Application of the Extended Grunwald-Winstein Equation to Solvolyses of n-Propyl Chloroformate
Kyong, Jin Burm; Won, Hoshik; Kevill, Dennis N.
2005-01-01
Abstract: Application of the extended Grunwald-Winstein equation to solvolyses of n-propyl chloroformate in a variety of pure and binary solvents indicates an addition-elimination pathway in the majority of the solvents but an ionization pathway in the solvents of highest ionizing power and lowest nucleophilicity. For methanolysis, a solvent deuterium isotope effect of 2.17 is compatible with the incorporation of general-base catalysis into the substitution process. Activation parameters are ...
Application of the Extended Grunwald-Winstein Equation to Solvolyses of n-Propyl Chloroformate
Directory of Open Access Journals (Sweden)
Dennis N. Kevill
2005-01-01
Full Text Available Abstract: Application of the extended Grunwald-Winstein equation to solvolyses of n-propyl chloroformate in a variety of pure and binary solvents indicates an addition-elimination pathway in the majority of the solvents but an ionization pathway in the solvents of highest ionizing power and lowest nucleophilicity. For methanolysis, a solvent deuterium isotope effect of 2.17 is compatible with the incorporation of general-base catalysis into the substitution process. Activation parameters are consistent with the duality of mechanism. Very modest positive salt effects are observed on adding chloride or bromide salts to the ethanolysis.
International Nuclear Information System (INIS)
Choi, Ho June; Koo, In Sun
2012-01-01
The specific rates of sovolysis of 4-methylthiophene-2-carbonyl chloride (1) have been determined in 26 pure and binary solvents at 25.0 .deg. C. Product selectivities are reported for solvolyses of 1 in aqueous ethanol and methanol binary mixtures. Comparison of the specific rates of solvolyses of 1 with those for p-methoxybenzoyl chloride (2) in terms of linear free energy relationships (LFER) are helpful in mechanistic considerations, as is also treatment in terms of the extended Grunwald-Winstein equation. It is proposed that the solvolyses of 1 in binary aqueous solvent mixtures proceed through an S N 1 and/or ionization (I) pathway rather than through an associative S N 2 and/or addition-elimination (A-E) pathway
Kevill, Dennis Neil; Kim, Chang-Bae; D'Souza, Malcolm John
2018-03-01
A Grunwald-Winstein treatment of the specific rates of solvolysis of α-bromoisobutyrophenone in 100% methanol and in several aqueous ethanol, methanol, acetone, 2,2,2-trifluoroethanol (TFE), and 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP) mixtures gives a good logarithmic correlation against a linear combination of N T (solvent nucleophilicity) and Y Br (solvent ionizing power) values. The l and m sensitivity values are compared to those previously reported for α-bromoacetophenone and to those obtained from parallel treatments of literature specific rate values for the solvolyses of several tertiary mesylates containing a C(=O)R group attached at the α-carbon. Kinetic data obtained earlier by Pasto and Sevenair for the solvolyses of the same substrate in 75% aqueous ethanol (by weight) in the presence of silver perchlorate and perchloric acid are analyzed using multiple regression analysis.
International Nuclear Information System (INIS)
Shore, B.W.
1981-01-01
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence
Differential equations extended to superspace
Energy Technology Data Exchange (ETDEWEB)
Torres, J. [Instituto de Fisica, Universidad de Guanajuato, A.P. E-143, Leon, Guanajuato (Mexico); Rosu, H.C. [Instituto Potosino de Investigacion Cientifica y Tecnologica, A.P. 3-74, Tangamanga, San Luis Potosi (Mexico)
2003-07-01
We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)
Differential equations extended to superspace
International Nuclear Information System (INIS)
Torres, J.; Rosu, H.C.
2003-01-01
We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)
Some Aspects of Extended Kinetic Equation
Directory of Open Access Journals (Sweden)
Dilip Kumar
2015-09-01
Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Evolution equations for extended dihadron fragmentation functions
International Nuclear Information System (INIS)
Ceccopieri, F.A.; Bacchetta, A.
2007-03-01
We consider dihadron fragmentation functions, describing the fragmentation of a parton in two unpolarized hadrons, and in particular extended dihadron fragmentation functions, explicitly dependent on the invariant mass, M h , of the hadron pair. We first rederive the known results on M h -integrated functions using Jet Calculus techniques, and then we present the evolution equations for extended dihadron fragmentation functions. Our results are relevant for the analysis of experimental measurements of two-particle-inclusive processes at different energies. (orig.)
A Note of Extended Proca Equations and Superconductivity
Directory of Open Access Journals (Sweden)
Christianto V.
2009-01-01
Full Text Available It has been known for quite long time that the electrodynamics of Maxwell equations can be extended and generalized further into Proca equations. The implications of in- troducing Proca equations include an alternative description of superconductivity, via extending London equations. In the light of another paper suggesting that Maxwell equations can be written using quaternion numbers, then we discuss a plausible exten- sion of Proca equation using biquaternion number. Further implications and experi- ments are recommended.
Rate and Product Studies of 1-Adamantylmethyl Haloformates Under Solvolytic Conditions
International Nuclear Information System (INIS)
Park, Kyoungho; Lee, Yelin; Lee Yongwoo; Kyong, Jin Burm; Kevill, Dennis N.
2012-01-01
Reactions of 1-adamantylmethyl chloroformate (1-AdCH 2 OCOCl, 1) and 1-adamantylmethyl fluoroformate (1-AdCH 2 OCOF, 2) in hydroxylic solvents have been studied. Application of the extended Grunwald-Winstein (G-W) equation to solvolyses of 1 in a variety of pure and binary solvents indicates an addition-elimination pathway in the majority of the solvents except an ionization pathway in the solvents of relatively low nucleophilcity and high ionizing power. The solvolyses of 2 show an addition-elimination pathway in all of the mixed solvents. The leaving group effects (k F /k Cl ), the kinetic solvent isotope effects (KSIEs, k MeOH /k MeOD ), and the enthalpy and entropy of activation for the solvolyses of 1 and 2 were also calculated. The selectivity values (S) for each solvent composition are reported and discussed. These observations are compared with those previously reported for other alkyl haloformate esters
Correlation of the Rates of Solvolysis of Neopentyl Chloroformate—A Recommended Protecting Agent
D’Souza, Malcolm J.; Carter, Shannon E.; Kevill, Dennis N.
2011-01-01
The specific rates of solvolysis of neopentyl chloroformate (1) have been determined in 21 pure and binary solvents at 45.0 °C. In most solvents the values are essentially identical to those for ethyl and n-propyl chloroformates. However, in aqueous-1,1,1,3,3,3-hexafluoro-2-propanol mixtures (HFIP) rich in fluoroalcohol, 1 solvolyses appreciably faster than the other two substrates. Linear free energy relationship (LFER) comparison of the specific rates of solvolysis of 1 with those for phenyl chloroformate and those for n-propyl chloroformate are helpful in the mechanistic considerations, as is also the treatment in terms of the Extended Grunwald-Winstein equation. It is proposed that the faster reaction for 1 in HFIP rich solvents is due to the influence of a 1,2-methyl shift, leading to a tertiary alkyl cation, outweighing the only weak nucleophilic solvation of the cation possible in these low nucleophilicity solvents. PMID:21541050
Correlation of the rates of solvolysis of neopentyl chloroformate-a recommended protecting agent.
D'Souza, Malcolm J; Carter, Shannon E; Kevill, Dennis N
2011-02-15
The specific rates of solvolysis of neopentyl chloroformate (1) have been determined in 21 pure and binary solvents at 45.0 °C. In most solvents the values are essentially identical to those for ethyl and n-propyl chloroformates. However, in aqueous-1,1,1,3,3,3-hexafluoro-2-propanol mixtures (HFIP) rich in fluoroalcohol, 1 solvolyses appreciably faster than the other two substrates. Linear free energy relationship (LFER) comparison of the specific rates of solvolysis of 1 with those for phenyl chloroformate and those for n-propyl chloroformate are helpful in the mechanistic considerations, as is also the treatment in terms of the Extended Grunwald-Winstein equation. It is proposed that the faster reaction for 1 in HFIP rich solvents is due to the influence of a 1,2-methyl shift, leading to a tertiary alkyl cation, outweighing the only weak nucleophilic solvation of the cation possible in these low nucleophilicity solvents.
Optimal control problem for the extended Fisher–Kolmogorov equation
Indian Academy of Sciences (India)
In this paper, the optimal control problem for the extended Fisher–Kolmogorov equation is studied. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.
Extended irreversible thermodynamics and the Jeffreys type constitutive equations
International Nuclear Information System (INIS)
Serdyukov, S.I.
2003-01-01
A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of the internal energy, the specific volume, and their material time derivatives. On the basis of this postulate, entropy balance equations and phenomenological equations are obtained, which directly lead to the Jeffreys type constitutive equations
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren
2012-01-01
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...
Extended Thermodynamics: a Theory of Symmetric Hyperbolic Field Equations
Müller, Ingo
2008-12-01
Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear first order differential equations. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. This presentation is based upon the book [1] of which the author of this paper is a co-author. For more details about the motivation and exploitation of the basic principles the interested reader is referred to that reference. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Physicists may prefer to appreciate the success of extended thermodynamics in light scattering and to work on the open problems concerning the modification of the Navier-Stokes-Fourier theory in rarefied gases as predicted by extended thermodynamics of 13, 14, and more moments.
Integro-differential equation approach extended to larger nuclei
International Nuclear Information System (INIS)
Adam, R.M.; Sofianos, S.A.; Fiedeldey, H.; Fabre de la Ripelle, M.
1992-01-01
We extend the integro-differential equation approach (IDEA) from few-nucleon to closed-shell and closed-subshell nuclei and outline the analytical methods required for the calculation of the density functions, which enter into the integro-differential equations. These contain all the physics for a system of fermions associated with the Pauli principle. In order to test the accuracy of the IDEA comparisons are made of the binding energies of 4 He, 12 C and 16 O obtained with effective potentials using the hypercentral approximation (HCA) providing a variational solution without correlations, the IDEA which fully includes the two-body correlations, the S-states integro-differential equation (SIDE) valid for potentials operating only on pairs in the S-state and those calculated by several variational or perturbative methods in the literature. (author)
Optimal control problem for the extended Fisher–Kolmogorov equation
Indian Academy of Sciences (India)
by methods of optimal control, such as chemical engineering and vehicle ... ern optimal control theories and applied models are not only represented by .... Obviously, equation (2.5) is an ordinary differential equation and according to ODE.
The integrability of an extended fifth-order KdV equation with Riccati ...
Indian Academy of Sciences (India)
method was extended to investigate variable coefficient NLEEs, which included gener- alized KdV equation, generalized modified KdV equation and generalized Boussinesq equation [11,12]. It is well known that KdV equation models a variety of nonlinear phenomena, including ion-acoustic waves in plasmas and shallow ...
Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach
International Nuclear Information System (INIS)
Dai Chaoqing; Zhang Jiefang
2006-01-01
In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.
Reconstruction of extended sources for the Helmholtz equation
Kress, Rainer; Rundell, William
2013-01-01
The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our
Extended sine-Gordon Equation Method and Its Application to Maccari's System
International Nuclear Information System (INIS)
Song Lina; Zhang Hongqing
2005-01-01
An extended sine-Gordon equation method is proposed to construct exact travelling wave solutions to Maccari's equation based upon a generalized sine-Gordon equation. It is shown that more new travelling wave solutions can be found by this new method, which include bell-shaped soliton solutions, kink-shaped soliton solutions, periodic wave solution, and new travelling waves.
International Nuclear Information System (INIS)
Yomba, Emmanuel
2005-01-01
By using a modified extended Fan's sub-equation method, we have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerative Jacobi elliptic wave function-like solutions for the (2 + 1)-dimensional dispersive long wave equation. The most important achievement of this method lies on the fact that, we have succeeded in one move to give all the solutions which can be previously obtained by application of at least four methods (method using Riccati equation, or first kind elliptic equation, or auxiliary ordinary equation, or generalized Riccati equation as mapping equation)
Adiabatic invariants of the extended KdV equation
Energy Technology Data Exchange (ETDEWEB)
Karczewska, Anna [Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra (Poland); Rozmej, Piotr, E-mail: p.rozmej@if.uz.zgora.pl [Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra (Poland); Infeld, Eryk [National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa (Poland); Rowlands, George [Department of Physics, University of Warwick, Coventry, CV4 7A (United Kingdom)
2017-01-30
When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present a direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from constant values are indeed small. - Highlights: • We suggest a new and simple method for calculating adiabatic invariants of second order wave equations. • It is easy to use and we hope that it will be useful if published. • Interesting numerics included.
Cellular solutions for the Poisson equation in extended systems
International Nuclear Information System (INIS)
Zhang, X.; Butler, W.H.; MacLaren, J.M.; van Ek, J.
1994-01-01
The Poisson equation for the electrostatic potential in a solid is solved using three different cellular techniques. The relative merits of these different approaches are discussed for two test charge densities for which an analytic solution to the Poisson equation is known. The first approach uses full-cell multiple-scattering theory and results in the famililar structure constant and multipole moment expansion. This solution is shown to be valid everywhere inside the cell, although for points outside the muffin-tin sphere but inside the cell the sums must be performed in the correct order to yield meaningful results. A modification of the multiple-scattering-theory approach yields a second method, a Green-function cellular method, which only requires the solution of a nearest-neighbor linear system of equations. A third approach, a related variational cellular method, is also derived. The variational cellular approach is shown to be the most accurate and reliable, and to have the best convergence in angular momentum of the three methods. Coulomb energies accurate to within 10 -6 hartree are easily achieved with the variational cellular approach, demonstrating the practicality of the approach in electronic structure calculations
Integrability of an extended (2+1)-dimensional shallow water wave equation with Bell polynomials
International Nuclear Information System (INIS)
Wang Yun-Hu; Chen Yong
2013-01-01
We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method. (general)
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
Reconstruction of extended sources for the Helmholtz equation
International Nuclear Information System (INIS)
Kress, Rainer; Rundell, William
2013-01-01
The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our inclusions are interior forces with compact support and our data consist of a single measurement of near-field Cauchy data on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler ‘equivalent point source’ problem, and which uses a Newton scheme to improve the corresponding initial approximation. (paper)
Reconstruction of extended sources for the Helmholtz equation
Kress, Rainer
2013-02-26
The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our inclusions are interior forces with compact support and our data consist of a single measurement of near-field Cauchy data on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler \\'equivalent point source\\' problem, and which uses a Newton scheme to improve the corresponding initial approximation. © 2013 IOP Publishing Ltd.
Correlation of the Rates of Solvolysis of Neopentyl Chloroformate—A Recommended Protecting Agent
Directory of Open Access Journals (Sweden)
Dennis N. Kevill
2011-02-01
Full Text Available The specific rates of solvolysis of neopentyl chloroformate (1 have been determined in 21 pure and binary solvents at 45.0 °C. In most solvents the values are essentially identical to those for ethyl and n-propyl chloroformates. However, in aqueous-1,1,1,3,3,3-hexafluoro-2-propanol mixtures (HFIP rich in fluoroalcohol, 1 solvolyses appreciably faster than the other two substrates. Linear free energy relationship (LFER comparison of the specific rates of solvolysis of 1 with those for phenyl chloroformate and those for n-propyl chloroformate are helpful in the mechanistic considerations, as is also the treatment in terms of the Extended Grunwald-Winstein equation. It is proposed that the faster reaction for 1 in HFIP rich solvents is due to the influence of a 1,2-methyl shift, leading to a tertiary alkyl cation, outweighing the only weak nucleophilic solvation of the cation possible in these low nucleophilicity solvents.
Soliton-like solutions to the GKdV equation by extended mapping method
International Nuclear Information System (INIS)
Wu Ranchao; Sun Jianhua
2007-01-01
In this note, many new exact solutions of the generalized KdV equation, such as rational solutions, periodic solutions like Jacobian elliptic and triangular functions, soliton-like solutions, are constructed by symbolic computation and the extended mapping method, with the auxiliary ordinary equation replaced by a more general one
International Nuclear Information System (INIS)
Kong Cuicui; Wang Dan; Song Lina; Zhang Hongqing
2009-01-01
In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.
International Nuclear Information System (INIS)
Hartwig, J. T.; Stokman, J. V.
2013-01-01
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Self-consistence equations for extended Feynman rules in quantum chromodynamics
International Nuclear Information System (INIS)
Wielenberg, A.
2005-01-01
In this thesis improved solutions for Green's functions are obtained. First the for this thesis essential techniques and concepts of QCD as euclidean field theory are presented. After a discussion of the foundations of the extended approach for the Feynman rules of QCD with a systematic approach for the 4-gluon vertex a modified renormalization scheme for the extended approach is developed. Thereafter the resummation of the Dyson-Schwinger equations (DSE) by the appropriately modified Bethe-Salpeter equation is discussed. Then the leading divergences for the 1-loop graphs of the resummed DSE are determined. Thereafter the equation-of-motion condensate is defined as result of an operator-product expansion. Then the self-consistency equations for the extended approaches are defined and numerically solved. (HSI)
International Nuclear Information System (INIS)
Feng Qinghua
2013-01-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established. (general)
International Nuclear Information System (INIS)
Darmani, G.; Setayeshi, S.; Ramezanpour, H.
2012-01-01
In this paper an efficient computational method based on extending the sensitivity approach (SA) is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations (DDEs), the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach. (general)
Classification of kink type solutions to the extended derivative nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Wyller, J.; Fla, T.; Juul Rasmussen, J.
1998-01-01
The Raman Extended Derivative Non Linear Schrodinger (R-EDNLS) equation which models single mode propagation in optical fibers, is shown to possess travelling and stationary kink envelope solutions of monotonic and oscillatory type. These structures have been called optical shocks in analogy...
International Nuclear Information System (INIS)
Shang Yadong
2008-01-01
The extended hyperbolic functions method for nonlinear wave equations is presented. Based on this method, we obtain a multiple exact explicit solutions for the nonlinear evolution equations which describe the resonance interaction between the long wave and the short wave. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for S and L, (b) the solitary wave solutions of kink-type for S and bell-type for L, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for S and L, (d) the singular travelling wave solutions, (e) periodic travelling wave solutions of triangle function types, and solitary wave solutions of rational function types. The variety of structure to the exact solutions of the long-short wave equation is illustrated. The methods presented here can also be used to obtain exact solutions of nonlinear wave equations in n dimensions
International Nuclear Information System (INIS)
Yomba, Emmanuel
2005-01-01
An extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs). The key idea of this method is to take full advantage of the general elliptic equation involving five parameters which has more new solutions and whose degeneracies can lead to special sub-equations involving three parameters. More new solutions are obtained for KdV-MKdV, Broer-Kaup-Kupershmidt (BKK) and variant Boussinesq equations. Then we present a technique which not only gives us a clear relation among this general elliptic equation and other sub-equations involving three parameters (Riccati equation, first kind elliptic equation, auxiliary ordinary equation, generalized Riccati equation and so on), but also provides an approach to construct new exact solutions to NLPDEs
Spin and energy evolution equations for a wide class of extended bodies
International Nuclear Information System (INIS)
Racine, Etienne
2006-01-01
We give a surface integral derivation of the leading-order evolution equations for the spin and energy of a relativistic body interacting with other bodies in the post-Newtonian expansion scheme. The bodies can be arbitrarily shaped and can be strongly self-gravitating. The effects of all mass and current multipoles are taken into account. As part of the computation one of the 2PN potentials parametrizing the metric is obtained. The formulae obtained here for spin and energy evolution coincide with those obtained by Damour, Soffel and Xu for the case of weakly self-gravitating bodies. By combining an Einstein-Infeld-Hoffman-type surface integral approach with multipolar expansions we extend the domain of validity of these evolution equations to a wide class of strongly self-gravitating bodies. This paper completes in a self-contained way a previous work by Racine and Flanagan on translational equations of motion for compact objects
Design of TIR collimating lens for ordinary differential equation of extended light source
Zhan, Qianjing; Liu, Xiaoqin; Hou, Zaihong; Wu, Yi
2017-10-01
The source of LED has been widely used in our daily life. The intensity angle distribution of single LED is lambert distribution, which does not satisfy the requirement of people. Therefore, we need to distribute light and change the LED's intensity angle distribution. The most commonly method to change its intensity angle distribution is the free surface. Generally, using ordinary differential equations to calculate free surface can only be applied in a point source, but it will lead to a big error for the expand light. This paper proposes a LED collimating lens based on the ordinary differential equation, combined with the LED's light distribution curve, and adopt the method of calculating the center gravity of the extended light to get the normal vector. According to the law of Snell, the ordinary differential equations are constructed. Using the runge-kutta method for solution of ordinary differential equation solution, the curve point coordinates are gotten. Meanwhile, the edge point data of lens are imported into the optical simulation software TracePro. Based on 1mm×1mm single lambert body for light conditions, The degrees of collimating light can be close to +/-3. Furthermore, the energy utilization rate is higher than 85%. In this paper, the point light source is used to calculate partial differential equation method and compared with the simulation of the lens, which improve the effect of 1 degree of collimation.
A relativistic extended Fermi-Thomas-like equation for a self-gravitating system of fermions
International Nuclear Information System (INIS)
Merloni, A.; Ruffini, R.; Torroni, V.
1998-01-01
The authors extend previous results of a Fermi-Thomas model, describing self-gravitating fermions in their ground state, to a relativistic gravitational theory in Minkowski space. In such a theory the source term of the gravitational potential depends both on the pressure and the density of the fluid. It is shown that, in correspondence of this relativistic treatment, still a Fermi-Thomas-like equation can be derived for the self-gravitating system, though the non-linearities are much more complex. No Fermi-Thomas-like equation can be obtained in the General Relativistic treatment. The canonical results for neutron stars and white dwarfs are recovered and also some erroneous statements in the scientific literature are corrected
Seadawy, Aly R.; Kumar, Dipankar; Chakrabarty, Anuz Kumar
2018-05-01
The (2+1)-dimensional hyperbolic and cubic-quintic nonlinear Schrödinger equations describe the propagation of ultra-short pulses in optical fibers of nonlinear media. By using an extended sinh-Gordon equation expansion method, some new complex hyperbolic and trigonometric functions prototype solutions for two nonlinear Schrödinger equations were derived. The acquired new complex hyperbolic and trigonometric solutions are expressed by dark, bright, combined dark-bright, singular and combined singular solitons. The obtained results are more compatible than those of other applied methods. The extended sinh-Gordon equation expansion method is a more powerful and robust mathematical tool for generating new optical solitary wave solutions for many other nonlinear evolution equations arising in the propagation of optical pulses.
Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum
2014-01-01
In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.
International Nuclear Information System (INIS)
Wang Baodong; Song Lina; Zhang Hongqing
2007-01-01
In this paper, we present a new elliptic equation rational expansion method to uniformly construct a series of exact solutions for nonlinear partial differential equations. As an application of the method, we choose the (2 + 1)-dimensional Burgers equation to illustrate the method and successfully obtain some new and more general solutions
Geometrical-integrability constraints and equations of motion in four plus extended super spaces
International Nuclear Information System (INIS)
Chau, L.L.
1987-01-01
It is pointed out that many equations of motion in physics, including gravitational and Yang-Mills equations, have a common origin: i.e. they are the results of certain geometrical integrability conditions. These integrability conditions lead to linear systems and conservation laws that are important in integrating these equations of motion
Energy Technology Data Exchange (ETDEWEB)
Singhatanadgid, Pairod; Jommalai, Panupan [Chulalongkorn University, Bangkok (Thailand)
2016-05-15
The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.
Existence and Uniqueness of Solution of Schrodinger equation in extended Colombeau algebra
Directory of Open Access Journals (Sweden)
Fariba Fattahi
2014-09-01
Full Text Available In this paper, we establish the existence and uniquenessresult of the linear Schr¨odinger equation with Marchaudfractional derivative in Colombeau generalized algebra.The purpose of introducing Marchaud fractional derivativeis regularizing it in Colombeau sense.
Extended common-image-point gathers for anisotropic wave-equation migration
Sava, Paul C.; Alkhalifah, Tariq Ali
2010-01-01
In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes
International Nuclear Information System (INIS)
Zhang Jiefang; Dai Chaoqing; Zong Fengde
2007-01-01
In this paper, with the variable separation approach and based on the general reduction theory, we successfully generalize this extended tanh-function method to obtain new types of variable separation solutions for the following Nizhnik-Novikov-Veselov (NNV) equation. Among the solutions, two solutions are new types of variable separation solutions, while the last solution is similar to the solution given by Darboux transformation in Hu et al 2003 Chin. Phys. Lett. 20 1413
Mohammed, Ahmed; Zeleke, Aklilu
2015-01-01
We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.
van Breukelen, B.M.
2007-01-01
The Rayleigh equation relates the change in isotope ratio of an element in a substrate to the extent of substrate consumption via a single kinetic isotopic fractionation factor (α). Substrate consumption is, however, commonly distributed over several metabolic pathways each potentially having a
OpenMx: An Open Source Extended Structural Equation Modeling Framework
Boker, Steven; Neale, Michael; Maes, Hermine; Wilde, Michael; Spiegel, Michael; Brick, Timothy; Spies, Jeffrey; Estabrook, Ryne; Kenny, Sarah; Bates, Timothy; Mehta, Paras; Fox, John
2011-01-01
OpenMx is free, full-featured, open source, structural equation modeling (SEM) software. OpenMx runs within the "R" statistical programming environment on Windows, Mac OS-X, and Linux computers. The rationale for developing OpenMx is discussed along with the philosophy behind the user interface. The OpenMx data structures are…
Peng, Wei-Qi; Tian, Shou-Fu; Zou, Li; Zhang, Tian-Tian
2018-01-01
In this paper, the extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms is investigated, whose particular cases are the Hirota equation, the Sasa-Satsuma equation and Lakshmanan-Porsezian-Daniel equation by selecting some specific values on the parameters of higher-order terms. We first study the stability analysis of the equation. Then, using the ansatz method, we derive its bright, dark solitons and some constraint conditions which can guarantee the existence of solitons. Moreover, the Ricatti equation extension method is employed to derive some exact singular solutions. The outstanding characteristics of these solitons are analyzed via several diverting graphics.
Extended common-image-point gathers for anisotropic wave-equation migration
Sava, Paul C.
2010-01-01
In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes anisotropy parameters (Gray et al., 2001). In particular, it is important to construct subsurface velocity models using techniques that are consistent with the methods used for imaging. Generally speaking, there are two possible strategies for velocity estimation from surface seismic data in the context of wavefield-based imaging (Sava et al., 2010). One possibility is to formulate an objective function in the data space, prior to migration, by matching the recorded data with simulated data. Techniques in this category are known by the name of waveform inversion. Another possibility is to formulate an objective function in the image space, after migration, by measuring and correcting image features that indicate model inaccuracies. Techniques in this category are known as wave-equation migration velocity analysis (MVA).
An extended step characteristic method for solving the transport equation in general geometries
International Nuclear Information System (INIS)
DeHart, M.D.; Pevey, R.E.; Parish, T.A.
1994-01-01
A method for applying the discrete ordinates method to solve the Boltzmann transport equation on arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the geometrical shape of a mesh element characteristic of a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. By using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons. Results for a number of test problems have been compared with solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well as numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN-II computer programs
International Nuclear Information System (INIS)
Jiang Bin; Zhang Yejing; Wang Yufei; Liu Anjin; Zheng Wanhua
2012-01-01
We present the extended Dirichlet-to-Neumann wave vector eigenvalue equation (DtN-WVEE) method to calculate the equi-frequency contour (EFC) of square lattice photonic crystals (PhCs). With the extended DtN-WVEE method and Snell's law, the effective refractive index of the mode with a circular EFC can be obtained, which is further validated with the refractive index weighted by the electric field or magnetic field. To further verify the EFC calculated by the DtN-WVEE method, the finite-difference time-domain method is also used. Compared with other wave vector eigenvalue equation methods that calculate EFC directly, the size of the eigenmatrix used in the DtN-WVEE method is much smaller, and the computation time is significantly reduced. Since the DtN-WVEE method solves wave vectors for given arbitrary frequencies, it can also find applications in studying the optical properties of a PhC with dispersive, lossy and magnetic materials. (paper)
International Nuclear Information System (INIS)
Lee, Jeong Hun; Park, Tong Kyu; Jeon, Seong Su
2014-01-01
The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core
Energy Technology Data Exchange (ETDEWEB)
Lee, Jeong Hun; Park, Tong Kyu; Jeon, Seong Su [FNC Technology Co., Ltd., Yongin (Korea, Republic of)
2014-05-15
The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core.
Lu, Dianchen; Seadawy, A. R.; Arshad, M.; Wang, Jun
In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by utilizing modified extended direct algebraic method. Soliton solutions in different forms such as bell and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc are obtained, which have large applications in different branches of physics and other areas of applied sciences. The obtained solutions are also presented graphically. Furthermore, many other nonlinear evolution equations arising in mathematical physics and engineering can also be solved by this powerful, reliable and capable method. The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica equation and (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the reliability and effectiveness of the current method.
Imanidis, Georgios; Luetolf, Peter
2006-07-01
An extended model for iontophoretic enhancement of transdermal drug permeation under constant voltage is described based on the previously modified Nernst-Planck equation, which included the effect of convective solvent flow. This model resulted in an analytical expression for the enhancement factor as a function of applied voltage, convective flow velocity due to electroosmosis, ratio of lipid to aqueous pathway passive permeability, and weighted average net ionic valence of the permeant in the aqueous epidermis domain. The shift of pH in the epidermis compared to bulk caused by the electrical double layer at the lipid-aqueous domain interface was evaluated using the Poisson-Boltzmann equation. This was solved numerically for representative surface charge densities and yielded pH differences between bulk and epidermal aqueous domain between 0.05 and 0.4 pH units. The developed model was used to analyze the experimental enhancement of an amphoteric weak electrolyte measured in vitro using human cadaver epidermis and a voltage of 250 mV at different pH values. Parameter values characterizing the involved factors were determined that yielded the experimental enhancement factors and passive permeability coefficients at all pH values. The model provided a very good agreement between experimental and calculated enhancement and passive permeability. The deduced parameters showed (i) that the pH shift in the aqueous permeation pathway had a notable effect on the ionic valence and the partitioning of the drug in this domain for a high surface charge density and depending on the pK(a) and pI of the drug in relation to the bulk pH; (ii) the magnitude and the direction of convective transport due to electroosmosis typically reflected the density and sign, respectively, of surface charge of the tissue and its effect on enhancement was substantial for bulk pH values differing from the pI of epidermal tissue; (iii) the aqueous pathway predominantly determined passive
Manning, Robert M.
2004-01-01
The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.
International Nuclear Information System (INIS)
Lampin, E.; Cristiano, F.; Lamrani, Y.; Colombeau, B.
2004-01-01
We present simulations of B TED based on a complete calculation of the extended defect growth/shrinkage during annealing. The Si self-interstitial supersaturation calculated at the extended defect depth is coupled to the set of equations for the B kick-out diffusion through a generation/recombination term in the diffusion equation of the Si self-interstitials. The simulations are compared to the measurements performed on a Si wafer containing several B marker layers, where the amount of TED varies from one peak to the other. The good agreement obtained on this experiment is very promising for the application of these calculations to the case of ultra-shallow B + implants
International Nuclear Information System (INIS)
Wang Qi; Chen Yong; Zhang Hongqing
2005-01-01
With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz, in which periodic solutions of nonlinear partial differential equations that can be expressed as a finite Laurent series of some of 12 Jacobi elliptic functions, is more powerful than exiting Jacobi elliptic function methods and is very powerful to uniformly construct more new exact periodic solutions in terms of rational formal Jacobi elliptic function solution of nonlinear partial differential equations. As an application of the method, we choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition
Energy Technology Data Exchange (ETDEWEB)
Shlivinski, A., E-mail: amirshli@ee.bgu.ac.il [Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105 (Israel); Lomakin, V., E-mail: vlomakin@eng.ucsd.edu [Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0407 (United States)
2016-03-01
Scattering or coupling of electromagnetic beam-field at a surface discontinuity separating two homogeneous or inhomogeneous media with different propagation characteristics is formulated using surface integral equation, which are solved by the Method of Moments with the aid of the Gabor-based Gaussian window frame set of basis and testing functions. The application of the Gaussian window frame provides (i) a mathematically exact and robust tool for spatial-spectral phase-space formulation and analysis of the problem; (ii) a system of linear equations in a transmission-line like form relating mode-like wave objects of one medium with mode-like wave objects of the second medium; (iii) furthermore, an appropriate setting of the frame parameters yields mode-like wave objects that blend plane wave properties (as if solving in the spectral domain) with Green's function properties (as if solving in the spatial domain); and (iv) a representation of the scattered field with Gaussian-beam propagators that may be used in many large (in terms of wavelengths) systems.
International Nuclear Information System (INIS)
Shaing, K. C.
2007-01-01
In Part I [Phys. Fluids B 2, 1190 (1990)] and Part II [Phys. Plasmas 12, 082508 (2005)], it was emphasized that the equilibrium plasma viscous forces when applied for the magnetohydrodynamic (MHD) modes are only rigorously valid at the mode rational surface where m-nq=0. Here, m is the poloidal mode number, n is the toroidal mode number, and q is the safety factor. This important fact has been demonstrated explicitly by calculating the viscous forces in the plateau regime in Parts I and II. Here, the effective viscous forces in the banana regime are calculated for MHD modes by solving the linear drift kinetic equation that is driven by the plasma flows first derived in Part I. At the mode rational surface, the equilibrium plasma viscous forces are reproduced. However, it is found that away from the mode rational surface, the viscous forces for MHD modes decrease, a behavior similar to that observed in the viscous forces for the plateau regime. The proper form of the momentum equation that is appropriate for the modeling of the MHD modes is also discussed
A new fourth-order Fourier-Bessel split-step method for the extended nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Nash, Patrick L.
2008-01-01
Fourier split-step techniques are often used to compute soliton-like numerical solutions of the nonlinear Schroedinger equation. Here, a new fourth-order implementation of the Fourier split-step algorithm is described for problems possessing azimuthal symmetry in 3 + 1-dimensions. This implementation is based, in part, on a finite difference approximation Δ perpendicular FDA of 1/r (∂)/(∂r) r(∂)/(∂r) that possesses an associated exact unitary representation of e i/2λΔ perpendicular FDA . The matrix elements of this unitary matrix are given by special functions known as the associated Bessel functions. Hence the attribute Fourier-Bessel for the method. The Fourier-Bessel algorithm is shown to be unitary and unconditionally stable. The Fourier-Bessel algorithm is employed to simulate the propagation of a periodic series of short laser pulses through a nonlinear medium. This numerical simulation calculates waveform intensity profiles in a sequence of planes that are transverse to the general propagation direction, and labeled by the cylindrical coordinate z. These profiles exhibit a series of isolated pulses that are offset from the time origin by characteristic times, and provide evidence for a physical effect that may be loosely termed normal mode condensation. Normal mode condensation is consistent with experimentally observed pulse filamentation into a packet of short bursts, which may occur as a result of short, intense irradiation of a medium
Extending the D’alembert solution to space–time Modified Riemann–Liouville fractional wave equations
International Nuclear Information System (INIS)
Godinho, Cresus F.L.; Weberszpil, J.; Helayël-Neto, J.A.
2012-01-01
In the realm of complexity, it is argued that adequate modeling of TeV-physics demands an approach based on fractal operators and fractional calculus (FC). Non-local theories and memory effects are connected to complexity and the FC. The non-differentiable nature of the microscopic dynamics may be connected with time scales. Based on the Modified Riemann–Liouville definition of fractional derivatives, we have worked out explicit solutions to a fractional wave equation with suitable initial conditions to carefully understand the time evolution of classical fields with a fractional dynamics. First, by considering space–time partial fractional derivatives of the same order in time and space, a generalized fractional D’alembertian is introduced and by means of a transformation of variables to light-cone coordinates, an explicit analytical solution is obtained. To address the situation of different orders in the time and space derivatives, we adopt different approaches, as it will become clear throughout this paper. Aspects connected to Lorentz symmetry are analyzed in both approaches.
Rankin, Brian D; Fox, Jeremy W; Barrón-Ortiz, Christian R; Chew, Amy E; Holroyd, Patricia A; Ludtke, Joshua A; Yang, Xingkai; Theodor, Jessica M
2015-08-07
Species selection, covariation of species' traits with their net diversification rates, is an important component of macroevolution. Most studies have relied on indirect evidence for its operation and have not quantified its strength relative to other macroevolutionary forces. We use an extension of the Price equation to quantify the mechanisms of body size macroevolution in mammals from the latest Palaeocene and earliest Eocene of the Bighorn and Clarks Fork Basins of Wyoming. Dwarfing of mammalian taxa across the Palaeocene/Eocene Thermal Maximum (PETM), an intense, brief warming event that occurred at approximately 56 Ma, has been suggested to reflect anagenetic change and the immigration of small bodied-mammals, but might also be attributable to species selection. Using previously reconstructed ancestor-descendant relationships, we partitioned change in mean mammalian body size into three distinct mechanisms: species selection operating on resident mammals, anagenetic change within resident mammalian lineages and change due to immigrants. The remarkable decrease in mean body size across the warming event occurred through anagenetic change and immigration. Species selection also was strong across the PETM but, intriguingly, favoured larger-bodied species, implying some unknown mechanism(s) by which warming events affect macroevolution. © 2015 The Author(s).
Energy Technology Data Exchange (ETDEWEB)
Zhen, Hui-Ling; Tian, Bo, E-mail: tian-bupt@163.com; Wang, Yu-Feng; Sun, Wen-Rong; Liu, Li-Cai [State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 (China)
2014-07-15
The extended Zakharov-Kuznetsov (eZK) equation for the magnetized two-ion-temperature dusty plasma is studied in this paper. With the help of Hirota method, bilinear forms and N-soliton solutions are given, and soliton propagation is graphically analyzed. We find that the soliton amplitude is positively related to the nonlinear coefficient A, while inversely related to the dispersion coefficients B and C. We obtain that the soliton amplitude will increase with the mass of the jth dust grain and the average charge number residing on the dust grain decreased, but the soliton amplitude will increase with the equilibrium number density of the jth dust grain increased. Upon the introduction of the periodic external forcing term, both the weak and developed chaotic motions can occur. Difference between the two chaotic motions roots in the inequality between the nonlinear coefficient l{sub 2} and perturbed term h{sub 1}. The developed chaos can be weakened with B or C decreased and A increased. Periodic motion of the perturbed eZK equation can be observed when there is a balance between l{sub 2} and h{sub 1}.
Müller, Ingo
1993-01-01
Physicists firmly believe that the differential equations of nature should be hyperbolic so as to exclude action at a distance; yet the equations of irreversible thermodynamics - those of Navier-Stokes and Fourier - are parabolic. This incompatibility between the expectation of physicists and the classical laws of thermodynamics has prompted the formulation of extended thermodynamics. After describing the motifs and early evolution of this new branch of irreversible thermodynamics, the authors apply the theory to mon-atomic gases, mixtures of gases, relativistic gases, and "gases" of phonons and photons. The discussion brings into perspective the various phenomena called second sound, such as heat propagation, propagation of shear stress and concentration, and the second sound in liquid helium. The formal mathematical structure of extended thermodynamics is exposed and the theory is shown to be fully compatible with the kinetic theory of gases. The study closes with the testing of extended thermodynamics thro...
Rational extended thermodynamics
Müller, Ingo
1998-01-01
Ordinary thermodynamics provides reliable results when the thermodynamic fields are smooth, in the sense that there are no steep gradients and no rapid changes. In fluids and gases this is the domain of the equations of Navier-Stokes and Fourier. Extended thermodynamics becomes relevant for rapidly varying and strongly inhomogeneous processes. Thus the propagation of high frequency waves, and the shape of shock waves, and the regression of small-scale fluctuation are governed by extended thermodynamics. The field equations of ordinary thermodynamics are parabolic while extended thermodynamics is governed by hyperbolic systems. The main ingredients of extended thermodynamics are • field equations of balance type, • constitutive quantities depending on the present local state and • entropy as a concave function of the state variables. This set of assumptions leads to first order quasi-linear symmetric hyperbolic systems of field equations; it guarantees the well-posedness of initial value problems and f...
Allen, Rob
2016-09-01
Structures within molecules and nuclei have relationships to astronomical patterns. The COBE cosmic scale plots, and large scale surveys of galaxy clusters have patterns also repeating and well known at atomic scales. The Induction, Strong Force, and Nuclear Binding Energy Periods within the Big Bang are revealed to have played roles in the formation of these large scale distributions. Equations related to the enormous patterns also model chemical bonds and likely nucleus and nucleon substructures. ratios of the forces that include gravity are accurately calculated from the distributions and shapes. In addition, particle masses and a great many physical constants can be derived with precision and accuracy from astrophysical shapes. A few very basic numbers can do modelling from nucleon internals to molecules to super novae, and up to the Visible Universe. Equations are also provided along with possible structural configurations for some Cold Dark Matter and Dark Energy.
2016-06-08
Ideal Magnetohydrodynamics,” J. Com- put. Phys., Vol. 153, No. 2, 1999, pp. 334–352. [14] Tang, H.-Z. and Xu, K., “A high-order gas -kinetic method for...notwithstanding any other provision of law , no person shall be subject to any penalty for failing to comply with a collection of information if it does...Riemann-solver-free spacetime discontinuous Galerkin method for general conservation laws to solve compressible magnetohydrodynamics (MHD) equations. The
Le, Duy Michael; Sørensen, Hanne R; Knudsen, Niels Ole; Schjoerring, Jan K; Meyer, Anne S
2014-01-01
Mineral elements present in lignocellulosic biomass feedstocks may accumulate in biorefinery process streams and cause technological problems, or alternatively can be reaped for value addition. A better understanding of the distribution of minerals in biomass in response to pretreatment factors is therefore important in relation to development of new biorefinery processes. The objective of the present study was to examine the levels of mineral elements in pretreated wheat straw in response to systematic variations in the hydrothermal pretreatment parameters (pH, temperature, and treatment time), and to assess whether it is possible to model mineral levels in the pretreated fiber fraction. Principal component analysis of the wheat straw biomass constituents, including mineral elements, showed that the recovered levels of wheat straw constituents after different hydrothermal pretreatments could be divided into two groups: 1) Phosphorus, magnesium, potassium, manganese, zinc, and calcium correlated with xylose and arabinose (that is, hemicellulose), and levels of these constituents present in the fiber fraction after pretreatment varied depending on the pretreatment-severity; and 2) Silicon, iron, copper, aluminum correlated with lignin and cellulose levels, but the levels of these constituents showed no severity-dependent trends. For the first group, an expanded pretreatment-severity equation, containing a specific factor for each constituent, accounting for variability due to pretreatment pH, was developed. Using this equation, the mineral levels could be predicted with R(2) > 0.75; for some with R(2) up to 0.96. Pretreatment conditions, especially pH, significantly influenced the levels of phosphorus, magnesium, potassium, manganese, zinc, and calcium in the resulting fiber fractions. A new expanded pretreatment-severity equation is proposed to model and predict mineral composition in pretreated wheat straw biomass.
DEFF Research Database (Denmark)
Le, Duy Michael; Sørensen, Hanne Risbjerg; Knudsen, Niels Ole
2014-01-01
Background: Mineral elements present in lignocellulosic biomass feedstocks may accumulate in biorefinery process streams and cause technological problems, or alternatively can be reaped for value addition. A better understanding of the distribution of minerals in biomass in response to pretreatment...... factors is therefore important in relation to development of new biorefinery processes. The objective of the present study was to examine the levels of mineral elements in pretreated wheat straw in response to systematic variations in the hydrothermal pretreatment parameters (pH, temperature......) Silicon, iron, copper, aluminum correlated with lignin and cellulose levels, but the levels of these constituents showed no severity-dependent trends. For the first group, an expanded pretreatment-severity equation, containing a specific factor for each constituent, accounting for variability due...
Marjani, Azam
2016-07-01
For biomolecules and cell particles purification and separation in biological engineering, besides the chromatography as mostly applied process, aqueous two-phase systems (ATPS) are of the most favorable separation processes that are worth to be investigated in thermodynamic theoretically. In recent years, thermodynamic calculation of ATPS properties has attracted much attention due to their great applications in chemical industries such as separation processes. These phase calculations of ATPS have inherent complexity due to the presence of ions and polymers in aqueous solution. In this work, for target ternary systems of polyethylene glycol (PEG4000)-salt-water, thermodynamic investigation for constituent systems with three salts (NaCl, KCl and LiCl) has been carried out as PEG is the most favorable polymer in ATPS. The modified perturbed hard sphere chain (PHSC) equation of state (EOS), extended Debye-Hückel and Pitzer models were employed for calculation of activity coefficients for the considered systems. Four additional statistical parameters were considered to ensure the consistency of correlations and introduced as objective functions in the particle swarm optimization algorithm. The results showed desirable agreement to the available experimental data, and the order of recommendation of studied models is PHSC EOS > extended Debye-Hückel > Pitzer. The concluding remark is that the all the employed models are reliable in such calculations and can be used for thermodynamic correlation/predictions; however, by using an ion-based parameter calculation method, the PHSC EOS reveals both reliability and universality of applications.
International Nuclear Information System (INIS)
Creutz, M.
1976-01-01
After some disconnected comments on the MIT bag and string models for extended hadrons, I review current understanding of extended objects in classical conventional relativistic field theories and their quantum mechanical interpretation
DEFF Research Database (Denmark)
Krueger, Joel; Szanto, Thomas
2016-01-01
beyond the neurophysiological confines of organisms; some even argue that emotions can be socially extended and shared by multiple agents. Call this the extended emotions thesis (ExE). In this article, we consider different ways of understanding ExE in philosophy, psychology, and the cognitive sciences...
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
Nonlinear integrodifferential equations as discrete systems
Tamizhmani, K. M.; Satsuma, J.; Grammaticos, B.; Ramani, A.
1999-06-01
We analyse a class of integrodifferential equations of the `intermediate long wave' (ILW) type. We show that these equations can be formally interpreted as discrete, differential-difference systems. This allows us to link equations of this type with previous results of ours involving differential-delay equations and, on the basis of this, propose new integrable equations of ILW type. Finally, we extend this approach to pure difference equations and propose ILW forms for the discrete lattice KdV equation.
Franceschi, Alessandro
2014-01-01
This book is a clear, detailed and practical guide to learn about designing and deploying you puppet architecture, with informative examples to highlight and explain concepts in a focused manner. This book is designed for users who already have good experience with Puppet, and will surprise experienced users with innovative topics that explore how to design, implement, adapt, and deploy a Puppet architecture. The key to extending Puppet is the development of types and providers, for which you must be familiar with Ruby.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
DEFF Research Database (Denmark)
Carrara-Augustenborg, Claudia
2012-01-01
There is no consensus yet regarding a conceptualization of consciousness able to accommodate all the features of such complex phenomenon. Different theoretical and empirical models lend strength to both the occurrence of a non-accessible informational broadcast, and to the mobilization of specific...... brain areas responsible for the emergence of the individual´s explicit and variable access to given segments of such broadcast. Rather than advocating one model over others, this chapter proposes to broaden the conceptualization of consciousness by letting it embrace both mechanisms. Within...... such extended framework, I propose conceptual and functional distinctions between consciousness (global broadcast of information), awareness (individual´s ability to access the content of such broadcast) and unconsciousness (focally isolated neural activations). My hypothesis is that a demarcation in terms...
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
Solution of differential equations by application of transformation groups
Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.
1968-01-01
Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.
Tarhini, Ali; Hone, Kate; Liu, Xiaohui; Tarhini, Takwa
2017-01-01
In this study, we examine the effects of individual-level culture on the adoption and acceptance of e-learning tools by students in Lebanon using a theoretical framework based on the Technology Acceptance Model (TAM). To overcome possible limitations of using TAM in developing countries, we extend TAM to include "subjective norms" (SN)…
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
Singular solitons and other solutions to a couple of nonlinear wave equations
International Nuclear Information System (INIS)
Inc Mustafa; Ulutaş Esma; Biswas Anjan
2013-01-01
This paper addresses the extended (G'/G)-expansion method and applies it to a couple of nonlinear wave equations. These equations are modified the Benjamin—Bona—Mahoney equation and the Boussinesq equation. This extended method reveals several solutions to these equations. Additionally, the singular soliton solutions are revealed, for these two equations, with the aid of the ansatz method
Equations of motion derived from a generalization of Einstein's equation for the gravitational field
International Nuclear Information System (INIS)
Mociutchi, C.
1980-01-01
The extended Einstein's equation, combined with a vectorial theory of maxwellian type of the gravitational field, leads to: a) the equation of motion; b) the equation of the trajectory for the static case of spherical symmetry, the test particle having a rest mass other than zero, and c) the propagation of light on null geodesics. All the basic tests of the theory given by Einstein's extended equation. Thus, the new theory of gravitation suggested by us is competitive. (author)
Tricomi, FG
2013-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff
Barbu, Viorel
2016-01-01
This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.
Extended biorthogonal matrix polynomials
Directory of Open Access Journals (Sweden)
Ayman Shehata
2017-01-01
Full Text Available The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B n (x, k and K(A,B n (x, k are discussed. For the matrix polynomials J(A,B n (x, k, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.
Behaviour of the extended Toda lattice
Wattis, Jonathan A. D.; Gordoa, Pilar R.; Pickering, Andrew
2015-11-01
We consider the first member of an extended Toda lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the Korteweg-de Vries and potential Boussinesq equations.
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
1978-01-01
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Auth.)
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Author)
Transport equation solving methods
International Nuclear Information System (INIS)
Granjean, P.M.
1984-06-01
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Directory of Open Access Journals (Sweden)
Vitanov Nikolay K.
2018-03-01
Full Text Available We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation
Vitanov, Nikolay K.; Dimitrova, Zlatinka I.
2018-03-01
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.
On polynomial solutions of the Heun equation
International Nuclear Information System (INIS)
Gurappa, N; Panigrahi, Prasanta K
2004-01-01
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)
Indian Academy of Sciences (India)
regarding nature of forces hold equally for liquids, even though the ... particle. Figure A. A fluid particle is a very small imaginary blob of fluid, here shown sche- matically in .... picture gives important information about the flow field. ... Bernoulli's equation is derived assuming ideal flow, .... weight acting in the flow direction S is.
International Nuclear Information System (INIS)
Gross, F.
1986-01-01
Relativistic equations for two and three body scattering are discussed. Particular attention is paid to relativistic three body kinetics because of recent form factor measurements of the Helium 3 - Hydrogen 3 system recently completed at Saclay and Bates and the accompanying speculation that relativistic effects are important for understanding the three nucleon system. 16 refs., 4 figs
Differential Equations as Actions
DEFF Research Database (Denmark)
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
Equations for formally real meadows
Bergstra, J.A.; Bethke, I.; Ponse, A.
2015-01-01
We consider the signatures Σm = (0,1,−,+,⋅,−1) of meadows and (Σm,s) of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom
A note on the extended dispersionless Toda hierarchy
Lee, Niann-Chern; Tu, Ming-Hsien
2013-04-01
We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological ℂP1 model. Using the dispersionless Hirota equations, we compute the twopoint functions and express the result in terms of Catalan numbers
Tangent Lines without Derivatives for Quadratic and Cubic Equations
Carroll, William J.
2009-01-01
In the quadratic equation, y = ax[superscript 2] + bx + c, the equation y = bx + c is identified as the equation of the line tangent to the parabola at its y-intercept. This is extended to give a convenient method of graphing tangent lines at any point on the graph of a quadratic or a cubic equation. (Contains 5 figures.)
On the existence of solutions for functional differential equations
International Nuclear Information System (INIS)
Walo Omana, R.
1994-12-01
The aim of the paper is to extend the Granas Topological Transversality Method used in boundary value problems for functional differential equations for first and second order, to the case of n-th order functional differential equations. 15 refs
Differential Equations Compatible with KZ Equations
International Nuclear Information System (INIS)
Felder, G.; Markov, Y.; Tarasov, V.; Varchenko, A.
2000-01-01
We define a system of 'dynamical' differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the 'dual' variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions
An integral transform of the Salpeter equation
International Nuclear Information System (INIS)
Krolikowski, W.
1980-03-01
We find a new form of relativistic wave equation for two spin-1/2 particles, which arises by an integral transformation (in the position space) of the wave function in the Salpeter equation. The non-locality involved in this transformation is extended practically over the Compton wavelength of the lighter of two particles. In the case of equal masses the new equation assumes the form of the Breit equation with an effective integral interaction. In the one-body limit it reduces to the Dirac equation also with an effective integral interaction. (author)
Hamiltonian dynamics of extended objects
Capovilla, R.; Guven, J.; Rojas, E.
2004-12-01
We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler Lagrange equations.
Hamiltonian dynamics of extended objects
International Nuclear Information System (INIS)
Capovilla, R; Guven, J; Rojas, E
2004-01-01
We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler-Lagrange equations
Hamiltonian dynamics of extended objects
Energy Technology Data Exchange (ETDEWEB)
Capovilla, R [Departamento de FIsica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo Postal 14-740, 07000 Mexico, DF (Mexico); Guven, J [School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4 (Ireland); Rojas, E [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apdo Postal 70-543, 04510 Mexico, DF (Mexico)
2004-12-07
We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler-Lagrange equations.
How Far Can Extended Knowledge Be Extended?
DEFF Research Database (Denmark)
Wray, K. Brad
2018-01-01
by an artifact, like a notebook or telescope. The chapter illustrates this by applying Pritchard’s account of extended knowledge to collaborating scientists. The beliefs acquired through collaborative research cannot satisfy both of Pritchard’s conditions of creditability. Further, there is evidence......Duncan Pritchard (2010) has developed a theory of extended knowledge based on the notion of extended cognition initially developed by Clark and Chalmers (1998). Pritchard’s account gives a central role to the notion of creditability, which requires the following two conditions to be met: (i...... that scientists are not prepared to take responsibility for the actions of the scientists with whom they collaborate....
On a functional equation related to the intermediate long wave equation
International Nuclear Information System (INIS)
Hone, A N W; Novikov, V S
2004-01-01
We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin-Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain. (letter to the editor)
Integral equation for Coulomb problem
International Nuclear Information System (INIS)
Sasakawa, T.
1986-01-01
For short range potentials an inhomogeneous (homogeneous) Lippmann-Schwinger integral equation of the Fredholm type yields the wave function of scattering (bound) state. For the Coulomb potential, this statement is no more valid. It has been felt difficult to express the Coulomb wave function in a form of an integral equation with the Coulomb potential as the perturbation. In the present paper, the author shows that an inhomogeneous integral equation of a Volterra type with the Coulomb potential as the perturbation can be constructed both for the scattering and the bound states. The equation yielding the binding energy is given in an integral form. The present treatment is easily extended to the coupled Coulomb problems
Nonlinear structures for extended Korteweg–de Vries equation in ...
Indian Academy of Sciences (India)
Posted on 26 August 2016. The domain part of the email address of all email addresses used by the office of Indian Academy of Sciences, including those of the staff, the journals, various programmes, and Current Science, has changed from 'ias.ernet.in' (or 'academy.ias.ernet.in') to 'ias.ac.in'. Thus, for example, the ...
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
Alzahrani, Hasnaa H.
2016-01-01
A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge
A Note on the Application of the Extended Bernoulli Equation
1999-02-01
SANTANDER SPAIN DEPARTMENTO DE QUIMICA FISICA FACULTAD DE CIENCIAS QUIMICAS UNIVERSIDAD COMPLUTENSE DE MADRID V G BAONZA M TARAVTLLO M CACERAS...UNIVERSIDAD DE OVIEDO FACULTAD DE QUIMICA DEPARTMENTO DE QUIMICA FISICA Y ANALITICA E FRANCISCO AVENIDA JULIAN CLAVERIA S/N 33006 - OVIEDO SPAIN
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
Alzahrani, Hasnaa H.
2016-07-26
A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.
Nonlinear structures for extended Korteweg–de Vries equation in ...
Indian Academy of Sciences (India)
The presence of immobile nanodust grains changes the general properties of the ...... rational-type solutions, which may be helpful to explain the creation of very .... investigate the behaviour of nonlinear structures in the Earth's ionosphere ...
New exact solutions for two nonlinear equations
International Nuclear Information System (INIS)
Wang Quandi; Tang Minying
2008-01-01
In this Letter, we investigate two nonlinear equations given by u t -u xxt +3u 2 u x =2u x u xx +uu xxx and u t -u xxt +4u 2 u x =3u x u xx +uu xxx . Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended
Isomorphism of Intransitive Linear Lie Equations
Directory of Open Access Journals (Sweden)
Jose Miguel Martins Veloso
2009-11-01
Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.
Extended Enterprise performance Management
Bobbink, Maria Lammerdina; Hartmann, Andreas
2014-01-01
The allegiance of partnering organisations and their employees to an Extended Enterprise performance is its proverbial sword of Damocles. Literature on Extended Enterprises focuses on collaboration, inter-organizational integration and learning to avoid diminishing or missing allegiance becoming an
Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations
Institute of Scientific and Technical Information of China (English)
SHEN Shou-Feng
2005-01-01
In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.
Perspectives on extended Deterrence
International Nuclear Information System (INIS)
Tertrais, Bruno; Yost, David S.; Bunn, Elaine; Lee, Seok-soo; Levite, Ariel e.; Russell, James A.; Hokayem, Emile; Kibaroglu, Mustafa; Schulte, Paul; Thraenert, Oliver; Kulesa, Lukasz
2010-05-01
In November 2009, the Foundation for Strategic Research (Fondation pour la recherche strategique, FRS) convened a workshop on 'The Future of extended Deterrence', which included the participation of some of the best experts of this topic, from the United States, Europe, the Middle East and East Asia, as well as French and NATO officials. This document brings together the papers prepared for this seminar. Several of them were updated after the publication in April 2010 of the US Nuclear Posture Review. The seminar was organized with the support of the French Atomic energy Commission (Commissariat a l'energie atomique - CEA). Content: 1 - The future of extended deterrence: a brainstorming paper (Bruno Tertrais); 2 - US extended deterrence in NATO and North-East Asia (David S. Yost); 3 - The future of US extended deterrence (Elaine Bunn); 4 - The future of extended deterrence: a South Korean perspective (Seok-soo Lee); 5 - Reflections on extended deterrence in the Middle East (Ariel e. Levite); 6 - extended deterrence, security guarantees and nuclear weapons: US strategic and policy conundrums in the Gulf (James A. Russell); 7 - extended deterrence in the Gulf: a bridge too far? (Emile Hokayem); 8 - The future of extended deterrence: the case of Turkey (Mustafa Kibaroglu); 9 - The future of extended deterrence: a UK view (Paul Schulte); 10 - NATO and extended deterrence (Oliver Thraenert); 11 - extended deterrence and assurance in Central Europe (Lukasz Kulesa)
Radar equations for modern radar
Barton, David K
2012-01-01
Based on the classic Radar Range-Performance Analysis from 1980, this practical volume extends that work to ensure applicability of radar equations to the design and analysis of modern radars. This unique book helps you identify what information on the radar and its environment is needed to predict detection range. Moreover, it provides equations and data to improve the accuracy of range calculations. You find detailed information on propagation effects, methods of range calculation in environments that include clutter, jamming and thermal noise, as well as loss factors that reduce radar perfo
Hamiltonian structure of linearly extended Virasoro algebra
International Nuclear Information System (INIS)
Arakelyan, T.A.; Savvidi, G.K.
1991-01-01
The Hamiltonian structure of linearly extended Virasoro algebra which admits free bosonic field representation is described. An example of a non-trivial extension is found. The hierarchy of integrable non-linear equations corresponding to this Hamiltonian structure is constructed. This hierarchy admits the Lax representation by matrix Lax operator of second order
On energy conservation in extended magnetohydrodynamics
International Nuclear Information System (INIS)
Kimura, Keiji; Morrison, P. J.
2014-01-01
A systematic study of energy conservation for extended magnetohydrodynamic models that include Hall terms and electron inertia is performed. It is observed that commonly used models do not conserve energy in the ideal limit, i.e., when viscosity and resistivity are neglected. In particular, a term in the momentum equation that is often neglected is seen to be needed for conservation of energy
A quantum extended Kalman filter
International Nuclear Information System (INIS)
Emzir, Muhammad F; Woolley, Matthew J; Petersen, Ian R
2017-01-01
In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements. (paper)
A quantum extended Kalman filter
Emzir, Muhammad F.; Woolley, Matthew J.; Petersen, Ian R.
2017-06-01
In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.
On implicit abstract neutral nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)
2016-04-15
In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.
Extended icosahedral structures
Jaric, Marko V
1989-01-01
Extended Icosahedral Structures discusses the concepts about crystal structures with extended icosahedral symmetry. This book is organized into six chapters that focus on actual modeling of extended icosahedral crystal structures. This text first presents a tiling approach to the modeling of icosahedral quasiperiodic crystals. It then describes the models for icosahedral alloys based on random connections between icosahedral units, with particular emphasis on diffraction properties. Other chapters examine the glassy structures with only icosahedral orientational order and the extent of tra
Quantum Gross-Pitaevskii Equation
Directory of Open Access Journals (Sweden)
Jutho Haegeman, Damian Draxler, Vid Stojevic, J. Ignacio Cirac, Tobias J. Osborne, Frank Verstraete
2017-07-01
Full Text Available We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.
Evolution equations for Killing fields
International Nuclear Information System (INIS)
Coll, B.
1977-01-01
The problem of finding necessary and sufficient conditions on the Cauchy data for Einstein equations which insure the existence of Killing fields in a neighborhood of an initial hypersurface has been considered recently by Berezdivin, Coll, and Moncrief. Nevertheless, it can be shown that the evolution equations obtained in all these cases are of nonstrictly hyperbolic type, and, thus, the Cauchy data must belong to a special class of functions. We prove here that, for the vacuum and Einstein--Maxwell space--times and in a coordinate independent way, one can always choose, as evolution equations for the Killing fields, a strictly hyperbolic system: The above theorems can be thus extended to all Cauchy data for which the Einstein evolution problem has been proved to be well set
Extending Database Integration Technology
National Research Council Canada - National Science Library
Buneman, Peter
1999-01-01
Formal approaches to the semantics of databases and database languages can have immediate and practical consequences in extending database integration technologies to include a vastly greater range...
On the relation between elementary partial difference equations and partial differential equations
van den Berg, I.P.
1998-01-01
The nonstandard stroboscopy method links discrete-time ordinary difference equations of first-order and continuous-time, ordinary differential equations of first order. We extend this method to the second order, and also to an elementary, yet general class of partial difference/differential
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Equating error in observed-score equating
van der Linden, Willem J.
2006-01-01
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and population of test takers. But it is argued that if the goal of equating is to adjust the scores of
Construction of a Roe linearization for the ideal MHD equations
International Nuclear Information System (INIS)
Cargo, P.; Gallice, G.; Raviart, P.A.
1996-01-01
In [3], Munz has constructed a Roe linearization for the equations of gas dynamics in Lagrangian coordinates. We extend this construction to the case of the ideal magnetohydrodynamics equations again in Lagrangian coordinates. As a consequence we obtain a Roe linearization for the MHD equations in Eulerian coordinates. (author)
Properties of meromorphic solutions to certain differential-difference equations
Directory of Open Access Journals (Sweden)
Xiaoguang Qi
2013-06-01
Full Text Available We consider the properties of meromorphic solutions to certain type of non-linear difference equations. Also we show the existence of meromorphic solutions with finite order for differential-difference equations related to the Fermat type functional equation. This article extends earlier results by Liu et al [12].
Stochastic Differential Equations and Kondratiev Spaces
Energy Technology Data Exchange (ETDEWEB)
Vaage, G.
1995-05-01
The purpose of this mathematical thesis was to improve the understanding of physical processes such as fluid flow in porous media. An example is oil flowing in a reservoir. In the first of five included papers, Hilbert space methods for elliptic boundary value problems are used to prove the existence and uniqueness of a large family of elliptic differential equations with additive noise without using the Hermite transform. The ideas are then extended to the multidimensional case and used to prove existence and uniqueness of solution of the Stokes equations with additive noise. The second paper uses functional analytic methods for partial differential equations and presents a general framework for proving existence and uniqueness of solutions to stochastic partial differential equations with multiplicative noise, for a large family of noises. The methods are applied to equations of elliptic, parabolic as well as hyperbolic type. The framework presented can be extended to the multidimensional case. The third paper shows how the ideas from the second paper can be extended to study the moving boundary value problem associated with the stochastic pressure equation. The fourth paper discusses a set of stochastic differential equations. The fifth paper studies the relationship between the two families of Kondratiev spaces used in the thesis. 102 refs.
Extended family medicine training
Slade, Steve; Ross, Shelley; Lawrence, Kathrine; Archibald, Douglas; Mackay, Maria Palacios; Oandasan, Ivy F.
2016-01-01
Abstract Objective To examine trends in family medicine training at a time when substantial pedagogic change is under way, focusing on factors that relate to extended family medicine training. Design Aggregate-level secondary data analysis based on the Canadian Post-MD Education Registry. Setting Canada. Participants All Canadian citizens and permanent residents who were registered in postgraduate family medicine training programs within Canadian faculties of medicine from 1995 to 2013. Main outcome measures Number and proportion of family medicine residents exiting 2-year and extended (third-year and above) family medicine training programs, as well as the types and numbers of extended training programs offered in 2015. Results The proportion of family medicine trainees pursuing extended training almost doubled during the study period, going from 10.9% in 1995 to 21.1% in 2013. Men and Canadian medical graduates were more likely to take extended family medicine training. Among the 5 most recent family medicine exit cohorts (from 2009 to 2013), 25.9% of men completed extended training programs compared with 18.3% of women, and 23.1% of Canadian medical graduates completed extended training compared with 13.6% of international medical graduates. Family medicine programs vary substantially with respect to the proportion of their trainees who undertake extended training, ranging from a low of 12.3% to a high of 35.1% among trainees exiting from 2011 to 2013. Conclusion New initiatives, such as the Triple C Competency-based Curriculum, CanMEDS–Family Medicine, and Certificates of Added Competence, have emerged as part of family medicine education and credentialing. In acknowledgment of the potential effect of these initiatives, it is important that future research examine how pedagogic change and, in particular, extended training shapes the care family physicians offer their patients. As part of that research it will be important to measure the breadth and uptake of
Center for Extended Magnetohydrodynamics Modeling
Energy Technology Data Exchange (ETDEWEB)
Ramos, Jesus [Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
2017-02-14
This researcher participated in the DOE-funded Center for Extended Magnetohydrodynamics Modeling (CEMM), a multi-institutional collaboration led by the Princeton Plasma Physics Laboratory with Dr. Stephen Jardin as the overall Principal Investigator. This project developed advanced simulation tools to study the non-linear macroscopic dynamics of magnetically confined plasmas. The collaborative effort focused on the development of two large numerical simulation codes, M3D-C1 and NIMROD, and their application to a wide variety of problems. Dr. Ramos was responsible for theoretical aspects of the project, deriving consistent sets of model equations applicable to weakly collisional plasmas and devising test problems for verification of the numerical codes. This activity was funded for twelve years.
Pseudodifferential equations over non-Archimedean spaces
Zúñiga-Galindo, W A
2016-01-01
Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph presents a very general theory of parabolic-type equations and their Markov processes motivated by their connection with models of complex hierarchic systems. The Gelfand-Shilov method for constructing fundamental solutions using local zeta functions is developed in a p-adic setting and several particular equations are studied, such as the p-adic analogues of the Klein-Gordon equation. Pseudodifferential equations for complex-valued functions on non-Archimedean local fields are central to contemporary harmonic analysis and mathematical physics and their theory reveals a deep connection with probability and number theory. The results of this book extend and complement the material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001), which emphasize spectral theory and evolution equations in a single variable, and Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and applica...
Calculation of propellant gas pressure by simple extended corresponding state principle
Bin Xu; San-jiu Ying; Xin Liao
2016-01-01
The virial equation can well describe gas state at high temperature and pressure, but the difficulties in virial coefficient calculation limit the use of virial equation. Simple extended corresponding state principle (SE-CSP) is introduced in virial equation. Based on a corresponding state equation, including three characteristic parameters, an extended parameter is introduced to describe the second virial coefficient expressions of main products of propellant gas. The modified SE-CSP second ...
The Extended Enterprise concept
DEFF Research Database (Denmark)
Larsen, Lars Bjørn; Vesterager, Johan; Gobbi, Chiara
1999-01-01
This paper provides an overview of the work that has been done regarding the Extended Enterprise concept in the Common Concept team of Globeman 21 including references to results deliverables concerning the development of the Extended Enterprise concept. The first section presents the basic concept...... picture from Globeman21, which illustrates the Globeman21 way of realising the Extended Enterprise concept. The second section presents the Globeman21 EE concept in a life cycle perspective, which to a large extent is based on the thoughts and ideas behind GERAM (ISO/DIS 15704)....
Subsurface offset behaviour in velocity analysis with extended reflectivity images
Mulder, W.A.
2012-01-01
Migration velocity analysis with the wave equation can be accomplished by focusing of extended migration images, obtained by introducing a subsurface offset or shift. A reflector in the wrong velocity model will show up as a curve in the extended image. In the correct model, it should collapse to a
Subsurface offset behaviour in velocity analysis with extended reflectivity images
Mulder, W.A.
2013-01-01
Migration velocity analysis with the constant-density acoustic wave equation can be accomplished by the focusing of extended migration images, obtained by introducing a subsurface shift in the imaging condition. A reflector in a wrong velocity model will show up as a curve in the extended image. In
Sum rules in extended RPA theories
International Nuclear Information System (INIS)
Adachi, S.; Lipparini, E.
1988-01-01
Different moments m k of the excitation strength function are studied in the framework of the second RPA and of the extended RPA in which 2p2h correlations are explicitly introduced into the ground state by using first-order perturbation theory. Formal properties of the equations of motion concerning sum rules are derived and compared with those exhibited by the usual 1p1h RPA. The problem of the separation of the spurious solutions in extended RPA calculations is also discussed. (orig.)
International Nuclear Information System (INIS)
Appelquist, T.; Terning, J.
1994-01-01
An extended technicolor model is constructed. Quark and lepton masses, spontaneous CP violation, and precision electroweak measurements are discussed. Dynamical symmetry breaking is analyzed using the concept of the big MAC (most attractive channel)
International Nuclear Information System (INIS)
Anon.
1984-01-01
Mine layouts, new machines and techniques, research into problem areas of ground control and so on, are highlighted in this report on extending mine life. The main resources taken into account are coal mining, uranium mining, molybdenum and gold mining
International Nuclear Information System (INIS)
Yan Zhenya
2005-01-01
A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer-Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations
The dialogically extended mind
DEFF Research Database (Denmark)
Fusaroli, Riccardo; Gangopadhyay, Nivedita; Tylén, Kristian
2014-01-01
A growing conceptual and empirical literature is advancing the idea that language extends our cognitive skills. One of the most influential positions holds that language – qua material symbols – facilitates individual thought processes by virtue of its material properties. Extending upon this model...... relate our approach to other ideas about collective minds and review a number of empirical studies to identify the mechanisms enabling the constitution of interpersonal cognitive systems....
Extending Mondrian Memory Protection
2010-11-01
a kernel semaphore is locked or unlocked. In addition, we extended the system call interface to receive notifications about user-land locking...operations (such as calls to the mutex and semaphore code provided by the C library). By patching the dynamically loadable GLibC5, we are able to test... semaphores , and spinlocks. RTO-MP-IST-091 10- 9 Extending Mondrian Memory Protection to loading extension plugins. This prevents any untrusted code
2016-06-06
number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 06-06-2016 2. REPORT TYPE Interim Report 3. DATES COVERED ... Corrosion Testing of Traditional and Extended Life Coolants 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Hansen, Gregory A. T...providing vehicle specific coolants. Several laboratory corrosion tests were performed according to ASTM D1384 and D2570, but with a 2.5x extended time
Tailoring discrete quantum walk dynamics via extended initial conditions
Energy Technology Data Exchange (ETDEWEB)
Valcarcel, German J de; Roldan, Eugenio [Departament d' Optica, Universitat de Valencia, Dr Moliner 50, 46100-Burjassot, Spain, EU (Spain); Romanelli, Alejandro, E-mail: german.valcarcel@uv.es, E-mail: eugenio.roldan@uv.es, E-mail: alejo@fing.edu.uy [Instituto de Fisica, Facultad de IngenierIa, Universidad de la Republica, CC 30, CP 11000, Montevideo (Uruguay)
2010-12-15
We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schroedinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.
Tailoring discrete quantum walk dynamics via extended initial conditions
International Nuclear Information System (INIS)
Valcarcel, German J de; Roldan, Eugenio; Romanelli, Alejandro
2010-01-01
We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schroedinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.
Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations
Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril
2011-01-01
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the loc...
A fractional Dirac equation and its solution
International Nuclear Information System (INIS)
Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru
2010-01-01
This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
Spectral theories for linear differential equations
International Nuclear Information System (INIS)
Sell, G.R.
1976-01-01
The use of spectral analysis in the study of linear differential equations with constant coefficients is not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour of the solutions. The spectral analysis, via the Jordan canonical form, will not only lead to a representation theorem for a basis of solutions, but will also give a rather precise statement of the (exponential) growth rates of various solutions. Various attempts have been made to extend this analysis to linear differential equations with time-varying coefficients. The most complete such extensions is the Floquet theory for equations with periodic coefficients. For time-varying linear differential equations with aperiodic coefficients several authors have attempted to ''extend'' the Foquet theory. The precise meaning of such an extension is itself a problem, and we present here several attempts in this direction that are related to the general problem of extending the spectral analysis of equations with constant coefficients. The main purpose of this paper is to introduce some problems of current research. The primary problem we shall examine occurs in the context of linear differential equations with almost periodic coefficients. We call it ''the Floquet problem''. (author)
Handbook of integral equations
Polyanin, Andrei D
2008-01-01
This handbook contains over 2,500 integral equations with solutions as well as analytical and numerical methods for solving linear and nonlinear equations. It explores Volterra, Fredholm, WienerHopf, Hammerstein, Uryson, and other equations that arise in mathematics, physics, engineering, the sciences, and economics. This second edition includes new chapters on mixed multidimensional equations and methods of integral equations for ODEs and PDEs, along with over 400 new equations with exact solutions. With many examples added for illustrative purposes, it presents new material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions.
An extended Henon-Heiles system
International Nuclear Information System (INIS)
Hone, A.N.W.; Novikov, V.; Verhoeven, C.
2008-01-01
We consider an integrable system of partial differential equations possessing a Lax pair with an energy-dependent Lax operator of second order, of the type described by Antonowicz and Fordy. By taking the travelling wave reduction of this system we show that the integrable case (ii) Henon-Heiles system can be extended by adding an arbitrary number of non-polynomial (rational) terms to the potential
Extended Deterministic Mean-Field Games
Gomes, Diogo A.
2016-04-21
In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.
Extended Deterministic Mean-Field Games
Gomes, Diogo A.; Voskanyan, Vardan K.
2016-01-01
In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.
Quantum Potential and Symmetries in Extended Phase Space
Directory of Open Access Journals (Sweden)
Sadollah Nasiri
2006-06-01
Full Text Available The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique. This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space. It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation. The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one. The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.
Multi-component WKI equations and their conservation laws
Energy Technology Data Exchange (ETDEWEB)
Qu Changzheng [Department of Mathematics, Northwest University, Xi' an 710069 (China) and Center for Nonlinear Studies, Northwest University, Xi' an 710069 (China)]. E-mail: qu_changzheng@hotmail.com; Yao Ruoxia [Department of Computer Sciences, East China Normal University, Shanghai 200062 (China); Department of Computer Sciences, Weinan Teacher' s College, Weinan 715500 (China); Liu Ruochen [Department of Mathematics, Northwest University, Xi' an 710069 (China)
2004-10-25
In this Letter, a two-component WKI equation is obtained by using the fact that when curvature and torsion of a space curve satisfy the vector modified KdV equation, a graph of the curve satisfies the two-component WKI equation, which is a natural generalization to the WKI equation. It is shown that the two-component WKI equation can be solved in terms of the extended WKI scheme, and it admits an infinite number of conservation laws. In the same vein, a n-component generalization to the WKI equation is proposed.
Japyassú, Hilton F; Laland, Kevin N
2017-05-01
There is a tension between the conception of cognition as a central nervous system (CNS) process and a view of cognition as extending towards the body or the contiguous environment. The centralised conception requires large or complex nervous systems to cope with complex environments. Conversely, the extended conception involves the outsourcing of information processing to the body or environment, thus making fewer demands on the processing power of the CNS. The evolution of extended cognition should be particularly favoured among small, generalist predators such as spiders, and here, we review the literature to evaluate the fit of empirical data with these contrasting models of cognition. Spiders do not seem to be cognitively limited, displaying a large diversity of learning processes, from habituation to contextual learning, including a sense of numerosity. To tease apart the central from the extended cognition, we apply the mutual manipulability criterion, testing the existence of reciprocal causal links between the putative elements of the system. We conclude that the web threads and configurations are integral parts of the cognitive systems. The extension of cognition to the web helps to explain some puzzling features of spider behaviour and seems to promote evolvability within the group, enhancing innovation through cognitive connectivity to variable habitat features. Graded changes in relative brain size could also be explained by outsourcing information processing to environmental features. More generally, niche-constructed structures emerge as prime candidates for extending animal cognition, generating the selective pressures that help to shape the evolving cognitive system.
A microscopic derivation of stochastic differential equations
International Nuclear Information System (INIS)
Arimitsu, Toshihico
1996-01-01
With the help of the formulation of Non-Equilibrium Thermo Field Dynamics, a unified canonical operator formalism is constructed for the quantum stochastic differential equations. In the course of its construction, it is found that there are at least two formulations, i.e. one is non-hermitian and the other is hermitian. Having settled which framework should be satisfied by the quantum stochastic differential equations, a microscopic derivation is performed for these stochastic differential equations by extending the projector methods. This investigation may open a new field for quantum systems in order to understand the deeper meaning of dissipation
Improved Durand-equation for multiple application
International Nuclear Information System (INIS)
Weber, M.
1986-01-01
The applicability of Durand's equation could be improved for general use by applying suitable parameters representing the grain-size distribution. Thus, the Durand equation cannot only describe polydisperse (pseudo)-homogeneous or heterogeneous transportation, but also solid-fluid mixtures containing a certain amount of fine particles. Even non-Newtonian influences can be taken into account. The applicability of the extended Durand equation for polydisperse mixtures will be demonstrated by measurement data. With respect to this, the transition between pseudohomogeneous and heterogeneous transport has been considered on the basis of measured concentration profiles
Extending quantum mechanics entails extending special relativity
International Nuclear Information System (INIS)
Aravinda, S; Srikanth, R
2016-01-01
The complementarity between signaling and randomness in any communicated resource that can simulate singlet statistics is generalized by relaxing the assumption of free will in the choice of measurement settings. We show how to construct an ontological extension for quantum mechanics (QMs) through the oblivious embedding of a sound simulation protocol in a Newtonian spacetime. Minkowski or other intermediate spacetimes are ruled out as the locus of the embedding by virtue of hidden influence inequalities. The complementarity transferred from a simulation to the extension unifies a number of results about quantum non-locality, and implies that special relativity has a different significance for the ontological model and for the operational theory it reproduces. Only the latter, being experimentally accessible, is required to be Lorentz covariant. There may be certain Lorentz non-covariant elements at the ontological level, but they will be inaccessible at the operational level in a valid extension. Certain arguments against the extendability of QM, due to Conway and Kochen (2009) and Colbeck and Renner (2012), are attributed to their assumption that the spacetime at the ontological level has Minkowski causal structure. (paper)
Integrodifferential equation approach. Pt. 1
International Nuclear Information System (INIS)
Oehm, W.; Sofianos, S.A.; Fiedeldey, H.; South Africa Univ., Pretoria. Dept. of Physics); Fabre de la Ripelle, M.; South Africa Univ., Pretoria. Dept. of Physics)
1990-02-01
A single integrodifferential equation in two variables, valid for A nucleons interacting by pure Wigner forces, which has previously only been solved in the extreme and uncoupled adiabatic approximations is now solved exactly for three- and four-nucleon systems. The results are in good agreement with the values obtained for the binding energies by means of an empirical interpolation formula. This validates all our previous conclusions, in particular that the omission of higher (than two) order correlations in our four-body equation only produces a rather small underbinding. The integrodifferential equation approach (IDEA) is here also extended to spin-dependent forces of the Malfliet-Tjon type, resulting in two coupled integrodifferential equations in two variables. The exact solution and the interpolated adiabatic approximation are again in good agreement. The inclusion of the hypercentral part of the two-body interaction in the definition of the Faddeev-type components again leads to substantial improvement for fully local potentials, acting in all partial waves. (orig.)
Wave-equation dispersion inversion
Li, Jing
2016-12-08
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Generalized Ordinary Differential Equation Models.
Miao, Hongyu; Wu, Hulin; Xue, Hongqi
2014-10-01
Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.
Introduction to differential equations
Taylor, Michael E
2011-01-01
The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis. The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponen
Uraltseva, N N
1995-01-01
This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrödinger equation). The book will be useful to researchers and graduate students working in p
Eckalbar, John C.
2002-01-01
Illustrates how principles and intermediate microeconomic students can gain an understanding for strategic price setting by playing a relatively large oligopoly game. Explains that the game extends to a continuous price space and outlines appropriate applications. Offers the Mathematica code to instructors so that the assumptions of the game can…
International Nuclear Information System (INIS)
Akama, Keiichi
1988-01-01
Starting with the space-time action of the transversally extended string, we derive its world-sheet action, which is that of a gravitational and gauge theory with matter fields on the world-sheet, with additional effects of the second fundamental quantity. (author)
Extended artistic appreciation.
Wilson, Robert A
2013-04-01
I propose that in at least some cases, objects of artistic appreciation are best thought of not simply as causes of artistic appreciation, but as parts of the cognitive machinery that drives aesthetic appreciation. In effect, this is to say that aesthetic appreciation operates via extended cognitive systems.
Towards Extended Vantage Theory
Glaz, Adam
2010-01-01
The applicability of Vantage Theory (VT), a model of (colour) categorization, to linguistic data largely depends on the modifications and adaptations of the model for the purpose. An attempt to do so proposed here, called Extended Vantage Theory (EVT), slightly reformulates the VT conception of vantage by capitalizing on some of the entailments of…
A procedure to construct exact solutions of nonlinear fractional differential equations.
Güner, Özkan; Cevikel, Adem C
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.
Third order differential equations with delay
Directory of Open Access Journals (Sweden)
Petr Liška
2015-05-01
Full Text Available In this paper, we study the oscillation and asymptotic properties of solutions of certain nonlinear third order differential equations with delay. In particular, we extend results of I. Mojsej (Nonlinear Analysis 68, 2008 and we improve conditions on the property B of N. Parhi and S. Padhi (Indian J. Pure Appl. Math., 33, 2002.
A Unified Introduction to Ordinary Differential Equations
Lutzer, Carl V.
2006-01-01
This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)
International Nuclear Information System (INIS)
Lebedev, D.R.
1979-01-01
Benney's equations of motion of incompressible nonviscous fluid with free surface in the approximation of long waves are analyzed. The connection between the Lie algebra of Hamilton plane vector fields and the Benney's momentum equations is shown
Fractional Schroedinger equation
International Nuclear Information System (INIS)
Laskin, Nick
2002-01-01
Some properties of the fractional Schroedinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom (fractional 'Bohr atom') and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schroedinger equations
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
International Nuclear Information System (INIS)
Ichiguchi, Katsuji
1998-01-01
A new reduced set of resistive MHD equations is derived by averaging the full MHD equations on specified flux coordinates, which is consistent with 3D equilibria. It is confirmed that the total energy is conserved and the linearized equations for ideal modes are self-adjoint. (author)
Gaussian solitary waves for the logarithmic-KdV and the logarithmic-KP equations
International Nuclear Information System (INIS)
Wazwaz, Abdul-Majid
2014-01-01
We investigate the logarithmic-KdV equation for more Gaussian solitary waves. We extend this work to derive the logarithmic-KP (Kadomtsev–Petviashvili) equation. We show that both logarithmic models are characterized by their Gaussian solitons. (paper)
Generalized latent variable modeling multilevel, longitudinal, and structural equation models
Skrondal, Anders; Rabe-Hesketh, Sophia
2004-01-01
This book unifies and extends latent variable models, including multilevel or generalized linear mixed models, longitudinal or panel models, item response or factor models, latent class or finite mixture models, and structural equation models.
Jacobian elliptic function expansion solutions of nonlinear stochastic equations
International Nuclear Information System (INIS)
Wei Caimin; Xia Zunquan; Tian Naishuo
2005-01-01
Jacobian elliptic function expansion method is extended and applied to construct the exact solutions of the nonlinear Wick-type stochastic partial differential equations (SPDEs) and some new exact solutions are obtained via this method and Hermite transformation
Extended Irreversible Thermodynamics
Jou, David
2010-01-01
This is the 4th edition of the highly acclaimed monograph on Extended Irreversible Thermodynamics, a theory that goes beyond the classical theory of irreversible processes. In contrast to the classical approach, the basic variables describing the system are complemented by non-equilibrium quantities. The claims made for extended thermodynamics are confirmed by the kinetic theory of gases and statistical mechanics. The book covers a wide spectrum of applications, and also contains a thorough discussion of the foundations and the scope of the current theories on non-equilibrium thermodynamics. For this new edition, the authors critically revised existing material while taking into account the most recent developments in fast moving fields such as heat transport in micro- and nanosystems or fast solidification fronts in materials sciences. Several fundamental chapters have been revisited emphasizing physics and applications over mathematical derivations. Also, fundamental questions on the definition of non-equil...
International Nuclear Information System (INIS)
Pavel Bona
2000-01-01
The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics which is related here to some other (more general, but also to more special and 'approximative') theories. Quantum mechanics is here primarily reformulated in an equivalent form of a Poisson system on the phase space consisting of density matrices, where the 'observables', as well as 'symmetry generators' are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfjoint operators. It is shown in this paper that inclusion of additional ('nonlinear') symmetry generators (i. e. 'Hamiltonians') into this reformulation of (linear) quantum mechanics leads to a considerable extension of the theory: two kinds of quantum 'mixed states' should be distinguished, and operator - valued functions of density matrices should be used in the role of 'nonlinear observables'. A general framework for physical theories is obtained in this way: By different choices of the sets of 'nonlinear observables' we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) quantum mechanics, or nonlinear extensions of quantum mechanics; also various 'quasiclassical approximations' to quantum mechanics are all sub theories of the presented extension of quantum mechanics - a version of the extended quantum mechanics. A general interpretation scheme of extended quantum mechanics extending the usual statistical interpretation of quantum mechanics is also proposed. Eventually, extended quantum mechanics is shown to be (included into) a C * -algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. The presentation includes an analysis of problems connected with differentiation on infinite-dimensional manifolds, as well as a solution of some problems connected with the work with only densely defined unbounded
Humbert, Richard
2010-01-01
A force acting on just part of an extended object (either a solid or a volume of a liquid) can cause all of it to move. That motion is due to the transmission of the force through the object by its material. This paper discusses how the force is distributed to all of the object by a gradient of stress or pressure in it, which creates the local…
Extending Critical Performativity
DEFF Research Database (Denmark)
Spicer, André; Alvesson, Mats; Kärreman, Dan
2016-01-01
In this article we extend the debate about critical performativity. We begin by outlining the basic tenets of critical performativity and how this has been applied in the study of management and organization. We then address recent critiques of critical performance. We note these arguments suffer...... of public importance; engaging with non-academic groups using dialectical reasoning; scaling up insights through movement building; and propagating deliberation...
A Novel Biped Pattern Generator Based on Extended ZMP and Extended Cart-Table Model
Directory of Open Access Journals (Sweden)
Guangbin Sun
2015-07-01
Full Text Available This paper focuses on planning patterns for biped walking on complex terrains. Two problems are solved: ZMP (zero moment point cannot be used on uneven terrain, and the conventional cart-table model does not allow vertical CM (centre of mass motion. For the ZMP definition problem, we propose the extended ZMP (EZMP concept as an extension of ZMP to uneven terrains. It can be used to judge dynamic balance on universal terrains. We achieve a deeper insight into the connection and difference between ZMP and EZMP by adding different constraints. For the model problem, we extend the cart-table model by using a dynamic constraint instead of constant height constraint, which results in a mathematically symmetric set of three equations. In this way, the vertical motion is enabled and the resultant equations are still linear. Based on the extended ZMP concept and extended cart-table model, a biped pattern generator using triple preview controllers is constructed and implemented simultaneously to three dimensions. Using the proposed pattern generator, the Atlas robot is simulated. The simulation results show the robot can walk stably on rather complex terrains by accurately tracking extended ZMP.
International Nuclear Information System (INIS)
Capozziello, Salvatore; De Laurentis, Mariafelicia
2011-01-01
Extended Theories of Gravity can be considered as a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein’s theory, is aimed to address conceptual and experimental problems recently emerged in astrophysics, cosmology and High Energy Physics. In particular, the goal is to encompass, in a self-consistent scheme, problems like inflation, dark energy, dark matter, large scale structure and, first of all, to give at least an effective description of Quantum Gravity. We review the basic principles that any gravitational theory has to follow. The geometrical interpretation is discussed in a broad perspective in order to highlight the basic assumptions of General Relativity and its possible extensions in the general framework of gauge theories. Principles of such modifications are presented, focusing on specific classes of theories like f(R)-gravity and scalar–tensor gravity in the metric and Palatini approaches. The special role of torsion is also discussed. The conceptual features of these theories are fully explored and attention is paid to the issues of dynamical and conformal equivalence between them considering also the initial value problem. A number of viability criteria are presented considering the post-Newtonian and the post-Minkowskian limits. In particular, we discuss the problems of neutrino oscillations and gravitational waves in extended gravity. Finally, future perspectives of extended gravity are considered with possibility to go beyond a trial and error approach.
Geometric construction of extended supergravity
International Nuclear Information System (INIS)
Kostelecky, V.A.
1982-01-01
This work describes the explict construction of the locally SO(4)-invariant, on-shell de Sitter supergravity. First, aspects of classical differential geometry used in the construction of local gauge theories are reviewed. Emphasis is placed on fiber bundles and their uses in Yang-Mills and Einstein theories. Next, the extension of the formalism to differential supergeometry is outlined. Applications to extended supergravities are discussed. Finally, the O(4) deSitter supergravity is obtained by considering a bundle of frames constructed using the orthosymplectic superalgebra osp(4/4). The structure group of this bundle is Sl(2C) x SO(4) and the tangent space to the base supermanifold is homeomorphic to the coset osp(4/4)/sl(2C) x so(4). Constraints taken into the Bianchi identifies yield a realization of the superalgebra in the function space of connections, vielbeins, curvatures and torsions of the bundle. Auxiliary fields, transformation laws and equations of motion are determined. Consistency of the realization is verified, proving closure of the algebra. The associated Poincare supergravity is obtained by a contraction
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
Extending Teach and Repeat to Pivoting Wheelchairs
Directory of Open Access Journals (Sweden)
Guillermo Del Castillo
2003-02-01
Full Text Available The paper extends the teach-and-repeat paradigm that has been successful for the control of holonomic robots to nonholonomic wheelchairs which may undergo pivoting action over the course of their taught movement. Due to the nonholonomic nature of the vehicle kinematics, estimation is required -- in the example given herein, based upon video detection of wall-mounted cues -- both in the teaching and the tracking events. In order to accommodate motion that approaches pivoting action as well as motion that approaches straight-line action, the estimation equations of the Extended Kalman Filter and the control equations are formulated using two different definitions of a nontemporal independent variable. The paper motivates the need for pivoting action in real-life settings by reporting extensively on the abilities and limitations of estimation-based teach-and-repeat action where pivoting and near-pivoting action is disallowed. Following formulation of the equations in the near-pivot mode, the paper reports upon experiments where taught trajectories which entail a seamless mix of near-straight and near-pivot action are tracked.
On reduction and exact solutions of nonlinear many-dimensional Schroedinger equations
International Nuclear Information System (INIS)
Barannik, A.F.; Marchenko, V.A.; Fushchich, V.I.
1991-01-01
With the help of the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra AO(n) the rank n and n-1 maximal subalgebras of the extended isochronous Galileo algebra, the rank n maximal subalgebras of the generalized extended classical Galileo algebra AG(a,n) the extended special Galileo algebra AG(2,n) and the extended whole Galileo algebra AG(3,n) are described. By using the rank n subalgebras, ansatze reducing the many dimensional Schroedinger equations to ordinary differential equations is found. With the help of the reduced equation solutions exact solutions of the Schroedinger equation are considered
Explorations of the extended ncKP hierarchy
International Nuclear Information System (INIS)
Dimakis, Aristophanes; Mueller-Hoissen, Folkert
2004-01-01
A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy (ncKP hierarchy) by a set of evolution equations in the Moyal-deformation parameters is further explored. Formulae are derived to compute these equations efficiently. Reductions of the xncKP hierarchy are treated, in particular to the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of the Sato formalism for the KP hierarchy is carried over to the generalized framework. In particular, the well-known bilinear identity theorem for the KP hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions of the ncKP equation are also solutions of the first few deformation equations. This is shown to be related to the existence of certain families of algebraic identities
International Nuclear Information System (INIS)
Chen, R.L.W.
1981-01-01
The use of an equation obtained earlier for computing electrical conductivity may be extended to partially ionized gases which depart from the Saha equation, provided the electron velocity distributions do not deviate from the Maxwellian distribution. (author)
A Priori Regularity of Parabolic Partial Differential Equations
Berkemeier, Francisco
2018-05-13
In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.
Electron transfer dynamics: Zusman equation versus exact theory
International Nuclear Information System (INIS)
Shi Qiang; Chen Liping; Nan Guangjun; Xu Ruixue; Yan Yijing
2009-01-01
The Zusman equation has been widely used to study the effect of solvent dynamics on electron transfer reactions. However, application of this equation is limited by the classical treatment of the nuclear degrees of freedom. In this paper, we revisit the Zusman equation in the framework of the exact hierarchical equations of motion formalism, and show that a high temperature approximation of the hierarchical theory is equivalent to the Zusman equation in describing electron transfer dynamics. Thus the exact hierarchical formalism naturally extends the Zusman equation to include quantum nuclear dynamics at low temperatures. This new finding has also inspired us to rescale the original hierarchical equations and incorporate a filtering algorithm to efficiently propagate the hierarchical equations. Numerical exact results are also presented for the electron transfer reaction dynamics and rate constant calculations.
Extended Testability Analysis Tool
Melcher, Kevin; Maul, William A.; Fulton, Christopher
2012-01-01
The Extended Testability Analysis (ETA) Tool is a software application that supports fault management (FM) by performing testability analyses on the fault propagation model of a given system. Fault management includes the prevention of faults through robust design margins and quality assurance methods, or the mitigation of system failures. Fault management requires an understanding of the system design and operation, potential failure mechanisms within the system, and the propagation of those potential failures through the system. The purpose of the ETA Tool software is to process the testability analysis results from a commercial software program called TEAMS Designer in order to provide a detailed set of diagnostic assessment reports. The ETA Tool is a command-line process with several user-selectable report output options. The ETA Tool also extends the COTS testability analysis and enables variation studies with sensor sensitivity impacts on system diagnostics and component isolation using a single testability output. The ETA Tool can also provide extended analyses from a single set of testability output files. The following analysis reports are available to the user: (1) the Detectability Report provides a breakdown of how each tested failure mode was detected, (2) the Test Utilization Report identifies all the failure modes that each test detects, (3) the Failure Mode Isolation Report demonstrates the system s ability to discriminate between failure modes, (4) the Component Isolation Report demonstrates the system s ability to discriminate between failure modes relative to the components containing the failure modes, (5) the Sensor Sensor Sensitivity Analysis Report shows the diagnostic impact due to loss of sensor information, and (6) the Effect Mapping Report identifies failure modes that result in specified system-level effects.
Modified Darboux transformations with foreign auxiliary equations
International Nuclear Information System (INIS)
Schulze-Halberg, Axel
2011-01-01
We construct a new type of first-order Darboux transformations for the stationary Schroedinger equation. In contrast to the conventional case, our Darboux transformations support arbitrary (foreign) auxiliary equations. We show that among other applications, our formalism can be used to systematically construct Darboux transformations for Schroedinger equations with energy-dependent potentials, including a recent result (Lin et al., 2007) as a special case. -- Highlights: → We generalize the Darboux transformation for the Schroedinger equation. → By admitting arbitrary auxiliary functions, we provide a new tool for generating solutions. → As a special case we recover a recent result on energy-dependent potentials. → We extend the latter result to very general energy-dependence.
Modelling conjugation with stochastic differential equations
DEFF Research Database (Denmark)
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik
2010-01-01
Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared......Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...... using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared...
Extended Wordsearches in Chemistry
Cotton, Simon
1998-04-01
Students can be encouraged to develop their factual knowledge by use of puzzles. One strategy described here is the extended wordsearch, where the wordsearch element generates a number of words or phrases from which the answers to a series of questions are selected. The wordsearch can be generated with the aid of computer programs, though in order to make them suitable for students with dyslexia or other learning difficulties, a simpler form is more appropriate. These problems can be employed in a variety of contexts, for example, as topic tests and classroom end-of-lesson fillers. An example is provided in the area of calcium chemistry. Sources of suitable software are listed.
Classical extended superconformal symmetries
International Nuclear Information System (INIS)
Viswanathan, R.R.
1990-10-01
Super-covariant differential operators are defined in two dimensions which map supersymmetry doublets to other doublets. The possibility of constructing a closed algebra among the fields appearing in such operators is explored. Such an algebra exists for Grassmann-odd differential operators. A representation for these operators in terms of free-field doublets is constructed. An explicit closed algebra involving fields of spin 2 and 5/2, in addition to the stress tensor and the supersymmetry generator, is constructed from such a free-field representation as an example of a non-linear extended superconformal algebra. (author). 9 refs
Probabilistic Representations of Solutions to the Heat Equation
Indian Academy of Sciences (India)
In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition , is given by the convolution of with the heat kernel (Gaussian density). Our results also extend the ...
Differential geometry techniques for sets of nonlinear partial differential equations
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle
Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.
2013-01-01
We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Zhu Jiamin; Ma Zhengyi
2007-01-01
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions
New solitary wave solutions to the modified Kawahara equation
International Nuclear Information System (INIS)
Wazwaz, Abdul-Majid
2007-01-01
In this work we use the sine-cosine method, the tanh method, the extended tanh method, and ansatze of hyperbolic functions for analytic treatment for the modified Kawahara equation. New solitons solutions and periodic solutions are formally derived. The change of the parameters, that will drastically change the characteristics of the equation, is examined. The employed approaches are reliable and manageable
Generalized ordinary differential equations not absolutely continuous solutions
Kurzweil, Jaroslav
2012-01-01
This book provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. It contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.
International Nuclear Information System (INIS)
Zhalij, Alexander
2002-01-01
We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A 0 (t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field
Schiesser, William E
2014-01-01
Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the com
Functional equations with causal operators
Corduneanu, C
2003-01-01
Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.
Extended resolvent and inverse scattering with an application to KPI
Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.; Prinari, B.
2003-08-01
We present in detail an extended resolvent approach for investigating linear problems associated to 2+1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schrödinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given.
Extended resolvent and inverse scattering with an application to KPI
International Nuclear Information System (INIS)
Boiti, M.; Pempinelli, F.; Pogrebkov, A.K.; Prinari, B.
2003-01-01
We present in detail an extended resolvent approach for investigating linear problems associated to 2+1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schroedinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given
Hojman's theorem of the third-order ordinary differential equation
International Nuclear Information System (INIS)
Hong-Sheng, Lü; Hong-Bin, Zhang; Shu-Long, Gu
2009-01-01
This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results. (general)
Exp-function method for solving fractional partial differential equations.
Zheng, Bin
2013-01-01
We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.
Solution of the stellar structure equations in Eulerian coordinates
International Nuclear Information System (INIS)
Deupree, R.G.
1976-01-01
The equations of hydrostatic and thermal equilibrium, assuming only radiative energy transport and spherical symmetry, are solved in Eulerian coordinates by a suitable modification of the Henyey method. An Eulerian approach may possibly be more suitably extended to more spatial dimensions than the usual Lagrangian procedure. The principle advantage of this method is that the equations of hydrostatic and thermal equilibrium and Poisson's equation may be solved simultaneously
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Degenerate nonlinear diffusion equations
Favini, Angelo
2012-01-01
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asympt...
Building Context with Tumor Growth Modeling Projects in Differential Equations
Beier, Julie C.; Gevertz, Jana L.; Howard, Keith E.
2015-01-01
The use of modeling projects serves to integrate, reinforce, and extend student knowledge. Here we present two projects related to tumor growth appropriate for a first course in differential equations. They illustrate the use of problem-based learning to reinforce and extend course content via a writing or research experience. Here we discuss…
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Solving Ordinary Differential Equations
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Reactimeter dispersion equation
A.G. Yuferov
2016-01-01
The aim of this work is to derive and analyze a reactimeter metrological model in the form of the dispersion equation which connects reactimeter input/output signal dispersions with superimposed random noise at the inlet. It is proposed to standardize the reactimeter equation form, presenting the main reactimeter computing unit by a convolution equation. Hence, the reactimeter metrological characteristics are completely determined by this unit hardware function which represents a transient re...
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
International Nuclear Information System (INIS)
Aslan İsmail
2014-01-01
The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 (2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic, trigonometric and rational, which have not been reported before. (general)
Extended Holomorphic Anomaly in Gauge Theory
Krefl, Daniel
2011-01-01
The partition function of an N=2 gauge theory in the Omega-background satisfies, for generic value of the parameter beta=-eps_1/eps_2, the, in general extended, but otherwise beta-independent, holomorphic anomaly equation of special geometry. Modularity together with the (beta-dependent) gap structure at the various singular loci in the moduli space completely fixes the holomorphic ambiguity, also when the extension is non-trivial. In some cases, the theory at the orbifold radius, corresponding to beta=2, can be identified with an "orientifold" of the theory at beta=1. The various connections give hints for embedding the structure into the topological string.
International Nuclear Information System (INIS)
Laenen, E.
1995-01-01
We propose a new evolution equation for the gluon density relevant for the region of small x B . It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multigluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed α s . We find that the effects of multigluon correlations on the deep-inelastic structure function are small. (orig.)
Integral equation hierarchy for continuum percolation
International Nuclear Information System (INIS)
Given, J.A.
1988-01-01
In this thesis a projection operator technique is presented that yields hierarchies of integral equations satisfied exactly by the n-point connectedness functions in a continuum version of the site-bond percolation problem. The n-point connectedness functions carry the same structural information for a percolation problem as then-point correlation functions do for a thermal problem. This method extends the Potts model mapping of Fortuin and Kastelyn to the continuum by exploiting an s-state generalization of the Widom-Rowlinson model, a continuum model for phase separation. The projection operator technique is used to produce an integral equation hierarchy for percolation similar to the Born-Green heirarchy. The Kirkwood superposition approximation (SA) is extended to percolation in order to close this hierarchy and yield a nonlinear integral equation for the two-point connectedness function. The fact that this function, in the SA, is the analytic continuation to negative density of the two-point correlation function in a corresponding thermal problem is discussed. The BGY equation for percolation is solved numerically, both by an expansion in powers of the density, and by an iterative technique due to Kirkwood. It is argued both analytically and numerically, that the BYG equation for percolation, unlike its thermal counterpart, shows non-classical critical behavior, with η = 1 and γ = 0.05 ± .1. Finally a sequence of refinements to the superposition approximations based in the theory of fluids by Rice and Lekner is discussed
Extending juvenility in grasses
Energy Technology Data Exchange (ETDEWEB)
Kaeppler, Shawn; de Leon Gatti, Natalia; Foerster, Jillian
2017-04-11
The present invention relates to compositions and methods for modulating the juvenile to adult developmental growth transition in plants, such as grasses (e.g. maize). In particular, the invention provides methods for enhancing agronomic properties in plants by modulating expression of GRMZM2G362718, GRMZM2G096016, or homologs thereof. Modulation of expression of one or more additional genes which affect juvenile to adult developmental growth transition such as Glossy15 or Cg1, in conjunction with such modulation of expression is also contemplated. Nucleic acid constructs for down-regulation of GRMZM2G362718 and/or GRMZM2G096016 are also contemplated, as are transgenic plants and products produced there from, that demonstrate altered, such as extended juvenile growth, and display associated phenotypes such as enhanced yield, improved digestibility, and increased disease resistance. Plants described herein may be used, for example, as improved forage or feed crops or in biofuel production.
International Nuclear Information System (INIS)
Goddard, Peter
1990-01-01
The algebra of the group of conformal transformations in two dimensions consists of two commuting copies of the Virasoro algebra. In many mathematical and physical contexts, the representations of ν which are relevant satisfy two conditions: they are unitary and they have the ''positive energy'' property that L o is bounded below. In an irreducible unitary representation the central element c takes a fixed real value. In physical contexts, the value of c is a characteristic of a theory. If c < 1, it turns out that the conformal algebra is sufficient to ''solve'' the theory, in the sense of relating the calculation of the infinite set of physically interesting quantities to a finite subset which can be handled in principle. For c ≥ 1, this is no longer the case for the algebra alone and one needs some sort of extended conformal algebra, such as the superconformal algebra. It is these algebras that this paper aims at addressing. (author)
Extended Poisson Exponential Distribution
Directory of Open Access Journals (Sweden)
Anum Fatima
2015-09-01
Full Text Available A new mixture of Modified Exponential (ME and Poisson distribution has been introduced in this paper. Taking the Maximum of Modified Exponential random variable when the sample size follows a zero truncated Poisson distribution we have derived the new distribution, named as Extended Poisson Exponential distribution. This distribution possesses increasing and decreasing failure rates. The Poisson-Exponential, Modified Exponential and Exponential distributions are special cases of this distribution. We have also investigated some mathematical properties of the distribution along with Information entropies and Order statistics of the distribution. The estimation of parameters has been obtained using the Maximum Likelihood Estimation procedure. Finally we have illustrated a real data application of our distribution.
International Nuclear Information System (INIS)
Bruyere, M.; Vallee, A.; Collette, C.
1986-09-01
Extended fuel cycle length and burnup are currently offered by Framatome and Fragema in order to satisfy the needs of the utilities in terms of fuel cycle cost and of overall systems cost optimization. We intend to point out the consequences of an increased fuel cycle length and burnup on reactor safety, in order to determine whether the bounding safety analyses presented in the Safety Analysis Report are applicable and to evaluate the effect on plant licensing. This paper presents the results of this examination. The first part indicates the consequences of increased fuel cycle length and burnup on the nuclear data used in the bounding accident analyses. In the second part of this paper, the required safety reanalyses are presented and the impact on the safety margins of different fuel management strategies is examined. In addition, systems modifications which can be required are indicated
Manca, V.; Salibra, A.; Scollo, Giuseppe
1990-01-01
Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either
Alternative equations of gravitation
International Nuclear Information System (INIS)
Pinto Neto, N.
1983-01-01
It is shown, trough a new formalism, that the quantum fluctuation effects of the gravitational field in Einstein's equations are analogs to the effects of a continuum medium in Maxwell's Electrodynamics. Following, a real example of the applications of these equations is studied. Qunatum fluctuations effects as perturbation sources in Minkowski and Friedmann Universes are examined. (L.C.) [pt
Energy Technology Data Exchange (ETDEWEB)
Yagi, M. [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Horton, W. [Texas Univ., Austin, TX (United States). Inst. for Fusion Studies
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite {beta} that we solve the perpendicular component of Ohm`s law to conserve the physical energy while ensuring the relation {del} {center_dot} j = 0.
International Nuclear Information System (INIS)
Yagi, M.; Horton, W.
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that we solve the perpendicular component of Ohm's law to conserve the physical energy while ensuring the relation ∇ · j = 0
International Nuclear Information System (INIS)
Yagi, M.; Horton, W.
1994-01-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that the perpendicular component of Ohm's law be solved to ensure ∇·j=0 for energy conservation
African Journals Online (AJOL)
The currently proposed model compaction equation was derived from data sourced from the. Niger Delta and it relates porosity to depth for sandstones under hydrostatic pressure condition. The equation is useful in predicting porosity and compaction trend in hydrostatic sands of the. Niger Delta. GEOLOGICAL SETTING OF ...
M. Hazewinkel (Michiel)
1995-01-01
textabstractDedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an
The generalized Fermat equation
Beukers, F.
2006-01-01
This article will be devoted to generalisations of Fermat’s equation xn + yn = zn. Very soon after the Wiles and Taylor proof of Fermat’s Last Theorem, it was wondered what would happen if the exponents in the three term equation would be chosen differently. Or if coefficients other than 1 would
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
Solutions of hyperbolic equations with the CIP-BS method
International Nuclear Information System (INIS)
Utsumi, Takayuki; Koga, James; Yamagiwa, Mitsuru; Yabe, Takashi; Aoki, Takayuki
2004-01-01
In this paper, we show that a new numerical method, the Constrained Interpolation Profile - Basis Set (CIP-BS) method, can solve general hyperbolic equations efficiently. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution owing to the constraints from the spatial derivatives of the master equations. Then, introducing scalar products, the linear and nonlinear partial differential equations are uniquely reduced to the ordinary differential equations for values and spatial derivatives at the grid points. The method gives stable, less diffusive, and accurate results. It is successfully applied to the continuity equation, the Burgers equation, the Korteweg-de Vries equation, and one-dimensional shock tube problems. (author)
A new formulation of equations of compressible fluids by analogy with Maxwell's equations
International Nuclear Information System (INIS)
Kambe, Tsutomu
2010-01-01
A compressible ideal fluid is governed by Euler's equation of motion and equations of continuity, entropy and vorticity. This system can be reformulated in a form analogous to that of electromagnetism governed by Maxwell's equations with source terms. The vorticity plays the role of magnetic field, while the velocity field plays the part of a vector potential and the enthalpy (of isentropic flows) plays the part of a scalar potential in electromagnetism. The evolution of source terms of fluid Maxwell equations is determined by solving the equations of motion and continuity. The equation of sound waves can be derived from this formulation, where time evolution of the sound source is determined by the equation of motion. The theory of vortex sound of aeroacoustics is included in this formulation. It is remarkable that the forces acting on a point mass moving in a velocity field of an inviscid fluid are analogous in their form to the electric force and Lorentz force in electromagnetism. The significance of the reformulation is interpreted by examples taken from fluid mechanics. This formulation can be extended to viscous fluids without difficulty. The Maxwell-type equations are unchanged by the viscosity effect, although the source terms have additional terms due to viscosities.
Invariant quantum mechanical equations of motion
Energy Technology Data Exchange (ETDEWEB)
Wigner, E. P. [Princeton University, Princeton, NJ (United States)
1963-01-15
One of the last few years’ most important developments in theoretical physics is the recognition that it is useful to extend to complex numbers the definition domain of intrinsically real variables, such as energy or angular momentum. This leads one to review many subjects which were considered to be closed. It should not have surprised me, therefore, when Dr. Salam asked me to report, at this seminar, on equations for elementary particles which are not believed to exist in nature, such as particles with imaginary mass. Even though the equations which describe such particles will play no role in the theory as long as the variables such as energy or angular momentum have physically meaningful values, that is, as long as they are real, they may play a significant role when the definition domain of these variables is extended.
Dynamics of electrically charged extended bodies: classical and quantum systems
International Nuclear Information System (INIS)
Aaberge, T.
1987-01-01
The author present generalizations of classical mechanics and quantum mechanics that make it possible to describe N charged extended bodies.In particular, we are able to write down a set of coupled equations for the system of N bodies plus field. The theory is based on a theory for the description of N charged chemical fluid components
The extended (G/G)-expansion method and travelling wave ...
Indian Academy of Sciences (India)
Home; Journals; Pramana – Journal of Physics; Volume 82; Issue 6. The extended (′/)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity. Zaiyun Zhang Jianhua Huang Juan Zhong Sha-Sha Dou Jiao Liu Dan Peng Ting Gao. Research Articles ...
The extended (G/G)-expansion method and travelling wave ...
Indian Academy of Sciences (India)
In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law non-linearity by the extended (′/)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters.
An extended rational thermodynamics model for surface excess fluxes
Sagis, L.M.C.
2012-01-01
In this paper, we derive constitutive equations for the surface excess fluxes in multiphase systems, in the context of an extended rational thermodynamics formalism. This formalism allows us to derive Maxwell–Cattaneo type constitutive laws for the surface extra stress tensor, the surface thermal
Multiphase averaging of periodic soliton equations
International Nuclear Information System (INIS)
Forest, M.G.
1979-01-01
The multiphase averaging of periodic soliton equations is considered. Particular attention is given to the periodic sine-Gordon and Korteweg-deVries (KdV) equations. The periodic sine-Gordon equation and its associated inverse spectral theory are analyzed, including a discussion of the spectral representations of exact, N-phase sine-Gordon solutions. The emphasis is on physical characteristics of the periodic waves, with a motivation from the well-known whole-line solitons. A canonical Hamiltonian approach for the modulational theory of N-phase waves is prescribed. A concrete illustration of this averaging method is provided with the periodic sine-Gordon equation; explicit averaging results are given only for the N = 1 case, laying a foundation for a more thorough treatment of the general N-phase problem. For the KdV equation, very general results are given for multiphase averaging of the N-phase waves. The single-phase results of Whitham are extended to general N phases, and more importantly, an invariant representation in terms of Abelian differentials on a Riemann surface is provided. Several consequences of this invariant representation are deduced, including strong evidence for the Hamiltonian structure of N-phase modulational equations
Symbolic dynamics of the Lorenz equations
International Nuclear Information System (INIS)
Fang Hai-ping; Hao Bailin.
1994-07-01
The Lorenz equations are investigated in a wide range of parameters by using the method of symbolic dynamics. First, the systematics of stable periodic orbits in the Lorenz equations is compared with that of the one-dimensional cubic map, which shares the same discrete symmetry with the Lorenz model. The systematics is then ''corrected'' in such a way as to encompass all the known periodic windows of the Lorenz equations with only one exception. Second, in order to justify the above approach and to understand the exceptions, another 1D map with a discontinuity is extracted from an extension of the geometric Lorenz attractor and its symbolic dynamics is constructed. All this has to be done in light of symbolic dynamics of two-dimensional maps. Finally, symbolic dynamics for the actual Poincare return map of the Lorenz equations is constructed in a heuristic way. New periodic windows of the Lorenz equations and their parameters can be predicted from this symbolic dynamics in combination with the 1D cubic map. The extended geometric 2D Lorenz map and the 1D antisymmetric map with a discontinuity describe the topological aspects of the Lorenz equations to high accuracy. (author). 44 refs, 17 figs, 8 tabs
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Supersymmetric quasipotential equations
International Nuclear Information System (INIS)
Zaikov, R.P.
1981-01-01
A supersymmetric extension of the Logunov-Tavkhelidze quasipotential approach is suggested. The supersymmetric Bethe- Salpeter equation is an initial equation. The transition from the four-time to the two-time Green function is made in the super- center-of-mass system. The two-time Green function has no inverse function in the whole spinor space. The resolvent operator if found using the Majorana character of the spinor wave function. The supersymmetric quasipotential equation is written. The consideration is carried out in the framework of the theory of chiral scalar superfields [ru
Local instant conservation equations
International Nuclear Information System (INIS)
Delaje, Dzh.
1984-01-01
Local instant conservation equations for two-phase flow are derived. Derivation of the equation starts from the recording of integral laws of conservation for a fixed reference volume, containing both phases. Transformation of the laws, using the Leibniz rule and Gauss theory permits to obtain the sum of two integrals as to the volume and integral as to the surface. Integrals as to the volume result in local instant differential equations, in particular derivatives for each phase, and integrals as to the surface reflect local instant conditions of a jump on interface surface
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
Mechanics of extended masses in general relativity
International Nuclear Information System (INIS)
Harte, Abraham I
2012-01-01
The 'external' or 'Right' motion of extended bodies is studied in general relativity. Compact material objects of essentially arbitrary shape, spin, internal composition and velocity are allowed as long as there is no direct (non-gravitational) contact with other sources of stress-energy. Physically reasonable linear and angular momenta are proposed for such bodies and exact equations describing their evolution are derived. Changes in the momenta depend on a certain 'effective metric' that is closely related to a non-perturbative generalization of the Detweiler-Whiting R-field originally introduced in the self-force literature. If the effective metric inside a self-gravitating body can be adequately approximated by an appropriate power series, the instantaneous gravitational force and torque exerted on it is shown to be identical to the force and torque exerted on an appropriate test body moving in the effective metric. This result holds to all multipole orders. The only instantaneous effect of a body's self-field is to finitely renormalize the 'bare' multipole moments of its stress-energy tensor. The MiSaTaQuWa expression for the gravitational self-force is recovered as a simple application. A gravitational self-torque is obtained as well. Lastly, it is shown that the effective metric in which objects appear to move is approximately a solution to the vacuum Einstein equation if the physical metric is an approximate solution to Einstein's equation linearized about a vacuum background. (paper)
Sun's pole-equator flux differences
Energy Technology Data Exchange (ETDEWEB)
Belvedere, G [Istituto di Astronomia dell' Universita di Catania, Italy; Paterno, L [Osservatorio Astrofisico di Catania, Italy
1977-04-01
The possibility that large flux differences between the poles and the equator at the bottom of the solar convective zone are compatible with the small differences observed at the surface is studied. The consequences of increasing the depth of the convective zone due to overshooting are explored. A Boussinesq model is used for the convective zone and it is assumed that the interaction of the global convection with rotation is modelled through a convective flux coefficient whose perturbed part is proportional to the local Taylor number. The numerical integration of the equations of motion and energy shows that coexistence between large pole-equator flux differences at the bottom and small ones at the surface is possible if the solar convective zone extends to a depth of 0.4 R(Sun). The angular velocity distribution inside the convective zone is in agreement with the ..cap alpha omega..-dynamo theories of the solar cycle.
Singularities of Type-Q ABS Equations
Directory of Open Access Journals (Sweden)
James Atkinson
2011-07-01
Full Text Available The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Tsintsadze, Nodar L.; Tsintsadze, Levan N.
2008-01-01
A general derivation of the charging equation of a dust grain is presented, and indicated where and when it can be used. A problem of linear fluctuations of charges on the surface of the dust grain is discussed.
Equations For Rotary Transformers
Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.
1988-01-01
Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.
Problems in differential equations
Brenner, J L
2013-01-01
More than 900 problems and answers explore applications of differential equations to vibrations, electrical engineering, mechanics, and physics. Problem types include both routine and nonroutine, and stars indicate advanced problems. 1963 edition.
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
Nonlinear differential equations
International Nuclear Information System (INIS)
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
International Nuclear Information System (INIS)
El-Jaick, Lea Jaccoud; Figueiredo, Bartolomeu D.B.
2009-01-01
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schroedinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials. (author)
A novel numerical flux for the 3D Euler equations with general equation of state
Toro, Eleuterio F.
2015-09-30
Here we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on general meshes; (ii) we use a general equation of state; and (iii) we achieve high order of accuracy in both space and time through application of the semi-discrete ADER methodology on general meshes. The resulting methods are systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems. Formal high accuracy is assessed through convergence rates studies for schemes of up to 4th order of accuracy in both space and time on unstructured meshes.
Institute of Scientific and Technical Information of China (English)
张解放; 吴锋民
2002-01-01
We study an approach to constructing multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Jimbo-Miwa (JM) equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3+1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations. Starting from these linear and bilinear partial differential equations, some multiple soliton solutions for the (3+1)-dimensional JM equation are obtained by introducing a class of formal solutions.
Traveling solitary wave solutions to evolution equations with nonlinear terms of any order
International Nuclear Information System (INIS)
Feng Zhaosheng
2003-01-01
Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Structural Equations and Causation
Hall, Ned
2007-01-01
Structural equations have become increasingly popular in recent years as tools for understanding causation. But standard structural equations approaches to causation face deep problems. The most philosophically interesting of these consists in their failure to incorporate a distinction between default states of an object or system, and deviations therefrom. Exploring this problem, and how to fix it, helps to illuminate the central role this distinction plays in our causal thinking.
Equations of radiation hydrodynamics
International Nuclear Information System (INIS)
Mihalas, D.
1982-01-01
The purpose of this paper is to give an overview of the role of radiation in the transport of energy and momentum in a combined matter-radiation fluid. The transport equation for a moving radiating fluid is presented in both a fully Eulerian and a fully Lagrangian formulation, along with conservation equations describing the dynamics of the fluid. Special attention is paid to the problem of deriving equations that are mutually consistent in each frame, and between frames, to 0(v/c). A detailed analysis is made to show that in situations of broad interest, terms that are formally of 0(v/c) actually dominate the solution, demonstrating that it is esential (1) to pay scrupulous attention to the question of the frame dependence in formulating the equations; and (2) to solve the equations to 0(v/c) in quite general circumstances. These points are illustrated in the context of the nonequilibrium radiation diffusion limit, and a sketch of how the Lagrangian equations are to be solved will be presented
Quantum linear Boltzmann equation
International Nuclear Information System (INIS)
Vacchini, Bassano; Hornberger, Klaus
2009-01-01
We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then review important limiting forms of the quantum linear Boltzmann equation, such as the case of quantum Brownian motion and pure collisional decoherence, as well as the application to matter wave optics. Finally, we point to the incorporation of quantum degeneracies and self-interactions in the gas by relating the equation to the dynamic structure factor of the ambient medium, and we provide an extension of the equation to include internal degrees of freedom.
Covariant field equations in supergravity
Energy Technology Data Exchange (ETDEWEB)
Vanhecke, Bram [KU Leuven, Institute for Theoretical Physics, Leuven (Belgium); Ghent University, Faculty of Physics, Gent (Belgium); Proeyen, Antoine van [KU Leuven, Institute for Theoretical Physics, Leuven (Belgium)
2017-12-15
Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Covariant field equations in supergravity
International Nuclear Information System (INIS)
Vanhecke, Bram; Proeyen, Antoine van
2017-01-01
Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Differential Equation over Banach Algebra
Kleyn, Aleks
2018-01-01
In the book, I considered differential equations of order $1$ over Banach $D$-algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. In noncommutative Banach algebra, initial value problem for linear homogeneous equation has infinitely many solutions.
On the exact solutions of high order wave equations of KdV type (I)
Bulut, Hasan; Pandir, Yusuf; Baskonus, Haci Mehmet
2014-12-01
In this paper, by means of a proper transformation and symbolic computation, we study high order wave equations of KdV type (I). We obtained classification of exact solutions that contain soliton, rational, trigonometric and elliptic function solutions by using the extended trial equation method. As a result, the motivation of this paper is to utilize the extended trial equation method to explore new solutions of high order wave equation of KdV type (I). This method is confirmed by applying it to this kind of selected nonlinear equations.
Directory of Open Access Journals (Sweden)
M. Arshad
Full Text Available In this manuscript, we constructed different form of new exact solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations by utilizing the modified extended direct algebraic method. New exact traveling wave solutions for both equations are obtained in the form of soliton, periodic, bright, and dark solitary wave solutions. There are many applications of the present traveling wave solutions in physics and furthermore, a wide class of coupled nonlinear evolution equations can be solved by this method. Keywords: Traveling wave solutions, Elliptic solutions, Generalized coupled Zakharov–Kuznetsov equation, Dispersive long wave equation, Modified extended direct algebraic method
On the solvability of initial-value problems for nonlinear implicit difference equations
Directory of Open Access Journals (Sweden)
Ha Thi Ngoc Yen
2004-07-01
Full Text Available Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differential-algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems.
Poincaré-MacMillan Equations of Motion for a Nonlinear Nonholonomic Dynamical System
Amjad, Hussain; Syed Tauseef, Mohyud-Din; Ahmet, Yildirim
2012-03-01
MacMillan's equations are extended to Poincaré's formalism, and MacMillan's equations for nonlinear nonholonomic systems are obtained in terms of Poincaré parameters. The equivalence of the results obtained here with other forms of equations of motion is demonstrated. An illustrative example of the theory is provided as well.
ON ENTIRE SOLUTIONS OF TWO TYPES OF SYSTEMS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Lingyun GAO
2017-01-01
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential (difference) equations to the systems of differential-difference equations.
Regularity criteria for the 3D magneto-micropolar fluid equations via ...
Indian Academy of Sciences (India)
3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations. Keywords. Magneto-micropolar fluid equations; regularity criteria; direction of velocity. 2010 Mathematics Subject Classification. 35Q35, 76W05 ...
Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation
International Nuclear Information System (INIS)
Hernandez-Heredero, Rafael; Reyes, Enrique G
2009-01-01
We announce two new structures associated with the Camassa-Holm (CH) equation: a Lie algebra of nonlocal symmetries, and a Darboux transformation for this important equation, which we construct using only our symmetries. We also extend our results to the associated Camassa-Holm equation introduced by J Schiff (1998 Physica D 121 24-43). (fast track communication)
Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation
Energy Technology Data Exchange (ETDEWEB)
Hernandez-Heredero, Rafael [Departamento de Matematica Aplicada, EUIT de Telecomunicacion, Universidad Politecnica de Madrid, Campus Sur Ctra de Valencia Km. 7. 28031, Madrid (Spain); Reyes, Enrique G [Departamento de Matematica y Ciencia de la Computacion, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago (Chile)], E-mail: rafahh@euitt.upm.es, E-mail: ereyes@fermat.usach.cl
2009-05-08
We announce two new structures associated with the Camassa-Holm (CH) equation: a Lie algebra of nonlocal symmetries, and a Darboux transformation for this important equation, which we construct using only our symmetries. We also extend our results to the associated Camassa-Holm equation introduced by J Schiff (1998 Physica D 121 24-43). (fast track communication)
Irreducibility conditions for extended superfields
International Nuclear Information System (INIS)
Sokatchev, E.
1981-05-01
The irreducible supermultiplets contained in an extended superfield are presented as sets of covariant derivatives of the superfield. Differential irreducibility constraints are easily obtained from this decomposition. (author)
New Formulation of the Governing Equations for Analyzing Outrigger Structures
International Nuclear Information System (INIS)
Er, G.-K.
2010-01-01
In this paper, an easily comprehensible solution procedure is proposed for the analysis of outrigger-braced structures. The idea is based on the compatibility of the columns' axial deformation. The unknowns are selected to be the axial forces in the columns. The resulted governing equations and the equations for the optimum analysis of the outrigger locations are different from the conventional ones, but numerical analysis shows that the results obtained with the new equations are same as those obtained with conventional equations. The relations between the new equations and the conventional ones are also figured out. The new procedure of formulating the governing equations can be easily extended to more complicated cases of outrigger-braced structures.
Systems of evolution equations and the singular perturbation method
International Nuclear Information System (INIS)
Mika, J.
Several fundamental theorems are presented important for the solution of linear evolution equations in the Banach space. The algorithm is deduced extending the solution of the system of singularly perturbed evolution equations into an asymptotic series with respect to a small positive parameter. The asymptotic convergence is shown of an approximate solution to the accurate solution. Singularly perturbed evolution equations of the resonance type were analysed. The special role is considered of the asymptotic equivalence of P1 equations obtained as the first order approximation if the spherical harmonics method is applied to the linear Boltzmann equation, and the diffusion equations of the linear transport theory where the small parameter approaches zero. (J.B.)
Optimal control of stochastic difference Volterra equations an introduction
Shaikhet, Leonid
2015-01-01
This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Minimal length, Friedmann equations and maximum density
Energy Technology Data Exchange (ETDEWEB)
Awad, Adel [Center for Theoretical Physics, British University of Egypt,Sherouk City 11837, P.O. Box 43 (Egypt); Department of Physics, Faculty of Science, Ain Shams University,Cairo, 11566 (Egypt); Ali, Ahmed Farag [Centre for Fundamental Physics, Zewail City of Science and Technology,Sheikh Zayed, 12588, Giza (Egypt); Department of Physics, Faculty of Science, Benha University,Benha, 13518 (Egypt)
2014-06-16
Inspired by Jacobson’s thermodynamic approach, Cai et al. have shown the emergence of Friedmann equations from the first law of thermodynamics. We extend Akbar-Cai derivation http://dx.doi.org/10.1103/PhysRevD.75.084003 of Friedmann equations to accommodate a general entropy-area law. Studying the resulted Friedmann equations using a specific entropy-area law, which is motivated by the generalized uncertainty principle (GUP), reveals the existence of a maximum energy density closed to Planck density. Allowing for a general continuous pressure p(ρ,a) leads to bounded curvature invariants and a general nonsingular evolution. In this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective. The existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature k. As an example we study the evolution of the equation of state p=ωρ through its phase-space diagram to show the existence of a maximum energy which is reachable in a finite time.
Quadratic Diophantine equations
Andreescu, Titu
2015-01-01
This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Boussinesq evolution equations
DEFF Research Database (Denmark)
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...
Extending cosmology: the metric approach
Mendoza, S.
2012-01-01
Comment: 2012, Extending Cosmology: The Metric Approach, Open Questions in Cosmology; Review article for an Intech "Open questions in cosmology" book chapter (19 pages, 3 figures). Available from: http://www.intechopen.com/books/open-questions-in-cosmology/extending-cosmology-the-metric-approach
Extended cognition and epistemic luck
Carter, J.A.
2013-01-01
When extended cognition is extended into mainstream epistemology, an awkward tension arises when considering cases of environmental epistemic luck. Surprisingly, it is not at all clear how the mainstream verdict that agents lack knowledge in cases of environmental luck can be reconciled with
Equations of mathematical physics
Tikhonov, A N
2011-01-01
Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri
Iteration of adjoint equations
International Nuclear Information System (INIS)
Lewins, J.D.
1994-01-01
Adjoint functions are the basis of variational methods and now widely used for perturbation theory and its extension to higher order theory as used, for example, in modelling fuel burnup and optimization. In such models, the adjoint equation is to be solved in a critical system with an adjoint source distribution that is not zero but has special properties related to ratios of interest in critical systems. Consequently the methods of solving equations by iteration and accumulation are reviewed to show how conventional methods may be utilized in these circumstances with adequate accuracy. (author). 3 refs., 6 figs., 3 tabs
Systematic Equation Formulation
DEFF Research Database (Denmark)
Lindberg, Erik
2007-01-01
A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....
Partial differential equations
Agranovich, M S
2002-01-01
Mark Vishik's Partial Differential Equations seminar held at Moscow State University was one of the world's leading seminars in PDEs for over 40 years. This book celebrates Vishik's eightieth birthday. It comprises new results and survey papers written by many renowned specialists who actively participated over the years in Vishik's seminars. Contributions include original developments and methods in PDEs and related fields, such as mathematical physics, tomography, and symplectic geometry. Papers discuss linear and nonlinear equations, particularly linear elliptic problems in angles and gener
Li, Tatsien
2017-01-01
This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
Scale-invariant extended inflation
International Nuclear Information System (INIS)
Holman, R.; Kolb, E.W.; Vadas, S.L.; Wang, Y.
1991-01-01
We propose a model of extended inflation which makes use of the nonlinear realization of scale invariance involving the dilaton coupled to an inflaton field whose potential admits a metastable ground state. The resulting theory resembles the Jordan-Brans-Dicke version of extended inflation. However, quantum effects, in the form of the conformal anomaly, generate a mass for the dilaton, thus allowing our model to evade the problems of the original version of extended inflation. We show that extended inflation can occur for a wide range of inflaton potentials with no fine-tuning of dimensionless parameters required. Furthermore, we also find that it is quite natural for the extended-inflation period to be followed by an epoch of slow-rollover inflation as the dilaton settles down to the minimum of its induced potential
Analysis of wave equation in electromagnetic field by Proca equation
International Nuclear Information System (INIS)
Pamungkas, Oky Rio; Soeparmi; Cari
2017-01-01
This research is aimed to analyze wave equation for the electric and magnetic field, vector and scalar potential, and continuity equation using Proca equation. Then, also analyze comparison of the solution on Maxwell and Proca equation for scalar potential and electric field, both as a function of distance and constant wave number. (paper)
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Nonlinear electrostatic wave equations for magnetized plasmas - II
DEFF Research Database (Denmark)
Dysthe, K. B.; Mjølhus, E.; Pécseli, H. L.
1985-01-01
For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent (electrosta......For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent...... (electrostatic) cut-off implies that various cases must be considered separately, leading to equations with rather different properties. Various equations encountered previously in the literature are recovered as limiting cases....
On reductions of the discrete Kadomtsev-Petviashvili-type equations
Fu, Wei; Nijhoff, Frank W.
2017-12-01
The reduction by restricting the spectral parameters k and k\\prime on a generic algebraic curve of degree N is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for the generic positive integer N≥slant 2 is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations as special degenerations. Some new integrable lattice models such as the discrete Sawada-Kotera, Kaup-Kupershmidt and Hirota-Satsuma equations in extended form are given as examples within the framework.
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials
International Nuclear Information System (INIS)
Contreras-Astorga, Alonso; Schulze-Halberg, Axel
2014-01-01
We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)
The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials
Energy Technology Data Exchange (ETDEWEB)
Contreras-Astorga, Alonso, E-mail: aloncont@iun.edu; Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, E-mail: xbataxel@gmail.com [Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)
2014-10-15
We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)].
Analytical approach for the Floquet theory of delay differential equations.
Simmendinger, C; Wunderlin, A; Pelster, A
1999-05-01
We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.
Real gas equation-of-state capability at Sandia Livermore
International Nuclear Information System (INIS)
Clark, G.L.
1978-03-01
A library of FORTRAN routines is described which model the equations-of-state for gases commonly encountered in compressible gas dynamics applications. The present library includes thermodynamic properties packages as well as compressibility models, both for pure gases and mixtures. Tables are included to allow hand calculation of equation-of-state problems for the gases hydrogen, helium, neon, argon, oxygen, air, and nitrogen. Generally the tables extend to several thousand atmospheres, with temperatures from the cryogenic realm to 400 K
Application of the bifurcation method to the modified Boussinesq equation
Directory of Open Access Journals (Sweden)
Shaoyong Li
2014-08-01
Firstly, we give a property of the solutions of the equation, that is, if $1+u(x, t$ is a solution, so is $1-u(x, t$. Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.
Novel loop-like solitons for the generalized Vakhnenko equation
International Nuclear Information System (INIS)
Zhang Min; Ma Yu-Lan; Li Bang-Qing
2013-01-01
A non-traveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method. The solution includes an arbitrary function of an independent variable. Based on the solution, two hyperbolic functions are chosen to construct new solitons. Novel single-loop-like and double-loop-like solitons are found for the equation
On geometric approach to Lie symmetries of differential-difference equations
International Nuclear Information System (INIS)
Li Hongjing; Wang Dengshan; Wang Shikun; Wu Ke; Zhao Weizhong
2008-01-01
Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach
The relationship among the solutions of two auxiliary ordinary differential equations
International Nuclear Information System (INIS)
Liu Xiaoping; Liu Chunping
2009-01-01
In a recent article [Phys. Lett. A 356 (2006) 124], Sirendaoreji extended their auxiliary equation method by introducing a new auxiliary ordinary differential equation (NAODE) and its 14 solutions. Then the author studied some nonlinear evolution equations (NLEEs) and got more exact travelling wave solutions. In this paper, we will show that the 14 solutions of the NAODE are actually the same as the solutions obtained by original auxiliary equation method, and they are only different in the form.
Reduced equations for finite beta tearing modes in tokamaks
International Nuclear Information System (INIS)
Izzo, R.; Monticello, D.A.; DeLucia, J.; Park, W.; Ryu, C.M.
1985-01-01
The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have del x B = 0 satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio epsilon of the torus and for βapprox.epsilon or smaller. This is demonstrated by deriving a reduced set of MHD equations that are correct to fifth order in epsilon. These equations contain the exact equilibrium relation and, as such, can be used to find three-dimensional stellarator equilibria. In addition, if a subsidiary ordering in eta, the resistivity, is made, the equations of Glasser, Greene, and Johnson [Phys. Fluids 8, 875 (1967); 19, 567 (1967)] are recovered. This set of reduced equations has been coded by extending the initial value code hIlo [Phys. Fluids 26, 3066 (1983)]. Results obtained for both ideal and resistive linear stability from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. Good agreement is shown for both zero and finite-beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter
Reduced equations for finite beta tearing modes in tokamaks
International Nuclear Information System (INIS)
Izzo, R.; Monticello, D.A.; DeLucia, J.; Park, W.; Ryu, C.M.
1984-10-01
The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have nabla vector.B vector = O satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio epsilon of the torus, and for β approx. epsilon or smaller. We demonstrate this by deriving a reduced set of MHD equations that are correct to 5th order in epsilon. These equations contain the exact equilibrium relation and as such can be used to find 3-D stellarator equilibria. In addition, if a subsidiary ordering in eta, the resistivity, is made, the equations of Glasser, Greene, and Johnson are recovered. This set of reduced equations has been coded by extending the initial value code, HILO. Results obtained, for both ideal and resistive linear stability, from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. The agreement is shown to be excellent for both zero and finite beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter
Indian Academy of Sciences (India)
The Raychaudhuri equation is central to the understanding of gravitational attraction in ... of K Gödel on the ideas of shear and vorticity in cosmology (he defines the shear. (eq. (8) in [1]) .... which follows from the definition of the scale factor l.
Generalized reduced magnetohydrodynamic equations
International Nuclear Information System (INIS)
Kruger, S.E.
1999-01-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
Indian Academy of Sciences (India)
research, teaching and practice related to the analysis and design ... its variants, are present in a large number of ma- chines used in daily ... with advanced electronics, sensors, control systems and computing ... ted perfectly well with the rapidly developing comput- .... velopment of the Freudenstein equation using Figure 3.
Differential Equation of Equilibrium
African Journals Online (AJOL)
user
ABSTRACT. Analysis of underground circular cylindrical shell is carried out in this work. The forth order differential equation of equilibrium, comparable to that of beam on elastic foundation, was derived from static principles on the assumptions of P. L Pasternak. Laplace transformation was used to solve the governing ...
Equational binary decision diagrams
J.F. Groote (Jan Friso); J.C. van de Pol (Jaco)
2000-01-01
textabstractWe incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and
Directory of Open Access Journals (Sweden)
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Structural Equation Model Trees
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
ANTHROPOMETRIC PREDICTIVE EQUATIONS FOR ...
African Journals Online (AJOL)
Keywords: Anthropometry, Predictive Equations, Percentage Body Fat, Nigerian Women, Bioelectric Impedance ... such as Asians and Indians (Pranav et al., 2009), ... size (n) of at least 3o is adjudged as sufficient for the ..... of people, gender and age (Vogel eta/., 1984). .... Fish Sold at Ile-Ife Main Market, South West Nigeria.
Indian Academy of Sciences (India)
However, one can associate the term with any solution of nonlinear partial differential equations (PDEs) which (i) represents a wave of permanent form, (ii) is localized ... In the past several decades, many methods have been proposed for solving nonlinear PDEs, such as ... space–time fractional derivative form of eq. (1) and ...
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
Guiding center drift equations
International Nuclear Information System (INIS)
Boozer, A.H.
1979-03-01
The quations for particle guiding center drift orbits are given in a new magnetic coordinate system. This form of the equations not only separates the fast motion along the lines from the slow motion across, but also requires less information about the magnetic field than many other formulations of the problem
dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
in real-life situations, it is important to find their exact solutions. Further, in ... But only little work is done on the high-dimensional equations. .... Similarly, to determine the values of d and q, we balance the linear term of the lowest order in eq.
Stochastic nonlinear beam equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
Savoy, L. G.
1988-01-01
Describes a study of students' ability to balance equations. Answers to a test on this topic were analyzed to determine the level of understanding and processes used by the students. Presented is a method to teach this skill to high school chemistry students. (CW)
Viability of bull semen extended with commercial semen extender ...
African Journals Online (AJOL)
Andrea Raseona
stored at 24 °C. Sperm motility parameters, morphology, and viability were analysed ... body size, slow average daily weight gain, decreased fertility, extended .... were determined by counting a total of 200 spermatozoa per each stained slide.
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Quantum equations from Brownian motions
International Nuclear Information System (INIS)
Rajput, B.S.
2011-01-01
Classical Schrodinger and Dirac equations have been derived from Brownian motions of a particle, it has been shown that the classical Schrodinger equation can be transformed to usual Schrodinger Quantum equation on applying Heisenberg uncertainty principle between position and momentum while Dirac Quantum equation follows it's classical counter part on applying Heisenberg uncertainly principle between energy and time without applying any analytical continuation. (author)
The Camassa-Holm equation as an incompressible Euler equation: A geometric point of view
Gallouët, Thomas; Vialard, François-Xavier
2018-04-01
The group of diffeomorphisms of a compact manifold endowed with the L2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is a natural extension of the classical L2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa-Holm (CH) equation in one dimension. Geometrically, we present an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S1, solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on R2 which preserves a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler-Arnold equation for the Hdiv right-invariant metric are length minimizing geodesics for sufficiently short times.
Cosmological dynamics of extended chameleons
International Nuclear Information System (INIS)
Tamanini, Nicola; Wright, Matthew
2016-01-01
We investigate the cosmological dynamics of the recently proposed extended chameleon models at both background and linear perturbation levels. Dynamical systems techniques are employed to fully characterize the evolution of the universe at the largest distances, while structure formation is analysed at sub-horizon scales within the quasi-static approximation. The late time dynamical transition from dark matter to dark energy domination can be well described by almost all extended chameleon models considered, with no deviations from ΛCDM results at both background and perturbation levels. The results obtained in this work confirm the cosmological viability of extended chameleons as alternative dark energy models.
Extended asymptotic functions - some examples
International Nuclear Information System (INIS)
Todorov, T.D.
1981-01-01
Several examples of extended asymptotic functions of two variables are given. This type of asymptotic functions has been introduced as an extension of continuous ordinary functions. The presented examples are realizations of some Schwartz distributions delta(x), THETA(x), P(1/xsup(n)) and can be multiplied in the class of the asymptotic functions as opposed to the theory of Schwartz distributions. The examples illustrate the method of construction of extended asymptotic functions similar to the distributions. The set formed by the extended asymptotic functions is also considered. It is shown, that this set is not closed with respect to addition and multiplication
Cosmological dynamics of extended chameleons
Energy Technology Data Exchange (ETDEWEB)
Tamanini, Nicola [Institut de Physique Théorique, CEA-Saclay, CNRS UMR 3681, Université Paris-Saclay, F-91191 Gif-sur-Yvette (France); Wright, Matthew, E-mail: nicola.tamanini@cea.fr, E-mail: matthew.wright.13@ucl.ac.uk [Department of Mathematics, University College London, Gower Street, London, WC1E 6BT (United Kingdom)
2016-04-01
We investigate the cosmological dynamics of the recently proposed extended chameleon models at both background and linear perturbation levels. Dynamical systems techniques are employed to fully characterize the evolution of the universe at the largest distances, while structure formation is analysed at sub-horizon scales within the quasi-static approximation. The late time dynamical transition from dark matter to dark energy domination can be well described by almost all extended chameleon models considered, with no deviations from ΛCDM results at both background and perturbation levels. The results obtained in this work confirm the cosmological viability of extended chameleons as alternative dark energy models.
Elements of partial differential equations
Sneddon, Ian Naismith
1957-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Directory of Open Access Journals (Sweden)
Müller Ingo
1999-01-01
Full Text Available The Navier-Stokes-Fourier theory of viscous, heat-conducting fluids provides parabolic equations and thus predicts infinite pulse speeds. Naturally this feature has disqualified the theory for relativistic thermodynamics which must insist on finite speeds and, moreover, on speeds smaller than $c$. The attempts at a remedy have proved heuristically important for a new systematic type of thermodynamics: Extended thermodynamics. That new theory has symmetric hyperbolic field equations and thus it provides finite pulse speeds. Extended thermodynamics is a whole hierarchy of theories with an increasing number of fields when gradients and rates of thermodynamic processes become steeper and faster. The first stage in this hierarchy is the 14-field theory which may already be a useful tool for the relativist in many applications. The 14 fields -- and further fields -- are conveniently chosen from the moments of the kinetic theory of gases. The hierarchy is complete only when the number of fields tends to infinity. In that case the pulse speed of non-relativistic extended thermodynamics tends to infinity while the pulse speed of relativistic extended thermodynamics tends to $c$, the speed of light. In extended thermodynamics symmetric hyperbolicity -- and finite speeds -- are implied by the concavity of the entropy density. This is still true in relativistic thermodynamics for a privileged entropy density which is the entropy density of the rest frame for non-degenerate gases.
On generalized fractional vibration equation
International Nuclear Information System (INIS)
Dai, Hongzhe; Zheng, Zhibao; Wang, Wei
2017-01-01
Highlights: • The paper presents a generalized fractional vibration equation for arbitrary viscoelastically damped system. • Some classical vibration equations can be derived from the developed equation. • The analytic solution of developed equation is derived under some special cases. • The generalized equation is particularly useful for developing new fractional equivalent linearization method. - Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.
Equations of motion for a rotor blade, including gravity, pitch action and rotor speed variations
DEFF Research Database (Denmark)
Kallesøe, Bjarne Skovmose
2007-01-01
This paper extends Hodges-Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms...... in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade...
Teachers and Technology: Development of an Extended Theory of Planned Behavior
Teo, Timothy; Zhou, Mingming; Noyes, Jan
2016-01-01
This study tests the validity of an extended theory of planned behaviour (TPB) to explain teachers' intention to use technology for teaching and learning. Five hundred and ninety two participants completed a survey questionnaire measuring their responses to eight constructs which form an extended TPB. Using structural equation modelling, the…
Thermodynamic admissibility of the extended Pom-Pom model for branched polymers
Soulages, J.; Hütter, M.; Öttinger, H.C.
2006-01-01
The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M.
An extended integrable fractional-order KP soliton hierarchy
International Nuclear Information System (INIS)
Li Li
2011-01-01
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.
An extended integrable fractional-order KP soliton hierarchy
Energy Technology Data Exchange (ETDEWEB)
Li Li, E-mail: li07099@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2011-01-17
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
Solving Linear Differential Equations
Nguyen, K.A.; Put, M. van der
2010-01-01
The theme of this paper is to 'solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field K. Representations of semi-simple Lie algebras and differential Galo is theory are the main tools. The results extend
equateIRT: An R Package for IRT Test Equating
Directory of Open Access Journals (Sweden)
Michela Battauz
2015-12-01
Full Text Available The R package equateIRT implements item response theory (IRT methods for equating different forms composed of dichotomous items. In particular, the IRT models included are the three-parameter logistic model, the two-parameter logistic model, the one-parameter logistic model and the Rasch model. Forms can be equated when they present common items (direct equating or when they can be linked through a chain of forms that present common items in pairs (indirect or chain equating. When two forms can be equated through different paths, a single conversion can be obtained by averaging the equating coefficients. The package calculates direct and chain equating coefficients. The averaging of direct and chain coefficients that link the same two forms is performed through the bisector method. Furthermore, the package provides analytic standard errors of direct, chain and average equating coefficients.
Symmetric extendibility of quantum states
Nowakowski, Marcin L.
2015-01-01
Studies on symmetric extendibility of quantum states become especially important in a context of analysis of one-way quantum measures of entanglement, distilabillity and security of quantum protocols. In this paper we analyse composite systems containing a symmetric extendible part with a particular attention devoted to one-way security of such systems. Further, we introduce a new one-way monotone based on the best symmetric approximation of quantum state. We underpin those results with geome...
Topological defects in extended inflation
International Nuclear Information System (INIS)
Copeland, E.J.; Kolb, E.W.; Chicago Univ., IL; Liddle, A.R.
1990-04-01
We consider the production of topological defects, especially cosmic strings, in extended inflation models. In extended inflation, the Universe passes through a first-order phase transition via bubble percolation, which naturally allows defects to form at the end of inflation. The correlation length, which determines the number density of the defects, is related to the mean size of bubbles when they collide. This mechanism allows a natural combination of inflation and large-scale structure via cosmic strings. 18 refs
Quasi-extended asymptotic functions
International Nuclear Information System (INIS)
Todorov, T.D.
1979-01-01
The class F of ''quasi-extended asymptotic functions'' is introduced. It contains all extended asymptotic functions as well as some new asymptotic functions very similar to the Schwartz distributions. On the other hand, every two quasiextended asymptotic functions can be multiplied as opposed to the Schwartz distributions; in particular, the square delta 2 of an asymptotic function delta similar to Dirac's delta-function, is constructed as an example
Some problems with extended inflation
International Nuclear Information System (INIS)
Weinberg, E.J.
1989-01-01
The recently proposed extended inflation scenario is examined. Upper bounds on the Brans-Dicke parameter ω are obtained by requiring that the recovery from the supercooled regime be such that the presently observed Universe could have emerged. These bounds are well below the present-day experimental limits, implying that one must use models which have a potential to fix the present value of the Brans-Dicke-like scalar field. The implications for extended inflation in such models are discussed
Topological defects in extended inflation
International Nuclear Information System (INIS)
Copeland, E.J.; Kolb, E.W.; Liddle, A.R.
1990-01-01
We consider the production of topological defects, especially cosmic strings, in extended-inflation models. In extended inflation, the Universe passes through a first-order phase transition via bubble percolation, which naturally allows defects to form at the end of inflation. The correlation length, which determines the number density of the defects, is related to the mean size of the bubbles when they collide. This mechanism allows a natural combination of inflation and large-scale structure via cosmic strings
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Moh’d Khier Al-Srihin
2017-01-01
Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.
Periodicity and positivity of a class of fractional differential equations.
Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim
2016-01-01
Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.
Extended likelihood inference in reliability
International Nuclear Information System (INIS)
Martz, H.F. Jr.; Beckman, R.J.; Waller, R.A.
1978-10-01
Extended likelihood methods of inference are developed in which subjective information in the form of a prior distribution is combined with sampling results by means of an extended likelihood function. The extended likelihood function is standardized for use in obtaining extended likelihood intervals. Extended likelihood intervals are derived for the mean of a normal distribution with known variance, the failure-rate of an exponential distribution, and the parameter of a binomial distribution. Extended second-order likelihood methods are developed and used to solve several prediction problems associated with the exponential and binomial distributions. In particular, such quantities as the next failure-time, the number of failures in a given time period, and the time required to observe a given number of failures are predicted for the exponential model with a gamma prior distribution on the failure-rate. In addition, six types of life testing experiments are considered. For the binomial model with a beta prior distribution on the probability of nonsurvival, methods are obtained for predicting the number of nonsurvivors in a given sample size and for predicting the required sample size for observing a specified number of nonsurvivors. Examples illustrate each of the methods developed. Finally, comparisons are made with Bayesian intervals in those cases where these are known to exist
Solórzano, S.; Mendoza, M.; Succi, S.; Herrmann, H. J.
2018-01-01
We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretization, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one- and two-dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of three-dimensional open systems is also illustrated.
DEFF Research Database (Denmark)
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...... (EME)is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures...... of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent...
Classical Diophantine equations
1993-01-01
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...
Flavored quantum Boltzmann equations
International Nuclear Information System (INIS)
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean
2010-01-01
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
Causal electromagnetic interaction equations
International Nuclear Information System (INIS)
Zinoviev, Yury M.
2011-01-01
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Equations of multiparticle dynamics
International Nuclear Information System (INIS)
Chao, A.W.
1987-01-01
The description of the motion of charged-particle beams in an accelerator proceeds in steps of increasing complexity. The first step is to consider a single-particle picture in which the beam is represented as a collection on non-interacting test particles moving in a prescribed external electromagnetic field. Knowing the external field, it is then possible to calculate the beam motion to a high accuracy. The real beam consists of a large number of particles, typically 10 11 per beam bunch. It is sometimes inconvenient, or even impossible, to treat the real beam behavior using the single particle approach. One way to approach this problem is to supplement the single particle by another qualitatively different picture. The commonly used tools in accelerator physics for this purpose are the Vlasov and the Fokker-Planck equations. These equations assume smooth beam distributions and are therefore strictly valid in the limit of infinite number of micro-particles, each carrying an infinitesimal charge. The hope is that by studying the two extremes -- the single particle picture and the picture of smooth beam distributions -- we will be able to describe the behavior of our 10 11 -particle system. As mentioned, the most notable use of the smooth distribution picture is the study of collective beam instabilities. However, the purpose of this lecture is not to address this more advanced subject. Rather, it has the limited goal to familiarize the reader with the analytical tools, namely the Vlasov and the Fokker-Planck equations, as a preparation for dealing with the more advanced problems at later times. We will first derive these equations and then illustrate their applications by several examples which allow exact solutions
Electroweak evolution equations
International Nuclear Information System (INIS)
Ciafaloni, Paolo; Comelli, Denis
2005-01-01
Enlarging a previous analysis, where only fermions and transverse gauge bosons were taken into account, we write down infrared-collinear evolution equations for the Standard Model of electroweak interactions computing the full set of splitting functions. Due to the presence of double logs which are characteristic of electroweak interactions (Bloch-Nordsieck violation), new infrared singular splitting functions have to be introduced. We also include corrections related to the third generation Yukawa couplings
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Damped nonlinear Schrodinger equation
International Nuclear Information System (INIS)
Nicholson, D.R.; Goldman, M.V.
1976-01-01
High frequency electrostatic plasma oscillations described by the nonlinear Schrodinger equation in the presence of damping, collisional or Landau, are considered. At early times, Landau damping of an initial soliton profile results in a broader, but smaller amplitude soliton, while collisional damping reduces the soliton size everywhere; soliton speeds at early times are unchanged by either kind of damping. For collisional damping, soliton speeds are unchanged for all time
Fun with Differential Equations
Indian Academy of Sciences (India)
IAS Admin
tion of ® with ¼=2. One can use the uniqueness of solutions of differential equations to prove the addition formulae for sin(t1 +t2), etc. But instead of continuing with this thought process, let us do something more interesting. Now we shall consider another system. Fix 0 < < 1. I am looking for three real-valued functions x(t), ...
Mathematics and Maxwell's equations
International Nuclear Information System (INIS)
Boozer, Allen H
2010-01-01
The universality of mathematics and Maxwell's equations is not shared by specific plasma models. Computations become more reliable, efficient and transparent if specific plasma models are used to obtain only the information that would otherwise be missing. Constraints of high universality, such as those from mathematics and Maxwell's equations, can be obscured or lost by integrated computations. Recognition of subtle constraints of high universality is important for (1) focusing the design of control systems for magnetic field errors in tokamaks from perturbations that have little effect on the plasma to those that do, (2) clarifying the limits of applicability to astrophysics of computations of magnetic reconnection in fields that have a double periodicity or have B-vector =0 on a surface, as in a Harris sheet. Both require a degree of symmetry not expected in natural systems. Mathematics and Maxwell's equations imply that neighboring magnetic field lines characteristically separate exponentially with distance along a line. This remarkably universal phenomenon has been largely ignored, though it defines a trigger for reconnection through a critical magnitude of exponentiation. These and other examples of the importance of making distinctions and understanding constraints of high universality are explained.
Directory of Open Access Journals (Sweden)
M. Paul Gough
2008-07-01
Full Text Available LandauerÃ¢Â€Â™s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the Ã¢Â€Â˜Why now?Ã¢Â€Â™ question we wonder Ã¢Â€Â˜What next?Ã¢Â€Â™ as we expect the information equation of state to tend towards w = 0 in the future.c
Generalized reduced MHD equations
International Nuclear Information System (INIS)
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson
Kinetic equation for spin-polarized plasmas
International Nuclear Information System (INIS)
Cowley, S.C.; Kulsrud, R.M.; Valeo, E.
1984-07-01
The usual kinetic description of a plasma is extended to include variables to describe the spin. The distribution function, over phase-space and the new spin variables, provides a sufficient description of a spin-polarized plasma. The evolution equation for the distribution function is given. The equations derived are used to calculate depolarization due to four processes, inhomogeneous fields, collisions, collisions in inhomogeneous fields, and waves. It is found that depolarization by field inhomogeneity on scales large compared with the gyroradius is totally negligible. The same is true for collisional depolarization. Collisions in inhomogeneous fields yield a depolarization rate of order 10 -4 S -1 for deuterons and a negligible rate for tritons in a typical fusion reactor design. This is still sufficiently small on reactor time scales. However, small amplitude magnetic fluctuations (of order one gauss) resonant with the spin precession frequency can lead to significant depolarization (depolarises triton in ten seconds and deuteron in a hundred seconds.)
Interference-exact radiative transfer equation
DEFF Research Database (Denmark)
Partanen, Mikko; Haÿrynen, Teppo; Oksanen, Jani
2017-01-01
Maxwell's equations with stochastic or quantum optical source terms accounting for the quantum nature of light. We show that both the nonlocal wave and local particle features associated with interference and emission of propagating fields in stratified geometries can be fully captured by local damping...... and scattering coefficients derived from the recently introduced quantized fluctuational electrodynamics (QFED) framework. In addition to describing the nonlocal optical interference processes as local directionally resolved effects, this allows reformulating the well known and widely used radiative transfer...... equation (RTE) as a physically transparent interference-exact model that extends the useful range of computationally efficient and quantum optically accurate interference-aware optical models from simple structures to full optical devices....
Determination of source terms in a degenerate parabolic equation
International Nuclear Information System (INIS)
Cannarsa, P; Tort, J; Yamamoto, M
2010-01-01
In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation
Generalized differential transform method to differential-difference equation
International Nuclear Information System (INIS)
Zou Li; Wang Zhen; Zong Zhi
2009-01-01
In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Pade technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.
Differential equations and integrable models: the SU(3) case
International Nuclear Information System (INIS)
Dorey, Patrick; Tateo, Roberto
2000-01-01
We exhibit a relationship between the massless a 2 (2) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schroedinger equation. This forms part of a more general correspondence involving A 2 -related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators phi 12 , phi 21 and phi 15 . This is checked against previous results obtained using the thermodynamic Bethe ansatz
International Nuclear Information System (INIS)
Zhi Hongyan
2009-01-01
In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.
Transformation of nonlinear discrete-time system into the extended observer form
Kaparin, V.; Kotta, Ü.
2018-04-01
The paper addresses the problem of transforming discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input-output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.
Extension of Gibbs-Duhem equation including influences of external fields
Guangze, Han; Jianjia, Meng
2018-03-01
Gibbs-Duhem equation is one of the fundamental equations in thermodynamics, which describes the relation among changes in temperature, pressure and chemical potential. Thermodynamic system can be affected by external field, and this effect should be revealed by thermodynamic equations. Based on energy postulate and the first law of thermodynamics, the differential equation of internal energy is extended to include the properties of external fields. Then, with homogeneous function theorem and a redefinition of Gibbs energy, a generalized Gibbs-Duhem equation with influences of external fields is derived. As a demonstration of the application of this generalized equation, the influences of temperature and external electric field on surface tension, surface adsorption controlled by external electric field, and the derivation of a generalized chemical potential expression are discussed, which show that the extended Gibbs-Duhem equation developed in this paper is capable to capture the influences of external fields on a thermodynamic system.
Computing generalized Langevin equations and generalized Fokker-Planck equations.
Darve, Eric; Solomon, Jose; Kia, Amirali
2009-07-07
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.
FMTLxLyLz DIMENSIONAL EQUAT DIMENSIONAL EQUATION ...
African Journals Online (AJOL)
eobe
plant made of 12mm thick steel plate was used in de steel plate ... water treatment plant. ... ameters affecting filtration processes were used to derive an equation usin ..... system. However, in deriving the equation onl terms are incorporated.
Reduction operators of Burgers equation.
Pocheketa, Oleksandr A; Popovych, Roman O
2013-02-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special "no-go" case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf-Cole transformation to a parameterized family of Lie reductions of the linear heat equation.
Directory of Open Access Journals (Sweden)
Domingues M. O.
2013-12-01
Full Text Available We present a new adaptive multiresoltion method for the numerical simulation of ideal magnetohydrodynamics. The governing equations, i.e., the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time discretization uses a compact Runge–Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution.
Extended hamiltonian formalism and Lorentz-violating lagrangians
Directory of Open Access Journals (Sweden)
Don Colladay
2017-09-01
Full Text Available A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler–Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.
Generalized variational formulations for extended exponentially fractional integral
Directory of Open Access Journals (Sweden)
Zuo-Jun Wang
2016-01-01
Full Text Available Recently, the fractional variational principles as well as their applications yield a special attention. For a fractional variational problem based on different types of fractional integral and derivatives operators, corresponding fractional Lagrangian and Hamiltonian formulation and relevant Euler–Lagrange type equations are already presented by scholars. The formulations of fractional variational principles still can be developed more. We make an attempt to generalize the formulations for fractional variational principles. As a result we obtain generalized and complementary fractional variational formulations for extended exponentially fractional integral for example and corresponding Euler–Lagrange equations. Two illustrative examples are presented. It is observed that the formulations are in exact agreement with the Euler–Lagrange equations.
Unraveling gravity beyond Einstein with extended test bodies
International Nuclear Information System (INIS)
Puetzfeld, Dirk; Obukhov, Yuri N.
2013-01-01
The motion of test bodies in gravity is tightly linked to the conservation laws. This well-known fact in the context of General Relativity is also valid for gravitational theories which go beyond Einstein's theory. Here we derive the equations of motion for test bodies for a very large class of gravitational theories with a general nonminimal coupling to matter. These equations form the basis for future systematic tests of alternative gravity theories. Our treatment is covariant and generalizes the classic Mathisson–Papapetrou–Dixon result for spinning (extended) test bodies. The equations of motion for structureless test bodies turn out to be surprisingly simple, despite the very general nature of the theories considered.
Extended hamiltonian formalism and Lorentz-violating lagrangians
Colladay, Don
2017-09-01
A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler-Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.
Bacteriospermia in extended porcine semen.
Althouse, Gary C; Lu, Kristina G
2005-01-15
Bacteriospermia is a frequent finding in freshly extended porcine semen and can result in detrimental effects on semen quality and longevity if left uncontrolled. The primary source of bacterial contamination is the boar. Other sources that have been identified include environment, personnel, and the water used for extender preparation. A 1-year retrospective study was performed on submissions of extended porcine semen for routine quality control bacteriological screening at the University of Pennsylvania. Out of 250 sample submissions, 78 (31.2%) tested positive for bacterial contamination. The most popular contaminants included Enterococcus spp. (20.5%), Stenotrophomonas maltophilia (15.4%), Alcaligenes xylosoxidans (10.3%), Serratia marcescens (10.3%), Acinetobacter lwoffi (7.7%), Escherichia coli (6.4%), Pseudomonas spp. (6.4%), and others (23.0%). Prudent individual hygiene, good overall sanitation, and regular monitoring can contribute greatly in controlling bacterial load. Strategies that incorporate temperature-dependent bacterial growth and hyperthermic augmentation of antimicrobial activity are valuable for effective control of susceptible bacterial loads. Aminoglycosides remain the most popular antimicrobial class used in porcine semen extenders, with beta-lactam and lincosamide use increasing. With the advent of more novel antimicrobial selection and semen extender compositions in swine, prudent application and understanding of in vitro pharmacodynamics are becoming paramount to industry success in the use of this breeding modality.
Extending Newton's law from nonlocal-in-time kinetic energy
International Nuclear Information System (INIS)
Suykens, J.A.K.
2009-01-01
We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics
Auxiliary equation method for solving nonlinear partial differential equations
International Nuclear Information System (INIS)
Sirendaoreji,; Jiong, Sun
2003-01-01
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
Jiang, Yanlin; von Davier, Alina A.; Chen, Haiwen
2012-01-01
This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating…
Extended Year, Extended Contracts: Increasing Teacher Salary Options.
Gandara, Patricia
1992-01-01
Reports on an attempt to raise teacher salaries through an extended contract made possible through year-round school schedules. Teacher satisfaction with the 1987 experiment in three California schools (the Orchard Plan) has been high. Elements that have contributed to job satisfaction are discussed. (SLD)
Asymptotics for Estimating Equations in Hidden Markov Models
DEFF Research Database (Denmark)
Hansen, Jørgen Vinsløv; Jensen, Jens Ledet
Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore a class of estimating equations is considered...
Spinodal equation of state for rutile TiO2
DEFF Research Database (Denmark)
Francisco, E.; Bermejo, M.; Baonza, V. Garcia
2003-01-01
We present a general computational scheme to extend the spinodal equation of state [Garcia Baonza , Phys. Rev. B 51, 28 (1995)] to the interpretation of the cell parameters response to hydrostatic pressure in orthogonal lattices. As an important example, we analyze the pressure (p)-volume (V)-tem...
The roots to an equation in particle slowing down theory
International Nuclear Information System (INIS)
Sjoestrand, N.G.
1979-08-01
Previous work on the roots to an equation arising in studies on anisotropic neutron scattering has been extended to include parameter values of interest for a problem put forward by M.M.R. Williams. Detailed numerical results are given. (author)
Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity
Directory of Open Access Journals (Sweden)
Hocine Ayadi
2018-02-01
Full Text Available In this article, we prove the existence and the regularity of distributional solutions for a class of nonlinear anisotropic elliptic equations with $p_i(x$ growth conditions, degenerate coercivity and $L^{m(\\cdot}$ data, with $m(\\cdot$ being small, in appropriate Lebesgue-Sobolev spaces with variable exponents. The obtained results extend some existing ones [8,10].
Langevin equation with the deterministic algebraically correlated noise
International Nuclear Information System (INIS)
Ploszajczak, M.; Srokowski, T.
1995-01-01
Stochastic differential equations with the deterministic, algebraically correlated noise are solved for a few model problems. The chaotic force with both exponential and algebraic temporal correlations is generated by the adjoined extended Sinai billiard with periodic boundary conditions. The correspondence between the autocorrelation function for the chaotic force and both the survival probability and the asymptotic energy distribution of escaping particles is found. (author)
Dimerization of Carboxylic Acids: An Equation of State Approach
DEFF Research Database (Denmark)
Tsivintzelis, Ioannis; Kontogeorgis, Georgios; Panayiotou, Costas
2017-01-01
The association term of the nonrandom hydrogen bonding theory, which is an equation of state model, is extended to describe the dimerization of carboxylic acids in binary mixtures with inert solvents and in systems of two different acids. Subsequently, the model is applied to describe the excess...
Using "Tracker" to Prove the Simple Harmonic Motion Equation
Kinchin, John
2016-01-01
Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.
Extended cognition in science communication.
Ludwig, David
2014-11-01
The aim of this article is to propose a methodological externalism that takes knowledge about science to be partly constituted by the environment. My starting point is the debate about extended cognition in contemporary philosophy and cognitive science. Externalists claim that human cognition extends beyond the brain and can be partly constituted by external devices. First, I show that most studies of public knowledge about science are based on an internalist framework that excludes the environment we usually utilize to make sense of science and does not allow the possibility of extended knowledge. In a second step, I argue that science communication studies should adopt a methodological externalism and accept that knowledge about science can be partly realized by external information resources such as Wikipedia. © The Author(s) 2013.
Partial differential equations
Levine, Harold
1997-01-01
The subject matter, partial differential equations (PDEs), has a long history (dating from the 18th century) and an active contemporary phase. An early phase (with a separate focus on taut string vibrations and heat flow through solid bodies) stimulated developments of great importance for mathematical analysis, such as a wider concept of functions and integration and the existence of trigonometric or Fourier series representations. The direct relevance of PDEs to all manner of mathematical, physical and technical problems continues. This book presents a reasonably broad introductory account of the subject, with due regard for analytical detail, applications and historical matters.
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Elliptic partial differential equations
Han, Qing
2011-01-01
Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things abo
dimensional Jaulent–Miodek equations
Indian Academy of Sciences (India)
(2+1)-dimensional Jaulent–Miodek equation; the first integral method; kinks; ... and effective method for solving nonlinear partial differential equations which can ... of the method employed and exact kink and soliton solutions are constructed ...
Equationally Noetherian property of Ershov algebras
Dvorzhetskiy, Yuriy
2014-01-01
This article is about equationally Noetherian and weak equationally Noetherian property of Ershov algebras. Here we show two canonical forms of the system of equations over Ershov algebras and two criteria of equationally Noetherian and weak equationally Noetherian properties.
Exclusion Bounds for Extended Anyons
Larson, Simon; Lundholm, Douglas
2018-01-01
We introduce a rigorous approach to the many-body spectral theory of extended anyons, that is quantum particles confined to two dimensions that interact via attached magnetic fluxes of finite extent. Our main results are many-body magnetic Hardy inequalities and local exclusion principles for these particles, leading to estimates for the ground-state energy of the anyon gas over the full range of the parameters. This brings out further non-trivial aspects in the dependence on the anyonic statistics parameter, and also gives improvements in the ideal (non-extended) case.
Adjustable extender for instrument module
International Nuclear Information System (INIS)
Sevec, J.B.; Stein, A.D.
1975-01-01
A blank extender module used to mount an instrument module in front of its console for repair or test purposes has been equipped with a rotatable mount and means for locking the mount at various angles of rotation for easy accessibility. The rotatable mount includes a horizontal conduit supported by bearings within the blank module. The conduit is spring-biased in a retracted position within the blank module and in this position a small gear mounted on the conduit periphery is locked by a fixed pawl. The conduit and instrument mount can be pulled into an extended position with the gear clearing the pawl to permit rotation and adjustment of the instrument
Extended Lagrangian formalism for rheonomic systems with variable mass
Directory of Open Access Journals (Sweden)
Mušicki Đorđe
2017-01-01
Full Text Available In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Mušicki, 2004, which began with the modification of the mechanics of such systems (V. Vujičić, 1987, is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.
Proceedings of the workshop on nonlinear MHD and extended MHD
International Nuclear Information System (INIS)
1998-01-01
Nonlinear MHD simulations have proven their value in interpreting experimental results over the years. As magnetic fusion experiments reach higher performance regimes, more sophisticated experimental diagnostics coupled with ever expanding computer capabilities have increased both the need for and the feasibility of nonlinear global simulations using models more realistic than regular ideal and resistive MHD. Such extended-MHD nonlinear simulations have already begun to produce useful results. These studies are expected to lead to ever more comprehensive simulation models in the future and to play a vital role in fully understanding fusion plasmas. Topics include the following: (1) current state of nonlinear MHD and extended-MHD simulations; (2) comparisons to experimental data; (3) discussions between experimentalists and theorists; (4) /equations for extended-MHD models, kinetic-based closures; and (5) paths toward more comprehensive simulation models, etc. Selected papers have been indexed separately for inclusion in the Energy Science and Technology Database
Proceedings of the workshop on nonlinear MHD and extended MHD
Energy Technology Data Exchange (ETDEWEB)
NONE
1998-12-01
Nonlinear MHD simulations have proven their value in interpreting experimental results over the years. As magnetic fusion experiments reach higher performance regimes, more sophisticated experimental diagnostics coupled with ever expanding computer capabilities have increased both the need for and the feasibility of nonlinear global simulations using models more realistic than regular ideal and resistive MHD. Such extended-MHD nonlinear simulations have already begun to produce useful results. These studies are expected to lead to ever more comprehensive simulation models in the future and to play a vital role in fully understanding fusion plasmas. Topics include the following: (1) current state of nonlinear MHD and extended-MHD simulations; (2) comparisons to experimental data; (3) discussions between experimentalists and theorists; (4) /equations for extended-MHD models, kinetic-based closures; and (5) paths toward more comprehensive simulation models, etc. Selected papers have been indexed separately for inclusion in the Energy Science and Technology Database.
Ultra Deep Wave Equation Imaging and Illumination
Energy Technology Data Exchange (ETDEWEB)
Alexander M. Popovici; Sergey Fomel; Paul Sava; Sean Crawley; Yining Li; Cristian Lupascu
2006-09-30
In this project we developed and tested a novel technology, designed to enhance seismic resolution and imaging of ultra-deep complex geologic structures by using state-of-the-art wave-equation depth migration and wave-equation velocity model building technology for deeper data penetration and recovery, steeper dip and ultra-deep structure imaging, accurate velocity estimation for imaging and pore pressure prediction and accurate illumination and amplitude processing for extending the AVO prediction window. Ultra-deep wave-equation imaging provides greater resolution and accuracy under complex geologic structures where energy multipathing occurs, than what can be accomplished today with standard imaging technology. The objective of the research effort was to examine the feasibility of imaging ultra-deep structures onshore and offshore, by using (1) wave-equation migration, (2) angle-gathers velocity model building, and (3) wave-equation illumination and amplitude compensation. The effort consisted of answering critical technical questions that determine the feasibility of the proposed methodology, testing the theory on synthetic data, and finally applying the technology for imaging ultra-deep real data. Some of the questions answered by this research addressed: (1) the handling of true amplitudes in the downward continuation and imaging algorithm and the preservation of the amplitude with offset or amplitude with angle information required for AVO studies, (2) the effect of several imaging conditions on amplitudes, (3) non-elastic attenuation and approaches for recovering the amplitude and frequency, (4) the effect of aperture and illumination on imaging steep dips and on discriminating the velocities in the ultra-deep structures. All these effects were incorporated in the final imaging step of a real data set acquired specifically to address ultra-deep imaging issues, with large offsets (12,500 m) and long recording time (20 s).
International Nuclear Information System (INIS)
Thaller, B.
1992-01-01
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics
International Nuclear Information System (INIS)
Sydoriak, S.G.
1976-01-01
Although criteria for cryostatic stability of superconducting magnets cooled by pool boiling of liquid helium have been widely discussed the same cannot be said for magnets cooled by natural convection or forced flow boiling in channels. Boiling in narrow channels is shown to be qualitatively superior to pool boiling because the recovery heat flux equals the breakaway flux for narrow channels, whereas the two are markedly different in pool boiling. A second advantage of channel boiling is that it is well understood and calculable; pool peak nucleate boiling heat flux has been adequately measured only for boiling from the top of an immersed heated body. Peak boiling from the bottom is much less and (probably) depends strongly on the extent of the bottom surface. Equations are presented by which one can calculate the critical boiling heat flux for parallel wall vertical channels subject to either natural convection or forced flow boiling, with one or both walls heated. The one-heated-wall forced flow equation is discussed with regard to design of a spiral wound solenoid (pancake magnet) having a slippery insulating tape between the windings
Solving Nonlinear Coupled Differential Equations
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Completely integrable operator evolutionary equations
International Nuclear Information System (INIS)
Chudnovsky, D.V.
1979-01-01
The authors present natural generalizations of classical completely integrable equations where the functions are replaced by arbitrary operators. Among these equations are the non-linear Schroedinger, the Korteweg-de Vries, and the modified KdV equations. The Lax representation and the Baecklund transformations are presented. (Auth.)
International Nuclear Information System (INIS)
Kalinowski, M.W.; Szymanowski, L.
1982-03-01
A generalization of the Truesdell F-equations is proposed and some solutions to them - generalized Fox F-functions - are found. It is also shown that a non-linear difference-differential equation, which does not belong to the Truesdell class, nevertheless may be transformed into the standard F-equation. (author)
On the Saha Ionization Equation
Indian Academy of Sciences (India)
Abstract. We revisit the Saha Ionization Equation in order to highlightthe rich interdisciplinary content of the equation thatstraddles distinct areas of spectroscopy, thermodynamics andchemical reactions. In a self-contained discussion, relegatedto an appendix, we delve further into the hidden message ofthe equation in terms ...
An existence theorem for a type of functional differential equation with infinite delay
Izsak, F.
We prove an existence theorem for a functional differential equation with infinite delay using the Schauder fixpoint theorem. We extend a result in [19] applying the fixed point procedure in an appropriate function space.
Directory of Open Access Journals (Sweden)
Guanwei Chen
2014-01-01
Full Text Available We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.
Oscillating particle-like solutions of nonlinear Klein-Gordon equation
International Nuclear Information System (INIS)
Bogolubsky, I.L.
1976-01-01
A denumerable set of oscillating spherically-symmetric particle-like solutions of the Klein-Gordon equation with cubic nonlinearity is found. Extended particles modelled by them turn out to be slightly radiating and long-lived
Exact solutions of time-fractional heat conduction equation by the fractional complex transform
Directory of Open Access Journals (Sweden)
Li Zheng-Biao
2012-01-01
Full Text Available The Fractional Complex Transform is extended to solve exactly time-fractional differential equations with the modified Riemann-Liouville derivative. How to incorporate suitable boundary/initial conditions is also discussed.
Global existence of a generalized solution for the radiative transfer equations
International Nuclear Information System (INIS)
Golse, F.; Perthame, B.
1984-01-01
We prove global existence of a generalized solution of the radiative transfer equations, extending Mercier's result to the case of a layer with an initially cold area. Our Theorem relies on the results of Crandall and Ligett [fr
On the analytical solution of Fornberg–Whitham equation with the ...
Indian Academy of Sciences (India)
This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence ..... The authors are grateful to the financial support extended by the King Fahd University of. Petroleum ...
International Nuclear Information System (INIS)
Williams, M.M.R.
2005-01-01
The integral equation derived by Nieuwenhuizen and Luck for transmission of radiation through an optically thick diffusive medium is reconsidered in the light of radiative transfer theory and extended to slabs of arbitrary thickness. (author)
Directory of Open Access Journals (Sweden)
Bui The Anh
2007-01-01
Full Text Available We extend the classical Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations. Finally, the result obtained is applied to derive necessary and sufficient conditions for the stability of positive system.
Nonlinear equations and optimisation
Watson, LT; Bartholomew-Biggs, M
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! In one of the papers in this collection, the remark that ""nothing at all takes place in the universe in which some rule of maximum of minimum does not appear"" is attributed to no less an authority than Euler. Simplifying the syntax a little, we might paraphrase this as Everything is an optimization problem. While this might be something of an overstatement, the element of exaggeration is certainly reduced if we consider the extended form: Every
DEFF Research Database (Denmark)
Karpman, V.I.; Shagalov, A.G.; Juul Rasmussen, J.
2002-01-01
The behavior of steady quasisoliton solutions to the extended third-order nonlinear Schrodinger (NLS) equation is studied in two cases: (i) when the coefficients in the equation approach the Hirota conditions, and (ii) near the limit of the regular NLS equation. (C) 2002 Published by Elsevier...
Extended memory management under RTOS
Plummer, M.
1981-01-01
A technique for extended memory management in ROLM 1666 computers using FORTRAN is presented. A general software system is described for which the technique can be ideally applied. The memory manager interface with the system is described. The protocols by which the manager is invoked are presented, as well as the methods used by the manager.
Geometrical interpretation of extended supergravity
International Nuclear Information System (INIS)
Townsend, P.K.; Nieuwenhuizen, P.van
1977-01-01
SO 2 extended supergravity is shown to be a geometrical theory, whose underlying gauge group is OSp(4,2). The couplings which gauge the SO 2 symmetry as well as the accompanying cosmological and masslike terms are directly obtained, and the usual SO 2 model is obtained after a Wigner-Inoenue group contraction. (Auth.)
Spin-4 extended conformal algebras
International Nuclear Information System (INIS)
Kakas, A.C.
1988-01-01
We construct spin-4 extended conformal algebras using the second hamiltonian structure of the KdV hierarchy. In the presence of a U(1) current a family of spin-4 algebras exists but the additional requirement that the spin-1 and spin-4 currents commute fixes the algebra uniquely. (orig.)
Departies: conceptualizing extended youth parties
DEFF Research Database (Denmark)
Fjær, Eivind Grip; Tutenges, Sébastien
2017-01-01
Every year, millions of young people travel away from home to party for days or weeks on end in permissive environments, such as music festivals, dance parties, and nightlife resorts. The studies that have been conducted on these extended youth parties have focused primarily on specific risk...
Applying and extending Oracle Spatial
Simon Gerard Greener, Siva Ravada
2013-01-01
This book is an advanced practical guide to applying and extending Oracle Spatial.This book is for existing users of Oracle and Oracle Spatial who have, at a minimum, basic operational experience of using Oracle or an equivalent database. Advanced skills are not required.
Extended unemployment and UI benefits
Robert G. Valletta; Katherine Kuang
2010-01-01
During the current labor market downturn, unemployment duration has reached levels well above its previous highs. Analysis of unemployment data suggests that extended unemployment insurance benefits have not been important factors in the increase in the duration of unemployment or in the elevated unemployment rate.
Engage, Enhance, and Extend Learning!
Keren-Kolb, Liz
2013-01-01
Educators often say that technology is more than a gimmick or add-on, and that it should engage, enhance, or extend learning in ways that traditional tools do not. Yet they seldom stop to define these terms, and they can be confusing, especially for teachers and preservice teachers. Recently, while collaborating on an English language arts and…
True amplitude wave equation migration arising from true amplitude one-way wave equations
Zhang, Yu; Zhang, Guanquan; Bleistein, Norman
2003-10-01
to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data when the one-way wavefields are replaced by their ray theoretic approximations. This extension enhances the original WEM method. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output obtained using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as currently formulated. Research on extending the method is ongoing at this time.
Reduction of infinite dimensional equations
Directory of Open Access Journals (Sweden)
Zhongding Li
2006-02-01
Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.
Directory of Open Access Journals (Sweden)
Taouil Hajer
2012-08-01
Full Text Available This paper is devoted to the helices processes, i.e. the solutions H : ℝ × Ω → ℝd, (t, ω ↦ H(t, ω of the helix equation egin{eqnarray} H(0,o=0; quad H(s+t,o= H(s,Phi(t,o +H(t,oonumber end{eqnarray} H ( 0 ,ω = 0 ; H ( s + t,ω = H ( s, Φ ( t,ω + H ( t,ω where Φ : ℝ × Ω → Ω, (t, ω ↦ Φ(t, ω is a dynamical system on a measurable space (Ω, ℱ. More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined by Φ, are investigated. Ce papier est consacré aux hélices, c’est-à-dire les solutions H : ℝ × Ω → ℝd, (t, ω ↦ H(t, ω de l’équation fonctionnelle egin{eqnarray} H(0,o=0; quad H(s+t,o= H(s,Phi(t,o +H(t,o onumber end{eqnarray} H ( 0 ,ω = 0 ; H ( s + t,ω = H ( s, Φ ( t,ω + H ( t,ω où Φ : ℝ × Ω → Ω, (t, ω ↦ Φ(t, ω est un système dynamique défini sur un espace mesurable (Ω, ℱ. Plus présisément, nous déterminons d’abord les hélices dominées puis nous caractérisons les hélices non différentiables. Dans ce dernier cas, l’hélice de Wiener joue un rôle important. Nous précisons aussi quelques relations des hélices avec les cocycles définis par Φ.
Growth of meromorphic solutions of higher-order linear differential equations
Directory of Open Access Journals (Sweden)
Wenjuan Chen
2009-01-01
Full Text Available In this paper, we investigate the higher-order linear differential equations with meromorphic coefficients. We improve and extend a result of M.S. Liu and C.L. Yuan, by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen, and the extended Winman-Valiron theory which proved by J. Wang and H.X. Yi. In addition, we also consider the nonhomogeneous linear differential equations.
Extended recency effect extended: blocking, presentation mode, and retention interval.
Glidden, L M; Pawelski, C; Mar, H; Zigman, W
1979-07-01
The effect of blocking of stimulus items on the free recall of EMR adolescents was examined. In Experiment 1 a multitrial free-recall list of 15 pictures was presented either simultaneously in groups of 3, or sequentially, one at a time. Consistent ordering was used in both conditions, so that on each trial, each item in each set of 3 pictures was presented contiguously with the other 2 items from that set. In addition, recall came immediately or after a filled or unfilled delay of 24.5 seconds. Results showed that simultaneous presentation led to higher recall, subjective organization, and clustering than did sequential presentation, but analysis of serial-position curves showed a much reduced extended recency effect in comparison with previous studies. Experiment 2 was designed to determine whether the cause of the reduced extended recency was the use of pictures rather than words as stimuli. Stimuli were presented either as pictures, as pictures with auditory labels, or as words with auditory labels, with both simultaneous and consistent ordering for all conditions. Results indicated a strong extended recency effect for all groups, eliminating presentation mode as a causal factor in the data of Experiment 1. We concluded that blocking leads to increased organization and recall over a variety of presentation modes, rates, and block sizes.
A quantum field theory of the extended electron
International Nuclear Information System (INIS)
Salesi, Giovanni; Recami, Erasmo; Universidade Estadual de Campinas, SP
1993-12-01
In a recent paper, the classical model of Barut and Zanghi (BZ) for the electron spin which interpreted the Zitterbewegung (zbw) motion along helical paths and its quantum version have been investigated by using the language of Clifford algebras. In also doing, a new non-linear Dirac-like equation (NDE) was derived. We want to readdress the whole subject, and complete it, by adopting - for the sake of physical clarity - the ordinary tensorial language. In particular, we re-derive here the NDE for the electron quantum field, show it to be associated with a new conserved probability current, and stress its importance for a quantum field theory of spin 1/2 fermions. Actually, we propose this equation in substitution for the Dirac equation, which comes from the former by averaging over a zbw cycle. We then derive a new equation of motion for the quantum field velocity, which will allow us to regard the electron as an extended object, with a classically intelligible internal structure (thus overcoming some known, long-standing problems). We carefully the solutions of the NDE; with special attention to those implying (at the classical limit) light-like helical motions, since these appear to be the most adequate equations for the electron description, from the kinematical and physical points of view, and do cope with the electron electromagnetic properties (such as Coulomb field and intrinsic magnetic moment). (author). 18 refs
A quantum field theory of the extended electron
Energy Technology Data Exchange (ETDEWEB)
Salesi, Giovanni [Universita Statale di Catania (Italy). Dipt. di Fisica; Recami, Erasmo [Universita Statale di Bergamo, Dalmine, BG (Italy). Facolta di Ingegneria; [Universidade Estadual de Campinas, SP (Brazil). Dept. de Matematica Aplicada
1993-12-01
In a recent paper, the classical model of Barut and Zanghi (BZ) for the electron spin which interpreted the Zitterbewegung (zbw) motion along helical paths and its quantum version have been investigated by using the language of Clifford algebras. In also doing, a new non-linear Dirac-like equation (NDE) was derived. We want to readdress the whole subject, and complete it, by adopting - for the sake of physical clarity - the ordinary tensorial language. In particular, we re-derive here the NDE for the electron quantum field, show it to be associated with a new conserved probability current, and stress its importance for a quantum field theory of spin 1/2 fermions. Actually, we propose this equation in substitution for the Dirac equation, which comes from the former by averaging over a zbw cycle. We then derive a new equation of motion for the quantum field velocity, which will allow us to regard the electron as an extended object, with a classically intelligible internal structure (thus overcoming some known, long-standing problems). We carefully the solutions of the NDE; with special attention to those implying (at the classical limit) light-like helical motions, since these appear to be the most adequate equations for the electron description, from the kinematical and physical points of view, and do cope with the electron electromagnetic properties (such as Coulomb field and intrinsic magnetic moment). (author). 18 refs.
Semi-continuous and multigroup models in extended kinetic theory
International Nuclear Information System (INIS)
Koller, W.
2000-01-01
The aim of this thesis is to study energy discretization of the Boltzmann equation in the framework of extended kinetic theory. In case that external fields can be neglected, the semi- continuous Boltzmann equation yields a sound basis for various generalizations. Semi-continuous kinetic equations describing a three component gas mixture interacting with monochromatic photons as well as a four component gas mixture undergoing chemical reactions are established and investigated. These equations reflect all major aspects (conservation laws, equilibria, H-theorem) of the full continuous kinetic description. For the treatment of the spatial dependence, an expansion of the distribution function in terms of Legendre polynomials is carried out. An implicit finite differencing scheme is combined with the operator splitting method. The obtained numerical schemes are applied to the space homogeneous study of binary chemical reactions and to spatially one-dimensional laser-induced acoustic waves. In the presence of external fields, the developed overlapping multigroup approach (with the spline-interpolation as its extension) is well suited for numerical studies. Furthermore, two formulations of consistent multigroup approaches to the non-linear Boltzmann equation are presented. (author)
Functional theory of extended Coulomb systems
International Nuclear Information System (INIS)
Martin, R.M.; Ortiz, G.
1997-01-01
A consistent formulation is presented for a functional theory of extended quantum many-particle systems with long-range Coulomb interactions, which extends the density-functional theory of Hohenberg and Kohn to encompass the theory of dielectrics formulated in terms of electric fields and polarization. We show that a complete description of insulators in the thermodynamic limit requires a functional of density and macroscopic polarization; nevertheless, for any insulator the state with zero macroscopic electric field can be considered a reference state that is a functional of the density alone. Dielectric phenomena involve the behavior of the material in the presence of macroscopic electric fields that induce changes of the macroscopic polarization from its equilibrium value in the reference state. In the thermodynamic limit there is strictly no ground state and constraints must be placed upon the electronic wave functions in order to have a well-defined energy functional; within these constrained subspaces the Hohenberg-Kohn theorems can be generalized in terms of the density and the change in the macroscopic polarization. The essential role of the polarization is shown by an explicit example of two potentials that lead to the same periodic density in a crystal, but different macroscopic electric fields and polarization. In the Kohn-Sham approach both the kinetic and the exchange-correlation energy are shown to depend upon the changes in polarization; this leads to generalized Kohn-Sham equations with a nonlocal operator. The effect can be traced to the polarization of the average exchange-correlation hole itself in the presence of macroscopic fields, which is essential for an exact description of static dielectric phenomena. copyright 1997 The American Physical Society
p-Euler equations and p-Navier-Stokes equations
Li, Lei; Liu, Jian-Guo
2018-04-01
We propose in this work new systems of equations which we call p-Euler equations and p-Navier-Stokes equations. p-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p - 1)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier-Stokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier-Stokes equations in Rd for γ = p and p ≥ d ≥ 2 through a compactness criterion.
Generalized quantal equation of motion
International Nuclear Information System (INIS)
Morsy, M.W.; Embaby, M.
1986-07-01
In the present paper, an attempt is made for establishing a generalized equation of motion for quantal objects, in which intrinsic self adjointness is naturally built in, independently of any prescribed representation. This is accomplished by adopting Hamilton's principle of least action, after incorporating, properly, the quantal features and employing the generalized calculus of variations, without being restricted to fixed end points representation. It turns out that our proposed equation of motion is an intrinsically self-adjoint Euler-Lagrange's differential equation that ensures extremization of the quantal action as required by Hamilton's principle. Time dependence is introduced and the corresponding equation of motion is derived, in which intrinsic self adjointness is also achieved. Reducibility of the proposed equation of motion to the conventional Schroedinger equation is examined. The corresponding continuity equation is established, and both of the probability density and the probability current density are identified. (author)
Alternatives to the Dirac equation
International Nuclear Information System (INIS)
Girvin, S.M.; Brownstein, K.R.
1975-01-01
Recent work by Biedenharn, Han, and van Dam (BHvD) has questioned the uniqueness of the Dirac equation. BHvD have obtained a two-component equation as an alternate to the Dirac equation. Although they later show their alternative to be unitarily equivalent to the Dirac equation, certain physical differences were claimed. BHvD attribute the existence of this alternate equation to the fact that their factorizing matrices were position-dependent. To investigate this, we factor the Klein-Gordon equation in spherical coordinates allowing the factorizing matrices to depend arbitrarily upon theta and phi. It is shown that despite this additional freedom, and without involving any relativistic covariance, the conventional four-component Dirac equation is the only possibility
Discrete q-derivatives and symmetries of q-difference equations
Energy Technology Data Exchange (ETDEWEB)
Levi, D [Dipartimento di Fisica, Universita Roma Tre and INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome (Italy); Negro, J [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain); Olmo, M A del [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain)
2004-03-12
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation.
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
Cima, A.; Gasull, A.; Mañosas, F.
2017-12-01
In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.
Wave Partial Differential Equation
Szöllös, Alexandr
2009-01-01
Práce se zabývá diferenciálními rovnicemi, jejich využitím při analýze vedení, experimenty s vedením a možnou akcelerací výpočtu v GPU s využitím prostředí nVidia CUDA. This work deals with diffrential equations, with the possibility of using them for analysis of the line and the possibility of accelerating the computations in GPU using nVidia CUDA. C
Gomez, Humberto
2016-06-01
The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter Λ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting Λ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the Λ algorithm.
Scaling of differential equations
Langtangen, Hans Petter
2016-01-01
The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...
Parabolized stability equations
Herbert, Thorwald
1994-01-01
The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.
Pomeau, Yves; Piasecki, Jarosław
2017-11-01
The existence of atoms has been long predicted by philosophers and scientists. The development of thermodynamics and of the statistical interpretation of its concepts at the end of the nineteenth century and in the early years of the twentieth century made it possible to bridge the gap of scales between the macroscopic world and the world of atoms. Einstein and Smoluchowski showed in 1905 and 1906 that the Brownian motion of particles of measurable size is a manifestation of the motion of atoms in fluids. Their derivation was completely different from each other. Langevin showed in 1908 how to put in a coherent framework the subtle effect of the randomness of the atomic world, responsible for the fluctuating force driving the motion of the Brownian particle and the viscosity of the "macroscopic" flow taking place around the same Brownian particle. Whereas viscous forces were already well understood at this time, the "Langevin" force appears there for the first time: it represents the fluctuating part of the interaction between the Brownian particle and the surrounding fluid. We discuss the derivation by Einstein and Smoluchowski as well as a previous paper by Sutherland on the diffusion coefficient of large spheres. Next we present Langevin's short note and explain the fundamental splitting into a random force and a macroscopic viscous force. This brings us to discuss various points, like the kind of constraints on Langevin-like equations. We insist in particular on the one arising from the time-reversal symmetry of the equilibrium fluctuations. Moreover, we discuss another constraint, raised first by Lorentz, which implies that, if the Brownian particle is not very heavy, the viscous force cannot be taken as the standard Stokes drag on an object moving at uniform speed. Lastly, we examine the so-called Langevin-Heisenberg and/or Langevin-Schrödinger equation used in quantum mechanics.
Scale calculus and the Schroedinger equation
International Nuclear Information System (INIS)
Cresson, Jacky
2003-01-01
This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,ε) such that f(t,ε)→f(t) when ε goes to zero. This led to several new notions as representations: fractal functions and ε-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schroedinger equation via the classical Newton's equation of dynamics using the scale operator. Under special assumptions we recover the classical Schroedinger equation and we discuss the relevance of these assumptions
Spectrum of the ballooning Schroedinger equation
International Nuclear Information System (INIS)
Dewar, R.L.
1997-01-01
The ballooning Schroedinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel-Kramers-Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second order differential equation with coefficients periodic in the independent variable θ k is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods: e.g. the apparent discontinuity between quantization formulae for open-quotes trappedclose quotes and open-quotes passingclose quotes modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over θ k
Interacting fields of arbitrary spin and N > 4 supersymmetric self-dual Yang-Mills equations
International Nuclear Information System (INIS)
Devchand, Ch.; Ogievetsky, V.
1996-06-01
We show that the self-dual Yang-Mills equations afford supersymmetrization to systems of equations invariant under global N-extended super-Poincare transformations for arbitrary values of N, without the limitation (N ≤ 4) applicable to standard non-self-dual Yang-Mills theories. These systems of equations provide novel classically consistent interactions for vector supermultiplets containing fields of spin up to N-2/2. The equations of motion of the component fields of spin greater than 1/2 are interacting variants of the first-order Dirac-Fierz equations for zero rest-mass fields of arbitrary spin. The interactions are governed by conserved currents which are constructed by an iterative procedure. In (arbitrarily extended) chiral superspace, the equations of motion for the (arbitrarily large) self-dual supermultiplet are shown to be completely equivalent to the set of algebraic supercurvature defining the self-dual superconnection. (author). 25 refs
New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod
Seadawy, Aly R.; Manafian, Jalil
2018-03-01
This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.
The extended bigraded Toda hierarchy
International Nuclear Information System (INIS)
Carlet, Guido
2006-01-01
We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ε-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [4]. Using R-matrix theory we give the bi-Hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups W-tilde (N) (A N+M-1 ) of the A series, defined by Dubrovin and Zhang (1998 Compos. Math. 111 167)
An extended Harry Dym hierarchy
International Nuclear Information System (INIS)
Ma Wenxiu
2010-01-01
An extended Harry Dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the Harry Dym hierarchy associated with the pseudo-differential operator L = u∂ + u 0 + u 1 ∂ -1 + .... The corresponding Lax presentation possesses a self-consistent source involving squared eigenfunctions. The resulting extended Harry Dym hierarchy is reduced to the Harry Dym hierarchy with self-consistent sources under the n-reduction, L n = (L n ) ≥2 , and the k-constrained Harry Dym hierarchy under the k-constraint, L k = (L k ) ≥2 + Σ N i=1 q i ∂ -1 r i ∂ 2 . A few particular examples are computed, together with their Lax pairs.
A high order solver for the unbounded Poisson equation
DEFF Research Database (Denmark)
Hejlesen, Mads Mølholm; Rasmussen, Johannes Tophøj; Chatelain, Philippe
2013-01-01
. The method is extended to directly solve the derivatives of the solution to Poissonʼs equation. In this way differential operators such as the divergence or curl of the solution field can be solved to the same high order convergence without additional computational effort. The method, is applied......A high order converging Poisson solver is presented, based on the Greenʼs function solution to Poissonʼs equation subject to free-space boundary conditions. The high order convergence is achieved by formulating regularised integration kernels, analogous to a smoothing of the solution field...... and validated, however not restricted, to the equations of fluid mechanics, and can be used in many applications to solve Poissonʼs equation on a rectangular unbounded domain....
Approximate Solution of LR Fuzzy Sylvester Matrix Equations
Directory of Open Access Journals (Sweden)
Xiaobin Guo
2013-01-01
Full Text Available The fuzzy Sylvester matrix equation AX~+X~B=C~ in which A,B are m×m and n×n crisp matrices, respectively, and C~ is an m×n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.
Continuous symmetric reductions of the Adler-Bobenko-Suris equations
International Nuclear Information System (INIS)
Tsoubelis, D; Xenitidis, P
2009-01-01
Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three-point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi 'generating partial differential equations' is established and an auto-Baecklund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1 δ=0 members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents
Numerical solution of plasma fluid equations using locally refined grids
International Nuclear Information System (INIS)
Colella, P.
1997-01-01
This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasma under consideration is typical of those used for the processing of semiconductors. The governing equations consist of a drift-diffusion model of the electrons and an isothermal model of the ions coupled by Poisson's equation. A discretization of the equations is given for a uniform spatial grid, and a time-split integration scheme is developed. The algorithm is then extended to accommodate locally refined grids. This extension involves the advancement of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. A brief discussion of a software implementation is followed by a presentation of numerical results
Black holes from extended inflation
International Nuclear Information System (INIS)
Hsu, S.D.H.; Lawrence Berkeley Lab., CA
1990-01-01
It is argued that models of extended inflation, in which modified Einstein gravity allows a graceful exit from the false vacuum, lead to copious production of black holes. The critical temperature of the inflationary phase transition must be >10 8 GeV in order to avoid severe cosmological problems in a universe dominated by black holes. We speculate on the possibility that the interiors of false vacuum regions evolve into baby universes. (orig.)
Locating and extending livelihoods research
DEFF Research Database (Denmark)
Prowse, Martin
2008-01-01
Much poverty and development research is not explicit about its methodology or philosophical foundations. Based on the extended case method of Burawoy and the epistemological standpoint of critical realism, this paper discusses a methodological approach for reflexive inductive livelihoods researc...... that overcomes the unproductive social science dualism of positivism and social constructivism. The approach is linked to a conceptual framework and a menu of research methods that can be sequenced and iterated in light of research questions....
Extended producer responsibility in oligopoly
Hiroaki Ino
2007-01-01
I investigate the optimal environmental tax under a policy based on extended producer responsibility (EPR) in oligopoly markets. I introduce the recycling market and explicitly consider how these policies affect the incentive for recycling. I derive the optimal tax rule, which depends on the weighted sum of the markup in the product market and the markdown in the recycling market. In contrast to the existing works that emphasize that the optimal tax rate is lower than the marginal external da...
On the invariance properties of the Klein–Gordon equation with ...
Indian Academy of Sciences (India)
Abstract. Here we attempt to find the nature of the external electromagnetic field such that the KG equation with external electromagnetic field is invariant. Lie's extended group method is applied to obtain the class of external electromagnetic field which admits the invariance of the KG equation. Though, the field potential ...
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)
2002-01-01
We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.
Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method
Directory of Open Access Journals (Sweden)
De-Gang Wang
2012-01-01
Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.
A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.
Regularity criteria for the 3D magneto-micropolar fluid equations via ...
Indian Academy of Sciences (India)
We consider sufficient conditions to ensure the smoothness of solutions to 3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations.
Using Automated Essay Scores as an Anchor When Equating Constructed Response Writing Tests
Almond, Russell G.
2014-01-01
Assessments consisting of only a few extended constructed response items (essays) are not typically equated using anchor test designs as there are typically too few essay prompts in each form to allow for meaningful equating. This article explores the idea that output from an automated scoring program designed to measure writing fluency (a common…
The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations
International Nuclear Information System (INIS)
Sheng Zhang
2006-01-01
More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained
A difference-equation formalism for the nodal domains of separable billiards
Energy Technology Data Exchange (ETDEWEB)
Manjunath, Naren; Samajdar, Rhine [Indian Institute of Science, Bangalore 560012 (India); Jain, Sudhir R., E-mail: srjain@barc.gov.in [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)
2016-09-15
Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in Samajdar and Jain (2014). The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.
Directory of Open Access Journals (Sweden)
San-Yang Liu
2014-01-01
Full Text Available Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.
Constraints based analysis of extended cybernetic models.
Mandli, Aravinda R; Venkatesh, Kareenhalli V; Modak, Jayant M
2015-11-01
The cybernetic modeling framework provides an interesting approach to model the regulatory phenomena occurring in microorganisms. In the present work, we adopt a constraints based approach to analyze the nonlinear behavior of the extended equations of the cybernetic model. We first show that the cybernetic model exhibits linear growth behavior under the constraint of no resource allocation for the induction of the key enzyme. We then quantify the maximum achievable specific growth rate of microorganisms on mixtures of substitutable substrates under various kinds of regulation and show its use in gaining an understanding of the regulatory strategies of microorganisms. Finally, we show that Saccharomyces cerevisiae exhibits suboptimal dynamic growth with a long diauxic lag phase when growing on a mixture of glucose and galactose and discuss on its potential to achieve optimal growth with a significantly reduced diauxic lag period. The analysis carried out in the present study illustrates the utility of adopting a constraints based approach to understand the dynamic growth strategies of microorganisms. Copyright © 2015 Elsevier Ireland Ltd. All rights reserved.
3-D resistive MHD calculations for tokamak plasmas: beyond the simple reduced set of equations
International Nuclear Information System (INIS)
Carreras, B.A.; Garcia, L.; Hender, T.C.; Hicks, H.R.; Holmes, J.A.; Lynch, V.E.; Masden, B.F.
1983-01-01
Numerical studies of the resistive stability of tokamak plasmas in cylindrical geometry have been performed using: (1) the full set of resistive Magnetohydrodynamic (MHD) equations and (2) an extended version of the reduced set of resistive MHD equations including diamagnetic and electron temperature effects. In particular, the nonlinear interaction of tearing modes of many helicities has been investigated. The numerical results confirm many of the features uncovered previously using the simple reduced equations. (author)
Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation
Directory of Open Access Journals (Sweden)
Mitsuo Kato
2018-01-01
Full Text Available A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.
Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College, Bisha, P.O. Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com
2007-08-27
By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.
Thandapani, Ethiraju; Kannan, Manju; Pinelas, Sandra
2016-01-01
In this paper, we present some sufficient conditions for the oscillation of all solutions of a second order forced impulsive delay differential equation with damping term. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. The results obtained in this paper extend and generalize some of the the known results for forced impulsive differential equations. An example is provided to illustrate the main result.
EAES: Extended Advanced Encryption Standard with Extended Security
Abul Kalam Azad; Md. Yamin Mollah
2018-01-01
Though AES is the highest secure symmetric cipher at present, many attacks are now effective against AES too which is seen from the review of recent attacks of AES. This paper describes an extended AES algorithm with key sizes of 256, 384 and 512 bits with round numbers of 10, 12 and 14 respectively. Data block length is 128 bits, same as AES. But unlike AES each round of encryption and decryption of this proposed algorithm consists of five stages except the last one which consists of four st...
Extended I-Love relations for slowly rotating neutron stars
Gagnon-Bischoff, Jérémie; Green, Stephen R.; Landry, Philippe; Ortiz, Néstor
2018-03-01
Observations of gravitational waves from inspiralling neutron star binaries—such as GW170817—can be used to constrain the nuclear equation of state by placing bounds on stellar tidal deformability. For slowly rotating neutron stars, the response to a weak quadrupolar tidal field is characterized by four internal-structure-dependent constants called "Love numbers." The tidal Love numbers k2el and k2mag measure the tides raised by the gravitoelectric and gravitomagnetic components of the applied field, and the rotational-tidal Love numbers fo and ko measure those raised by couplings between the applied field and the neutron star spin. In this work, we compute these four Love numbers for perfect fluid neutron stars with realistic equations of state. We discover (nearly) equation-of-state independent relations between the rotational-tidal Love numbers and the moment of inertia, thereby extending the scope of I-Love-Q universality. We find that similar relations hold among the tidal and rotational-tidal Love numbers. These relations extend the applications of I-Love universality in gravitational-wave astronomy. As our findings differ from those reported in the literature, we derive general formulas for the rotational-tidal Love numbers in post-Newtonian theory and confirm numerically that they agree with our general-relativistic computations in the weak-field limit.
Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations
Athanassoulis, Agissilaos
2018-03-01
We consider the semiclassical limit of nonlinear Schrödinger equations with initial data that are well localized in both position and momentum (non-parametric wavepackets). We recover the Wigner measure (WM) of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. WMs have been used to create effective models for wave propagation in: random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the WM are often ill-posed on the physically interesting spaces of initial data. In this paper we are able to select the measure-valued solution of the 1 + 1 dimensional Vlasov-Poisson equation which correctly captures the semiclassical limit, thus finally resolving the non-uniqueness in the seminal result of Zhang et al (2012 Comm. Pure Appl. Math. 55 582-632). The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.
Introduction to partial differential equations
Borthwick, David
2016-01-01
This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.Within each section the author creates a narrative that answers the five questions: (1) What is the scientific problem we are trying to understand? (2) How do we model that with PDE? (3) What techniques can we use to analyze the PDE? (4) How do those techniques apply to this equation? (5) What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.