scholarly journals Causal variational principles in the infinite-dimensional setting: Existence of minimizers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christoph Langer
Keyword(s):  
Variational Principles ◽  
Polish Space ◽  
Dimension Theory ◽  
Regular Measure ◽  
Lagrange Equations ◽  
Locally Compact ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Existence Of Minimizers

Abstract We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established.

scholarly journals Causal variational principles in the σ-locally compact setting: Existence of minimizers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Felix Finster ◽  
Christoph Langer
Keyword(s):  
Borel Measure ◽  
Variational Principles ◽  
Hausdorff Spaces ◽  
Lagrange Equations ◽  
Locally Compact ◽  
Compact Range ◽  
Existence Of Minimizers ◽  
Lower Semi ◽  
Regular Borel Measure ◽  
Compact Subsets

AbstractWe prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy.

The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

2015 ◽  
Vol 70 (11) ◽  
pp. 913-917
Author(s):  
Wei Liu ◽  
Yafeng Liu ◽  
Shujuan Yuan
Keyword(s):  
Coordinate System ◽  
Spectral Problem ◽  
Integrable System ◽  
Evolution Equations ◽  
Lagrange Equations ◽  
Lax Pairs ◽  
Infinite Dimensional ◽  
Hamilton System ◽  
Finite Dimensional ◽  
Dimensional Soliton

AbstractIn this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.

scholarly journals Sums of finite-dimensional spaces

1969 ◽  
Vol 1 (3) ◽  
pp. 357-361 ◽  
Author(s):  
B.R. Wenner
Keyword(s):  
Metric Spaces ◽  
Dimension Theory ◽  
General Hypothesis ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Locally Countable

Analogues are developed to the sum theorems in the dimension theory of metric spaces. It is shown that, within the class of metric spaces, any locally countable, σ-locally finite, or closure-preserving sum of finite-dimensional sets is countable-dimensional. Similar results are obtained under the more general hypothesis of countable-dimensional rather than finite-dimensional sets.

A characterization of piecewise linear homology manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 271-274 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Piecewise Linear ◽  
Subsequent Paper ◽  
Locally Compact ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Set ◽  
Topological Manifolds ◽  
Homology Manifolds

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

scholarly journals Fiberwise KK-equivalence of continuous fields of C*-algebras

2008 ◽  
Vol 3 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Marius Dadarlat
Keyword(s):  
Compact Space ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Hilbert Cube ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Finite Dimensional ◽  
Compact Metrizable Space ◽  
Continuous Fields

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

scholarly journals Invariance of infinite-dimensional classes of spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 76-83
Author(s):  
B. R. Wenner
Keyword(s):  
Metric Spaces ◽  
Central Area ◽  
Discrete Point ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional

AbstractThe central area of investigation is in the isolation of conditions on mappings which leave invariant the classes of locally finite-dimensional metric spaces and strongly countable-dimensional metric spaces. Examples of such properties are open and closed with discrete point-inverses, open and finite-to-one, or open, closed, and countable-to-one.

Symmetries, conservation laws and variational principles for vector field theories†

1996 ◽  
Vol 120 (2) ◽  
pp. 369-384 ◽  
Author(s):  
Ian M. Anderson ◽  
Juha Pohjanpelto
Keyword(s):  
Conservation Laws ◽  
Variational Principles ◽  
Field Theories ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Differential Identities ◽  
Infinite Dimensional ◽  
Independent Variables ◽  
Generalized Symmetries ◽  
And Variational Principles

The interplay between symmetries, conservation laws, and variational principles is a rich and varied one and extends well beyond the classical Noether's theorem. Recall that Noether's first theorem asserts that to every r dimensional Lie algebra of (generalized) symmetries of a variational problem there are r conserved quantities for the corresponding Euler-Lagrange equations. Noether's second theorem asserts that infinite dimensional symmetry algebras (depending upon arbitrary functions of all the independent variables) lead to differential identities for the Euler-Lagrange equations.

Ring properties related to symmetric rings

2014 ◽  
Vol 24 (07) ◽  
pp. 935-967
Author(s):  
Da Woon Jung ◽  
Tai Keun Kwak ◽  
Min Jung Lee ◽  
Yang Lee
Keyword(s):  
Basic Structure ◽  
Ring Theory ◽  
Polynomial Rings ◽  
Minimal Order ◽  
Module Theory ◽  
Finite Rings ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional

The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.

LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

2021 ◽  
pp. 1-41
Author(s):  
VOLODYMYR MAZORCHUK ◽  
RAFAEL MRÐEN
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Direct Sum ◽  
Levi Decomposition ◽  
Finite Dimensional ◽  
Semisimple Part

Abstract For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

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