The First Main Theorem

2014 ◽  
pp. 25-90
Author(s):  
Junjiro Noguchi ◽  
Jörg Winkelmann
Keyword(s):  
First Main Theorem

scholarly journals Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions

2020 ◽  
Author(s):  
Yosuke Kawamoto ◽  
Hirofumi Osada ◽  
Hideki Tanemura
Keyword(s):  
Finite Volume ◽  
Stochastic Dynamics ◽  
Matrix Theory ◽  
Dirichlet Forms ◽  
Interaction Potentials ◽  
Brownian Motions ◽  
Infinite Systems ◽  
Point Field ◽  
First Main Theorem ◽  
Class Interaction

Abstract The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms $ (\mathcal {E}^{\mathsf {upr}},\mathcal {D}^{\mathsf {upr}})$ ( E u p r , D u p r ) and $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of $ {C_{0}^{3}} $ C 0 3 -class.

On the first main theorem on a finite manifold

1969 ◽  
Vol 110 (4) ◽  
pp. 319-334 ◽  
Author(s):  
Shigeo Ozaki ◽  
Teiichi Higuchi
Keyword(s):  
First Main Theorem

scholarly journals OPTIMIZATION OF MATRIX SEMIRINGS FOR CLASSIFICATION SYSTEMS

2011 ◽  
Vol 84 (3) ◽  
pp. 492-503 ◽  
Author(s):  
D. Y. GAO ◽  
A. V. KELAREV ◽  
J. L. YEARWOOD
Keyword(s):  
Discrete Event Systems ◽  
Discrete Event ◽  
Classification Systems ◽  
Second Main Theorem ◽  
Matrix Rings ◽  
Multiple Classifiers ◽  
Generating Sets ◽  
Event Systems ◽  
First Main Theorem ◽  
Structural Matrix

AbstractThe max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

scholarly journals PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

2017 ◽  
Vol 82 (4) ◽  
pp. 1317-1355
Author(s):  
PHILIPP SCHLICHT
Keyword(s):  
Baire Space ◽  
Inaccessible Cardinal ◽  
Second Main Theorem ◽  
Perfect Set ◽  
Uncountable Cardinal ◽  
First Main Theorem ◽  
Definable Functions ◽  
The Second Main Theorem

AbstractWe extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from elements ofλOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal aboveλ. We further obtain some related results about definable functions onλλand consequences of resurrection axioms for definable subsets ofλλ.

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