scholarly journals Coefficients of an analytic function subordination class determined by rotations

1996 ◽  
Vol 60 (2) ◽  
pp. 245-254
Author(s):  
Seok Chan Kim
Keyword(s):  
Analytic Function ◽  
Convex Hull ◽  
Extreme Points ◽  
Closed Convex Hull ◽  
Image Position ◽  
Positive Integers ◽  
Compact Subsets

AbstractLet A denote the set of all functions analytic in U = {z: |;z| < 1} equipped with the topology of unifrom convergence on compact subsets of U. For F ∈ A define Let s(F) and s(F) denote the closed convex hull of s(F) and the set of extreme points of , respectively. Let R denote the class of all F ∈ A such that = {Fx}: |x| = 1} where Fx = F(xz).We prove that |An| ≤ |AMN| for all positive integers M and N, and for . We also prove that if , then F is a univelaent halfplane mapping.

scholarly journals Some results on isometric composition operators on Lipschitz spaces

2021 ◽  
Author(s):  
Abraham Rueda Zoca
Keyword(s):  
Convex Hull ◽  
Composition Operator ◽  
Metric Spaces ◽  
Composition Operators ◽  
Extreme Points ◽  
Necessary Condition ◽  
Closed Convex Hull ◽  
Lipschitz Spaces ◽  
Complete Characterisation

AbstractGiven two metric spaces M and N we study, motivated by a question of N. Weaver, conditions under which a composition operator $$C_\phi :{\mathrm {Lip}}_0(M)\longrightarrow {\mathrm {Lip}}_0(N)$$ C ϕ : Lip 0 ( M ) ⟶ Lip 0 ( N ) is an isometry depending on the properties of $$\phi $$ ϕ . We obtain a complete characterisation of those operators $$C_\phi $$ C ϕ in terms of a property of the function $$\phi $$ ϕ in the case that $$B_{{\mathcal {F}}(M)}$$ B F ( M ) is the closed convex hull of its preserved extreme points. Also, we obtain necessary condition for $$C_\phi $$ C ϕ being an isometry in the case that M is geodesic.

Some more characterizations of Banach spaces containing l1

1976 ◽  
Vol 80 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Richard Haydon
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Subset ◽  
Compact Convex ◽  
Extreme Points ◽  
Compact Convex Subset ◽  
Closed Convex Hull ◽  
Separability Assumption

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.

On the Krein-Milman theorem in CAT(κ) spaces

2018 ◽  
Vol 34 (3) ◽  
pp. 401-404
Author(s):  
BANCHA PANYANAK ◽  
Keyword(s):  
Convex Hull ◽  
Convex Subset ◽  
Compact Convex ◽  
Extreme Points ◽  
General Setting ◽  
Compact Convex Subset ◽  
Closed Convex Hull ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset

Let κ > 0 and (X, ρ) be a complete CAT(κ) space whose diameter smaller than ... It is shown that if K is a nonempty compact convex subset of X, then K is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT(κ) spaces.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals Monochromatic solutions to equations with unit fractions

1991 ◽  
Vol 43 (3) ◽  
pp. 387-392 ◽  
Author(s):  
Tom C. Brown ◽  
Voijtech Rödl
Keyword(s):  
Solutions To Equations ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

The Min-Max Load Criteria as a Measure of Machining Fixture Performance

1994 ◽  
Vol 116 (4) ◽  
pp. 500-507 ◽  
Author(s):  
E. C. DeMeter
Keyword(s):  
Convex Hull ◽  
Linear Model ◽  
Model Validation ◽  
Contact Region ◽  
Extreme Points ◽  
Simulation Experiments ◽  
Machining Operations ◽  
The Impact ◽  
External Loads ◽  
The Continuum

Spherical-tipped locators and clamps are often used for the restraint of castings during machining. For structurally rigid castings, contact region deformation and micro-slippage are the predominant modes of workpiece displacement. In turn contact region deformation and micro-slippage are heavily influenced by contact region loading. This paper presents a linear model for predicting the impact of locator and clamp placement on workpiece displacement throughout a series of machining operations. It illustrates how the continuum of external loads exerted on a workpiece during machining can be bounded within a convex hull, and how the extreme points of this hull are used within the model. Finally it describes the simulation experiments which were used for model validation.

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