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...

Remark. Apropos of reduction mod p: If V is a Qp-vector space and G ⊂ GL(V ) is a compact subgroup, then there exists a G-fixed lattice in V for the following reason. Pick any lattice L ⊂ V . Then the G-stabilizer of L is open and of finite index. So Λ = ∑ g∈G gL ⊂ V is also a lattice, and it is definitely G-stable. The same works with coefficients in any finite extension of Qp, or even in Qp (...

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...

We prove a variant of the well-known Reidemeister-Schreier Theorem for finitely L-presented groups. More precisely, we prove that each finite index subgroup of a finitely L-presented group is itself finitely L-presented. Our proof is constructive and it yields a finite L-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely L-presented ...

There are many notions of discrete convexity of polyominoes (namely hvconvex [1], Q-convex [2], L-convex polyominoes [5]) and each one has been deeply studied. One natural notion of convexity on the discrete plane leads to the definition of the class of hv-convex polyominoes, that is polyominoes with consecutive cells in rows and columns. In [1] and [6], it has been shown how to reconstruct in ...

In the present paper we prove a strong form of Arnold diffusion. Let T2 be the two torus and B2 be the unit ball around the origin in R2. Fix ρ > 0. Our main result says that for a “generic” time-periodic perturbation of an integrable system of two degrees of freedom H0(p) + εH1(θ, p, t), θ ∈ T, p ∈ B, t ∈ T = R/Z, with a strictly convex H0, there exists a ρ-dense orbit (θ , p , t)(t) in T2×B2×...