Related papers: The probability that a subspace contains a positiv…

Given two vectors in Euclidean space, how unlikely is it that a random vector has a larger inner product with the shorter vector than with the longer one? When the random vector has independent, identically distributed components, we…

The probability behavior in the m+1 relativistic space-time is considered. The probability, which is defined by the relativistic m+1-vector of the probability density, is investigated.

An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

We investigate the probability that a random quadratic form in ${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result…

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$,…

Quantitative bounds for random embeddings of $\mathbb{R}^{k}$ into Lorentz sequence spaces are given, with improved dependence on $\varepsilon$.

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

In this paper we show that the closure of a random sample for a k-dimensional random vector is almost surely a deterministic set of all heavy points of the distribution. A heavy point is defined to be a point for which all its neighborhoods…

We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and…

A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real…

Randomness is a ubiquitous phenomenon that is practically accompanied by physical events described by probability theory. However, probability by definition in the theory is a nonnegative scalar quantity. Here, we propose the concept of…

The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…

We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as…

We consider positivity properties of vector bundles over conic neighbourhoods of sections of submersions over $1$-convex manifolds.

Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in $\mathbb{R}^n$ obtained from the partitions of the fixed positive integer $n$. These distributions arise naturally when…

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a…

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the…

In this paper we find the generating function for the number of vertices which have $k$ elements in their subtree and use this generating function to calculate the probability that a vertex has a size $k$ subtree. We also show how this same…

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. Define a real-valued weight function on the $1$-dimensional vector spaces of $V$ such that the sum of all weights is zero. Let the weight of a subspace $S$ be…