Integral operator approaches for scattered data fitting on spheres

We introduce and study a family of integral operators in the Kantorovich sense acting on functions defined on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we obtain general convergence results in particular cases of Orlicz spaces, as Lp−spaces, interpolation spaces and exponential spaces. Finally we construct some concrete examples of our operators and we show some graphical representations.

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called Ψ‐fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

For decreasing service centers’ selection risks in emergency facility location selection, it is crucial to have selected candidate service centers within deeply detailed facility location selection model. To achieve this, a new approach developed in this article involves two stages. In the first stage, the fuzzy multi-attribute group decision making (MAGDM) model for evaluation of the selection of candidate service centers is created. For the aggregation of experts’ assessments of candidate service centers (with respect to attributes) aggregation operators’ approach is used. Experts’ assessments are presented in fuzzy terms with semantic form of triangular fuzzy numbers. For the deeply detailed facility location selection modeling and for the intellectual activity of experts in their evaluations, pairwise interactions between attributes of MAGDM model are considered in the construction of the second-order additive triangular fuzzy valued fuzzy measure (TFVFM). The associated triangular fuzzy probability averaging (As-TFPA) aggregation operators’ family is constructed with respect to TFVFM. Analytical properties of the As-TFPA operators are studied. The new operators are certain extensions of the well-known Choquet integral operator. The extensions, in contrast to the Choquet aggregation, consider all possible pair-wise interactions of the attributes by introducing associated fuzzy probabilities of a TFVFM. At the end of the first stage, a candidate service center’s selection index is defined as As-TFPA operator’s aggregation value on experts’ assessments with respect to attributes. At the second stage, fuzzy multi-objective facility location set covering problem (MOFLSCP) is created for facility location selection optimal planning with new criteria: (1) maximization of candidate service centers selection index and classical two criteria, (2) minimization of the total cost needed to open service centers and (3) minimization of number of agents needed to operate the opened service centers. For the constructed two-stage methodology a simulation example of emergency service facility location planning for a city is considered. The example gives the Pareto fronts obtained by As-TFPA operators, the Choquet integral-TFCA operator and well-known TOPSIS approach, for optimal selecting candidate sites for the servicing of demand points. The comparative analysis identifies that the differences in the Pareto solutions, obtained by using As-TFPA operators and TFCA operator or TOPSIS aggregation, are also caused by the fact that TFCA operator or TOPSIS approach considers the pair interaction indexes for only one consonant structure of attributes. While new As-TFPA aggregations provide all pairwise interactions for all consonant structures.

The Construction of Solutions for Boundary Value Problems by Function Theoretic Methods

Part I: Transmutations, Integral Equations and Special Functions.- Some Recent Developments in the Transmutation Operator Approach.- Transmutation Operators and Their Applications.- Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications.- Second Type Neumann Series Related to Nicholson's and to Dixon-Ferrar Formula.- On Some Generalizations of the Properties of the Multidimensional Generalized Erdelyi-Kober Operators and Their Applications.- Alternative Approach to Miller-Paris Transformations and Their Extensions.- Transmutation Operators For Ordinary Dunkl-Darboux Operators.- Theorems on Restriction of Fourier-Bessel and Multidimensional Bessel Transforms to Spherical Surfaces.- Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis.- Polynomial Quantization on Line Bundles.- Fourier-Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations.- Inversion of Hyperbolic B-Potentials.- One-Dimensional and Multi-Dimensional Integral Transforms of Buschman-Erdelyi Type with Legendre Functions in Kernels.- Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems.- Part II: Transmutations in ODEs, Direct and Inverse Problems.- On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators.- Expansion in Terms of Appropriate Functions and Transmutations.- Transmutation Operators as a Solvability Concept of Abstract Singular Equations.- On the Bessel-Wright Operator and Transmutation with Applications.- On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments.- Transmutation Operators Boundary Value Problems.- Solution of Inverse Problems for Differential Operators with Delay.- Part III: Transmutations for Partial and Fractional Differential Equations.- Transmutations of the Composed Erdelyi-Kober Fractional Operators and Their Applications.- Distributed Order Equations in Banach Spaces with Sectorial Operators.- Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications.- Strong Solutions of Semilinear Equations with Lower Fractional Derivatives.- Mean Value Theorems and Properties of Solutions of Linear Differential Equations.- Transmutations for Multi-Term Fractional Operators.- Fractional Bessel Integrals and Derivatives on Semi-axes.- The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay.- Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient.

General expressions for the matrix elements of the discrete SN-equivalent integral transport operator are derived in slab geometry. Their asymptotic behavior versus cell optical thickness is investigated both for a homogeneous slab and for a heterogeneous slab characterized by a periodic material discontinuity wherein each optically thick cell is surrounded by two optically thin cells in a repeating pattern. In the case of a homogeneous slab, the asymptotic analysis conducted in the thick-cell limit for a highly scattering medium shows that the discretized integral transport operator approaches a tridiagonal matrix possessing a diffusion-like coupling stencil. It is further shown that this structure is approached at a fast exponential rate with increasing cell thickness when the arbitrarily high order transport method of the nodal type and zero-order spatial approximation (AHOT-N0) formalism is employed to effect the spatial discretization of the discrete ordinates transport operator. In the case of periodically heterogeneous slab configurations, the asymptotic behavior is realized by pushing apart the cells’ optical thicknesses; i.e., the thick cells are made thicker while the thin cells are made thinner at a prescribed rate. We show that in this limit the discretized integral transport operator is approximated by a pentadiagonal structure. Notwithstanding, the discrete operator is amenable to algebraic transformations leading to a matrix representation still asymptotically approaching a tridiagonal structure at a fast exponential rate bearing close resemblance to the diffusive operator.The results of the asymptotic analysis of the integral transport matrix are then used to gain insight into the excellent convergence properties of the adjacent-cell preconditioner (AP) acceleration scheme. Specifically, the AP operator exactly captures the asymptotic structure acquired by the integral transport matrix in the thick-cell limit for homogeneous slabs of pure-scatterer or partial-scatterer material, and for periodically heterogeneous slabs hosting purely scattering materials. In the above limits the integral transport matrix reduces to a diffusive structure consistent with the diffusive matrix template used to construct the AP. In the case of periodically heterogeneous slabs containing absorbing materials, the AP operator partially captures the asymptotic structure acquired by the integral transport matrix. The inexact agreement is due either to discrepancies in the equations for the boundary cells or to the nondiffusive structure acquired by the integral transport matrix. These findings shed light on the immediate convergence, i.e., convergence in two iterations, displayed by the AP acceleration scheme in the asymptotic limit for slabs hosting purely scattering materials, both in the homogeneous and periodically heterogeneous cases. For periodically heterogeneous slabs containing absorbing materials, immediate convergence is achieved by modifying the original recipe for constructing the AP so that the correct asymptotic structure of the integral transport matrix coincides with the AP operator in the asymptotic limit.

Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions.

A brief survey is given on using transformation operators in the inverse spectral theory of integral and integro-differential operators possessing a convolutional term to be recovered. The central place of this approach is occupied by reducing the inverse problem to solving some nonlinear equation, which can be solved globally. We illustrate this scheme on several examples, among which there are: one-dimensional perturbation of the convolution operator, Sturm–Liouville-type integro-differential operators and an integro-differential Dirac system.

Publisher Copyright: IEEE

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a 'reconstruction' formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.

Semilinear fractional differential equations based on a new integral operator approach

We consider a $T$-periodically perturbed autonomous functional differential equation of neutral type. We assume the existence\nof a $T$-periodic limit cycle $x_0$ for the unperturbed autonomous system. We also assume that the linearized unperturbed equation\naround the limit cycle has the characteristic multiplier $1$ of geometric multiplicity $1$ and algebraic multiplicity greater than $1$.\nThe paper deals with the existence of a branch of $T$-periodic solutions emanating from the limit cycle. The problem of finding such\na branch is converted into the problem of finding a branch of zeros of\na suitably defined bifurcation equation\n$P(x,\\varepsilon) +\\varepsilon Q(x, \\varepsilon)=0$.\nThe main task of the paper is to define a novel equivalent integral operator having the property that the $T$-periodic adjoint Floquet solutions\nof the unperturbed linearized operator correspond to those of the equation $P'(x_0(\\theta),0)=0$, $\\theta\\in[0,T]$. Once this is done it is possible to express\nthe condition for the existence of a branch of zeros for the bifurcation equation in terms of a multidimensional Malkin bifurcation function.

This paper mainly focuses on the least square regularized regression learning algorithm in a setting of unbounded sampling. Our task is to establish learning rates by means of integral operators. By imposing a moment hypothesis on the unbounded sampling outputs and a function space condition associated with marginal distribution ρ X , we derive learning rates which are consistent with those in the bounded sampling setting.

Letk be an inverse Fourier transform of a real valued bounded and summable functionK, and let {λjτ (τ > 0)} denote the eigenvalues of the Hermitian integral operator (Wk(τ)ϕ)(t) = ∫0τk (t −s)ϕ(s)ds (ϕ ∈L2(0,τ)). The well known Kac, Murdock and Szego formula asserts that $$\mathop {\lim }\limits_{\tau \to \infty } \tau ^{ - 1} \sum\limits_{j = 1}^\infty {[\lambda _j^{(\tau )} ]^3 = (2\pi )} ^{ - 1} \int {_{ - \infty }^\infty [K(x)]^5 dx (s = 2,3, \cdot \cdot \cdot ,)} $$ . The main aim of the present paper is to extend this formula to the case of a complex-valued matrix functionK. We achieve this extension by developing an operator approach which is valid for a wide class of convolution type operators.