Upward directedness of the Rudin-Keisler ordering of p-points

1990 ◽  
Vol 55 (2) ◽  
pp. 449-456
Author(s):  
Claude Laflamme
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Related Question ◽  
The Other ◽  
Other Hand ◽  
New Construction

It has been proven by Blass [1973] that any two P-points which have a P-point as a common upper bound in the Rudin-Keisler (RK) ordering necessarily have a common lower bound (necessarily a P-point). Hence two nonisomorphic Ramsey ultrafilters have neither a common lower nor a common upper bound which is a P-point. So in a model of CH for example (or MA, P(c),…), the RK ordering restricted to P-points is neither upward nor downward directed, since it is well known that nonisomorphic Ramsey ultrafilters exist in such models. On the other hand, we will see that in the model for “near coherence of filters” (NCF) produced by Blass and Shelah [1985], the RK ordering of P-points is upward, hence downward directed. This shows that the question of directedness of the RK ordering of P-points, upward or downward, cannot be decided in ZFC.There is a related question, asked by Blass in [1973], whether two P-points which have a common lower bound necessarily have a common upper bound which is a P-point. Our main result establishes the independence of this statement relative to ZFC. Its consistency will follow as soon as we show that the RK ordering of P-points is upward directed in the NCF model mentioned above, which we do in §2. But its independence will require a new construction, and will be given in §3.

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

scholarly journals Relative annihilators in semilattices

1973 ◽  
Vol 9 (2) ◽  
pp. 169-185 ◽  
Author(s):  
J.C. Varlet
Keyword(s):  
Lower Bound ◽  
Principal Ideal ◽  
The Other ◽  
Fixed Element ◽  
Other Hand ◽  
The One ◽  
Implicative Semilattices

An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.

Automorphism groups with some finiteness conditions

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Konrad Pióro
Keyword(s):  
Lower Bound ◽  
Binary Operation ◽  
Automorphism Groups ◽  
The Other ◽  
Finiteness Conditions ◽  
Other Hand ◽  
The Right

AbstractAll considered groups are torsion or do not contain infinitely generated subgroups. If such a groupNext, we show that ifThe Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, if

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

scholarly journals Nano topology induced by Lattices

2019 ◽  
Vol 38 (5) ◽  
pp. 197-204
Author(s):  
M. Lellis Thivagar ◽  
V. Sutha Devi
Keyword(s):  
Lower Bound ◽  
19Th Century ◽  
Upper Bound ◽  
Lattice Theory ◽  
Ordered Set ◽  
Equivalence Classes ◽  
The Other ◽  
Five Elements ◽  
Lower And Upper Approximations ◽  
The 19Th Century

Lattice is a partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. Dedekind worked on lattice theory in the 19th century. Nano topology explored by Lellis Thivagar et.al. can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. This is named as Nano topology, because of its size and what ever may be the size of universe it has atmost five elements in it. The elements of Nano topology are called the Nano open sets. This paper is to study the nano topology within the context of lattices. In lattice, there is a special class of joincongruence relation which is defined with respect to an ideal. We have defined the nano approximations of a set with respect to an ideal of a lattice. Also some properties of the approximations of a set in a lattice with respect to ideals are studied. On the other hand, the lower and upper approximations have also been studied within the context various algebraic structures.

Set-theoretical basis for real numbers

1950 ◽  
Vol 15 (4) ◽  
pp. 241-247
Author(s):  
Hao Wang
Keyword(s):  
Upper Bound ◽  
Theoretical Basis ◽  
The Other ◽  
Real Numbers ◽  
Full Generality ◽  
Ordered Field ◽  
Other Hand

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

scholarly journals Triebel-Lizorkin capacity and hausdorff measure in metric spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 617-624
Author(s):  
Nijjwal Karak
Keyword(s):  
Hausdorff Measure ◽  
Upper Bound ◽  
Metric Spaces ◽  
Measure Zero ◽  
Gauge Function ◽  
The Other ◽  
Other Hand ◽  
Zero Capacity

AbstractWe provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff h-measure zero for a suitable gauge function h.

scholarly journals Effectiveness of transposed inverse sets in faber regions

1983 ◽  
Vol 6 (2) ◽  
pp. 285-295 ◽  
Author(s):  
J. A. Adepoju ◽  
M. Nassif
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Main Result Theorem ◽  
Simple Set ◽  
Other Hand ◽  
Basic Set ◽  
Result Theorem ◽  
The One ◽  
The Given ◽  
Class Of Functions

The effectiveness properties, in Faber regions, of the transposed inverse of a given basic set of polynominals, are investigated in the present paper. A certain inevitable normalizing substitution, is first formulated, to be undergone by the given set to ensure the existence of the transposed inverse in the Faber region. The first main result of the present work (Theorem 2.1), on the one hand, provides a lower bound of the class of functions for which the normalized transposed inverse set is effective in the Faber region. On the other hand, the second main result (Theorem 5.2) asserts the fact that the normalized transposed inverse set of a simple set of polynomials, which is effective in a Faber region, should not necessarily be effective there.

Limit Design of Perforated Cylindrical Shells per ASME Code

1977 ◽  
Vol 99 (4) ◽  
pp. 646-651 ◽  
Author(s):  
J. S. Porowski ◽  
W. J. O’Donnell ◽  
J. R. Farr
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
The Other ◽  
Safety Factors ◽  
Design Standards ◽  
Asme Code ◽  
Optimal Spacing ◽  
Circular Holes ◽  
Margin Of Safety ◽  
Actual Limit

Limit pressures are evaluated for cylindrical perforated shells containing circular holes arranged in the various penetration patterns typically found in practice. Statically admissible discontinuous fields of stress are used to obtain rigorous lower-bound limit pressures. These solutions are shown to be quite close to the actual limit pressures based on the efficiency of the discontinuous fields of stress and a comparison with upper-bound solutions. The optimal spacing parameters are obtained over the interesting range of ligament efficiencies for both the diamond (romboidal) and rectangular penetration patterns. The results of this work show that for most penetration configurations, the present ASME standards require substantially thicker shells than would be needed to maintain the same safety factors that are used for unperforated shells. On the other hand, the British and German standards allow the use of thinner shells than would be needed to maintain the safety factors for unperforated shells. Revised design curves which provide the same margin of safety for all configurations are derived and are proposed for use in design standards.

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