This paper is concerned with the following Schrödinger equation:\[\begin{cases}-\triangle u + V(x)u = f(x, u), &\textrm{for $x \in \mathbb{R}^{N}$}, \\u(x) \to 0, &\textrm{as $|x| \to \infty$},\end{cases}\] where the potential $V$ and $f$ are periodic with respect to $x$ and $0$ is a boundary point of the spectrum $\sigma(-\triangle+V)$. By a new technique for showing the boundedness of Cerami sequences, we are able to obtain the existence of nontrivial solutions with mild assumptions on $f$.

Following a construction of Gross and Reeder, we define simple supercuspidal representations of $\operatorname{GL}(n)$ over a $\mathfrak{p}$-adic field. We show that they have conductor $\mathfrak{p}^{n+1}$. We then give a general formula for the matrix coefficient attached to a new vector, and make it completely explicit when $n=2$.

We provide the explicit formula for the Bellman function corresponding to the weak-type $L^{p,\infty}\to L^{q,\infty}$ estimates for the dyadic maximal operator on $\mathbb{R}^n$. Actually, we do this in the more general setting of maximal operators associated with a tree-like structure on a nonatomic probability space. The approach rests on a clever combination of some novel optimization and combinatorial arguments.

This paper is devoted to the study the first-order behavior of the value function of a parametric optimal control problem with linear constraints and a nonconvex cost function. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the value function to a parametric optimal control problem.

The Mumford-Tate conjecture is first proved for CM abelian varieties by H. Pohlmann [Ann. Math., 1968]. In this note we give another proof of this result and extend it to CM motives.

Let $G$ be a graph, $u$ be a vertex of $G$, and $B(u)$ (or $B_G(u)$) be the set of $u$ with all its neighbors in $G$. A sequence $(B_1,B_2,...,B_n)$ of subsets of an $n$-set $S$ is a neighborhood sequence if there exists a graph $G$ with a vertex set $S$ and a permutation $(v_1,v_2,...,v_n)$ of $S$ such that $B(v_i)=B_i$ for $i=1,2,...,n$. Define $Aut(B_1,B_2,...,B_n)$ as theset $\{f: f$ is a permutation of $V(G)$ and $(f(B_1),f(B_2),...,f(B_n))$ is a permutation of $B_1$, $B_2$, ...,$B_n\}$. In this paper, we first prove that, for every finite group $\Gamma$, there exists a neighborhood sequence $(B_1,B_2,...,B_n)$ such that $\Gamma$ is group isomorphic to $Aut(B_1,B_2,...,B_n)$. Second, we show that, for each finite group $\Gamma$, there exists a neighborhood sequence $(B_1,B_2,...,B_n)$ such that, for eachsubgroup $H$ of $\Gamma$, $H$ is group isomorphic to $Aut(E_1,E_2,...,E_t)$ for some neighborhood sequence $(E_1,E_2,...,E_t)$ where $E_i \subseteq B_{j_i}$ and $j_1 \lt j_2 \lt \cdots \lt j_t$. Finally, we give some classes of graphs $G$ withneighborhood sequence $(B_1,B_2,...,B_n)$ satisfying that $Aut(G)$ and $Aut(B_1,B_2,...,B_n)$ are different.

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S^m_{\rho,\delta}$ with $0\leq \delta\lt 1, 0\lt \rho\leq 1, \delta \leq \rho$ and $-(n+1)\lt m \leq - (n+1)(1-\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^\theta} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx \lt \infty$$ for some $\theta\in [0,\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\mathbb R^n)$ introduced by Goldberg [9].

As a consequence, when $\rho=1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen [21].

In this paper, we construct infinitely many bi-invariantmetrics on the Hamiltonian diffeomorphism group and study their basic properties and corresponding generalizations of the Hofer inequality and Sikorav one.

In this paper, we study the geometry of anti-invariant Riemanniansubmersions from a Kähler manifold onto a Riemannian manifold. Wefirst determine the base space when the total space of ananti-invariant Riemannian submersion is Einstein and then weinvestigate new conditions for anti-invariant Riemannian submersionsto be Clairaut submersions. We also focus on the geometry of Clairaut anti-invariant submersions.

We consider a mathematical problem for quasistatic contact between anelectro elastic-viscoplastic body and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. We employ the electro elastic-viscoplastic with damage constitutive law for the material. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a system of an elliptic variational inequality for the displacement, a parabolic variational inequality for the damage and a variational equality for the electric stress. We establish a variational formulation for the model and we give the wear conditions for the existence of a unique weak solution to the problem. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.

Let $N$ denote a sufficiently large even integer. In this paper itis proved that for $0.941 \leq \theta \leq 1$, the equation$$N=p+P_2,\hspace{10mm} p\leq N^{\theta}$$is solvable, where $p$ is a prime and $P_2$ is an almost prime withat most two prime factors. The range $0.941 \leq \theta \leq 1$ extended the previous one $0.945 \leq \theta \leq 1$.

Slant and Legendre curves are considered on Berger $su(2)$ and are characterized through the scalar product between the normal at the curve and the vertical vector field; in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of these curves is obtained as well as their curvature and torsion. For the slant non-Legendre case we derive a Lancret-type invariant. By using the exponential map we obtain remarkable classes of curves on $S^3(1)$; in the helix case, and taking into account a B.-Y. Chen characterization of Legendre curves, we get a $1$-parameter family of curves in relationship with the spectrum of the quantum harmonic oscillator. These curves, called by us quantum spherical curves, and their mates, provided by integer multiples of $\pi$, belong to antipodal Hopf fibres.

This paper considers the following nonhomogeneous fourth order elliptic equations of Kirchhoff type:\begin{eqnarray*}\begin{cases}\displaystyle\triangle ^2u-(a+b\int_{\text R^N}|\nabla u|^2dx)\triangle u+V(x)u= f(x,u)+h(x),~\text{in}~ \text R^N,\\ \displaystyle u\in H^2(\text R^N), \end{cases}\end{eqnarray*}where constants $a\gt 0,~b\geq0$. Under certain assumptions on $V(x)$, $f(x,u)$ and $h(x)$, we show the existence and multiplicity of solutions by the Ekeland$^{,}$s variational principle and the Mountain Pass Theorem in the critical theory.

Let $K :[0,T_0) \to \Bbb F$ be a locally integrable function, and $C :X\to X$ a bounded linear operator on a Banach space $X$ over the field $\Bbb F$(=$\Bbb R$ or $\Bbb C$). In this paper, we will deduce some basic properties of a nondegenerate local $K$-convoluted $C$-semigroup on $X$ and some generation theorems of local $K$-convoluted $C$-semigroups on $X$ with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $K$-convoluted $C$-semigroup on $X$ with subgenerator $A$ and the unique existence of solutions of the abstract Cauchy problem:\[ \textrm{ACP}(A,f,x) \qquad \begin{cases} u'(t) = A u(t)+f(t) &\textrm{for a.e. $t \in (0,T_0)$},\\ u(0)=x \end{cases}\]when $K$ is a kernel on $[0,T_0)$, $C :X\to X$ an injection, and $A:\text{D}(A)\subset X\to X$ a closed linear operator in $X$ such that $CA\subset AC$. Here $0\lt T_0\leq\infty$, $x\in X$, and $f\in\text{L}_{loc}^{1}([0,T_0),X)$.

For $0 \lt p \lt 1$, Haberl and Ludwig defined symmetric and asymmetric $L_p$-intersection bodies. In this paper, we introduce general $L_p$-intersection bodies and study their properties. In particular, we obtain the extremal values of their volume andestablish a Brunn-Minkowski type inequality for them.

Let $C_k$ (respectively, $S_k$) denote a cycle (respectively, astar) with $k$ edges. For graphs $F$, $G$ and $H$, a$(G,H)$-decomposition of $F$ is a partition of the edge set of $F$into copies of $G$ and copies of $H$ with at least one copy of $G$and at least one copy of $H$. In this paper, necessary andsufficient conditions for the existence of the$(C_k,S_k)$-decomposition of multicrowns are given.

A vertex-deleted subgraph of a graph $G$ is a card. A dacard consists of a card and the degree of the missing vertex. The degree-associated reconstruction number of a graph $G$, denoted $\textrm{drn}(G)$, is the minimum number of dacards that suffice to reconstruct $G$. The adversary degree-associated reconstruction number $\textrm{adrn}(G)$ is the least $k$ such that every set of $k$ dacards determines $G$. The analogous parameters for degree-associated edge reconstruction are $\textrm{dern}(G)$ and $\textrm{adern}(G)$. We determine these four parameters for all complete multipartite graphs andtheir complements. The answer is usually $2$ for all four parameters,but there are exceptions in each case.

In this paper, based on the extended versions of the Farkas lemma for convex systems introduced recently in [9], we establish an extended version of a so called Hahn-Banach-Lagrange theorem introduced by Stephan Simons in [22]. This generalized version of the Hahn-Banach-Lagrange theorem holds in locally convex Hausdorff topological vector spaces under a Slater-type constraint qualification condition and with the relaxing of the lower semi-continuity of some functions involved and the closedness of the constrained sets. The version, in turn, yields extended versions of the Mazur-Orlicz theorem, the sandwich theorem, and the Hahn-Banach theorem concerning extended sublinear functions. It is then shown that all the generalized versions of the Farkas lemma for cone-convex/sublinear-convex systems in [9] and the new extended Hahn-Banach-Lagrange theorem just obtained are equivalent together. A class of composite problems involving sublinear-convex mappings is considered at the end of the paper. Here the main results of the paper are applied to get a strong duality result and optimality conditions for the class of problems. Moreover, a formula for the conjugate of the supremum of a family (possibly infinite, not lower semi-continuous) of convex functions is then derived from the duality result to show the generality and the significance of the class of problems in consideration.

In this note, we solve the inverse nodal problem and Ambarzumyanproblem for the $p$-Laplacian coupled with periodic or anti-periodicboundary conditions. We also extend some results in a previous paperto $p$-Laplacian with $L^1$ potentials, and for arbitrary linearseparated boundary conditions. There we prove a generalizedRiemann-Lebesgue Lemma which is of independent interest.