On a Transformation Group

1969 ◽  
Vol 21 ◽  
pp. 935-941 ◽  
Author(s):  
S. K. Kaul
Keyword(s):  
Finite Order ◽  
Metric Space ◽  
Transformation Group ◽  
Real Numbers ◽  
Compact Metric Space ◽  
Topological Transformation ◽  
Linear Fractional Transformations ◽  
Group Of Homeomorphisms

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.

Discontinuity Conditions on Transformation Groups

1972 ◽  
Vol 15 (3) ◽  
pp. 417-419
Author(s):  
D. V. Thompson
Keyword(s):  
Metric Space ◽  
Transformation Group ◽  
Convergence Condition ◽  
Transformation Groups ◽  
Compact Metric Space ◽  
Topological Transformation ◽  
Discontinuity Conditions ◽  
Group 1

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

scholarly journals Review of Suppes 1957 Proposals For Division by Zero

2021 ◽  
Author(s):  
James Anderson ◽  
Jan Bergstra
Keyword(s):  
Real Numbers ◽  
Integral Calculus ◽  
Arithmetical Functions ◽  
First Order ◽  
Advantages And Disadvantages ◽  
Definition Of ◽  
Type 3 ◽  
Order Predicate Calculus

We review the exposition of division by zero and the definition of total arithmetical functions in ``Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to ``the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory. We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations are of Extension Type at least 1. Totalisations of the differential and integral calculus have Extension Type at least 2. In particular, Meadows have Extension Type 1, Wheels have Extension Type 2, and Transreal numbers have Extension Type 3. It appears that Suppes was the modern originator of the idea that all real numbers divided by zero are equal to zero. This has Extension Type 0 and is, therefore, arbitrary.

scholarly journals A theorem on homeomorphism groups and products of spaces

1969 ◽  
Vol 1 (1) ◽  
pp. 137-141 ◽  
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Cantor Set ◽  
Compact Metric Space ◽  
Compact Metric Spaces ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the Cantor set, C, onto itself. Let p: C → M be a map of C onto a compact metric space M, and let G(p, M) be is a group.The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn (xn) → y Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C) The present paper exhibits a structure theorem connecting these characterizing subgroups of H(C) and products of spaces: Let M1 and M2 be compact metric spaces. Then there are standard maps p: C → M1 × M2 and pi: C → Mi, i = 1, 2, such that G(p, M1 × M2) = G(p1, M1) ∩ G(p2, M2).

Type 2 Diabetes Overtakes Type 1 in Hispanic Girls

2008 ◽  
Vol 38 (15) ◽  
pp. 18
Author(s):  
SHERRY BOSCHERT
Keyword(s):  
Type 2 Diabetes

Appropriateness and Timing of Kidney and/or Pancreas Transplants in Type 1 and Type 2 Diabetes

2001 ◽  
Vol 8 (1) ◽  
pp. 70-82 ◽  
Author(s):  
Amy L. Friedman
Keyword(s):  
Type 2 Diabetes

The efficacy and safety of LY2963016 among patients with Type 1 and Type 2 diabetes previously treated with insulin glargine

2015 ◽  
Vol 10 (S 01) ◽  
Author(s):  
D Dahl ◽  
LB Lacaya ◽  
RK Pollom ◽  
LL Ilag ◽  
JS Zielonka
Keyword(s):  
Type 2 Diabetes ◽  
Insulin Glargine ◽  
Efficacy And Safety ◽  
Previously Treated

Molecular analysis of FVIII gene in severe HA patients of Costa Rica

2010 ◽  
Vol 30 (S 01) ◽  
pp. S150-S152
Author(s):  
G. Jiménez-Cruz ◽  
M. Mendez ◽  
P. Chaverri ◽  
P. Alvarado ◽  
W. Schröder ◽  
...  
Keyword(s):  
Factor Viii ◽  
Coagulation Factor ◽  
Costa Rican ◽  
Factor Viii Gene ◽  
Intron 22 Inversion ◽  
Intron 1 ◽  
Polymerase Chain ◽  
Inversion Analysis

SummaryHaemophilia A (HA) is X-chromosome linked bleeding disorders caused by deficiency of the coagulation factor VIII (FVIII). It is caused by FVIII gene intron 22 inversion (Inv22) in approximately 45% and by intron 1 inversion (Inv1) in 5% of the patients. Both inversions occur as a result of intrachromosomal recombination between homologous regions, in intron 1 or 22 and their extragenic copy located telomeric to the FVIII gene. The aim of this study was to analyze the presence of these mutations in 25 HA Costa Rican families. Patients, methods: We studied 34 HA patients and 110 unrelated obligate members and possible carriers for the presence of Inv22or Inv1. Standard analyses of the factor VIII gene were used incl. Southern blot and long-range polymerase chain reaction for inversion analysis. Results: We found altered Inv22 restriction profiles in 21 patients and 37 carriers. It was found type 1 and type 2 of the inversion of Inv22. During the screening for Inv1 among the HA patient, who were Inv22 negative, we did not found this mutation. Discussion: Our data highlight the importance of the analysis of Inv22 for their association with development of inhibitors in the HA patients and we are continuous searching of Inv1 mutation. This knowledge represents a step for genetic counseling and prevention of the inhibitor development.

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