Related papers: Analysis of the Ratio $D(n)/n$

In this note we explore duality in reverse convex optimization with reverse convex inequality constraints. While we are examining the special case of a finite index set of the inequality constraints, we are primarily interested in the…

This note examines the infinite divisibility of density-based transformations of normal random variables. We characterize a class of density-based transformations of normal variables which produces non-infinitely divisible distributions. We…

Deep neural networks (DNNs) have become an enabling component for a myriad of artificial intelligence applications. DNNs have shown sometimes superior performance, even compared to humans, in cases such as self-driving, health applications,…

Some limit theorems of the type $\int_{\Omega}f_n dm_n -- --> \int_{\Omega}f dm$ are presented for scalar, (vector), (multi)-valued sequences of m_n-integrable functions f_n. The convergences obtained, in the vector and multivalued…

Let x(n) be a recurrence relation. The main purpose of this article is to determine a recurrence for powers of x(n).

We revisit the problem of radial pulsations of neutron stars by computing four general-relativistic polytropic models, in which "density" and "adiabatic index" are involved with their discrete meanings: (i) "rest-mass density" or (ii)…

Divergence estimators based on direct approximation of density-ratios without going through separate approximation of numerator and denominator densities have been successfully applied to machine learning tasks that involve distribution…

Let $\mathcal{L}$ be the closure of the set of all real numbers $\alpha$, such that there exist infinitely many integers $n$, such that $\alpha=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower…

We consider combinatorial properties of the Mills' ratio, and explore the interplay between a continued fraction expansion for the Mills' ratio, the Laplace polynomials and a new family of combinatorial identities.

We introduce the notion of information ratio $\text{Ir}(H/G)$ between two (simple, undirected) graphs $G$ and $H$, defined as the supremum of ratios $k/n$ such that there exists a mapping between the strong products $G^k$ to $H^n$ that…

Abstract differential-algebraic equations (ADAEs) of a semilinear type are studied. Theorems on the existence and uniqueness of solutions and the maximal interval of existence, on the global solvability of the ADAEs, the boundedness of…

Let $d_N=ND_N(\omega)$ be the discrepancy of the Van der Corput sequence in base $2$. We improve on the known bounds for the number of indices $N$ such that $d_N\leq \log N/100$. Moreover, we show that the summatory function of $d_N$…

For a positive integer $n$, if $\sigma(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $\sigma(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about…

Some examples are easier for humans to classify than others. The same should be true for deep neural networks (DNNs). We use the term example perplexity to refer to the level of difficulty of classifying an example. In this paper, we…

For each element of certain families of integer sequences, we study the term-wise ratios of the Hankel transforms of three sequences related to that element by series reversion. In each case, the ratios define well-known sequences, and in…

The data processing inequality (DPI) is a scalar inequality satisfied by distinguishability measures on density matrices. For some distinguishability measures, saturation of the scalar DPI implies an operator equation relating the arguments…

In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…

We study the properties of a general continued fraction of Ramanujan. In some certain cases we evaluate it completely.

In the standard formulation of the occupancy problem one considers the distribution of r balls in n cells, with each ball assigned independently to a given cell with probability 1/n. Although closed form expressions can be given for the…