scholarly journals Maximal topologies

1973 ◽  
Vol 15 (3) ◽  
pp. 279-290 ◽  
Author(s):  
Asit Baran Raha
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Compact Spaces ◽  
Other Hand ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Necessary And Sufficient

This article is devoted to studying maximal π spaces where π = Lindelöf, countably compact, connected, lightly compact or pseudocompact. Necessary and sufficient conditions for Lindelöf or countably compact spaces to be maximal Lindelöf or maximal countably compact have been obtained. On the other hand only necessary conditions for maximal π spaces have been deduced where π = connected, lightly compact or pseudocompact.

Fair bets and inductive probabilities

1955 ◽  
Vol 20 (3) ◽  
pp. 263-273 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Inductive Probability ◽  
Other Hand ◽  
Frequency Interpretation ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

scholarly journals Two theorems on generalised metric spaces

1997 ◽  
Vol 55 (2) ◽  
pp. 197-206
Author(s):  
Sergey Svetlichny
Keyword(s):  
Compact Space ◽  
Distance Function ◽  
Weak Topology ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Compact Spaces ◽  
Countably Compact Space ◽  
Countably Compact ◽  
Necessary And Sufficient

We prove that any compact space, and even any countably compact space having the weak topology with respect to a sequence of symmetrisable subspaces, is metrisable. This generalises results of Arhangel'skii and Nedev on metrisability of symmetrisable compact spaces. Also we define and study contraction functions on generalised metric spaces whose topology can be described in terms of a ‘distance function’ which is not quite a metric. In particular we present necessary and sufficient conditions for a space of countable pseudo-character to be submetrisable in terms of real-valued contraction functions on this space.

On the Periodicity of Compositions of Entire Functions

1966 ◽  
Vol 18 ◽  
pp. 724-730 ◽  
Author(s):  
Fred Gross
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
General Problem ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Periodic Entire Function

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Hideto Nakashima
Keyword(s):  
Sufficient Conditions ◽  
Rational Functions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Symmetric Cones ◽  
The Real ◽  
Tube Domains ◽  
Homogeneous Cone ◽  
Necessary And Sufficient ◽  
Relative Invariants

AbstractIn this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].

Normal and osculating maps for submanifolds of RN

1990 ◽  
Vol 114 (1-2) ◽  
pp. 39-55 ◽  
Author(s):  
I. Cattaneo Gasparini ◽  
G. Romani
Keyword(s):  
Isometric Immersion ◽  
Normal Bundle ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Precise Formulation ◽  
Parallel Case ◽  
Necessary And Sufficient

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

AN W ALGEBRAS AND CLASSIFICATION

1992 ◽  
Vol 07 (36) ◽  
pp. 3419-3423
Author(s):  
LIU CHAO ◽  
BOYU HOU
Keyword(s):  
Lie Algebra ◽  
Regular Element ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arbitrary Degree ◽  
Wznw Model ◽  
Necessary And Sufficient

The necessary and sufficient conditions for the existence of a regular element of arbitrary degree under arbitrary integral gradation of the Lie algebra g is presented. Such elements, while chosen as constraints in WZNW model, give rise to a W-algebra. It is then found that there might be some isomorphic relations between different W-algebras. The necessary conditions for such isomorphisms to appear are also given. Up to the A4 cases these conditions are checked to be sufficient.

Embedding theorems and the spectra of certain differential operators

1991 ◽  
Vol 434 (1892) ◽  
pp. 643-656 ◽  
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Interest ◽  
Embedding Theorems ◽  
Necessary And Sufficient

This paper is concerned with problems of the form n Ʃ k =0 (─1) k ( ρ 2 k y ( k ))( k ) = λ r 2 y on R , ry ∈ L 2 ( R ) and gives conditions on the coefficients sufficient to ensure that the spectrum is discrete; necessary conditions are also given. In certain circumstances, necessary and sufficient conditions for discreteness are given, thus extending the celebrated Molcanov criterion. These results follow from embedding theorems which have independent interest.

scholarly journals ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

2011 ◽  
Vol 84 (2) ◽  
pp. 238-254 ◽  
Author(s):  
ROSWITHA HOFER ◽  
PETER KRITZER
Keyword(s):  
Uniform Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Future Research ◽  
Short Summary ◽  
The One ◽  
Halton Sequences ◽  
Necessary And Sufficient

AbstractWe discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

scholarly journals On a Periodic Solution of the 4-Body Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Jian Chen ◽  
Bingyu Li
Keyword(s):  
Periodic Solution ◽  
Angular Velocity ◽  
Periodic Motion ◽  
Equilateral Triangle ◽  
Sufficient Conditions ◽  
Constant Angular Velocity ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
The Masses ◽  
Necessary And Sufficient

We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular velocity about a resting particle. We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.

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