New lower bounds for partial k‐parallelisms

Some Lower Bounds for Estrada Index

Lower Bounds for Dynamic Partial Sums

Let G be a group. The partial sums problem asks to maintain an array A[1 . . n] of group elements, initialized to zeroes (a.k.a. the identity), under the following operations: update(k,∆): modify A[k]← ∆, where ∆ ∈ G. query(k): returns the partial sum ∑k i=1A[i]. For concreteness, let us work on a machine with w-bits words (w ≥ lg n), and take G to be Z/2wZ, i.e. integer arithmetic on machine w...

New Lower Bounds for -Nets

Following groundbreaking work by Haussler and Welzl (1987), the use of small -nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest -nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Könemann, and Sharpe)...

Upper and lower bounds for numerical radii of block shifts

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine ...

Some lower bounds for the $L$-intersection number of graphs

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...