UNIFORM PROCEDURES IN UNCOUNTABLE STRUCTURES

2018 ◽  
Vol 83 (2) ◽  
pp. 529-550 ◽  
Author(s):  
NOAM GREENBERG ◽  
ALEXANDER G. MELNIKOV ◽  
JULIA F. KNIGHT ◽  
DANIEL TURETSKY
Keyword(s):  
Metric Spaces ◽  
Space Theory ◽  
Computable Categoricity ◽  
The Third ◽  
Scott Family ◽  
Computable Metric Space ◽  
General Program ◽  
Effective Algebra ◽  
Image Position ◽  
Countable Case

AbstractThis article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

scholarly journals Improving KKM Theory on General Metric Type Spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Sehie Park
Keyword(s):  
Convex Space ◽  
Metric Spaces ◽  
Space Theory ◽  
Type Space ◽  
Generalized Metric ◽  
Abstract Convex Space ◽  
Type Spaces

Abstract A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.

scholarly journals Dislocation Arrangement in a Thick LEO GaN Film on Sapphire

2000 ◽  
Vol 5 (S1) ◽  
pp. 97-103
Author(s):  
Kathleen A. Dunn ◽  
Susan E. Babcock ◽  
Donald S. Stone ◽  
Richard J. Matyi ◽  
Ling Zhang ◽  
...  
Keyword(s):  
High Resolution ◽  
Electron Diffraction ◽  
Threading Dislocations ◽  
X Ray Diffraction ◽  
Burgers Vector ◽  
X Ray ◽  
The Third ◽  
Dislocation Arrangement ◽  
Image Position ◽  
Gan Film

Diffraction-contrast TEM, focused probe electron diffraction, and high-resolution X-ray diffraction were used to characterize the dislocation arrangements in a 16µm thick coalesced GaN film grown by MOVPE LEO. As is commonly observed, the threading dislocations that are duplicated from the template above the window bend toward (0001). At the coalescence plane they bend back to lie along [0001] and thread to the surface. In addition, three other sets of dislocations were observed. The first set consists of a wall of parallel dislocations lying in the coalescence plane and nearly parallel to the substrate, with Burgers vector (b) in the (0001) plane. The second set is comprised of rectangular loops with b = 1/3 [110] (perpendicular to the coalescence boundary) which originate in the coalescence boundary and extend laterally into the film on the (100). The third set of dislocations threads laterally through the film along the [100] bar axis with 1/3<110>-type Burgers vectors These sets result in a dislocation density of ∼109 cm−2. High resolution X-ray reciprocal space maps indicate wing tilt of ∼0.5º.

scholarly journals Preface

2015 ◽  
Vol 27 (4) ◽  
pp. 459-459
Author(s):  
ACHIM JUNG ◽  
GUO-QIANG ZHANG
Keyword(s):  
Lattice Theory ◽  
Process Algebra ◽  
Metric Spaces ◽  
Domain Theory ◽  
Conference Series ◽  
The Third ◽  
Language Semantics ◽  
And Mathematics ◽  
October 17 ◽  
Partial Logics

The International Symposium on Domain Theory (ISDT) is a conference series intended to be a forum for researchers in domain theory and its applications. Topics include topological and logical aspects of domains; categories of domains and powerdomains; continuous posets and their representations; partial orders, lattice theory and metric spaces; types, process algebra and concurrency; non-classical and partial logics; programming language semantics; applications in computer science and mathematics. This conference series was founded by Yingming Liu, Yixiang Chen, Klaus Keimel, and Guo-Qiang Zhang. All ISDT events have taken place in China. The first ISDT was held in Shanghai, October 17–24, 1999; the second ISDT was held in Chengdu, October 22–26, 2001; the third ISDT occurred in Xi'an, China, May 10–14, 2004; the fourth ISDT was held in Changsha, June 2–6, 2006; and the fifth ISDT took place in Shanghai, September 11–14, 2009.

scholarly journals Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Beth Boardman ◽  
Troy Harden ◽  
Sonia Martínez
Keyword(s):  
Degrees Of Freedom ◽  
Metric Spaces ◽  
Dynamic Environments ◽  
Random Trees ◽  
Asymptotically Optimal ◽  
Euclidean Metric ◽  
The Third ◽  
Dubins Vehicle ◽  
Improved Performance ◽  
Dubins Vehicles

Three algorithms that improve the performance of the asymptotically optimal Rapidly exploring Random Tree (RRT*) are presented in this paper. First, we introduce the Goal Tree (GT) algorithm for motion planning in dynamic environments where unexpected obstacles appear sporadically. The GT reuses the previous RRT* by pruning the affected area and then extending the tree by drawing samples from a shadow set. The shadow is the subset of the free configuration space containing all configurations that have geodesics ending at the goal and are in conflict with the new obstacle. Smaller, well defined, sampling regions are considered for Euclidean metric spaces and Dubins' vehicles. Next, the Focused-Refinement (FR) algorithm, which samples with some probability around the first path found by an RRT*, is defined. The third improvement is the Grandparent-Connection (GP) algorithm, which attempts to connect an added vertex directly to its grandparent vertex instead of parent. The GT and GP algorithms are both proven to be asymptotically optimal. Finally, the three algorithms are simulated and compared for a Euclidean metric robot, a Dubins' vehicle, and a seven degrees-of-freedom manipulator.

THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 435-445 ◽  
Author(s):  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO ◽  
YOUNG JAE SIM
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Sharp Inequality ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third ◽  
Image Position

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra

1971 ◽  
Vol 23 (3) ◽  
pp. 445-450 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Hilbert Space ◽  
Quotient Space ◽  
Principal Result ◽  
Irreducible Representations ◽  
C Algebra ◽  
Fell Topology ◽  
The Third ◽  
Definition Of ◽  
Image Position

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the formwhere the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals On Appell's Function P (θ, φ)

1932 ◽  
Vol 3 (1) ◽  
pp. 53-55
Author(s):  
Pierre Humbert
Keyword(s):  
Third Order ◽  
The Third ◽  
Direct Generalization ◽  
Circular Functions ◽  
Image Position

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansionwhere j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

scholarly journals F-Metric, F-Contraction and Common Fixed-Point Theorems with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 586 ◽  
Author(s):  
Awais Asif ◽  
Muhammad Nazam ◽  
Muhammad Arshad ◽  
Sang Og Kim
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
The Third ◽  
Set Valued Mappings ◽  
Common Fixed Point Theorems

In this paper, we noticed that the existence of fixed points of F-contractions, in F -metric space, can be ensured without the third condition (F3) imposed on the Wardowski function F : ( 0 , ∞ ) → R . We obtain fixed points as well as common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in F -metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented. The application shows the role of fixed-point theorems in dynamic programming, which is widely used in computer programming and optimization. Our results extend and generalize the previous results in the existing literature.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

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