We extend the interval and fuzzy gH-differentiability to consider interval and fuzzy valued functions of several variables and to include directional gH-differentiability; the proposed setting is ...more general than the existing definitions in the literature and allows a unified view of total and direction gH-differentiability and for the computation of partial gH-derivatives, directional gH-derivative and level wise gH-differentiability in the fuzzy valued case. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well known tangency between a differentiable function and its tangent plane. The proposed new setting allows an analysis of conditions for local optimality (dominance with respect to interval and level-wise partial orders well known in the literature) in terms of directional gH-derivatives, including concepts of local convexity, and to formulate KKT-like conditions for non-dominated solutions in constrained optimization problems.
Machine learning interpretation has a well-established discussion in various areas. Whereas most interpretation is made after the modelling, one may try to obtain the informative by itself model. ...Mainly such models are represented by time-series models such as AR(auto-regression). For time series, we may also obtain the expression in closed form, which has the advantage of being interpretable. Transferring from single to multiple dimensions increases the search space drastically. Thus, we must reduce it for the initial assumption search. The paper proposes the algorithm of algebraic expression discovery using evolutionary optimization. The initial assumption is made using the Fourier series decomposition. Several examples of the algorithm's work based on known functions and arctic ocean ice concentration data are shown.
Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse ...angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan.
Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in Rd. Their structures are studied through a detailed analysis of classical Hahn ...polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relationship between the index set of orthogonal polynomials and the lattice set in the polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral properties of the orthogonal polynomials are also discussed.
In this paper, given an integral domain U, we investigate the main properties of a relation ←mod which is based on the interrelation between subdomains of U and finitely generated unitary submodules ...of U. We shall characterize it in terms of a second relation ≺⋄ between n-tuples (u1,…,un) of elements of U and subdomains D of U defined by the vanishing in (u1,…,un) of some polynomial p(Z1,…,Zn) belonging to a specific subset of the polynomial ring in several variables DZ1,…,Zn. Such an equivalence shall be used in order to introduce three specific collections of subdomains XU, BU and PU, whose algebraic properties present a close connection with geometrical and combinatorial properties induced by ←mod.
On the other hand, the characterization of the subdomains of XU leads to the more general problem of finding a map Ψ associating with a subdomain D of U a collection Ψ(D) of subdomains of KU such that the intersection of some or of any member of Ψ(D) gives D. In this perspective, in the present paper we shall study two further collections of subdomains of U, denoted respectively by EU and LU, whose main properties are related to those of the families PU and BU.
Finally, our investigation of all the aforementioned subdomain families shall be also related to the study of pairs (e,ξ), where e∈U∖{0} and ξ is an idempotent ring endomorphism of U whose kernel agrees with the ideal of U generated by e. We shall exhibit several results concerning the membership of ξ(U) and of ξ(U)e to the above subdomain families.
This paper is about the characterization of the rate of best polynomial approximation on a domain in Rd by multivariate polynomials of degree n via a suitable modulus of continuity or smoothness. We ...shall be concerned with the situations: approximation on convex polytopes and approximation on (not necessarily convex) smooth algebraic domains. We shall introduce appropriate moduli of smoothness and prove the corresponding Jackson theorem. The usual weak converses are also true.
We study the Cesàro means of the orthogonal polynomial expansions (OPEs) with respect to the weight function ∏i=1d|xi|2κi on the unit sphere Sd−1⊂Rd for all parameters κ1,⋯,κd>−12. We obtain sharp ...pointwise estimates for the corresponding Cesàro kernels, which were previously known when all parameters are nonnegative. We settle the problem for the case when min1≤j≤dκj<0. Our estimates allow us to establish sharp results on Cesàro summability with less restriction on the parameters. We also establish similar results for the corresponding weighted OPEs on the unit ball and on the simplex.
Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of ...the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.
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