We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a useful machinery for the study of completeness in these classes of surfaces. In pa...

We determine when the topological spaces |G| naturally associated with a graph G and its ends are metrizable or compact. In the most natural topology, |G| is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone’s theorem that metric spaces are paracompact. We show that |G| is compact in the most natural topology if and only if no finite vertex s...

this text carries out some ideas about exact and p− exact sequences of transformationsemigroups. some theorems like the short five lemma (lemma 1.3 and lemma 2.3) are valid here as inexact sequences of r − modules.

The referred equations in this Supplement are labeled as (S1.1), (S2.1) and so on, whereas labels such as (1), (2), etc. refer to equations the main text. We also include here additional lemmas and corollaries (and when necessary, their proofs), sometimes within the proof of an assertion in the main text. The proof of each assertion in the main text ends with a , and the proof of each additiona...

The main aim of this paper is to deal with a fuzzy version of Farkas lemma involving trapezoidal fuzzy numbers. In turns to that the fuzzy linear programming and duality theory on these problems can be used to provide a constructive proof for Farkas lemma. Keywords Farkas Lemma, Fuzzy Linear Programming, Duality, Ranking Functions.

In this paper, we attempt to investigate the connection between various types of ideals (for examples $(m, n)$-ideal, bi-ideal, interior ideal, quasi ideal, prime ideal and maximal ideal) of an ordered semigroup $(S,cdot ,leq)$ and the correspond hyperideals of its EL-hyperstructure $(S,*)$ (if exists). Moreover, we construct the class of EL-$Gamma$-semihypergroup associated to a partially orde...

The contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly p...

By means of a more precise treatment of the congruence relation between n-extensions we show that it is possible to avoid the Schanuel Lemma for proving some results on the functors Ext. An n-extension E from D to C is an exact sequence of the form (1) 0 C El E2 * * En D 0 in an abelian category. The idea to use congruence classes n-extensions as elements of Ext' (D C) is due to Yoneda ([7], [8...

We begin with three technical lemmas. Lemma 2 Let y ̸ = 0 and 0 < λ 1 ≤ ∥X T y∥ ∞. We have ⟨ y λ 1 − θ * 1 , θ * 1 ⟩ ≥ 0. Proof Since the Euclidean projection of y λ1 onto {θ : ∥X T θ∥ ∞ ≤ 1} is θ * 1 , it follows from Lemma 1 that ⟨θ * 1 − y λ 1 , θ − θ * 1 ⟩ ≥ 0, ∀θ : ∥X T θ∥ ∞ ≤ 1. (46) As 0 ∈ {θ : ∥X T θ∥ ∞ ≤ 1}, we have Eq. (45). Lemma 3 Let y ̸ = 0 and 0 < λ 1 ≤ ∥X T y∥ ∞. If θ * 1 parallel...