The graph grabbing game on Km,n-trees

The graph grabbing game is a two-player game on weighted connected graphs where all weights are non-negative. Two players, Alice and Bob, alternately remove a non-cut vertex from the graph (i.e., the resulting graph is still connected) and get the weight assigned to the vertex, where the starting player is Alice. Each player’s aim is to maximize his/her outcome when all vertices have been taken, and Alice wins the game if she gathered at least half of the total weight. Seacrest and Seacrest (2017) proved that Alice has a winning strategy for every weighted tree with even order, and conjectured that the same statement holds for every weighted connected bipartite graph with even order. In this paper, we prove that Alice wins the game on a type of a connected bipartite graph with even order called a $K_{m,n}$-tree.     

Multivariable codes in principal ideal polynomial quotient rings with applications to additive modular bivariate codes over F4

In this work, we study the structure of multivariable modular codes over finite chain rings when the ambient space is a principal ideal ring. We also provide some applications to additive modular codes over the finite field ${\mathbb{F}}_4$.     

Subspace partitions of Fqn containing direct sums

Let $q$ be a prime power and $n$ be a positive integer. A subspace partition of $V={\mathbb{F}}_q^n$, the vector space of dimension $n$ over ${\mathbb{F}}_q$, is a collection $\Pi$ of subspaces of $V$ such that each nonzero vector of $V$ is contained in exactly one subspace in $\Pi$; the multiset of dimensions of subspaces in $\Pi$ is then called a Gaussian partition of $V$. We say that $\Pi$contains a direct sum if there exist subspaces $W_1,\dots ,W_k\in \Pi$ such that $W_1\oplus ?\oplus W_k=V$. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers $a_1$ and $a_2$ with $n>a_1>a_2\ge 1$, our main theorem shows that if $\Pi$ is a subspace partition of ${\mathbb{F}}_q^n$ with $m_i$ subspaces of dimension $a_i$ for $i=1,2$, then $\Pi$ contains a direct sum when $a_1{x}_1+a_2{x}_2=n$ has a solution $(x_1,x_2)$ for some integers $x_1,x_2\ge 0$ and $m_2$ belongs to the union $I$ of two natural intervals. The lower bound of $I$ captures all subspace partitions with dimensions in $\{a_1,a_2\}$ that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when $m_2?I$ or when the condition on the existence of a nonnegative integral solution $(x_1,x_2)$ is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when $q\to 1$) as well as subspace and set partitions.     

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 $M$.     

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.     

Quasineutral limit of the two-fluid Euler–Poisson system in a bounded domain of R3

The quasineutral limit of the two-fluid Euler–Poisson system (one for ions and another for electrons) in a bounded domain of ${\mathbb{R}}^3$ is rigorously proved by investigating the existence and the stability of boundary layers. The non-penetration boundary condition for velocities and Dirichlet boundary condition for electric potential are considered.     

On the zeroth stable A1-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.     

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.     

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.     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of ${\mathbb{P}}^1$ in $A_g$ and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus $T_g$ inside $A_g$. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.     

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.     

Hod up to ADR+Θ is measurable

Suppose M is a transitive class size model of $A{D}_{\mathbb{R}}+\text{“Θ is regular”}$. M is a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$ if (i) $\mathbb{R},Ord?M$ (ii) there is $\mu \in M$ such that $M?\text{“}\mathit{\mu }\text{ is a normal }\mathbb{R}\text{-complete measure on Θ”}$ and (iii) for any transitive class size $N?M$ such that $\mathbb{R}?N$, $N?\text{“there is no }\mathbb{R}\text{-complete measure on Θ”}$. Continuing Trang's work in [8], we compute HOD of a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$.     

An update on non-Hamiltonian 54-tough maximal planar graphs

Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant and Owens (1995). In addition, we present two generalizations of a similar result of Tká? who considered $1$-tough maximal planar graphs (Tká?, 1996); we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in both mentioned papers.     

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.