Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

Existence of incomplete canonical Kirkman packing designs

For $u,v$ positive integers with $u\equiv v\equiv 4\left(mod6\right)$, let ICKPD$(u,v)$ denote a canonical Kirkman packing of order $u$ missing one of order $v$. In this paper, it is shown that the necessary condition for existence of an ICKPD$(u,v)$, namely $u\ge 3v+4$, is sufficient with a definite exception $(u,v)=(16,4)$, and except possibly when $v>76$, $v\equiv 4\left(mod12\right)$ and $u\in \{3v+4,3v+10\}$.     

Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations:${\mathrm{\Delta }}^m{u}_i(k)+P_i(k,u_1(k),u_2(k),\dots ,u_n(k))=0,k\in \{0,1,\dots ,N\},i=1,2,\dots ,n$together with Sturm–Liouville boundary conditions${\mathrm{\Delta }}^j{u}_i(0)=0,0⩽j⩽m-3\text{,}$$\zeta {\mathrm{\Delta }}^{m-2}{u}_i(0)-\eta {\mathrm{\Delta }}^{m-1}{u}_i(0)=0,\omega {\mathrm{\Delta }}^{m-2}{u}_i(N+1)+\delta {\mathrm{\Delta }}^{m-1}{u}_i(N+1)=0\text{,}$where $m⩾2,N⩾m-1,\zeta >0,\omega >0,\eta ⩾0,\delta ⩾\omega ,\zeta \omega (N+1)+\zeta \delta +\eta \omega >0$. By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on $\{0,1,\dots ,N+m\}$. Examples are also included to illustrate the results obtained.     

Quasigroups satisfying Stein’s third law with a specified number of idempotents

Quasigroups satisfying Stein’s third law (QSTL for short) have been associated with other types of combinatorial configurations, such as cyclic orthogonal arrays. These have been studied quite extensively over the years by various researchers, including Curt Lindner. An idempotent model of a QSTL of order $v$ (briefly QSTL$\left(v\right)$), corresponds to a perfect Mendelsohn design of order $v$ with block size four (briefly a $(v,4,1)$-PMD) and these are known to exist if and only if $v\equiv 0,1\left(\mathrm{mod}4\right)$, except for $v=4,8$. There is a QSTL$\left(4\right)$ with two idempotents and it is known that a QSTL$\left(8\right)$ contains either $0$ or $4$ idempotents. In this paper, we formally investigate the existence of a QSTL$\left(v\right)$ with a specified number $n$ of idempotent elements, briefly denoted by QSTL$(v,n)$. The necessary conditions for the existence of a QSTL$(v,n)$ are $v\equiv 0,1\left(\mathrm{mod}4\right)$, $0\le n\le v$, and $v?n$ is even. We show that these conditions are also sufficient with few definite exceptions and a handful of possible exceptions. Holey perfect Mendelsohn designs of type $4^n{u}^1$ with block size four (HPMD$\left(4^n{u}^1\right)$ for short) are useful to establish the spectrum of QSTL$(v,n)$. In particular, we show that for $0\le u\le 8$, an HPMD$\left(4^n{u}^1\right)$ exists if and only if $n\ge max(4,?u/2?+1)$, except possibly $(n,u)=(12,1)$.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let $c{?}_k\left(n\right)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c{?}_3\left(n\right)$ and $c{?}_9\left(n\right)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that

$c{?}_3\left(3^{2k}n+\frac{7?3^{2k}+1}{8}\right)\equiv 0(mod{3}^{4k+5}).$

We give two different proofs to the congruences satisfied by $c{?}_9\left(n\right)$. One of the proofs uses a relation between $c{?}_9\left(n\right)$ and $c{?}_3\left(n\right)$ due to Kolitsch, for which we provide a new proof in this paper.     

Orthogonally Resolvable Matching Designs

An orthogonally resolvable matching design OMD$(n,k)$ is a partition of the edges of the complete graph $K_n$ into matchings of size $k$, called blocks, such that the blocks can be resolved in two different ways. Such a design can be represented as a square array whose cells are either empty or contain a matching of size $k$, where every vertex appears exactly once in each row and column. In this paper we show that an OMD$(n,k)$ exists if and only if $n\equiv 0(mod2k)$ except when $k=1$ and $n=4$ or $6$.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$