scholarly journals Compactness theorems for problems with unknown boundary

2021 ◽  
Vol 21 (1) ◽  
pp. 105-112
Author(s):  
A.G. Podgaev ◽  
◽  
T.D. Kulesh ◽  
Keyword(s):  
Phase Transitions ◽  
Approximate Solutions ◽  
Compactness Theorem ◽  
Unknown Boundary ◽  
Entire Domain ◽  
Higher Derivatives ◽  
Compactness Theorems

The compactness theorem is proved for sequences of functions that have estimates of the higher derivatives in each subdomain of the domain of definition, divided into parts by a sequence of some curves of class W_2^1. At the same time, in the entire domain of determining summable higher derivatives, these sequences do not have. These results allow us to make limit transitions using approximate solutions in problems with an unknown boundary that describe the processes of phase transitions.

Uniform inductive definability and infinitary languages

1976 ◽  
Vol 41 (1) ◽  
pp. 109-120
Author(s):  
Anders M. Nyberg
Keyword(s):  
Proof Theory ◽  
Completeness Theorem ◽  
Recursion Theory ◽  
Compactness Theorem ◽  
Consequence Relation ◽  
Admissible Sets ◽  
Inductive Definitions ◽  
Natural Connection ◽  
Recent Approach ◽  
Compactness Theorems

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA.Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.

scholarly journals Physics of nonlinear oscillations with nonlocal variables

2021 ◽  
Vol 2056 (1) ◽  
pp. 012010
Author(s):  
O A Volkova ◽  
M H Khamis Hassan ◽  
T F Kamalov
Keyword(s):  
Nonlinear Equations ◽  
Nonlinear Oscillations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Physical Processes ◽  
Suspension Point ◽  
Higher Derivatives

Abstract In cases where physical processes cannot be described by linear equations, and nonlinear equations are difficult to solve mathematically, we have to use approximate solutions to such problems. One such example is the description of the Kapitsa pendulum, which is a pendulum with a vibrating suspension point. In contrast to the previously known methods of describing such a problem, in this paper we propose to use additional variables in the form of higher derivatives, which allows us to obtain corrections that give a more detailed contribution to the description of this problem.

A short proof of the McCoy conjecture in higher-dimensional classical continuous models of Kac types

2017 ◽  
Vol 31 (10) ◽  
pp. 1750111 ◽  
Author(s):  
Assane Lo
Keyword(s):  
Phase Transitions ◽  
Correlation Functions ◽  
Short Proof ◽  
Higher Dimensional ◽  
Continuous Models ◽  
Higher Derivatives ◽  
Derivatives Of

We consider the pressure and correlation functions of d-dimensional classical continuous models of Kac type. We prove that if the kth moments of the potential exist, then the system cannot have phase transitions of order lower than k. We also obtain a better formula for the higher derivatives of the pressure that leads to more precise estimates of the truncated correlations.

Σ1-compactness and ultraproducts

1972 ◽  
Vol 37 (4) ◽  
pp. 668-672
Author(s):  
Nigel Cutland
Keyword(s):  
Completeness Theorem ◽  
Compactness Theorem ◽  
Admissible Sets ◽  
Theoretic Proof ◽  
Compactness Theorems ◽  
The Way

This paper is devoted to a description of the way in which ultraproducts can be used in proofs of various well-known Σ1-compactness theorems for infinitary languages ℒA associated with admissible sets A; the method generalises the ultra-product proof of compactness for finitary languages.The compactness theorems we consider are (§2) the Barwise Compactness Theorem for ℒA when A is countable admissible [1], and (§3) the Cofinality (ω) Compactness Theorem of Barwise and Karp [2] and [4]. Our proof of the Barwise theorem unfortunately has the defect that it relies heavily on the Completeness Theorem for ℒA. This defect has, however, been avoided in the case of the Cf(ω) Compactness Theorem, so we have a purely model-theoretic proof.

scholarly journals On a topology property for the moduli space of Kapustin–Witten equations

2019 ◽  
Vol 31 (5) ◽  
pp. 1119-1138
Author(s):  
Teng Huang
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Trivial Solution ◽  
Smooth Solution ◽  
Compactness Theorem ◽  
Dimensional Manifold ◽  
Link Type ◽  
Kähler Surface ◽  
Compactness Theorems ◽  
Simply Connected

AbstractIn this article, we study the Kapustin–Witten equations on a closed, simply connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use Taubes’ compactness theorem [C. H. Taubes, Compactness theorems for {\mathrm{SL}(2;\mathbb{C})} generalizations of the 4-dimensional anti-self dual equations, preprint 2014, https://arxiv.org/abs/1307.6447v4] to prove that if {(A,\phi)} is a smooth solution to the Kapustin–Witten equations and the connection A is closed to a generic ASD connection {A_{\infty}}, then {(A,\phi)} must be a trivial solution. We also prove that the moduli space of the solutions to the Kapustin–Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all generic. At last, we extend the results for the Kapustin–Witten equations to other equations on gauge theory such as the Hitchin–Simpson equations and the Vafa–Witten on a compact Kähler surface.

Electron Microscopy of Graphite Intercalation Compounds

1982 ◽  
Vol 40 ◽  
pp. 544-545
Author(s):  
G. Timp ◽  
L. Salamanca-Riba ◽  
L.W. Hobbs ◽  
G. Dresselhaus ◽  
M.S. Dresselhaus
Keyword(s):  
Electron Microscopy ◽  
Phase Transitions ◽  
Domain Wall ◽  
Intercalation Compounds ◽  
Lattice Plane ◽  
Wall Model ◽  
Graphite Intercalation Compounds ◽  
Fundamental Symmetry ◽  
Graphite Intercalation

Electron microscopy can be used to study structures and phase transitions occurring in graphite intercalations compounds. The fundamental symmetry in graphite intercalation compounds is the staging periodicity whereby each intercalate layer is separated by n graphite layers, n denoting the stage index. The currently accepted model for intercalation proposed by Herold and Daumas assumes that the sample contains equal amounts of intercalant between any two graphite layers and staged regions are confined to domains. Specifically, in a stage 2 compound, the Herold-Daumas domain wall model predicts a pleated lattice plane structure.

Multiple phase transitions investigated by high-speed TEM

1990 ◽  
Vol 48 (4) ◽  
pp. 550-551
Author(s):  
Oleg Bostanjoglo ◽  
Peter Thomsen-Schmidt
Keyword(s):  
Phase Transitions ◽  
High Speed ◽  
Short Exposure ◽  
Semiconductor Film ◽  
Time Resolved ◽  
Short Exposure Time ◽  
Multiple Phase ◽  
Model Substance ◽  
History Of ◽  
Electron Image

Thin GexTe1-x (x = 0.15-0.8) were studied as a model substance of a composite semiconductor film, in addition being of interest for optical storage material. Two complementary modes of time-resolved TEM were used to trace the phase transitions, induced by an attached Q-switched (50 ns FWHM) and frequency doubled (532 nm) Nd:YAG laser. The laser radiation was focused onto the specimen within the TEM to a 20 μm spot (FWHM). Discrete intermediate states were visualized by short-exposure time doubleframe imaging /1,2/. The full history of a transformation was gained by tracking the electron image intensity with photomultiplier and storage oscilloscopes (space/time resolution 100 nm/3 ns) /3/. In order to avoid radiation damage by the probing electron beam to detector and specimen, the beam is pulsed in this continuous mode of time-resolved TEM,too.Short events ( <2 μs) are followed by illuminating with an extended single electron pulse (fig. 1c)

Evolution of Phase Transitions

2006 ◽  
Author(s):  
Rohan Abeyaratne ◽  
James K. Knowles
Keyword(s):  
Phase Transitions
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