The Projective Antecedent of the Three Reflection Theorem

1991 ◽  
Vol 34 (2) ◽  
pp. 265-274
Author(s):  
F. A. Sherk
Keyword(s):  
Projective Geometry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Metric Geometry ◽  
Natural Question ◽  
Complete Answer ◽  
Special Cases ◽  
Reflection Theorem ◽  
Necessary And Sufficient

AbstractA complete answer is given to the question: Under what circumstances is the product of three harmonic homologies in PG(2, F) again a harmonic homology ? This is the natural question to ask in seeking a generalization to projective geometry of the Three Reflection Theorem of metric geometry. It is found that apart from two familiar special cases, and with one curious exception, the necessary and sufficient conditions on the harmonic homologies produce exactly the Three Reflection Theorem.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

scholarly journals H-function with complex parameters I: existence

2001 ◽  
Vol 25 (9) ◽  
pp. 571-586
Author(s):  
Fadhel A. Al-Musallam ◽  
Vu Kim Tuan
Keyword(s):  
Sufficient Conditions ◽  
Integral Transforms ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Type Integral ◽  
Necessary And Sufficient

AnH-function with complex parameters is defined by a Mellin-Barnes type integral. Necessary and sufficient conditions under which the integral defining theH-function converges absolutely are established. Some properties, special cases, and an application to integral transforms are given.

THE QUASITRIANGULAR STRUCTURES FOR ω-SMASH COPRODUCT HOPF ALGEBRAS

2009 ◽  
Vol 08 (05) ◽  
pp. 673-687 ◽  
Author(s):  
ZHENGMING JIAO
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Necessary And Sufficient

In this paper, the quasitriangular structures of ω-smash coproduct Hopf algebras Bω ⋈ H as constructed by Caenepeel, Ion, Militaru and Zhu were studied. Necessary and sufficient conditions for ω-smash coproduct Hopf algebras to be quasitriangular Hopf algebras are given in terms of properties of their components. As applications of our results, some special cases are discussed. Especially, The quasitriangular structures for D(H)* and H4ω ⋈ kℤ2 are constructed.

scholarly journals A note on the intersection of free maximal ideals

1969 ◽  
Vol 10 (1-2) ◽  
pp. 204-206 ◽  
Author(s):  
Stewart M. Robinson
Keyword(s):  
Compact Support ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Structure ◽  
Maximal Ideals ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Ring Of Functions

In [2] we proved that if X admits a complete uniform structure, the intersection of the free maximal ideals in C(x) is precisely Ck(X), the ring of functions with compact support. In the present paper we are able to sharpen this result somewhat and give necessary and sufficient conditions on a space X so that this conclusion holds. Both our previous result and that of Kohls for p-spaces follow as special cases of our theorem.

scholarly journals On the Solutions of Some Linear Complex Quaternionic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Cennet Bolat ◽  
Ahmet İpek
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sylvester Equation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Linear Complex ◽  
The Real ◽  
Special Cases ◽  
Necessary And Sufficient

Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

On the asphericity of one-relator relative presentations

1994 ◽  
Vol 124 (4) ◽  
pp. 713-728 ◽  
Author(s):  
Martin Edjvet
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Necessary And Sufficient

We study the 1-relator relative presentation 〈H, x|xaxbx−1c〉 where H is a group, a, b, c ∈ H, x ∉ H and b, c ≠ 1. We give necessary and sufficient conditions for this presentation to be aspherical apart from two outstanding special cases which remain open.

Graphs with cyclomatic number three having panconnected square

2017 ◽  
Vol 09 (05) ◽  
pp. 1750067
Author(s):  
G. L. Chia ◽  
W. Hemakul ◽  
S. Singhun
Keyword(s):  
Sufficient Conditions ◽  
Discrete Math ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Cyclomatic Number ◽  
Number Formula ◽  
Necessary And Sufficient ◽  
Hamiltonian Properties ◽  
Square Of A Graph

The square of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding edges joining those pairs of vertices whose distance from each other in [Formula: see text] is two. If [Formula: see text] is connected, then the cyclomatic number of [Formula: see text] is defined as [Formula: see text]. Graphs with cyclomatic number not more than [Formula: see text] whose square are panconnected have been characterized, among other things, in [G. L. Chia, S. H. Ong and L. Y. Tan, On graphs whose square have strong Hamiltonian properties, Discrete Math. 309 (2009) 4608–4613, G. L. Chia, W. Hemakul and S. Singhun, Graphs with cyclomatic number two having panconnected square, Discrete Math. 311 (2011) 850–855]. Here, we show that if [Formula: see text] has cyclomatic number [Formula: see text] and [Formula: see text] is panconnected, then [Formula: see text] is one of the eight families of graphs, [Formula: see text], defined in the paper. Further, we obtain necessary and sufficient conditions for three larger families of graphs (which contains [Formula: see text] as special cases) whose square are panconnected.

On ω-consistency and related properties

1956 ◽  
Vol 21 (3) ◽  
pp. 246-252 ◽  
Author(s):  
Steven Orey
Keyword(s):  
Natural Number ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Model ◽  
Special Cases ◽  
Necessary And Sufficient

1. This paper grew out of an attempt to answer a question raised in [4]. Let a logic L containing “numerals” z1, z2, … and a certain statement N(x) (intended to express the proposition that x is a natural number) be called ω-inconsistent if there is a statement such that ⊦ F(zk) for k = 1, 2, …, and ⊦ ∼(x)·N(x) ⊃ F(x); then it is evident that L cannot have a model in which N(x) is satisfied by the images of the numerals and nothing else if L is ω-inconsistent.Question: If L is ω-consistent, i.e. not ω-inconsistent, must there be such a model? Calling a model of the kind just described a special model, we ask for necessary and sufficient conditions on L to insure the existence of a special model. We give several sets of such conditions, applicable to a certain very inclusive class of logics, in Theorem 1 and Theorems 3 and 4. Theorem 2 shows that a logic may be ω-consistent but still not have a special model.This paper was close to completion when [3] appeared. For systems with only denumerably many symbols our results include Henkin's, for, by adjoining a new predicate N(x) to each of the systems considered in [3] which have only a denumerable number of constant symbols and then adding as an axiom (x)N(x), these systems become special cases of the systems we consider. It is easily seen that Henkin's Theorem 7 essentially proves the equivalence of conditions (2) and (3) in our Theorem 1, and Theorem 3 of [3] corresponds to our Theorem 2. Incidentally, our argument of Theorem 2 could also serve to prove Henkin's Theorem 6.

Sign in / Sign up

Export Citation Format

Share Document