Integral representation of local functionals depending on vector fields

Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

On the Truncated Multidimensional Moment Problems in Cn

We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.

Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

Multivalent Prestarlike Functions with Respect to Symmetric Points

A class of p-valent functions of complex order is defined with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefficient estimates and the solution to the Fekete–Szegő problem are obtained for the defined function class. Further, we extended the results using quantum calculus. Several consequences of our main results are pointed out.

Combinatorics of KP hierarchy structural constants

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on $$\tau $$ τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

Search for identical regions in images using invariant moments

One of the ways to describe objects on images is to identify some of their characteristic points or points of attention. Areas of neighborhoods of attention points are described by descriptors (lots of signs) in such way that they can be identified and compared. These signs are used to search for identical points in other images. The article investigates and establishes the possibility of searching for arbitrary local image regions by descriptors constructed with using invariant moments. A feature of the proposed method is that the calculation of the invariant moments of local areas is carried out with using the integral representation of the geometric moments of the image. Integral representation is a matrix with the same size as the image. The elements of the matrix is the sums of the geometric moments of individual pixels, which are located above and to the left with respect to the coordinates of this element. The number of matrices depends on the order of the geometric moments. For moments up to the second order (inclusively), there will be six such matrices. Calculation of one of six geometric moments of an arbitrary rectangular area of the image comes down up to 3 operations such as summation or subtraction of elements of the corresponding matrix located in the corners of this area. The invariant moments are calculated on base of six geometric moments. The search is performed by scanning the image coordinate grid with a window of a given size. In this case, the invariant moments and additional parameters are calculated and compared with similar parameters of the neighborhoods of the reference point of different size (taking into account the possible change in the image scale). The best option is selected according to a given condition. Almost all mass operations of the procedures for calculating the parameters of standards and searching of identical points make it possible explicitly perform parallel computations in the SIMD mode. As a result, the integral representation of geometric moments and the possibility of using parallel computations at all stages will significantly speed up the calculations and allow you to get good indicators of the search efficiency for identical points and the speed of work

On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

$$a_{1}$$ meson-nucleon coupling constant at finite temperature from the soft-wall AdS/QCD model

We study the temperature dependence of the $$a_1$$ a 1 meson-nucleon coupling constant in the framework of the soft-wall AdS/QCD model with thermal dilaton field. Profile functions for the axial-vector and fermion fields in the AdS-Schwarzschild metric are presented. It is constructed an interaction Lagrangian for the fermion-axial-vector-thermal dilaton fields system in the bulk of space-time. From this Lagrangian integral representation for the $$g_{a_1NN}$$ g a 1 N N coupling constant is derived. The temperature dependence of this coupling constant is numerically analyzed.

AdS one-loop partition functions from bulk and edge characters

We show that the one-loop partition function of any higher spin field in (d + 1)-dimensional Anti-de Sitter spacetime can be expressed as an integral transform of an SO(2, d) bulk character and an SO(2, d − 2) edge character. We apply this character integral formula to various higher-spin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters that lead to agreement with the predictions of HS/CFT holography. We also discuss the relation between the character integral representation and the Rindler-AdS thermal partition function.