Let $G$ be a graph with vertex set $V(G) = \{v_1, v_2, \ldots, v_n\}$ and edge set $E(G)$. For any vertex $v_i \in V(G)$, let $d_i$ denote the degree of $v_i$. The normalized Laplacian matrix of the graph $G$ is the matrix $\mathcal{L} = (\mathcal{L}_{ij})$ given by\[\mathcal{L}_{ij} =\begin{cases}1 &\textrm{if $i = j$ and $d_{i} \neq 0$} \\-\frac{1}{\sqrt{d_{i} \,d_{j}}} &\textrm{if $v_i v_j \in E(G)$} \\0 &\textrm{otherwise}.\end{cases}\] In this paper, we obtain some bounds on the second smallest normalized Laplacian eigenvalue of tree $T$ in terms of graph parameters and characterize the extremal trees. Utilizing these results we present some lower bounds on the normalized Laplacian energy (or Randić energy) of tree $T$ and characterize trees for which the bound is attained.

A subset $K$ of vertices of a graph $G$ is gated if for every vertex $x \in V(G)$ there exists a gate $v \in K$ which is on a shortest path between $x$ and any vertex $u$ of $K$. We give a characterization of gated sets in an arbitrary graph $G$ and several necessary conditions. This characterization yields very nice results in the case of weakly modular graphs, which are also presented. We also show that the trees are precisely the graphs which present a convex geometry with respect to the gated convexity.

We study the existence of simple $(q, k, \lambda)$ BIBDs when the number of elements is a prime power $q$ and $\{c_1, c_2\} \cap \{1, 2\}$ is not empty, where $c_1 = \gcd(k, q-1)$ and $c_2 = \gcd(k-1, q-1)$. We show that in many situations the necessary conditions $\lambda (q-1) \equiv 0 \mod (k-1)$, $\lambda q(q-1) \equiv 0 \mod k(k-1)$, and $\lambda \leq \binom{q-2}{k-2}$ are also sufficient for the existence of a simple $(q, k, \lambda)$ BIBD. These new results improve the valid range of simple BIBDs.

The well known partition function $p(n)$ has a long research history, where $p(n)$ denotes the number of solutions of the equation $n = a_1 + \cdots + a_k$ with integers $1 \leq a_1 \leq \cdots \leq a_k$. In this paper, we investigate a new partition function. For any real number $r \gt 1$, let $p_r(n)$ be the number of solutions of the equation $n = \lfloor \sqrt[r]{a_1} \rfloor + \cdots + \lfloor \sqrt[r]{a_k} \rfloor$ with integers $1 \leq a_1 \leq \cdots \leq a_k$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. In this paper, it is proved that $\exp(c_1 n^{r/(r+1)}) \leq p_r(n) \leq \exp(c_2n^{r/(r+1)})$ for two positive constants $c_1$ and $c_2$ (depending only $r$).

We study a curve flow for convex closed plane curves. It is described by a third order linear equation for the radius of curvature of the evolving curve. It is shown that under the flow the evolving curve stays convex, bounds fixed area, length, and has fixed center. However, its curvature may blow up in finite time.

If the curvature of this flow does not blow up before time $2\pi$, then the flow will exist smoothly for all time and is periodic in time with period $2\pi$. In particular, the flow does not have a limiting curve unless the initial curve is a circle.

In this note we prove that all the Pontryagin classes of para-Sasakian space forms vanish.

In this work, we investigate the nonlocal time-delayed andreaction-diffusion epidemic model studied by Guo et al. [Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latentperiod and non-local infections, J. Math. Biol. 65 (2012) 1387–410]. In thecase that the coefficients are independent of the spatial variable, we obtainthe global stability of the disease-free equilibrium and the unique endemicequilibrium, which partially answers the open problem proposed by Guo et al.

In this paper, we study the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and multiplicative white noise. We consider the existence of random attractors in a weighted space $l_\rho^2 \times l_\rho^2$ for this system, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

We study a constraint minimization problem on $S_c = \{ u \in H^1(\mathbb{R}^N), |u|_2^2 = c, c \in (0, c^*) \}$ for the following $L^2$-critical Kirchhoff type functional:\begin{align*} E_\alpha(u) &= \frac{a}{2} \int_{\mathbb{R}^N} |\nabla u|^2 \, dx + \frac{b}{4} \left( \int_{\mathbb{R}^N} |\nabla u|^2 \, dx \right)^2 + \frac{1}{\alpha+2} \int_{\mathbb{R}^N} V(x) |u|^{\alpha+2} \, dx \\ &\quad - \frac{N}{2N+8} \int_{\mathbb{R}^2} |u|^{\frac{2N+8}{N}} \, dx,\end{align*}where $N \leq 3$, $a, b \gt 0$ are constants, $\alpha \in [0, \frac{8}{N})$ and $V(x) \in L^\infty(\mathbb{R}^N)$ is a suitable potential. We prove that the problem has at least one minimizer if $\alpha \in [2, \frac{8}{N})$ and the energy of the minimization problem is negative. Moreover, some non-existence results are obtained when the energy of the problem equals to zero.

Let $\operatorname{WCC}(X)$ be the collection of all non-empty, weakly compact, convex subsets of a Banach space $X$ endowed with the Hausdorff metric $h$. Weak topology $\mathcal{T}_{w}$ will be defined on $\operatorname{WCC}(X)$. We shall prove that every weakly compact ($\mathcal{T}_{w}$-compact) convex subset $\mathcal{K} \subset (\operatorname{WCC}(X), \mathcal{T}_{w})$ has an extreme point. We also show that there exists strongly bounded ($h$-bounded), closed ($h$-closed) convex subsets which are not weakly closed (i.e., not $\mathcal{T}_{w}$-closed).

We study an evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity\[\begin{cases} u''(x) + \lambda f(u) = 0, \quad -1 \lt x \lt 1, \\ u(-1) = u(1) = 0, \\ f(u) = -\varepsilon u^{3} + \sigma u^{2} + \tau u + \rho,\end{cases}\]where $\lambda \gt 0$ is a bifurcation parameters, $\varepsilon \gt 0$ is an evolution parameter, and $\sigma$, $\rho \gt 0$, $\tau \geq 0$ are constants. In addition, we improve lower and upper bounds of the critical bifurcation value $\widetilde{\varepsilon}$ of the problem.

This paper is mainly concerned with the shock reflection phenomenon for convex scalar conservation law with Dirac measure source term. The Riemann solutions are constructed completely and then the impact of the strength of source term on the Riemann solutions is considered in detail. In order to illustrate the shock reflection phenomenon, the initial data with three pieces of constant states are considered and the interactions between a backward shock wave plus a stationary wave discontinuity and a rarefaction wave are displayed in all kinds of situations. Furthermore, the global solutions are constructed completely and the large time asymptotic states are obtained. In some certain situations, the shock reflection phenomenon is captured when a rarefaction wave interacts with a stationary wave discontinuity and then is divided into a transmitted rarefaction wave and a reflected shock wave at the critical point. In addition, it is shown that the Riemann solutions are stable with respect to the particular small perturbations of the initial data.

The constraint qualification Q-CCCQ plays an important role in quasiconvex programming and has been developed by many authors to investigate the set containment problem, duality and optimality conditions for quasiconvex programming. In this paper, we consider an infinite quasiconvex inequality system defined by a family of proper lower semicontinuous quasiconvex functions $\{h_i : i \in I \}$ and establish some sufficient conditions for ensuring the Q-CCCQ in terms of the interior-point condition together with approximate continuity assumption of the function $i \mapsto h_i(x)$.

It has been realized that the proof of Theorem 5.1 in Section 5 is imcomplete. It was pointed out by Professor Jongsu Kim and Israel Evangelista. Here we give a correct proof of Theorem 5.1