We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.

This paper presents an original practical criterion of global stability analysis of nonlinear polynomial systems. This criterion derived from the application of the Lyapunov direct method with a quadratic function generalizes the famous Lyapunov stability condition for linear systems. Useful mathematical transformations have allowed the formulation of the obtained conditions as an LMI (Linear M...

In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods [37, 51]. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to x−1 on a finite interval [λmin , λmax ], 0 < λmin < λmax in the ‖ ·‖∞ norm. The con...

Many error inequalities in polynomial interpolation can be found in [1, 7]. These error bounds for interpolating polynomials are usually expressed by means of the norms ‖ · ‖p, 1≤ p ≤∞. Some new error inequalities (for corrected interpolating polynomials) are given in [10, 11]. The last mentioned inequalities are similar to error inequalities obtained in recent years in numerical integration an...

In the min-knapsack problem one aims at choosing a set of objects with minimum total cost and total profit above a given threshold. In this paper, we study a class of valid inequalities for min-knapsack known as bounded pitch inequalities, which generalize the well-known unweighted cover inequalities. While separating over pitch-1 inequalities is NPhard, we show that approximate separation over...

We study the mixing inequalities which were introduced by Günlük and Pochet (2001). We show that a mixing inequality which mixes n MIR inequalities has MIR rank at most n if it is a type I mixing inequality and at most n− 1 if it is a type II mixing inequality. We also show that these bounds are tight for n = 2. Given a mixed-integer set PI = P ∩ Z(I) where P is a polyhedron and Z(I) = {x ∈ Rn ...

Two-step MIR inequalities are valid inequalities derived from a facet of a simple mixedinteger set with three variables and one constraint. In this paper we investigate how to effectively use these inequalities as cutting planes for general mixed-integer problems. We study the separation problem for single constraint sets and show that it can be solved in polynomial time when the resulting ineq...

The violation of certain Bell inequalities allows for device-independent information processing secure against nonsignaling eavesdroppers. However, this only holds for the Bell network, in which two or more agents perform local measurements on a single shared source of entanglement. To overcome the practical constraints that entangled systems can only be transmitted over relatively short distan...

We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial order, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates....

We propose a polynomial algorithm for a separable convex optimization problem with linear constraints. We do not make any additional assumptions about the structure of the objective function except for polynomial computability. That is, the objective function can be non-differentiable. The running time of our algorithm is polynomial in the size of the input consisting of an instance of the prob...