Stability of a simple Levi–Civitá functional equation on non-unital commutative semigroups

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)?f(x)h(y)?f(y)|≤?? and |f(x+y)?g(x)h(y)?k(y)|≤??, respectively. We also study the bounded solutions of the simple Levi–Civitá functional inequality.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.     

Algorithms for low-dimensional topology

In this article, we re-introduce the so called “Arkaden–Faden–Lage” (briefly, AFL) representation of knots in three-dimensional space introduced by Kurt Reidemeister in the 1930s and show how it can be used to develop efficient algorithms to compute some important topological knot structures. In particular, we introduce an efficient algorithm to calculate the holonomic representation of knots introduced by V. Vassiliev and give the main ideas on how to use the AFL representations of knots to compute the Kontsevich integral. The methods introduced here are to our knowledge novel and can open new perspectives in the development of fast algorithms in low-dimensional topology.     

Generalized convolution quadrature with variable time stepping. Part II: Algorithm and numerical results

In this paper we address the implementation of the Generalized Convolution Quadrature (gCQ) presented and analyzed by the authors in a previous paper for solving linear parabolic and hyperbolic convolution equations. Our main goal is to overcome the current restriction to uniform time steps of Lubich's Convolution Quadrature (CQ). A major challenge for the efficient realization of the new method is the evaluation of high-order divided differences for the transfer function in a fast and stable way. Our algorithm is based on contour integral representation of the numerical solution and quadrature in the complex plane. As the main application we consider the wave equation in exterior domains, which is formulated as a retarded boundary integral equation. We provide numerical experiments to illustrate the theoretical results.     

A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation. This bound is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group (see Definition 2.7) and is sextuply exponential in the order of the differential equation. In this paper, we use Szántó’s algorithm of triangular representation for algebraic sets to analyze the complexity of computing the Galois group of a linear differential equation and we give a new bound which is triple exponential in the order of the given differential equation.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.     

On powers of Stieltjes moment sequences,II

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy–Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.     

Majorizing Sequences for Nonlinear Fredholm–Hammerstein Integral Equations

We use the method of majorizing sequences to study the applicability of Newton's method to solve nonlinear Fredholm–Hammerstein integral equations. For this, we use center convergence conditions on points different from the starting point of Newton's method, which is the point usually used by other authors until now when center conditions are required. In addition, the theoretical significance of the method is used to draw conclusions about the existence and uniqueness of solutions and about the region in which they are located. As a result, we modify the domain of starting points for Newton's method.     

What are the last digits of …?

We propose a class assignment where students are asked to construct and implement an efficient algorithm to calculate the last digits of a positive integral power of a positive integer. The mathematical prerequisites for this assignment are very limited: knowledge of remainder calculus and the binary representation of a positive integer. The periodicity of the last digits is studied by means of the Euler totient function and the Carmichael function.     

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.     

A Robust Spectral Method for Solving Heston’s Model

In this paper, we consider the Heston’s volatility model (Heston in Rev. Financ. Stud. 6: 327–343, 1993]. We simulate this model using a combination of the spectral collocation method and the Laplace transforms method. To approximate the two dimensional PDE, we construct a grid which is the tensor product of the two grids, each of which is based on the Chebyshev points in the two spacial directions. The resulting semi-discrete problem is then solved by applying the Laplace transform method based on Talbot’s idea of deformation of the contour integral (Talbot in IMA J. Appl. Math. 23(1): 97–120, 1979).     

A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time

We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17th problem. The main idea is to make use of the randomness contained in the input itself.     

The Besicovitch–Federer projection theorem is false in every infinite-dimensional Banach space

The theme of the article is the study of the unipotent part of Arthur’s trace formula for general linear groups. The case of regular (or “regular by blocks”) unipotent orbits has been treated in another paper (cf. [10]). Here we are interested in the contribution of Richardson orbits that are induced by Levi subgroups with two-by-two distinct blocks. In this case, the contribution is remarkably given by a global unipotent weighted orbital integral. As a corollary, we get integral formulas for some of Arthur’s global coefficients. We also present a new construction of Arthur’s local unipotent weighted orbital integral and an explicit computation of some of them.     

Method of Integral Equations for the Three-Dimensional Problem of Wave Reflection from an Irregular Surface

The problem on the reflection of the field of a plane H-polarized three-dimensional electromagnetic wave from a perfectly conducting interface between media which contains a local perfectly conducting inhomogeneity is considered. To construct a numerical algorithm, the boundary value problem for the system of Maxwell equations in an infinite domain with irregular boundary is reduced to a system of singular integral equations, which is solved by the approximation–collocation method. The elements of the resulting complex matrix are calculated by a specially developed algorithm. The solution of the system of singular integral equations is used to obtain an integral representation for the reflected electromagnetic field and computational formulas for the directional diagram of the reflected electromagnetic field in the far region.     

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.     

Efficient treatment of stationary free boundary problems

In the present paper we consider the efficient treatment of free boundary problems by shape optimization. We reformulate the free boundary problem as shape optimization problem. A second order shape calculus enables us to analyze the shape problem under consideration and to prove convergence of a Ritz–Galerkin approximation of the shape. We show that Newton's method requires only access to the underlying state function on the boundary of the domain. We compute these data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems.     

An integral for the scalar propagator

Using a convenient contour in the complex plane, an integral representation is obtained for the hypergeometric function, in the case when this function does not reduce to the polynomial case. As an application, a contour integral associated with the massive scalar propagator used in Feynman diagrams is discussed.     

Wavelet compression of integral operators arising from boudary-domain integral method

The boundary-domain integral method uses Green's functions to write integral representations of partial differential equations. Since Green's functions are non-local, the systems of linear equations arising from the discretization of integral representations are fully populated. Several algorithms have been proposed, which yield a data-sparse approximation of these systems, such as the fast multipole method, adaptive cross approximation and among others, wavelet compression. In the framework of solving the Navier-Stokes equations in velocity-vorticity form one may utilize the boundary-domain integral method for the solution of the kinematics equation to calculate the boundary vorticity values. Since the kinematics equation is a Poisson type equation, usually its integral representation is written with the Green's function for the Laplace operator. In this work, we introduce a false time into the equation and parabolize its nature. Thus, a time-dependent Green's function may be used. This provides a new parameter, the time step, which can be set to control the Green's function. The time-dependent Green's function is a global function, but by carefully choosing the time step, its behaviour is almost local. This makes it a good candidate for wavelet compression, yielding much better compression ratios at a given accuracy than when using the Green's function for the Laplace operator. However, as false time is introduced, several time steps must be solved in order to reach a converged solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new integral filter algorithm for unconstrained global optimization

In this paper, making use a exponential integral filter, a new algorithm for unconstrained global optimization is proposed. Compared with Yang’s absolute value type integral filter method (Yang et al., Appl Math Comput 18:173–180, 2007), this algorithm is more effective and more sensitive. Numerical results for some typical examples show that in most cases, this algorithm works effectively and reliably.