Compactification of perception pairs and spaces of group equivariant non-expansive operators

Comments on “Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua”, [Chaos, Solitons and Fractals. 168 (2023) 113203

Iterative methods for computing the nonlinear Moore–Penrose metric generalized inverse of Banach space operators: convergence, error bounds, and applications

UDC 517.53 The sigmoid function is a special function emulating biological neurons in terms of signal processing or sending alerts to various departments of the brain for responses. It is also good for machine learning, while the generalized discrete probability distribution series is useful for modeling infectious diseases, genetic modeling, clinical trials, risk analysis, optimal pricing, ecological and climatic modeling, and conservation biology. We explore the convex combination and several other types of behaviors of the sigmoid function associated with the generalized discrete probability distribution series in the space of univalent function theory. This is done by introducing and studying a new derivative operator. Several coefficient inequalities and consequences of various choices of the parameters involved are discussed. Graphically, by using the Python software, various convex combinations of $K_{\phi}$ and $g_{\phi}$ are presented in terms of disease and intervention, policy and policy intervention, investment return, disease spread, infection and intervention, free market and intervention, and cost and efficiency relations.

We consider an abstract symmetric Fock space and the Hilbert space generated symmetric polynomials on the space of absolutely summing sequences. In particular, the properties of the symmetric shift operator, such as linearity and unboundedness are investigated. Also, we find eigenvectors of this operator. We construct an operator in the symmetric Fock space, which is similar to the symmetric shift operator. In the paper we note that this operator is not hypercyclic.

This paper introduces and investigates the concept of the q-numerical range for tuples of bounded linear operators in Hilbert spaces. We establish various inequalities concerning the q-numerical radius associated with these operator tuples. Furthermore, we extend our study to define the q-numerical range in semi-Hilbert spaces and provide a proof of its convexity. Additionally, we explore several related results in this context.

Abstract This paper deals with the modified sampling Kantorovich operators. We start by presenting the primary notations of Orlicz spaces and fundamental auxiliary results. To show modular convergence of the corresponding operators in Orlicz spaces, we obtain well definiteness in Orlicz spaces and norm convergence in the space of continuous functions with compact support. In the last section, we present some examples of ρ -kernels which satisfy the corresponding assumptions and present some graphical representations.

Abstract We present an analytical framework to investigate phase dynamics in discrete-time quantum walks, with a focus on geometric phases in both single-and two-particle systems. Leveraging the Cayley-Hamilton theorem, we construct the evolution operator in momentum space and derive exact closed-form solutions for the full time evolution. This allows us to explicitly extract total, dynamic, and most notably, non-adiabatic geometric phases throughout the walk. In the two-particle case, we further examine how initial entanglement influences the accumulation and interplay of these phases. Our results offer a unified and exact approach to understanding the geometric structure underlying quantum walk dynamics, with particular emphasis on non-adiabatic effects.

Fredholm theory of continuous families of bounded linear operators in Krein spaces

Some linear maps preserving continuous frames for operators in Hilbert spaces

This paper presents dual new algorithms for answering the Equilibrium Problem and Fixed Point Problems in Banach spaces. By operating the Bregman distance, we make a sweeping statement projection-based ways to unimpressed boundaries in non-Euclidean spaces, mainly in instances where conventional ideas of Lipschitz continuity or monotonicity are restrictive. The firstly algorithm services the generalized resolvent operator and Bregman projections in reflexive Banach spaces, founding strong coming together below relaxed settings. The next algorithm, planned for Hilbert spaces, integrates mistake open-mindedness machineries to confirm constancy in the attendance of computational perturbations.

This study investigates the approximation properties of two different types of parameter-dependent generalizations of Stancu type operators. In the first step, it is determined that this operator defined on the interval [-1,1] is an operator of Korovkin type satisfying the theorem and its important properties are analyzed. Then, a new class of operators of Kantorovich type is defined using this operator and the study on the approximation properties of these operators is elaborated. Another important part of the study is to investigate the convergence properties of both classes of operators in Lp spaces. In this context, the effect of the operators on functions and their convergence properties are evaluated and the advantages of the newly defined operators over the classical approximations are demonstrated. In addition, graphs of the approximation of these operators are presented and the effects of the operators on the functions are visually analyzed. By presenting the theoretical analysis and visual results of both operators, the study provides important information about their convergence.

Given Hilbert space operators A,B, and X, let ▵A,B and δA,B denote, respectively, the elementary operators ▵A,B(X)=I−AXB and the generalised derivation δA,B(X)=AX−XB. This paper considers the structure of operators Dd1,d2m(I)=0 and Dd1,d2m compact, where m is a positive integer, D=▵ or δ, d1=▵A*,B* or δA*,B* and d2=▵A,B or δA,B. This is a continuation of the work performed by C. Gu for the case where ▵δA*,B*,δA,Bm(I)=0, and the author with I.H. Kim for the cases where ▵δA*,B*,δA,Bm(I)=0 or ▵δA*,B*,δA,Bm is compact, and δ▵A*,B*,▵A,Bm(I)=0 or δ▵A*,B*,δA,Bm is compact. Operators Dd1,d2m(I)=0 are examples of operators with a finite spectrum; indeed, the operators A,B have at most a two-point spectrum, and if Dd1,d2m is compact, then (the non-nilpotent operators) A,B are algebraic. Dd1,d2m(I)=0 implies Dd1,d2n(I)=0 for integers n≥m: the reverse implication, however, fails. It is proved that Dd1,d2m(I)=0 implies Dd1,d2(I)=0 if and only if of A and B (are normal and hence) satisfy a Putnam–Fuglede commutativity property.

Abstract The $$H^\infty $$ H ∞ -functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fields of science. An extension of the $$H^\infty $$ H ∞ -functional calculus also exists in the hypercomplex setting, where it is based on the notion of S -spectrum. Originally this was done for sectorial quaternionic operators, but then also generalized all the way to bisectorial fully Clifford operators. In the latter setting and in Hilbert spaces, this paper now characterizes the boundedness of the $$H^\infty $$ H ∞ -functional calculus through certain quadratic estimates. Due to substantial differences in the definitions of the S -spectrum and the S -resolvent operators, the proofs of quadratic estimates in this setting face additional challenges compared to the classical theory of complex operators.

In this paper, we consider a sequence of operators as a wavelet type extension of univariate generalized Kantorovich operators depending on a positive real parameter given in [3]. We establish quantitative estimates for the rate of convergence of these operators in the continuous functions space and $L^{p}$-spaces in terms of modulus of continuity and $K$-functionals, respectively. Furthermore, some inequalities such as Bernstein-Markov type for continuous functions and variation preservation type property of the operators when the involved function is of bounded variation are provided.

In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear operator associated with a special Grushin equation. Finally, we prove the linear operator related to the singularly perturbed Grushin equation is invertible when restricted to the complement of its approximate kernel space.

Regularity of Parabolic Ornstein–Uhlenbeck Equations via Boundedness of Fractional Muckenhoupt-Type Weighted Singular Operators in Variable Herz Spaces

Visualization grammars from ggplot2 to Vega-Lite are based on the Grammar of Graphics (GoG), our most comprehensive formal theory of visualization. The GoG helped expand the expressive gamut of visualization by moving beyond fixed chart types and towards a design space of composable operators. Yet, the resultant design space has surprising limitations, inconsistencies, and cliffs - even seemingly simple charts like mosaics, waffles, and ribbons fall out of scope of most GoG implementations. To author such charts, visualization designers must either rely on overburdened grammar developers to implement purpose-built mark types (thus reintroducing the issues of typologies) or drop to lower-level frameworks. In response, we present GoFish: a declarative visualization grammar that formalizes Gestalt principles (e.g., uniform spacing, containment, and connection) that have heretofore been complected in GoG constructs. These graphical operators achieve greater expressive power than their predecessors by enabling recursive composition: they can be nested and overlapped arbitrarily. Through a diverse example gallery, we demonstrate how graphical operators free users to arrange shapes in many different ways while retaining the benefits of high-level grammars like scale resolution and coordinate transform management. Recursive composition naturally yields an infinite design space that blurs the boundary between an expressive, low-level grammar and a concise, high-level one. In doing so, we point towards an updated theory of visualization, one that is open to an innumerable space of graphic representations instead of limited to a fixed set of "good" designs.

Topological structure of the space of composition operators on the Hardy space of Dirichlet series

Abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.