A pairwise almost disjoint subspace of kN of maximal dimension

We show that if k is an infinite field, then there exists a subspace $W?k^{\mathbb{N}}$ of dimension $|k{|}^{{\mathrm{?}}_0}$, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.     

A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

For a martingale $M$ starting at $x$ with final variance ${\sigma }^2$, and an interval $(a,b)$, let $\Delta =\frac{b?a}{\sigma }$ be the normalized length of the interval and let $\delta =\frac{|x?a|}{\sigma }$ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of $(a,b)$ by $M$ is at most $\frac{\sqrt{1+{\delta }^2}?\delta }{2\Delta }$ if ${\Delta }^2\le 1+{\delta }^2$ and at most $\frac{1}{1+{(\Delta +\delta )}^2}$ otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of $(a,b)$ for submartingales with the corresponding final distribution. Each of these two bounds is at most $\frac{\sigma }{2(b?a)}$, with equality in the first bound for $\delta =0$. The upper bound $\frac{\sigma }{2}$ on the length covered by $M$ during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound $\sigma$ on the expected maximum of $M$ above $x$, the Dubins & Schwarz sharp upper bound $\sigma \sqrt{2}$ on the expected maximal distance of $M$ from $x$, and the Dubins, Gilat & Meilijson sharp upper bound $\sigma \sqrt{3}$ on the expected diameter of $M$.     

d-matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A $d$-matching in a 3-uniform hypergraph $H$ is a matching of size $d$. Let $V_1,V_2$ be a partition of $n$ vertices such that $\left|V_1\right|=2d?1$ and $\left|V_2\right|=n?2d+1$. Denote by $E_3(2d?1,n?2d+1)$ the 3-uniform hypergraph with vertex set $V_1\cup V_2$ consisting of all those edges which contain at least two vertices of $V_1$. Let $H$ be a 3-uniform hypergraph of order $n\ge 9d^2$ such that $deg\left(u\right)+deg\left(v\right)>2\left[\left(\genfrac{}{}{0ex}{}{n?1}{2}\right)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)\right]$ for any two adjacent vertices $u,v\in V\left(H\right)$. In this paper, we prove $H$ contains a $d$-matching if and only if $H$ is not a subgraph of $E_3(2d?1,n?2d+1)$.     

2p-cycle decompositions of some regular graphs and digraphs

In this paper, we consider $2k$-cycle decomposition of $K_m\times K_n$ and directed $2k$-cycle decompositions of ${(K_m°{\overline{K}}_n)}^1$ and ${(K_m\times K_n)}^1$, where $°$ and $\times$ denote the wreath product and tensor product of graphs, respectively. Using the results obtained here, we prove that for $m,n\ge 3$, the obvious necessary conditions for the existence of a $C_{2k}$-decomposition of $K_m\times K_n$ are sufficient whenever $k\in \{p,2^{?}\},$ where $p$ is a prime and $?\ge 2$. Also, we show that the necessary conditions for the existence of ${\overrightarrow{C}}_{2p}$-decompositions of ${(K_m°{\overline{K}}_n)}^1$ and ${(K_m\times K_n)}^1$ are sufficient whenever $p$ is a prime, where ${\overrightarrow{C}}_{2p}$ denotes the directed cycle of length $2p$.     

Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

Characterization of the induced matching extendable graphs with 2n vertices and 3n edges

A graph $G$ is induced matching extendable or IM-extendable if every induced matching of $G$ is contained in a perfect matching of $G$. In 1998, Yuan proved that a connected IM-extendable graph on $2n$ vertices has at least $3n?2$ edges, and that the only IM-extendable graph with $2n$ vertices and $3n?2$ edges is $T\times K_2$ , where $T$ is an arbitrary tree on $n$ vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with $2n\ge 6$ vertices and $3n?1$ edges is $T\times K_2+e$, where $T$ is an arbitrary tree on $n$ vertices and $e$ is an edge connecting two vertices that lie in different copies of $T$ and have distance 3 between them in $T\times K_2$. In this paper, we introduced the definition of $Q$-joint graph and characterized the connected IM-extendable graphs with $2n\ge 4$ vertices and $3n$ edges.     

A degree condition for a graph to have (a,b)-parity factors

Let $a$ and $b$ be two positive integers such that $a\le b$ and $a\equiv b(mod2)$. A graph $F$ is an $(a,b)$-parity factor of a graph $G$ if $F$ is a spanning subgraph of $G$ and for all vertices $v\in V\left(F\right)$, $d_F\left(v\right)\equiv b(mod2)$ and $a\le d_F\left(v\right)\le b$. In this paper we prove that every connected graph $G$ with $n\ge b(a+b)(a+b+2)∕\left(2a\right)$ vertices has an $(a,b)$-parity factor if $na$ is even, $\delta \left(G\right)\ge (b?a)∕a+a$, and for any two nonadjacent vertices $u,v\in V\left(G\right)$, $max\{d_G\left(u\right),d_G\left(v\right)\}\ge \frac{an}{a+b}$. This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998).     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

Moments about the mean of the size of a self-conjugate (s,t)-core partition

Johnson proved that if $s,t$ are coprime integers, then the $r$th moment of the size of an $(s,t)$-core is a polynomial of degree $2r$ in $t$ for fixed $s$. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of ${\stackrel{?}{\mathfrak{S}}}_t∕{\mathfrak{S}}_t$ and $t$-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an $(s,t)$-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an $(s,t)$-core and the first nine moments about the mean of the size of an $(s,s+1)$-core using Maple. To get the moments about the mean of the size of a self-conjugate $(s,t)$-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if $s,t$ are coprime integers, then the $r$th moment about the mean of the size of a self-conjugate $(s,t)$-core is a quasipolynomial of period 2 and degree $2r$ in $t$ for fixed odd $s$. Then, based on a bijection of Ford, Mai and Sze between self-conjugate $(s,t)$-cores and lattice paths in $?\frac{s}{2}?\times ?\frac{t}{2}?$ rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate $(s,t)$-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate $(s,t)$-core.     

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 $c>0$.     

On q-Steiner systems from rank metric codes

In this paper we prove that rank metric codes with special properties imply the existence of $q$-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a $[2d,d,d]$ dually almost MRD code $C\le {\mathbb{F}}_{q^m}^{2d}$$(2d\le m)$ which has no code words of rank weight $d+1$ form a $q$-Steiner system $S{(d?1,d,2d)}_q$. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.     

Disjointness preserving C0-semigroups and local operators on ordered Banach spaces

We generalize results concerning ${\mathrm{C}}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving ${\mathrm{C}}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local ${\mathrm{C}}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.     

List r-hued chromatic number of graphs with bounded maximum average degrees

For integers $k,r>0$, a $(k,r)$-coloring of a graph $G$ is a proper coloring $c$ with at most $k$ colors such that for any vertex $v$ with degree $d\left(v\right)$, there are at least min$\{d\left(v\right),r\}$ different colors present at the neighborhood of $v$. The $r$-hued chromatic number of $G$, ${\chi }_r\left(G\right)$, is the least integer $k$ such that a $(k,r)$-coloring of $G$ exists. The list$r$-hued chromatic number${\chi }_{L,r}\left(G\right)$ of $G$ is similarly defined. Thus if $\Delta \left(G\right)\ge r$, then ${\chi }_{L,r}\left(G\right)\ge {\chi }_r\left(G\right)\ge r+1$. We present examples to show that, for any sufficiently large integer $r$, there exist graphs with maximum average degree less than 3 that cannot be $(r+1,r)$-colored. We prove that, for any fraction $q<\frac{14}{5}$, there exists an integer $R=R\left(q\right)$ such that for each $r\ge R$, every graph $G$ with maximum average degree $q$ is list $(r+1,r)$-colorable. We present examples to show that for some $r$ there exist graphs with maximum average degree less than 4 that cannot be $r$-hued colored with less than $\frac{3r}{2}$ colors. We prove that, for any sufficiently small real number $?>0$, there exists an integer $h=h\left(?\right)$ such that every graph $G$ with maximum average degree $4??$ satisfies ${\chi }_{L,r}\left(G\right)\le r+h\left(?\right)$. These results extend former results in Bonamy et al. (2014).