The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.
{"title":"Imaginary cone and reflection subgroups of Coxeter groups","authors":"M. Dyer","doi":"10.4064/dm773-6-2019","DOIUrl":"https://doi.org/10.4064/dm773-6-2019","url":null,"abstract":"The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2012-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets","authors":"Dae San Kim, H. Kim","doi":"10.4064/DM487-0-1","DOIUrl":"https://doi.org/10.4064/DM487-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"15 1","pages":"1-49"},"PeriodicalIF":1.8,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An envelope in a category is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone-v{C}ech compactification of a topological space, or universal enveloping algebra of a Lie algebra. Dually, a refinement generalizes operations of "interior enrichment", like bornologification (or saturation) of a locally convex space, or simply connected covering of a Lie group. In this paper we define envelopes and refinements in abstract categories and discuss the conditions under which these constructions exist and are functors. The aim of the exposition is to build a fundament for duality theories of non-commutative groups based on the idea of envelope. The advantage of this approach is that in the arising theories the analogs of group algebras are Hopf algebras. At the same time the classical Fourier and Gelfand transforms are interpreted as envelopes with respect to the prearranged classes of algebras.
{"title":"Envelopes and refinements in categories, with applications to functional analysis","authors":"S. Akbarov","doi":"10.4064/dm702-12-2015","DOIUrl":"https://doi.org/10.4064/dm702-12-2015","url":null,"abstract":"An envelope in a category is a construction that generalizes the operations of \"exterior completion\", like completion of a locally convex space, or Stone-v{C}ech compactification of a topological space, or universal enveloping algebra of a Lie algebra. Dually, a refinement generalizes operations of \"interior enrichment\", like bornologification (or saturation) of a locally convex space, or simply connected covering of a Lie group. In this paper we define envelopes and refinements in abstract categories and discuss the conditions under which these constructions exist and are functors. The aim of the exposition is to build a fundament for duality theories of non-commutative groups based on the idea of envelope. The advantage of this approach is that in the arising theories the analogs of group algebras are Hopf algebras. At the same time the classical Fourier and Gelfand transforms are interpreted as envelopes with respect to the prearranged classes of algebras.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"513 1","pages":"1-188"},"PeriodicalIF":1.8,"publicationDate":"2011-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if f is locally distributionally integrable over the real line and psi is an element of D(R) is a test function, then f psi is distributionally integrable, and the formula = (dist)integral(infinity)(-infinity) f(x)psi(x)dx, defines a distribution f is an element of D'(R) that has distributional point values almost everywhere and actually f(x) = f(x) almost everywhere. The indefinite distributional integral F(x) = (dist) integral(x)(a) f(t)dt corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to f (x). The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals (in the Cesaro sense), mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.
{"title":"A general integral","authors":"R. Estrada, J. Vindas","doi":"10.4064/dm483-0-1","DOIUrl":"https://doi.org/10.4064/dm483-0-1","url":null,"abstract":"We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. \u0000Our integral has the property that if f is locally distributionally integrable over the real line and psi is an element of D(R) is a test function, then f psi is distributionally integrable, and the formula = (dist)integral(infinity)(-infinity) f(x)psi(x)dx, \u0000defines a distribution f is an element of D'(R) that has distributional point values almost everywhere and actually f(x) = f(x) almost everywhere. \u0000The indefinite distributional integral F(x) = (dist) integral(x)(a) f(t)dt corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to f (x). \u0000The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals (in the Cesaro sense), mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"483 1","pages":"1-49"},"PeriodicalIF":1.8,"publicationDate":"2011-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wir beweisen explizite, d.h., nicht-asymptotische Fehlerabschatzungen fur Markov-Ketten-Monte-Carlo-Methoden. Ziel ist es, den Erwartungswert einer Funktion f bzgl. eines Mases π zu berechnen. Verschiedene Konvergenzeigenschaften von Markov-Ketten implizieren verschiedene Fehlerabschatzungen. Fur gleichmasig ergodische und reversible Markov-Ketten beweisen wir eine untere und eine obere Fehlerschranke bzgl. der L2-Norm von f. Wenn eine L2-Spektrallucke existiert, die eine schwachere Konvergenzeigenschaft als die gleichmasige Ergodizitat darstellt, dann zeigen wir eine obere Fehlerschranke bzgl. der Lp-Norm fur p>2. Das so genannte Aufwarmen ist ein gebrauchliches Mittel um den Algorithmus abzustimmen. Wir schlagen ein Rezept fur die Wahl der Aufarmzeit vor und begrunden dieses ausfuhrlich. Die Fehlerabschatzungen werden anschliesend auf das Problem der Integration bzgl. einer nicht normierten Dichte angewendet. Genauer gesagt betrachten wir die Integration bzgl. log-konkaver Dichtefunktionen und die Integration uber konvexe Korper. Durch die Verwendung des Metropolis-Algorithmus und des Hit-and-run-Algorithmus zeigen wir, dass diese beiden Probleme "polynomial tractable" sind.
{"title":"Explicit error bounds for Markov chain Monte Carlo","authors":"Daniel Rudolf","doi":"10.4064/dm485-0-1","DOIUrl":"https://doi.org/10.4064/dm485-0-1","url":null,"abstract":"Wir beweisen explizite, d.h., nicht-asymptotische Fehlerabschatzungen fur Markov-Ketten-Monte-Carlo-Methoden. Ziel ist es, den Erwartungswert einer Funktion f bzgl. eines Mases π zu berechnen. Verschiedene Konvergenzeigenschaften von Markov-Ketten implizieren verschiedene Fehlerabschatzungen. Fur gleichmasig ergodische und reversible Markov-Ketten beweisen wir eine untere und eine obere Fehlerschranke bzgl. der L2-Norm von f. Wenn eine L2-Spektrallucke existiert, die eine schwachere Konvergenzeigenschaft als die gleichmasige Ergodizitat darstellt, dann zeigen wir eine obere Fehlerschranke bzgl. der Lp-Norm fur p>2. Das so genannte Aufwarmen ist ein gebrauchliches Mittel um den Algorithmus abzustimmen. Wir schlagen ein Rezept fur die Wahl der Aufarmzeit vor und begrunden dieses ausfuhrlich. Die Fehlerabschatzungen werden anschliesend auf das Problem der Integration bzgl. einer nicht normierten Dichte angewendet. Genauer gesagt betrachten wir die Integration bzgl. log-konkaver Dichtefunktionen und die Integration uber konvexe Korper. Durch die Verwendung des Metropolis-Algorithmus und des Hit-and-run-Algorithmus zeigen wir, dass diese beiden Probleme \"polynomial tractable\" sind.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"12 1","pages":"1-93"},"PeriodicalIF":1.8,"publicationDate":"2011-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for $N$-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_N$ of all unitary equivalence classes of such $N$-tuples are established and certain ideals in $CDD_N$ are distinguished. It is proved that infinite operations in $CDD_N$ may be reconstructed from the direct sum operation of a pair. textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense) uniqueness is established. The issue of classification of ideals in $CDD_N$ (up to isomorphism) is discussed. A model for $CDD_N$ is described and its concrete realization is presented. A new partial order of $N$-tuples of operators is introduced and its fundamental properties are established. Extremal importance of unitary disjointness of $N$-tuples and the way how it `tidies up' the structure of $CDD_N$ are emphasized.
{"title":"Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces","authors":"Piotr Niemiec","doi":"10.4064/dm482-0-1","DOIUrl":"https://doi.org/10.4064/dm482-0-1","url":null,"abstract":"An textit{ideal} of $N$-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for $N$-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_N$ of all unitary equivalence classes of such $N$-tuples are established and certain ideals in $CDD_N$ are distinguished. It is proved that infinite operations in $CDD_N$ may be reconstructed from the direct sum operation of a pair. textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense) uniqueness is established. The issue of classification of ideals in $CDD_N$ (up to isomorphism) is discussed. A model for $CDD_N$ is described and its concrete realization is presented. A new partial order of $N$-tuples of operators is introduced and its fundamental properties are established. Extremal importance of unitary disjointness of $N$-tuples and the way how it `tidies up' the structure of $CDD_N$ are emphasized.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"482 1","pages":"1-106"},"PeriodicalIF":1.8,"publicationDate":"2011-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.
{"title":"Lie systems: theory, generalisations, and applications","authors":"J. Cariñena, J. Lucas","doi":"10.4064/dm479-0-1","DOIUrl":"https://doi.org/10.4064/dm479-0-1","url":null,"abstract":"Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"479 1","pages":"1-162"},"PeriodicalIF":1.8,"publicationDate":"2011-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A be a Banach algebra. Then there are two natural products on the second dual A′′ of A arising from left and right translations by elements of A; they are called the Arens products; we denote these products by and ♦, respectively. For definitions and discussions of these products, see [2], [4], and [5], for example. We briefly recall the definitions. For a ∈ A, λ ∈ A′, and Φ ∈ A′′, define λ · a and a · λ in A′ by
{"title":"Second duals of measure algebras","authors":"H. Dales, A. Lau, D. Strauss","doi":"10.4064/DM481-0-1","DOIUrl":"https://doi.org/10.4064/DM481-0-1","url":null,"abstract":"Let A be a Banach algebra. Then there are two natural products on the second dual A′′ of A arising from left and right translations by elements of A; they are called the Arens products; we denote these products by and ♦, respectively. For definitions and discussions of these products, see [2], [4], and [5], for example. We briefly recall the definitions. For a ∈ A, λ ∈ A′, and Φ ∈ A′′, define λ · a and a · λ in A′ by","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"481 1","pages":"1-121"},"PeriodicalIF":1.8,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Minimal Non-finitely Based Monoids","authors":"Edmond W. H. Lee, Jian-Rong Li","doi":"10.4064/DM475-0-1","DOIUrl":"https://doi.org/10.4064/DM475-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"475 1","pages":"1-65"},"PeriodicalIF":1.8,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we construct and study certain classes of infinite dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group of weighted diffeomorphisms on a Banach space. Further, we also construct certain types of weighted mapping groups. Both the weighted diffeomorphism groups and the weighted mapping groups are shown to be regular Lie groups in Milnor's sense. We also discuss semidirect products of the former groups. Moreover, we study the integrability of Lie algebras of certain vector fields.
{"title":"Weighted diffeomorphism groups of Banach spaces and weighted mapping groups","authors":"B. Walter","doi":"10.4064/dm484-0-1","DOIUrl":"https://doi.org/10.4064/dm484-0-1","url":null,"abstract":"In this work, we construct and study certain classes of infinite dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group of weighted diffeomorphisms on a Banach space. Further, we also construct certain types of weighted mapping groups. Both the weighted diffeomorphism groups and the weighted mapping groups are shown to be regular Lie groups in Milnor's sense. We also discuss semidirect products of the former groups. Moreover, we study the integrability of Lie algebras of certain vector fields.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"484 1","pages":"1-126"},"PeriodicalIF":1.8,"publicationDate":"2010-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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