On the homological and algebraical properties of some Feichtinger algebras

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.

TRAILER AS AN ECTOCOMPONENT OF CINEMA DISCOURSE: NEUROLINGVOSEMIOTIC QUALIFICATION OF A LANGUAGE ORGANIZATION

The article is devoted to the problem of neurolinguosemiotic qualification of the language organization of trailers as autonomous components of film discourse. The author of this article draws the following conclusions: 1) the trailer belongs to the external components of film discourse, which is due to the presence of semantic and functional correlations with the concept of «advertising»; 2) between the concepts of «trailer text» and «advertising text» the relationship of the so-called approximate identity, because they have identical structural elements, primarily of a reference nature; 3) the text of the trailer is more complex compared to the advertising text, because the linguosemiotic structure of the first introduced a larger number of typological types of signs; 4) the linguosemiotic structure of the advertising text, in which there are dialogic and monologue signs, has the form of meaningfully and formally interconnected remarks of actors who can perform different roles (average consumers of goods / services, professionals, celebrities, etc.), with the final thesis announcer (rarely – titrated on the screen graphically); 5) the linguosemiotic structure of the trailer text is a combination of autonomous multifunctional monologue signs, nonlinearly related to each other by subject (signs of actor, expert, witness and announcer); 6) the vectors of influence of linguosemiotic units of advertising texts and trailer texts do not coincide, because in the former the metaprograms of centrifugal / centripetal motivation, activity, desired modality, comparison focus are actualized, and in the latter – metaprograms of time orientation, breakdown size, way of thinking.

CENTRE OF BANACH ALGEBRA VALUED BEURLING ALGEBRAS

We 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.

Thirlwall's law is not a tautology, but some empirical tests of it nearly are

This article examines the charge that Thirlwall's law is a theoretical tautology. It shows that a certain approach to empirical testing of that law can sometimes – under conditions analysed here – result in econometric estimates that reflect an approximate identity or ‘near-tautology’. Nevertheless, other methods of empirically testing the law are not subject to the near-tautology critique, and hence the theory itself is not a tautology. Econometric estimates for the US and Mexico reveal that the near-tautology critique applies to data for the former but not the latter; the difference in these results is explained by exactly the reasons discussed here. The article offers an alternative interpretation of Thirlwall's law as implying a benchmark for analysing whether national income, rather than relative prices, is the main adjusting factor in response to current-account imbalances in the long run.

ARENS REGULARITY OF PROJECTIVE TENSOR PRODUCT

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra $A\widehat{\otimes}B$ is Arens regular.

ReLU Network with Bounded Width Is a Universal Approximator in View of an Approximate Identity

Deep neural networks have shown very successful performance in a wide range of tasks, but a theory of why they work so well is in the early stage. Recently, the expressive power of neural networks, important for understanding deep learning, has received considerable attention. Classic results, provided by Cybenko, Barron, etc., state that a network with a single hidden layer and suitable activation functions is a universal approximator. A few years ago, one started to study how width affects the expressiveness of neural networks, i.e., a universal approximation theorem for a deep neural network with a Rectified Linear Unit (ReLU) activation function and bounded width. Here, we show how any continuous function on a compact set of Rnin,nin∈N can be approximated by a ReLU network having hidden layers with at most nin+5 nodes in view of an approximate identity.

Johnson Pseudo-Contractibility and Pseudo-Amenability of θ-Lau Product

Given Banach algebras A and B and θ ∈ ∆(B). We shall study the Johnson pseudo-contractibility and pseudo-amenability of the θ-Lau product A×θ B. We show that if A ×θ B is Johnson pseudo-contractible, then both A and B are Johnson pseudo-contractible and A has a bounded approximate identity. In some particular cases, a complete characterization of Johnson pseudo-contractibility of A ×θ B is given. Also, we show that pseudo-amenability of A ×θ B implies the approximate amenability of A and pseudo-amenability of B.

Forcing axioms and coronas of C∗-algebras

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

APROKSIMASI FUNGSI KONTINU TERBATAS DENGAN KONVOLUSI

Convolution is a mathematical operation on two functions that produces a new function that can be seen as a modified version of one of its original functions. The convolution operator has no identity element. However, it has an approximate identity. It can be found as a sequence of gk such that convolution of f and gk converges to f for k→∞. It implies that convolution can be used to approximate a function. In this article, we have proven basic theorems about approximation function by convolution for a bounded function in C(Rd). Konvolusi adalah suatu operasi pada dua fungsi dan menghasilkan suatu fungsi baru yang dapat dipandang sebagai versi modifikasi dari salah satu fungsi aslinya. Operasi konvolusi tidak memiliki unsur identitas. Namun, operasi konvolusi memiliki identitas hampiran, yakni dapat ditemukannya suatu barisan fungsi gk sehingga konvolusi dari f dan gk konvergen ke f untuk k→∞. Hal ini mengakibatkan konvolusi dapat digunakan untuk aproksimasi fungsi. Pada artikel ini dibuktikan teorema-teorema yang mendasari aproksimasi fungsi dengan konvolusi bagi fungsi terbatas di C(Rd) .