Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps

We consider jump-type stochastic differential equations with drift, diffusion, and jump terms. Logarithmic derivatives of densities for the solution process are studied, and Bismut–Elworthy–Li-type formulae are obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markov property of the process.     

On perfect hashing of numbers with sparse digit representation via multiplication by a constant

Consider the set of vectors over a field having non-zero coefficients only in a fixed sparse set and multiplication defined by convolution, or the set of integers having non-zero digits (in some base b) in a fixed sparse set. We show the existence of an optimal (or almost-optimal, in the latter case) ‘magic’ multiplier constant that provides a perfect hash function which transfers the information from the given sparse coefficients into consecutive digits. Studying the convolution case we also obtain a result of non-degeneracy for Schur functions as polynomials in the elementary symmetric functions in positive characteristic.     

Euler Pseudoprime Polynomials and Strong Pseudoprime Polynomials

It is usual to emphasize the analogy between the integers and polynomials with coefficients in a finite field, comparing different notions in the two points of view. We introduce a particular rank one Drinfeld module to get an exponentiation for polynomials and then define the notions of Euler pseudoprimes and strong pseudoprimes for polynomials with coefficients in a finite field. As for the integers, we have Solovay–Strassen and Miller–Rabin tests for polynomials.     

Orbits of rational n-sets of projective spaces under the action of the linear group

Let k=Fq be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space PN over k, and we compute the generating function for the numbers of PGLN+1(k)-orbits of these n-sets. For N=1,2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.     

Fourier-Haar coefficients and properties of continuous functions

It is well known that if the Fourier-Haar coefficients have a certain order or if a certain series composed of the Fourier-Haar coefficients of a function f(x) ∈ C(0, 1) converges, then the function has a certain form. In the present paper, we prove that not only the Fourier-Haar coefficients, but also the difference of these coefficients possess these properties.     

Binomial and factorial congruences for Fq[t]

We present several elementary theorems, observations and questions related to the theme of congruences satisfied by binomial coefficients and factorials modulo primes (or prime powers) in the setting of polynomial ring over a finite field. When we look at the factorial of n or the binomial coefficient ‘n choose m’ in this setting, though the values are in a function field, n and m can be usual integers, polynomials or mixed. Thus there are several interesting analogs of the well-known theorems of Lucas, Wilson etc. with quite different proofs and new phenomena.     

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n=m=2 we determine the maximum number of critical periods and their location; to the case n=m=3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

The Drazin inverse as a gradient

The coefficients in the expansion of adj(λI ? A) are expressed as gradients, and some new representations are given for the Drazin inverse of a matrix over an arbitrary field. These results are then combined to express the Drazin inverse as a gradient of a function of the entries of the matrix.     

Coefficients of half-integral weight modular forms

In this paper, we study the distribution of the coefficients a(n) of half-integral weight modular forms modulo odd integers M. As a consequence, we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in Ono and Skinner (Fourier coefficients of half-integral weight modular forms modulo ?, Ann. of Math. 147 (1998) 453-470). Moreover, we find a simple criterion for proving cases of Newman's conjecture for the partition function.     

On metaplectic forms over function fields

We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight ${\frac{3}{2}}$ functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula. This formula is then applied to L-series coming from elliptic curves over function fields.     

The Fourier coefficients of functions with bounded variation

We consider Fourier coefficients for functions of bounded variation with respect to general orthonormal systems (GONS). We show that in general case the sequence of coefficients may have an arbitrarily prescribed order of vanishing. In this connection we seek for a class of GONS such that Fourier coefficients of a function from V(0, 1) satisfy the same inequality as in the case of classical systems (trigonometric, Walsh, and Haar ones). In the present paper we study this problem and related issues.     

Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains

We establish the global Hessian estimate in Orlicz spaces for a fourth-order parabolic system with discontinuous tensor coefficients in a non-smooth domain under the assumptions that the coefficients have small weak BMO semi-norms, the boundary of a domain is δ-Reifenberg flat for δ>0 small and the given Young function satisfies some moderate growth condition. As a corollary we obtain an optimal global W2,p regularity for such a system.     

Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network

In this paper, a novel hybrid method based on fuzzy neural network for approximate solution of fuzzy linear systems of the form Ax = Bx + d, where A and B are two square matrices of fuzzy coefficients, x and d are two fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution, a simple and fast algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.     

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.     

Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d _T that represents a possible solution to this problem. Indeed, d _T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with ? ⁿ -valued filtering functions. Furthermore, we prove a result showing the relationship between d _T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.     

A new subclass of meromorphic functions with positive coefficients

In the present paper, we consider the new subclass of Meromorphic functions with positive coefficients and obtain a necessary and sufficient condition for a function f to be in this class. We obtain distortion properties, radius of convexity, starlikeness, convex linear combinations for the function f in this class.     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.     

Simpleness of Leavitt path algebras with coefficients in a commutative semiring

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, establishing some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a commutative semiring S, extending the well-known characterizations when S is a field or a commutative ring. We also present a complete characterization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S.     

On center sets of ODEs determined by moments of their coefficients

The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.