arXiv subjects

Falguni Roy

Publications and source records attributed to Falguni Roy.

9 recordsLinked to original sources

arXiv2025

Characterizations of structured Bohemian matrices and their Inner and Outer Bohemian Inverses

In this paper, we systematically define and characterize various classes of Bohemian matrices with respect to the population $\mathbb{P}=\{0, \pm 1\}$, focusing on their inner and outer Bohemian inverses. The classes under consideration include rank-one Bohemian matrices, as well as higher-rank Bohemian matrix classes, specifically Classes I, II, and III. For rank-one Bohemian matrices, a complete description of the outer Bohemian inverse sets is provided along with their cardinalities. Additionally, new insights into inner Bohemian inverses and their cardinalities are given. Characterizations of the complete set of inner inverses for the Class III matrices and full-row rank Class II matrices are obtained. Furthermore, we study the sets of rank-one outer inverses for Classes I and II, and examine the sets of rank $r$ outer inverses for full-row rank Class II matrices of rank $r$. Moreover, the set of outer inverses is completely characterized for rank-two full-row rank Class III matrices. In particular, we provide an explicit formula for the cardinality of the set of outer Bohemian inverses for rank-two full-row rank Class I matrices.

math.NA↗

arXiv2025

The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in\mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.

math.FA↗

arXiv2025

A Closer Look on Gender Stereotypes in Movie Recommender Systems and Their Implications with Privacy

The movie recommender system typically leverages user feedback to provide personalized recommendations that align with user preferences and increase business revenue. This study investigates the impact of gender stereotypes on such systems through a specific attack scenario. In this scenario, an attacker determines users' gender, a private attribute, by exploiting gender stereotypes about movie preferences and analyzing users' feedback data, which is either publicly available or observed within the system. The study consists of two phases. In the first phase, a user study involving 630 participants identified gender stereotypes associated with movie genres, which often influence viewing choices. In the second phase, four inference algorithms were applied to detect gender stereotypes by combining the findings from the first phase with users' feedback data. Results showed that these algorithms performed more effectively than relying solely on feedback data for gender inference. Additionally, we quantified the extent of gender stereotypes to evaluate their broader impact on digital computational science. The latter part of the study utilized two major movie recommender datasets: MovieLens 1M and Yahoo!Movie. Detailed experimental information is available on our GitHub repository: https://github.com/fr-iit/GSMRS

cs.IR↗

arXiv2024

A Deep Dive into Fairness, Bias, Threats, and Privacy in Recommender Systems: Insights and Future Research

Recommender systems are essential for personalizing digital experiences on e-commerce sites, streaming services, and social media platforms. While these systems are necessary for modern digital interactions, they face fairness, bias, threats, and privacy challenges. Bias in recommender systems can result in unfair treatment of specific users and item groups, and fairness concerns demand that recommendations be equitable for all users and items. These systems are also vulnerable to various threats that compromise reliability and security. Furthermore, privacy issues arise from the extensive use of personal data, making it crucial to have robust protection mechanisms to safeguard user information. This study explores fairness, bias, threats, and privacy in recommender systems. It examines how algorithmic decisions can unintentionally reinforce biases or marginalize specific user and item groups, emphasizing the need for fair recommendation strategies. The study also looks at the range of threats in the form of attacks that can undermine system integrity and discusses advanced privacy-preserving techniques. By addressing these critical areas, the study highlights current limitations and suggests future research directions to improve recommender systems' robustness, fairness, and privacy. Ultimately, this research aims to help develop more trustworthy and ethical recommender systems that better serve diverse user populations.

cs.IR↗

arXiv2024

Drazin and group invertibility in algebras spanned by two idempotents

For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq)^{m-1}=(pq)^{m}$ but $(pq)^{m-2}p\neq (pq)^{m-1}p$ for some $m(\geq2)\in\mathbb{N}.$ Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of $(αp+q)$ under two different assumptions, $(pq)^{m-1}=(pq)^m$ and $λ(pq)^{m-1}=(pq)^m,$ here $α$ is a non-zero and $λ$ is a non-unit real or complex number.

math.FA↗

arXiv2024

Drazin and g-Drazin invertibility of combinations of three Banach algebra elements

Consider a complex unital Banach algebra $\mathcal{A}.$ For $x_1,x_2,x_3\in\mathcal{A},$ in this paper, we establish that under certain assumptions on $x_1,x_2,x_3$, Drazin (resp. g-Drazin) invertibility of any three elements among $x_1,x_2,x_3$ and $x_1+x_2+x_3\text{ }(\text{or }x_1x_2+x_1x_3+x_2x_3)$ ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents $p,q\in\mathcal{A},$ this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of $$λ_1p+γ_1q-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right)$$ and $$λ_1-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right),$$ where $γ_1,λ_i\in\mathbb{C}$ for $i=1,2,\cdots,m,$ with $λ_1γ_1\neq0.$ Furthermore, for $x_1,x_2$, we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among $x_1,x_2$ and $x_1+x_2$ indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that $x_1x_2=α(x_1+x_2)$ for some $α\in\mathbb{C}$. Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among $x_1,x_2$ and $x_1+x_2$, by using the other two elements and their Drazin (resp. g-Drazin) inverse.

math.FA↗

arXiv2023

Additive and multiplicative properties of Drazin inverse under new weakly commutativity condition

Given a complex Banach space $X$, let $\mathcal{B}(X)$ be the collection of all bounded linear operators on $X.$ For $A,B\in\mathcal{B}(X)$ we define $A,B$ are $A$-weakly commutative if there exists $C\in\mathcal{B}(X)$ satisfying $$AB=CA\text{ and }BA=AC.$$ The objective of this paper is to study the Drazin invertibility of $A+B$ and $AB$, when $A, B\in\mathcal{B}(X)^D$ are $A\text{ and }B$-weakly commutative. Consequently, this extends some of the additive and multiplicative results established by Huanyin and Marjan Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for a distinct family of elements. Moreover, we also establish these additive and multiplicative properties for g-Drazin inverses in a complex Banach algebra.

math.FA↗

arXiv2023

A $W$-weighted generalization of $\{1,2,3,1^{k}\}$-inverse for rectangular matrices

This paper presents a novel extension of the $\{1,2,3,1^{k}\}$-inverse concept to complex rectangular matrices, denoted as a $W$-weighted $\{1,2,3,1^{k}\}$-inverse (or $\{1',2',3',{1^{k}}'\}$-inverse), where the weight $W \in \mathbb{C}^{n \times m}$. The study begins by introducing a weighted $\{1,2,3\}$-inverse (or $\{1',2',3'\}$-inverse) along with its representations and characterizations. The paper establishes criteria for the existence of $\{1',2',3'\}$-inverses and extends the criteria to $\{1'\}$-inverses. It is further demonstrated that $A\in \mathbb{C}^{m \times n}$ admits a $\{1',2',3',{1^{k}}'\}$-inverse if and only if $r(WAW)=r(A)$, where $r(\cdot)$ is the rank of a matrix. The work additionally establishes various representations for the set $A\{ 1',2',3',{1^{k}}'\}$, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations for the set $A\{ 1,2,3,{1^{k}}\}$. $\{ 1',2',3',{1^{k}}'\}$-inverse is shown to be unique if and only if it has index $0$ or $1$, reducing it to the weighted core inverse. Moreover, the paper investigates properties and characterizations of $\{1',2',3',{1^{k}}'\}$-inverses, which then results in new insights into the characterizations of the set $A\{ 1,2,3,{1^{k}}\}$.

math.NA↗

arXiv2023

New upper bounds for the $q$-numerical radius of Hilbert space operators

This article introduces several new upper bounds for the $q$-numerical radius of bounded linear operators on complex Hilbert spaces. Our results refine some of the existing upper bounds in this field. The $q$-numerical radius inequalities of products and commutators of operators follow as special cases. Finally, some new inequalities for the $q$-numerical radius of $2 \times 2$ operator matrices are established.

math.FA↗