Lower bounds on Davenport–Schinzel sequences via rectangular Zarankiewicz matrices

Lower Bounds on Davenport-Schinzel Sequences via Rectangular Zarankiewicz Matrices

An order-s Davenport-Schinzel sequence over an n-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length s+2. The main problem is to determine the maximum length of such a sequence, as a function of n and s. When s is fixed this problem has been settled (see Agarwal, Sharir, and Shor [1], Nivasch [12] and Pettie [15]) but when s is a function of n, very li...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

On Lower and Upper Bounds of Matrices

k=1 |ak|. Hardy’s inequality can be interpreted as the lp operator norm of the Cesàro matrix C, given by cj,k = 1/j, k ≤ j, and 0 otherwise, is bounded on lp and has norm ≤ p/(p − 1) (The norm is in fact p/(p − 1)). It is known that the Cesàro operator is not bounded below, or the converse of inequality (1.1) does not hold for any positive constant. However, if one assumes C acting only on non-...

Lower Bounds of Copson Type for Hausdorff Matrices on Weighted Sequence Spaces

Let = be a non-negative matrix. Denote by the supremum of those , satisfying the following inequality: where , , and also is increasing, non-negative sequence of real numbers. If we used instead of The purpose of this paper is to establish a Hardy type formula for , where is Hausdorff matrix and A similar result is also established for where In particular, we apply o...

Lower bounds on Nullstellensatz proofs via designs

The Nullstellensatz proof system is a proof system for propositional logic based on algebraic identities in fields. Prior work has proved lower bounds of the degree Nullstellensatz refutations using combinatorial constructions called designs. This paper surveys the use of designs for Nullstellensatz lower bounds. We give a new, more general definition of designs. We present an explicit construc...