Searcharxiv⌕ Search

arXiv subjects

Sourav Mandal

Publications and source records attributed to Sourav Mandal.

5 recordsLinked to original sources
arXiv2026

Unlocking novel topological structures via rough families

Very recently, the notion of rough family has been introduced in [Leonetti, P., J. Convex Anal. 32(4):1083-1090, 2025] to explore rough ideal convergence in topological spaces where the limit of a sequence may not be unique. This raises the question of whether $T_2$ topological spaces can be characterized using rough families. In this article, we prove that a topological space is $T_2$ if and only if it can never be a rough topological space. In this context, we first introduce the notions of rough interior and rough closure of a set from the perspective of a rough family, which leads to the definition of rough open sets (rough closed sets). As a consequence, we generate a new topology, termed rough topology, as well as rough homeomorphism. Our main contribution presents the novelty of this new class; in particular, we explicitly construct several examples which ensure that two non-homeomorphic spaces can be roughly homeomorphic under certain roughness. Additionally, we extend the concepts of compactness as well as connectedness, where our findings diverge from existing literature in these areas, in a nutshell, providing new insights and perspectives.

math.GN↗
arXiv2025

On Characterizing Potential Friends of 20

Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $Ω(n)$ and $ω(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$.

math.GM↗
arXiv2025

A note on necessary conditions for a friend of 10

Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ where $f$ is the smallest odd positive integer greater than $1$ and less than or equal to $\min\{ 2a+1,p-1\}$, provided $5^{2a}\mid \mid N$. Further, there exist prime factors $p$ and $q$ (not necessarily distinct) of $N$ such that $p\equiv1 \pmod {10}$ and $q\equiv 1\pmod 6$. Besides, we prove that if a Fermat prime $F_k$ divides $N$ then $N$ must have a prime factor congruent to $1$ modulo $2F_k$. Also, if we consider the form of $N$ as $N=5^{2a}m^2$ then $m$ is non square-free. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-4$ and if $Ω(m)\leq K$ then $N< 5\cdot 6^{(2^{K-2a+1}-1)^2}$ where $Ω(n)$ and $ω(n)$ denote the total number of prime factors and the number of distinct prime factors of the integer $n$ respectively.

math.NT↗
arXiv2021

When degree of roughness is a neighborhood over locally solid Riesz spaces

In this paper we introduce the notion of rough weighted $\mathcal{I}_τ$-limit points set and weighted $\mathcal{I}_τ$-cluster points set in a locally solid Riesz space which are more generalized version of rough weighted $\mathcal{I}$-limit points set and weighted $\mathcal{I}$-cluster points set in a $θ$-metric space respectively. Successively to compare with the following important results of Fridy [Proc. Amer. Math. Soc. {118} (4) (1993), 1187-1192] and Das [Topology Appl. {159} (10-11) (2012), 2621-2626], respectively be stated as \begin{description} \item[(i)] Any number sequence $x=\{x_{n}\}_{n\in \mathbb{N}},$ the statistical cluster points set of $x$ is closed, \item[(ii)] In a topological space the $\mathcal{I}$-cluster points set is closed, \end{description} we show that in general, the weighted $\mathcal{I}_τ$-cluster points set in a locally solid Riesz space may not be closed. The resulting summability method unfollows some previous results in the direction of research works of Aytar [Numer. Funct. Anal. Optim. {29} (3-4) (2008) 291-303], D$\ddot{\mbox{u}}$ndar [Numer. Funct. Anal. Optim. {37} (4) (2016) 480-491], Ghosal [Math. Slovaca {70} (3) (2020) 667-680] and Savaş, Et [Period. Math. Hungar. 71 (2015) 135-145].

math.GN↗