Integrable Functions Research Articles

SIMULATION OF THE PROPAGATION OF A PLANE ELECTROMAGNETIC WAVE IN AN INHOMOGENEOUS NONABSORBING MEDIUM

Analytical solutions of the Ishimaru's parabolic equation for the coherence function of the electromagnetic field, which describe the temporal properties of the pulse at the output of a heterogeneous non-dissipative medium, are considered. A generalization of the approach used in the Ishimaru model to describe the time evolution of a monochromatic electromagnetic pulse enveloping in homogeneous non-dissipative media is obtained for the case of non-homogeneous non-dissipative media. Thus, an attempt was made to take into account the influence of the heterogeneity of the medium on the shape of the resulting pulse. When solving the given problem, the difficulties associated with the calculation of the continuous integral arising in the space of diffusion trajectories were overcome. This made it possible to obtain an explicit expression for the Green's function of the task and to build a computational algorithm based on which a number of numerous experiments were conducted. The analysis of the work was carried out based on the apparatus of quadratic integral functionals based on the solutions of differential stochastic equations. In the paper, the invariant temporal properties of the envelope of monochromatic electromagnetic pulses recorded after passing through a flat layer of a scattering heterogeneous medium, i.e., properties that remain unchanged when the parameters of the medium vary, in particular, the distribution of the concentration of scattering centers. The dynamics of the formation of time plumes of the scattered wave, in which the tail part is located in the peripheral time area, were analyzed. When propagating at the speed of transformation of the wave front, it reflects the appearance of the layers of the scattering region and its longitudinal shape. It is noted that the development of the proposed approximation approach to the processes affecting the time delay of electromagnetic pulses can be the accounting for the attenuation of radiation during its propagation in an inhomogeneous absorbing medium.

O gipersingulyarnykh operatorakh, svyazannykh s peridinamikoy

For a hypersingular integral operator of the Calderón–Zygmund type associated with problems of peridynamics, we find a Hilbert space that is taken by this operator to the space of square integrable periodic functions.

Inner product spaces and Fourier Series Approximation

The aim of this paper is to give the basic notions of the theory of Fourier Series in connection with Inner production spaces. The preliminary concepts and fundamentals of normed spaces, inner product spaces and the Fourier series are being discussed. The space of all periodic complex integrable functions forms an inner product space. Basic trigonometric functions Sin x and Cos x will form a basis for this linear space. A special emphasis is given in deriving the formulae for the Fourier coefficients. The convergence properties of Fourier series are also studied.

Poisk parametrov modeli s nailuchshey lokal'noy upravlyaemost'yu

We study the problem of optimal choice of model parameters with respect to anyfunctional. Locally controllable affine systems and integral functionals depending on the programcontrol are considered. Local controllability of affine systems with nonnegative inputs isproved in the case where the columns multiplying the controls form a positive basis. For suchsystems, we introduce the local controllability coefficient and pose the problem of its maximizationdepending on the choice of model parameters. As an example, we consider a very simplifiedmodel of an underwater vehicle and study the problem of finding an arrangement of its controlpropellers in which the energy consumption of the vehicle is minimal.

On a new approach in the space of measurable functions

In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.

Approximation and interpolation with neural network

In this paper we show that multilayer feedforward networks with one single hidden layer.and certain types of activation functions can approximate univariant continuous functions defined on a compact set. arbitrarily well. In particular, our results contain some usual activation functions such as sigmoidal functions, RELU functions and threshold functions. Besides, since interpolation problems are highly related to approximation problem, we demonstrate that a wide range of functions have the ability to interpolate and generalize our results to functions which are not polynomial on R. Compared to existing results by numerous work, our methods are more intuitive and less technical. Lastly, the paper discusses the possibility of combining interpolation property and approximating property together, and demonstrates that given any Riemann integrable functions on a compact set in R, with several points on its graph, the finite combination of monotone sigmoidal functions can pass through these points and approximate the given function arbitrarily well with respect to L^1 (dx) (in the sense of Riemann integral) when the number of points getting large.

Generalized Abel equations and applications to translation invariant Radon transforms

Abstract Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, R j \mathcal{R}_{j} , of interest in URT, which integrates a square integrable function of compact support, 𝑓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that R j \mathcal{R}_{j} is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from R j ⁢ f \mathcal{R}_{j}f data with added pseudo-random noise.

Bessel functions and Kloosterman integrals on GL(n)

This paper will focus on the proof of local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [5][6][7], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [25][26] and G. Stevens' approach [35] on estimating certain GL(n) generalized Kloosterman sums.

Almost Everywhere Convergence of T Means with Respect to the Vilenkin System of Integrable Functions

Almost Everywhere Convergence of T Means with Respect to the Vilenkin System of Integrable Functions

Моделювання вимушених гармонічних коливань прямокутного фундамента на водонасиченій шаруватій основі

A numerical study of the dynamic contact problem is carried out by the method of orthogonal polynomials. А symbolic solution was found for the double transformants of the vertical displacements of the layered two-phase poroelastic liquid saturated base under the action of arbitrary vertical contact pressures. Numerical analysis was performed for the Rayleigh function and the components of displacements in the kernels of integrable functions. An example of calculating double integrals necessary to determine the contact pressures of two-phase soil under the base of the foundation is given.

Generating probability distributions on intervals and spheres: Convex decomposition

This work studies the convexity of a class of positive definite probability measures with density functionsρ=cλ∑n=0∞g(n)Cnλwλ, that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f+,f−)∈Eλ′×Eλ′ up to normalization, in the sense that f=f+−f−, where Eλ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class Eλ′, but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.

Almost everywhere convergence of varying-parameter setting Cesaro means of Fourier series on the group of 2-adic integers

In this paper we prove that the maximal operator of Cesaro-means for one-dimensional Fourier series on the group of 2-adic integers is of weak type (L^{1}, L^{1}). Moreover, we prove the almost everywhere convergence of Cesaro means of integrable functions.

Syzygies, constant rank, and beyond

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

Infinite pointwise lineability: general criteria and applications

In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set of infinite differentiable and integrable functions, for any order p≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\ge 1$$\\end{document}, on RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^N$$\\end{document} which are unbounded in a pre-fixed set.

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.

Integration on minimal -superspace and emergence of space

We investigate the possibilities of integration on the minimal -superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on despite the fact that the part of the minimal -superspace is

Homogenization in perforated domains at the critical scale

We describe the asymptotic behaviour of the minimal heterogeneous d-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set Ω⊆Rd, with d≥2. Two parameters are involved: ɛ, the radius of the ball, and δ, the length scale of the heterogeneity of the medium. We prove that this capacity behaves as C|logɛ|1−d, where C=C(λ) is an explicit constant depending on the parameter λ≔limɛ→0|logδ|/|logɛ|.We determine the Γ-limit of oscillating integral functionals subjected to Dirichlet boundary conditions on periodically perforated domains. Our first result is used to study the behaviour of the functionals near the perforations which, in this instance, are balls of radius ɛ. We prove that an additional strange term arises involving C(λ).

Multipliers of Variationally Mcshane Integrable Functions in Locally Convex Space

Abstract We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a σ-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space X. We also show: in case X is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case X is representable by semi-norm.

Applying New Algorithms for Numerical Integration on the Sphere in the Far Field of Sound Pressure

For some sound sources, the function of the square of sound pressure amplitudes on the sphere in the far field is an integrable function or can be integrated with geometrical simplifications, so an exact or approximated analytical expression for the sound power can be calculated. However, often the sound pressure on the sphere in the far field can only be defined in discrete points, for which a numerical integration is required for the calculation of the sound power. In this paper, two new algorithms, Anchored Radially Projected Integration on Spherical Triangles (ARPIST) and Spherical Quadrature Radial Basis Function (SQRBF), for surface numerical integration are used to calculate the sound power from the sound pressures on the sphere surface in the far field, and their solutions are compared with the analytical and the finite element method solution. If function values are available at any location on a sphere, ARPIST has a greater accuracy and stability than SQRBF while being faster and easier to implement. If function values are available only at user-prescribed locations, SQRBF can directly calculate weights while ARPIST needs data interpolation to obtain function values at predefined node locations, which reduces the accuracy and increases the calculation time.

A generalization of the Moore and Yang integral and interval probability density functions

ABSTRACT Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions.