arXiv subjects

Alexander Kushkuley

Publications and source records attributed to Alexander Kushkuley.

9 recordsLinked to original sources

arXiv2026

Absolute values and tensor powers of irreducible characters

Let $ \chi $ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(|\chi|/\chi(1))^{t} $ in terms of character ratios $ (|\chi(g)|/\chi(1))^{t}, \; g \in G, \; t \geq 0 $. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the subject of growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations. Similar type of reasoning can be applied to some questions related to commuting probability. In particular, we obtain an analogue of Frobenis formula for the probability of an "event" $ |\chi( [x,y]^{-1}g)| = r $

math.NA↗

arXiv2025

Identities of Irreducible Representations and Gassmann Equivalence

Identities of complex irreducible representations of finite groups can be explicitly constructed from character value sets. Among other things, these identities determine representations up to Gassmann equivalency. Some examples of identities related to spherical space forms and to representations of finite $p$-groups are presented. Some old results on irreducible representations with the same identities are revisited

math.RT↗

arXiv2025

Some Remarks on Commuting Probability

We introduce a weighted sum of irreducible character ratios as an estimator for commutator probabilities. The estimator yields Frobenius formula when applied to a regular representation

math.NA↗

arXiv2023

Some Remarks on Random Vectors and $O(n)$-Invariants

Computations involving invariant random vectors are directly related to the theory of invariants (cf. e.g \cite{Weing_1}). Some simple observations along these lines are presented in this paper. We note in particular that sum of elements of the standard basis of $ O(n)$-invariants is equal to the expectation of a random Veronese tensor up to a known scalar multiplier.

math.NA↗

arXiv2023

Improving Recommendation Relevance by simulating User Interest

Most if not all on-line item-to-item recommendation systems rely on estimation of a distance like measure (rank) of similarity between items. For on-line recommendation systems, time sensitivity of this similarity measure is extremely important. We observe that recommendation "recency" can be straightforwardly and transparently maintained by iterative reduction of ranks of inactive items. The paper briefly summarizes algorithmic developments based on this self-explanatory observation. The basic idea behind this work is patented in a context of online recommendation systems.

math.NA↗

arXiv2021

A Remark on Random Vectors and Irreducible Representations

The expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on a unit sphere in $\mathbb{R}^n $ is equal to $1/n$. We show that this is a characteristic property of random unit vectors defined on invariant probability subspaces of irreducible representations of compact Lie groups. We also discuss a relation of this fact to some properties of random invariant tensors

math.PR↗

arXiv2019

Heavy Hitters and Bernoulli Convolutions

A very simple event frequency approximation algorithm that is sensitive to event timeliness is suggested. The algorithm iteratively updates categorical click-distribution, producing (path of) a random walk on a standard $n$-dimensional simplex. Under certain conditions, this random walk is self-similar and corresponds to a biased Bernoulli convolution. Algorithm evaluation naturally leads to estimation of moments of biased (finite and infinite) Bernoulli convolutions.

math.NA↗

arXiv2017

Block Approximation of Tall Sparse Matrices and Block-Givens Rotations

Estimation of top singular values is one of the widely used techniques and one of the intensively researched problems in Numerical Linear Algebra and Data Science. We consider here two general questions related to this problem: How top singular values are affected by zeroing out a sparse rectangular block of a matrix? How much top singular values differ from top column norms of a tall sparse non-negative matrix ?

math.NA↗

arXiv2014

A Note on Lerner Index, Cross-Elasticity and Revenue Optimization Invariants

We study common properties of retail pricing models within a general framework of calculus of variations. In particular, we observe that for any demand model, optimal de-seasoned revenue rate divided by price elasticity is time invariant. We also obtain a generalization of a well known inverse relationship between price elasticity of demand and Lerner index. These invariance results are illustrated by two contrasting examples of markdown optimization and optimal continuous replenishment

math.OC↗