A robust version of the output controller design for discrete-time systems is introduced. Instead of a single stable point a stable polytope (or simplex) is preselected in the coefficient space of closed-loop characteristic polynomials. A constructive procedure for generating stable simplexes is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedu...

It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give a...

The efficient numerical treatment of high-dimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energy-norm based sparse gr...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

Consider the problem of minimizing a polynomial $f$ over compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates positivity polynomials due Putinar and Schmüdgen. When $\mathbf{X}$ is unit ball or standard simplex, we show that Schmüdgen-type...

We give a result that implies an improvement by a factor or IogIog n in the hypercube bounds for the geometric problems of hatched planar point location I trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched arc t...

Hypercube estimators for the mean vector in a general linear model include algebraic equivalents to penalized least squares estimatorswith quadratic penalties and to submodel least squares estimators. Penalized least squares estimators necessarily break down numerically for certain penalty matrices. Equivalent hypercube estimators resist this source of numerical instability. Under conditions, a...

In this paper, we consider the problem of embedding a given graph into a fixed-size hypercube. This work is related to other recent research on hypercube embedding. In [1], Bhat describes an order IVl log2 IVI algorithm for testing whether a given graph G = (V, E) is exactly an n-cube. Deciding whether a graph has a distance-preserving embedding into a hypercube can be done in polynomial time [...

We provide probabilistic lower bounds for the star discrepancy of Latin hypercube samples. These bounds are sharp in the sense that they match the recent probabilistic upper bounds for the star discrepancy of Latin hypercube samples proved in [M. Gnewuch, N. Hebbinghaus. Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples. Preprint 2016.]. Toge...

We show that by exchanging any two independent edges in any shortest cycle of the n-cube (n 2 3), its diameter decreases by one unit. This leads us to define a new class of n-regular graphs, denoted Ten, with Zn vertices and diameter n 1, which has the (n l)-cube as subgraph. Other properties of TQ,, such as connectivity and the lengths of the disjoints paths are also investigated. Moreover, we...