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A translation surface is obtained by taking plane polygons and gluing their edges by translations. We ask which subgroups of the Veech group of a primitive translation surface can be realised via a translation covering. For many primitive surfaces we prove that partition stabilising congruence subgroups are the Veech group of a covering surface. We also address the coverings via their monodromy groups and present examples of cyclic coverings in short orbits, i.e. with large Veech groups.
cyclic covering --- monodromy group --- Kongruenzgruppe --- zyklische Überlagerung --- Translationsüberlagerung --- translation coveringVeechgruppe --- congruence subgroup --- Monodromiegruppe --- Veech group
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Improving learning and teaching at schools or universities may start with choosing evidence-based interventions and practices, but does not end there. To ensure sustainable changes to programs in educational practice, interventions need to address complex issues related to theories, research designs, and measurements. This book presents typical but often overlooked problems in intervention research in educational practice. These problems are embedded in various educational areas such as, amongst others, school effectiveness, instructional design or motivational aspects of teacher trainings.
education --- educational interventions --- self-congruence intervention --- teacher --- volitional competences --- motive implementation --- learning skills --- developmental measurement approaches --- self-assessment --- reflective writing --- Empirische Bildungsforschung
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In der aktuellen Kontaktlinguistik wurden überwiegend Sprachen in Kontakt betrachtet, die genetisch nicht verwandt und strukturell distant sind. Der vorliegende Sammelband setzt sich zum Ziel, mit einem Schwerpunkt im Bereich der romanischen und slavischen Sprachen die Bedeutung von bereits zu Beginn des Kontakts vorhandenen Effekten der strukturellen Kongruenz zwischen den beteiligten Sprachen zu beleuchten. Dabei spielen ererbte wie auch typologische Ähnlichkeiten eine Rolle. Modern contact linguistics has primarily focused on contact between languages that are genetically unrelated and structurally distant. This compendium of articles looks instead at the effects of pre-existing structural congruency between the affected languages at the time of their initial contact, using the Romance and Slavic languages as examples. In contact of this kind, both genetic and typological similarities play a part.
Sociolinguistics --- Dialectology --- Languages in contact. --- Language and languages --- Typology (Linguistics) --- Variation. --- Typology (Linguistics). --- Grammar, Comparative and general --- Linguistic typology --- Linguistics --- Linguistic universals --- Characterology of speech --- Language diversity --- Language subsystems --- Language variation --- Linguistic diversity --- Variation in language --- Areal linguistics --- Typology --- Classification --- Congruence. --- Convergence. --- Language Contact. --- Romance Languages. --- Slavic Languages.
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The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature "Proof-by-Picture" method and its conversion to transparencies. Key features: * Includes the latest findings from a pioneering researcher in the field * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems * Additional information provided by the author online at: http://www.maths.umanitoba.ca/homepages/gratzer.html/ The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices.
Congruence lattices. --- Lattice theory. --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lattice theory --- Algebra. --- Logic, Symbolic and mathematical. --- Distribution (Probability theory. --- Number theory. --- Order, Lattices, Ordered Algebraic Structures. --- Mathematical Logic and Foundations. --- Probability Theory and Stochastic Processes. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Metamathematics --- Syllogism --- Mathematical analysis --- Ordered algebraic structures. --- Mathematical logic. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Treillis, Théorie des --- Ordres (mathematiques) --- Treillis
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This is a self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presenting the major results of the last 70 years on congruence lattices of finite lattices, featuring the author's signature Proof-by-Picture method. Key features: * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and over 140 illustrations * This new edition includes two new parts on Planar Semimodular Lattices and The Order of Principle Congruences, covering the research of the last 10 years The book is appropriate for a one-semester graduate course in lattice theory, and it is a practical reference for researchers studying lattices. Reviews of the first edition: "There exist a lot of interesting results in this area of lattice theory, and some of them are presented in this book. [This] monograph…is an exceptional work in lattice theory, like all the contributions by this author. … The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. Moreover, the author provides a series of companion lectures which help the reader to approach the Proof-by-Picture sections." (Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica, Vol. LII (1), 2007) "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. [I]n addition to giving the full formal details of the proofs, the author chooses a somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the combinatorial (as opposed to algebraic) nature of many of the presented results. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." —Mathematical Reviews.
Mathematics. --- Algebra. --- Ordered algebraic structures. --- Mathematical logic. --- Number theory. --- Probabilities. --- Order, Lattices, Ordered Algebraic Structures. --- Mathematical Logic and Foundations. --- Probability Theory and Stochastic Processes. --- Number Theory. --- Probability --- Statistical inference --- Number study --- Numbers, Theory of --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Math --- Logic, Symbolic and mathematical. --- Distribution (Probability theory. --- Algebra --- Mathematics --- Mathematical analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Congruence lattices. --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk
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There are over a million jazz recordings, but only a few hundred tunes have been recorded repeatedly. Why did a minority of songs become jazz standards? Why do some songs--and not others--get rerecorded by many musicians? Shaping Jazz answers this question and more, exploring the underappreciated yet crucial roles played by initial production and markets--in particular, organizations and geography--in the development of early twentieth-century jazz. Damon Phillips considers why places like New York played more important roles as engines of diffusion than as the sources of standards. He demonstrates why and when certain geographical references in tune and group titles were considered more desirable. He also explains why a place like Berlin, which produced jazz abundantly from the 1920's to early 1930's, is now on jazz's historical sidelines. Phillips shows the key influences of firms in the recording industry, including how record companies and their executives affected what music was recorded, and why major companies would rerelease recordings under artistic pseudonyms. He indicates how a recording's appeal was related to the narrative around its creation, and how the identities of its firm and musicians influenced the tune's long-run popularity. Applying fascinating ideas about market emergence to a music's commercialization, Shaping Jazz offers a unique look at the origins of a groundbreaking art form.
Jazz --- Accordion and piano music (Jazz) --- Clarinet and piano music (Jazz) --- Cornet and piano music (Jazz) --- Double bass and piano music (Jazz) --- Jazz duets --- Jazz ensembles --- Jazz music --- Jazz nonets --- Jazz octets --- Jazz quartets --- Jazz quintets --- Jazz septets --- Jazz sextets --- Jazz trios --- Jive (Music) --- Saxophone and piano music (Jazz) --- Vibraphone and piano music (Jazz) --- Wind instrument and piano music (Jazz) --- Xylophone and piano music (Jazz) --- African Americans --- Music --- Third stream (Music) --- Washboard band music --- Social aspects --- History and criticism --- E-books --- History and criticism. --- Social aspects. --- African Americans. --- Berlin. --- German jazz. --- Milenburg Joys. --- New York. --- Victorian-era firms. --- Weimar Germany. --- adoption narratives. --- anti-jazz sentiments. --- authenticity. --- black musicians. --- bottled water. --- consumers. --- critics. --- cultural elites. --- cultural markets. --- cultural objects. --- cultural products. --- diffusion. --- discographical canon. --- disconnected cities. --- disconnectedness. --- geographic mobility. --- geography. --- green technology. --- identity sequences. --- identity threats. --- identity. --- incumbents. --- jazz music. --- jazz recordings. --- jazz standards. --- jazz. --- legitimacy. --- markets. --- mobility networks. --- musicians. --- nanotechnology. --- organizational role identities. --- product appeal. --- production. --- pseudonyms. --- race. --- re-recording. --- reception. --- record company deception. --- record company. --- record labels. --- recording industry. --- recording location. --- social congruence. --- social systems. --- sociological congruence. --- software.
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This work introduces tools from the field of category theory that make it possible to tackle a number of representation problems that have remained unsolvable to date (e.g. the determination of the range of a given functor). The basic idea is: if a functor lifts many objects, then it also lifts many (poset-indexed) diagrams.
Functor theory --- Partially ordered sets --- Lattice theory --- Congruence lattices --- Algebra, Boolean --- Algebraic logic --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Algebra --- Algebra, Boolean. --- Algebraic logic. --- Boolean algebra --- Boole's algebra --- Mathematics. --- Algebra. --- Category theory (Mathematics). --- Homological algebra. --- K-theory. --- Ordered algebraic structures. --- Mathematical logic. --- Category Theory, Homological Algebra. --- General Algebraic Systems. --- Order, Lattices, Ordered Algebraic Structures. --- Mathematical Logic and Foundations. --- K-Theory. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Algebraic structures, Ordered --- Structures, Ordered algebraic --- Algebraic topology --- Homology theory --- Homological algebra --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Mathematical analysis --- Math --- Science --- Logic, Symbolic and mathematical.
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In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic.
Hodge theory. --- Logarithms. --- Logs (Logarithms) --- Algebra --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Algebraic group. --- Algebraic variety. --- Analytic manifold. --- Analytic space. --- Annulus (mathematics). --- Arithmetic group. --- Atlas (topology). --- Canonical map. --- Classifying space. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Complex manifold. --- Complex number. --- Congruence subgroup. --- Conjecture. --- Connected component (graph theory). --- Continuous function. --- Convex cone. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential form. --- Direct image functor. --- Divisor. --- Elliptic curve. --- Equivalence class. --- Existential quantification. --- Finite set. --- Functor. --- Geometry. --- Hodge structure. --- Homeomorphism. --- Homomorphism. --- Inverse function. --- Iwasawa decomposition. --- Local homeomorphism. --- Local ring. --- Local system. --- Logarithmic. --- Maximal compact subgroup. --- Modular curve. --- Modular form. --- Moduli space. --- Monodromy. --- Monoid. --- Morphism. --- Natural number. --- Nilpotent orbit. --- Nilpotent. --- Open problem. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Point at infinity. --- Proper morphism. --- Pullback (category theory). --- Quotient space (topology). --- Rational number. --- Relative interior. --- Ring (mathematics). --- Ring homomorphism. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Smooth morphism. --- Special case. --- Strong topology. --- Subgroup. --- Subobject. --- Subset. --- Surjective function. --- Tangent bundle. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transversality (mathematics). --- Two-dimensional space. --- Vector bundle. --- Vector space. --- Weak topology.
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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.
Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Abelian group. --- Automorphism. --- Big O notation. --- Bijection. --- Binary number. --- Bisection. --- Borel set. --- C0. --- Calculation. --- Cantor set. --- Cartesian coordinate system. --- Combination. --- Compass-and-straightedge construction. --- Congruence subgroup. --- Conjecture. --- Conjugacy class. --- Continuity equation. --- Convex lattice polytope. --- Convex polytope. --- Coprime integers. --- Counterexample. --- Cyclic group. --- Diameter. --- Diophantine approximation. --- Diophantine equation. --- Disjoint sets. --- Disjoint union. --- Division by zero. --- Embedding. --- Equation. --- Equivalence class. --- Ergodic theory. --- Ergodicity. --- Factorial. --- Fiber bundle. --- Fibonacci number. --- Fundamental domain. --- Gauss map. --- Geometry. --- Half-integer. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperplane. --- Ideal triangle. --- Intersection (set theory). --- Interval exchange transformation. --- Inverse function. --- Inverse limit. --- Isometry group. --- Lattice (group). --- Limit set. --- Line segment. --- Linear algebra. --- Linear function. --- Line–line intersection. --- Main diagonal. --- Modular group. --- Monotonic function. --- Multiple (mathematics). --- Orthant. --- Outer billiard. --- Parallelogram. --- Parameter. --- Partial derivative. --- Penrose tiling. --- Permutation. --- Piecewise. --- Polygon. --- Polyhedron. --- Polytope. --- Product topology. --- Projective geometry. --- Rectangle. --- Renormalization. --- Rhombus. --- Right angle. --- Rotational symmetry. --- Sanity check. --- Scientific notation. --- Semicircle. --- Sign (mathematics). --- Special case. --- Square root of 2. --- Subsequence. --- Summation. --- Symbolic dynamics. --- Symmetry group. --- Tangent. --- Tetrahedron. --- Theorem. --- Toy model. --- Translational symmetry. --- Trapezoid. --- Triangle group. --- Triangle inequality. --- Two-dimensional space. --- Upper and lower bounds. --- Upper half-plane. --- Without loss of generality. --- Yair Minsky.
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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.
Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.
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