arXiv subjects

Amit Ujlayan

Publications and source records attributed to Amit Ujlayan.

9 recordsLinked to original sources

arXiv2026

Inner Products and Banach Algebra structures on Bicomplex Numbers and Their Associated Spaces

In this paper, we introduce various types of inner products and norms on the bicomplex number system $\mathbb{C}_2$, the bicomplex vector space ${\mathbb{C}_2}^{n}$, the space of bicomplex matrices ${C_2}^{m \times n}$, and the space of bicomplex polynomials $\mathbb{C}_2[\xi]_n$. We investigate the relationships among these inner products and norms, and establish several results. Furthermore, we prove that $\mathbb{C}_2$ and ${\mathbb{C}_2}^{n}$ are Banach algebras and Hilbert spaces. These results provide a unified framework for the study of inner product structures and normed linear spaces over bicomplex numbers and their associated spaces.

math.RT↗

arXiv2025

Some Separation Axioms in Topological Spaces

In this paper, we introduced the concepts of new separation axioms called $ SC^* $-separation axioms and $ H^* $-separation axioms by using $ SC^* $ and $ H^* $-open sets in topological spaces. The $ SC^* $-separation axioms include $ SC^* $-$ C_0 $, $ SC^* $-$ C_1 $, weakly $ SC^* $-$ C_0 $, and weakly $ SC^* $-$ C_1 $ spaces, while the $ H^* $-separation axioms include $ H^* $-$ T_{\frac{1}{2}} $, $ H^* $-$ T_b $, and $ H^* $-$ T_d $ spaces. Also, we obtained several properties of such spaces.

math.GN↗

arXiv2025

H*-Normal Spaces and Some Related Functions

This paper introduces and explores functions defined on $ H^* $-normal spaces through the framework of $ H^* $-open sets. We extend the concept of $ H^* $-normality and investigate its connections with $ g $-normal and classical normal spaces. Additionally, we generalize $ H^* $-closed and $ H^* $-generalized closed functions, analyzing their fundamental properties. Several characterizations of $ H^* $-normal spaces are established, along with preservation theorems that highlight the structural significance of these functions.

math.GN↗

arXiv2025

Almost SC*-normal spaces

This paper introduces and investigates a new class of almost normal spaces, referred to as almost SC*-normal spaces, which are defined using SC*-open sets. Building on the work of A. Chandrakala and K. Bala Deepa Arasi, we explore several properties of these spaces within the framework of topology. Moreover, we obtain some new characterizations and preservation theorems of almost SC*-normal spaces.

math.GN↗

arXiv2025

SC*-Regular spaces and some functions

This paper introduces a novel class of topological spaces, termed SC*-regular spaces, which are defined using SC*-open sets. We explore their fundamental properties and examine their connections with existing regularity concepts, such as regular, almost, softly, weakly, alpha, zeta, and generalized-regular spaces respectively. Furthermore, we examined and define, analyze generalized SC*-closed sets and SC*-generalized closed functions, establishing key properties and preservation theorems. Several characterizations of SC*-regular spaces are also presented, providing new insights into generalized regularity in topology.

math.GN↗

arXiv2018

Automated Classification of Hand-grip action on Objects using Machine Learning

Brain computer interface is the current area of research to provide assistance to disabled persons. To cope up with the growing needs of BCI applications, this paper presents an automated classification scheme for handgrip actions on objects by using Electroencephalography (EEG) data. The presented approach focuses on investigation of classifying correct and incorrect handgrip responses for objects by using EEG recorded patterns. The method starts with preprocessing of data, followed by extraction of relevant features from the epoch data in the form of discrete wavelet transform (DWT), and entropy measures. After computing feature vectors, artificial neural network classifiers used to classify the patterns into correct and incorrect handgrips on different objects. The proposed method was tested on real dataset, which contains EEG recordings from 14 persons. The results showed that the proposed approach is effective and may be useful to develop a variety of BCI based devices to control hand movements.

q-bio.NC↗

arXiv2017

Analytical aspects of matrix interpolation problems and its applications

In this paper, the $mn$-dimensional space of tensor-product polynomials of two variables, of degree at most $(m-1)+(n-1)$, is considered. A theory of two-variate polynomials is developed by establishing the algebra and basic algebraic properties with respect to the usual addition, scalar multiplication, and a newly defined algebraic operation in the considered space. Further, the existence of the considered space is established with respect to the matrix interpolation problem (MIP), $P(i,j)=a_{ij}$ for all $1 \leq i \leq m$, $1 \leq j \leq n$, corresponds to a given matrix $(a_{ij})_{m \times n}$ in the space of $m \times n$ order real matrices. The poisedness of the MIP is proved and three formulae are presented to construct the respective polynomial in the considered space. After that, using construction formulae, a polynomial map from the space of $m \times n$ order real matrices to the considered space is defined. Some properties of the polynomial map are investigated and some isomorphic structures between the spaces are installed. It is proved that the considered space is isomorphic to the space of $m \times n$ order real matrices with respect to the algebra structure. The polynomials in the considered space with respect to the MIP's for the given matrices also preserve the geometric properties of the matrices such as transpose, symmetry, and skew-symmetry. Some examples are included to demonstrate and verify the results.

math.GM↗

arXiv2017

A Bivariate Polynomial Problem for Matrices

This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate polynomial subspaces (BVPSs) to be an isomorphism in some finite-dimensional BVPSs. In the process of solving, the article deals with the existence, uniqueness, and construction of the polynomials in some finite-dimensional BVPSs concerning the solution of the proposed problem. To this end, a relationship is established between the proposed problem and a class of Lagrange bivariate polynomial interpolation problems (LBVPIPs). As a result, the existence of a standard and a new class of finite-dimensional BVPSs of various total degrees has been established in which the proposed problem always possesses a unique solution. In addition, some formulas are derived to construct the needed polynomials in these BVPSs. Further, the possible applicability of the proposed problem is discussed in LBVPIPs, focusing on the finite rectangular schemes of bivariate interpolation points on the natural Cartesian grid. At last, some numerical examples are considered to justify the theoretical findings.

math.GM↗

arXiv2017

A new fractional derivative and its fractional integral with some example

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with respect to a parameter varying over unit interval. Thus it could also be regarded as a fractional derivative. Reason of calling it as deformable derivative is because of its intrinsic property of continuously deforming function to derivative. This is substantiated by its linear connection to function and its derivative. Besides discussing some of its basic properties, we discover the forms of Rolle's, Mean Value and Taylor's theorems. The fundamental theorem of calculus for this fractional derivative could be taken as definition of its fractional integral. As a theoretical application some fractional differential equations are solved.

math.CA↗