Approximate Identity Research Articles

On the Importance of Coefficient Alpha for Measurement Research: Loading Equality Is Not Necessary for Alpha's Utility as a Scale Reliability Index.

The population relationship between coefficient alpha and scale reliability is studied in the widely used setting of unidimensional multicomponent measuring instruments. It is demonstrated that for any set of component loadings on the common factor, regardless of the extent of their inequality, the discrepancy between alpha and reliability can be arbitrarily small in any considered population and hence practically ignorable. In addition, the set of parameter values where this discrepancy is negligible is shown to possess the same dimensionality as that of the underlying model parameter space. The article contributes to the measurement and related literature by pointing out that (a) approximate or strict loading identity is not a necessary condition for the utility of alpha as a trustworthy index of scale reliability, and (b) coefficient alpha can be a dependable reliability measure with any extent of inequality in the component loadings.

Pathological Set of Initial Data for Scaling-Supercritical Nonlinear Schrödinger Equations

Abstract The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains a superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity, which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.

Değişken üslü Lebesgue uzaylarında konvolüsyonların bazı özellikleri

A convolution in the variable exponent Lebesgue spaces is defined and the possibility its approximation by finite linear combinations of Steklov means is proved. Moreover, the convergence of the special convolutions sequence constructed via approximate identity to the original function is showed.

Korovkin-type theorems and local approximation problems

Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces).Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász–Mirakyan operators, Gauss–Weierstrass operators and Bernstein–Schnabl operators on convex subsets of Hilbert spaces.Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded 2π− periodic functions on R and with an application related to sequences of convolution operators generated by positive approximate identities.

Maps preserving two-sided zero products on Banach algebras

Let A and B be Banach algebras with bounded approximate identities and let Φ:A→B be a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ is a weighted Jordan homomorphism provided that A is zero product determined and weakly amenable. These conditions are in particular fulfilled when A is the group algebra L1(G) with G any locally compact group. We also study a more general type of continuous linear maps Φ:A→B that satisfy Φ(a)Φ(b)+Φ(b)Φ(a)=0 whenever ab=ba=0. We show in particular that if Φ is surjective and A is a C⁎-algebra, then Φ is a weighted Jordan homomorphism.

Weakly compact multipliers and φ-amenable Banach algebras

Let 𝒜 be a Banach algebra and let J be a closed ideal of 𝒜 such that φ|J≠0 for some nonzero character φ on 𝒜. We obtain some relations between the existence of compact and weakly compact multipliers on J and on 𝒜 in some sense. Then we apply these results to hypergroup algebra L1(K) when K is a locally compact hypergroup. In particular, for a closed ideal J in L1(K) we prove that K is compact if and only if there is f∈J such that φ1(f)≠0 and the multiplication operator λf:g↦g∗f is weakly compact on J. Using this, we study Arens regularity of J whenever it has a bounded left approximate identity. Finally, we apply these results on some abstract Segal algebras with respect to the L1(K).

Codimension-Two Bifurcations and Bifurcation Controls in a Discrete Biological System with Weak Allee Effect

In this paper, the codimension-two bifurcations and bifurcation controls of a discrete predatory system with weak Allee effect on prey are investigated. After analyzing the qualitative properties of all the fixed points, we explore the existences of 1:2 and 1:3 strong resonances at the unique interior fixed point. Combining the adjoint operator method with an approximate identity transformation, we obtain the normal forms of these strong resonances. Meanwhile, we calculate the approximate expressions of bifurcation curves around the resonance point, which include Neimark–Sacker and period doubling bifurcation curves. And we give the conditions under which these bifurcations are supercritical or subcritical. To study global bifurcations near the resonance point, we use an ordinary differential equation to approximate the original discrete system by the time-one approximation flow theory. In biology, bifurcations and unstable fluctuations may be harmful for the breeding of biological population. Then we extend the hybrid control strategy to control 1:2 and 1:3 strong resonances. Numerical simulations are in good agreement with our theoretical results. By using the MatContM package, we perform numerical bifurcation analyses to further support our theoretical results, in which the bifurcation points and the normal form coefficients are completely consistent with our theories. According to the results, we finally analyze the impact of weak Allee effect on the system. And a reasonable biological explanation is given for these conclusions.

Approximate Identities in Algebras of Compact Operators on Frechet Spaces

Bounded approximate identity(bai) is a key concept in the theory of amenability of algebras. In this paper, we show that algebra of compact operators on Frechet space X has both the right and left locally bounded approximate identities. Sufficient conditions for the existence of these identities are established based on the geometry properties of the Frechet space X and its dual space X' respectively.

Flett potentials associated with differential-difference Laplace operators

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1−1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on R. Here, k ≥ 0 and m∈N\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385–451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(R) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.

Spin weighted wavelets on the sphere—Frames

A new construction of spin weighted wavelets on the sphere is presented in the paper. The wavelets are similar to those derived from approximate identities but the constraints of the definition are slightly weaker. They allow for quite a big freedom in wavelet construction so that even spin weighted wavelets with explicit representations exist. The price to be paid is that no direct reconstruction is given and the wavelet transform must be inverted with frame methods. The set of wavelet coefficients is a priori continuous but it is shown how it can be discretized so that no information about the analyzed signal get lost.

Кінодискурс як дворівнева система складників: функційний погляд

The article is devoted to the problem of modeling the structure of film discourse from a functional point of view. The author explained the essence of the concept of “film discourse” and clarified its relationship with related terms. The article denies the expediency of identifying such concepts as “telecine cinematographic discourse” and “cinematographic discourse” with film discourse. The material proves the hyperonymic status of film discourse. It has been noticed that scientists use the following terms in film discourse studies: film text, film dialogue, film replica, film trailer, film announcement and film slogan. It has been determined that film dialogues, film replicas and film slogans are among them of a purely verbal nature. Movie texts, along with movie trailers and movie announcements, accumulate verbal and other semiotic signs. It is established that the connections between the groups of these concepts show the transition from the largest to the smallest: 1) film text – film dialogue – film replica; 2) film trailer – film announcement – film slogan. The expediency of applying a functional approach to modeling the structure of film discourse based on the provisions of the methodology of discourse analysis and the idea of the dyadic nature of the structure of discourse as such is substantiated. However, it is also suggested to take into account the two levels of communication in film discourse (external level – collective author / audience; internal level – addressee character and addressee character) and to distinguish between ectoand entocomponents. Entocomponents are film texts with film dialogues and film lines. These are products of communicative activity that have a certain meaning and semiotic form that can convey this meaning. They are also marked by integrity, completeness and perform a variety of discourse functions, in addition to advertising. Ectocomponents are film trailers, film announcements and film slogans, because they are the products of communicative activities that have their meaning and semiotic form, while being relatively complete and concise with a focus on the implementation of primarily advertising function. It is determined that ento- and ectocomponents in the structure of film discourse interact with each other on the principle of hierarchical subordination. Ectocomponents are subordinated to ethnic components, despite the fact that external components have a wider functional palette. It is proved that the relations between ento- and ectocomponents are established. approximate identity, because movie trailers and movie announcements are almost always created on the basis of pre-borrowed frames, episodes and cues / dialogues from the main movie text. This was confirmed by a comparative analysis of the language organization of the film trailer and the film text of the documentary “Your Vasyl”

On Banach algebras defined by multipliers

In this paper, we investigate a Banach algebra AT, where A is a Banach algebra and T is a left (right) multiplier on A. We study some concepts on AT such as n-weak amenability, cyclic amenability, biflatness, biprojectivity and Arens regularity. For the group algebra L1(G) of an infinite compact group G, it is shown that there is a multiplier T such that L1(G)T has not a bounded approximate identity. For ?1(S), where S is a regular semigroup with a finite number of idempotents, we show that there is a multiplier T such that Arens regularity of ?1(S)T implies that S is compact.

On Pseudo‐contractibility and Ultra Central Approximate Identity of Some Semigroup Algebras

In this paper, we investigate the existence of an ultra central approximate identity for a Banach algebra A and its second dual A∗∗. Also, we prove that for a left cancellative regular semigroup S, ℓ1(S)∗∗ has an ultra central approximate identity if and only if S is a group. As an application, we show that for a left cancellative semigroup S, ℓ1(S)∗∗ is pseudo‐contractible if and only if S is a finite group. We also study this property for φ‐Lau product Banach algebras and the module extension Banach algebras.

Some properties of convolution in symmetric spaces and approximate identity

This paper deals with the symmetric space of functions and its subspace where continuous functions are dense is considered. Main properties of convolution which plays a vital role in harmonic analysis, as in other areas of mathematics are established in this space. Following the classical case, it is proved that the convolution can be approximated by linear combinations of shifts in a subspace of the considered space. An approximate identity for the convolution is also considered in that subspace.

Invariance and normality of projections in the dual of Banach algebras

We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.

Variational characterizations of weighted Hardy spaces and weighted spaces

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

On the homological and algebraical properties of some Feichtinger algebras

Abstract Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space B min in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of B min with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that B min is a symmetric Segal subalgebra of L 1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of B min on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which A min $\mathcal{A}_{\min}$ with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product.

CENTRE OF BANACH ALGEBRA VALUED BEURLING ALGEBRAS

AbstractWe prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$ . As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.

The inclusion theorems for generalized variable exponent grand Lebesgue spaces

In this paper, we discuss and investigate the existence of the inclusion $ L^{p(.),\theta }\left( \mu \right) \subseteq L^{q(.),\theta }\left( \nu \right) $, where $\mu $ and $\nu $ are two finite measures on $\left( X,\Sigma \right) .$ Moreover, we show that the generalized variable exponent grand Lebesgue space$ L^{p(.),\theta }\left( \Omega \right) $ has a potential-type approximate identity, where $\Omega $ is a bounded open subset of $ \mathbb{R}^{d}.$

The unbounded Kasparov product by a differentiable module

In this paper we investigate the unbounded Kasparov product between a differentiable module and an unbounded cycle of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. Our assumptions on the differentiable module are weak and we do in particular not require that it satisfies any kind of smooth projectivity conditions. The algebras that we work with are furthermore not required to possess a smooth approximate identity. The lack of an adequate projectivity condition on our differentiable module entails that the usual class of unbounded Kasparov modules is not flexible enough to accommodate the unbounded Kasparov product and it becomes necessary to twist the commutator condition by an automorphism. We show that the unbounded Kasparov product makes sense in this twisted setting and that it recovers the usual interior Kasparov product after taking bounded transforms. Since our unbounded cycles are twisted (or modular) we are not able to apply the work of Kucerovsky for recognizing unbounded representatives for the bounded Kasparov product, instead we rely directly on the connection criterion developed by Connes and Skandalis. In fact, since we do not impose any twisted Lipschitz regularity conditions on our unbounded cycles, even the passage from an unbounded cycle to a bounded Kasparov module requires a substantial amount of extra care.