Polynomial Hulls of Sets Invariant Under an Action of the Special Unitary Group

1988 ◽  
Vol 40 (5) ◽  
pp. 1256-1271
Author(s):  
John T. Anderson
Keyword(s):  
Continuous Functions ◽  
Difficult Problem ◽  
Ideal Space ◽  
Analytic Structure ◽  
Compact Sets ◽  
Polynomially Convex ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
Image Position

If K is a compact subset of Cn, will denote the polynomial hull of K: arises in the study of uniform algebras as the maximal ideal space of the algebra P(K) of uniform limits on K of polynomials (see [3]). The condition (K is polynomially convex) is a necessary one for uniform approximation on K of continuous functions by polynomials (P(K) = C(K)). If K is not polynomially convex, the question of existence of analytic structure in is of particular interest. For n = 1, is the union of K and the bounded components of C\K. The determination of in dimensions greater than one is a more difficult problem. Among the special classes of compact sets K whose polynomial hulls have been determined are those invariant under certain group actions on Cn.

SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

2020 ◽  
Vol 102 (2) ◽  
pp. 186-195
Author(s):  
WEILIANG WANG ◽  
LU LI
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Continuous Functions ◽  
Lipschitz Functions ◽  
Image Position

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set $$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$ where $\unicode[STIX]{x1D70F}_{1}$, $\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with $\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all $x,y\in [0,1]$.

scholarly journals Systems of derivations on topological algebras of power series

1975 ◽  
Vol 19 (3) ◽  
pp. 358-370
Author(s):  
Henry J. Schultz
Keyword(s):  
Power Series ◽  
Regularity Condition ◽  
Banach Algebras ◽  
Binomial Coefficient ◽  
Continuous Functions ◽  
Analytic Structure ◽  
Topological Algebras ◽  
Linear Maps ◽  
The Many ◽  
Image Position

If Do, D1, … are linear maps from an algebra A to an algebra B, both over the complexes, then {Do, D1, …} is a system of derivations if for all a, b in A and for all nonnegative integers k, we have Where C(k, i) is the binomial coefficient k!/i! (k—i)!. By (1.1) we see that Do must be a homomorphism and in case Do = I, where I is the identity map, D1 is a derivation and, for k ≧ 2, the Dk are higher derivations in the sense of Jacobson (1964), page 191. Gulick (1970), Theorem 4.2, proved that if A is a commutative regular semi-simple F-algebra with identity and {DO, D1, …} is a system of derivations from A to B = C(S(A)), the algebra of all continuous functions on the spectrum of A, where Dox = x, then the Dk are all continuous. Carpenter (1971), Theorem 5, shows that the regularity condition is unnecessary and Loy (1973) generalizes this a bit further. One of the many interesting features of systems of derivations is that they help determine analytic structure in Banach algebras (see for example, Miller (to appear)).

The determination of convex bodies from the mean of random sections

1992 ◽  
Vol 112 (2) ◽  
pp. 419-430 ◽  
Author(s):  
Paul Goodey ◽  
Wolfgang Weil
Keyword(s):  
Convex Bodies ◽  
Random Set ◽  
Compact Sets ◽  
Random Line ◽  
Random Sections ◽  
The Mean ◽  
Linear Sections ◽  
Image Position ◽  
Interior Points

Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = K ∩ E is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull

A remark on topologies for characteristic functions

1970 ◽  
Vol 68 (3) ◽  
pp. 703-705 ◽  
Author(s):  
David Kendall ◽  
John Lamperti
Keyword(s):  
Fourier Transforms ◽  
General Setting ◽  
Continuous Functions ◽  
Characteristic Functions ◽  
Compact Sets ◽  
Product Topology ◽  
Sequential Convergence ◽  
Borel Probability ◽  
Image Position ◽  
Bounded Continuous Functions

Let M1 denote the set of Borel probability measures on the real line and M1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M1 and M1 is in fact a homeomorphism, provided M1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, ifwhere μn ∈ M1 (and so φn ∈ M1), then φn(λ)→φ0(λ) uniformly on all compact sets if and only if μn ⇒ μ0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φn(λ)→φ0 pointwise (for all λ), the conclusion that μn ⇒ μ0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative, even in a much more general setting.

Graph Theory and Probability

1959 ◽  
Vol 11 ◽  
pp. 34-38 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Explicit Construction ◽  
Difficult Problem ◽  
Complete Subgraph ◽  
Image Position ◽  
Set Of Points

A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) — 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.) The determination of g(n) seems a very difficult problem; the best inequalities for g(n) are (3)It is not even known that g(n)1/n tends to a limit. The lower bound in (1) has been obtained by combinatorial and probabilistic arguments without an explicit construction.

Tribology of rail bogie

2018 ◽  
Vol 77 (4) ◽  
pp. 230-240
Author(s):  
D. P. Markov
Keyword(s):  
Contact Fatigue ◽  
Basic Element ◽  
Difficult Problem ◽  
Railway Track ◽  
Rolling Stock ◽  
Kinematic Parameters ◽  
Approximate Estimation ◽  
Railway Bogie ◽  
Classical Calculation

Railway bogie is the basic element that determines the force, kinematic, power and other parameters of the rolling stock, and its movement in the railway track has not been studied enough. Classical calculation of the kinematic and dynamic parameters of the bogie's motion with the determination of the position of its center of rotation, the instantaneous axes of rotation of wheelsets, the magnitudes and directions of all forces present a difficult problem even in quasi-static theory. The paper shows a simplified method that allows one to explain, within the limits of one article, the main kinematic and force parameters of the bogie movement (installation angles, clearance between the wheel flanges and side surfaces of the rails), wear and contact damage to the wheels and rails. Tribology of the railway bogie is an important part of transport tribology, the foundation of the theory of wheel-rail tribosystem, without which it is impossible to understand the mechanisms of catastrophic wear, derailments, contact fatigue, cohesion of wheels and rails. In the article basic questions are considered, without which it is impossible to analyze the movement of the bogie: physical foundations of wheel movement along the rail, types of relative motion of contacting bodies, tribological characteristics linking the force and kinematic parameters of the bogie. Kinematics and dynamics of a two-wheeled bogie-rail bicycle are analyzed instead of a single wheel and a wheelset, which makes it clearer and easier to explain how and what forces act on the bogie and how they affect on its position in the rail track. To calculate the motion parameters of a four-wheeled bogie, it is represented as two two-wheeled, moving each on its own rail. Connections between them are replaced by moments with respect to the point of contact between the flange of the guide wheel and the rail. This approach made it possible to give an approximate estimation of the main kinematic and force parameters of the motion of an ideal bogie (without axes skewing) in curves, to understand how the corners of the bogie installation and the gaps between the flanges of the wheels and rails vary when moving with different speeds, how wear and contact injuries arise and to give recommendations for their assessment and elimination.

ON THREADING CONTINUOUS FUNCTIONS THROUGH COMPACT SETS

1982 ◽  
Vol 8 (2) ◽  
pp. 455
Author(s):  
Akemann ◽  
Bruckner
Keyword(s):  
Continuous Functions ◽  
Compact Sets

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals Determination of mechanical properties by nanoindentation independently of indentation depth measurement

2012 ◽  
Vol 27 (19) ◽  
pp. 2551-2560 ◽  
Author(s):  
Gaylord Guillonneau ◽  
Guillaume Kermouche ◽  
Sandrine Bec ◽  
Jean-Luc Loubet
Keyword(s):  
Mechanical Properties ◽  
Indentation Depth ◽  
Depth Measurement ◽  
Image Position

Abstract

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