Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient $f(z)$ of the critical nonlinearity. For sufficiently small $\lambda ,\mu >0$, there are at least k positive solutions of the semilinear elliptic systems$\{\begin{array}{c}?\mathrm{\Delta }u=\lambda g(z){|u|}^{p?2}u+\frac{\alpha }{\alpha +\beta }f(z){|u|}^{\alpha ?2}u{|v|}^{\beta }\text{in}\Omega ;\hfill \\ ?\mathrm{\Delta }v=\mu h(z){|v|}^{p?2}v+\frac{\beta }{\alpha +\beta }f(z){|u|}^{\alpha }{|v|}^{\beta ?2}v\text{in}\Omega ;\hfill \\ u=v=0\text{on}?\Omega ,\hfill \end{array}$ where $0\in \Omega ?{\mathbb{R}}^N$ is a bounded domain, $\alpha >1$, $\beta >1$ and $2<p<\alpha +\beta =2^{?}$ for $N>4$.     

A note on two geometric paths with few crossings for points labeled by integers in the plane

Let $S$ be a set of $n$ points in the plane in general position such that the integers $1,2,\dots ,n$ are assigned to the points bijectively. Set $h$ be an integer with $1\le h<n(n+1)∕2$. In this paper we consider the problem of finding two vertex-disjoint simple geometric paths consisting of all points of $S$ such that the sum of labels of the points in one path is equal to $h$ and the paths have as few crossings as possible. We prove that there exists such a pair of paths with at most two crossings between them.     

A sharp bound on the expected number of upcrossings of an L2-bounded Martingale

For a martingale $M$ starting at $x$ with final variance ${\sigma }^2$, and an interval $(a,b)$, let $\Delta =\frac{b?a}{\sigma }$ be the normalized length of the interval and let $\delta =\frac{|x?a|}{\sigma }$ be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of $(a,b)$ by $M$ is at most $\frac{\sqrt{1+{\delta }^2}?\delta }{2\Delta }$ if ${\Delta }^2\le 1+{\delta }^2$ and at most $\frac{1}{1+{(\Delta +\delta )}^2}$ otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of $(a,b)$ for submartingales with the corresponding final distribution. Each of these two bounds is at most $\frac{\sigma }{2(b?a)}$, with equality in the first bound for $\delta =0$. The upper bound $\frac{\sigma }{2}$ on the length covered by $M$ during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound $\sigma$ on the expected maximum of $M$ above $x$, the Dubins & Schwarz sharp upper bound $\sigma \sqrt{2}$ on the expected maximal distance of $M$ from $x$, and the Dubins, Gilat & Meilijson sharp upper bound $\sigma \sqrt{3}$ on the expected diameter of $M$.     

Sequences of integers with missing quotients and dense points without neighbors

Let $A$ be a pre-defined set of rational numbers. We say that a set of natural numbers $S$ is an $A$-quotient-free set if no ratio of two elements in $S$ belongs to $A$. We find the maximal asymptotic density and the maximal upper asymptotic density of $A$-quotient-free sets when $A$ belongs to a particular class.It is known that in the case $A=\{p,q\}$, where $p$, $q$ are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.     

Optimal spanners for axis-aligned rectangles

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the $L_1$-distance. The dilation of a pair of points is then defined as the length of the shortest rectilinear path between them that stays within the union of the rectangles and the connecting segments, divided by their $L_1$-distance. The dilation of the graph is the maximum dilation over all pairs of points in the union of the rectangles.We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain four results on this problem: (i) for arbitrary graphs, the problem is NP-hard; (ii) for trees, we can solve the problem by linear programming on $\mathrm{O}(n^2)$ variables and constraints; (iii) for paths, we can solve the problem in time $\mathrm{O}(n^3\log n)$; (iv) for rectangles sorted vertically along a path, the problem can be solved in $\mathrm{O}(n^2)$ time, and a $(1+\varepsilon )$-approximation can be computed in linear time.     

Global bifurcation of periodic solutions in nonlinear evolution problems with periodic forcing

This paper discusses the global bifurcation of $2\pi$-periodic solutions of $u_t=P(\lambda ,x,?)u+f(\lambda ,t,u)$ with a homogeneous Dirichlet boundary condition, where $P(\lambda ,x,?)$ is linear elliptic and the nonlinearity $f$ is $2\pi$-periodic in $t$.The main differences from existing theories devoted to this type of problem can roughly be summarized as follows: (i) the bifurcation analysis makes no use of evolution operators or related concepts (Poincaré maps, Floquet multipliers, etc.); (ii) the bifurcation/nonbifurcation points are characterized through an associated stationary problem; (iii) the functional setting allows for nonlinearities $f$ exhibiting time discontinuities.Among other things, the results include various partial generalizations of the “bifurcation from the principal eigenvalue” theorem, which, unlike the classical version, do not require linear parameter dependence.     

Some bounds and limits in the theory of Riemann’s zeta function

For any real $a>0$ we determine the supremum of the real $\sigma$ such that $\zeta (\sigma +it)=a$ for some real $t$. For $0<a<1$, $a=1$, and $a>1$ the results turn out to be quite different.We also determine the supremum $E$ of the real parts of the ‘turning points’, that is points $\sigma +it$ where a curve $\mathrm{Im}\zeta (\sigma +it)=0$ has a vertical tangent. This supremum $E$ (also considered by Titchmarsh) coincides with the supremum of the real $\sigma$ such that ${\zeta }^{\prime }(\sigma +it)=0$ for some real $t$.We find a surprising connection between the three indicated problems: $\zeta \left(s\right)=1$, ${\zeta }^{\prime }\left(s\right)=0$ and turning points of $\zeta \left(s\right)$. The almost extremal values for these three problems appear to be located at approximately the same height.     

Symplectic subspaces of symplectic Grassmannians

Let $V$ be a non-degenerate symplectic space of dimension $2n$ over the field $\mathbb{F}$ and for a natural number $l<n$ denote by $C_l\left(V\right)$ the incidence geometry whose points are the totally isotropic $l$-dimensional subspaces of $V$. Two points $U,W$ of $C_l\left(V\right)$ will be collinear when $W\subset U^{\perp }$ and $dim(U\cap W)=l-1$ and then the line on $U$ and $W$ will consist of all the $l$-dimensional subspaces of $U+W$ which contain $U\cap W$. The isomorphism type of this geometry is denoted by $C_{n,l}\left(\mathbb{F}\right)$. When $\mathrm{char}\left(\mathbb{F}\right)\ne 2$ we classify subspaces $S$ of $C_l\left(\mathbb{F}\right)$ where $S\cong C_{m,k}\left(\mathbb{F}\right)$.