Subspace Of Space Research Articles

On Non-linear Mappings Preserving the Semi-inner Product

We say that a smooth normed space X has a property (SL), if every mapping f:X rightarrow X preserving the semi-inner product on X is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space ell _p, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space ell _p have the property (SL) for every 1< p < infty . Finally, using a variant of the Gowers–Maurey space, we construct an infinite-dimensional uniformly smooth Banach space X such that every smooth Banach space isomorphic to X has the property (SL).

On subspaces of spaces C_p(X) isomorphic to spaces c_{0} and ell _{q} with the topology induced from mathbb {R}^{mathbb {N}}

The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space mathbb {R}^X) is denoted by C_p(X). In this paper we continue the systematic study of sequences spaces c_{0} and ell _{q} (for 0<qle infty ) with the topology induced from mathbb {R}^{mathbb {N}} (denoted by (c_{0})_p and (ell _{q})_{p}, respectively) and their role in the theory of C_p(X) spaces. For every infinite Tychonoff space X we construct a subspace F of C_p(X) that is isomorphic to (c_{0})_p; if X contains an infinite compact subset, then the copy F of (c_{0})_p is closed in C_p(X). It follows that C_p(X) contains a copy of (ell _{q})_{p} for every 0<qle infty . We prove that for any infinite compact space X the space C_p(X) contains no closed copy of (ell _{q})_{p} for qin (0, 1]cup {infty } and no complemented copy for 0<qle infty . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.

Uniformly convex subspaces of measures with the kantorovich norm

In this paper, we consider signed Borel measures on a compact metric space. We study the uniform convexity of the Kantorovich norm on subspaces of the whole space of signed measures. We construct an example of an infinite-dimensional subspace of measures on which the Kantorovich norm is uniformly convex. We also obtain an example of an infinite compact set $(X, \rho)$ such that all uniformly convex subspaces of the space of measures on $X$ are finite-dimensional.

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Shift-invariant spline subspaces of Sobolev spaces and their order of approximation

Closed shift-invariant subspaces of Hilbert space are the core idea of important classes of signal and image analysis methods such as wavelets, shearlets, radial basis functions and many more. In 1994 de Boor, DeVore and Ron [4] gave a full characterization of the order of approximation in such shift-invariant subspaces of L2(Rn), n∈N. Ron and Holtz extended this work in [10] to Sobolev norms, and gave specific criteria on the generating functions to guarantee certain orders of approximation. However, not for all generating functions these criteria are easy to verify. We here give new criteria which are designed for generating functions given explicitly in Fourier domain. We show by various examples on spline functions that in these cases our new criteria give better estimates on the order of approximation.

Characterization of some closed linear subspaces of Morrey spaces and approximation

Characterization of some closed linear subspaces of Morrey spaces and approximation

The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $$e_1,e_2$$ , let $$X_i$$ denote the set of $$e_i$$ -dimensional subspaces of a fixed finite vector space $$V=({\mathbb F}_q)^{e_1+e_2}$$ . Let $$Y_i$$ be a non-empty subset of $$X_i$$ and let $$\alpha _i = |Y_i|/|X_i|$$ . We give a positive lower bound, depending only on $$\alpha _1,\alpha _2,e_1,e_2,q$$ , for the proportion of pairs $$(S_1,S_2)\in Y_1\times Y_2$$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

Alpha magnitude

Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth's definition, we introduce alpha magnitude and investigate some of its key properties. Heuristic observations lead us to conjecture a relationship with the Minkowski dimension of compact subspaces of Euclidean space. Finally, alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and we thus propose it as a new measure for the estimation of fractal dimensions of real-world data sets that is easily computable.

The structure of subspaces in Orlicz spaces lying between $$L^1$$ and $$L^2$$

The structure of subspaces in Orlicz spaces lying between $$L^1$$ and $$L^2$$

The Nordic Interactive and Comparative Spaces Within the Arena of Education

The paper proposes three educational spaces, an operational space, an interactive space and a comparative space. The focus is on the interactive space in order to shed light on various modes of Nordic educational cooperation, with a brief discussion on the rather unique Nordic comparative space. Based on different data and sources, the paper proposes that within the interactive space, practical, scientific, administrative and policy sub-spaces connect across the Nordic countries. The interactive space is seen as socially constructed, multidimensional, and centring on communication.

Linear Diophantine Fuzzy Subspaces of a Vector Space

The notion of a linear diophantine fuzzy set as a generalization of a fuzzy set is a mathematical approach that deals with vagueness in decision-making problems. The use of reference parameters associated with validity and non-validity functions in linear diophantine fuzzy sets makes it more applicable to model vagueness in many real-life problems. On the other hand, subspaces of vector spaces are of great importance in many fields of science. The aim of this paper is to combine the two notions. In this regard, we consider the linear diophantine fuzzification of a vector space by introducing and studying the linear diophantine fuzzy subspaces of a vector space. First, we studied the behaviors of linear diophantine fuzzy subspaces of a vector space under a linear diophantine fuzzy set. Second, and by means of the level sets, we found a relationship between the linear diophantine fuzzy subspaces of a vector space and the subspaces of a vector space. Finally, we discuss the linear diophantine fuzzy subspaces of a quotient vector space.

Basicity of the systems of sines and cosines with linear phases in grand-Sobolev spaces

In this work systems of sines 1∪ {sin (n − β)t}n≥1 and 1∪ {cos (n − β)t}n≥1 cosines are considered, where β is real parameter. Criterion for the completeness, minimality and basicity of these systems with respect to the parameter β in one subspace of grand-Sobolev space are found.

On the nonhomogeneous wavelet bi-frames for reducing subspaces of Hs(K)

Ahmad and Shiekh in Filomat 34: 6(2020), 2091-2099 have constructed dual wavelet frames in Sobolev spaces on local fields of positive characteristic. We continued the study and provided the characterization of nonhomogeneous wavelet bi-frames. First of all we introduce the reducing subspaces of Sobolev spaces over local fields of prime characteristics and then provide the way to characterize the nonhomogeneous wavelet bi-frames over such fields. Our results are better than those established by Ahmad and Shiekh.

Metrizable subspaces of representation spaces

We show that various representation spaces for continua contain a homeomorphic copy of the harmonic sequence, so even though it is not a T0 space, it contains infinite metrizable subspaces. It remains an open question whether this representation space (topologized using continuous surjections) contains a closed subspace homeomorphic to the harmonic sequence. In fact, the representation space for continua topologized using various kinds of mappings (open, confluent, or monotone mappings, for instance) can be shown to have some closed points, but it remains open in many cases whether there are more than two closed points in most instances. Examples of closed points include the pseudo-arc and the universal pseudo-solenoid. Along the way, we find a few related results about metrizable subspaces of representation spaces for continua.

Fault Detection and Isolation for a Class of Nonlinear Systems Based on Gerschgorin Theorem and Optimization Approach

This article is concerned with the fault detection and isolation (FDI) problem for a class of nonlinear systems described by the T–S fuzzy models. Based on the concept of minimum unobservability subspace and geometric property of factor space, a set of FDI filters where each residual is only affected by one fault and completely decoupled from other faults is designed. Furthermore, in the decoupling space, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }/H_{-}$ </tex-math></inline-formula> performance indexes are provided to enhance the sensitivity of residual to faults and robustness to disturbances. In particular, to solve the nonconvex filter design problem caused by introducing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{-}$ </tex-math></inline-formula> index, the Gerschgorin theorem is first used to linearize the corresponding filter design conditions in the outer region of a ball. Then, the FDI filter design problem is converted into a convex optimization one, which is solved via the linear matrix inequality (LMI) control Toolbox, and the advantages and effectiveness of the proposed FDI method are verified through two simulation examples.

Applications of Siegel’s lemma to a system of linear forms and its minimal points

Consider a real matrix $\Theta$ consisting of rows $(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(\Theta)$ and $\widehat{w}(\Theta)$ respectively. For $m=1$, a sharp lower bound for the ratio $w(\Theta)/\widehat{w}(\Theta)$ was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to $\Theta$. Our conditional result extends to general $m>1$ (but may not be optimal in this case). Moreover, our hypothesis is always satisfied in particular for $m=1, n=2$ and thereby unconditionally confirms a previous observation of Jarn\'ik. We formulate our results in the more general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number $\ell$ of consecutive best approximation vectors are linearly independent. Our method is based on Siegel's Lemma.

On extensions of bilinear maps

Abstract The paper deals with extension of bounded bilinear maps. It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces. This leads to a full characterization for extension of bounded bilinear maps on the Cartesian product of arbitrary subspaces of Hilbert spaces. Applications concerning projective tensor products are also investigated.

A Class of Association Schemes and Their Automorphisms

In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups.

Joint Discriminative Latent Subspace Learning for Image Classification

Latent subspace learning aims to produce a latent representation for better reconstruction and classification from high-dimensional data through exploiting the optimal subspace. Current latent subspace learning methods commonly have three problems: 1) The discriminative property is ignored when learning the latent subspace, 2) The redundancy exists between the latent subspace and the prediction space, 3) There is no unified latent subspace that exploits knowledge jointly from the raw space, latent subspace, and label space. In this paper, we formulate the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</b> oint <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</b> iscriminative <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</b> atent <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</b> ubspace <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</b> earning (JDLSL) problem to address these issues, and provide its optimization solution. JDLSL learns image representation from two aspects: a) the joint learning of latent subspaces for data reconstruction and prediction, b) the joint learning of label space and latent subspace for data reconstruction. To integrate knowledge from the joint learning, we organize the sparsity-induced latent subspace, where row-sparsity and column sparsity are simultaneously imposed. We provide the theoretical proof for the discriminativity learning ability of the sparsity-induced latent subspace. Extensive experiments and comparisons with the state-of-the-art showed that the proposed method has better performance. JDLSL shows a competitive performance with deep features compared to deep learning architectures, reflecting it potential integrating with deep learning.

High-dimensional tennis balls

We show that there exist constants \(\alpha,\epsilon&gt;0\) such that for every positive integer \(n\) there is a continuous odd function \(f:S^m\to S^n\), with \(m\geq \alpha n\), such that the \(\epsilon\)-expansion of the image of \(f\) does not contain a great circle. This result is motivated by a conjecture of Vitali Milman about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to \(\ell_2^n\).Mathematics Subject Classifications: 46B09, 60C05Keywords: Anti-Ramsey, antipodal subsphere