Lower Bounds for the Multivariate Normal Mills' Ratio

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...

Some Lower Bounds for Estrada Index

Lower bounds for multivariate approximation by affine-invariant dictionaries

The problem of approximating locally smooth multivariate functions by linear combinations of elements from an affine-invariant redundant dictionary is considered. Augmenting recent upper bound results for approximation, we establish lower bounds on the performance of such schemes. The lower bounds are tight to within a logarithmic factor in the number of elements used in the approximation. Usin...

On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space Ed,α of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d every cubature rule which reduces the initial error necessarily uses at least d+ 1 function values. Since this holds independently of the number of permutation-...

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 ...