We prove that for a uniformly convex Lagrangian system L on a compact manifold M , almost all energy levels contain a periodic orbit. We also prove that below Mañé’s critical value of the lift of the Lagrangian to the universal cover, cu(L), almost all energy levels have conjugate points. We prove that if the energy level [E = k] is of contact type and M 6= T then the free time action functiona...

This paper sheds a new light on submodular function minimization and maximization from the viewpoint of discrete convex analysis. L-convex functions and M-concave functions constitute subclasses of submodular functions on an integer interval. Whereas L-convex functions can be minimized efficiently on the basis of submodular (set) function minimization algorithms, M-concave functions are identif...

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. Does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? A compact...

The classical notion of persistency, originated in 60s, has been extensively studied up to now. An action of a processing system is said to be persistent if, whenever it becomes enabled, it remains enabled until executed. And a system is said to be persistent if each of its actions is persistent. This notion, introduced by Karp/Miller, is one of the most frequently discussed issues in the Petri...

Let K ⊂ R d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I L (K) = conv(K ∩L), where L is a randomly translated and rotated copy of the integer lattice Z d. We estimate the expected number of vertices of I L (K), whose behaviour is similar...

A class of discrete convex functions that can efficiently be minimized has been considered by Murota. Among them are L\-convex functions, which are natural extensions of submodular set functions. We first consider the problem of minimizing an L\-convex function with a linear inequality constraint having a positive normal vector. We propose a polynomial algorithm to solve it based on a binary se...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

For n ≥ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection Ku can be translated inside Lu for every direction u, while the volumes of K and L satisfy Vn(K) > Vn(L). A more general construction is then given for n-dimensional convex bodies K and L such that the orthogonal projection Kξ can be translated inside Lξ for every k-dimensional subspace ξ of R...