On Bessel Functions Related with Certain Classes of Analytic Functions with respect to Symmetrical Points

In the present investigation, subclasses of analytic functions with respect to symmetrical points which are defined by the generalized Bessel functions of the first kind of order μ are introduced. Furthermore, some alluring geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results are studied. Moreover, some consequences of our results are also given.

The induced basis axioms for a closed G-V fuzzy matroid

In this paper, we research a class of axioms in closed G-V fuzzy matroids. The main research method is to transform fuzzy matroids into matroids. First, we study many properties of the basis family of induced matroids, and define a new mapping which can reflect the relationship between bases of induced matroids of a G-V fuzzy matroid. Second, we discuss the new mapping, and reveal the relationship and properties among the fundamental sequence, the induced basis family and the new mapping of a G-V fuzzy matroid. From these relationships and properties, we extract four key attributes: normativity property, inclusion property, exchange property, and right surjection. Finally, we propose and prove “the induced basis axioms for a closed G-V fuzzy matroid” by these key attributes. With the help of these axioms, a closed G-V fuzzy matroid can be uniquely determined by a finite number sequence, a subset family and a mapping on this subset family when they satisfy above four attributes, and vice versa.

On Training Neural Network Decoders of Rate Compatible Polar Codes via Transfer Learning

Neural network decoders (NNDs) for rate-compatible polar codes are studied in this paper. We consider a family of rate-compatible polar codes which are constructed from a single polar coding sequence as defined by 5G new radios. We propose a transfer learning technique for training multiple NNDs of the rate-compatible polar codes utilizing their inclusion property. The trained NND for a low rate code is taken as the initial state of NND training for the next smallest rate code. The proposed method provides quicker training as compared to separate learning of the NNDs according to numerical results. We additionally show that an underfitting problem of NND training due to low model complexity can be solved by transfer learning techniques.

Fast Computation of Polynomial Data Points Over Simplicial Face Values

Polynomial functions [Formula: see text] of degree [Formula: see text] have a form in the Bernstein basis defined over [Formula: see text]-dimensional simplex [Formula: see text]. The Bernstein coefficients exhibit a number of special properties. The function [Formula: see text] can be optimised by the smallest and largest Bernstein coefficients (enclosure bounds) over [Formula: see text]. By a proper choice of barycentric subdivision steps of [Formula: see text], we prove the inclusion property of Bernstein enclosure bounds. To this end, we provide an algorithm that computes the Bernstein coefficients over subsimplices. These coefficients are collected in an [Formula: see text]-dimensional array in the field of computer-aided geometric design. Such a construct is typically classified as a patch. We show that the Bernstein coefficients of [Formula: see text] over the faces of a simplex coincide with the coefficients contained in the patch.

Multiprojective spaces and the arithmetically Cohen–Macaulay property

In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.