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.     

Delay differential logistic equation with linear harvesting

In this paper, we consider a logistic delay equation with a linear delay harvesting term of the following form:$\dot{N}(t)=r(t)N(t)\left[a-b_{0}N(t)-\sum _{k=1}^m{b}_{k}N(h_k(t))\right]-\sum _{l=1}^n{c}_l(t)N(g_l(t)),t⩾0\text{,}$$N(t)=\phi (t),t<0,N(0)=N_0$in both cases when $b_0=0$ and $b_0\ne 0$. We present some results on the boundedness and positiveness of the solutions of this equation without the condition that $N_0$ is upper bounded by some constant which is necessary to the corresponding results in [L. Berezansky, E. Braverman, L. Idels, Delay differential logistic equation with harvesting, Math. Comput. Modelling 40 (2004) 1509–1525], and our results extend these known results.     

Periodicity in a nonlinear discrete predator–prey system with state dependent delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator–prey systemwhere $a_i,c_j,d_i:Z\to R^{+}$ are positive $\omega$-periodic, $\omega$ is a fixed positive integer. $b_1,b_2:Z\to R^{+}$ are $\omega$-periodic and ${\sum }_{k=0}^{\omega -1}{b}_i(k)>0$. ${\tau }_i,{\sigma }_j,{\rho }_i:Z\times R\times R\to R(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are $\omega$-periodic with respect to their first arguments, respectively. ${\alpha }_i,{\beta }_j,{\gamma }_i(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are positive constants.     

Further results on logarithmic terraces

Let p be an odd prime, and let x be a primitive root of p. Suppose that we write the elements of ${\mathbb{Z}}_{p-1}$ as $1,2,\dots ,p-1,$ and that, wherever we evaluate $x^l(\mathrm{mod}p)$, we always write it as one of $1,2,\dots ,p-1.$ Let $?=(l_1,\dots ,l_{p-1})$ be a terrace for ${\mathbb{Z}}_{p-1}$. Then $?$ is said to be a logarithmic terrace if $\mathbf{e}=(e_1,\dots ,e_{p-1})$, defined by $e_i\equiv x^{l_i}(\mathrm{mod}p)$, is also a terrace for ${\mathbb{Z}}_{p-1}$. We study properties of logarithmic terraces, in particular investigating terraces which are simultaneously logarithmic for two different primitive roots.     

Positive solution for system of nonlinear first-order PBVPs on time scales

In this paper we are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale $\mathbb{T}$where $f_i:[0,T]\times [0,+\infty {)}^n\to \mathbb{R}$ is continuous and there exists a constant $M_i>0$ such that$M_i{x}_i-f_i(t,x_1,x_2,\dots ,x_n)⩾0\text{for}(x_1,x_2,\dots ,x_n)\in [0,+\infty {)}^n,t\in [0,T]\text{.}$Some existence criteria of positive solution are established by using a fixed point theorem for operators on cone.     

On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4

Faulhuber and Steinerberger conjectured that the logarithmic derivative of ${?}_4$ has the property that $y^2{?}_4^{\prime }(y)/{?}_4(y)$ is strictly decreasing and strictly convex. In this small note, we prove this conjecture.     

The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small [5]. It has been conjectured by Nagylaki and Lou [2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in [6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.     

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.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.