Kantorovich Type Research Articles

Q-Deformed and delta-parametrized A-generalized logistic function induced Banach space valued multivariate multi layer neural network approximations

Abstract. Here we research the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or RN, N ∈ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. Mathematics Subject Classification (2010): 41A17, 41A25, 41A30, 41A36. Keywords: Multi-layer approximation, q-deformed and λ-parametrized A- generalized logistic function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, iterated approximation.

Generalizations of the Kantorovich and Wielandt Inequalities with Applications to Statistics

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models.

ψ$$ \psi $$‐Bernstein–Kantorovich operators

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

On the convergence properties of generalized Szász–Kantorovich type operators involving Frobenious–Euler–Simsek-type polynomials

<p>This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.</p>

Towards optimal transport for quantum densities

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: Topics in Optimal Transportation, Amer. Math. Soc. 2003]).

Multiple general sigmoids based Banach space valued neural network multivariate approximation

Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by several different among themselves general sigmoid functions. This is done on the purpose to activate as many as possible neurons. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. We finish with related \(L_{p}\) approximations.

Some density results by deep Kantorovich type neural network operators

In this paper, we prove density results by using deep Kantorovich type neural network operators. Firstly, we define a two layer neural network operator and prove the density results in the spaces C(I) and Lp(I) for p≥1, where I:=[−1,1]. Then we extend it to a multi-layer neural network operator and prove the corresponding density results. Our study provides a generalizations of the well known single layer Kantorovich type neural network operator in terms of its deeper version.

Voronovskaja type theorem for some nonpositive Kantorovich type operators

In this paper we will study a Voronovskaja type theorem and a simultaneous approximation result for a new class of generalized Bernstein operators. The new operators are obtained using a generalization of Kantorovich's method, namely, we will introduce a sequence of operators $K_n^l=D^l\circ B_{n+l}\circ I^l$, where $B_{n+l}$ are Bernstein operators, $D^{l}f=f^{(l)}+a_{l-1}f^{(n-1)}+\dots+a_1f'+a_0f$ is a differential operator with constant coefficients $a_j,\ j\in\{0,\dots,l-1\}$ and $I^{l}$ a corresponding antiderivative operator such that $D^{l}\circ I^{l}=Id$.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Riemann–Liouville fractional integral type exponential sampling Kantorovich series

In the present paper, we introduce a new family of sampling Kantorovich type operators using fractional-type integrals. We study approximation properties of newly constructed operators and give a rate of convergence via a suitable modulus of continuity. Furthermore, we obtain an asymptotic formula considering locally regular functions. Secondly, we deal with logarithmic weighted spaces. By using a certain weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem considering the remainder of Mellin–Taylor’s formula. Moreover, we give a relation between generalized exponential sampling operators and newly constructed operators. Finally, we present some examples of kernels satisfying the obtained results. The results are examined by illustrative numerical table and graphical representations.

Approximation by Szasz–Kantorovich type operators associated with d-symmetric d-orthogonal polynomials of Brenke type

Approximation by Szasz–Kantorovich type operators associated with d-symmetric d-orthogonal polynomials of Brenke type

Order of Approximation by a New Univariate Kantorovich Type Operator

In order to approximate Lebesgue integrable functions on [0, 1], a sequence of linear positive integral operators of Kantorovich type Lσ<sσ>f (x) with a parameter sσ is introduced. The estimates for rates of approximation for functions with a specific smoothness are proved using the appropriate modulus of continuity.

Charlier Polinomlarıyla İlişkili Kantorovich Tipi Operatörler Sınıfının Yaklaşım Özellikleri

In this paper, we introduce a kind of Charlier polynomial-based Szász-Kantorovich type operator. We begin by using Korovkin's theorem to demonstrate the uniform convergence of these operators. Second, using mathematical techniques like Peetre’s K-functional notion and the common modulus of the operators, we evaluate the order of convergence of the operators. Third, we use the Voronovskaya type approximation theorem to derive an asymptotic formula for the operator we gave. Finally, we give a numerical example using Maple 2022.

Kinetic friction of structurally superlubric 2D material interfaces

The ultra-low kinetic friction Fk of 2D structurally superlubric interfaces, connected with the fast motion of the incommensurate moiré pattern, is often invoked for its linear increase with velocity v0 and area AS, but never seriously addressed and calculated so far. Here we do that, exemplifying with a twisted graphene layer sliding on top of bulk graphite – a demonstration case that could easily be generalized to other systems. Neglecting quantum effects and assuming a classical Langevin dynamics, we derive friction expressions valid in two temperature regimes. At low temperatures the nonzero sliding friction velocity derivative dFk/dv0 is shown by Adelman–Doll–Kantorovich type approximations to be equivalent to that of a bilayer whose substrate is affected by an analytically derived effective damping parameter, replacing the semi-infinite substrate. At high temperatures, friction grows proportional to temperature as analytically required by fluctuation–dissipation. The theory is validated by non-equilibrium molecular dynamics simulations with different contact areas, velocities, twist angles and temperatures. Using 6°-twisted graphene on Bernal graphite as a prototype we find a shear stress of measurable magnitude, from 25kPa at low temperature to 260kPa at room temperature, yet only at high sliding velocities such as 100m/s. However, it will linearly drop many orders of magnitude below measurable values at common experimental velocities such as 1μm/s, a factor 10−8 lower. The low but not ultra-low “engineering superlubric” friction measured in existing experiments should therefore be attributed to defects and/or edges, whose contribution surpasses by far the negligible moiré contribution.

Convergence Properties of a Kantorovich Type of Szász Operators Involving Negative Order Genocchi Polynomials

The goal of this research is to construct a generalization of a Kantorovich type of Szász operators involving negative-order Genocchi polynomials. With the aid of Korovkin’s theorem, modulus of continuity, Lipschitz class, and Peetre’s K-functional the approximation properties and convergence rate of these operators are established. To illustrate how operators converge to a certain function, we present some examples.

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

Abstract Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ N , N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Neural network interpolation operators of multivariate functions

In this paper, we introduce a type of multivariate neural network interpolation operators Fn,σ(f) activated by some newly defined sigmoidal functions, and give both the direct and the converse results of the approximation by Fn,σ(f) for multivariate continuous functions. We also introduce a Kantorovich type variant of Fn,σ(f), and establish both the direct theorem and the converse theorem of approximation by the Kantorovich type operators in Lp spaces with 1≤p<∞. Finally, we give some numerical examples to demonstrate the validity of the obtained results, and apply our operators to the image super-resolution reconstruction.

Approximation by the Extended Neural Network Operators of Kantorovich Type

Based on the idea of integral averaging and function extension, an extended Kantorovich-type neural network operator is constructed, and its error estimate of approximating continuous functions is obtained by using the modulus of continuity. Furthermore, by introducing the normalization factor, the approximation property of the new version of the extended Kantorovich-type neural network (normalized extended Kantorovich-type neural network) operator is obtained in Lp[−1,1]. The numerical examples show that this newly proposed neural network operator has a better approximation performance than the classical one, especially at the endpoints of a compact interval.

On bivariate Kantorovich exponential sampling series

In this paper, we introduce and analyze the approximation properties of bivariate generalization for the family of Kantorovich type exponential sampling series. We derive the basic convergence result and Voronovskaya type theorem for the proposed sampling series. Using logarithmic modulus of smoothness, we establish the quantitative estimate of order of convergence for the Kantorovich type exponential sampling series. Furthermore, we study the convergence results for the generalized Boolean sum (GBS) operator associated with bivariate Kantorovich exponential sampling series. At the end, we provide a few examples of kernels to which the presented theory can be applied along with the graphical representation and error estimates.