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Anupam Mondal

Publications and source records attributed to Anupam Mondal.

16 recordsLinked to original sources
arXiv2026

A combinatorial nerve theorem for effective homology computation

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, $X$ is a simplicial complex with covering subcomplexes $A_1, \dots ,A_k$, that is, $X= \cup_{i=1}^k A_i$ and $\mathcal{N}(X)$ is the nerve of $X$ with respect to its covering. Let $\mathcal{W}_α$ be a given gradient vector field on $A_α(=\cap_{i \in α} A_i)$ for each $α\in \mathcal{N}(X)$. Then, we use the mere information of the gradient trajectories in $A_α$ for each $α\in \mathcal{N}(X)$ to explicitly compute the homology groups of $X$. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of $X$. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.

math.CO↗
arXiv2026

Bond Disproportionation, Ligand Holes, and Persistent Spin Textures in Ag$_2$BiO$_3$

The origin of the proposed bond disproportionated insulating state of the non-centrosymmetric ($Pnn2$) phase of Ag$_{2}$BiO$_{3}$ is explored using first principles electronic structure calculations. The novel insulating state is elucidated by first considering the initially proposed centosymmetric metallic ($Pnna$) phase of Ag$_{2}$BiO$_{3}$. Our calculations reveal that the valence skipping Bi$^{4+}$ ions in this phase are better described as Bi$^{3+}\underline{L}$ with completely filled Bi-(6$s$) states and a ligand hole. However, phonon calculations indicate that the metallic ($Pnna$) state is dynamically unstable. Structural stability is achieved through breathing distortions of the oxygen octahedra, resulting in two inequivalent Bi sites and a reduction of symmetry to the $Pnn2$ phase. Electronic structure calculations further reveal that the $Pnn2$ phase is a bond disproportionated insulator where the nominal charge state of Bi is described by : 2[Bi$^{3+}\underline{L}$ (Bi$^{4+}$)] $\rightarrow$ Bi$^{3+}\underline{L}^{2-δ}$ (Bi1$^{5+}$) + Bi$^{3+}\underline{L}^δ$ (Bi2$^{3+}$), highlighting the crucial role of ligand holes in driving the insulating state. Next we have investigated the electronic structure of Ag$_{2}$BiO$_{3}$ in the insulating ($Pnn2$) phase including spin-orbit coupling. Our density functional theory (DFT ) calculations complemented by ${\bf k.p}$ model Hamiltonian analysis reveal persistent spin-textures around the $X$ and $Y$ high symmetry points of the orthorhombic Brillouin zone imposed by non-symmorphic symmetry, positioning Ag$_{2}$BiO$_{3}$ as a promising candidate for spintronic applications.

cond-mat.mtrl-sci↗
arXiv2026

Interplay of Valley, Orbital, Spin, and Layer Degrees of Freedom in Ta$_2$CS$_2$ MXene

We show that the MXene Ta$_2$CS$_2$ provides an excellent platform for hosting multiple coupled degrees of freedom, viz., valley, spin, orbital, and layer. The interplay among these degrees of freedom gives rise to a range of intriguing properties in reciprocal space, including valley-orbital and orbital-layer coupling. In the presence of spin-orbit interaction, these couplings lead to valley-dependent and layer-dependent spin splitting of the electronic bands. We further show that the intrinsic electric polarization in Ta$_2$CS$_2$ introduces an additional tuning parameter, enabling control over these coupled degrees of freedom and resulting in switchable valley-dependent orbital moments and Zeeman-like spin splitting. We demonstrate that these nontrivial orbital and spin textures manifest in the orbital and spin Hall effects, respectively. Our results establish noncentrosymmetric MXenes as a promising platform for exploring the interplay among multiple degrees of freedom, their tunability, and the resulting orbital and spin transport phenomena in these two-dimensional materials, thereby paving the way for next-generation spin-orbitronic devices.

cond-mat.str-el↗
arXiv2026

Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators

Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators.

math.CO↗
arXiv2026

A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex

We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic matchings on the independence complexes of specific subgraphs. As an application, we determine the homotopy type of the independence complexes of the family of chordal graphs and of a class of graphs generalising the comparability graphs of grid posets in an algorithmic and combinatorial manner via discrete Morse theory, some of which were previously obtained by sophisticated homotopy theoretic techniques. Even when the homotopy type is not easily determinable, our construction may be applied to obtain a pre-processing framework for efficient homology computation.

math.CO↗
arXiv2026

$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset

We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours.

math.CO↗
arXiv2026

Bottleneck Transformer-Based Approach for Improved Automatic STOI Score Prediction

In this study, we have presented a novel approach to predict the Short-Time Objective Intelligibility (STOI) metric using a bottleneck transformer architecture. Traditional methods for calculating STOI typically requires clean reference speech, which limits their applicability in the real world. To address this, numerous deep learning-based nonintrusive speech assessment models have garnered significant interest. Many studies have achieved commendable performance, but there is room for further improvement. We propose the use of bottleneck transformer, incorporating convolution blocks for learning frame-level features and a multi-head self-attention (MHSA) layer to aggregate the information. These components enable the transformer to focus on the key aspects of the input data. Our model has shown higher correlation and lower mean squared error for both seen and unseen scenarios compared to the state-of-the-art model using self-supervised learning (SSL) and spectral features as inputs.

eess.AS↗
arXiv2025

On the Deployment of RIS-mounted UAV Networks

Reconfigurable intelligent surfaces (RIS) enable smart wireless environments by dynamically controlling signal propagation to enhance communication and localization. Unmanned aerial vehicles (UAVs) can act as flying base stations and thus, improve system performance by avoiding signal blockages. In this paper, we propose a gradient ascent and coordinate search based method to determine the optimal location for a system that consists of a UAV and a RIS, where the UAV serves cellular users (CUs) and the RIS serves device-to-device (D2D) pairs. In particular, by optimizing the net throughput for both the D2D pairs and the CUs, the suggested method establishes the ideal location for the RIS-mounted UAV. We consider both line of sight (LoS) and non-LoS paths for the RIS and UAV to calculate the throughput while accounting for blockages in the system. The numerical results show that the proposed method performs better than the existing approaches in terms of both the net throughput and the user fairness.

eess.SY↗
arXiv2025

A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs

Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract simplicial complex of disconnected graphs of order $n$ (which was previously done by Victor Vassiliev using classical topological methods) using discrete Morse theoretic techniques, which are purely combinatorial in nature. The techniques involve the construction (and verification) of a discrete gradient vector field on the complex. However, the verification part relies on a claim that doesn't seem to hold. In this note, we provide a couple of counterexamples against this specific claim. We also provide an alternative proof of the bigger claim that the constructed discrete vector field is indeed a gradient vector field. Our proof technique relies on a key observation which is not specific to the problem at hand, and thus is applicable while verifying a constructed discrete vector field is a gradient one in general.

math.CO↗
arXiv2024

Topology of matching complexes of complete graphs via discrete Morse theory

Bouc (1992) first studied the topological properties of $M_n$, the matching complex of the complete graph of order $n$, in connection with Brown complexes and Quillen complexes. Björner et al. (1994) showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, and conjectured that this connectivity bound is sharp. Shareshian and Wachs (2007) settled the conjecture by inductively showing that the $ν_n$-dimensional homology group of $M_n$ is nontrivial, with Bouc's calculation of $H_1(M_7)$ serving as the pivotal base step. In general, the topology of $M_n$ is not very well-understood, even for a small $n$. In the present article, we look into the topology of $M_n$, and $M_7$ in particular, in the light of discrete Morse theory as developed by Forman (1998). We first construct a gradient vector field on $M_n$ (for $n \ge 5$) that doesn't admit any critical simplices of dimension up to $ν_n-1$, except one unavoidable $0$-simplex, which also leads to the aforementioned $(ν_n-1)$-connectedness of $M_n$ in a purely combinatorial way. However, for an efficient homology computation by discrete Morse theoretic techniques, we are required to work with a gradient vector field that admits a low number of critical simplices, and also allows an efficient enumeration of gradient paths. An optimal gradient vector field is one with the least number of critical simplices, but the problem of finding an optimal gradient vector field, in general, is an NP-hard problem (even for $2$-dimensional complexes). We improve the gradient vector field constructed on $M_7$ in particular to a much more efficient (near-optimal) one, and then with the help of this improved gradient vector field, compute the homology groups of $M_7$ in an efficient and algorithmic manner. We also augment this near-optimal gradient vector field to one that we conjecture to be optimal.

math.CO↗
arXiv2024

Discrete Morse theory and the topology of matching complexes of complete graphs

We denote the matching complex of the complete graph with $n$ vertices by $M_n$. Bouc first studied the topological properties of $M_n$ in connection with the Quillen complex. Later Björner, Lovász, Vrećica, and Živaljević showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, but in general the topology of $M_n$ is not very well-understood even for smaller natural numbers. Forman developed discrete Morse theory, which has various applications in diverse fields of studies. In this article, we develop a discrete Morse theoretic technique to capture deeper structural topological properties of $M_n$. We show that $M_n$ is \emph{geometrically} $(ν_n-1)$-connected, where the notion of geometrical $k$-connectedness as defined in this article, is stronger than that of homotopical $k$-connectedness. Previously, Björner et al. showed that $M_8$ is simply connected, but not 2-connected. The technique developed here helped us determine that $M_8$ is in fact homotopy equivalent to a wedge of 132 spheres of dimension 2.

math.CO↗
arXiv2023

On Elser's conjecture and the topology of $U$-nucleus complex

Dorpalen-Barry et al. proved Elser's conjecture about sign of Elser's number by interpreting them as certain sums of reduced Euler characteristics of an abstract simplicial complex known as $U$-nucleus complex. We prove a conjecture posed by them regarding the homology of $U$-nucleus complex.

math.CO↗
arXiv2020

Covering the Plane by a Sequence of Circular Disks with a Constraint

We are interested in the following problem of covering the plane by a sequence of congruent circular disks with a constraint on the distance between consecutive disks. Let $(\mathcal{D}_n)_{n \in \mathbb N}$ be a sequence of closed unit circular disks such that $\cup_{n \in \mathbb{N}} \mathcal{D}_n = \mathbb {R}^2$ with the condition that for $n \ge 2$, the center of the disk $\mathcal{D}_n$ lies in $\mathcal{D}_{n-1}$. What is a "most economical" or an optimal way of placing $\mathcal{D}_n$ for all $n \in \mathbb{N}$? We answer this question in the case where no "sharp" turn is allowed, i.e. if $C_n$ is the center of the disk $\mathcal{D}_n$, then for all $n \ge 2$, % $\angle C_{n-1}C_nC_{n+1}$ is not very small. We also consider a related problem. We wish to find out an optimal way to cover the plane with unit circular disks with the constraint that each disk contains the centers of at least two other disks. We find out the answer in the case when the centers of the disks form a two-dimensional lattice.

math.MG↗
arXiv2018

Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called König-Egerváry if the sum of its independence number ($α(G)$) and matching number ($μ(G)$) equals $|V(G)|$. It is known that bipartite graphs are König-Egerváry. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2α(G)$ (proved by Short in 2016). A characterization of unicyclic non-König-Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $α(G) - μ(G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.

math.CO↗
arXiv2017

Tracing Linguistic Relations in Winning and Losing Sides of Explicit Opposing Groups

Linguistic relations in oral conversations present how opinions are constructed and developed in a restricted time. The relations bond ideas, arguments, thoughts, and feelings, re-shape them during a speech, and finally build knowledge out of all information provided in the conversation. Speakers share a common interest to discuss. It is expected that each speaker's reply includes duplicated forms of words from previous speakers. However, linguistic adaptation is observed and evolves in a more complex path than just transferring slightly modified versions of common concepts. A conversation aiming a benefit at the end shows an emergent cooperation inducing the adaptation. Not only cooperation, but also competition drives the adaptation or an opposite scenario and one can capture the dynamic process by tracking how the concepts are linguistically linked. To uncover salient complex dynamic events in verbal communications, we attempt to discover self-organized linguistic relations hidden in a conversation with explicitly stated winners and losers. We examine open access data of the United States Supreme Court. Our understanding is crucial in big data research to guide how transition states in opinion mining and decision-making should be modeled and how this required knowledge to guide the model should be pinpointed, by filtering large amount of data.

cs.CL↗