Existence and non-existence results for strong external difference families

We consider strong external difference families (SEDFs); these are external difference families satisfying additional conditions on the patterns of external differences that occur, and were first defined in the context of classifying optimal strong algebraic manipulation detection codes. We establish new necessary conditions for the existence of $(n,m,k,\lambda )$-SEDFs; in particular giving a near-complete treatment of the $\lambda =2$ case. For the case $m=2$, we obtain a structural characterization for partition type SEDFs (of maximum possible $k$ and $\lambda$), showing that these correspond to Paley partial difference sets. We also prove a version of our main result for generalized SEDFs, establishing non-trivial necessary conditions for their existence.     

Toughness,binding number and restricted matching extension in a graph

A connected graph $G$ with at least $2m+2n+2$ vertices is said to satisfy the property $E(m,n)$ if $G$ contains a perfect matching and for any two sets of independent edges $M$ and $N$ with $\left|M\right|=m$ and $\left|N\right|=n$ with $M\cap N=?$, there is a perfect matching $F$ in $G$ such that $M?F$ and $N\cap F=?$. In particular, if $G$ is $E(m,0)$, we say that $G$ is $m$-extendable. One of the authors has proved that every $m$-tough graph of even order at least $2m+2$ is $m$-extendable (Plummer, 1988). Chen (1995) and Robertshaw and Woodall (2002) gave sufficient conditions on binding number for $m$-extendability. In this paper, we extend these results and give lower bounds on toughness and binding number which guarantee $E(m,n)$.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.     

Parameters of connectivity in (a,b)-linear graphs

For some a and b positive rational numbers, a simple graph with n vertices and $m=an-b$ edges is an $(a,b)$-linear graph, when $n>2b$. We characterize non-empty classes of $(a,b)$-linear graphs and determine those which contain connected graphs. For non-empty classes, we build sequences of $(a,b)$-linear graphs and sequences of connected $(a,b)$-linear graphs. Furthermore, for each of these sequences where every graph is bounded by a constant, we show that its correspondent sequence of diameters diverges, while its correspondent sequence of algebraic connectivities converges to zero.     

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number $\mathrm{adrn}(G)$ is the least k such that every set of k dacards determines G. We determine $\mathrm{adrn}(D_{m,n,p})$, where the double-broom $D_{m,n,p}$ with $p\ge 2$ is the tree with $m+n+p$ vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine $\mathrm{adrn}(D_{m,n,p})$ for all $m,n,p$. For $2\le m\le n$, usually $\mathrm{adrn}(D_{m,n,p})=m+2$, except $\mathrm{adrn}(D_{m,m+1,p})=m+1$ and $\mathrm{adrn}(D_{m,m+2,p})=m+3$. There are exceptions when $(m,n)=(2,3)$ or $p=4$. For $m=1$ the usual value is 4, with exceptions when $p\in \{2,3\}$ or $n=2$.     

Balanced diagonals in frequency squares

We say that a diagonal in an array is $\lambda$-balanced if each entry occurs $\lambda$ times. Let $L$ be a frequency square of type $F(n;\lambda )$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m=n∕\lambda \}$ occurs $\lambda$ times per row and $\lambda$ times per column. We show that if $m?3$, $L$ contains a $\lambda$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m?4$ and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.     

Lower bounds for the Hilbert number of polynomial systems

Let $H(m)$ denote the maximal number of limit cycles of polynomial systems of degree m. It is called the Hilbert number. The main part of Hilbert?s 16th problem posed in 1900 is to find its value. The problem is still open even for $m=2$. However, there have been many interesting results on the lower bound of it for $m?2$. In this paper, we give some new lower bounds of this number. The results obtained in this paper improve all existing results for all $m?7$ based on some known results for $m=3,4,5,6$. In particular, we obtain that $H(m)$ grows at least as rapidly as $\frac{1}{2\ln 2}{(m+2)}^2\ln (m+2)$ for all large m.     

A panconnectivity theorem for bipartite graphs

Let $G$ be a simple $m\times n$ bipartite graph with $m\ge n$. We prove that if the minimum degree of $G$ satisfies $\delta \left(G\right)\ge m∕2+1$, then $G$ is bipanconnected: for every pair of vertices $x,y$, and for every appropriate integer $2\le ?\le 2n$, there is an $x,y$-path of length $?$ in $G$.     

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition, where $m\in [n,\infty )$ and $K\ge 0$ are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition and on compact manifolds equipped with $(?K,m)$-super Ricci flows.     

Waring’s problem for integral quaternions

A (Lipschitz) integral quaternion is a Hamiltonian quaternion of the form $a+bi+cj+dk$ with all of $a,b,c,d\in \mathbb{Z}$. In 1946, Niven showed that the integral quaternions expressible as a sum of squares of integral quaternions are precisely those for which $2\mid b,c,d$; moreover, all of these are expressible as sums of three squares. Now let $m$ be an integer with $m>2$, and suppose $2^r\parallel m$. We show that if $r=0$ (i.e., $m$ is odd), then all integral quaternions are sums of $m$th powers, while if $r>0$, then the integral quaternions that are sums of $m$th powers are precisely those for which $2^r\mid b,c,d$ and $2^{r+1}\mid b+c+d$. Moreover, all of these are expressible as a sum of $g\left(m\right)$$m$th powers, where $g\left(m\right)$ is a positive integer depending only on $m$.     

Homomorphism complexes and k-cores

For any fixed graph $G$, we prove that the topological connectivity of the graph homomorphism complex Hom($G,K_m$) is at least $m?D\left(G\right)?2$, where $D\left(G\right)={max}_{H?G}\delta \left(H\right)$, for $\delta \left(H\right)$ the minimum degree of a vertex in a subgraph $H$. This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree $\Delta \left(G\right)$ was used in place of $D\left(G\right)$, and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, $\chi \left(G\right)\le D\left(G\right)+1$, as $\chi \left(G\right)=min\{m:\text{Hom}(G,K_m)\ne ?\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c∕n$ for a fixed constant $c>0$.