Isomorphism Of Spaces Research Articles

On Density and Bishop–Phelps–Bollobás-Type Properties for the Minimum Norm

We study the set MA(X,Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {MA}}(X,Y)$$\\end{document} of operators between Banach spaces X and Y that attain their minimum norm, and the set QMA(X,Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {QMA}}(X,Y)$$\\end{document} of operators that quasi attain their minimum norm. We characterize the Radon–Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets MA(X,Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {MA}}(X,Y)$$\\end{document} and QMA(X,Y)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {QMA}}(X,Y)$$\\end{document}. We show that every infinite-dimensional Banach space X has an isomorphic space Y, such that not every operator from X to Y quasi attains its minimum norm. We introduce and study Bishop–Phelps–Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

The Banach–Mazur distance between isomorphic spaces of continuous functions is not always an integer

For over half a century, Aleksander Pełczyński's question, whether the Banach–Mazur distance between any two isomorphic C(K) spaces is an integer, has remained open; as usual, K is a compact Hausdorff space and C(K) denotes the Banach space of all continuous real-valued functions on K, provided with the maximum norm. We answer this question in the negative. Moreover, we prove that the Banach–Mazur distance between isomorphic spaces C(K) and C0(L) is not always an integer; here L is a locally compact but noncompact Hausdorff space and C0(L) denotes the Banach space of all continuous real-valued functions vanishing at infinity on L, furnished with the maximum norm.

Mirror map for Landau-Ginzburg models with nonabelian groups

BHK mirror symmetry as introduced by Berglund–Hübsch and Marc Krawitz between Landau–Ginzburg (LG) models has been the topic of much study in recent years. An LG model is determined by a potential function and a group of symmetries. BHK mirror symmetry is only valid when the group of symmetries is comprised of the so-called diagonal symmetries. Recently, an extension to BHK mirror symmetry to include nonabelian symmetry groups has been conjectured. In this article, we provide a mirror map at the level of state spaces between the LG A-model state space and the LG B-model state space for the mirror model predicted by the BHK mirror symmetry extension for nonabelian LG models. We introduce two technical conditions, the Diagonal Scaling Condition, and the Equivariant Φ condition, under which a bi-degree preserving isomorphism of state spaces (the mirror map) is guaranteed to exist, and we prove that the condition is always satisfied if the permutation part of the group is cyclic of prime order.

Symmetry and Invariant Bases in Finite Element Exterior Calculus

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan’s classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms.

О конструкции канонической формы на расслоении реперов

The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism play­ing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct sum­mand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-fra­me, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.

Ergodic decompositions of Dirichlet forms under order isomorphisms

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.

Solution of a mixed nonaxisymmetric problem of the theory of elasticity for anisotropic bodies of revolution

The paper developed a technique for solving mixed nonaxisymmetric problems of the theory of elasticity for bounded bodies of revolution made of a transversely isotropic material under the action of surface forces specified according to a cyclic law. The technique involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states, conjugated by isomorphism, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The internal state includes the components of the tensor of stresses, deformations, and the displacement vector. The boundary state includes efforts and displacements at the boundary of the body. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of the boundary state isomorphic to it. The basis is formed on the basis of the general solution of the boundary value problem of elastostatics for a transversely isotropic body of revolution. Orthogonalization of state spaces is carried out, where the internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external forces is used. Finally, finding the desired state is reduced to solving an infinite system of algebraic equations for the Fourier coefficients. The solution of the problem with mixed boundary conditions for a circular in plan cylinder of transversely isotropic coarse dark gray siltstone with anisotropy axis coinciding with the geometric axis of symmetry is presented. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of problem solutions and graphical visualization of the results are presented.

Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations

Abstract In this article, we introduce and study sequence spaces of Cesàro–Nörlund operators of order n associated with a sequence of Orlicz functions. We obtain some topological properties and Schauder basis of these sequence spaces. Moreover, we compute the α-, β- and γ-duals and the matrix transformations of these newly formed sequence spaces. Finally, we prove that these sequence spaces are of Banach–Saks type p and have a weak fixed-point property.

Segal–Bargmann Transforms Associated to a Family of Coupled Supersymmetries

The Segal–Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal–Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal–Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Segal–Bargmann transforms for a specific class of coupled SUSYs which includes the quantum harmonic oscillator as a special case. We will show that the associated Segal–Bargmann spaces are distinct from the usual Segal–Bargmann space: their associated weight functions are no longer Gaussian and are spanned by stricter subsets of the holomorphic polynomials. The coupled SUSY Segal–Bargmann spaces provide new examples of reproducing kernel Hilbert spaces.

Remarks on pseudo-vertex-transitive graphs with small diameter

Let Γ denote a Q-polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Γ. For a vertex x∈X and for 0≤i≤D, let Ei⁎(x) denote the projection matrix to the ith subconstituent space of Γ with respect to x. The Terwilliger algebra T(x) of Γ with respect to x is the semisimple subalgebra of MatX(C) generated by A,E0⁎(x),E1⁎(x),…,ED⁎(x). Let V denote a C-vector space consisting of complex column vectors with rows indexed by X. We say Γ is pseudo-vertex-transitive whenever for any vertices x,y∈X, there exists a C-vector space isomorphism ρ:V→V such that (ρA−Aρ)V=0 and (ρEi⁎(x)−Ei⁎(y)ρ)V=0 for all 0≤i≤D. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter D∈{2,3,4}. For D=2, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For D=3, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For D=4, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.

Banach-Stone Theorem for isometries on spaces of vector-valued differentiable functions

Let Q⊆Rm, K⊆Rn be open sets, p,q∈N, 1≤r<∞ and let E,F be Banach spaces. Denote by C⁎p(Q,E)r the space of all f∈Cp(Q,E) with bounded derivatives of order ≤p, endowed with the norm‖f‖=sups∈Q⁡‖[(‖∂λf(s)‖E)λ∈Λ]‖r, where ‖⋅‖r denotes the ℓr norm on RΛ, Λ={λ:|λ|≤p}. Let T:C⁎p(Q,E)r→C⁎q(K,F)r be a linear surjective isometry. Then m=n and p=q and there are a Cp-diffeomorphism τ:K→Q and Banach space isomorphisms V(t):E→F so thatTf(t)=V(t)f(τ(t)) if f∈C⁎p(Q,E),t∈K. The result holds in a more general setting. The proof establishes a direct link between isometries and biseparating maps.

Multiplicative maps of matrices over rings

Let R be an arbitrary associative ring (not necessarily with an identity). A sufficient condition is given to determine when a multiplicative isomorphism on the matrix ring is additive without the assumption of idempotents. We formulate a version for matrix rings over rings with identities, and then present two applications. One is a new characterization of the automorphisms of the matrix algebra over an arbitrary field by means of multiplicative preservers, and the other one is a new characterization of topologically isomorphisms of finite dimensional real Banach spaces with dimensions no less than 2 by the existence of multiplicative isomorphisms between their rings of all bounded linear operators.

Mathcal{N} = 4 SYM, Argyres-Douglas theories, and an exact graded vector space isomorphism

In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates mathfrak{su}(2) mathcal{N} = 4 super Yang-Mills (SYM) to a theory we call “(3, 2)”. This latter theory is an exactly marginal diagonal SU(2) gauging of three D3(SU(2)) Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs branch. However, this transformation breaks half the supersymmetry of the SYM theory as well as its full mathcal{N} = 2 SU(2)F flavor symmetry. Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small mathcal{N} = 4 super-Virasoro algebra at c = −9 get mapped to the mathcal{A}(6) stress tensor and some of its conformal descendants, while the extra supersymmetry currents on the mathcal{N} = 4 side get mapped to higher-dimensional fermionic currents and their descendants on the mathcal{A}(6) side. We prove these relations are facets of an exact graded vector space isomorphism (GVSI) between these two VOAs. This GVSI respects the U(1)r charge of the parent 4D theories. We briefly sketch how more general mathfrak{su}(N) mathcal{N} = 4 SYM theories are related to an infinite class of AD theories via generalizations of our example. We conclude by showing that, in this class of theories, the mathcal{A}(6) VOA saturates a new inequality on the number of strong generators.

Baire bijections of extreme points generated by small-bound isomorphisms of simplex spaces

We show that small-bound isomorphisms of spaces of affine continuous functions on Choquet simplices with Lindel?f boundaries induce Baire measurable bijections of their sets of extreme points.

An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

In this paper, we further develop the study of rings of integro-differential-delay operators considered as noncommutative polynomial algebras satisfying standard calculus identities. Within the algebraic analysis approach, we show that transformations and reductions of linear differential time-delay systems can be interpreted as homomorphisms and isomorphisms of finitely presented left modules over an algebra of integro-differential-delay operators. In particular, we show how Fiagbedzi-Pearson's transformation can be found again and generalized. This transformation maps the solutions of a first-order differential linear system with state and input delays to the solutions of a purely state-space linear system. Fiagbedzi-Pearson's transformation reduces to the well-known Artstein's reduction when the system has no state delay and yields an isomorphism of the solution spaces.

Civilizational Project as Subject of Research: Logic of Meanings and Optics of Analysis

The super-powerful polysemy of discourse about civilization is seen not as a defect in language and speech, but as an objective, irreducible characterization of the subject itself and the discourse about it. This excludes the possibility of universal or consolidated definitions. The solution is offered in a special kind of semantic empathy — in the acceptance of the discourse about civilization in the presumption of its fullness, in the meaningful intuitions of understanding. The principle of "civilizational multiplicity" or "civilization of civilizations" leads to the ideas of the multiplicity of civilizational projects, including the need to understand the Russian project of civilizational development as a "project of projects", that is, not only a mega-, but also a meta-project. Modern times are seen as "times of projects" with corresponding outlets in the problems of postmodernity, postmodern, and postmodernism. The temporal contradictions of the project, the paradoxes of dynamics and statics are considered. Is a project a future in the present and/or a present in the future? The ontological duality of the project is similar: process and thing, life and product, plan and embodiment. The discourse on civilization is considered as a systemic object. Reconstruction of the common problem-thematic space is carried out by the method of contour mapping. The isomorphism of historical time and space of analysis is substantiated: a picture as an image and as a motion picture, as a sequence of frames. A reverse perspective technique is introduced — reverse research and retro analysis, in which the genesis is supplemented by "autopsy" techniques

Structure of the Lipschitz free p-spaces $${\mathcal{F}}_p({\mathbb{Z}}^d)$$ and $${\mathcal{F}}_p({\mathbb{R}}^d)$$ for $$0&lt;p\le 1$$

Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for \(0<p\le 1\) over the Euclidean spaces \({\mathbb{R}}^d\) and \({\mathbb{Z}}^d\). To that end, we show that \({\mathcal{F}}_p({\mathbb{R}}^d)\) admits a Schauder basis for every \(p\in (0,1]\), thus generalizing the corresponding result for the case \(p=1\) by Hájek and Pernecká (J Math Anal Appl 416(2):629–646, 2014, Theorem 3.1) and answering in the positive a question that was raised by Albiac et al. in (J Funct Anal 278(4):108354, 2020). Explicit formulas for the bases of \({\mathcal{F}}_p({\mathbb{R}}^d)\) and its isomorphic space \({\mathcal{F}}_p([0,1]^d)\) are given. We also show that the well-known fact that \({\mathcal{F}}({\mathbb{Z}})\) is isomorphic to \(\ell _{1}\) does not extend to the case when \(p<1\), that is, \({\mathcal{F}}_{p}({\mathbb{Z}})\) is not isomorphic to \(\ell _{p}\) when \(0<p<1\).

The Maxim of Probabilism, with special regard to Reichenbach

It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations.

Unbounded Wiener\u2013Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

Non-Linear Inner Structure of Topological Vector Spaces

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. On the other hand, it was proved in the inner structure literature that isomorphisms of vector spaces and translations preserve the sets of inner points and outer points. In this manuscript, we show that in general, affine maps and convex maps do not preserve inner points. Finally, by making use of the inner structure, we find a simple proof of the fact that a convex and absorbing set is a neighborhood of 0 in the finest locally convex vector topology. In fact, we show that in a convex set with internal points, the subset of its inner points coincides with the subset of its internal points, which also coincides with its interior with respect to the finest locally convex vector topology.