Regular Subspaces Research Articles

On direct sums and quotient spaces of near-vector spaces

In this paper we study the direct sums of subspaces and some constructions of quotient spaces of near-vector spaces, as defined by André. In particular, for near-vector spaces constructed by taking copies of finite fields, we characterise the quasi-kernels of their quotient spaces, find their cardinality and determine when they are regular. In the case of non-regular quotient spaces, we show how they decompose into maximal regular subspaces. We show how the theory of finite-dimensional near-vector spaces constructed from finite fields allows us to reconstruct near-vector spaces with certain quotient spaces.

On independence in near-vector spaces and their matroids

In this paper we study and characterise independence in near-vector spaces constructed using copies of finite fields. We show that for regular near-vector spaces of this nature, independence is equivalent to the notion of independence in the associated vector space. We prove that for the construction where the number of maximal regular subspaces coincides with the dimension, there are more elements outside of the quasi-kernel which can generate the entire space than previously thought. As a highlight, we completely characterise independence for this space and define matroids for finite field constructions and those using copies of a proper finite near-field.

On Soft Normed Quasilinear Spaces

In this study, we investigate some properties of soft quasi-sequences and present new results. We then study the completeness of soft normed quasilinear space and present an analog of convergence and boundness results of soft quasi sequences in soft normed quasilinear spaces. Moreover, we define regular and singular subspaces of soft quasilinear spaces and draw several conclusions related to these notions. Afterward, we provide examples of these results in soft normed quasilinear spaces generalizing well-known results in soft linear normed spaces. Additionally, we offer new results concerning soft quasi subspaces of soft normed quasilinear spaces. Finally, we discuss the need for further research.

Toward an intuitive understanding of the structure of near-vector spaces

In this article we analyze the definition André proposed for near-vector spaces to make it more transparent. We also study the class of near-vector spaces over division rings and give a characterization of regularity that gives a new insight into the decomposition of near-vector spaces into regular subspaces. We explicitly describe span and deduce a new characterization of subspaces.

Isometries between completely regular vector-valued function spaces

In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of $$C_0(X,E)$$ and $$C_0(Y,F)$$ where X and Y are locally compact Hausdorff spaces and E and F are normed spaces, not assumed to be either strictly convex or complete. We show that for a class of normed spaces F satisfying a newly defined property related to their T-sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between A and B whenever A and B are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.

Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F(ℝn) and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, ‖u‖q = where 1 ≤ q ≤ ∞, we stated that (F(ℝn), ‖u‖q) is a complete normed space. Furthermore, we introduced an inner product in this space for the case q = 2. The inner product must be in the form. For u, v ∈ F(ℝn). We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F(ℝn). Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

Interpolation of a regular subspace complementing the span of a radially singular function

We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^{\alpha }$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the sub

Orthogonal sums in Kreĭn spaces

Infinite orthogonal families of regular subspaces of a Kreĭn space exhibit a wide range of behavior. An elementary method is used to show that conditions on sums of projections produce behavior similar to that of orthogonal closed subspaces of a Hilbert space.

On reducible state variables of logical control networks

Abstract Reducible state variables are important for the study of reduced-order observer and control design of logical control networks (LCNs). This paper studies the number of reducible state variables of LCNs. Based on the upper bound of reducible state variables, several sufficient/necessary conditions are presented for the number of reducible state variables. In addition, using the regular subspace theory, the coordinate transformation as well as the swap matrix are constructed, and some new criteria are proposed to calculate reducible state variables.

Effective intervals and regular Dirichlet subspaces

It is shown in Li and Ying (2019) that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.

J-Self-Adjoint Projections in Krein Spaces

Let ℋ be a Krein space with fundamental symmetry J. Starting with a canonical block-operator matrix representation of J, we study the regular subspaces of ℋ. We also present block-operator matrix representations of the J-self-adjoint projections for the regular subspaces of ℋ, as well as for the regular complements of the isotropic part in a pseudo-regular subspace of ℋ.

Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

Regular Dirichlet extensions of one-dimensional Brownian motion

L’extension régulière de Dirichlet est la notion duale de celle de sous-espace de Dirichlet régulier. Le but principal de cet article est de caractériser toutes les extensions régulières de Dirichlet du mouvement brownien unidimensionnel et d’explorer leurs structures. On montre que chaque extension régulière du mouvement brownien unidimensionnel de Dirichlet peut essentiellement se décomposer en intervalles invariants disjoints au plus dénombrables et en un ensemble $\mathcal{E}$-polaire relatif à cette extension régulière de Dirichlet. Sur chaque intervalle invariant, l’extension régulière de Dirichlet est caractérisée de manière unique par une fonction d’échelle dans une classe donnée. Pour explorer la structure de l’extension régulière de Dirichlet, on applique l’idée introduite dans (Ann. Probab. 45 (2017) 857–872), on formule les formes de Dirichlet de trace et atteint le processus de reprise associé à la restriction à chaque intervalle invariant du complément orthogonal de $H_{e}^{1}(\mathbb{R})$ dans l’espace de Dirichlet étendu de l’extension régulière de Dirichlet. En conséquence, nous trouvons une réponse à un problème de longue date, à savoir si une forme pure de Dirichlet avec sauts comporte des sous-espaces propres de Dirichlet réguliers.

New applications of extremely regular function spaces

Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an $\varepsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of $\ell_1$.

Closed-Form Estimation of Normal Modes From a Partially Sampled Water Column

The output of a vertical linear array is used to infer about the parameters of the normal mode model that describes acoustic propagation in a shallow water. Existing subspace algorithms perform singular vector decomposition of the array data matrix to estimate the sampled model functions. Estimates are exact only if the sensing array is totally covering the water column. We design a new subspace algorithm free from this very restrictive requirement. We use two short hydrophone arrays and activate a monochromatic source at different depths. Estimates of both the modal functions and the wave numbers are obtained in a fully automatic and search-free manner. The algorithm can be qualified as truly high resolution in the sense that, while using short sensing arrays, estimation error becomes arbitrarily low if observation noise is arbitrarily low. This method compares advantageously to existing subspace techniques, as well as transform-domain techniques that require impulsive sources, among other constraints. With two (eigen and singular) vector decompositions, the proposed technique has the complexity of a regular subspace algorithm.

Intrinsic metric and one-dimensional diffusions

One-dimensional strongly local and regular Dirichlet forms which are irreducible can always be characterized by the so-called scale function s and speed measure m. In this paper we derive the intrinsic metric of such a Dirichlet form in terms of s and m. As an application, we give a new characterization of regular Dirichlet subspaces and extensions of Brownian motion, comparing to Fang et al. (2005) and Li and Ying (2019). Finally two examples on volume doubling property of regular Dirichlet subspaces of Brownian motion are presented.

Regular subspaces of regular Dirichlet forms

This paper focuses on the problem of the existence and representation of regular Dirichlet subspaces of a regular Dirichlet form, which was raised by the second author when studying the characterization of Killing transform of Markov processes by Dirichlet forms, and has been studied for about 20 years. A brief survey on recent progress on this problem in the frame of 1-dimensional symmetric diffusions is presented.

An Extension Theorem for a Sequence of Krein Space Contractions

Consider Krein spaces U and Y and let Hk and Kk be regular subspaces of U and Y, respectively, such that Hk⊂Hk+1 and Kk⊂Kk+1 (k∈N). For each k∈N, let Ak:Hk→Kk be a contraction. We derive necessary and sufficient conditions for the existence of a contraction B:U→Y such that BHk=Ak. Some interesting results are proved along the way.

Nonlinear isometries between function spaces

We demonstrate that any surjective isometry T:A→B not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form T(f)=T(0)+Re[μ⋅(f∘τ)]+iIm[ν⋅(f∘ρ)], where μ and ν are continuous and unimodular, there exists a clopen set K with ν=μ on K and ν=−μ on Kc, and τ and ρ are homeomorphisms.

Measurability of Random Attractors for Quasi Strong-to-Weak Continuous Random Dynamical Systems

In order to obtain the measurability of a random attractor, the RDS is usually required to be continuous which, however, is hard to verify in many applications. In this paper, we introduce a quasi strong-to-weak (abbrev. quasi-S2W) continuity and establish a new existence theorem for random attractors. It is shown that such continuity is equivalent to the closed-graph property for mappings taking values in weakly compact spaces. Moreover, it is inheritable: if a mapping is quasi-S2W continuous in some space, then so it is automatically in more regular subspaces. Also, a mapping with such continuity must be measurable. These results enable us to study random attractors in regularity spaces without further proving the system’s continuity. In addition, applying the core idea to bi-spatial random attractor theory we establish new existence theorems ensuring that the bi-spatial attractors are measurable in regularity spaces. As an application, for a stochastic reaction–diffusion equation with general conditions we study briefly the random attractor in $$H^1(\mathbb {R}^d)$$ , the $$(L^2(\mathbb {R}^d), H^1(\mathbb {R}^d) )$$ -random attractor and the $$(L^2(\mathbb {R}^d),L^p(\mathbb {R}^d))$$ -random attractor, $$p>2$$ , $$d\in \mathbb {N}$$ .