In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN ...

λ1 + . . .+ λm = 1, then we say that y is an affine combination of y1, . . . ,ym ∈Y . If, in addition, λi ≥ 0 for 1 ≤ i ≤ m, then we say that y is a convex combination of y1, . . . ,ym ∈ Y . A convex set is any subset of Rn that is closed under the operation of taking convex combinations. In fact, it can be shown that a subset X is convex if and only if for all x0,x1 ∈ X and 0 ≤ λ ≤ 1, the poin...

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions defined over the real space. In this paper, we consider a ...

Nakamura [N] introduced the G-Hilbert scheme G-Hilb C3 for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-HilbC3. This note calculates A-Hilb C3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilate...

We prove that a Banach space X has normal structure provided it contains a finite codimensional subspace Y such that all spreading models for Y have normal structure. We show that a Banach space X is strictly convex if the set of fixed points of any nonexpansive map defined in any convex subset C C X is convex and give a sufficient condition for uniform convexity of a space in terms of nonexpan...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

Nakamura [N] introduced the G-Hilbert scheme for a finite subgroup G ⊂ SL(3, C), and conjectured that it is a crepant resolution of the quotient C3/G. He proved this for a diagonal Abelian group A by introducing an explicit algorithm that calculates A-HilbC3. This note calculates A-HilbC3 much more simply, in terms of fun with continued fractions plus regular tesselations by equilateral triangl...

L-convex functions are nonlinear discrete functions on integer points that are computationally tractable in optimization. In this paper, a discrete Hessian matrix and a local quadratic expansion are defined for L-convex functions. We characterize L-convex functions in terms of the discrete Hessian matrix and the local quadratic expansion.

We introduce two classes of discrete quasiconvex functions, called quasi M-convex and L-convex functions, by generalizing the concepts of M-convexity and L-convexity due to Murota (1996, 1998). We investigate the structure of quasi M-convex and L-convex functions with respect to level sets, and show that various greedy algorithms work for the minimization of quasi M-convex and L-convex function...

If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected “existence vs. prevalence” phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. The main result represents a new connection between...