Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials $f\left(X\right)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f\left(a\right)$ and $f\left(b\right)$. In particular, we obtain irreducibility criteria for the case where $f\left(a\right)$ and $f\left(b\right)$ have few prime factors, and $f$ is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.     

Mapping properties of the zero-balanced hypergeometric functions

In this paper, the order of convexity of $z_2{F}_1(a,b;c;z)$ is given under certain conditions on the positive real parameters a, b and c. We show also that the image domains of the unit disc $\mathbb{D}$ under some shifted zero-balanced hypergeometric functions $z_2{F}_1(a,b;a+b;z)$ are convex and bounded by two horizontal lines. This solves a problem raised by Ponnusamy and Vuorinen in [10].     

Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles

For any graph $G$ with $a,b\in V\left(G\right)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V\left(S(G,a,b)\right)$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles.     

Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let $g(f):={\sum }_{j=1}^k{b}_j(z)f(z+c_j)$ be a shift polynomial of f with small meromorphic coefficients $b_j$ in the sense of $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided α, β, $b_0$ are similar small meromorphic functions, we consider zero distribution of $f^n{(g(f))}^s?b_0$, resp. of $g(f)?\alpha f^n?\beta$.     

Toughness condition for the existence of all fractional (a,b,k)-critical graphs

In this article, we obtain a sufficient condition related to toughness $\tau \left(G\right)$ for a graph to be all fractional $(a,b,k)$-critical. We prove that if $\tau \left(G\right)\ge \frac{(b^2?1)+ak}{a}$ for some nonnegative integers $a,b,k$, then $G$ is all fractional $(a,b,k)$-critical. Our result improves the known results in Liu and Zhang (2008) and Liu and Cai (2009).     

Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

Generalized traveling-wave solutions of nonlinear reaction–diffusion equations with delay and variable coefficients

The paper presents a number of new exact solutions to nonlinear reaction–diffusion equations with delay of the form $c\left(x\right)u_t={\left[a\left(x\right)u_x\right]}_x+b\left(x\right)F(u,w),w=u(x,t-\tau ),$where $\tau >0$ is the delay time, and $F(u,w)$ is an arbitrary function of two arguments. Solutions are sought in the form of a generalized traveling-wave, $u=U\left(z\right)$ with $z=t+\theta \left(x\right)$. It is shown that one of the two functional coefficients $a\left(x\right)$ and $b\left(x\right)$ of the equation considered can be specified arbitrarily. Examples of delay reaction–diffusion equations and their solutions are given. New exact solutions of few other nonlinear delay PDEs are also obtained.     

On strictly Deza graphs with parameters (n,k,k−1,a)

A non-empty $k$-regular graph $\Gamma$ on $n$ vertices is called a Deza graph if there exist constants $b$ and $a$ $(b\ge a)$ such that any pair of distinct vertices of $\Gamma$ has either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\Gamma$ and are written as the quadruple $(n,k,b,a)$. If a Deza graph has diameter 2 and is not strongly regular, then it is called a strictly Deza graph. In the present paper, we investigate strictly Deza graphs whose parameters $(n,k,b,a)$ satisfy the conditions $k=b+1$ and $\frac{k(k-1)-a(n-1)}{b-a}>1$.     

No-three-in-line problem on a torus: Periodicity

Let ${\tau }_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m\times n$ with no three collinear points. The value ${\tau }_{m,n}$ is known for the case where $gcd(m,n)$ is prime. It is also known that ${\tau }_{m,n}\le 2gcd(m,n)$. In this paper we generalize some of the known tools for determining ${\tau }_{m,n}$ and also show some new. Using these tools we prove that the sequence ${\left({\tau }_{z,n}\right)}_{n\in \mathbb{N}}$ is periodic for all fixed $z>1$. In general, we do not know the period; however, if $z=p^a$ for $p$ prime, then we can bound it. We prove that ${\tau }_{p^a,p^{(a?1)p+2}}=2p^a$ which implies that the period for the sequence is $p^b$, where $b$ is at most $(a?1)p+2$.     

The covering radius of the Reed–Muller code RM(2,7) is 40

It was proved by J. Schatz that the covering radius of the second order Reed–Muller code $RM(2,6)$ is 18 (Schatz (1981)). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for the covering radius of $RM(2,n)$, $n=8,9,10$.     

Constructing sparse Davenport–Schinzel sequences

For any sequence $u$, the extremal function $Ex(u,j,n)$ is the maximum possible length of a $j$-sparse sequence with $n$ distinct letters that avoids $u$. We prove that if $u$ is an alternating sequence $abab\dots$ of length $s$, then $Ex(u,j,n)=\Theta \left(s{n}^2\right)$ for all $j\ge 2$ and $s\ge n$, answering a question of Wellman and Pettie (2018) and extending the result of Roselle and Stanton that $Ex(u,2,n)=\Theta \left(s{n}^2\right)$ for any alternation $u$ of length $s\ge n$ (Roselle and Stanton, 1971).Wellman and Pettie also asked how large must $s\left(n\right)$ be for there to exist $n$-block $DS(n,s\left(n\right))$ sequences of length $\Omega \left(n^{2-o\left(1\right)}\right)$. We answer this question by showing that the maximum possible length of an $n$-block $DS(n,s\left(n\right))$ sequence is $\Omega \left(n^{2-o\left(1\right)}\right)$ if and only if $s\left(n\right)=\Omega \left(n^{1-o\left(1\right)}\right)$. We also show related results for extremal functions of forbidden 0–1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters.     

Separable discrete functions: Recognition and sufficient conditions

A discrete function of $n$ variables is a mapping $g:X_1\times \dots \times X_n\to A$, where $X_1,\dots ,X_n$, and $A$ are arbitrary finite sets. Function $g$ is called separable if there exist $n$ functions $g_i:X_i\to A$ for $i=1,\dots ,n$, such that for every input $x_1,\dots ,x_n$ the function $g(x_1,\dots ,x_n)$ takes one of the values $g_1\left(x_1\right),\dots ,g_n\left(x_n\right)$. Given a discrete function $g$, it is an interesting problem to ask whether $g$ is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of $n$ variables is known only for $n=2$. In this paper we will show that a slightly more general recognition problem, when $g$ is not fully but only partially defined, is NP-complete for $n\ge 3$. We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for $n\ge 4$.The general recognition problem contains the above mentioned special case for $n=2$. This case is well-studied in the context of game theory, where (separable) discrete functions of $n$ variables are referred to as (assignable) $n$-person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of $n$ variables for any $n$, thus generalizing the above result known for $n=2$. Our proof is constructive. Using a graph-based discrete algorithm we show how for a given weakly totally tight (partially defined) discrete function $g$ of $n$ variables one can construct separating functions $g_1,\dots ,g_n$ in polynomial time with respect to the size of the input function.