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Towards a proof of the zig-zag conjecture

Nov 29, 2022ByProf. Eknath Ghate (TIFR, Mumbai)Seminar
Galois representations are fundamental tools with which to study many problems in number theory. They are most famously attached to modular forms and primes p. An outstanding problem regarding them is to determine their reductions modulo p in various settings. In the global setting, the shape of the reduction is connected to congruences for the form.
In the local crystalline setting, a particularly difficult case to treat is that of exceptional weights. My zig-zag conjecture predicts that, in this case, the reduction should be given by an explicit alternating sequence of irreducible and reducible representations.
In this talk, I will state the conjecture and announce a proof on inertia for large weights. I will also give a brief outline of the proof, which uses some delicate limiting arguments in a certain blow-up space to reduce the argument to the shape of the reductions of semi-stable representations.

An introduction to modular tensor categories

Dec 16th, 2019ByDr. Deepak Naidu, Department of Mathematical Sciences, Northern Illinois University.Seminar
Modular tensor categories were introduced by V. Turaev in 1992, but some aspects were anticipated earlier by K.-H. Rehren (1990) and G. Moore and N. Seiberg (1989). A modular tensor category (MTC) is a braided tensor category (e.g., the category of representations of a group) satisfying certain finiteness and nondegeneracy conditions. The strong finiteness axioms imposed on a MTC imply that the resulting theory is more algebraic than categorical in nature.
Modular tensor categories arise in several diverse areas such as quantum group theory, low-dimensional topology, vertex operator algebras, and conformal field theory. For example, MTCs give rise to invariants of closed oriented 3-manifolds and links in such manifolds.
In my talk, I will explain how every MTC gives rise to a representation of the modular group SL(2, Z), justifying the nomenclature "modular". I will also explain how every braided tensor category gives rise to representations of the braid group, justifying the nomenclature "braided". Throughout my talk, I will focus on examples, rather than precise definitions.
The kernel of the representation of SL(2, Z) arising from a MTC is known to be a congruence subgroup. Time permitting, I will describe my recent work in this direction, involving Drinfeld doubles of finite abelian groups and dihedral groups.

Distinguished representations over finite fields

Dec 02, 2019ByProf. U.K. Anandavardhanan, IIT Mumbai.Seminar
An irreducible representation of a finite group G_1 is said to be distinguished with respect to a subgroup G_2 if it has a G_2-invariant vector. The notion of distinction is well studied when G_1=G(\mathbb F_q^2) and G_2=G(\mathbb F_q), where G is an algebraic group defined over the finite field F_q. In this talk, we survey this theme and cover some recent results which are joint with Nadir Matringe.

Uniqueness of Hahn-Banach Extension and Related Geometric Properties of Banach Spaces (Symposium on Geometry of Banach Spaces)

Dec 02, 2019BySoumitra Daptari, Senior Research Fellow, IIT HyderabadSeminar
The classical Hahn-Banach Theorem assures if Y is a subspace of a normed vector space X and f be a linear form defined on Y , dominated by a sublinear functional p, defined on X, then f can be extended to whole X dominated by p. Coming for the cases when this extension exists uniquely one can see that the situation is very rear. It is well known that X is strictly convex if and only if every subspace of X has Unique Hahn-Banach extension property.
This global condition on X forces the space to be smooth. This motivates us to ask the following question. Given a normed space X and a subspace Y of X, when do the elements in Y have unique norm preserving extension to the whole space X? This property is known as property-U, R. R. Phelps answered this question in his paper [1], It is observed that Y has property-U iff Y ? is a Haar subspace of X .
Asvald Lima, Eva Oja, characterized this property and its variants (property-SU, M-ideal etc.) in terms of many geometrical properties of the space. Our aim is to identify the subspaces of L1-preduals which have property-SU. Our current investigation includes when a stronger geometric property translates through the property-SU. We have identified a few subspaces of finite-dimensional vector spaces which satisfy this property.

Composition Operators Between Segel Bargmann Spaces (Symposium on Geometry of Banach Spaces)

Dec 02, 2019BySudip Ranjan Bhuia, Senior Research Fellow, IIT HyderabadSeminar
We study the boundedness and compactness of linear combination of two composition operators de ned between two Segal-Bargmann spaces. Here we will discuss if the linear combination is bounded (Compact, resp.) then individually both the composition operator are bounded (Compact, resp.).
More precisely, let E1 and E2 be two complex Hilbert spaces. We show that for two mappings  : E1 ! E2 and : E1 ! E2 and for a; b 2 C n f0g, the composition operator aC + bC : H(E2) ! H(E1) is bounded (compact, resp.) if and only if both the composition operators C and C are bounded (compact, resp.). We also characterized the boundedness an compactness of the operators interms of the function theoretic properies of the inducing maps, which generalizes the results over the Fock space in [2]. In addition, we investigate the norm of the linear sum of two composition operators

Weyl's Theorem for Commuting Tuple of Paranormal Operators (Symposium on Geometry of Banach Spaces)

Dec 02, 2019ByNeeru Bala, Senior Research Fellow, IIT HyderabadSeminar
In this talk, we show that a commuting pair of -paranormal operators having quasitriangular property satisfy the Weyl's theorem-I, that is T (T) n TW (T) = 00(T) and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is T (T) n !(T) = 00(T). Also we prove that if f is an analytic function in a neighbourhood of T (T), then Weyl's theorem-II holds for f(T), where T is a pair of commuting paranormal operators.

The Non-commutative Analogue of Korovkin's Sets and Peak Points (Symposium on Geometry of Banach Spaces)

Dec 02, 2019ByP. Shankar, Post Doctorate Fellow, ISI BangaloreSeminar
In this talk, We introduce quasi hyperrigid sets in C*-algebras which are weaker than hyperrigid sets. We also introduce weak boundary representations and study the relation between boundary representations and weak boundary representations for operator systems of C*-algebras.
We prove an analogue of Saskin's theorem relating quasi hyperrigid operator systems and weak boundary representations for operator systems of C*-algebras. we introduce the notion of a weak peak point for an operator system in a C*-algebra which is a non-commutative analogue of peak point.
We study the relation between weak peak points, weak boundary representations and boundary representations. We give characterization for hyperrigid operator systems in terms of orthogonality properties of Hilbert modules. We study hyperrigidity of operator systems in C*-algebras in the context of tensor products of C*-algebras.

Strong Proximinality and Continuity of Metric Projection in Lindenstrauss Spaces (Symposium on Geometry of Banach Spaces)

Dec 02, 2019ByC R Jayanarayan, Assistant Professor, IIT Palakkad.Seminar
In this talk, we discuss the strong proximinality of subspaces and closed unit ball of subspaces of Lindenstrauss spaces. We give a characterization of the strong proximinality of (closed unit ball of) finite co-dimensional subspaces of Lindenstrauss spaces terms of a differentiability concept. We also discuss the continuity of metric projection in Lindenstrauss spaces.

Greedy Approximations (Symposium on Geometry of Banach Spaces)

Dec 02, 2019ByDivya Khurana, Post-Doctorate fellow, IISc Bangalore.Seminar
The 'Thresholding Greedy Algorithm' was introduced by Konyagin and Temlyakov in order to study a special basis in Banach spaces. This basis is known as the greedy basis and can be characterized by unconditional and democratic properties of the basis. Some of the classical Banach spaces fail to have unconditional basis. This fact motivated many researchers to introduce and study some weaker versions of the greedy basis, namely, quasi-greedy basis, almost greedy basis and partially greedy basis.
Almost greedy basis can be characterized by quasi-greediness and democratic property and partially greedy basis can be characterized by quasi-greediness and conservative property. In this talk, we will discuss some new characterizations of almost greedy and partially greedy basis. We will also discuss generalizations of greedy basis and its relatives. This is joint work with Stephen J. Dilworth.

On non-separable Gurariy space (Symposium on Geometry of Banach Spaces)

Dec 02, 2019ByAryaman Sensarma, Visiting Scientist ISI BangaloreSeminar
In this talk, we discuss about (non-separable) Gurariy spaces. We present that, a (non-separable) Banach space is a Gurariy space if and only if every separable a.i.-ideal in X is isometric to the separable Gurariy space G. We also obtain a similar characterization of L1-predual spaces in terms of ideals. Along the way, we show that the family of ideals/a.i.-ideals in a Banach space is closed under increasing limits. We show that if X is almost isometric to a Gurariy space, then it is also a Gurariy space, thus answering a question of Prof. T. S. S.R. K. Rao. This is a joint work with Prof. P. Bandyopadhyay and Prof. S. Dutta.

Geometric Properties of Certain Modular Spaces (Symposium on Geometry of Banach Spaces)

Dec 01, 2019ByAtanu Manna, Assistant Professor of Mathematics, Indian Institute of Carpet TechnologySeminar
Geometric properties of certain kind of Banach function and sequence spaces are current interest of study by many mathematicians. This interest is of course due to various applications of these geometric properties in fixed point theory, optimization theory, differential, integral equations and other areas of mathematical analysis. In this presentation, the notion of modular spaces, in particular Orlicz, Musielak- Orlicz, Cesàro, Orlicz-Cesààro, Musielak-Orlicz-Cesàro and ʌ-sequence spaces are recalled.
Some definitions of important geometric properties such as Kadec-Klee property (property (H)), uniform Kadec-Klee property (UKK), coordinate wise uniform Kadec-Klee property (UKKc), uniform Opial property (UOP), property (β) of Rolewicz, property (k-β) for fixed integer k ≥ 1, weak uniform normal structure (WUNS) etc. and their connection with each other are also recapitulated. After giving a brief survey of the work done on the geometric properties of certain modular spaces, we mostly try to display our outcomes and interest. The following major observations are made:
- ʌ-sequence spaces ʌp and ʌ^p possess the uniform Opial property, coordinate wise uniform Kadec-Klee property, ( )-property of Rolewicz, and weak uniform normal structure. - ʌ-sequence spaces p and ^p have extreme points. - Musielak-Orlicz spaces l((En)) generated by a Musielak-Orlicz function  and a sequence (En) of nite dimensional spaces En, n 2 N, equipped with both the Luxemburg and the Amemiya norm possesses (k- ) property for xed integer k  1. - Similar conclusions are drawn for Cesaro spaces (cesp), generalized Cesaro spaces (ces^p), Musielak-Orlicz-Cesaro spaces (ces), Cesaro sequence spaces of order (ces( ; p)) and Cesaro di erence sequence spaces of order m(O(m)^p).
The outcomes strengthen several earlier known results. It is pertinent to mention that entire presentation is based on the recent contributions, referring to [1] & [2] of the author.

On Small Combination of Slices and Diameter two properties in Banach Spaces (Symposium on Geometry of Banach Spaces)

Dec 01, 2019BySudeshna Basu,Ramakrishna Mission Vivekananda Educational and Research Institute
In this work, we present a chronological description of the notion of Small Combination of Slices (SCS) in Banach Spaces and various aspects of this geometric property. We discuss the relationship of Small Combination of Slices and its relation to other geometric properties like dentability and Radon Nikodym property. We also discuss this property in the context of spaces of operators.
Diameter 2 properties which sits at the other end of the spectrum, was first introduced by O. Nygaard and D. Werner and subsequently studied extensively by J. Guerrero, G. Lopez-Perez, and A Zoca among others. We prove that the dual of an M ideal of a Banach space X has w*-strong D2P(w*- Slice 2P) if has w*-strong D2P(w*-Slice 2P). We explore similar results in the context of C(K)-spaces and duals of strict ideals. We establish a connection between the SCS properties and several versions of diameter 2 properties.

Existence of Linear Extension Operators and its Implications (Symposium on Geometry of Banach Spaces)

Dec 01, 2019ByM. A. Sofi, Department of Mathematics, Central University of KashmirSeminar
The Hahn Banach extension theorem associates to each continuous linear functional g on a subspace M of a normed linear space a continuous linear functional f on X. On the other hand, a consequence of Bartle Graves theorem tells us that the resulting map F from to associating to each a suitable is continuous. The question whether F can also be chosen to be linear- in which case F shall be called an 'extension operator'- turns out to be very interesting and in fact leads to Hilbert spaces as the only class of Banach spaces enjoying this property.
On the other hand, a nonlinear analogue of Hahn Banach theorem is provided by an old result of McShane asserting that Lipschitz maps defined on a subspace of a metric space extends to a Lipschitz map on the whole space.
The purpose of this talk is to present the current state of art involving certain nonlinear analogues of this phenomenon where we shall concentrate mainly on the Lipschitz analogue of this problem involving the existence of a (continuous) linear 'extension operator' between appropriate spaces of Lipschitz functions. Time permitting, we shall also indicate some applications of these ideas to certain questions that arise naturally in the discussion.

Chebesv Subspaces of C*-Algebra (Symposium on Geometry of Banach Spaces)

Dec 01, 2019ByM. N. N. Namboodiri, Department of Mathematics, Cochin University of ScienceSeminar
The classical results are proved mainly using the lattice theoretic properties of scalar functions and the topology involved. But most of the pioneering results were proved using constructive hard analysis techniques. Excellent surveys are due to Ivan Singer[6], Karl-Georg Steffens refer[2], H. Berens and G. G. Lorentz can be found in [2] to cite a few important ones.
During 1930's P. P. Korovkin from the Russian school of analysis initiated the study of unifying many approximation processes such as Bernstein polynomial approximation, Fejér trigonometric polynomial approximation etc., using positivity of linear operators on spaces of continuous functions. His celebrated theorems known as Korovkin type theorems have attained wide range of attention that also led to Korovkin type theorems and Korovkin sets in general Banach algebras and C*-algebras in particular.
One of the three major results of Korovkin connects Čebyšev systems and Korovkin's test sets in certain special cases. Extensive surveys of Korovkin type theorems have also appeared. This lecture is about Čebyšev systems and some related questions.

An Elliptic Behaviour in Operator Algebras (Symposium on Geometry of Banach Spaces)

Dec 01, 2019ByAnil Kumar Karn, Reader, NISER, BhubaneswarSeminar
In this talk, we shall discuss the notion of absolute compatibility for a pair of positive elements in a C*-algebra. We shall present a geometric description of this notion in the case of M2. We shall also discuss a characterization of absolute compatibility in the case of a von Neumann algebra.

Biduals of Spaces of Operators (Symposium on Geometry of Banach Spaces)

Dec 01, 2019ByT S S R K Rao, Visiting Professor, Ashoka UniversitySeminar
An interesting question that takes its motivation from Hilbert space operator theory, is to study spaces of operators between non-re exive Banach spaces, whose biual in a natural way is again a space of op-erators. We exhibit several classes of Banach spaces which have this property, when the range space is a space of continuous functions.

T-duality - past, present and future

Sep 04, 2019ByProf. Peter Bouwknegt, Director, ANU Mathematical Sciences Institute (HisWebsite)Seminar
In this talk, Prof Bouwknegt will review a geometric analogue of the Fourier transform, which arises in String Theory under the name of T-duality. In particular, he will discuss global aspects of T-duality and mention some recent generalisations. T-duality has important applications in different areas of mathematics, such as in differential geometry, algebraic topology, operator algebras, non-commutative geometry, as well as in physics.

Arakelov Geometry of Modular Curves X_0(p^2)

Aug 02, 2019ByDr. Chitrabhanu Chaudhari, IISER PuneSeminar
We shall explore the geometry of the Modular curve X_0(p^2) and it's regular minimal model over the ring of integers, which is an arithmetic surface. After a base change we shall show that the regular minimal model is semi-simple. Arakelov has introduced an intersection pairing for divisors on arithmetic surfaces. We shall derive an expression for the Arakelov self-intersection of the relative dualising sheaf on the regular minimal model of X_0(p^2). As a consequence, we shall give some number theoretic applications for this computation. This is joint work with Dr. Debargha Banerjee and Dr. Diganta Borah.

Betti Numbers of Gaussian Excursions in the Sparse Regime

Nov 19, 2018ByDr.Gugan ThoppeSeminar
Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further have a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools.

Algorithmic techniques for Polynomial Rings over Noetherian commutative rings.

Oct 31, 2018ByDr. Maria FrancisSeminar
Polynomial rings over Noetherian commutative rings have applications in several areas like cryptography, control theory, coding theory and algebraic geometry. For example, in lattice based cryptography, most arithmetic operations are over integers, and in control theory parametric equations with polynomials themselves as coefficients are very common. Algorithmic techniques for polynomial rings over fields are well studied with Groebner bases being one of the fundamental tools in computational ideal theory. Even though various approaches have been proposed to extend Groebner bases theory to polynomial rings over rings, these techniques have only looked at extending basic definitions and concepts.

Inverses of graphs and reciprocal eigenvalue properties

Oct 22, 2018ByDr. Abhay Kumar SinghSeminar

Graded free resolution and its applications

Oct 22, 2018ByDr. Neeraj KumarSeminar
The concept of free resolution, introduced by D. Hilbert in 1890, played a vital role in solving a famous 19th century open problem:``fundamental problem of invariant theory''. Now the minimal free resolution appears in many diverse topics such as algebraic geometry, combinatorics, computational algebra, invariant theory, number theory, non-commutative algebra etc. Our basic setting will be finitely generated graded modules(vector spaces) over a graded commutative ring. We shall introduce homological invariants such as projective dimension, depth, regularity and Betti numbers, associated with the minimal graded free resolution of modules. Then we define the Hilbert series and the Poincar\'e-Betti series of modules. Let $k$ be a field. Given a bigraded $k$-algebra $R$, one can associate a graded $k$-algebra, diagonal subalgebra to it. I will present recent results on Cohen-Macaulay and Koszul property of diagonal subalgebras of bigraded algebras. I will also present some results on the regularity and syzygies of monomial ideals. At the end, I will present some problems concerning Koszul algebras and Cohen-Macaulay property for future investigation

Castelnuovo-Mumford regularity and its asymptotic behaviour

Oct 22, 2018ByDr. Dipankar GhoshSeminar
Castelnuovo-Mumford regularity is a kind of universal bound for important invariants. It provides links between local cohomology theory and the syzygies of finitely generated graded modules over a polynomial ring over a field. In this seminar, I will speak about this invariant and its importance in commutative algebra and algebraic geometry. At the end, I will discuss some of my results on asymptotic linear bounds of Castelnuovo-Mumford regularity involving powers of homogeneous ideals in a polynomial ring over a field.

Algebraic K-theory and homology stability

Oct 22, 2018ByDr.Husney Parvez SarwarSeminar
We shall begin with the homotopy invariance property of K-theory. After reviewing monoids and monoid algebras, we present some results which are monoid version of the homotopy invariance property in K-theory. This answers a question of Gubeladze. Next, we will discuss the monoid version of Weibel's vanishing conjecture and some results in this direction. Finally, we will talk about the homology stability for groups. Here we present a result which improves homology stability for symplectic groups. If the time permits, some application of the homology stability will be given to the hermitian K-theory.

On the Symbol-Pair Distance of Repeated-Root Con-stacyclic Codes of Prime Power Lengths

Oct 15, 2018ByDr. Abhay Kumar SinghSeminar
Let p be a prime, and λ be a nonzero element of the finite field Fpm. The λ-constacyclic codes of length ps over Fpm are linearly ordered under set-theoretic inclusion, i.e., 0 ≤ i ≤ ps of the chain ring [Fpm/〈xps −λ〉]. they are the ideals 〈(x − λ0)i〉, This structure also gives the du- als of such λ-constacyclic codes, which are λ-1-constacyclic codes, and their hulls, which in turn provides neccessary and sufficient conditions for the exis- tence of self-dual, self- orthogonal, dual containing, and LCD λ-constacyclic code. Then this structure is used to establish the symbol-pair distances of all such λ-constacyclic codes. Among others, we identified all MDS symbol- pair constacyclic codes of length ps which satisfied the Singleton Bound for symbol-pair codes, i.e., |C| = pm(n-dsp(C)+2).1

On the design of experiments with ordered treatments

Oct 15, 2018ByDr. Satya Prakash SinghSeminar
There are many situations where one expects an ordering among $K\geq2$ experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly, very little work has been done in the context of the design of experiments. Here, a principled approach to the design of experiments with ordered treatments is provided. In particular, we propose two classes of designs which are optimal for testing different types of hypotheses. The theoretical findings are supplemented with thorough numerical experimentation. It is shown that there is a substantial gain in power when an experiment is both designed and analyzed using methods which account for order restrictions.

On the structure and distances of repeated-root constacyclic codes

Oct 15, 2018ByDr. Anuradha SharmaSeminar
There are many situations where one expects an ordering among $K\geq2$ experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly, very little work has been done in the context of the design of experiments. Here, a principled approach to the design of experiments with ordered treatments is provided. In particular, we propose two classes of designs which are optimal for testing different types of hypotheses. The theoretical findings are supplemented with thorough numerical experimentation. It is shown that there is a substantial gain in power when an experiment is both designed and analyzed using methods which account for order restrictions.

Local Convergence of higher order iterative methods for solving nonlinear equations in Banach spaces

Oct 15, 2018ByDr. Maroju PrashanthSeminar
First part, we present a local convergence study of a fifth order iterative method proposed in [1] in order to approximate a locally unique root of a nonlinear equation. The analysis is discussed under the assumption that first order Fréchet derivative satisfies the Lipschitz continuity condition. Finally, we provide computable radii and error bounds based on the Lipschitz constant analyzed in the theoretical results. Some of numerical examples are worked out and compare these results with existing methods results. Second part, Third-order stirling-like method for finding a fixed point for nonlinear equa- tions in Banach spaces is proposed. The local convergence of this method for finding the fixed points of nonlinear equations in Banach spaces is discussed. The convergence established un- der the assumption that the first order Fréchet derivative satisfies the Lipschitz continuity condition. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We consider the numerical examples and calculate the existence and uniqueness region of convergence balls even we fail to apply the results in [3–6] due to the F is not contraction on Ω. References [1] A. Cordero, J.L. Hueso, E. Martnez, J.R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Applied Mathematics Letters 25, 23692374, (2012). [2] L.B. Rall, Computational Solution of Nonlinear Operator Equations, E. Robert Krieger, New York, 1969. [3] L. B. Rall, Convergence of Stirling’s Method in Banach Spaces, Aequationes Math. 12(1975), 12-20. [4] Argyros, I.K., Stirlings method and fixed points of nonlinear operator equations in Banach spaces, Bull. Inst. Math. Acad. Sin. (N.S.) 23(1995), 13-20 . [5] Parhi, S.K., Gupta, D.K, Semilocal convergence of Stirlings method under Hölder continuous first derivative in Banach spcaes, Int. J. Comput. Math. 87(2010), 2752-2759. 1 [6] Parhi, S.K., Gupta, D.K, Relaxing convergence conditions for Stirlings method, Math. Meth- ods Appl. Sci. 33(2010), 224-232. 2

On logarithmic coefficients of univalent functions

Oct 15, 2018ByDr. Vasu Deva Rao AlluSeminar
Let S denote the class of analytic and univalent (i.e., one-to-one) functions f in the unit disk D := {z ∈ C : |z| < 1} satisfying f(0) = 0 = f ′(0) − 1. The logarithmic coefficients γ n of f ∈ S are defined by log f(z) zn, for n = 1,2,.... If f ∈ S, then it is known that |γ 1 | ≤ 1 and |γ 2 | ≤ (1+2e−2)/2. These bounds are sharp. Finding sharp upper bounds for |γ n | for n ≥ 3 is an open and long standing problem. In this talk, we present some sharp coefficient inequalities concerning γ 3 when f ∈ S. We also present the sharp logarithmic coefficients γ n , γ 4 and γ 5 for n = 1,2,3 for some of the well-known subclasses of S. 1 z = 2 ∞∑ n=1 γ n

A unified approach to stochastic comparisons of multivariate mixture models

Oct 15, 2018ByDr. Sameen NaqviSeminar
In recent times, significant advances have been made in the study of multivariate mixtures. One such advancement is towards comparison of various multivariate mixture models in terms of stochastic orders. However, the results that have been obtained until now have some limitations in terms of their applications. In this talk, we shall consider a more general scenario and make stochastic comparisons between random vectors corresponding to two multivariate mixture models. These stochastic comparisons will be made with respect to multivariate hazard rate, reversed hazard rate and likelihood ratio orders. As a consequence, we will unify various results on stochastic comparisons of multivariate mixtures available in the literature.

Iterative and Fixed Point Technique for Solving Variational Inequalities and Equilibrium Problems

Oct 11, 2018ByDr. Shuja Haider RizviSeminar
In the last decades, the theory of variational inequalities has been developed very significantly for its wide range of applications in many research areas. It is known that the optimization problems and many others important mathematical problems are closely related to variational inequality problems (in short, VIPs) and equilibrium problems (in short, EPs). Therefore the theory of variational inequality and equilibrium problems are also been developed and are very interesting topics of current studies.

High resolution WENO schemes for compressible flows

Oct 10, 2018ByDr. NagarajuSeminar
A modified third and fifth order weighted essentially non-oscillatory (WENO) schemes are proposed for the solution of inviscid compressible flows (Euler equations), where a new global-smoothness indicator is devised which shows an improved solution behavior over many existing WENO schemes, for the problems which contain discontinuities and attains optimal-order of accuracy at the critical points. For fifth order WENO scheme the smoothness indicators are constructed based on L1 measure. The WENO scheme was designed in such a way that least weights should be given to the most discontinuous sub- stencil, increasing their weights creates oscillations that could prevent the WENO scheme from attaining the expected order of accuracy. Assigning larger weight to the sub-stencils containing the discontinuity has been studied and suggested improvements in the scheme that could capture the smooth solutions even after increasing the weights of the lesser smooth sub-stencils. Further, proposed a new problem independent discontinuity locater for hybridization.

Analysis of Penalty Methods for Constrained Optimization Problems

Constrained optimization problems are frequently encountered in a vast range of applications in the areas of science, engineering, economics and many more. The penalty method has been considered as one of the most efficient methods to solve constrained optimization problems. In my presentation, I will focus on two of my research works in which penalty methods have been applied to solve a nonsmooth vector optimization problem and a variational problem.
In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so- called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surfaces, we call these admissible. There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterizes admissibility. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).
In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so- called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surfaces, we call these admissible. There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterizes admissibility. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).
Many graph colorings are motivated by frequency assignment problem which is a problem of assigning frequencies to transmitters in an optimal way and with no interference. Interference can occur if trans- mitters located sufficiently close to each other receive close frequencies. At the same time the demand for frequencies is outstripping the pace of technological change in expanding the usable spectrum. Fre- quency assignment problems can be modeled graph theoretically by taking stations as vertices of the graph and two vertices are adjacent if interference (up to some extent which is fixed for a given set of transmitters) is possible between them. In the present talk, we study two of such colorings, namely, radio k-colorigs of graphs and irreducible no-hole colorings of graphs, in detail. Also, we see, in brief, k-distance colorings of graphs and broadcast labelings of graphs.
A graph is said to be vertex-transitive (VT) if its automorphism group act transitively on its vertices. Laszlo Lovasz (1969) [3] conjectured that “Every finite connected VT graph contains a Hamiltonian cycle except the five known counterexamples.” In most of the known examples, one can easily find a hamiltonian cycle in a VT graph by considering a semiregular automorphism. Working in this direction, Dragan Marusic (1980) [5] conjectured that “(Polycirculant Conjecture) Every finite VT graph admits a semiregular automorphism.” Thus, a proof of polycirculant conjecture may provide an insight to the Lovasz Conjecture. Since then, polycirculant conjecture was proved for various special families of VT graphs. However, a complete solution is yet to be found. According to Marusic, a complete solution to polycirculant conjecture may require new tools and approaches. In this talk, we briefly discuss the current state of this conjecture and introduce a new notion of groupifiability of graphs to approach the problem. In this approach, we formulate an equivalent form of polycirculant conjecture and prove the conjecture for a large sub-class of VT graphs.
In this talk, we present and analyse a time-stepping DPG method for the heat equation. The motivation of this work is to develop a general theoretical DPG framework for the heat equation that can lead to robust approximations of singularly perturbed parabolic problems. For a time-stepping DPG method, we consider DPG approximation based on ultra-weak variational formulation in space and combine with the backward Euler time stepping approximation. Well-posedness and stable approximation properties are obtained from a precise analysis of the underlying time-discrete variational formulation at every time step. We prove appropriate convergence properties for field variables and derive a Céa-type error estimate. Extension of our results to treat more general linear parabolic problems will also be discussed. Finally some of the future research plans will be highlighted.
Hilbert function is an important algebraic object used extensively in several different areas of mathematics. This encodes the numerical behavior of a projective variety. On the other hand Gorenstein algebras are also very common and significant. An important open problem in commutative algebra is to characterize the Hilbert functions of Gorenstein K-algebras. Recently, jointly with M. E. Rossi, we resolved this problem in some cases (K-algebras of socle degree 4). Moreover, we applied these methods to classify local Gorenstein K-algebras with fixed Hilbert function which is another widely open problem on this topic. In this talk, we will discuss these new developments. This has applications in the study of the punctual Hilbert schemes parameterizing zero-dimensional subschemes of fixed degree in the projective space.
We talk about the mod 2 cohomology ring of the Grassmannian $\widetilde{G}_{n,3}$ of oriented 3-planes in $\mathbb{R}^n$. We first state the previously known results. Then we discuss the degrees of the indecomposable elements in the cohomology ring. We have an almost complete description of the cohomology ring. This description provides lower and upper bounds on the cup length of $\widetilde{G}_{n,3}$.
We explain the meaning of local to global principle and then we state several examples with this. We are interested in understanding the existence of Q-rational l-isognies of Abelian Varietie A of dimension 2 if there exists an F_p-rational l-isogeny for the local variety A_p for every prime p.
We focus on the necessay and sufficient condition for the boundedness of infinite toeplitz operator defined on l^2(Z) and also we find the spectrum of the same operator.
We will show that, any bounded convolution operator on L^2(R) is commutes with all translations on L^2(R). In fact, these are only the bounded operators which will commute with all translations.
We focus on the necessay and sufficient condition for the boundedness of infinite toeplitz operator defined on l^2(Z) and also we find the spectrum of the same operator.
Telecardiology is envisaged as a supplement to inadequate local cardiac care, especially, in infrastructure deficient communities. Yet the associated infrastructure constraints are often ignored while designing a traditional telecardiology system that simply records and transmits user electrocardiogram (ECG) signals to a professional diagnostic facility. Against this backdrop, we propose a two-tier telecardiology framework, where constraints on resources, such as power and bandwidth, are met by compressively sampling ECG signals, identifying anomalous signals, and transmitting only the anomalous signals. Specifically, we design practical compressive classifiers based on inherent properties of ECG signals, such as self-similarity and periodicity, and illustrate their efficacy by plotting receiver operating characteristics (ROC). Using such classifiers, we realize a resource-constrained telecardiology system, which, for the PhysioNet databases, allows no more than 0.5% undetected patients at an average downsampling factor of five, reducing the power requirement by 80% and bandwidth requirement by 83.4% compared to traditional telecardiology.
We introduce a class of operators, namely Abolutely minimum attaining operators defined on complex Hilbert spaces of arbitrary dimension and discuss some examples. We prove that every positive Absolutely minimum attaining operator is diagonalizable, by means of that we study the spectral properties of such operators and also we prove a characterization theorem for this class of operators.
Let A be a unital Banach algebra and G(A) be the set of invertible elements of A. An element a ∈ G(A) is said to satisfy BOBP (Biggest open ball property) if B  a, 1 ka−1k  is the biggest ball which is centered at a and is completely contained in G(A) i.e the boundary of the ball B  a, 1 ka−1k  necessarily intersects the set of non-invertible elements. We say a Banach algebra A satisfies BOBP if every a ∈ G(A) satisfies BOBP. We make an attempt on understanding as well as characterising all Banach Algebras which satisfy BOBP and which do not. To actualise the above mentioned objective, we discuss the observations made in some classical Banach Algebras.
This presentation focuses on the following question "When the level set of the condition spectrum has an empty interior?". Certain analytical results, examples and motivation from pseudo-spectra, connected to the above question, will be discussed.
Basic concepts of porous medium, modes of heat and mass transfer, stability analyses will be discussed. The stability of convection in a horizontal porous layer which is saturated with fluid and induced by horizontal temperature gradients subjected to horizontal mass flow is investigated by means of linear and nonlinear stability analysis. The effects of variable gravity field and vertical throughflow are also considered in this analysis. The nonlinear stability analysis part has been developed via energy functional. Shooting and Runge-Kutta methods have been used to solve eigenvalue problem in both cases. Comparison is made between linear and nonlinear stability results.
A finite collection of congruences is said to be a covering of integers if every integer satisfies at least one among these. The restriction that the moduli involved be distinct and satisfy certain property makes finding such covering systems nontrivial. Paul Erdos was fascinated by coverings, and along with Selfridge, Graham and others, proposed a series of problems and conjectures on the topic in his lifetime. One such problem asks whether there is a covering involving distinct odd moduli > 1. As discovered by Schinzel, this problem is closely related to an (somewhat relaxed version) irreducibility conjecture of Paul Turan which is based on Van der Warden's theorem that "almost all" polynomials having integer coefficients are irreducible over the rationals. We discuss these connections, as well as some of the past and recent developments made towards solving these problems.
It is well-known that a normalized Hecke eigenform of integral weight has algebraic Fourier coefficients. However, for half-integral weight modular forms, there are no analogous results known. In a joint work with Soma Purkait, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them.
Every normal operator in quternionic Hilbert space is unitarily equivalent to a multiplication operator induced by some suitable function.
Best approximation in normed linear space is a well studied subject and it has numerous applications in many other streams in analysis. The theory has some nice interplay with the geometry of the unit ball and the differentiability of the dual norm. If the best approximation is guaranteed for a given subspace, it is natural to study how this set of best approximation is located. Various strengthenings of best approximation comes in this context. Part I will be devoted towards some basic concepts and known results in this direction and Part II will discuss some of the speaker's recent observations.
Reproducing Kernel Hilbert Spaces (RKHS) are function spaces. They play a major role in many applicational settings - for instance, in Kernel Methods in Machine Learning, Sampling theory and Extremal problems. In this talk, after presenting the definition of an RKHS, we will see some simple examples of RKHS. Following this, we will deal with one non-trivial example of an RKHS and try to give some ideas of the proof.
In finite dimensional case, the condition spectrum judges the computational stability of solving a linear system. In a complex commutative Banach algebra with unit, the condition spectrum and almost multiplicative function have connection similar to what the spectrum and multiplicative function have. The connections are analogous to the Gelfand theory and Gleason-Kahane-Zelazko theorem. Some results are compared with eigenvalues and pseudo-spectra as well.
Our aim is to study a brief introduction to Topological dynamics. This talk will deal with basic concepts of Topological dynamical system.We study the definition and basic properties of Minimality,Topological Transitivity,wandering and non-wandering points,Topological conjugacy and Expansiveness.
Our aim is to study a brief introduction to Topological dynamics. This talk will deal with basic concepts of Topological dynamical system.We study the definition and basic properties of Minimality,Topological Transitivity,wandering and non-wandering points,Topological conjugacy and Expansiveness.
For data that can be sparsely generated, one can obtain good reconstructions from reduced number of measurements - thereby compressing the sensing process rather than the traditionally sensed data. A wealth of recent developments in applied mathematics, by the name of Compressed Sensing (CS) aim at achieving this objective. The backbone behind such recoveries is sparse modeling of target data by the elements of a given system function, called Dictionary or CS matrix. The resentation aims at exploring the basic theory and applications of dictionary design methods.
We introduce norm attaining and minimum attaining operators and special classes of these operators on a Hilbert space. Our aim is to characterize these classes and obtain structure of these operators. We present some results that we have obtained and possible future problems.
Stability of thermo-solutal convection induced by temperature and concentration gradients in a horizontal fluid saturated porous medium with non zero net mass flow is investigated balong with viscous dissipation and the Soret effect by means of linear stability analysis. These horizontal gradients induces Hadley circulation. Basic velocity is superposition of Hadley type flow and uniform flow. One term Galerkin approximation method is used to obtain the solution analytically. Critical Vertical Rayleigh number is obtained for assigned values of critical wave number, Horizontal thermal Rayleigh number, Gebhart number, Peclet number, horizontal and vertical Solutal Rayleigh number, Lewis number and Soret number. Physcial explanation is given for the onset of convection.
In 2006, J. Jorgenson and J. Kramer have constructed a certain "key-identity" relating the canonical and hyperbolic volume forms defined on a compact Riemann surface. Using this identity they transformed a problem in Arakelov theory to that of hyperbolic geometry, and derived bounds for the canonical Green's function. In this talk, we describe an extension of the key-identity to non-compact hyperbolic Riemann surfaces of finite volume, and its applications to modular curves.
Let f be a generalized modular function (GMF) of weight 0 and level N, such that its q-exponents c(n)(n >= 1) are all real, and div(f) = 0. We show the equidistribution of signs for c(p)(p prime), by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.
The present study deals with the effect of stress relaxation on the mechanism of instability of the thermo-solutal convection. The energy method is developed to discuss the non-linear stability of convection in horizontal porous layer subject to an inclined temperature gradient. In this study both linear and non-linear stability analysis are carried out for a large number of parameter values. The horizontal components of these gradients induce a Hadley circulation, which becomes unstable when vertical components are sufficiently large and this instability is analyzed using three dimensional normal modes. The vertical thermal Rayleigh number is treated as the eigen value. The system that constitutes the eigen value problem is solved by using shooting method for various modes of instability. Results are presented for various values of the governing parameters of the flow.
In recent years, sparse representation has emerged as a powerful tool for efficiently processing data in non-traditional ways. This is mainly due to the fact that natural data of interest tend to have sparse representation in some basis. A wealth of recent optimization techniques in applied mathematics, by the name of Compressive Sampling Theory (CST or CS theory) aim at providing the sparse description of such data in redundant bases. This talk briefly walks through the current developments in Sparse representation theory with emphasis on the construction of CS matrices, and shows how the developments could be used for applications in Computed Tomography.
From Hahn-Banach theorem it follows any closed bounded convex set in a normed linear space is a intersection of closed half spaces. It was S. Mazur who asked when it can be expressed as a intersection of closed balls. Several characterizations are known till today. Dr. Paul will try to cover the results for finite dimensional spaces. Several strengthening and weakening are also possible and if time permits Dr. Paul will try to discuss those results.
The talk will start with a short introduction to Fuzzy Set Theory, its need and the different connectives employed. Following this fuzzy inference systems, which employ fuzzy if-then rules, will be discussed in detail. Finally, some new results regarding their function approximation capabilities will be presented.
Fuzzy Implications are a generalisation of the classical two-valued implication to the infinite-valued setting. They play an important role both in the theory and application, as can be seen from their use in Multi-valued logic, Approximate Reasoning, Fuzzy Control, Image Processing and Data Analysis. While the analytical aspects of these operators have been well-studied, only some nascent attempts have been made on an algebraic study of them. In this talk, after a short introduction to these connectives, we present some of our latest results, wherein we propose a novel generation process of these operations and show that the set of all fuzzy implications form a monoid. This is the richest structure known so far on these operations and helps us in giving a natural ordering on this set.

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