We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finitedimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpo...

The main purpose of this note is to prove an upper bound on the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body. This bound improves on former bounds and narrows the gap towards a lattice point analogue of Minkowski’s second theorem on successive minima. Minkowski’s proof of his second theorem is rather lengthy and it was also criticis...

Monsky’s theorem from 1970 states that a square cannot be dissected into an odd number n of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky’s theorem to “constrained framed maps”; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for th...

1. A CORRECTION Theorem 3.6 in [1] contains an error in the stated bound. As is argued above the theorem, the statistical distance to the uniform distribution decreases by a factor O ( (m/ log m)−1/2 ) = O ((log n)1/2n−1/4) for every two iterations. This implies that the bound obtained for Theorem 3.6 should read O ( n1− t−1 2 1 4 (log n) t−1 2 1 2 ) . I am grateful to Justin Holmgren for point...

A positional game is essentially a generalization of Tic-Tac-Toe played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erdős– Selfridge theorem, which gives a simple criterion for the existence of a Breaker’s winning strategy on a finite hypergraph F . It has been shown that the bound in the Erdős–Selfridge theorem can be tight and that numerous extremal hypergr...

This paper considers a bi-criteria planning problems on single machine, with the goal of minimizing total square time duration and maximizing earliness. To solve this problem we have to find Pareto set. We introduced strong relation between lower bound, upper bound number efficient solutions via theorem which shows also that is near optimal solution if small.

Abstract In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on effective constant in the error term of Ideal Theorem.

We prove a sampling discretization theorem for the square norm of functions from finite dimensional subspace satisfying Nikol'skii's inequality with an upper bound on number points order dimension subspace.

We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a ...

We discuss the spectral subspace perturbation problem for a selfadjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tanΘ Theorem. We also extend the Davis–Kahan ta...