In this paper we prove that if a finite reflexive digraph admits Gumm operations, then it also admits a near unanimity operation. This is a generalization of similar results obtained earlier for posets and symmetric reflexive digraphs by the second author and his collaborators. In the special case of reflexive digraphs our new result confirms a conjecture of Valeriote that states that any finit...

In this paper X denotes a set, R, R1, R2 denote binary relations, x, y, z denote sets, and n, m, k denote natural numbers. Let us consider a binary relation R on X. Now we state the propositions: (1) fieldR ⊆ X. (2) If x, y ∈ R, then x, y ∈ X. Now we state the propositions: (3) Let us consider sets X, Y . Then (idX)Y = X ∩ Y . (4) 〈x, y〉 ∈ R |2 X if and only if x, y ∈ X and 〈x, y〉 ∈ R. (5) dom(...

(1) Let L be a non empty reflexive transitive relational structure and x, y be elements of L. If x ≤ y, then compactbelow(x)⊆ compactbelow(y). (2) For every non empty reflexive relational structure L and for every element x of L holds compactbelow(x) is a subset of CompactSublatt(L). (3) For every relational structure L and for every relational substructure S of L holds every subset of S is a s...

Let X be a NLS. Consider the set of all bounded linear functionals on X, and associate the norm defined above (as an exercise verify that it is indeed a norm). The resulting space, denoted X∗, is called the dual of X. It is another exercise to verify that X∗ is a Banach space (even if X isn’t). We can also consider the double dual X∗∗. It is easy to show that X is naturally imbedded in X∗∗ by t...

This paper follows the recent discussion on the sparse solution recovery with quasi-norms `q, q ∈ (0, 1) when the sensing matrix possesses a Restricted Isometry Constant δ2k (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if δ2k ≤ 1/2, any minimizer of the lq minimization, at lea...

We consider the scattering that is described by the equation (−∆ x + q(x, y) is a periodic function of y, q and f have compact supports with respect to x. We are interested in the solution satisfying the radiation condition at infinity and describe the asymptotic behavior of the solution as ǫ → 0. In addition, we find the asymptotic behavior of the scattering amplitude of the plain wave. Either...