Department of Mathematics

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

In this paper, we provide a direct proof of the fact that the symmetric embedding of B(H) into any of its bimodules induced by Markov maps is compact and establishes that this property characterizes type I factors. This is an extension of a result of D. Buchholz, C. D'Antoni, and R. Longo [J. Funct. Anal. 88(2), 233-250 (1990)]. © 2021 Author(s).

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Journal of Mathematical Physics

On symmetric embedding of type i factors

We consider approximability of two optimization problems called Minimum Open k-Monopoly (Min-Open-k-Monopoly) and Minimum Partial Open k-Monopoly (Min-P-Open-k-Monopoly), where k is a fixed positive integer. The objective, in Min-Open-k-Monopoly, is to find a minimum cardinality vertex set S⊆ V in a given graph G = (V,E) such that |N(v)∩S|≥12|N(v)|+k, for every vertex v ϵ V. On the other hand, given a graph G = (V,E), in Min-P-Open-k-Monopoly it is required to find a minimum cardinality vertex set S ⊆ V such that |N(v)∩S|≥1/2|N(v)|+k, for every v ϵ V. \ S. We prove that Min-Open-k-Monopoly and Min-P-Open-k-Monopoly are approximable within a factor of O(log n). Then, we show that these two problems cannot be approximated within a factor of (1/3−ϵ)lnn and (1/4−ϵ)lnn, respectively, for any ϵ&gt; 0, unless NP⊆ Dtime(nO(loglogn)). For 4-regular graphs, we prove that these two problems are APX-complete. Min-Open-1-Monopoly can be approximated within a factor of 2621≈1.2381 where as Min-P-Open-1-Monopoly can be approximated within a factor of 1.65153. For k ≥ 2, we also present approximation algorithms for these two problems for (2k + 2)-regular graphs. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Theory of Computing Systems

Approximability of open k-monopoly problems

Let K = Q(θ) be an algebraic number field with θ satisfying an irreducible polynomial x6- m having integer coefficient m. We explicitly provide an integral basis of K along with some examples. As an application, we show that when m is a squarefree integer such that m ≡ 2; 3 mod 4 and m ≢ ±1 mod 9, then K has a power integral basis. © 2021 Rocky Mountain Mathematics Consortium. All rights reserved.

Rocky Mountain Journal of Mathematics

Explicit integral basis of pure sextic fields

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute the second graded Betti numbers of JG, and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals. © 2020 Elsevier B.V.

Fulltext

Journal of Pure and Applied Algebra

Almost complete intersection binomial edge ideals and their Rees algebras

Scientists and Engineers in various fields have been immensely dependent on theory and numeric of differential equations to solve or understood diverse problems originating in various physical processes. Differential equations not only improve understanding of physical processes in engineering and technology but also touch various branches of Mathematics. These physical processes can arise in wide range of field like Biological systems, finance, fluid dynamics, industrial production etc.

Differential Equations and Dynamical Systems

Preface

As the fault-based attacks are becoming a more pertinent threat in today's era of edge computing/internet-of-things, there is a need to streamline the existing tools for better accuracy and ease of use, so that we can gauge the attacker's power and a proper countermeasure can be devised in the long run. In this regard, we propose a machine learning (ML) assisted tool that can be used in the context of a differential fault attack. In particular, finding the exact fault location by analysing the output difference (typically the XOR of the nonfaulty and the faulty ciphertexts) is somewhat nontrivial. During the literature survey, we notice that the Pearson's correlation coefficient dominantly is used for this purpose, and has almost become the defacto standard. While this method can yield good accuracy for certain cases, we argue that an ML-based method is more powerful in all the situations we experiment with. We substantiate our claim by showing the relative performances (we choose the commonly used multilayer perceptron as our ML tool) with two variants of Grain-128a (a stream cipher, and a stream cipher with authentication), the lightweight stream cipher LIZARD and the lightweight block cipher SIMON-32 (where the faults are injected at the fifth last rounds). Our results demonstrate that a common ML tool can outperform the correlation with the same training/testing data. We believe that our work extends the state-of-the-art by showing how traditional cryptographic methods can be replaced by a more powerful ML tool. © 2021 The Authors. CAAI Transactions on Intelligence Technology published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology and Chongqing University of Technology.

CAAI Transactions on Intelligence Technology

Differential fault location identification by machine learning

We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted χ[♯2](G). This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite planar graphs and subcubic K4-minor-free graphs have exact square chromatic number at most 4. We then turn our attention to the class of fullerene graphs, which are cubic planar graphs with face sizes 5 and 6. We characterize fullerene graphs with exact square chromatic number 3. Furthermore, supporting a conjecture of Chen, Hahn, Raspaud and Wang (that all subcubic planar graphs are injectively 5-colorable) we prove that any induced subgraph of a fullerene graph has exact square chromatic number at most 5. This is done by first proving that a minimum counterexample has to be on at most 80 vertices and then computationally verifying the claim for all such graphs. © 2021 Elsevier B.V.

Discrete Applied Mathematics

Exact square coloring of subcubic planar graphs

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1, …, TN from finite Cartesian product space X × · · · × X into X, where (X, d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Fractal and Fractional

Iterated functions systems composed of generalized θ-contractions

Abstract: Fractal interpolation provides an efficient way to describe a smooth or non-smooth structure associated with nature and scientific data. The aim of this paper is to introduce a bivariate Hermite fractal interpolation formula that generalizes the classical Hermite interpolation formula for two variables. It is shown here that the proposed Hermite fractal interpolation function and its derivatives of all orders are good approximations of original function even if the partial derivatives of original function are non-smooth in nature. © 2021, Pleiades Publishing, Ltd.

Numerical Analysis and Applications

Generalized Bivariate Hermite Fractal Interpolation Function

In this paper, we establish Lipschitz conditions for the norm of holomorphic mappings between the unit ball Bn in Cn and X, a complex normed space. This extends the work of Djordjević and Pavlović. © Académie des sciences, Paris and the authors, 2021. Some rights reserved.