Related papers: (2^n,2^n,2^n,1)-relative difference sets and their…

Recent experimental work has demonstrated the potential to combine the merits of diffractive and on-chip photonic information processing devices in a single chip by making use of planar (or slab) waveguides. Researchers have adapted key…

For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan…

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function…

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the…

Given an entire function $f$ of finite order $\rho$, let $L(z,f)=\sum_{j=0}^{m}b_{j}(z)f^{(k_{j})}(z+c_{j})$ be a linear delay-differential polynomial of $f$ with small coefficients in the sense of $O(r^{\lambda+\varepsilon})+S(r,f)$,…

We show that any function $f:\mathbb{H}^n\to\mathbb{H}$ with $f(z+c)=f(z)+c$, $z\in\mathbb{H}^n$, for some $c>0$ has a property that any limit function of a family $\{\frac{f(tz)}{t}\}_{t>0}$ when $t\to\infty$ is linear.

The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of…

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $\xi\in\mathbb{R}^n$ the function…

We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq…

In the paper, the author expresses the difference $2^m\bigl[\zeta\bigl(-m,\frac{1+x}{2}\bigr)-\zeta\bigl(-m,\frac{2+x}{2}\bigr)\bigr]$ in terms of a linear combination of the function $\Gamma(m+1){\,}_2F_1(-m,-x;1;2)$ for $m\in\mathbb{N}_0$…

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which…

Let $m$ be a positive integer and $p$ a prime. In this paper, we investigate the differential properties of the power mapping $x^{p^m+2}$ over $\mathbb{F}_{p^n}$, where $n=2m$ or $n=2m-1$. For the case $n=2m$, by transforming the derivative…

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory…

Fix an integer $h \geq 2$, and let $b_1, \ldots, b_h$ be (not necessarily distinct) positive integers with $\gcd(b_1, \ldots, b_h) = 1$. For any subset $A \subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1, \ldots,…

Let $a, b\in \mathbb{N}$ be relatively prime. We consider $(a-1)(b-1)/2$, which arises in the study of the $pq$-th cyclotomic polynomial, where $p,q$ are distinct primes. We prove two possible representations of $(a-1)(b-1)/2$ as…

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d)…

We present a mathematical model: dynamical systems over finite sets (DSF), and we show that Boolean and discrete genetic models are special cases of DFS. In this paper, we prove that a function defined over finite sets with different number…

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show…

Dillon observed that an APN function $F$ over $\mathbb{F}_2^{n}$ with $n$ greater than $2$ must satisfy the condition $\{F(x) + F(y) + F(z) + F(x + y + z) \,:\, x,y,z \in\mathbb{F}_2^n\}= \mathbb{F}_2^n$. Recently, Taniguchi (2023)…

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over $\mathbb{F}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over $\mathbb{F}_{2^r}$ is…