Asymptotic expansions of Jacobi functions
The author presents an asymptotic expansion of the Jacobi polynomials which is based on the fact, that these polynomials are special hypergeometric functions. He uses an integral representation of these functions and expands the integrand in a power series. He derives explicit error bounds on this expansion. (HSI)
Asymptotic and Exact Expansions of Heat Traces
Eckstein, Michał, E-mail: michal@eckstein.pl [Jagiellonian University, Faculty of Physics, Astronomy and Applied Computer Science (Poland); Zając, Artur, E-mail: artur.zajac@uj.edu.pl [Jagiellonian University, Faculty of Mathematics and Computer Science (Poland)
2015-12-15
We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.
Asymptotic behaviour of firmly non expansive sequences
We introduce the notion of firmly non expansive sequences in a Banach space and present several results concerning their asymptotic behaviour extending previous results and giving an affirmative answer to an open question raised by S. Reich and I. Shafir. Applications to averaged mappings are also given. (author). 16 refs
Zero bias transformation and asymptotic expansions
Jiao, Ying
2012-01-01
Let W be a sum of independent random variables. We apply the zero bias transformation to deduce recursive asymptotic expansions for $\\mathbb {E}[h(W)]$ in terms of normal expectations, or of Poisson expectations for integer-valued random variables. We also discuss the estimates of remaining errors.
Asymptotic expansion of the wavelet transform with error term
R. S. Pathak; Pathak, Ashish
2014-01-01
UsingWong's technique asymptotic expansion for the wavelet transform is derived and thereby asymptotic expansions for Morlet wavelet transform, Mexican Hat wavelet transform and Haar wavelet transform are obtained.
Asymptotic expansions for the Gaussian unitary ensemble
Haagerup, Uffe; Thorbjørnsen, Steen
2012-01-01
Let g : R ¿ C be a C8-function with all derivatives bounded and let trn denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value ¿{trn(g(Xn))} for a rather general class of random matrices Xn, including the Gaussian...... Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix Xn that where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients...... aj(g), j ¿ N, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Trn[f(Xn)], Trn[g(Xn)]}, where f is a function of the same kind as g, and Trn = n trn. Special focus is drawn to the case where and for ¿, µ in C\\R. In this case the mean and...
ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS
ZHU Xiao-feng; LI Xiu-chun
2006-01-01
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.
Asymptotic expansions for high-contrast linear elasticity
Poveda, Leonardo A.
2015-03-01
We study linear elasticity problems with high contrast in the coefficients using asymptotic limits recently introduced. We derive an asymptotic expansion to solve heterogeneous elasticity problems in terms of the contrast in the coefficients. We study the convergence of the expansion in the H1 norm. © 2015 Elsevier B.V.
Extended Analytic Device Optimization Employing Asymptotic Expansion
Mackey, Jonathan; Sehirlioglu, Alp; Dynsys, Fred
2013-01-01
Analytic optimization of a thermoelectric junction often introduces several simplifying assumptionsincluding constant material properties, fixed known hot and cold shoe temperatures, and thermallyinsulated leg sides. In fact all of these simplifications will have an effect on device performance,ranging from negligible to significant depending on conditions. Numerical methods, such as FiniteElement Analysis or iterative techniques, are often used to perform more detailed analysis andaccount for these simplifications. While numerical methods may stand as a suitable solution scheme,they are weak in gaining physical understanding and only serve to optimize through iterativesearching techniques. Analytic and asymptotic expansion techniques can be used to solve thegoverning system of thermoelectric differential equations with fewer or less severe assumptionsthan the classic case. Analytic methods can provide meaningful closed form solutions and generatebetter physical understanding of the conditions for when simplifying assumptions may be valid.In obtaining the analytic solutions a set of dimensionless parameters, which characterize allthermoelectric couples, is formulated and provide the limiting cases for validating assumptions.Presentation includes optimization of both classic rectangular couples as well as practically andtheoretically interesting cylindrical couples using optimization parameters physically meaningful toa cylindrical couple. Solutions incorporate the physical behavior for i) thermal resistance of hot andcold shoes, ii) variable material properties with temperature, and iii) lateral heat transfer through legsides.
Spectral Expansion for the Asymptotically Spectral Periodic Differential Operators
O. A. Veliev
2016-01-01
In this paper we investigate the spectral expansion for the asymptotically spectral differential operators generated in all real line by ordinary differential expression of arbitrary order with periodic matrix-valued coefficients
Asymptotic chaos expansions in finance theory and practice
Nicolay, David
2014-01-01
Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo. Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (suc...
Zero bias transformation and asymptotic expansions II : the Poisson case
Jiao, Ying
2009-01-01
We apply a discrete version of the methodology in \\cite{gauss} to obtain a recursive asymptotic expansion for $\\esp[h(W)]$ in terms of Poisson expectations, where $W$ is a sum of independent integer-valued random variables and $h$ is a polynomially growing function. We also discuss the remainder estimations.
Generalized heat kernel coefficients for a new asymptotic expansion
The method which allows for asymptotic expansion of the one-loop effective action W = lndetA is formulated. The positively defined elliptic operator A = U + M2 depends on the external classical fields taking values in the Lie algebra of the internal symmetry group G. Unlike the standard method of Schwinger - DeWitt, the more general case with the nongenerate mass matrix M = diag(m1, m2, ...) is considered. The first coefficients of the new asymptotic series are calculated and their relationship with the Seeley - DeWitt coefficients is clarified
Asymptotic expansions of Mellin convolution integrals: An oscillatory case
López, José L.; Pagola, Pedro
2010-01-01
In a recent paper [J.L. López, Asymptotic expansions of Mellin convolution integrals, SIAM Rev. 50 (2) (2008) 275-293], we have presented a new, very general and simple method for deriving asymptotic expansions of for small x. It contains Watson's Lemma and other classical methods, Mellin transform techniques, McClure and Wong's distributional approach and the method of analytic continuation used in this approach as particular cases. In this paper we generalize that idea to the case of oscillatory kernels, that is, to integrals of the form , with c[set membership, variant]R, and we give a method as simple as the one given in the above cited reference for the case c=0. We show that McClure and Wong's distributional approach for oscillatory kernels and the summability method for oscillatory integrals are particular cases of this method. Some examples are given as illustration.
A GLOBALLY UNIFORM ASYMPTOTIC EXPANSION OF THE HERMITE POLYNOMIALS
Shi Wei
2008-01-01
In this article, the author extends the validity of a uniform asymptotic ex-pansion of the Hermite polynomials HN(√2n+1α) to include all positive values of a.His method makes use of the rational functions introduced by Olde Daalhuis and Temme (SIAM J. Math. Anal., (1994), 25: 304-321). A new estimate for the remainder is given.
Asymptotic Expansions of Feynman Amplitudes in a Generic Covariant Gauge
Linhares, C. A.; Malbouisson, A. P. C.; Roditi, I.
2006-01-01
We show in this paper how to construct Symanzik polynomials and the Schwinger parametric representation of Feynman amplitudes for gauge theories in an unspecified covariant gauge. The complete Mellin representation of such amplitudes is then established in terms of invariants (squared sums of external momenta and squared masses). From the scaling of the invariants by a parameter we extend for the present situation a theorem on asymptotic expansions, previously proven for the case of scalar fi...
Asymptotic expansion and statistical description of turbulent systems
A new approach to studying turbulent systems is presented in which an asymptotic expansion of the general dynamical equations is performed prior to the application of statistical methods for describing the evolution of the system. This approach has been applied to two specific systems: anomalous drift wave turbulence in plasmas and homogeneous, isotropic turbulence in fluids. For the plasma case, the time and length scales of the turbulent state result in the asymptotic expansion of the Vlasov/Poisson equations taking the form of nonlinear gyrokinetic theory. Questions regarding this theory and modern Hamiltonian perturbation methods are discussed and resolved. A new alternative Hamiltonian method is described. The Eulerian Direct Interaction Approximation (EDIA) is slightly reformulated and applied to the equations of nonlinear gyrokinetic theory. Using a similarity transformation technique, expressions for the thermal diffusivity are derived from the EDIA equations for various geometries, including a tokamak. In particular, the unique result for generalized geometry may be of use in evaluating fusion reactor designs and theories of anomalous thermal transport in tokamaks. Finally, a new and useful property of the EDIA is pointed out. For the fluid case, an asymptotic expansion is applied to the Navier-Stokes equation and the results lead to the speculation that such an approach may resolve the problem of predicting the Kolmogorov inertial range energy spectrum for homogeneous, isotropic turbulence. 45 refs., 3 figs
Asymptotic expansions for high-contrast elliptic equations
Calo, Victor M.
2014-03-01
In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in [Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model Simul. 8 (2010) 1461-1483] where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the paper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high-and low-conductivity inclusions. © 2014 World Scientific Publishing Company.
Asymptotic expansion of the multi-orientable random tensor model
Fusy, Eric
2014-01-01
Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in N, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.
Relativistic Stars in Starobinsky gravity matched asymptotic expansion
Arapoğlu, Savaş; Ekşi, K Yavuz
2016-01-01
We study the structure of relativistic stars in $\\mathcal{R}+\\alpha \\mathcal{R}^{2}$ theory using the method of matched asymptotic expansion to handle the higher order derivatives in field equations arising from the higher order curvature term. We find solutions, parametrized by $\\alpha$, for uniform density stars matching to the Schwarzschild solution outside the star. We obtain the mass-radius relations and study the dependence of maximum mass on $\\alpha$. We find that $M_{\\max} \\propto \\alpha^{-3/2}$ for values of $\\alpha$ larger than $10~{\\rm km^2}$. For each $\\alpha$ the maximum mass configuration has the biggest compactness parameter ($\\eta = GM/Rc^2$) and we argue that the general relativistic stellar configuration corresponding to $\\alpha=0$ is the most compact among these.
Renormalization and asymptotic expansion of Dirac's polarized vacuum
Gravejat, Philippe; Séré, Eric
2010-01-01
We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant $\\alphaph$, provided that the ultraviolet cut-off behaves as $\\Lambda\\sim e^{3\\pi(1-Z_3)/2\\alphaph}\\gg1$. The renormalization parameter $0
Non-LTE transfer - 3. Asymptotic expansion for small epsilon
Radiative transfer with complete frequency redistribution in the full- or half-space with interior sources is considered in the limit of small epsilon (probability of collisional destruction). It is shown that a suitably scaled source function satisfied for epsilon → a singular equation. In the full-space case with a Doppler profile this reads S(tau) - B(au) =-(-Δ)sup(1/2)S(tau) where S and B are the (scaled) source function and thermal source function, (-Δ)sup(1/2) is the square-root of the negative of the Laplacian; this equation replaces the classical diffusion equation for coherent scattering. In the half-space case, the asymptotic expansion gives only the interior source function (vanishing at the surface) and there is an additional boundary layer. The numerical aspects are considered in a companion paper. (author)
Solution of internal erosion equations by asymptotic expansion
Dubujet P.
2012-07-01
Full Text Available One dimensional coupled soil internal erosion and consolidation equations are considered in this work for the special case of well determined sand and clay mixtures with a small proportion of clay phase. An enhanced modelling of the effect of erosion on elastic soil behavior was introduced through damage mechanics concepts. A modified erosion law was proposed. The erosion phenomenon taking place inside the soil was shown to act like a perturbation affecting the classical soil consolidation equation. This interpretation has enabled considering an asymptotic expansion of the coupled erosion consolidation equations in terms of a perturbation parameter linked to the maximum expected internal erosion. A robust analytical solution was obtained via direct integration of equations at order zero and an adequate finite difference scheme that was applied at order one.
Naoto Kunitomo; Akihiko Takahashi
2003-01-01
This paper reviews the asymptotic expansion approach based on Malliavin-Watanabe Calculus in Mathematical Finance. We give the basic formulation of the asymptotic expansion approach and discuss its power and usefulness to solve important problems arised in nance. As illustrations we use three major problems in nance and give some useful formulae and new results including numerical analyses.
For the free models of statistical mechanics on torus, including the Ising model, dimer model and Gaussian model, exact asymptotic expansions of free energy, internal energy and specific heat in the vicinity of the critical point are found. It is shown that there is a direct relation between the terms of the expansion and Kronecker's double series. The latter can be expressed in terms of the elliptic θ-functions in all orders of the asymptotic expansion. (author)
On the ambiguity of field correlators represented by asymptotic perturbation expansions
Caprini, Irinel; Fischer, Jan; Vrkoč, Ivo
2009-01-01
Starting from the divergence pattern of perturbation expansions in Quantum Field Theory and the (assumed) asymptotic character of the series, we address the problem of ambiguity of a function determined by the perturbation expansion. We consider functions represented by an integral of the Laplace-Borel type along a general contour in the Borel complex plane. Proving a modified form of the Watson lemma, we obtain a large class of functions having the same asymptotic perturbation expansion. Som...
Allaire, Grégoire; Briane, Marc; Vanninathan, Muthusamy
2016-01-01
in press International audience In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation [5] where the homogenized equation can be improve...
Li-qun Cao; De-chao Zhu; Jian-Lan Luo
2002-01-01
In this paper, we will discuss the asymptotic behaviour for a class of hyper bolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of composite media. A complete multiscale asymptotic expansion and its rigorous verification will be reported.
HUANG; Yunqing; SHU; Shi; YU; Haiyuan
2004-01-01
In this paper, we discuss the error estimation of the linear finite element solution on criss-cross mesh. Using space orthogonal decomposition techniques, we obtain an asymptotic expansion and superconvergence results of the finite element solution. We first prove that the asymptotic expansion has different forms on the two kinds of nodes and then derive a high accuracy combination formula of the approximate derivatives.
A New Computational Scheme for Computing Greeks by the Asymptotic Expansion Approach
Matsuoka, Ryosuke; Takahashi, Akihiko; Uchida, Yoshihiko
2005-01-01
We developed a new scheme for computing "Greeks"of derivatives by an asymptotic expansion approach. In particular, we derived analytical approximation formulae for deltas and Vegas of plain vanilla and av-erage European call options under general Markovian processes of underlying asset prices. Moreover, we introduced a new variance reduction method of Monte Carlo simulations based on the asymptotic expansion scheme. Finally, several numerical examples under CEV processes con?rmed the validity...
Asymptotic expansions of integral means and applications to the ratio of gamma functions
Elezović, Neven; Vukšić, Lenka
2013-01-01
Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form $B(A(x))=C(x)$, where $B$ and $C$ have known asymptotic expansions. The results are illustrated by calculation of some important integral means connected with gamma and digamma functions.
Asymptotic Expansions in the CLT in Free Probability
Chistyakov, G P
2011-01-01
We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.
The renormalization method based on the Taylor expansion and applications for asymptotic analysis
Liu, Cheng-shi
2016-01-01
Based on the Taylor expansion, we propose a renormalization method for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the...
THE HIGHER ASYMPTOTIC EXPANSIONS FINDING FOR BOUNDARY VALUE PROBLEM OF THE ZOM MODEL
Kovalenko A. V.
2013-12-01
Full Text Available In this article authors propose the asymptotic solution of the boundary value problem modeling the transport of salt ions in the cell electrodialysis desalination unit. The domain of the camera desalting broken into two subdomains: electroneutrality and space charge. Subdomains has own asymptotic expansion. The subdomain of the space charge has unique solvability of the current approach used by the solvability condition of the next approximation
On the asymptotic expansion of the curvature of perturbations of the $L_{2}$ connection
Amit
2013-01-01
We establish that the Hitchin connection is a perturbation of the $L_{2}$-connection. We notice that such a formulation of the Hitchin connection does not necessarily require the manifold in question possessing a rigid family of Kähler structures. We then proceed to calculate the asymptotic expansion of general perturbations of the $L_{2}$-connection, and see when under certain assumptions such perturbations are flat and projectively flat. During the calculations we also found an asymptotic e...
Asymptotic expansions for large closed and loss queueing networks
Yaakov Kogan
2002-01-01
Full Text Available Loss and closed queueing network models have long been of interest to telephone and computer engineers and becoming increasingly important as models of data transmission networks. This paper describes a uniform approach that has been developed during the last decade for asymptotic analysis of large capacity networks with product form of the stationary probability distribution. Such a distribution has an explicit form up to the normalization constant, or the partition function. The approach is based on representing the partition function as a contour integral in complex space and evaluating the integral using the saddle point method and theory of residues. This paper provides an introduction to the area and a review of recent work.
Borisov, D
2009-01-01
We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant but can rotate along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We also show that for sufficiently thin rod this set contains any prescribed number of the first eigenvalues and give the complete asymptotic expansions for the associated eigenfunctions in C^k-norms.
Matched asymptotic expansions and the numerical treatment of viscous-inviscid interaction
Veldman, AEP
2001-01-01
The paper presents a personal view on the history of viscous-inviscid interaction methods, a history closely related to the evolution of the method of matched asymptotic expansions. The main challenge in solving Prandtl's boundary-layer equations has been to overcome the singularity at a point of st
An Enhanced Asymptotic Expansion for the Stability of Nonlinear Elastic Structures
Christensen, Claus Dencker; Byskov, Esben
2010-01-01
A new, enhanced asymptotic expansion applicable to stability of structures made of nonlinear elastic materials is established. The method utilizes “hyperbolic” terms instead of the conventional polynomial terms, covers full kinematic nonlinearity and is applied to nonlinear elastic Euler columns...... with two different types of cross-section. Comparison with numerical results show that our expansion provides more accurate predictions of the behavior than usual expansions. The method is based on an extended version of the principle of virtual displacements that covers cases with auxiliary conditions...
The renormalization method based on the Taylor expansion and applications for asymptotic analysis
Liu, Cheng-shi
2016-01-01
Based on the Taylor expansion, we propose a renormalization method for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the same time, the mathematical foundation of the method is simple and the logic of the method is very clear, therefore, it is very easy in practice. As application, we obtain the uniform valid asymptotic solutions to some problems including vector field, boundary layer and boundary value problems of nonlinear wave equations. Moreover, we discuss the normal form theory and reduction equations of dynamical systems. Furthermore, by combining the topological deformation and the RG method, a modified method namely the homotopy r...
Honkonen, J; Nalimov, M Yu
2002-01-01
Large order asymptotic behaviour of renormalization constants in the minimal subtraction scheme for the $\\phi ^4$ $(4-\\epsilon)$ theory is discussed. Well-known results of the asymptotic $4-\\epsilon $ expansion of critical indices are shown to be far from the large order asymptotic value. A {\\em convergent} series for the model $\\phi ^4$ $(4-\\epsilon)$ is then considered. Radius of convergence of the series for Green functions and for renormalisation group functions is studied. The results of the convergent expansion of critical indices in the $4-\\epsilon $ scheme are revalued using the knowledge of large order asymptotics. Specific features of this procedure are discussed.
Large order asymptotic behaviour of renormalization constants in the minimal subtraction scheme for the φ4 (4 - ε) theory is discussed. Well-known results of the asymptotic 4 - ε expansion of critical indices are shown to be far from the large order asymptotic value. A convergent series for the model φ4 (4 - ε) is then considered. Radius of convergence of the series for Green functions and for renormalisation group functions is studied. The results of the convergent expansion of critical indices in the 4 - ε scheme are revalued using the knowledge of large order asymptotics. Specific features of this procedure are discussed (Authors)
Guoping Xu; Harry Zheng
2010-01-01
In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is...
APPLICATION OF A HIGH-ORDER ASYMPTOTIC EXPANSION SCHEME TO LONG-TERM CURRENCY OPTIONS
Kohta Takehara; Masashi Toda; Akihiko Takahashi
2011-01-01
Recently, not only academic researchers but also many practitioners have used the methodology so-called ``an asymptotic expansion method" in their proposed techniques for a variety of financial issues. e.g. pricing or hedging complex derivatives under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory (Watanabe [1987], Yoshida [1992a,b]) in Malliavin calculus and essentially based on the framework initiated by Kunitomo and Takahashi [2003...
Fujikoshi, Yasunori; Shimizu, Ryoichi
1989-01-01
This paper deals with the distribution of $\\mathbf{X} = \\sum^{1/2}\\mathbf{Z}$, where $\\mathbf{Z}: p \\times 1$ is distributed as $N_p(0, I_p), \\sum$ is a positive definite random matrix and $\\mathbf{Z}$ and $\\sum$ are independent. Assuming that $\\sum = I_p + BB'$, we obtain an asymptotic expansion of the distribution function of $\\mathbf{X}$ and its error bound, which is useful in the situation where $\\sum$ tends to $I_p$. A stronger version of the expansion is also given. The results are appl...
Generalized asymptotic expansions for coupled wavenumbers in fluid-filled cylindrical shells
Kunte, M. V.; Sarkar, Abhijit; Sonti, Venkata R.
2010-12-01
Analytical expressions are found for the coupled wavenumbers in an infinite fluid-filled cylindrical shell using the asymptotic methods. These expressions are valid for any general circumferential order ( n). The shallow shell theory (which is more accurate at higher frequencies) is used to model the cylinder. Initially, the in vacuo shell is dealt with and asymptotic expressions are derived for the shell wavenumbers in the high- and the low-frequency regimes. Next, the fluid-filled shell is considered. Defining a relevant fluid-loading parameter μ, we find solutions for the limiting cases of small and large μ. Wherever relevant, a frequency scaling parameter along with some ingenuity is used to arrive at an elegant asymptotic expression. In all cases, Poisson's ratio ν is used as an expansion variable. The asymptotic results are compared with numerical solutions of the dispersion equation and the dispersion relation obtained by using the more general Donnell-Mushtari shell theory ( in vacuo and fluid-filled). A good match is obtained. Hence, the contribution of this work lies in the extension of the existing literature to include arbitrary circumferential orders ( n).
Vijay Prakash, S.; Sonti, Venkata R.
2016-02-01
Nonlinear acoustic wave propagation in an infinite rectangular waveguide is investigated. The upper boundary of this waveguide is a nonlinear elastic plate, whereas the lower boundary is rigid. The fluid is assumed to be inviscid with zero mean flow. The focus is restricted to non-planar modes having finite amplitudes. The approximate solution to the acoustic velocity potential of an amplitude modulated pulse is found using the method of multiple scales (MMS) involving both space and time. The calculations are presented up to the third order of the small parameter. It is found that at some frequencies the amplitude modulation is governed by the Nonlinear Schrödinger equation (NLSE). The first objective here is to study the nonlinear term in the NLSE. The sign of the nonlinear term in the NLSE plays a role in determining the stability of the amplitude modulation. Secondly, at other frequencies, the primary pulse interacts with its higher harmonics, as do two or more primary pulses with their resultant higher harmonics. This happens when the phase speeds of the waves match and the objective is to identify the frequencies of such interactions. For both the objectives, asymptotic coupled wavenumber expansions for the linear dispersion relation are required for an intermediate fluid loading. The novelty of this work lies in obtaining the asymptotic expansions and using them for predicting the sign change of the nonlinear term at various frequencies. It is found that when the coupled wavenumbers approach the uncoupled pressure-release wavenumbers, the amplitude modulation is stable. On the other hand, near the rigid-duct wavenumbers, the amplitude modulation is unstable. Also, as a further contribution, these wavenumber expansions are used to identify the frequencies of the higher harmonic interactions. And lastly, the solution for the amplitude modulation derived through the MMS is validated using these asymptotic expansions.
Calise, Anthony J.; Melamed, Nahum
1993-01-01
In this paper we develop a general procedure for constructing a matched asymptotic expansion of the Hamilton-Jacobi-Bellman equation based on the method of characteristics. The development is for a class of perturbation problems whose solution exhibits two-time-scale behavior. A regular expansion for problems of this type is inappropriate since it is not uniformly valid over a narrow range of the independent variable. Of particular interest here is the manner in which matching and boundary conditions are enforced when the expansion is carried out to first order. Two cases are distinguished - one where the left boundary condition coincides with or lies to the right of the singular region and one where the left boundary condition lies to the left of the singular region. A simple example is used to illustrate the procedure, and its potential application to aeroassisted plane change is described.
Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions
We present the first complete derivation of the well-known asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol
A note on asymptotic expansions for Markov chains using operator theory
Jensen, J.L.
1987-01-01
We consider asymptotic expansions for sums Sn on the form Sn = fhook0(X0) + fhook(X1, X0) + ... + fhook(Xn, Xn-1), where Xi is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on fhook(Xi, Xi-1), a simple representation of the cha...... characteristic function of Sn is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function x → E(g(Xi)exp[itfhook(Xi, x)]|Xi-1 = x). © 1987....
The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants
Borot, Gaetan; Eynard, Bertrand
2010-01-01
We establish the relation between two objects: an integrable system related to Painlev\\'e II equation, and the symplectic invariants of a certain plane curve S(TW). This curve describes the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This shows that the s -> -infinity asymptotic expansion of Tracy-Widow law F_{GUE}(s), governing the distribution of the maximal eigenvalue in hermitian random matrices, is given by sympl...
Dissipative dynamics of fission in the framework of asymptotic expansion of Fokker-Planck equation
The dynamics of fission has been formulated by generalising the asymptotic expansion of the Fokker-Planck equation in terms of the strength of the fluctuations where the diffusion coefficients depend on the stochastic variables explicitly. The prescission neutron multiplicities and mean kinetic energies of the evaporated neutrons have been calculated and compared with the respective experimental data over a wide range of excitation energy and compound nuclear mass. The mean and the variance of the total kinetic energies of the fission fragments have been calculated and compared with the experimental values. (orig.)
A formal asymptotic expansion of a solution of the initial problem for a singularly perturbed differential-operational nonlinear equation in a small parameter has been constructed in the critical case. Splash functions of and boundary functions have been estimated of found and assessment of the residual member of the expansion has been obtained
Tian, Jianxiang; Mulero, A
2016-01-01
Despite the fact that more that more than 30 analytical expressions for the equation of state of hard-disk fluids have been proposed in the literature, none of them is capable of reproducing the currently accepted numeric or estimated values for the first eighteen virial coefficients. Using the asymptotic expansion method, extended to the first ten virial coefficients for hard-disk fluids, fifty-seven new expressions for the equation of state have been studied. Of these, a new equation of state is selected which reproduces accurately all the first eighteen virial coefficients. Comparisons for the compressibility factor with computer simulations show that this new equation is as accurate as other similar expressions with the same number of parameters. Finally, the location of the poles of the 57 new equations shows that there are some particular configurations which could give both the accurate virial coefficients and the correct closest packing fraction in the future when higher virial coefficients than the t...
Matched asymptotic expansion for caged black holes: regularization of the post-Newtonian order
The 'dialogue of multipoles' matched asymptotic expansion for small black holes in the presence of compact dimensions is extended to the post-Newtonian order for arbitrary dimensions. Divergences are identified and are regularized through the matching constants, a method valid to all orders and known as Hadamard's partie finie. It is closely related to 'subtraction of self-interaction' and shows similarities with the regularization of quantum field theories. The black hole's mass and tension (and the 'black hole Archimedes effect') are obtained explicitly at this order, and a Newtonian derivation for the leading term in the tension is demonstrated. Implications for the phase diagram are analysed, finding agreement with numerical results and extrapolation shows hints for Sorkin's critical dimension-a dimension where the transition turns second order
Cardone, G; Panasenko, G P
2012-01-01
The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e. in a connected union of thin rectangles with heights of order $\\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed: it contains some Poiseuille type flows in the channels (rectangles) with some boundary layers correctors in the neighborhoods of the bifurcations of the channels. The estimates for the difference of the exact solution and its asymptotic approximation are proved.
Datta, Nilanjana; Hsieh, Min-Hsiu; Oppenheim, Jonathan
2016-05-01
State redistribution is the protocol in which given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol of coherent state merging as a primitive.
Masato Shinjo
2015-12-01
Full Text Available The Hankel determinant appears in representations of solutions to several integrable systems. An asymptotic expansion of the Hankel determinant thus plays a key role in the investigation of asymptotic analysis of such integrable systems. This paper presents an asymptotic expansion formula of a certain Casorati determinant as an extension of the Hankel case. This Casorati determinant is then shown to be associated with the solution to the discrete hungry Lotka–Volterra (dhLV system, which is an integrable variant of the famous prey–predator model in mathematical biology. Finally, the asymptotic behavior of the dhLV system is clarified using the expansion formula for the Casorati determinant.
Long-range asymptotic expansion of the diagonal Born–Oppenheimer correction
Graphical abstract: We derived formulas for coefficients A6, A8, A10 determining long-range asymptotic behavior of the adiabatic correction to the potential energy. The formulas were used to compute the coefficients for hydrogen molecule and helium dimers. Abstract: Formulas for the coefficients A6, A8, and A10 determining the long-range asymptotic behavior Ead(R)∼-A6R-6-A8R-8-A10R-10 of the diagonal Born–Oppenheimer (adiabatic) correction Ead(R) to the potential energy of a diatomic molecule are derived using two standard definitions of Ead(R). The first one is based on the explicit separation of the center-of-mass and rotational coordinates from the total Hamiltonian of a system, while the second definition uses the Born–Handy expression in a laboratory system of coordinates. Expressions for the asymptotic coefficients resulting from both definitions are proved to be equivalent. The obtained formulas are used to compute the asymptotics of the adiabatic correction for the ground state of the hydrogen molecule and for the helium dimer in the lowest quintet and singlet states. In the latter case basis sets up to 8-tuple zeta quality were used to adequately account for the electron correlation effects.
On the asymptotic expansion of the curvature of perturbations of the $L_{2}$ connection
De, Amit
We establish that the Hitchin connection is a perturbation of the $L_{2}$-connection. We notice that such a formulation of the Hitchin connection does not necessarily require the manifold in question possessing a rigid family of Kähler structures. We then proceed to calculate the asymptotic expan...
Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach
Cyril Grunspan
2011-01-01
We invert the Black-Scholes formula. We consider the cases low strike, large strike, short maturity and large maturity. We give explicitly the ?rst 5 terms of the expansions. A method to compute all the terms by induction is also given. At the money, we have a closed form formula for implied lognormal volatility in terms of a power series in call price.
Bao, Zhigang
2011-01-01
We study the renormalized real sample covariance matrix $H=X^TX/\\sqrt{MN}-\\sqrt{M/N}$ with $N/M\\rightarrow0$ in this paper, where $X=[X_{jk}]_{M\\times N}$ is an $M\\times N$ real random matrix with entries sharing the same first five order moments, and we assume $\\mathbb{E}|X_{jk}|^5\\leq\\infty$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\\mathbb{E}\\{m_N(z)\\}$, different from the Wigner matrix case, it is shown that the order of the first order term of the remainder may depend on the ratio $N/M$. Then for some fine test function, a central limit theorem for the linear eigenvalue statistics of $H$ is established. We show that the variance of the limiting normal distribution coincides with the case of a real Gaussian Wigner matrix, under the effect of central limit theorem and asymptotic independence for the matrix entries.
Bezyaev Vladimir Ivanovich
2014-09-01
Full Text Available The authors present an efficient algorithm different from the previously known to construct the asymptotics of solutions of nonautonomous systems of ordinary differential equations with meromorphic matrix. Schrödinger equation, Dirac system, Lippman-Schwinger equation and other equations of quantum mechanics with spherically symmetric and meromorphic potentials may be reduced to such systems. The Schrödinger equation and the Dirac system describe the stationary states of an electron in a Coulomb field with a fixed point charge in the description of the relativistic and nonrelativistic hydrogen atom. The Lippman-Schwinger equation of scattering theory describes the results of collision and interaction of quantum-mechanical particles in mathematical language after these particles have already diverged a long way from one another and ceased to interact. The observed algorithm supplements the known results and allows you to approach the analysis of the problems of this type with a fairly simple and at the same time, a universal point of view.
We consider one-dimensional Fokker–Planck and Schrödinger equations with a potential that approaches a periodic function at spatial infinity. We extend the low-energy expansion method, which was introduced in previous papers, to be applicable to such asymptotically periodic cases. Using this method, we study the low-energy behavior of the Green function. (paper)
Nasution, Muhammad Ridlo Erdata
2014-06-01
A new asymptotic expansion homogenization analysis is proposed to analyze 3-D composite in which thermomechanical and finite thickness effects are considered. Finite thickness effect is captured by relieving periodic boundary condition at the top and bottom of unit-cell surfaces. The mathematical treatment yields that only 2-D periodicity (i.e. in in-plane directions) is taken into account. A unit-cell representing the whole thickness of 3-D composite is built to facilitate the present method. The equivalent in-plane thermomechanical properties of 3-D orthogonal interlock composites are calculated by present method, and the results are compared with those obtained by standard homogenization method (with 3-D periodicity). Young\\'s modulus and Poisson\\'s ratio obtained by present method are also compared with experiments whereby a good agreement is particularly found for the Young\\'s modulus. Localization analysis is carried out to evaluate the stress responses within the unit-cell of 3-D composites for two cases: thermal and biaxial tensile loading. Standard finite element (FE) analysis is also performed to validate the stress responses obtained by localization analysis. It is found that present method results are in a good agreement with standard FE analysis. This fact emphasizes that relieving periodicity in the thickness direction is necessary to accurately simulate the real free-traction condition in 3-D composite. © 2014 Elsevier Ltd.
Dettmann, Carl P.
2002-01-01
Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order 64 in the noise strength. Comparison with the usual asymptotic expansions obtained from integrals and with a previous calculation of the electrostatic potential of exactly selfsimilar fractal charge distributions, suggests a remarkably accurate form for the late terms in the expansion, with parameters determined independently from the...
Mirela Kohr
2004-09-01
Full Text Available We study the low Reynolds number flow of an incompressible Newtonian fluid of infinite expanse past a cylinder of arbitrary cross section by using the method of matched asymptotic expansions. The analysis that will be made in this paper is equivalent to that developed by Power (1990 in order to solve the resulting inner (or Stokes problems with the completed double-layer boundary integral equation method (CDLBIEM due to Power and Miranda (1987. We will solve these problems by the boundary integral method developed by Hsiao and Kress (1985.
Profant, T.; Klusák, Jan; Ševeček, O.; Kotoul, M.; Hrstka, M.; Marcián, P.
Zurich : Trans Tech Publications, 2014 - (Šandera, P.), s. 745-748 ISBN 978-3-03785-934-6. ISSN 1013-9826. - (Key Engineering Materials. 592-593). [MSMF 7 - International Conference on Materials Structure & Micromechanics of Fracture /7./. Brno (CZ), 01.07.2013-03.07.2013] Institutional support: RVO:68081723 Keywords : Orthotropic bi-material notch * two-state integral * non-singular stress terms * T-stress * matched asymptotic expansion Subject RIV: JL - Materials Fatigue, Friction Mechanics http://www.scientific.net/KEM.592-593.745
Two-loop two-point functions with masses asymptotic expansions and Taylor series, in any dimension
Broadhurst, D J; Tarasov, O V
1993-01-01
In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and small $q^2$, in $d$ dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Pad\\'{e} approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon self-energy, for all $d$, and achieve highly accelerated convergence of its expansions in powers of $q^2/m^2$ or $m^2/q^2$, for $d=4$.
In this paper we apply to gravitational waves (GW) from the inspiral phase of binary systems a recently derived frequentist methodology to calculate analytically the error for a maximum likelihood estimate of physical parameters. We use expansions of the covariance and the bias of a maximum likelihood estimate in terms of inverse powers of the signal-to-noise ration (SNR)s where the square root of the first order in the covariance expansion is the Cramer Rao lower bound (CRLB). We evaluate the expansions, for the first time, for GW signals in noises of GW interferometers. The examples are limited to a single, optimally oriented, interferometer. We also compare the error estimates using the first two orders of the expansions with existing numerical Monte Carlo simulations. The first two orders of the covariance allow us to get error predictions closer to what is observed in numerical simulations than the CRLB. The methodology also predicts a necessary SNR to approximate the error with the CRLB and provides new insight on the relationship between waveform properties, SNR, dimension of the parameter space and estimation errors. For example the timing match filtering can achieve the CRLB only if the SNR is larger than the Kurtosis of the gravitational wave spectrum and the necessary SNR is much larger if other physical parameters are also unknown.
王德辉
2007-01-01
This paper is concerned with the distributional properties of a median unbiased estimator of ARCH(0,1) coefficient. The exact distribution of the estimator can be easily derived, however its practical calculations are too heavy to implement,even though the middle range of sample sizes. Since the estimator is shown to have asymptotic normality, asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements. Accuracies of expansion formulas are evaluated numerically, and the results of which show that we can effectively use the expansion as a fine approximatioh of the distribution with rapid calculations. Derived expansion are applied to testing hypothesis of stationarity, and an implementation for a real data set is illustrated.
A powerful new technique, using a two-dimensional position-sensitive multichannel plate detector coupled with a spherical-sector electron analyzer, allows rapid data collection of doubly- and triply-differential electron spectra. This device is used to collect spectra, doubly differential in electron energy and emission angle, for the electron loss to the continuum cross section, where an electron originally bound to the incoming projectile is excited just over the ionization limit into the continuum of the ion, for O5+ projectiles incident on helium and argon gas targets at energies of 41, 82, and 105 MeV at the Holifield Heavy Ion Research Facility at Oak Ridge National Laboratory. By making a multipole expansion of the cross section in the projectile frame, and appropriately including the effects of the electron analyzer, a general characterization of the electron loss to the continuum cusp (ELC) shape is made. These results are directly compared with the recent theory of J. Burgdoerfer, which derives the multipole content for the ELC cross section, and excellent agreement is found for the case of argon targets at the highest two energies, where the theory is expected to be more reliable. The lack of good agreement with helium target data suggests the need for better treatment of electron-electron interactions within the theoretical framework
Nonstandard asymptotic analysis
Berg, Imme
1987-01-01
This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes topics of general interest, like the remarkable behaviour of Taylor polynomials of elementary functions. Noting that within nonstandard analysis, "small", "large", and "domain of validity of asymptotic behaviour" have a precise meaning, a nonstandard alternative to classical asymptotics is developed. Special emphasis is given to applications in numerical approximation by convergent and divergent expansions: in the latter case a clear asymptotic answer is given to the problem of optimal approximation, which is valid for a large class of functions including many special functions. The author's approach is didactical. The book opens with a large introductory chapter which can be read without much knowledge of nonstandard analysis. Here the main features of the t...
Numerical Asymptotic Solutions Of Differential Equations
Thurston, Gaylen A.
1992-01-01
Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.
Chako, N. [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires
1968-07-01
We have applied the method of stationary phase to evaluate double and multiple integrals of the type: (A) U(k) = g(x)e{sup ik{phi}}{sup (x)} d(x), (x)=(x{sub 1},..., x{sub n}) for large values of the parameter k. In the first part we have established in a rigorous manner the stationary phase method to double and multiple integrals of type (A). Furthermore we have obtained an asymptotic expansion of (A), if the amplitude and phase functions can be developed in a canonical form near the vicinity of critical or stationary points of the integral. This development contains as particular cases all those which are important in physical applications, especially, to diffraction and scattering of electromagnetic and corpuscular waves by optical systems, diffracting bodies and potential scatterers. In the second part we have considered the problem of convergence of the expansion of the principal contribution to the integral in the asymptotic sense of Poincare. The proof is based on the increasing method used in mathematical analysis. The third part is devoted to the derivation of various asymptotic series due to different types of critical or stationary points associated with the amplitude and phase functions. In the fourth part we have generalized the method to multiple integrals and to the case where the parameter k enter implicitly in the phase function The latter type of integrals extend the scope of the former type to a number of important physical problems; for instance, to the propagation of waves in dispersive and absorbing media. In the last chapter we have made a study and compared the results obtained by the application of the stationary phase method to the integrals (double) of diffraction and the results derived by using the Young-Rubinowicz method. Result of our analysis shows the equivalence of the two methods of approach to the problems of diffraction based, on one hand, on the Fresnel-Kirchhoff theory and, on the other hand, the Young-Rubinowicz theory
Asymptotic behavior of atomic momentals
Thakkar, Ajit J.
1987-05-01
Knowledge of the large and small momentum transfer behavior of the electron momentum distribution is an important ingredient in the analysis of experimental isotropic Compton profiles. This behavior ultimately rests upon the asymptotic behavior of atomic momentals (momentum space orbitals). The small momentum Maclaurin expansion and the large momentum asymptotic expansion of atomic momentals with arbitrary angular momentum quantum number are derived in this paper. Their implications for momentum densities and Compton profiles are derived and discussed.
Asymptotically hyperbolic connections
Fine, Joel; Krasnov, Kirill; Scarinci, Carlos
2015-01-01
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-d...
Exponential asymptotics and gravity waves
Chapman, S. J.; Vanden-Broeck, J.
2006-01-01
The problem of irrotational inviscid incompressible free-surface flow is examined in the limit of small Froude number. Since this is a singular perturbation, singularities in the flow field (or its analytic continuation) such as stagnation points, or corners in submerged objects or on rough beds, lead to a divergent asymptotic expansion, with associated Stokes lines. Recent techniques in exponential asymptotics are employed to observe the switching on of exponentially small gravity waves acro...
Asymptotic analysis and boundary layers
Cousteix, Jean
2007-01-01
This book presents a new method of asymptotic analysis of boundary-layer problems, the Successive Complementary Expansion Method (SCEM). The first part is devoted to a general comprehensive presentation of the tools of asymptotic analysis. It gives the keys to understand a boundary-layer problem and explains the methods to construct an approximation. The second part is devoted to SCEM and its applications in fluid mechanics, including external and internal flows. The advantages of SCEM are discussed in comparison with the standard Method of Matched Asymptotic Expansions. In particular, for the first time, the theory of Interactive Boundary Layer is fully justified. With its chapter summaries, detailed derivations of results, discussed examples and fully worked out problems and solutions, the book is self-contained. It is written on a mathematical level accessible to graduate and post-graduate students of engineering and physics with a good knowledge in fluid mechanics. Researchers and practitioners will estee...
Asymptotically flat and regular Cauchy data
Dain, S
2002-01-01
I describe the construction of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate. I emphasize the motivations and the main ideas behind the proofs.
8. Asymptotically Flat and Regular Cauchy Data
Dain, Sergio
I describe the construction of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers of a radial coordinate. I emphasize the motivations and the main ideas behind the proofs.
EMC effect: asymptotic freedom with nuclear targets
General features of the EMC effect are discussed within the framework of quantum chromodynamics as expressed via the operator product expansion and asymptotic freedom. These techniques are reviewed with emphasis on the target dependence. 22 references
Dirichlet eigenvalues of asymptotically flat triangles
Ourmières-Bonafos, Thomas
2015-01-01
This paper is devoted to the study of the eigenpairs of the Dirichlet Laplacian on a family of triangles where two vertices are fixed and the altitude associated with the third vertex goes to zero. We investigate the dependence of the eigenvalues on this altitude. For the first eigenvalues and eigenfunctions, we obtain an asymptotic expansion at any order at the scale cube root of this altitude due to the influence of the Airy operator. Asymptotic expansions of the eigenpairs are provided, ex...
Asymptotic perturbation theory of waves
Ostrovsky, Lev
2014-01-01
This book is an introduction to the perturbation theory for linear and nonlinear waves in dispersive and dissipative media. The main focus is on the direct asymptotic method which is based on the asymptotic expansion of the solution in series of one or more small parameters and demanding finiteness of the perturbations; this results in slow variation of the main-order solution. The method, which does not depend on integrability of basic equations, is applied to quasi-harmonic and non-harmonic periodic waves, as well as to localized waves such as solitons, kinks, and autowaves. The basic theor
Buividovich, P V
2015-01-01
We discuss the feasibility of applying Diagrammatic Monte-Carlo algorithms to the weak-coupling expansions of asymptotically free quantum field theories, taking the large-$N$ limit of the $O(N)$ sigma-model as the simplest example where exact results are available. We use stereographic mapping from the sphere to the real plane to set up the perturbation theory, which results in a small bare mass term proportional to the coupling $\\lambda$. Counting the powers of coupling associated with higher-order interaction vertices, we arrive at the double-series representation for the dynamically generated mass gap in powers of both $\\lambda$ and $\\log(\\lambda)$, which converges quite quickly to the exact non-perturbative answer. We also demonstrate that it is feasible to obtain the coefficients of these double series by a Monte-Carlo sampling in the space of Feynman diagrams. In particular, the sign problem of such sampling becomes milder at small $\\lambda$, that is, close to the continuum limit.
Exponential asymptotics of the Voigt functions
Paris, R. B.
2015-06-01
We obtain the asymptotic expansion of the Voigt functionss K( x, y) and L( x, y) for large (real) values of the variables x and y, paying particular attention to the exponentially small contributions. A Stokes phenomenon is encountered as with x > 0 fixed. Numerical examples are presented to demonstrate the accuracy of these new expansions.
Large degree asymptotics of generalized Bessel polynomials
López, J.L.; Temme, N.M.
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in t
Numerical integration of asymptotic solutions of ordinary differential equations
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Asymptotic safety: A simple example
We use the Gross-Neveu model in 2f expansion where the model is known to be renormalizable to all orders. In this limit, the fixed-point action as well as all universal critical exponents can be computed analytically. As asymptotic safety has become an important scenario for quantizing gravity, our description of a well-understood model is meant to provide for an easily accessible and controllable example of modern nonperturbative quantum field theory.
Asymptotically hyperbolic black holes in Horava gravity
Janiszewski, Stefan
2014-01-01
Solutions of Hořava gravity that are asymptotically Lifshitz are explored. General near boundary expansions allow the calculation of the mass of these spacetimes via a Hamiltonian method. Both analytic and numeric solutions are studied which exhibit a causal boundary called the universal horizon, and are therefore black holes of the theory. The thermodynamics of an asymptotically Anti-de Sitter Hořava black hole are verified.
陈刚; 王梦婕
2014-01-01
通过对χ2分布概率密度函数的自变量进行标准化变换,将其展开成如下形式：2nχ2( x；n)=1+r1(t)n +r2(t)n +r3(t)n n +r4(t)n2éëùûφ(t)+o 1n2(),其中n为自由度,φ(t)为标准正态分布的密度函数,ri(t)(1≤i≤4)均为关于t的多项式。从该展开式得到χ2分布密度函数的一个近似计算公式。进一步建立φ( t)的幂系数积分递推关系,得到χ2分布函数的渐近展开式。最后通过数值计算验证了这些结果在实际应用中的有效性。%Through the transformation of the independent variable of χ2 distribution probability density function,degree of freedom of which is n,the equation can be expanded as follows: 2nχ2(x;n)=f(t;n)= 1+r1(t)n +r2(t)n +r3(t)n n +r4(t)n2éë ùûφ(t)+o 1n2( ) ,here,φ(t) is a density function of standard normal distribution;ri(t) is a 3i order polynomial of t(1≤i≤4). An approximate formula can be obtained from the expansion of the distribution density function. We further establish the integral recurrence relations of the power coefficients of the standard normal density function and obtain the asymptotic expansion of the distribution function ofχ2 . Finally,the effectiveness of these results in practical application was verified by the numerical calculations.
Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case
Zhang JM; Cao LH
2010-01-01
We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation where and are integers, and have asymptotic expansions of the form , , for large values of , , and .
Asymptotic solution for EI Nino-southern oscillation of nonlinear model
MO Jia-qi; LIN Wan-tao
2008-01-01
A class of nonlinear coupled system for E1 Nino-Southern Oscillation (ENSO) model is considered. Using the asymptotic theory and method of variational iteration, the asymptotic expansion of the solution for ENSO models is obtained.
Extended asymptotic theory of unstable resonator modes.
Li, Y Q; Sung, C C
1990-10-20
The modes in an unstable resonator can be computed within the limit of a large Fresnel number using the asymptotic expansion of the diffraction integral, as shown by Horwitz, Butts, and Avizonis. The expansion is not valid for the points of interest around or beyond the shadow boundary of the output light. We use a better numerical representation, which extends the regions of use. The comparison of several cases with earlier work shows that the asymptotic theory can be successfully applied for all parameters without restrictions. PMID:20577410
Asymptotically Safe Dark Matter
Sannino, Francesco; Shoemaker, Ian M.
2015-01-01
We introduce a new paradigm for dark matter (DM) interactions in which the interaction strength is asymptotically safe. In models of this type, the coupling strength is small at low energies but increases at higher energies, and asymptotically approaches a finite constant value. The resulting...... searches are the primary ways to constrain or discover asymptotically safe dark matter....
Large Degree Asymptotics of Generalized Bessel Polynomials
López, J. L.; Temme, Nico
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\\pm i/n$ are derived, and a new expansion in terms of modified Bessel fu...
Asymptotic analysis of the Nörlund and Stirling polynomials
Mark Daniel Ward
2012-04-01
Full Text Available We provide a full asymptotic analysis of the N{\\"o}rlund polynomials and Stirling polynomials. We give a general asymptotic expansion---to any desired degree of accuracy---when the parameter is not an integer. We use singularity analysis, Hankel contours, and transfer theory. This investigation was motivated by a need for such a complete asymptotic description, with parameter 1/2, during this author's recent solution of Wilf's 3rd (previously Unsolved Problem.
Large Deviations and Asymptotic Methods in Finance
Gatheral, Jim; Gulisashvili, Archil; Jacquier, Antoine; Teichmann, Josef
2015-01-01
Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find th...
Asymptotic analysis, Working Note No. 1: Basic concepts and definitions
Garbey, M. [Universite Claude Bernard Lyon 1, 69 - Villeurbanne (France). Lab. d`Analyse Numerique; Kaper, H.G. [Argonne National Lab., IL (United States)
1993-07-01
In this note we introduce the basic concepts of asymptotic analysis. After some comments of historical interest we begin by defining the order relations O, o, and O{sup {number_sign}}, which enable us to compare the asymptotic behavior of functions of a small positive parameter {epsilon} as {epsilon} {down_arrow} 0. Next, we introduce order functions, asymptotic sequences of order functions and more general gauge sets of order functions and define the concepts of an asymptotic approximation and an asymptotic expansion with respect to a given gauge set. This string of definitions culminates in the introduction of the concept of a regular asymptotic expansion, also known as a Poincare expansion, of a function f : (0, {epsilon}{sub o}) {yields} X, where X is a normed vector space of functions defined on a domain D {epsilon} R{sup N}. We conclude the note with the asymptotic analysis of an initial value problem whose solution is obtained in the form of a regular asymptotic expansion.
Asymptotic estimates for generalized Stirling numbers
Chelluri, R.; Richmond, L.B.; Temme, Nico
2000-01-01
Uniform asymptotic expansions are given for the Stirling numbers of the first kind for integral arguments and for the second kind as defined for real arguments by Flajolet and Prodinger. The logconcavity of the resulting real valued function of Flajolet and Prodinger is established for a range including the classical integral domain.
Asymptotic estimates for generalized Stirling numbers
Chelluri, R.; Richmond, L.B.; Temme, N.M.
1999-01-01
Uniform asymptotic expansions are given for the Stirling numbers of the first kind for integral arguments and for the second kind as defined for real arguments by Flajolet and Prodinger. The logconcavity of the resulting real valued function of Flajolet and Prodinger is established for a range inclu
Asymptotics and Borel summability
Costin, Ovidiu
2008-01-01
Incorporating substantial developments from the last thirty years into one resource, Asymptotics and Borel Summability provides a self-contained introduction to asymptotic analysis with special emphasis on topics not covered in traditional asymptotics books. The author explains basic ideas, concepts, and methods of generalized Borel summability, transseries, and exponential asymptotics. He provides complete mathematical rigor while supplementing it with heuristic material and examples, so that some proofs may be omitted by applications-oriented readers.To give a sense of how new methods are us
On asymptotic extension dimension
Repovš, Dušan; Zarichnyi, Mykhailo
2011-01-01
The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper metric space and extension dimension of its Higson corona.
ASYMPTOTIC QUANTIZATION OF PROBABILITY DISTRIBUTIONS
Klaus P(o)tzelberger
2003-01-01
We give a brief introduction to results on the asymptotics of quantization errors.The topics discussed include the quantization dimension,asymptotic distributions of sets of prototypes,asymptotically optimal quantizations,approximations and random quantizations.
Asymptotic Resource Usage Bounds
Albert E.; Alonso D.; Arenas P.; Genaim S.; Puebla G.
2009-01-01
When describing the resource usage of a program, it is usual to talk in asymptotic terms, such as the well-known “big O” notation, whereby we focus on the behaviour of the program for large input data and make a rough approximation by considering as equivalent programs whose resource usage grows at the same rate. Motivated by the existence of non-asymptotic resource usage analyzers, in this paper, we develop a novel transformation from a non-asymptotic cost function (which can be produced by ...
Bousso, Raphael
2016-01-01
We show that known entropy bounds constrain the information carried off by radiation to null infinity. We consider distant, planar null hypersurfaces in asymptotically flat spacetime. Their focussing and area loss can be computed perturbatively on a Minkowski background, yielding entropy bounds in terms of the energy flux of the outgoing radiation. In the asymptotic limit, we obtain boundary versions of the Quantum Null Energy Condition, of the Generalized Second Law, and of the Quantum Bousso Bound.
Asymptotically Safe Dark Matter
Sannino, Francesco
2014-01-01
We introduce a new paradigm for dark matter interactions according to which the interaction strength is asymptotically safe. In models of this type, the interaction strength is small at low energies but increases at higher energies towards a finite constant value of the coupling. The net effect is to partially offset direct detection constraints without affecting thermal freeze-out at higher energies. High-energy collider and indirect annihilation searches are the primary ways to constrain or discover asymptotically safe dark matter.
Asymptotic properties of difference schemes of maximum odd accuracy
The asymptotic estimates of the difference Green function and difference step function are obtained for difference schemes of maximum odd accuracy (2k-1), k = 0(1nh-1), h is step of the net. The problem is reduced to construction of asymptotic expansion of some integrals. Principal estimates are obtained by application of the saddle point method. The saddle points determining asymptotic expansion are situated near finite radius circle and they become close to each other when h → 0. These asymptotic estimates give that the numerical solution convergences to the solution of the continuous problem with the rate 0(hsup(N+α) 1n1nh-1). The width of zone over which an isolated discontinuity spreads out is proportional to 1nh-1
Quasi-extended asymptotic functions
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 delta2 of an asymptotic function delta similar to Dirac's delta-function, is constructed as an example
Theory of asymptotic operation. A summary of basic principles
Tkachov, Fyodor V.
1997-01-01
This summary of several talks given in 1990-1993 discusses the problem of asymptotic expansions of multiloop Feynman diagrams in masses and external momenta - a central problem in perturbative quantum field theory. Basic principles of the theory of asymptotic operation -- the most powerful tool for that purpose -- are discussed. Its connection with the conventional methods is explained (the BPHZ theory, the method of leading logarithmic approximation etc.). The problem of non-euclidean asympt...
Berezin-Toeplitz quantization and its kernel expansion
Ma, Xiaonan
2012-01-01
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.
Puschnigg, Michael
1996-01-01
The aim of cyclic cohomology theories is the approximation of K-theory by cohomology theories defined by natural chain complexes. The basic example is the approximation of topological K-theory by de Rham cohomology via the classical Chern character. A cyclic cohomology theory for operator algebras is developed in the book, based on Connes' work on noncommutative geometry. Asymptotic cyclic cohomology faithfully reflects the basic properties and features of operator K-theory. It thus becomes a natural target for a Chern character. The central result of the book is a general Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. Besides this, the book contains numerous examples and calculations of asymptotic cyclic cohomology groups.
Skin Effect in Electromagnetism and Asymptotic Behaviour of Skin Depth for High Conductivity
Péron, Victor; Caloz, Gabriel; Dauge, Monique; Faou, Erwan
2010-01-01
International audience We study a three-dimensional model for the skin effect in electromagnetism. The 3-D case of the Maxwell equations in harmonic regime on a domain composed of a dielectric and of a highly conducting material is considered. We derive an asymptotic expansion with respect to a small parameter related to high conductivity. This expansion is theoretically justified at any order. The asymptotic expansion and numerical simulations in axisymmetric geometry exhibit the influenc...
A large part of physics consists of learning which asymptotic methods to apply where, yet physicists are not always taught asymptotics in a systematic way. Asymptotology is given using an example from aerodynamics, and a rent Phys. Rev. Letter Comment is used as a case study of one subtle way things can go wrong. It is shown that the application of local analysis leads to erroneous conclusions regarding the existence of a continuous spectrum in a simple test problem, showing that a global analysis must be used. The final section presents results on a more sophisticated example, namely the WKBJ solution of Mathieu equation. 13 refs., 2 figs
Jones, D S
1997-01-01
Many branches of science and engineering involve applications of mathematical analysis. An important part of applied analysis is asymptotic approximation which is, therefore, an active area of research with new methods and publications being found constantly. This book gives an introduction to the subject sufficient for scientists and engineers to grasp the fundamental techniques, both those which have been known for some time and those which have been discovered more recently. The asymptotic approximation of both integrals and differential equations is discussed and the discussion includes hy
Asymptotics for spherical needlets
Baldi, P.; Kerkyacharian, G.; Marinucci, D.; Picard, D.
We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high-frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.
Asymptotic freedom, asymptotic flatness and cosmology
Holographic RG flows in some cases are known to be related to cosmological solutions. In this paper another example of such correspondence is provided. Holographic RG flows giving rise to asymptotically-free β-functions have been analyzed in connection with holographic models of QCD. They are shown upon Wick rotation to provide a large class of inflationary models with logarithmically-soft inflaton potentials. The scalar spectral index is universal and depends only on the number of e-foldings. The ratio of tensor to scalar power depends on the single extra real parameter that defines this class of models. The Starobinsky inflationary model as well as the recently proposed models of T-inflation are members of this class. The holographic setup gives a completely new (and contrasting) view to the stability, naturalness and other problems of such inflationary models
Litim, Daniel F.; Sannino, Francesco
2014-01-01
We study the ultraviolet behaviour of four-dimensional quantum field theories involving non-abelian gauge fields, fermions and scalars in the Veneziano limit. In a regime where asymptotic freedom is lost, we explain how the three types of fields cooperate to develop fully interacting ultraviolet ...
Cristallini, Achille
2016-07-01
A new and intriguing machine may be obtained replacing the moving pulley of a gun tackle with a fixed point in the rope. Its most important feature is the asymptotic efficiency. Here we obtain a satisfactory description of this machine by means of vector calculus and elementary trigonometry. The mathematical model has been compared with experimental data and briefly discussed.
Asymptotic Enumeration of RNA Structures with Pseudoknots
Jin, Emma Y
2007-01-01
In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for $k$-noncrossing RNA structures. Our results are based on the generating function for the number of $k$-noncrossing RNA pseudoknot structures, ${\\sf S}_k(n)$, derived in \\cite{Reidys:07pseu}, where $k-1$ denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function $\\sum_{n\\ge 0}{\\sf S}_k(n)z^n$ and obtain for $k=2$ and $k=3$ the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary $k$ singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula ${\\sf S}_3(n) \\sim \\frac{10.4724\\cdot 4!}{n(n...
Hydrodynamics, resurgence and trans-asymptotics
Basar, Gokce
2015-01-01
The second-order hydrodynamical description of a homogeneous conformal plasma that undergoes a boost- invariant expansion is given by a single nonlinear ordinary differential equation, whose resurgent asymptotic properties we study, developing further the recent work of Heller and Spalinski [Phys. Rev. Lett. 115, 072501 (2015)]. Resurgence clearly identifies the non-hydrodynamic modes that are exponentially suppressed at late times, analogous to the quasi-normal-modes in gravitational language, organizing these modes in terms of a trans-series expansion. These modes are analogs of instantons in semi-classical expansions, where the damping rate plays the role of the instanton action. We show that this system displays the generic features of resurgence, with explicit quantitative relations between the fluctuations about different orders of these non-hydrodynamic modes. The imaginary part of the trans-series parameter is identified with the Stokes constant, and the real part with the freedom associated with init...
Optimistic Agents are Asymptotically Optimal
Sunehag, Peter; Hutter, Marcus
2012-01-01
We use optimism to introduce generic asymptotically optimal reinforcement learning agents. They achieve, with an arbitrary finite or compact class of environments, asymptotically optimal behavior. Furthermore, in the finite deterministic case we provide finite error bounds.
Asymptotic Flatness in Rainbow Gravity
Hackett, Jonathan
2005-01-01
A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is compatible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented.
Coarse geometry and asymptotic dimension
Grave, Bernd
2006-01-01
We consider asymptotic dimension of coarse spaces. We analyse coarse structures induced by metrisable compactifications. We calculate asymptotic dimension of coarse cell complexes. We calculate the asymptotic dimension of certain negatively curved spaces, e.g. for complete, simply connected manifolds with bounded, strictly negative sectional curvature.
Another Asymptotic Notation : "Almost"
Mondal, Nabarun; Ghosh, Partha P.
2013-01-01
Asymptotic notations are heavily used while analysing runtimes of algorithms. Present paper argues that some of these usages are non trivial, therefore incurring errors in communication of ideas. After careful reconsidera- tion of the various existing notations a new notation is proposed. This notation has similarities with the other heavily used notations like Big-Oh, Big Theta, while being more accurate when describing the order relationship. It has been argued that this notation is more su...
Higher order asymptotics for the Hirota equation via Deift–Zhou higher order theory
Huang, Lin, E-mail: huangl12@fudan.edu.cn [School of Mathematical Sciences, Fudan University, Shanghai 200433 (China); Xu, Jian, E-mail: jianxu02@gmail.com [College of Science, University of Shanghai for Science and Technology, Shanghai 200093 (China); Fan, En-gui, E-mail: faneg@fudan.edu.cn [School of Mathematical Sciences, Fudan University, Shanghai 200433 (China)
2015-01-02
In this paper, the Deift–Zhou higher order asymptotic theory is used to further establish the full asymptotic expansion for the solution of the Hirota equation to all order, as t→∞. The method is rigorous and does not rely on an a priori ansatz for the form of the solution. - Highlights: • Give RHP for Hirota equation. • Systemically apply Deift–Zhou theory. • Give full asymptotics for Hirota equation.
Gulisashvili, A.; Stein, E. M.
2009-01-01
We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas f...
Local heat kernel asymptotics for nonminimal differential operators
The method for computing the local coefficients in the asymptotical heat kernel expansion is extended to the case of nonminimal differential operators. The lowest expansion coefficients are calculated for the second-order nonminimal operators on oneforms over a riemannian space in arbitrary dimensions. Unlike the coefficients for the minimal second-order operators those for nonminimal ones turn out to be essentially dependent on the space dimension. (orig.)
Asymptotic freedom in Horava-Lifshitz gravity
D'Odorico, Giulio; Schutten, Marrit
2014-01-01
We use the Wetterich equation for foliated spacetimes to study the RG flow of projectable Horava-Lifshitz gravity coupled to n Lifshitz scalars. Using novel results for anisotropic heat kernels, the matter-induced beta functions for the gravitational couplings are computed explicitly. The RG flow exhibits an UV attractive anisotropic Gaussian fixed point where Newton's constant vanishes and the extra scalar mode decouples. This fixed point ensures that the theory is asymptotically free in the large-n expansion, indicating that projectable Horava-Lifshitz gravity is perturbatively renormalizable. Notably, the fundamental fixed point action does not obey detailed balance.
The asymptotic safety scenario and scalar field inflation
Rahmede, Christoph
2013-01-01
We study quantum gravity corrections to early universe cosmology as resulting within the asymptotic safety scenario. We analyse if it is possible to obtain accelerated expansion in the regime of the renormalisation group fixed point in a theory with Einstein-Hilbert gravity and a scalar field. We show how this phase impacts cosmological perturbations observed in the cosmic microwave background.
Gap asymptotics in a weakly bent leaky quantum wire
Exner, Pavel; Kondej, S.
2015-01-01
Roč. 48, č. 49 (2015), s. 495301. ISSN 1751-8113 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : singular Schroedinger operators * delta interaction * leaky quantum wires * weak perturbation * asymptotic expansion Subject RIV: BE - Theoretical Physics Impact factor: 1.583, year: 2014
Mathematical justification of Kelvin-Voigt beam models by asymptotic methods
Rodríguez-Arós, Á. D.; Viaño, J. M.
2012-06-01
The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method. The material behaviour is modelled by using a general Kelvin-Voigt constitutive law.
Duality and asymptotic geometries
Boonstra, H J; Skenderis, K
1997-01-01
We consider a series of duality transformations that leads to a constant shift in the harmonic functions appearing in the description of a configuration of branes. This way, for several intersections of branes, we can relate the original brane configuration which is asymptotically flat to a geometry of the type $adS_k \\xx E^l \\xx S^m$. The implications of our results for supersymmetry enhancement, M(atrix) theory at finite N, and for supergravity theories in diverse dimensions are discussed.
On the asymptotic distribution of the mean absolute deviation about the mean
Segers, Johan
2014-01-01
The mean absolute deviation about the mean is an alternative to the standard deviation for measuring dispersion in a sample or in a population. For stationary, ergodic time series with a finite first moment, an asymptotic expansion for the sample mean absolute deviation is proposed. The expansion yields the asymptotic distribution of the sample mean absolute deviation under a wide range of settings, allowing for serial dependence or an infinite second moment. Key words: central limit theorem;...
On the asymptotic distribution of the mean absolute deviation about the mean
Segers, Johan
2014-01-01
The mean absolute deviation about the mean is an alternative to the standard deviation for measuring dispersion in a sample or in a population. For stationary, ergodic time series with a finite first moment, an asymptotic expansion for the sample mean absolute deviation is proposed. The expansion yields the asymptotic distribution of the sample mean absolute deviation under a wide range of settings, allowing for serial dependence or an infinite second moment.
Asymptotic analysis and numerical modeling of mass transport in tubular structures
Cardone, G; Sirakov, Y
2009-01-01
In the paper the flow in a thin tubular structure is considered. The velocity of the flow stands for a coefficient in the diffusion-convection equation set in the thin structure. An asymptotic expansion of solution is constructed. This expansion is used further for justification of an asymptotic domain decomposition strategy essentially reducing the memory and the time of the code. A numerical solution obtained by this strategy is compared to the numerical solution obtained by a direct FEM computation.
Asymptotic Symmetries from finite boxes
Andrade, Tomas
2015-01-01
It is natural to regulate an infinite-sized system by imposing a boundary condition at finite distance, placing the system in a "box." This breaks symmetries, though the breaking is small when the box is large. One should thus be able to obtain the asymptotic symmetries of the infinite system by studying regulated systems. We provide concrete examples in the context of Einstein-Hilbert gravity (with negative or zero cosmological constant) by showing in 4 or more dimensions how the Anti-de Sitter and Poincar\\'e asymptotic symmetries can be extracted from gravity in a spherical box with Dirichlet boundary conditions. In 2+1 dimensions we obtain the full double-Virasoro algebra of asymptotic symmetries for AdS$_3$ and, correspondingly, the full Bondi-Metzner-Sachs (BMS) algebra for asymptotically flat space. In higher dimensions, a related approach may continue to be useful for constructing a good asymptotically flat phase space with BMS asymptotic symmetries.
Asymptotic symmetries from finite boxes
Andrade, Tomás; Marolf, Donald
2016-01-01
It is natural to regulate an infinite-sized system by imposing a boundary condition at finite distance, placing the system in a 'box.' This breaks symmetries, though the breaking is small when the box is large. One should thus be able to obtain the asymptotic symmetries of the infinite system by studying regulated systems. We provide concrete examples in the context of Einstein-Hilbert gravity (with negative or zero cosmological constant) by showing in 4 or more dimensions how the anti-de Sitter and Poincaré asymptotic symmetries can be extracted from gravity in a spherical box with Dirichlet boundary conditions. In 2 + 1 dimensions we obtain the full double-Virasoro algebra of asymptotic symmetries for AdS3 and, correspondingly, the full Bondi-Metzner-Sachs (BMS) algebra for asymptotically flat space. In higher dimensions, a related approach may continue to be useful for constructing a good asymptotically flat phase space with BMS asymptotic symmetries.
Regular Variation and Smile Asymptotics
Benaim, Shalom; Friz, Peter
2006-01-01
We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.
Extended asymptotic functions - some examples
Several examples of extended asymptotic functions are exposed. These examples will illustrate the notions introduced in another paper but at the same time they have a significance as realizations of some Schwartz disctibutions: delta(x), H(x), P(1/xsup(n)), etc. The important thing is that the asymptotic functions of these examples (which, on their part, are realizations of the above-mentioned distributions) can be multiplied in the class of the asymptotic functions as opposed to the theory of Schwartz distributions. Some properties of the set of all extended asymptotic functions are considered which are essential for the next step of this approach
Spectral asymptotics of a strong δ′ interaction supported by a surface
Highlights: • Attractive δ′ interactions supported by a smooth surface are considered. • Surfaces can be either infinite and asymptotically planar, or compact and closed. • Spectral asymptotics is determined by the geometry of the interaction support. - Abstract: We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive δ′ interaction supported by a smooth surface in R3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrödinger type operator with an effective potential expressed in terms of the interaction support curvatures
Asymptotic Efficiency in OLEDS
Nelson, Mitchell C
2015-01-01
Asymptotic efficiency (high output without droop) was recently reported for OLEDS in which a thin emitter layer is located at the anti-node in a resonant microcavity. Here we extend our theoretical analysis to treat multi-mode devices with isotropic emitter orientation. We recover our efficiency equations for the limiting cases with an isotropic emitter layer located at the anti-node where output is linear in current, and for an isotropic emitter located at the node where output can exhibit second order losses with an overall efficiency coefficient that depends on loss terms in competition with a cavity factor. Additional scenarios are described where output is driven by spontaneous emission, or mixed spontaneous and stimulated emission, with stimulated emission present in a loss mode, potentially resulting in cavity driven droop or output clamping, and where the emitter layer is a host-guest system.
On asymptotics for difference equations
Rafei, M.
2012-01-01
In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the
Selected asymptotic methods with applications to electromagnetics and antennas
Fikioris, George; Bakas, Odysseas N
2013-01-01
This book describes and illustrates the application of several asymptotic methods that have proved useful in the authors' research in electromagnetics and antennas. We first define asymptotic approximations and expansions and explain these concepts in detail. We then develop certain prerequisites from complex analysis such as power series, multivalued functions (including the concepts of branch points and branch cuts), and the all-important gamma function. Of particular importance is the idea of analytic continuation (of functions of a single complex variable); our discussions here include som
On the charge density and asymptotic tail of a monopole
Harland, Derek
2015-01-01
We propose a new definition for the abelian magnetic charge density of a non-abelian monopole, based on zero-modes of an associated Dirac operator. Unlike the standard definition of the charge density, this density is smooth in the core of the monopole. We show that this charge density induces a magnetic field whose expansion in powers of 1/r agrees with that of the conventional asymptotic magnetic field to all orders. We also show that the asymptotic field can be easily calculated from the spectral curve. Explicit examples are given for known monopole solutions.
Asymptotic Phase for Stochastic Oscillators
Thomas, Peter J.; Lindner, Benjamin
2014-12-01
Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined. We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of the Kolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators, even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.
Asymptotic Expansion and the LG/(Fano, General Type) Correspondence
Acosta, Pedro
2014-01-01
The celebrated LG/CY correspondence asserts that the Gromov-Witten theory of a Calabi-Yau (CY) hypersurface in weighted projective space is equivalent to its corresponding FJRW-theory (LG) via analytic continuation. It is well known that this correspondence fails in non-Calabi-Yau cases. The main obstruction is a collapsing or dimensional reduction of the state space of the Landau-Ginzburg model in the Fano case, and a similar collapsing of the state space of Gromov-Witten theory in the general type case. We state and prove a modified version of the cohomological correspondence that describes this collapsing phenomenon at the level of state spaces. This result confirms a physical conjecture of Witten-Hori-Vafa. The main purpose of this article is to provide a quantum explanation for the collapsing phenomenon. A key observation is that the corresponding Picard-Fuchs equation develops irregular singularities precisely at the points where the collapsing occurs. Our main idea is to replace analytic continuation w...
Toeplitz quantization and asymptotic expansions for real bounded symmetric domains
Engliš, Miroslav; Upmeier, H.
2011-01-01
Roč. 268, 3-4 (2011), s. 931-967. ISSN 0025-5874 R&D Projects: GA ČR(CZ) GA201/06/0128 Institutional research plan: CEZ:AV0Z10190503 Keywords : bounded symmetric domain * Toeplitz operator * star product Subject RIV: BA - General Mathematics Impact factor: 0.749, year: 2011 http://rd.springer.com/article/10.1007/s00209-010-0702-9
Asymptotic completeness and multiparticle structure in field theories
Previous proofs of asymptotic completeness and related results on scattering in field theories are restricted to P(φ)2 models in the 2- and 3-particle regions. In this paper, new cluster expansions that are well adapted to more direct proofs and generalizations of these results are presented. In contrast to previous ones, they are designed to provide direct graphical definitions of general irreducible kernels satisfying structure equations recently proposed and shown to be closely linked with asymptotic completeness and with the multiparticle structure of Green functions and collision amplitudes in general energy regions. The method can be applied as previously to P(φ)2 and can also be extended to theories involving renormalization which are controlled by phase-space analysis. It is here illustrated in detail for the Bethe-Salpeter kernel in φ24, in which case a new proof of its 4-particle decay (which yields asymptotic completeness in the 2-particle region) is given. (orig.)
Asymptotic corrections to the eigenvalue density of the GUE and LUE
Garoni, T. M.; Forrester, P. J.; Frankel, N. E.
2005-10-01
We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N ×N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.
Asymptotic structure of isolated systems
The main methods to formulate asymptotic flatness conditions are introduced and motivation and basic ideas are emphasized. Any asymptotic flatness condition proposed up to now describes space-times which behave somehow like Minkowski space, and a very explicit exposition of the structure at infinity of Minkowski space is given. This structure is used to describe the asymptotic behaviour of fields on Minkowski space in a frame-dependent way. The definition of null infinity for curved space-time according to Penrose is given and attempts to define spacelike infinity are outlined. The conformal bundle approach to the formulation of asymptotic behaviour is described and its relation to null and spacelike infinity is given, as far as known. (Auth.)
Asymptotic algebra of quantum electrodynamics
Herdegen, Andrzej
2004-01-01
The Staruszkiewicz quantum model of the long-range structure in electrodynamics is reviewed in the form of a Weyl algebra. This is followed by a personal view on the asymptotic structure of quantum electrodynamics.
Asymptotics of the filtration problem for suspension in porous media
Kuzmina Ludmila Ivanovna
2015-01-01
Full Text Available The mechanical-geometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media - a solid body with pores - channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. The particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macro-averaging of micro-stochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of suspended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. The terms of the asymptotic expansions satisfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.
2000-01-01
In this paper,the author studies the asymptotic accuracies of the one-term Edgeworth expansions and the bootstrap approximation for the studentized MLE from randomly censored exponential population.It is shown that the Edgeworth expansions and the bootstrap approximation are asymptotically close to the exact distribution of the studentized MLE with a rate.
Gradient expansion for anisotropic hydrodynamics
Florkowski, Wojciech; Spaliński, Michał
2016-01-01
We compute the gradient expansion for anisotropic hydrodynamics. The results are compared with the corresponding expansion of the underlying kinetic-theory model with the collision term treated in the relaxation time approximation. We find that a recent formulation of anisotropic hydrodynamics based on an anisotropic matching principle yields the first three terms of the gradient expansion in agreement with those obtained for the kinetic theory. This gives further support for this particular hydrodynamic model as a good approximation of the kinetic-theory approach. We further find that the gradient expansion of anisotropic hydrodynamics is an asymptotic series, and the singularities of the analytic continuation of its Borel transform indicate the presence of non-hydrodynamic modes.
The 6j-symbol: recursion, correlations and asymptotics
We study the asymptotic expansion of the {6j}-symbol using the Schulten-Gordon recursion relations. We focus on the particular case of the isosceles tetrahedron and we provide explicit formulas for up to the third-order corrections beyond the leading order. Moreover, in the framework of spinfoam models for 3D quantum gravity, we show how these recursion relations can be used to derive Ward-Takahashi-like identities between the expectation values of graviton-like spinfoam correlations.
Strong and ratio asymptotics for Laguerre polynomials revisited
Deaño, Alfredo; Huertas, Edmundo J.; Marcellán, Francisco
2013-01-01
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in t...
On the accuracy of the asymptotic theory for cylindrical shells
Niordson, Frithiof; Niordson, Christian
1999-01-01
We study the accuracy of the lowest-order bending theory of shells, derived from an asymptotic expansion of the three-dimensional theory of elasticity, by comparing the results of this shell theory for a cylindrical shell with clamped ends with the results of a solution to the three......-dimensional problem. The results are also compared with those of some commonly used engineering shell theories....
Two-parameter Asymptotics in Magnetic Weyl Calculus
Lein, Max
2008-01-01
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter \\eps, the case of small coupling $\\lambda$ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which ...
Airy asymptotics: the logarithmic derivative and its reciprocal
Kearney, Michael J [Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey, GU2 7XH (United Kingdom); Martin, Richard J [AHL, Man Investments Limited, Sugar Quay, Lower Thames Street, London, EC3R 6DU (United Kingdom)], E-mail: m.j.kearney@surrey.ac.uk
2009-10-23
We consider the asymptotic expansion of the logarithmic derivative of the Airy function Ai'(z)/Ai(z), and also its reciprocal Ai(z)/Ai'(z), as |z| {yields} {infinity}. We derive simple, closed-form solutions for the coefficients which appear in these expansions, which are of interest since they are encountered in a wide variety of problems. The solutions are presented as Mellin transforms of given functions; this fact, together with the methods employed, suggests further avenues for research.
Two-parameter Asymptotics in Magnetic Weyl Calculus
Lein, Max
2008-01-01
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter $\\eps$, the case of small coupling $\\lambda$ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two H\\"ormander class symbols are proven: (i) $\\eps \\ll 1$ and $\\lambda \\ll 1$, (ii) $\\eps \\ll 1$ and $\\lambda = 1$ as well as (iii) $\\eps = 1$ and $\\lambda \\ll 1$. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and ...
Two-parameter asymptotics in magnetic Weyl calculus
Lein, Max
2010-12-01
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ɛ, the case of small coupling λ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two Hörmander class symbols are proven as (i) ɛ ≪ 1 and λ ≪ 1, (ii) ɛ ≪ 1 and λ = 1, as well as (iii) ɛ = 1 and λ ≪ 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate the results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of fourth order in 1/c.
Two-parameter asymptotics in magnetic Weyl calculus
This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter ε, the case of small coupling λ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two Hoermander class symbols are proven as (i) ε<< 1 and λ<< 1, (ii) ε<< 1 and λ= 1, as well as (iii) ε= 1 and λ<< 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate the results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of fourth order in 1/c.
Edgeworth expansion for functionals of continuous diffusion processes
Podolskij, Mark; Yoshida, Nakahiro
2016-01-01
This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the second order Edgeworth expansion for power variation of...... diffusion processes. Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for studentized statistics of power variations....
Edgeworth expansion for functionals of continuous diffusion processes
Podolskij, Mark; Yoshida, Nakahiro
This paper presents new results on the Edgeworth expansion for high frequency functionals of continuous diffusion processes. We derive asymptotic expansions for weighted functionals of the Brownian motion and apply them to provide the Edgeworth expansion for power variation of diffusion processes....... Our methodology relies on martingale embedding, Malliavin calculus and stable central limit theorems for semimartingales. Finally, we demonstrate the density expansion for studentized statistics of power variations....
ASYMPTOTIC SOLUTION OF ACTIVATOR INHIBITOR SYSTEMS FOR NONLINEAR REACTION DIFFUSION EQUATIONS
Jiaqi MO; Wantao LIN
2008-01-01
A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.
Asymptotic behavior for a dissipative plate equation in $R^N$ with periodic coefficients
Eleni Bisognin
2008-03-01
Full Text Available In this work we study the asymptotic behavior of solutions of a dissipative plate equation in $mathbb{R}^N$ with periodic coefficients. We use the Bloch waves decomposition and a convenient Lyapunov function to derive a complete asymptotic expansion of solutions as $to infty$. In a first approximation, we prove that the solutions for the linear model behave as the homogenized heat kernel.
A Semigroup Expansion for Pricing Barrier Options
Takashi Kato
2014-01-01
Full Text Available This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.
Baez, J C; Egan, G F; Baez, John C.; Egan, Greg
2002-01-01
The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO(4) is replaced by the Euclidean group of isometries of R^3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks, including the Lorentzian 10j symbols, in terms of these degenerate spin networks.
Asymptotics for restricted integer compositions
Malandro, Martin E
2011-01-01
We study the compositions of an integer n where the part sizes of the compositions are restricted to lie in a finite set. We obtain asymptotic formulas for the number of such compositions, the total and average number of parts among all such compositions, and the total and average number of times a particular part size appears among all such compositions. Several of our asymptotics have the additional property that their absolute errors---not just their percentage errors---go to 0 as n goes to infinity. Along the way we also obtain recurrences and generating functions for calculating several of these quantities. Our asymptotic formulas come from the meromorphic analysis of our generating functions. Our results also apply to questions about certain kinds of tilings and rhythm patterns.
We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.
Exact and asymptotic results for insurance risk models with surplus-dependent premiums
Albrecher, Hansjörg; Palmowski, Zbigniew; Regensburger, Georg; Rosenkranz, Markus
2011-01-01
In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cram\\'er-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.
Asymptotic risks of Viterbi segmentation
Kuljus, Kristi
2010-01-01
We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation.
ASYMPTOTIC METHODS OF STATISTICAL CONTROL
Orlov A. I.
2014-10-01
Full Text Available Statistical control is a sampling control based on the probability theory and mathematical statistics. The article presents the development of the methods of statistical control in our country. It discussed the basics of the theory of statistical control – the plans of statistical control and their operational characteristics, the risks of the supplier and the consumer, the acceptance level of defectiveness and the rejection level of defectiveness. We have obtained the asymptotic method of synthesis of control plans based on the limit average output level of defectiveness. We have also developed the asymptotic theory of single sampling plans and formulated some unsolved mathematical problems of the theory of statistical control
Asymptotic freedom for nonrelativistic confinement
Some aspects of asymptotic freedom are discussed in the context of a simple two-particle nonrelativistic confining potential model. In this model, asymptotic freedom follows from the similarity of the free-particle and bound state radial wave functions at small distances and for the same angular momentum and the same large energy. This similarity, which can be understood using simple quantum mechanical arguments, can be used to show that the exact response function approaches that obtained when final state interactions are ignored. A method of calculating corrections to this limit is given, and explicit examples are given for the case of a harmonic oscillator
Comment on Asymptotically Safe Inflation
Tye, S -H Henry
2010-01-01
We comment on Weinberg's interesting analysis of asymptotically safe inflation (arXiv:0911.3165). We find that even if the gravity theory exhibits an ultraviolet fixed point, the energy scale during inflation is way too low to drive the theory close to the fixed point value. We choose the specific renormalization groupflow away from the fixed point towards the infrared region that reproduces the Newton's constant and today's cosmological constant. We follow this RG flow path to scales below the Planck scale to study the stability of the inflationary scenario. Again, we find that some fine tuning is necessary to get enough efolds of infflation in the asymptotically safe inflationary scenario.
Asymptotically free SU(5) models
The behaviour of Yukawa and Higgs effective charges of the minimal SU(5) unification model is investigated. The model includes ν=3 (or more, up to ν=7) generations of quarks and leptons and, in addition, the 24-plet of heavy fermions. A number of solutions of the renorm-group equations are found, which reproduce the known data about quarks and leptons and, due to a special choice of the coupling constants at the unification point are asymptotically free in all charges. The requirement of the asymptotical freedom leads to some restrictions on the masses of particles and on their mixing angles
Thermal expansion of fuel pellet is an important property which limits the lifetime of the fuels in reactors, because it affects both the pellet and cladding mechanical interaction and the gap conductivity. By fitting a number of available measured data, recommended equations have been presented and successfully used to estimate thermal expansion coefficient of the nuclear fuel pellet. However, due to large scatter of the measured data, non-consensus data have been omitted in formulating the equations. Also, the equation is strongly governed by the lack of appropriate experimental data. For those reasons, it is important to develop theoretical methodologies to better describe thermal expansion behaviour of nuclear fuel. In particular, first-principles and molecular dynamics simulations have been certainly contributed to predict reliable thermal expansion without fitting the measured data. Furthermore, the two theoretical techniques have improved on understanding the change of fuel dimension by describing the atomic-scale processes associated with lattice expansion in the fuels. (author)
Asymptotics of weighted random sums
Corcuera, José Manuel; Nualart, David; Podolskij, Mark
2014-01-01
In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We show that these sums converge in law to the integral of the...
Ruin problems and tail asymptotics
Rønn-Nielsen, Anders
The thesis Ruin Problems and Tail Asymptotics provides results on ruin problems for several classes of Markov processes. For a class of diffusion processes with jumps an explicit expression for the joint Laplace transform of the first passage time and the corresponding undershoot is derived. An...
Thermodynamics of asymptotically safe theories
Rischke, Dirk H.; Sannino, Francesco
2015-01-01
We investigate the thermodynamic properties of a novel class of gauge-Yukawa theories that have recently been shown to be completely asymptotically safe, because their short-distance behaviour is determined by the presence of an interacting fixed point. Not only do all the coupling constants freeze...
Cherniavski, V. M.; Shtemler, Yu. M.
2013-01-01
The potential flow of an incompressible inviscid heavy fluid over a light one is considered. The integral version of the method of matched asymptotic expansion is applied to the construction of the solution over long intervals of time. The asymptotic solution describes the flow in which a bubble rises with constant speed and the "tongue" is in free fall. The outer expansion is stationary, but the inner one depends on time. It is shown that the solution exists within the same range of Froude n...
Non-Fefferman-Graham asymptotics and holographic renormalization in New Massive Gravity
Cunliff, Colin
2013-01-01
The asymptotic behavior of new massive gravity (NMG) is analyzed for all values of the mass parameter satisfying the Breitenlohner-Freedman bound. The traditional Fefferman-Graham expansion fails to capture the dynamics of NMG, and new terms in the asymptotic expansion are needed to include the massive graviton modes. New boundary conditions are discovered for a range of values $-1<2m^2l^2<1$ at which non-Einstein modes decay more slowly than the Brown-Henneaux boundary conditions. The holographically renormalized stress tensor is computed for these modes, and the relevant counterterms are identified up to unphysical ambiguities.
Edgeworth expansion for the pre-averaging estimator
Podolskij, Mark; Veliyev, Bezirgen; Yoshida, Nakahiro
In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its...... asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions....
On Asymptotically Efficient Estimation in Semiparametric Models
Schick, Anton
1986-01-01
A general method for the construction of asymptotically efficient estimates in semiparametric models is presented. It improves and modifies Bickel's (1982) construction of adaptive estimates and obtains asymptotically efficient estimates under conditions weaker than those in Bickel.
On transfinite extension of asymptotic dimension
Radul, Taras
2006-01-01
We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension.
Born expansions for charged particle scattering
High-order terms in Born expansions of scattering amplitudes in powers of charge are frequently divergent when long-range Coulomb interactions are present asymptotically. Expansions which are free from these logarithmic divergences have been constructed recently. This paper illustrates these expansions with the simplest example, namely the non-relativistic Rutherford scattering of two charged particles. This approach represents an adequate framework for the calculation of transition amplitudes and a comprehensive starting point for the development of consistent perturbation approximations in multi-channel descriptions of strongly interacting atomic systems
Asymptotic functions and multiplication of distributions
Considered is a new type of generalized asymptotic functions, which are not functionals on some space of test functions as the Schwartz distributions. The definition of the generalized asymptotic functions is given. It is pointed out that in future the particular asymptotic functions will be used for solving some topics of quantum mechanics and quantum theory
Structure and asymptotic theory for nonlinear models with GARCH errors
Felix Chan
2015-01-01
Full Text Available Nonlinear time series models, especially those with regime-switching and/or conditionally heteroskedastic errors, have become increasingly popular in the economics and finance literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the processes or the associated asymptotic theory. In this paper, we derive sufficient conditions for strict stationarity and ergodicity of three different specifications of the first-order smooth transition autoregressions with heteroskedastic errors. This is essential, among other reasons, to establish the conditions under which the traditional LM linearity tests based on Taylor expansions are valid. We also provide sufficient conditions for consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator for a general nonlinear conditional mean model with first-order GARCH errors.
Asymptotic structure of isolated systems
I discuss the general ideas underlying the subject of ''asymptotics'' in general relativity and describe the current status of the concepts resulting from these ideas. My main concern will be the problem of consistency. By this I mean the question as to whether the geometric assumptions inherent in these concepts are compatible with the dynamics of the theory, as determined by Einstein's equations. This rather strong bias forces me to leave untouched several issues related to asymptotics, discussed in the recent literature, some of which are perhaps thought equally, or more important, by other workers in the field. In addition I shall, for coherence of presentation, mainly consider Einstein's equations in vacuo. When attention is confined to small neighbourhoods of null and spacelike infinity, this restriction is not important, but is surely relevant for more global issues. (author)
Asymptotic safety goes on shell
It is well known in quantum field theory that the off-shell effective action depends on the gauge choice and field parametrization used in calculating it. Nevertheless, the typical scheme in which the scenario of asymptotically safe gravity is investigated is an off-shell version of the functional renormalization group equation. Working with the Einstein-Hilbert truncation as a test bed, we develop a new scheme for the analysis of asymptotically safe gravity in which the on-shell part of the effective action is singled out and we show that the beta function for the essential coupling has no explicit gauge dependence. In order to reach our goal, we introduce several technical novelties, including a different decomposition of the metric fluctuations, a new implementation of the ghost sector and a new cut-off scheme. We find a nontrivial fixed point, with a value of the cosmological constant that is independent of the gauge-fixing parameters. (paper)
Asymptotic safety goes on shell
Benedetti, Dario
2012-01-01
It is well known in quantum field theory that the off-shell effective action depends on the gauge choice and field parametrization used in calculating it. Nevertheless, the typical scheme in which the scenario of asymptotically safe gravity is investigated is an off-shell version of the functional renormalization group equation. Working with the Einstein-Hilbert truncation as a test bed, we develop a new scheme for the analysis of asymptotically safe gravity in which the on-shell part of the effective action is singled out and we show that the beta function for the essential coupling has no explicit gauge dependence. In order to reach our goal, we introduce several technical novelties, including a different decomposition of the metric fluctuations, a new implementation of the ghost sector and a new cut-off scheme. We find a nontrivial fixed point, with a value of the cosmological constant that is independent of the gauge-fixing parameters.
Asymptotic analysis of Hoppe trees
Leckey, Kevin
2012-01-01
We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight $\\vartheta>0$, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For $\\vartheta=1$ the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length and number of leaves of the Hoppe tree with $n$ nodes as well as the depth of the last inserted node asymptotically as $n\\to \\infty$. Mainly expectations, variances and asymptotic distributions of these parameters are derived.
Asymptotic Excisions of Metric Spaces and Ideals of Asymptotic Coarse Roe Algebras
LI Jin-xiu; WANG Qin
2006-01-01
We introduce in this note the notions of asymptotic excision of proper metric spaces and asymptotic equivalence relation for subspaces of metric spaces, which are relevant in characterizing spatial ideals of the asymptotic coarse Roe algebras. We show that the lattice of the asymptotic equivalence classes of the subspaces of a proper metric space is isomorphic to the lattice of the spatial ideals of the asymptotic Roe algebra. For asymptotic excisions of the metric space, we also establish a Mayer-Vietoris sequence in K-theory of the asymptotic coarse Roe algebras.
Exponential asymptotics and capillary waves
Chapman, S. J.; Vanden-Broeck, J.
2002-01-01
Recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow (for example, stagnation points and corners) meet it. The amplitude of these waves is determined, and the implications are considered for many quite general flows....
Asymptotic algebra for charged particles and radiation
A C*-algebra of asymptotic fields which properly describes the infrared structure in quantum electrodynamics is proposed. The algebra is generated by the null asymptotic of electromagnetic field and the time asymptotic of charged matter fields which incorporate the corresponding Coulomb fields. As a consequence Gauss' law is satisfied in the algebraic setting. Within this algebra the observables can be identified by the principle of gauge invariance. A class of representations of the asymptotic algebra is constructed which resembles the Kulish-Faddeev treatment of electrically charged asymptotic fields. (orig.)
Renormalization constants and asymptotic behaviour in quantum electrodynamics
Using dimensional regularization a field theory contains at least one parameter less than the dimension of a mass. The Callan-Symanzik equations for the renormalization constants then become soluble entirely in terms of the coefficient functions. Explicit expressions are obtained for all the renormalization constants in Quantum Electrodynamics. At nonexceptional momenta the infrared behaviour and the three leading terms in the asymptotic expansion of any Greens function are controlled by the Callan-Symanzik equations. For the propagators the three leading terms are computed explicitly in terms of functions of α only. The gauge dependence of the electron propagator in momentum space is solved explicitly in all orders of perturbation theory. (Auth.)
Asymptotic scaling corrections in QCD with Wilson fermions from the 3-loop average plaquette
Alles, B.; FEO, A; Panagopoulos, H.
1998-01-01
We calculate the 3-loop perturbative expansion of the average plaquette in lattice QCD with N_f massive Wilson fermions and gauge group SU(N). The corrections to asymptotic scaling in the corresponding energy scheme are also evaluated. We have also improved the accuracy of the already known pure gluonic results at 2 and 3 loops.
Loss asymptotics for the single-server queue with complete rejection
Zwart, A.P.
2015-01-01
Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level K. A basic performance measure of this system is the probability PK that a customer gets rejected in steady state. This paper presents asymptotic expansions for PK as K ... If
Full text: Calculational methods and Reduce software are described for determining polyhomogeneous asymptotic expansions of solutions of Einstein's equations in null characteristic transport form. As an example, results concerning peeling of gravitational radiation in Null Quasi-Spherical (NQS) spacetimes are presented
Synchronization of complex chaotic systems in series expansion form
This paper studies the synchronization of complex chaotic systems in series expansion form by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of an error dynamics, and is applied to guiding the design of the secure communication. Finally, numerical results are studied for the Quantum-CNN oscillators synchronizing with unidirectional/bidirectional linear coupling to show the effectiveness of the proposed synchronization strategy
Synchronization of complex chaotic systems in series expansion form
Ge Zhengming [Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan (China)]. E-mail: zmg@cc.nctu.edu.tw; Yang Chenghsiung [Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan (China)
2007-12-15
This paper studies the synchronization of complex chaotic systems in series expansion form by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of an error dynamics, and is applied to guiding the design of the secure communication. Finally, numerical results are studied for the Quantum-CNN oscillators synchronizing with unidirectional/bidirectional linear coupling to show the effectiveness of the proposed synchronization strategy.
Asymptotic black hole quasinormal frequencies
Motl, Lubos; Neitzke, Andrew
2003-01-01
We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d greater than or equal to 4 and Reissner-Nordstrom black holes in d = 4, in the limit of infinite damping. For Schwarzschild in d greater than or equal to 4 we find that the asymptotic real part is THawkinglog(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d = 4. For Reissner-Nordstrom in d = 4 w...
Asymptotics of robust utility maximization
Knispel, Thomas
2012-01-01
For a stochastic factor model we maximize the long-term growth rate of robust expected power utility with parameter $\\lambda\\in(0,1)$. Using duality methods the problem is reformulated as an infinite time horizon, risk-sensitive control problem. Our results characterize the optimal growth rate, an optimal long-term trading strategy and an asymptotic worst-case model in terms of an ergodic Bellman equation. With these results we propose a duality approach to a "robust large deviations" criterion for optimal long-term investment.
Asymptotics for Associated Random Variables
Oliveira, Paulo Eduardo
2012-01-01
The book concerns the notion of association in probability and statistics. Association and some other positive dependence notions were introduced in 1966 and 1967 but received little attention from the probabilistic and statistics community. The interest in these dependence notions increased in the last 15 to 20 years, and many asymptotic results were proved and improved. Despite this increased interest, characterizations and results remained essentially scattered in the literature published in different journals. The goal of this book is to bring together the bulk of these results, presenting
S, Vijay Prakash; Sonti, Venkata R.
2016-07-01
Structural-acoustic waveguides of two different geometries are considered: a 2-D rectangular and a circular cylindrical geometry. The objective is to obtain asymptotic expansions of the fluid-structure coupled wavenumbers. The required asymptotic parameters are derived in a systematic way, in contrast to the usual intuitive methods used in such problems. The systematic way involves analyzing the phase change of a wave incident on a single boundary of the waveguide. Then, the coupled wavenumber expansions are derived using these asymptotic parameters. The phase change is also used to qualitatively demarcate the dispersion diagram as dominantly structure-originated, fluid-originated or fully coupled. In contrast to intuitively obtained asymptotic parameters, this approach does not involve any restriction on the material and geometry of the structure. The derived closed-form solutions are compared with the numerical solutions and a good match is obtained.
Modeling broadband poroelastic propagation using an asymptotic approach
Vasco, Donald W.
2009-05-01
An asymptotic method, valid in the presence of smoothly-varying heterogeneity, is used to derive a semi-analytic solution to the equations for fluid and solid displacements in a poroelastic medium. The solution is defined along trajectories through the porous medium model, in the manner of ray theory. The lowest order expression in the asymptotic expansion provides an eikonal equation for the phase. There are three modes of propagation, two modes of longitudinal displacement and a single mode of transverse displacement. The two longitudinal modes define the Biot fast and slow waves which have very different propagation characteristics. In the limit of low frequency, the Biot slow wave propagates as a diffusive disturbance, in essence a transient pressure pulse. Conversely, at low frequencies the Biot fast wave and the transverse mode are modified elastic waves. At intermediate frequencies the wave characteristics of the longitudinal modes are mixed. A comparison of the asymptotic solution with analytic and numerical solutions shows reasonably good agreement for both homogeneous and heterogeneous Earth models.
Asymptotic kinetic theory of magnetized plasmas: quasi-particle concept
The asymptotic kinetic theory of magnetized plasmas is elaborated within the context of general statistical approach and asymptotic methods, developed by M. Krylov and M. Bohol'ubov, for linear and non-linear dynamic systems with a rapidly rotating phase. The quasi-particles are introduced already on the microscopic level. Asymptotic expansions enable to close the description for slow processes, and to relate consistently particles and guiding centres to quasi-particles. The kinetic equation for quasi-particles is derived. It makes a basis for the reduced description of slow collective phenomena in the medium. The kinetic equation for quasi-particles takes into account self-consistent interaction fields, quasi-particle collisions and collective-fluctuation-induced relaxation of quasi-particle distribution function. The relationships between the distribution functions for particles, guiding centres and quasi-particles are derived taking into account fluctuations, which can be especially important in turbulent states. In this way macroscopic (statistical) particle properties can be obtained from those of quasi-particles in the general case of non-equilibrium. (authors)
Caustics, counting maps and semi-classical asymptotics
Ercolani, N M
2009-01-01
This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitean random matrices. The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion, (and its derivatives) are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are in fact specific rational functions of a distinguished irrational (algebraic) function of the generating function parameters. This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of parameter). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions. These results in turn provide new information about the asymptotics of ...
Asymptotic black hole quasinormal frequencies
Motl, L; Motl, Lubos; Neitzke, Andrew
2003-01-01
We give a simple derivation of the quasinormal frequencies of Schwarzschild black holes in d>=4 and non-extremal Reissner-Nordstrom black holes in d=4, in the limit of infinite damping. For Schwarzschild in d=4 the asymptotic real part of the frequency is (T_Hawking)log(1+2cos(pi.j)), where j is the spin of the perturbation; this confirms a result previously obtained by other means. For Schwarzschild in d>4 we find that the asymptotic real part is (T_Hawking)log(3) for scalar perturbations. For non-extremal Reissner-Nordstrom in d=4 we find a specific but generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for axial electromagnetic-gravitational perturbations; there is nevertheless a hint that the value (T_Hawking)log(2) may be special in this case. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane.
Asymptotically Free Gauge Theories. I
Wilczek, Frank; Gross, David J.
1973-07-01
Asymptotically free gauge theories of the strong interactions are constructed and analyzed. The reasons for doing this are recounted, including a review of renormalization group techniques and their application to scaling phenomena. The renormalization group equations are derived for Yang-Mills theories. The parameters that enter into the equations are calculated to lowest order and it is shown that these theories are asymptotically free. More specifically the effective coupling constant, which determines the ultraviolet behavior of the theory, vanishes for large space-like momenta. Fermions are incorporated and the construction of realistic models is discussed. We propose that the strong interactions be mediated by a "color" gauge group which commutes with SU(3)xSU(3). The problem of symmetry breaking is discussed. It appears likely that this would have a dynamical origin. It is suggested that the gauge symmetry might not be broken, and that the severe infrared singularities prevent the occurrence of non-color singlet physical states. The deep inelastic structure functions, as well as the electron position total annihilation cross section are analyzed. Scaling obtains up to calculable logarithmic corrections, and the naive lightcone or parton model results follow. The problems of incorporating scalar mesons and breaking the symmetry by the Higgs mechanism are explained in detail.
The maximum drag reduction asymptote
Choueiri, George H.; Hof, Bjorn
2015-11-01
Addition of long chain polymers is one of the most efficient ways to reduce the drag of turbulent flows. Already very low concentration of polymers can lead to a substantial drag and upon further increase of the concentration the drag reduces until it reaches an empirically found limit, the so called maximum drag reduction (MDR) asymptote, which is independent of the type of polymer used. We here carry out a detailed experimental study of the approach to this asymptote for pipe flow. Particular attention is paid to the recently observed state of elasto-inertial turbulence (EIT) which has been reported to occur in polymer solutions at sufficiently high shear. Our results show that upon the approach to MDR Newtonian turbulence becomes marginalized (hibernation) and eventually completely disappears and is replaced by EIT. In particular, spectra of high Reynolds number MDR flows are compared to flows at high shear rates in small diameter tubes where EIT is found at Re Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n° [291734].
Asymptotic safety goes on shell
Benedetti, Dario
2011-01-01
It is well known in quantum field theory that the off-shell effective action depends on the gauge choice and field parametrization used in calculating it. Nevertheless, the typical scheme in which the scenario of asymptotically safe gravity is investigated is an off-shell version of the functional renormalization group equation. Working with the Einstein-Hilbert truncation as a test bed, we develop a new scheme for the analysis of asymptotically safe gravity in which the on-shell part of the effective action is singled out and we show that the beta function for the essential coupling has no explicit gauge-dependence. In order to reach our goal, we introduce several technical novelties, including a different decomposition of the metric fluctuations, a new implementation of the ghost sector, and a new cut-off scheme. We find a non-trivial fixed point, with a value of the cosmological constant which is independent of the gauge-fixing parameters.
Asymptotic conservation laws in field theory
Anderson, Ian M.; Torre, Charles G.
1996-01-01
A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation...
Asymptotically Plane Wave Spacetimes and their Actions
Witt, Julian Le; Ross, Simon F.
2008-01-01
We propose a definition of asymptotically plane wave spacetimes in vacuum gravity in terms of the asymptotic falloff of the metric, and discuss the relation to previously constructed exact solutions. We construct a well-behaved action principle for such spacetimes, using the formalism developed by Mann and Marolf. We show that this action is finite on-shell and that the variational principle is well-defined for solutions of vacuum gravity satisfying our asymptotically plane wave falloff condi...
Asymptotics of near unit roots (in Russian)
Stanislav Anatolyev; Nikolay Gospodinov
2012-01-01
Sometimes the conventional asymptotic theory yields that the limiting distribution changes discontinuously, or that the asymptotic distribution does not approximate accurately the actual finite-sample distribution. In such situations one finds useful an asymptotic tool of drifting parameterizations where certain parameters are allowed to depend explicitly on the sample size. It proves useful, among other things, for impulse response analysis and forecasting of strongly dependent processes at ...
Asymptotic independence and a network traffic model
Maulik, Krishanu; Resnick, Sidney; Rootzén, Holger
2002-01-01
The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows u...
Exponential asymptotic stability for linear volterra equations
John A. D. Appleby
2000-01-01
This note studies the exponential asymptotic stability of the zero solution of the linear Volterra equation x˙ (t) = Ax(t) + t 0 K(t − s)x(s) ds by extending results in the paper of Murakami “Exponential Asymptotic Stability for scalar linear Volterra Equations”, Differential and Integral Equations, 4, 1991. In particular, when K isi ntegrable and has entries which do not change sign, and the equation has a uniformly asymptotically stable solution, exponential asympto...
We consider a two-dimensional model Schroedinger equation with logarithmic integral non-linearity. We find asymptotic expansions for its solutions (Airy polarons) that decay exponentially at the 'semi-infinity' and oscillate along one direction. These solutions may be regarded as new special functions, which are somewhat similar to the Airy function. We use them to construct global asymptotic solutions of Schroedinger equations with a small parameter and with integral non-linearity of Hartree type
Nikonov, Anatolij Viktorovič; Zakharov, D.D.
2015-01-01
Asymptotically accurate low-frequency models for isotropic elastic coatings and interlayers are developed. The main constraint is the requirement on contact conditions for the layer and the base that at least one of the boundary conditions must include the displacement component in an explicit form. The displacement and stress fields in the three-dimensional elastic system are sought in the form of asymptotic expansion into power series of a small parameter - the ratio between the half-thickn...
Supersymmetric asymptotic safety is not guaranteed
Intriligator, Kenneth; Sannino, Francesco
It was recently shown that certain perturbatively accessible, non-supersymmetric gauge-Yukawa theories have UV asymptotic safety, without asymptotic freedom: the UV theory is an interacting RG fixed point, and the IR theory is free. We here investigate the possibility of asymptotic safety in...... supersymmetric theories, and use unitarity bounds, and the a-theorem, to rule it out in broad classes of theories. The arguments apply without assuming perturbation theory. Therefore, the UV completion of a non-asymptotically free susy theory must have additional, non-obvious degrees of freedom, such as those of...
Why are tensor field theories asymptotically free?
Rivasseau, Vincent
2015-01-01
In this pedagogic letter we explain the combinatorics underlying the generic asymptotic freedom of tensor field theories. We focus on simple combinatorial models with a $1/p^2$ propagator and quartic interactions and on the comparison between the intermediate field representations of the vector, matrix and tensor cases. The transition from asymptotic freedom (tensor case) to asymptotic safety (matrix case) is related to the crossing symmetry of the matrix vertex whereas in the vector case, the lack of asymptotic freedom ("Landau ghost"), as in the ordinary scalar case, is simply due to the absence of any wave function renormalization at one loop.
Edgeworth expansion for the survival function estimator in the Koziol-Green model
SUN; Liuquan(孙六全); WU; Guofu(吴国富)
2002-01-01
In the KozioI-Green or proportional hazards random censorship model, the asymptotic accuracy of the estimated one-term Edgeworth expansion and the smoothed bootstrap approximation for the Studen tized Abdushukurov-Cheng-Lin estimator is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of the Studentized Abdushukurov-Cheng-Lin estimator than the normal approximation.
Thermodynamics of asymptotically safe theories
Rischke, Dirk H
2015-01-01
We investigate the thermodynamic properties of a novel class of gauge-Yukawa theories that have recently been shown to be completely asymptotically safe, because their short-distance behaviour is determined by the presence of an interacting fixed point. Not only do all the coupling constants freeze at a constant and calculable value in the ultraviolet, their values can even be made arbitrarily small for an appropriate choice of the ratio $N_c/N_f$ of fermion colours and flavours in the Veneziano limit. Thus, a perturbative treatment can be justified. We compute the pressure, entropy density, and thermal degrees of freedom of these theories to next-to-next-to-leading order in the coupling constants.
An Edgeworth expansion for symmetric statistics
Bentkus, V.; Götze, F.; van Zwet, W. R.
1997-01-01
We consider asymptotically normal statistics which are symmetric functions of N i.i.d. random variables. For these statistics we prove the validity of an Edgeworth expansion with remainder $O(N^{-1})$ under Cramér's condition on the linear part of the statistic and moment assumptions for all parts of the statistic. By means of a counterexample we show that it is generally not possible to obtain an Edgeworth expansion with remainder $o(N^{-1})$ without imposing additional assumptions on the...
Asymptotic spectral theory for nonlinear time series
Shao, Xiaofeng; Wu, Wei Biao
2007-01-01
We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.
Asymptotics of implied volatility far from maturity
Michael R., Tehranchi
2009-01-01
This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.
Complete WKB asymptotics of high frequency vibrations in a stiff problem
Babych, N
2008-01-01
Asymptotic behaviour of eigenvalues and eigenfunctions of a stiff problem is described in the case of the fourth-order ordinary differential operator. Considering the stiffness coefficient that depends on a small parameter epsilon and vanishes as epsilon tends to zero on a subinterval, we prove the existence of low and high frequency resonance vibrations. The low frequency vibrations admit the power series expansions on epsilon but this method is not applicable to the description of high frequency vibrations. However, the nonclassical asymptotics on epsilon of the high frequency vibrations were constructed using the WKB method.
Recent axiomatic results on the (non holonomic) analytic structure of the multiparticle S matrix and Green functions are reviewed and related general conjectures are described: (i) formal expansions of Green functions in terms of (holonomic) Feynman-type integrals in which each vertex represents an irreducible kernel, and (ii) ''graph by graph unitarity'' and other discontinuity formulae of the latter. These conjectures are closely linked with unitarity or asymptotic completeness equations, which they yield in a formal sense. In constructive field theory, a direct proof of the first conjecture (together with an independent proof of the second) would thus imply, as a first step, asymptotic completeness in that sense
Non-asymptotic fractional order differentiators via an algebraic parametric method
Liu, Dayan
2012-08-01
Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie\\'s modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations. © 2012 IEEE.
Universal asymptotic umbrella for hydraulic fracture modeling
Linkov, Aleksandr M
2014-01-01
The paper presents universal asymptotic solution needed for efficient modeling of hydraulic fractures. We show that when neglecting the lag, there is universal asymptotic equation for the near-front opening. It appears that apart from the mechanical properties of fluid and rock, the asymptotic opening depends merely on the local speed of fracture propagation. This implies that, on one hand, the global problem is ill-posed, when trying to solve it as a boundary value problem under a fixed position of the front. On the other hand, when properly used, the universal asymptotics drastically facilitates solving hydraulic fracture problems (both analytically and numerically). We derive simple universal asymptotics and comment on their employment for efficient numerical simulation of hydraulic fractures, in particular, by well-established Level Set and Fast Marching Methods.
Penrose type inequalities for asymptotically hyperbolic graphs
Dahl, Mattias; Sakovich, Anna
2013-01-01
In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\\bH^n$. The graphs are considered as subsets of $\\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article concerning the asymptotically Euclidean case.
1/N-expansion for the anharmonic oscillator
The properties of the 1/N-expansion for the anharmonic oscillator in quantum mechanics have been investigated. The first seven terms of the expansion for the energy of ground and first excited levels are obtained analytically. The high-order behaviour of the 1/N-expansion coefficients in closed form was found, the asymptotic series obtained being Borel summable. The formulae derived was used to find the first seven coefficients of the perturbative expansion in powers of the coupling constant in the case of the double-well potential for arbitrary number of components N. These exact expressions enable us to guess the large-order behaviour of the perturbative coefficients for N=0, 1, ... 4. An example of summing the asymptotic series in powers of 1/N applying the Pade-Borel method is given
Asymptotics of phase and wave functions
Zhaba, V I
2016-01-01
For single and twochannel nucleon-nucleon scattering the asymptotic form of the phase function for r->0 were taken into account for the asymptotic behavior of the wave function. Asymptotics of the wave function will not r^(l+1), and will have a more complex view and be also determined by the behavior of the potential near the origin. Have examined the cases for nonsingular (weakly singular) and strongly singular potentials. Were the numerical calculations of phase, amplitude and wave functions for the nucleon-nucleon potential Argonne v18. Considered 1S0-, 3P0-, 3P1- states of the np- system.
Asymptotic study of subcritical graph classes
Drmota, Michael; Kang, Mihyun; Kraus, Veronika; Rué, Juanjo
2010-01-01
We present a unified general method for the asymptotic study of graphs from the so-called "subcritical"$ $ graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class ${\\cG}=\\cup_n {\\cG_n}$ satisfies asymptotically the universal behaviour
Belov, P. A., E-mail: pavelbelov@gmail.com; Yakovlev, S. L., E-mail: yakovlev@cph10.phys.spbu.ru [St. Petersburg State University, Department of Computational Physics (Russian Federation)
2013-02-15
The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.
Langmuir probe study of plasma expansion in pulsed laser ablation
Hansen, T.N.; Schou, Jørgen; Lunney, J.G.
1999-01-01
Langmuir probes were used to monitor the asymptotic expansion of the plasma produced by the laser ablation of a silver target in a vacuum. The measured angular and temporal distributions of the ion flux and electron temperature were found to be in good agreement with the self-similar isentropic and...
Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions
BISWAJYOTI SAHA
2016-08-01
For an arithmetical function $f$ with absolutely convergent Ramanujan expansion, we derive an asymptotic formula for the $\\sum_{n\\leq N}$ f(n)$ with explicit error term. As a corollary we obtain new results about sum-of-divisors functions and Jordan’s totient functions.
Asymptotic-group analysis of algebraic equations
Shamrovskii, A. D.; I. V. Andrianov; J. Awrejcewicz
2004-01-01
Both the method of asymptotic analysis and the theory of extension group are applied to study the Descates equation. The proposed algorithm allows to obtain various variants of simplification and can be easily generalized to their algebraic and differential equations.
The Lorentzian proper vertex amplitude: Asymptotics
Engle, Jonathan; Zipfel, Antonia
2015-01-01
In previous work, the Lorentzian proper vertex amplitude for a spin-foam model of quantum gravity was derived. In the present work, the asymptotics of this amplitude are studied in the semi-classical limit. The starting point of the analysis is an expression for the amplitude as an action integral with action differing from that in the EPRL case by an extra `projector' term which scales linearly with spins only in the asymptotic limit. New tools are introduced to generalize stationary phase methods to this case. For the case of boundary data which can be glued to a non-degenerate Lorentzian 4-simplex, the asymptotic limit of the amplitude is shown to equal the single Feynman term, showing that the extra term in the asymptotics of the EPRL amplitude has been eliminated.
Asymptotic Likelihood Distribution for Correlated & Constrained Systems
Agarwal, Ujjwal
2016-01-01
It describes my work as summer student at CERN. The report discusses the asymptotic distribution of the likelihood ratio for total no. of parameters being h and 2 out of these being are constrained and correlated.
A quantum kinematics for asymptotically flat spacetimes
Campiglia, Miguel
2014-01-01
We construct a quantum kinematics for asymptotically flat spacetimes based on the Koslowski-Sahlmann (KS) representation. The KS representation is a generalization of the representation underlying Loop Quantum Gravity (LQG) which supports, in addition to the usual LQG operators, the action of `background exponential operators' which are connection dependent operators labelled by `background' $su(2)$ electric fields. KS states have, in addition to the LQG state label corresponding to 1 dimensional excitations of the triad, a label corresponding to a `background' electric field which describes 3 dimensional excitations of the triad. Asymptotic behaviour in quantum theory is controlled through asymptotic conditions on the background electric fields which label the {\\em states} and the background electric fields which label the {\\em operators}. Asymptotic conditions on the triad are imposed as conditions on the background electric field state label while confining the LQG spin net graph labels to compact sets. We...
Asymptotic fixed points for nonlinear contractions
Yong-Zhuo Chen
2005-06-01
Full Text Available Recently, W. A. Kirk proved an asymptotic fixed point theorem for nonlinear contractions by using ultrafilter methods. In this paper, we prove his theorem under weaker assumptions. Furthermore, our proof does not use ultrafilter methods.
Negative dimension in general and asymptotic topology
Maslov, V. P.
2006-01-01
We introduce the notion of negative topological dimension and the notion of weight for the asymptotic topological dimension. Quantizing of spaces of negative dimension is applied to linguistic statistics.
Kinematical bound in asymptotically translationally invariant spacetimes
Shiromizu, T; Tomizawa, S; Shiromizu, Tetsuya; Ida, Daisuke; Tomizawa, Shinya
2004-01-01
We present positive energy theorems in asymptotically translationally invariant spacetimes which can be applicable to black strings and charged branes. We also address the bound property of the tension and charge of branes.
Precise Asymptotics for Lévy Processes
Zhi Shui HU; Chun SU
2007-01-01
Let {X(t), t ≥ 0} be a Lévy process with EX(1)=0 and EX2(1)＜∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t≥0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d.random variables.
Asymptotic representation theorems for poverty indices
Lo, Gane Samb; Sall, Serigne Touba
2010-01-01
We set general conditions under which the general poverty index, which summarizes all the available indices, is asymptotically represented with some empirical processes. This representation theorem offers a general key, in most directions, for the asymptotics of the bulk of poverty indices and issues in poverty analysis. Our representation results uniformly hold on a large collection of poverty indices. They enable the continuous measure of poverty with longitudinal data.
AGB (asymptotic giant branch): Star evolution
Becker, S.A.
1987-01-01
Asymptotic giant branch stars are red supergiant stars of low-to-intermediate mass. This class of stars is of particular interest because many of these stars can have nuclear processed material brought up repeatedly from the deep interior to the surface where it can be observed. A review of recent theoretical and observational work on stars undergoing the asymptotic giant branch phase is presented. 41 refs.
Loop Quantum Gravity and Asymptotically Flat Spaces
Arnsdorf, Matthias
2000-01-01
After motivating why the study of asymptotically flat spaces is important in loop quantum gravity, we review the extension of the standard framework of this theory to the asymptotically flat sector based on the GNS construction. In particular, we provide a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. States in these Hilbert spaces can be interpreted as describing fluctuations around fiducial fixed backgrounds. When the backgr...
General smile asymptotics with bounded maturity
Francesco Caravenna; Jacopo Corbetta
2014-01-01
We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009), 1-12]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion ...