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Start Over Coverage 121-131

1. Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees

Creator:
Azarija, Jernej and Škrekovski, Riste
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
number of spanning trees and extremal graph
Language:
English
Description:
Let α(n) be the least number k for which there exists a simple graph with k vertices having precisely n ≥ 3 spanning trees. Similarly, define β(n) as the least number k for which there exists a simple graph with k edges having precisely n ≥ 3 spanning trees. As an n-cycle has exactly n spanning trees, it follows that α(n), β(n) ≤ n. In this paper, we show that α(n) ≤ 1⁄3 (n + 4) and β(n) ≤ 1⁄3 (n + 7) if and only if n /∈ {3, 4, 5, 6, 7, 9, 10, 13, 18, 22}, which is a subset of Euler’s idoneal numbers. Moreover, if n 6≡ 2 (mod 3) and n ≠ 25 we show that α(n) ≤ 1⁄4 (n + 9) and β(n) ≤ 1⁄4 (n + 13). This improves some previously estabilished bounds.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On Riesz homomorphisms in unital f-algebras

Creator:
Chil, Elmiloud
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
vector lattice, d-algebra, and f-algebra
Language:
English
Description:
The main topic of the first section of this paper is the following theorem: let A be an Archimedean f-algebra with unit element e, and T : A → A a Riesz homomorphism such that T 2 (f) = T(fT(e)) for all f ∈ A. Then every Riesz homomorphism extension Te of T from the Dedekind completion A δ of A into itself satisfies Te2 (f) = Te(fT(e)) for all f ∈ A δ . In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative d-algebras.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Trivial generators for nontrivial fibres

Creator:
Carlsson, Linus
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
holomorphic function, Banach algebra, and generator
Language:
English
Description:
Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in C n where the fibre is nontrivial, has to exceed n. This is shown not to be the case.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public