arXiv subjects

Arindam Biswas

Publications and source records attributed to Arindam Biswas.

36 records · Page 2Linked to original sources

arXiv2020

Asymptotic behaviour of minimal complements

The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-empty subsets $W,W'$ in a group $G$, the set $W'$ is said to be a complement to $W$ if $W\cdot W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The inverse problem asks which sets may or not occur as minimal complements. We show some new results on the inverse problem and investigate how the study of the inverse problem naturally gives rise to questions about the asymptotic behaviour of these sets, providing partial answers to some of them.

math.CO↗

arXiv2020

On non-minimal complements

The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon--Kravitz--Larson, Burcroff--Luntzlara and also that of the authors, shed light on some of the questions in this direction. These works have focussed mainly on the group of integers, or on abelian groups. In this work, our motivation is two-fold: (i) to show some new results on the inverse problem, (ii) to concentrate on the inverse problem in not necessarily abelian groups. As a by-product, we obtain new results on non-minimal complements in the group of integers and more generally, in any finitely generated abelian group of positive rank and in any free abelian group of positive rank. Moreover, we show the existence of uncountably many subsets in such groups which are "robust" non-minimal complements.

math.CO↗

arXiv2020

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

math.NT↗

arXiv2020

On minimal complements in groups

Let $W,W'\subseteq G$ be nonempty subsets in an arbitrary group $G$. The set $W'$ is said to be a complement to $W$ if $WW'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. We show that, if $W$ is finite then every complement of $W$ has a minimal complement, answering a problem of Nathanson. This also shows the existence of minimal $r$-nets for every $r\geqslant 0$ in finitely generated groups. Further, we give necessary and sufficient conditions for the existence of minimal complements of a certain class of infinite subsets in finitely generated abelian groups, partially answering another problem of Nathanson. Finally, we provide infinitely many examples of infinite subsets of abelian groups of arbitrary finite rank admitting minimal complements.

math.CO↗

arXiv2020

On additive co-minimal pairs

A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of $G$.

math.CO↗

arXiv2020

Infinite co-minimal pairs in the integers and integral lattices

Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $σ$, there exists a subset $A$ of $\mathbb{Z}^{2d}$ such that $A$ and $σ(A)$ form a co-minimal pair. The existence and construction of co-minimal pairs in the integers with both the subsets $A$ and $B$ ($A\neq B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.

math.NT↗

arXiv2019

Asymptotic complements in the integers

Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in $\mathbb{Z}$ and deal with a problem of Nathanson on their existence and their inexistence.

math.NT↗

arXiv2019

Flexibility and movability in Cayley graphs

Let $\mathbfΓ = (V,E)$ be a (non-trivial) finite graph with $λ: E \rightarrow \mathbb{R}_{+}$, an edge labelling of $\mathbfΓ$. Let $ρ: V\rightarrow \mathbb{R}^{2}$ be a map which preserves the edge labelling. The graph $\mathbfΓ$ is said to be flexible if there exists an infinite number of such maps (upto equivalence by rigid transformations) and it is said to be movable if there exists an infinite number of injective maps. We study movability of Cayley graphs and construct regular moving graphs of all degrees. Further, we give explicit constructions of "dense", movable graphs.

math.CO↗

arXiv2019

Summarizing User-generated Textual Content: Motivation and Methods for Fairness in Algorithmic Summaries

As the amount of user-generated textual content grows rapidly, text summarization algorithms are increasingly being used to provide users a quick overview of the information content. Traditionally, summarization algorithms have been evaluated only based on how well they match human-written summaries (e.g. as measured by ROUGE scores). In this work, we propose to evaluate summarization algorithms from a completely new perspective that is important when the user-generated data to be summarized comes from different socially salient user groups, e.g. men or women, Caucasians or African-Americans, or different political groups (Republicans or Democrats). In such cases, we check whether the generated summaries fairly represent these different social groups. Specifically, considering that an extractive summarization algorithm selects a subset of the textual units (e.g. microblogs) in the original data for inclusion in the summary, we investigate whether this selection is fair or not. Our experiments over real-world microblog datasets show that existing summarization algorithms often represent the socially salient user-groups very differently compared to their distributions in the original data. More importantly, some groups are frequently under-represented in the generated summaries, and hence get far less exposure than what they would have obtained in the original data. To reduce such adverse impacts, we propose novel fairness-preserving summarization algorithms which produce high-quality summaries while ensuring fairness among various groups. To our knowledge, this is the first attempt to produce fair text summarization, and is likely to open up an interesting research direction.

cs.IR↗

arXiv2019

A Cheeger type inequality in finite Cayley sum graphs

Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_Σ(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}η, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_Σ(G,S)$ and $η= 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.

math.CO↗

arXiv2019

On a cheeger type inequality in Cayley graphs of finite groups

Let $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}γ, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $γ= 2^{9}d^{6}(d+1)^{2}$.

math.GR↗

arXiv2019

On semilinear sets and asymptotically approximate groups

Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A \rbrace.$$ Nathanson considered the concept of an asymptotic approximate group. Let $r,l \in \mathbb{N}$. The set $A$ is said to be an $(r,l)$ approximate group in $G$ if there exists a subset $X$ in $G$ such that $|X|\leqslant l$ and $A^{r}\subseteq XA$. The set $A$ is an asymptotic $(r,l)$-approximate group if the product set $A^{h}$ is an $(r,l)$-approximate group for all sufficiently large $h$. Recently, Nathanson showed that every finite subset $A$ of an abelian group is an asymptotic $(r,l')$ approximate group (with the constant $l'$ explicitly depending on $r$ and $A$). We generalise the result and show that, in an arbitrary abelian group $G$, the union of $k$ (unbounded) generalised arithmetic progressions is an asymptotic $(r,(4rk)^k)$-approximate group.

math.NT↗

arXiv2019

Minimal additive complements in finitely generated abelian groups

Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--Sàndor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this article, we study subsets which are not eventually periodic. We introduce the notion of "spiked subsets" and give necessary and sufficient conditions for the existence of minimal complements for them. This provides a partial answer to a problem of Nathanson.

math.CO↗

arXiv2017

A New Bengali Readability Score

In this paper we have proposed methods to analyze the readability of Bengali language texts. We have got some exceptionally good results out of the experiments.

cs.CL↗

arXiv2016

A Supervised Authorship Attribution Framework for Bengali Language

Authorship Attribution is a long-standing problem in Natural Language Processing. Several statistical and computational methods have been used to find a solution to this problem. In this paper, we have proposed methods to deal with the authorship attribution problem in Bengali.

cs.CL↗

arXiv2016

Diameter Bound for Finite Simple Groups of Large Rank

Given a non-abelian finite simple group $G$ of Lie type, and an arbitrary generating set $S$, it is conjectured by Laszlo Babai that its Cayley graph $Γ(G,S)$ will have a diameter of $(\log |G|)^{O(1)}$. However, little progress has been made when the rank of $G$ is large. In this article, we shall show that if $G$ has rank $n$, and its base field has order $q$, then the diameter of $Γ(G,S)$ would be $q^{O(n(\log n + \log q)^3)}$.

math.GR↗

arXiv2016

A Simple Condition for the Existence of Transversals

Hall's Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of the sets in the family and the sizes of their intersections so that a transversal exists. Using this, we prove that in a bipartite graph $G$ (bipartition $\{A, B\}$), without 4-cycles, if $°(v) \geq \sqrt{2e|A|}$ for all $v \in A$, then $G$ has a matching of size $|A|$.

cs.DM↗

arXiv2014

Inter-Rater Agreement Study on Readability Assessment in Bengali

An inter-rater agreement study is performed for readability assessment in Bengali. A 1-7 rating scale was used to indicate different levels of readability. We obtained moderate to fair agreement among seven independent annotators on 30 text passages written by four eminent Bengali authors. As a by product of our study, we obtained a readability-annotated ground truth dataset in Bengali. .

cs.CL↗