Related papers: Quantization for uniform distributions on hexagona…

In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\sigma_n \mathbb{P}(S_n\!=\!j)\leq\eta$ where $\sigma_n$ denotes the…

This paper presents a methodology for finding numerically, by means of curve-following, all real solutions of a general system of $n$ nonlinear equations in $n$ unknowns, within a given $n$-dimensional box. The main idea behind our method…

Uniform probability distributions on $\ell_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a…

Let $k$ be a number field. We study how well can finite sets of $\mathcal O_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in…

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows…

For each k >= 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce m(k) = max[(k-1)!/ 2, 1] packings of h(k) equal disks inside a circle which we call "the curved hexagonal packings". The curved hexagonal packing of 7 disks…

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a…

This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in…

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether $n$ unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or…

We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a…

A probability distribution over {-1, 1}^n is (eps, k)-wise uniform if, roughly, it is eps-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (eps, k)-wise uniform distribution can…

We consider the problems of computing the average degree and the size of a given network in a distributed fashion under quantized communication. We present two distributed algorithms which rely on quantized operation (i.e., nodes process…

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is…

The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite dimensional Hilbert space. It allows us, for example, to define a distance…

We use computational experiments to find the rectangles of minimum perimeter into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. In many of the packings…

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the…

In this work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and…

We considered the problem of obtaining upper bounds for the mathematical expectation of the $q$-norm ($2\leqslant q \leqslant \infty$) of the vector which is uniformly distributed on the unit Euclidean sphere. We finish the paper with…

The standard method of transforming a continuous distribution on the line to the uniform distribution on the unit interval is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, in that, for each…

We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media…