The main result of this paper is the extension of the Schur–Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if ∑n j=1 ξj ∑n j=1 ηj for every n if and only if ξ = Qη for some orthostochastic matrix Q. When ξ and η are summable, requirin...

for each n ∈ N. Note that each On is open since it is a union of open sets (cf. Theorem 2.24(a) in Rudin). We claim that A = ⋂∞n=1 On. First, if a ∈ A, then a ∈ (a− 1/n, a+ 1/n) for each n ∈ N, so a ∈ On for each n ∈ N, and hence a ∈ ⋂∞n=1 On. Thus, A ⊆ ⋂∞n=1 On. Conversely, assume x ∈ R and x / ∈ A. Since A is closed, it follows that x is not a limit point of A, so there exists an ε > 0 such t...

A family of sets A is said to be union-closed if {A∪B : A, B ∈ A} ⊂ A. Frankl’s conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of P...

We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In [7] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegene...

Most readers are puzzled to find that the concepts of set and mapping were simply regarded as logical ones. With regards to the concept of set, this was a usual view in the late 19 th century, well represented for instance with Schröder, Peano and Russell. As Russell said [...], logic consists of three parts: the theory of propositions, the theory of classes, and the theory of relations. [...] ...

For an artinian ring R, the directed zero-divisor graph Γ(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for finite ring R, especially, it is proved that for any ring R, if there exists a source b in Γ(R) with b = 0, then |R| = 4 and R = {0, a, b, c}, where a and c are left identity elements and ba = 0 = bc. S...

Abstmct# A decomposition is given for fini*.e ordered sets P and is shown to bc a unique decomposition in the sense of Brylawski. Hence there exists a universal invariant g(P) for this decomposition , and we c(Dmpute g(P) explicitly. Some modifications of this decomposition are considered ; in particular, one which forms a bidecomposition toecther with disjoint union.

Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...

We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, wh...