Related papers: Gaps of Binary Numerical Semigroups and of Binary …

The finite ring Z_k = Z(+,.) mod p^k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units G_k in Z_k(.) has order (p-1)p^{k-1}, implying product structure G_k = A_k B_k. Here core A_k of order p-1 is an…

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…

For a prime p and base b, the digit function delta(r) = floor(br/p) partitions the residues {1, ..., p-1} into b contiguous bins. The collision count C(g) records how many residues share a bin with their image under multiplication by g. We…

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

The singleton and doubleton minors of a polymatroid $\rho$ encode a surprising amount of information about the structural complexity of $\rho$. Given any polymatroid $\rho$, we can subtract from it a maximally-separated polymatroid,…

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general…

For a set $\cM=\{-\mu,-\mu+1,\ldots, \lambda\}\setminus\{0\}$ with non-negative integers $\lambda,\mu<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(\lambda,\mu;q)$-\emph{covering…

We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces $$ [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2} $$ in the off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$…

Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary…

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that,…

Let $1\le p_0<p,q <q_0\le \infty$. Given a pair of weights $(w,\sigma)$ and a sparse family $\mathcal S$, we study the two weight inequality for the following bi-sublinear form \[ B(f, g)= \sum_{Q\in\mathcal S}\langle…

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial…

Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If…

Given a $(0,p)$-mixed characteristic complete discrete valued field $\mathcal{K}$ we define a class of finite field extensions called \emph{pseudo-perfect} extensions such that the natural restriction map on the mod-$p$ Milnor $K$-groups is…

We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

We study decomposable combinatorial labeled structures in the exp-log class, specifically, two examples of type a=1 and two examples of type a=1/2. Our approach is to establish how well existing theory matches experimental data. For…

In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $$…

Let $\mathcal{F}$ be a family of subsets of $[n]$ and $L$ be a subset of $[n]$. We say $\mathcal{F}$ is an $L$-differencing Sperner system if $|A\setminus B|\in L$ for any distinct $A,B\in\mathcal{F}$. Let $p$ be a prime and $q$ be a power…

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle…