Related papers: Dense sets of integers with prescribed representat…

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…

An upper mean here is a subadditive functional $\overline M$ defined on bounded functions on a commutative group which has, beside some natural requirements, the property we call restricted additivity: if $g(x)= f(x)+f(x+t)$, then…

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers…

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with…

For a fixed rational number g different from -1,0,1 and integers a and d the set N_g(a,d) of primes p for which the order of g(mod p) is congruent to a(mod d) is considered. It is shown, assuming the Generalized Riemann Hypothesis (GRH),…

Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by…

For $\alpha >0$, let $$\mathscr{A}=\{ a_1<a_2<a_3<\cdots\}$$ and $$\mathscr{L}=\{ \ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not~necessarily~different)}$$ be two sequences of positive integers with $\mathscr{A}(m)>(\log m)^\alpha $ for…

For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that…

For a finite group $G$, the representation dimension is the smallest integer realizable as the degree of a complex faithful representation of $G$. In this article, we compute representation dimension for some $p$-groups, their direct…

Let $U$ be a Lucas sequence, $p$ be prime, and $\rho_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid \rho_U(p)$, where $d\geq 1$ is a fixed integer. Our…

Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ functions with compact support on a forward geodesically complete Finsler manifold $(M, F)$, is dense…

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…

An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest…

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a…

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures;…

For $g\in \mathbb{N}$, let $G=\Sp(2g,\mathbb{Z})$ be the integral symplectic group and $S(g)$ be the set of all positive integers which can occur as the order of an element in $G$. In this paper, we show that $S(g)$ is a bounded subset of…

We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)\zeta(4+ix)}{7\cosh\pi\frac{x-t}{7}}\,dx, \qquad Z(t)=\frac{\Re…

For any positive integer $p\geq 3$, let $A$ be a proper subset of $\{0,1,\ldots, p-1\}$ with $\sharp A=s\geq 2$. Suppose $h: \{0,1,\ldots,s-1\}\to A$ is a one-to-one map which is strictly increasing with $A=\{h(0),h(1),\ldots,h(s-1)\}$. We…