Centre for Advanced Research in Applied Mathematics and Statistics, MAHE, Manipal, India
The present conference shall provide a platform for leading Mathematicians and Statisticians, working around the globe in the theme area to discuss several research issues and to introduce new innovations. The main goal of the conference is to bring experts, young researchers, and students together to present recent developments in this dynamic and important field. The conference also aims to stimulate research and support the interaction between the scientists by creating an environment for the participants to exchange ideas and to initiate collaborations and professional partnerships.
On behalf of the Scientific Committee, and the Local Organizing Committee, we have much pleasure in extending a cordial invitation to participate in this conference.
The theme of the conference shall focus on
Linear Algebra and Graph Theory are important branches of Mathematics having applications in each and every branch of Applied Sciences. The topic ‘Matrix Methods in Statistics’ is a branch of Linear Algebra and Matrix Theory containing a variety of challenging problems in Linear Statistical Models and Statistical Inference having applications in various branches of Applied Statistics such as Natural Sciences, Medicine, Economics, Electrical Engineering, Markov Chains, Digital Signal Processing, Pattern Recognition and Neural Network, to name a few. Advances in Combinatorial Matrix theory are motivated by a wide range of subjects such as Networks, Chemistry, Genetics, Bioinformatics, Computer Science, and Information Technology. The areas of Classical Matrix Theory and Combinatorial Matrix Theory interact with each other, which is evident from the interplay between Graphs and Matrices. Generalized Inverses of Matrices such as the Incidence Matrix and Laplacian Matrix are mathematically interesting and have great practical significance. Covariance Matrices play an important role in the study of uncertainty associated with data related to measurements, which is an important part of applied Mathematics and Statistics.
This conference is in sequel to the conferences CMTGIM 2012, ICLAA 2014, and ICLAA 2017 held in Manipal during January 2012, December 2014, and December 2017 respectively. The present conference shall provide a platform for leading Mathematicians and Statisticians, working around the globe in the theme area to discuss several research issues and to introduce new innovations. The main goal of the conference is to bring experts, young researchers, and students together to present recent developments in this dynamic and important field. The conference also aims to stimulate research and support the interaction between the scientists by creating an environment for the participants to exchange ideas and to initiate collaborations and professional partnerships.
The ICLAA 2021 will be held beside 28th IWMS, and will have some common sessions, in honor of Prof. C. R. Rao, on the common day December 15, 2021. We are sure that ICLAA 2021 will be a great success with the participation of prominent researchers from across the globe working in Matrix Theory, Graph Theory, Network Science, Statistics, and allied subjects.
Beside organizing invited talks from eminent speakers, we invite participants to submit their research articles in the sessions of contributory talks.
We intend to publish a special issue dedicated to the conferences ALAPS 2020, IWMS 2021, and ICLAA 2021 in a reputed journal. We thank ‘AKCE International Journal of Graphs and Combinatorics’ for its consent for publishing a special issue with the articles focused on common area of interest of the journal and conferences.
The guest editorial team invites research articles in the following mentioned focus area for the possible publication in the special issue dedicated to the conferences. The article need not be the one presented in the ALAPS 2020/ IWMS 2021/ ICLAA 2021 and similarly, the author need not be among speakers or participants of ALAPS 2020/ IWMS 2021/ ICLAA 2021, though we encourage the speakers to submit an article presented in the conference.
Focus Area of Special Issue: Keeping in view of the focus area of the journal and conference, the articles submitted are expected to be in the following broad area:
Guest Editors Team: Consists Scientific Advisory Committee (SAC) members
associated with
Journal: AKCE International Journal of Graphs and Combinatorics (An open-access journal)
Web Link: https://www.tandfonline.com/loi/uakc20.
Publisher: Taylor & Francis online
How to Submit: Author may submit an article within the focus area mentioned above, directly to the convener of SAC, who is the coordinator for the special issue and the proceedings ( km.prasad@manipal.edu; kmprasad63@gmail.com) with the suggestion of a guest editor to handle the paper.
Formatting: Author may refer to instructions for authors on the journal site with the link: https://www.tandfonline.com/action/authorSubmission?show=instructions&journalCode=uakc20
Submission fee & APC: There is no Submission Fee or Article Processing Charges (APC) for the articles submitted for the special issue.
Review Process
Important Dates
Receiving the articles: April 30, 2022
Completing the review process including necessary revision: September 30, 2022
Submitting the final manuscript in the necessary format: October 31, 2022
Publication of special issue: January 31, 2023 (subject to the convenience of AKCE Journal)
This volume is to cherish the beautiful path laid by the living legend
who completed 100 years of fruitful life in 2020 and in memories of young and dynamic
whom we missed in this year of 2021.
The proceedings proposal has been submitted to SPRINGER for the possible publication in a scopus indexed series.
Research articles in detail and expository articles are invited for the possible publication in the volume of proceedings in the focus area of ALAPS 2020, IWMS 2021, ICLAA 2021 and the work by Professor Rao. Though we appreciate the submission of work presented in any of these connected conferences, we also invite any good articles in the form of chapters which throw more light on the work of Professor Rao and help young scholars to understand Rao’s work and further advancements. We welcome articles in memory of Arbind Lal.
Editorial Team
Nature of article
We appreciate articles of following nature:
Timeline
Last Date for receiving the articles: April 30, 2022
Completion of Review, including necessary revision: October 31, 2022
Proof reading & Pre-production work by SPRINGER: December 31, 2022
Publication of edited volume: March 31, 2023
Number of pages
We appreciate the number of pages of article restricted to 20-25, but in some exceptional cases it could be more.
We do hope, we have an opportunity to meet in Manipal and enjoy the academic deliberation.
Ravindra B Bapat
Chair, Scientific Committee
ICLAA 2021 & IWMS 2021
Narayana Sabhahit (Registrar, MAHE)
Chair, Local Organizing Committee
ICLAA 2021 & IWMS 2021
Category | IWMS 2021 / ICLAA 2021 / IWMS 2021 + ICLAA 2021 |
---|---|
Foreign Nationals | $100 |
Indian Nationals, Research Scholars, Faculty and students | ₹1000+GST |
In-person participation is subject to prevailing travel conditions and Covid-19 protocols. Therefore all contributory speakers and other participants are required to make travel arrangements with caution. The people who have already registered for in-person attendance are allowed virtual participation.
Category | IWMS 2021 / ICLAA 2021 | IWMS 2021 + ICLAA 2021 |
---|---|---|
Foreign Nationals | $350 | $350 |
Indian Nationals, Research Scholars, and Faculty | ₹2000+GST | ₹3500+GST |
Indian Students having no funding support | ₹1000+GST | ₹2000+GST |
Category | IWMS 2021 / ICLAA 2021 | IWMS 2021 + ICLAA 2021 |
---|---|---|
Foreign Nationals | $200 | $200 |
Indian Nationals, Research Scholars, and Faculty | ₹1000+GST | ₹2000+GST |
The following are prevailing COVID protocols (as of November 23, 2021) to be followed at the MAHE Campus. At the time of the beginning of conferences, the protocol may change subject to the situation. Delegates coming from outstation are required to note the same.
Wish to see you at Manipal!!
Research Interests:
Perturbation theory for linear operators, Structured perturbation theory for matrices, Nonlinear eigenvalue problems, Multiparameter eigenvalue problems. Read more
Achievements:
Currently Prof. Rafikul Alam is working as a professor in the department of Mathematics, Indian Institute of Technology… Read more
Let $ U \subset \mathbb{C}$ be open and $T: U \longrightarrow \mathbb{C}^{n\times n}$ be holomorphic. Consider the nonlinear eigenvalue problem (NEP): $$ \text{ Find } \lambda \in U \text{ and a nonzero } v \in \mathbb{C}^n \text{ such that } T(\lambda) v = 0.$$ NEPs arise in many applications in science and engineering. Computing solution of an NEP is a challenging task and various methods for solving NEPs have been proposed in recent years. We present a trace-moment based method for computing eigenvalues of $T(\lambda).$ If a region in $U$ contains $\ell$ distinct eigenvalue $\lambda_1, \ldots, \lambda_\ell$ of $T(\lambda)$ then by utilizing trace-moments of $T(\lambda)$ we construct an $\ell\times \ell$ Hankel pencil $L(\lambda) := \widehat{H}- \lambda H$ such that $\lambda_1, \ldots, \lambda_\ell$ are simple eigenvalues of $ L(\lambda).$ Hence by solving the linear eigenvalue problem $L(\lambda) v =0$ we obtain the eigenvalues $\lambda_1, \ldots, \lambda_\ell$ of $T(\lambda).$
Research Interests:
Network Analysis, Graph Theory and its applications, Fuzzy logic and its applications to pattern recognition, Topology , Geometry… Read more
Achievements:
He has more than 250 publications and more than 20 books. Most of these books are prescribed as… Read more
Let $G$ be a simple graph of order $n$. The nullity $\eta(G)$ of $G$ is the algebraic multiplicity of $0$ as an eigen value of the adjacency matrix of $G$. A graph $G$ with $\eta(G) > 0$ is called a singular graph. For a bipartite graph $G$ (corresponding to an alterant hydrocarbon) if $\eta(G) > 0$, then it indicates that the molecule which such a graph represents is unstable. A characterization of graphs for which $\eta(G) > 0$ still remains open. In this talk we present a survey of results on $η(G)$ and indicate open problems and directions for further research.
Research Interests:
Ordered data analysis, Uni-variate and multivariate distribution theory, Reliability theory, Survival analysis, Applied probability, Stochastic orderings, Non-parametric statistics,… Read more
Achievements:
Professor N. Balakrishnan is a Distinguished University Professor at McMaster University, Hamilton, Ontario, Canada. He completed his PhD… Read more
Research Interests:
Matrices and Graphs, Nonnegative Matrices, Matrix Inequalities and Generalized Inverses Read more
Achievements:
He has published more than 140 research papers in these areas in reputed national and international journals and… Read more
Research Interests:
Mathematics: Convexity and Inequalities, Matrix Theory. Optimization Theory: Nonsmooth Analysis, Symbolic Computation. Applied Mathematics: Numerical Linear Algebra, Linear… Read more
Achievements:
Adi Ben-Israel received his PhD from Northwestern University, United States (1962). He has published over 5 books in… Read more
Research Interests:
Matrices, Graphs and the connections between the two Read more
Achievements:
Abraham Berman received his B.Sc. (1966) and M.Sc. (1968) in mathematics from the Technion. In 1970 he received… Read more
Let $S$ be a set of nonnegative numbers. A matrix $A$ is completely positive over $S$ if it can be decomposed as $A=BB^T$, where the entries of $B$ are in $S$. The talk is a survey of results on completely positivity over $S$ in the cases $S=R,\;S=Q\; S=Z$ and $S=\{0,1\}$.
Research Interests:
Operator Theory/Operator Algebras with special reference to Quantum Probability Read more
Achievements:
He received his PhD in Mathematics from Indian Statistical Institute. He was a Post-doctoral fellow at the University… Read more
Research Interests:
Economics: Information economics, Moral hazard problems and incentives. Econometrics: Large dimensional time series, Resampling in time series, Asymptotics… Read more
Achievements:
Professor completed his M. Stat. (1980) and PhD (1987) from the Indian Statistical Institute. He has been an… Read more
Research Interests:
Magic Squares, Latin squares, Philatelic Latin squares, Magic card puzzles Read more
Achievements:
Garry Ka Lok Chu has been teaching at Dawson College for more than a decade. He works as… Read more
Research Interests:
Generalized Inverses of Matrices and Linear Complimentarity Problem Read more
Achievements:
N.Eagambaram completed his M. Sc. in Statistics from Annamalai University in 1976. He worked as Lecturer in Statistics… Read more
Given a matrix $M$ and a vector $q$, the Linear Complementarity Problem(LCP($M,q$) is to find a solution $z$ to $$ [Mz + q \ge 0\, z^{T}(Mz+q) = , z \ge 0] $$ Lemke’s Algorithm applied to LCP($M,q+\theta p$) fetches solutions to the LCP for all $\theta \in [\theta_{0}, \infty)$ for some $\theta_{0}$ and $p > 0$. We express $\theta$ as a function of time index $t$. Solution $z$ obtained from Lemke’s Algorithm is a continuous function of $t$. A variant of Lemke’s Algorithm is proposed using the continuity property. Some results on characterization of $Q$-matrices are obtained.
Research Interests:
Matrix Theory, Discrete Mathematics, Graph Theory, Combinatorial Matrix Analysis Read more
Achievements:
He has written a book titled 'Totally Nonnegative Matrices' which was published by Princeton University Press. He has… Read more
The inverse eigenvalue problem for graphs has become a central research enterprise for the past 30 years with many exciting and interesting advances and related applications. One aspect of this problem is studying the fewest possible eigenvalues associated with a given graph.
This presentation will provide a broad survey of current results involving this curious and important graph parameter, which is typically labeled as $q(G)$ for a given graph $G$. In particular, I will discuss the cases when $q(G) \geq |G|-1$ and when $q(G)=2$, along with other intriguing results on $q$ for various families of graphs. I will also touch upon recent variants of the parameter $q$ connected with certain `strong’ properties of matrices.
Research Interests:
Algebra, Linear Algebra Read more
Achievements:
Distinguished Professor S. K. Jain served Ohio University, Department of Mathematics from 1970 to 2009. He also served… Read more
By using semigroup techniques, J. Araújo and F.C. Silva proved that a matrix B with coefficients in a division ring D is a product of conjugates of any matrix A with rank(B) smaller or equal to rank(A). We prove this result over an algebraically closed field, in an elementary way suitable even for undergraduate students.
Research Interests:
Algebraic Graph Theory: Linear Algebra and its Applications to Graph Theory, Adjacency and Laplacian Spectra of Graphs. Read more
Achievements:
Debajit Kalita received his MSc degree from Gauhati University and a Ph.D. degree from IIT Guwahati. He has… Read more
A block is said to be pendant if it has exactly one point of articulation. Suppose that we are given a collection blocks and we construct all possible connected graphs with these blocks keeping the number of pendant blocks fixed. In this talk, we describe the structure of the graphs that minimize the algebraic connectivity among all such graphs. As an application, we conclude that over all such graphs made with the given blocks, the algebraic connectivity is minimum for a graph whose block structure is a path.
Research Interests:
Positivity and analysis, representation theory of Lie algebras, combinatorics and discrete mathematics Read more
Achievements:
Prof. Apoorva Khare is currently working as an Associate Professor in Mathematics in the Indian Institute of Science,… Read more
Given a finite simple connected graph $G = (V,E)$, we introduce a novel invariant which we call its blowup-polynomial $p_G((n_v)_{v \in V})$. To do so, we compute the determinant of the distance matrix of the graph blowup, obtained by taking $n_v$ copies of the vertex $v$, and remove an exponential factor. First: we show that as a function of the sizes $n_v$, $p_G$ is a polynomial, is multi-affine, and is real-stable. Second: we show that the multivariate polynomial $p_G$ is intimately related to the characteristic polynomial $q_G$ of the distance matrix $D_G$, and that it fully recovers $G$ whereas $q_G$ does not. Third: we obtain a novel characterization of the complete multi-partite graphs, as precisely those whose “homogenized” blowup-polynomials are Lorentzian/strongly Rayleigh. (Joint with Projesh Nath Choudhury.)
Research Interests:
Theory and Applications of Nonnegative Matrices, Combinatorial Matrix Theory, Spectral Graph Theory Read more
Achievements:
Prof. Kirkland has been published more than 130 journal articles, and author of 4 books. He is the… Read more
We consider the spread of an infectious disease, modelled as a network of patches (to reflect a heterogeneous environment) with movement between patches. The invasibility of the disease can be measured by the network reproduction number $R_0$, which turns out to be the spectral radius of a certain nonnegative matrix. In this talk we work with an approximation to $R_0$; that approximation applies in the case that the time scale of movement is substantially larger than the time scale of the disease dynamics. We investigate how perturbations in the network structure affect the value of $R_0$, and discuss the changes to the network that yield the largest decrease in $R_0$. Throughout we use techniques from matrix analysis, and adopt perspectives from combinatorics.
Research Interests:
Applied Statistics, Linear Programming, Nonlinear Programming, Non-cooperative games, Stochastic games, Statistical Quality Control, Six Sigma, Quality Management. Read more
Achievements:
Based on his research and teaching interests, in applied statistics and matrix methods, he has published several research… Read more
Research Interests:
Spectral Graph Theory, Linear and Multi-linear Algebra Read more
Achievements:
He is a member of the editorial board of reputable journals like Linear Multilinear algebra. Also has a… Read more
Research Interests:
Matrix methods in statistics, Generalized inverses, Canonical correlations Read more
Achievements:
He was a Senior Researcher of the Academy of Finland in 1992--1995. His main research interest lies on… Read more
We consider the general linear model $y = X \beta + \varepsilon$ supplemented with the new unobservable random vector $ y_{*}$,coming from $ y_{*} = X_{*}\beta + \varepsilon_{*}$, where the covariance matrix of $y_{*}$ is known as well as the cross-covariance matrix between $y_{*}$ and $y$. A linear statistic $F y$ is called linearly sufficient for $X_{*} \beta$ if there exists a matrix $A$ such that $A F y$ is the best linear unbiased estimator, BLUE, for $X_{*} \beta$. The concept of linear sufficiency with respect to a predictable random vector is defined in the corresponding way but considering the best linear unbiased predictor, BLUP, instead of BLUE. In this paper, we consider the linear sufficiency of $F y$ with respect to $ y_{*}$, $X_{*} \beta$, and $\varepsilon_{*}$.
Research Interests:
Game theory, Linear and non-linear programming, matrix theory, applied statistics, operations research Read more
Achievements:
Being a dynamic emeritus professor at the University of Illinois at Chicago has published more than 60 remarkable… Read more
Research Interests:
Complementarity problems on symmetric cones, spectral properties of non-negative matrices and their generalizations Read more
Achievements:
He has published 14 research papers in reputed national and international journals. He has publications in linear algebra… Read more
Let $F=[f_{ij}]$ be an $n \times n$ symmetric matrix. Define $d_{ij}:=f_{ii}+f_{jj}-2f_{ij}$. Now, the matrix $D=[d_{ij}]$ is a called a Euclidean distance matrix (EDM). EDMs have several interesting properties. We introduce a simple generalization of a EDM. Fix $a,b>0$. Define $$E=[a^{2}g_{ii} + b^{2}g_{jj}-2ab g_{ij}],$$ where $G=\left[ g_{ij} \right]$ is a positive semidefinite matrix such that $g\mathbf{1}=0$, where $\mathbf{1}$ is the vector of all ones in $\mathbb{R}^{n}$. We call $E$ a generalized EDM. Despite $E$ being a non-symmetric matrix, many of the interesting properties of a EDM can be extended to a generalized EDM. All these properties will be discussed the talk.
Research Interests:
Read more
Achievements:
Bhaskara Rao Kopparty had Training in Statistics, did research in pure mathematics and mathematical economics and now is… Read more
Research Interests:
Geometry of Banach spaces, Vector Measures, Tensor Product spaces, L^1-predual theory, Choquet theory, Function algebras, and approximation theory Read more
Achievements:
Prof. TSSRK Rao had his undergraduate education in Vijayawada. He obtained B.Sc. and M.Sc. from Andhra University and… Read more
An important direction of investigation in Operator theory of Banach spaces, is, to perform standard Banach space theoretic operations on spaces of operators and ask if the resulting object is again a space of operators (possibly between different Banach spaces). In this talk, we tackle this problem for the well-known geometric operation, Birkhoff-James orthogonality.
For non-reflexive Banach spaces, $X,Y$, for a closed subspace $M$ of operators, we investigate Birkhoff-James orthogonality of an operator $T$ to $M$ with that of $T^{**}$ with an appropriate subspace of $M^{**}$.
Research Interests:
Designs, Combinatorics, Graphs, Cryptography, Discrete Mathematics, Graph Τheory, Discrete Geometry, Coding Theory, Coding Read more
Achievements:
Prof Sharad Sane was born in 1950, completed his MSc 1974 and received his PhD degree in 1979.… Read more
Ryser design can be described as a binary square matrix $A$ for which $A^tA$ equals the sum of a diagonal matrix and a multiple of $J$, the all $1$ matrix with the additional requirement that $A$ has at least two different column sums (equivalently two different row sums). This talk will discuss two different proofs of the Ryser-Woodall theorem (on Ryser designs), the first of which is by Ryser that is combinatorial in nature and the second by Ionin and M.S. Shrikhande that is linear algebraic. This talk will give an account of the known results on Ryser’s conjecture and some recent work in this direction.
Research Interests:
Multivariate Statistics, Growth Curve model with a linearly structured covariance matrix, Kronecker structured covariance matrix Read more
Achievements:
Martin Singull is a Professor, Head of Division of Mathematical Statistics at the Department of Mathematics, Linkoping University.… Read more
In both univariate and multivariate analysis, residuals are used for model diagnostics. In this presentation we consider the MANOVA and the GMANOVA-MANOVA models and different matrix residuals are established. The interpretation of the residuals is discussed and several properties are verified. A realistic, similar to some real studies, but artificial data will be analysed to illustrate how the residuals can be used in model validations.
Research Interests:
Infinite Linear Programming, Generalized Inverses of Operators over Hilbert Spaces, Nonnegative Generalized Inverses, Generalizations of Matrix Monotonicity, Linear… Read more
Achievements:
K. C. Sivakumar is a Professor at Department of Mathematics, Indian Institute of Technology Madras. He obtained his… Read more
Let $\mathcal{S}^n$ denote the space of real symmetric matrices of order $n \times n.$ A linear operator $T$ on $\mathcal{S}^n$ is said to be a $P$-operator, if the following implication holds for every $X \in \mathcal{S}^n$: $$XT(X)=T(X)X \preceq 0 \Longrightarrow X=0,$$ where, for $Y \in \mathcal{S}^n$ we use $Y \succeq 0$ to denote that $Y$ is positive semidefinite, i.e., $u^tYu \geq 0$ for all $u \in \mathbb{R}^n$. Also, $-Y \succeq 0$ is denoted by $Y \preceq 0$.
Let $A \in \mathbb{R}^{n \times n}$ be fixed. The following three maps have been well studied, in the context of the semidefinite linear complementarity problems (SDLCP).
The Lyapunov transformation $L_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, is defined by $$L_A(X)=AX+XA^T,$$ the Stein transformation $S_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, by $$S_A(X)= X-AXA^T$$ and the multiplicative transformation $M_A: \mathcal{S}^n \rightarrow \mathcal{S}^n$, by $$M_A(X)= AXA^T.$$ In [M. S. Gowda, Y. Song(2000), Theorem 5], it is proved that $L_A$ has the $P$-property iff $A$ is positive stable (meaning, all the eigenvalues of $A$ have positive real part). Also, in [M. S. Gowda, T. Parthasarathy(2000), Theorem 6], it is shown that $S_{A}$ has $P$-property iff $A$ is Schur stable (meaning, all the eigenvalues of $A$ lie in the open unit disk). Finally, it is shown in [P. Bhimashankaran, T. Parthasarathy, A. L. N. Murthy, G. S. R. Murthy(2012), Theorem 17] that $M_A$ has $P$-property iff $A$ is either positive definite or negative definite. It is important to observe that these results have some interesting and nontrivial implications to SDLCP.
A linear operator $T$ on $\mathcal{S}^n$ is a $P_{\#}$-operator [introduced in M. Rajesh Kanna, K. C. Sivakumar(2014), Definition 1.3], if for every $X \in \mathcal{S}^n$ we have: $$X \in R(T), XT(X)=T(X)X \preceq 0 \Longrightarrow X=0.$$ For $x,y \in \mathcal{S}^n,$ define $X\circ Y:=\frac{1}{2}(XY+YX).$ We say that the operator $T$ has the Jordan-$P$-property if $$X\circ L(X) \preceq 0 \Longrightarrow X=0.$$
$T$ is said to possess the Jordan-w-$P$-property if $$X\circ L(X) \preceq 0 \Longrightarrow L(X)=0.$$
Finally, $T$ is said to satisfy the w-$P$-property if $$XL(X) =L(X)X, ~X\circ L(X) \preceq 0 \Longrightarrow L(X)=0.$$ The three notions given in the previous para were proposed in [J Tao(2009)]. The results stated earlier for the Lyapunov, Stein and the multiplicative transformations, have been shown to have analogues for operators satisfying any of the four latter properties defined above (see, for instance [I. Jeyaraman, Kavita Bisht, K. C. Sivakumar(2017)] and [J. Tao(2009)]).
In this talk, first we give a brief survey. We then present some new relationships between operators belonging to these classes.
Let $T$ be a tree on $n$ vertices and let $\mathscr{L}^T_q$ be the $q-$analogue of its Laplacian. For a partition $\lambda \vdash n$, let the normalized immanant of $\mathscr{L}^T_q$ indexed by $\lambda$ be denoted as $\overline{\text{Imm}}_\lambda(\mathscr{L}^T_q)$. A string of inequalities among $\overline{\text{Imm}}_\lambda(\mathscr{L}^T_q)$ is known when $\lambda$ varies over hook partitions of $n$ as the size of the first part of $\lambda$ decreases. In this work, we show a similar sequence of inequalities when $\lambda$ varies over two row partitions of $n$ as the size of the first part of $\lambda$ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of $\mathfrak{S}$ indexed by two row partitions.
Our proof can be interpreted using the combinatorics of Riordan paths and our main lemma admits a nice probabilisitic interpretation involving peaks at odd heights in generalized Dyck paths or equivalently involving special descents in Standard Young Tableaux with two rows. As a corollary, we also get inequalities between $\overline{\text{Imm}}_\lambda(\mathscr{L}^{T_1}_q)$ and $\overline{\text{Imm}}_\lambda(\mathscr{L}^{T_1}_q)$ when $T_1$ and $T_2$ are comparable trees in the $\text{GTS}_n$ poset and when $\lambda_1$ and $\lambda_2$ are both two rowed partitions of $n$, with $\lambda_1$ having a larger first part than $\lambda_2$.
The adjacency matrix of the $n$-cube and the closely related tridiagonal matrix of Mark Kac have an elegant spectral theory that arise in a variety of applications. This paper defines $q$-analogs of these two matrices and studies their spectral theory.
We consider two applications: a $q$-analog of the random walk on the $n$-cube and a $q$-analog of the product formula for the number of rooted spanning trees of the $n$-cube.
Research Interests:
Matrices and statistics, with particular emphasis on canonical correlations, canonical efficiHadamard products, Matrix inequalities, Matrix partial orderings, Matrix… Read more
Achievements:
George P. H. Styan has a long and honourable career in Mathematics. He has served on the Editorial… Read more
carams.mahe@gmail.com