The shadow of a collection A of k-sets is defined as the collection of the (k − 1)-sets which are contained in at least one k-set of A. Given |A|, the size of the shadow is minimum when A is the family of the first k-sets in squashed order (by definition, a k-set A is smaller than a k-set B in the squashed order if the largest element of the symmetric difference of A and B is in B). We give a t...

‎semidiscrete finite element approximation of a hyperbolic type‎ ‎integro-differential equation is studied. the model problem is‎ ‎treated as the wave equation which is perturbed with a memory term.‎ ‎stability estimates are obtained for a slightly more general problem.‎ ‎these, based on energy method, are used to prove optimal order‎ ‎a priori error estimates.‎

We prove that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and describe its monolith. The endomorphism semiring is congruence simple if and only if the semilattice has both a least and a largest element.

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element. 1

and where the first inequality follows from the definition (2) of the weak-learning condition. Let λ∗ be a minimizer of the min-max expression. Unless the first entry of each-row of (Hλ∗ −B) is the largest, the right hand side of the min-max expression can be made arbitrarily large by choosing C ∈ Ceor appropriately. For example, if in some row i, the j0th element is strictly larger than the fi...

We consider a linear single-input single-output system on a Hilbert space X , with infinitesimal generator A, bounded control element b, and bounded observation element c. We address the problem of finding the largest feedback invariant subspace of X that is in the space c⊥ perpendicular to c. If b is not in c⊥, we show this subspace is c⊥. If b is in c⊥, a number of situations may occur, depen...