Abstract:
Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely convergent Fourier series. New properties of functions from these algebras are obtained. Some conditions which determine membership of f in F(R) are given. For many elementary functions f the problem of belonging f to F(R) can be resolved easily using these conditions. We prove that the Hilbert operator is a bijective isometric operator in the Banach spaces W_0, F(R), K(R)-A_1 (A_1 is the one-dimension space of constant functions). We also consider the classes M_k, which are similar to the Bochner classes F_k, and obtain integral representation of the Carleman transform of measures of M_k by integrals of some specific form.

Abstract:
We give solutions to Problems 2.21, 2.31 and 2.32, which were posed Borzow\'a-Moln\'arov\'a, Hal\v{c}inov\'a and Hutn\'ik in [{\it The smallest semicopula-based universal integrals I: properties and characterizations,} Fuzzy Sets and Systems (2014), http://dx.doi.org/ 10.1016/j.fss. 2014.09.0232014].

Abstract:
Some algebraic properties of integralsover configuration spaces are investigated in order to better understandquantization and the Connes-Kreimer algebraic approach to renormalization. In order to isolate the mathematical-physics interface toquantum field theory independent from the specifics of the variousimplementations, the sigma model of Kontsevich is investigated in moredetail. Due to the convergence of the configuration space integrals, themodel allows to study the Feynman rules independently, from an axiomaticpoint of view, avoiding the intricacies of renormalization, unavoidablewithin the traditional quantum field theory. As an application, a combinatorial approach to constructingthe coefficients of formality morphisms is suggested, as an alternative tothe analytical approach used by Kontsevich. These coefficients are "Feynman integrals", although not quite typical since they do converge. A second example of "Feynman integrals", defined asstate-sum model, is investigated. Integration is understood here as formalcategorical integration, or better as a duality structure on thecorresponding category. The connection with a related TQFT is mentioned,supplementing the Feynman path integral interpretation of Kontsevichformula. A categorical formulation for the Feynman path integralquantization is sketched, towards Feynman Processes, i.e. representations ofdg-categories with duality, thought of as complexified Markov processes.

Abstract:
Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive definite or hermitian positive definite, are available \cite{4}. Real scalar variables case of the Dirichlet models are generalized in various directions. One such generalization of the type-2 or inverted Dirichlet is looked into in this article. Matrix-variate analogue, when the matrices are hermitian positive definite, are worked out along with some properties which are mathematically and statistically interesting.

Abstract:
The purpose of this paper is to prove several results in approximation by complex Picard, Poisson-Cauchy, and Gauss-Weierstrass singular integrals with Jackson-type rate, having the quality of preservation of some properties in geometric function theory, like the preservation of coefficients' bounds, positive real part, bounded turn, starlikeness, and convexity. Also, some sufficient conditions for starlikeness and univalence of analytic functions are preserved.

Abstract:
We discuss the evaluation of certain d dimensional angular integrals which arise in perturbative field theory calculations. We find that the angular integral with n denominators can be computed in terms of a certain special function, the so-called H-function of several variables. We also present several illustrative examples of the general result and briefly consider some applications.

Abstract:
The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$, $\{X_t\}$ is a Poisson process, and $\{X_t\}$ and $\{Y_t\}$ are independent. An example of $Z$ of type $G$ with selfdecomposable mixing distribution is given.

Abstract:
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.

Abstract:
Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history. The original error was made when a (startlingly) famous mathematician incorrectly differentiated under the integral sign with some convergent integrals that depend on a paramater.