The Lattice of all Topologies is Complemented

1968 ◽  
Vol 20 ◽  
pp. 805-807 ◽  
Author(s):  
A. C. M. Van Rooij
Keyword(s):  
Affirmative Answer ◽  
Alternative Proof ◽  
Finite Set

In (3), J. Hartmanis raised the question whether the lattice of all topologies in a given set is complemented and gave the affirmative answer for the case of a finite set. H. Gaifman (2), has extended this result to denumerable sets. Using Gaifman's paper, Anne K. Steiner (4) has proved that the lattice is always complemented. Our aim in this article is to give an alternative proof, independent of Gaifman's results. So far, Steiner's proof has not been available to the author.

Implicational formulas in intuitionistic logic

1974 ◽  
Vol 39 (4) ◽  
pp. 661-664 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Equivalence Classes ◽  
Alternative Proof ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Free Generators ◽  
Finite Set ◽  
Image Position ◽  
The Right

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

scholarly journals Rates of convergence to normality for samples from a finite set of random variables

1996 ◽  
Vol 60 (3) ◽  
pp. 355-362 ◽  
Author(s):  
R. D. John ◽  
J. Robinson
Keyword(s):  
Finite Population ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Alternative Proof ◽  
Independent Variables ◽  
First Case ◽  
Finite Set

AbstractRates of convergence to normality of O(N-½) are obtained for a standardized sum of m random variables selected at random from a finite set of N random variables in two cases. In the first case, the sum is randomly normed and the variables are not restricted to being independent. The second case is an alternative proof of a result due to von Bahr, which deals with independent variables. Both results derive from a rate obtained by Höglund in the case of sampling from a finite population.

POSI-MODULAR SYSTEMS WITH MODULOTONE REQUIREMENTS UNDER PERMUTATION CONSTRAINTS

2010 ◽  
Vol 02 (01) ◽  
pp. 61-76 ◽  
Author(s):  
TOSHIMASA ISHII ◽  
KAZUHISA MAKINO
Keyword(s):  
Greedy Algorithm ◽  
Location Problem ◽  
Monotone Function ◽  
Modular Function ◽  
Discrete Math ◽  
Natural Generalization ◽  
Source Location ◽  
Alternative Proof ◽  
Modular Systems ◽  
Finite Set

Given a system (V, f, r) on a finite set V consisting of a posi-modular function f : 2V → ℝ and a modulotone function r : 2V → ℝ, we consider the problem of finding a minimum set R ⊆ V such that f(X) ≥ r(X) for all X ⊆ V - R. The problem, called the transversal problem, was introduced in [M. Sakashita, K. Makino, H. Nagamochi and S. Fujishige, Minimum transversals in posi-modular systems, SIAM J. Discrete Math.23 (2009) 858–871] as a natural generalization of the source location problem and external network problem with edge-connectivity requirements in undirected graphs and hypergraphs. By generalizing [H. Tamura, H. Sugawara, M. Sengoku and S. Shinoda, Plural cover problem on undirected flow networks, IEICE Trans.J81-A (1998) 863–869] for the source location problem, we show that the transversal problem can be solved by a simple greedy algorithm if r is π-monotone, where a modulotone function r is π-monotone if there exists a permutation π of V such that the function pr: V × 2V → ℝ associated with r satisfies pr(u, W) ≥ pr(v, W) for all W ⊆ V and u, v ∈ V with π(u) ≥ π(v). Here we show that any modulotone function r can be characterized by pr as r(X) = max {pr(v, W) | v ∈ X ⊆ V - W}. We also show the structural properties on the minimal deficient sets [Formula: see text] for the transversal problem for π-monotone function r, i.e., there exists a basic tree T for [Formula: see text] such that π(u) ≤ π(v) for all arcs (u,v) in T, which, as a corollary, gives an alternative proof for the correctness of the greedy algorithm for the source location problem. Furthermore, we show that a fractional version of the transversal problem can be solved by the algorithm similar to the one for the transversal problem.

scholarly journals Retract Rationality and Algebraic Tori

2019 ◽  
Vol 63 (1) ◽  
pp. 173-186 ◽  
Author(s):  
Federico Scavia
Keyword(s):  
Prime Number ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Algebraic Tori ◽  
Finite Set ◽  
Multiplicative Type

AbstractFor any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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