Space Topologies and Their Dual Space Topologies for Conventional Functional Space Topologies

In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(Rn) of functions of class C∞ on Rn which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C∞ with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.

The Dual of the Two-Variable Exponent Amalgam Spaces (Lq(),lp())(Ω)

Wiener amalgam spaces are a class of function spaces where the function’s local and global behavior can be easily distinguished. These spaces are ex-tensively used in Harmonic analysis that originated in the work of Wiener. In this paper: we first introduce a two-variable exponent amalgam space (Lq(),lp())(Ω). Secondly, we investigate some basic properties of these spaces, and finally, we study their dual.

A semantic and emotion-based dual latent variable generation model for a dialogue system

With the development of intelligent agents pursuing humanisation,artificial intelligence must consider emotion,the most basic spiritual need in human interaction.Traditional emotional dialogue systems usually use an external emotional dictionary to select appropriate emotional words to add to the response or concatenate emotional tags and semantic features in the decoding step to generate appropriate responses.However,selecting emotional words from a fixed emotional dictionary may result in loss of the diversity and consistency of the response.We propose a semantic and emotion-based dual latent variable generation model(Dual-LVG)for dialogue systems,which is able to generate appropriate emotional responses without an emotional dictionary.Different from previous work,the conditional variational autoencoder(CVAE)adopts the standard transformer structure.Then,Dual-LVG regularises the CVAE latent space by introducing a dual latent space of semantics and emotion.The content diversity and emotional accuracy of the generated responses are improved by learning emotion and semantic features respectively.Moreover,the average attention mechanism is adopted to better extract semantic features at the sequence level,and the semi-supervised attention mechanism is used in the decoding step to strengthen the fusion of emotional features of the model.Experimental results show that Dual-LVG can successfully achieve the effect of generating different content by controlling emotional factors.

THE DUAL OF NONCOMMUTATIVE LORENTZ SPACES

We prove some noncommutative analogues of the classical results about dual spaces of the classical Lorentz spaces.

Curvature Motion on Dual Hyperbolic Unit Sphere H^(2)_(0)

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere H2<sub style="margin-left:-8px;">0 in the dual Lorentzian space D31 with dual signature (+,+-) . We carry the obtained results to the Lorentzian line space R3<sub style="margin-left:-8px;">1 by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in R3<sub style="margin-left:-8px;">1.

Fock-Sobolev Spaces and Weighted Composition Operators among Them

Abstract： We characterize the boundedness and compactness of weighted compo-sition operators among some Fock-Sobolev spaces. We also estimate the norm and essential norm of these operators. Furthermore, we discuss the duality spaces of Fock-Sobolev spaces Fs^p,mwhen 0 〈 p 〈∞.

CONSIDER SAINT-VENANT'S PRINCIPLE BY MEANS OF CHAIN MODEL

A precise background theory of computational mechanics is formed. Saint_Venant's principle is discussed in chain model by means of this precise theory. The classical continued fraction is developed into operator continued fraction to be the constrictive formulation of the chain model. The decay of effect of a self_equilibrated system of forces in chain model is decided by the convergence of operator continued fraction, so the reasonable part of Saint_Venant's principle is described as the convergence of operator continued fraction. In case of divergence the effect of a self_equilibrated system of forces may be non_zero at even infinite distant sections, so Saint_Venant's principle is not a common principle.

Classical Fourier Analysis over Homogeneous Spaces of Compact Groups

This paper introduces a unified operator theory approach to the abstract Fourier analysis over homogeneous spaces of compact groups. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and μ be the normalized G-invariant measure on G/H associated to the Weil＇s formula. Then, we present a generalized abstract framework of Fourier analysis for the Hilbert function space L^2 （G / H, μ）.

ON SIMULTANEOUS WEAKLY-CHEBYSHEV SUBSPACES

In this paper, we shall introduce and characterize simultaneous quasi-Chebyshev （and weakly-Chebyshev） subspaces of normed spaces with respect to a bounded set S by using elements of the dual space.

Weakly Compact Sets and Riesz Representation Theorem in Musielak Sequence Spaces

In this work,we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function.Finally,as an immediate consequence of the criteria considered in this paper,the criteria of the weakly compact sets of Orlicz sequence spaces are deduced.

The Dual Spaces of Sets of Difference Sequences of Order m and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.

Duals of Hardy Spaces of B-valued Martingales for 0

The dual of B-valued martingale Hardy space Hr^s（p） （B） with small index 0 〈 r ≤ 1, which is associated with the conditional p-variation of B-valued martingale, is characterized. In order to obtain the results, a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.

Dual Space, Carleson Measure and Sequence Interpolation on A^p(φ)(0

Let D={z∈: |z|【1} and φ be a normal function on [0, 1). For p∈(0, 1) such a function φ is used to define a Bergman space A^p(φ) on D with weight φ~p(|·|)/(1-|·|~2). In this paper, the dual space of A^p(φ) is given, four characteristics of Carleson measure on A^p(φ) are obtained. Moreover, as an application, three sequence interpolation theorems in A^p(φ) are derived.

The Fine Spectra of the Difference Operator △ Over the Sequence Space bv_p,(1≤p<∞)

The sequence space bvp consisting of all sequences （xk） such that （xk -xk-1） belongs to the space gp has recently been introduced by Basar and Altay [Ukrainian Math. J., 55（1）, 136-147（2003）]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f-dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator A acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg＇s classification of the difference operator △ over the sequence space bvp has been determined, where 1≤p〈∞.

Norming Subspaces Isomorphic to l_1

Norming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to l1 provided that the following two conditions are satisfied： （1） X＊ contains a subspace isomorphic to l1; and （2） X＊ contains a separable norming subspace.

On Absolute N?rlund Spaces and Matrix Operators

In a more recent paper, the second author has introduced a space ｜Cα｜k as the set of all series by absolute summable using Cesaro matrix of order α 〉 -1. In the present paper we extend it to the absolute NSrlund space ｜Np^θ｜k taking Norlund matrix in place of Cesaro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.

Wavelets and Geometric Structure for Function Spaces

With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L 1 are studied.

Axioms in the Variety of eO-Algebras

The variety eO of extended Ockham algebras consists of those algebras （L; ∧, ∨, f, k, 0, 1） such that （L; ∧, ∨, 0, 1） is a bounded distributive lattice together with a dual endomorphism f on L and an endomorphism k on L such that fk = kf. In this paper we extend Urquhart＇s theorem to eO-algebras and we are in particular concerned with the subclass e2M of eO-algebras in which f^2 = id and k^2 = id. We show that there are 19 non-equivalent axioms in e2M and then order them by implication.

EXISTENCE OF WEAK SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, by using weak dissipative type conditions and weak noncompact conditions, we give the existence of weak solutions of differential equations in weak complete Banach spaces.