arXiv subjects

Kiran Kumar

Publications and source records attributed to Kiran Kumar.

4 recordsLinked to original sources

arXiv2026

Ergodicity in discrete-time quantum walks

We undertake a detailed analysis of ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result of our paper holds in dimension one, and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk, and the equidistribution of its dynamics in position space, which appears for the first time in the context of large-volume quantum ergodicity. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub ``No Repeating Graphs'', and we distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). Many examples are included to illustrate the criterion and to distinguish between the types of ergodicity.

math-ph↗

arXiv2026

Semicircle law for multi-parameter random simplicial complexes

In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct probabilities. For $n,d \in \mathbb{N}$ and $\mathbf{p}=(p_1,p_2,\ldots, p_d)\in (0,1]^d$, the multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ is constructed inductively. Starting with $n$ vertices, edges (1-cells) are included independently with probability $p_1$, yielding the Erd\H{o}s-R\'enyi graph $G(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$. We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p})$, under the assumptions $\min_{i=1,\ldots d-1}\liminf p_i >0$ and $np_d \rightarrow \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among their entries. We prove that the empirical spectral measures of both matrices converge weakly to the semicircle law in probability. We also prove that the limiting spectral measure of the signed and unsigned adjacency matrices of another associated model, the multi-parameter upper model, is the semicircle law, under the assumption $np_d(1-p_d) \rightarrow \infty$. Further, we establish convergence results for the number of $(d-1)$-cells and the number of maximal $(d-1)$-cells of multi-parameter simplicial complexes.

math.PR↗

arXiv2012

Map / Reduce Deisgn and Implementation of Apriori Alogirthm for handling voluminous data-sets

Apriori is one of the key algorithms to generate frequent itemsets. Analyzing frequent itemset is a crucial step in analysing structured data and in finding association relationship between items. This stands as an elementary foundation to supervised learning, which encompasses classifier and feature extraction methods. Applying this algorithm is crucial to understand the behaviour of structured data. Most of the structured data in scientific domain are voluminous. Processing such kind of data requires state of the art computing machines. Setting up such an infrastructure is expensive. Hence a distributed environment such as a clustered setup is employed for tackling such scenarios. Apache Hadoop distribution is one of the cluster frameworks in distributed environment that helps by distributing voluminous data across a number of nodes in the framework. This paper focuses on map/reduce design and implementation of Apriori algorithm for structured data analysis.

cs.DC↗

arXiv2012

Korovkin results and Frobenius optimal approximants for infinite dimensional bounded linear operators

The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type theorems are proved for convergence of completely positive maps with respect to weak, strong and uniform clustering of sequences of matrices of growing order. Such modes of convergence were originally considered for Toeplitz matrices (see [23],[26]). As an application, we translate the Korovkin-type approach used in the finite dimensional case, in the setting of preconditioning large linear systems with Toeplitz structure, into the infinite dimensional context of operators acting on separable Hilbert spaces. The asymptotic of these pre-conditioners are obtained and analyzed using the concept of completely positive maps. It is observed that any two limit points of the same sequence of pre-conditioners are the same modulo compact operators. Finally, we prove the generalized versions of the Korovkin type theorems in [23]. Keywords: Completely positive maps, Frobenius norm, Pre-conditioners.

math.FA↗