A bijective proof of Amdeberhan’s conjecture on the number of (s,s+2)-core partitions with distinct parts

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s?1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s,s+2)$-core partitions with distinct parts and a set of lattice paths.     

Cuspidal integrals for SL(3)∕Kϵ

We show that for the symmetric spaces $\mathrm{SL}(3,\mathbb{R})∕{\mathrm{SO}(1,2)}_e$ and $\mathrm{SL}(3,\mathbb{C})∕\mathrm{SU}(1,2)$ the cuspidal integrals are absolutely convergent. We further determine the behavior of the corresponding Radon transforms and relate the kernels of the Radon transforms to the different series of representations occurring in the Plancherel decomposition of these spaces. Finally we show that for the symmetric space $\mathrm{SL}(3,\mathbb{H})∕\mathrm{Sp}(1,2)$ the cuspidal integrals are not convergent for all Schwartz functions.     

The size of a graph is reconstructible from any n−2 cards

Let $G$ and $H$ be graphs of order $n$. The number of common cards of $G$ and $H$ is the maximum number of disjoint pairs $(v,w)$, where $v$ and $w$ are vertices of $G$ and $H$, respectively, such that $G?v?H?w$. We prove that if the number of common cards of $G$ and $H$ is at least $n?2$ then $G$ and $H$ must have the same number of edges when $n\ge 29$. This is the first improvement on the $25$-year-old result of Myrvold that if $G$ and $H$ have at least $n?1$ common cards then they have the same number of edges. It also improves on the result of Woodall and others that the numbers of edges of $G$ and $H$ differ by at most $1$ when they have $n?2$ common cards.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

The benefit of preemption with respect to the ℓp norm

We establish tight bounds on the benefit of preemption with respect to the ${?}_p$ norm minimization objective for identical machines and for two uniformly related machines (based on their speed ratio). This benefit of preemption is the supremum ratio between the optimal costs of non-preemptive and preemptive schedules.     

Nodal standing waves for a gauged nonlinear Schrödinger equation in R2

This paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear Schrödinger equation

$\{\begin{array}{c}?\mathrm{\Delta }u+\omega u+(\frac{h^2(\left|x\right|)}{{\left|x\right|}^2}+\underset{\left|x\right|}{\overset{+\infty }{\int }}\frac{h(s)}{s}{u}^2(s)ds)u=\lambda {\left|u\right|}^{p?2}u,x\in {\mathbb{R}}^2,\hfill \\ u(x)=u(|x|)\in H^1({\mathbb{R}}^2),\hfill \end{array}$

where $\omega ,\lambda >0$, $p>6$ and

$h(s)=\frac{1}{2}\underset{0}{\overset{s}{\int }}r{u}^2(r)dr$

is the so-called Chern–Simons term. We prove that for any positive integer k, the problem has a sign-changing solution ${u^{\lambda }}_k$ which changes sign exactly k times. Moreover, the energy of $u_k^{\lambda }$ is strictly increasing in k, and for any sequence $\{{\lambda }_n\}\to +\infty (n\to \infty )$, there exists a subsequence $\{{\lambda }_{n_s}\}$, such that ${({\lambda }_{n_s})}^{\frac{1}{p?2}}{u}_k^{{\lambda }_{n_s}}$ converges in $H^1({\mathbb{R}}^2)$ to $w_k$ as $s\to \infty$, where $w_k$ also changes sign exactly k times and solves the following equation

$?\mathrm{\Delta }u+\omega u=|u{|}^{p?2}u,u\in H^1({\mathbb{R}}^2).$

     

Multiplicity and concentration behaviour of positive solutions for Schrödinger-Kirchhoff type equations involving the p-Laplacian in RN

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type

$-{\in }^{p}M\left({\in }^{p-N}{\int }_{{\mathbb{R}}^N}{\left|?u\right|}^p\right){\Delta }_{p}u+V\left(x\right){\left|u\right|}^{p-2}u=f\left(u\right)$

in ${\mathbb{R}}^N$, where Δ_p is the p-Laplacian operator, 1 < p < N, M: ${\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and V: ${\mathbb{R}}^{\text{N}}\to {\mathbb{R}}^{+}$ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.     

On approximation of Ginzburg–Landau minimizers by S1-valued maps in domains with vanishingly small holes

We consider a two-dimensional Ginzburg–Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg–Landau parameter vs. hole radius) is motivated by a recently discovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg–Landau problems in the classes of ${\mathbb{S}}^1$-valued and $\mathbb{C}$-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.     

Curved fronts in the Belousov–Zhabotinskii reaction–diffusion systems in R2

In this paper we consider a diffusion system with the Belousov–Zhabotinskii (BZ for short) chemical reaction. Following Brazhnik and Tyson [4] and Pérez-Muñuzuri et al. [45], who predicted V-shaped fronts theoretically and discovered V-shaped fronts by experiments respectively, we give a rigorous mathematical proof of their results. We establish the existence of V-shaped traveling fronts in ${\mathbb{R}}^2$ by constructing a proper supersolution and a subsolution. Furthermore, we establish the stability of the V-shaped front in ${\mathbb{R}}^2$.     

Corrigendum to the paper “Ovoidal packings of PG(3,q) for even q”

We point out that the proof of Theorem 3.3 of Bagchi and Sastry (2013) contains a serious flaw. Accordingly, this theorem needs to be modified. In consequence, we also have to retract Corollary 3.4, Corollary 3.6 and Theorem 3.8 of Bagchi and Sastry (2013).     

Classification of α-Ricci flat graphs with girth at least five

Lin, Lu and Yau classified Ricci flat graphs with girth at least 5 in Lin et al., 2014 [7] and Cushing et al., 2018 [4,5]. In Lin et al., 2014, they defined $\alpha$-Ricci curvature for the definition of Ricci curvature. We will classify $\alpha$-Ricci flat graphs with girth at least 5 for all $\alpha \in (0,1)$.     

The Σ1-provability logic of HA

In this paper we introduce a modal theory ${\mathsf{iH}}_{\sigma }$ which is sound and complete for arithmetical ${\mathrm{\Sigma }}_1$-interpretations in $\mathsf{HA}$, in other words, we will show that ${\mathsf{iH}}_{\sigma }$ is the ${\mathrm{\Sigma }}_1$-provability logic of $\mathsf{HA}$. Moreover we will show that ${\mathsf{iH}}_{\sigma }$ is decidable. As a by-product of these results, we show that $\mathsf{HA}+\square \perp$ has de Jongh property.     

Voter model in a random environment in ?d

We consider the voter model with flip rates determined by {μ_e, e ∈ E_d}, where E_d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice ?d. Suppose that {μ_e, e ∈ E_d} are independent and identically distributed (i.i.d.) random variables satisfying μ_e≥1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: δ₀ and δ₁.