfer dynamics along the collector tube for system analysis. Moreover, presented as a reduced order bilinear state space model, the well established control theory for this class of systems can be applied. The approximation efficiency has been proven by several simulation tests, which have been performed considering parameters of the Acurex field with real external working conditions. Model accuracy has been evaluated by comparison to the analytical solution of the hyperbolic distributed model and its semi discretized approximation highlighting the benefits of using the proposed numerical scheme. Furthermore, model sensitivity to the different parameters of the gaussian interpolation has been studied.
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Approximation of ruin probabilities via Erlangized scale mixtures
DEFF Research Database (Denmark)
Peralta, Oscar; Rojas-Nandayapa, Leonardo; Xie, Wangyue
2018-01-01
In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér–Lundbergreserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale...... a simple methodology for constructing a sequence of distributions having the form Π⋆G with the purpose of approximating the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size...... distribution is modeled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed....
Subquadratic medial-axis approximation in $\\mathbb{R}^3$
Directory of Open Access Journals (Sweden)
Christian Scheffer
2015-09-01
Full Text Available We present an algorithm that approximates the medial axis of a smooth manifold in $\\mathbb{R}^3$ which is given by a sufficiently dense point sample. The resulting, non-discrete approximation is shown to converge to the medial axis as the sampling density approaches infinity. While all previous algorithms guaranteeing convergence have a running time quadratic in the size $n$ of the point sample, we achieve a running time of at most $\\mathcal{O}(n\\log^3 n$. While there is no subquadratic upper bound on the output complexity of previous algorithms for non-discrete medial axis approximation, the output of our algorithm is guaranteed to be of linear size.
Approximate models for the analysis of laser velocimetry correlation functions
International Nuclear Information System (INIS)
Robinson, D.P.
1981-01-01
Velocity distributions in the subchannels of an eleven pin test section representing a slice through a Fast Reactor sub-assembly were measured with a dual beam laser velocimeter system using a Malvern K 7023 digital photon correlator for signal processing. Two techniques were used for data reduction of the correlation function to obtain velocity and turbulence values. Whilst both techniques were in excellent agreement on the velocity, marked discrepancies were apparent in the turbulence levels. As a consequence of this the turbulence data were not reported. Subsequent investigation has shown that the approximate technique used as the basis of Malvern's Data Processor 7023V is restricted in its range of application. In this note alternative approximate models are described and evaluated. The objective of this investigation was to develop an approximate model which could be used for on-line determination of the turbulence level. (author)
Nonlinear Multigrid solver exploiting AMGe Coarse Spaces with Approximation Properties
DEFF Research Database (Denmark)
Christensen, Max la Cour; Villa, Umberto; Engsig-Karup, Allan Peter
spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method), were less successful due to lack of such good approximation...... properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on unstructured meshes has the ability to be as powerful/successful as FAS on geometrically refined meshes. For comparison, Newton’s method and Picard iterations with an inner state-of-the-art linear solver...... are compared to FAS on a nonlinear saddle point problem with applications to porous media flow. It is demonstrated that FAS is faster than Newton’s method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate...
Semiclassical approximation to time-dependent Hartree--Fock theory
International Nuclear Information System (INIS)
Dworzecka, M.; Poggioli, R.
1976-01-01
Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form
Approximation of entropy solutions to degenerate nonlinear parabolic equations
Abreu, Eduardo; Colombeau, Mathilde; Panov, Evgeny Yu
2017-12-01
We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space L^∞, whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and L^1-stability. We prove that the sequence of approximate solutions is strongly L^1-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.
Approximation results for neural network operators activated by sigmoidal functions.
Costarelli, Danilo; Spigler, Renato
2013-08-01
In this paper, we study pointwise and uniform convergence, as well as the order of approximation, for a family of linear positive neural network operators activated by certain sigmoidal functions. Only the case of functions of one variable is considered, but it can be expected that our results can be generalized to handle multivariate functions as well. Our approach allows us to extend previously existing results. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of neural network operators activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. In particular, we show that for C(1)-functions, the order of approximation for our operators with logistic and hyperbolic tangent functions here obtained is higher with respect to that established in some previous papers. The case of quasi-interpolation operators constructed with sigmoidal functions is also considered. Copyright © 2013 Elsevier Ltd. All rights reserved.
An electromagnetic multipole expansion beyond the long-wavelength approximation
Alaee, Rasoul; Rockstuhl, Carsten; Fernandez-Corbaton, I.
2018-01-01
The multipole expansion is a key tool in the study of light-matter interactions. All the information about the radiation of and coupling to electromagnetic fields of a given charge-density distribution is condensed into few numbers: The multipole moments of the source. These numbers are frequently computed with expressions obtained after the long-wavelength approximation. Here, we derive exact expressions for the multipole moments of dynamic sources that resemble in their simplicity their approximate counterparts. We validate our new expressions against analytical results for a spherical source, and then use them to calculate the induced moments for some selected sources with a non-trivial shape. The comparison of the results to those obtained with approximate expressions shows a considerable disagreement even for sources of subwavelength size. Our expressions are relevant for any scientific area dealing with the interaction between the electromagnetic field and material systems.