The Design of Space Based on Architectural Geometry
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Affine Springer Fibers and Affine Deligne-Lusztig Varieties
Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system. -
Dynamics for Discrete Subgroups of Sl 2(C)
DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]). -
Discussion on Application for Interior Space Design and the Application of Interior Design Style
3rd International Conference on Education, Management and Computing Technology (ICEMCT 2016) Discussion on Application for Interior Space Design and the Application of Interior Design Style Xiaodong Zhang Institute of Environmental art design, Hebei Institute of Fine Art, Shijiazhuang hebei, 050700, China Keywords: interior design, psychological space, design style Abstract. Interior design is the essence of the interior space design, including physical space design and psychological space design two levels. Interior design style is an important issue for interior designers and owners. Through analysis and Research on the relationships between design and interior design style of interior space, summed up the "interior space design to create a style of interior design" and "interior design style shape the psychological space two obvious characteristics. Interior Space Design and Its Content The Meaning of Interior Space Design. Interior space design is a creative activity of human transformation of living space, through the interior space design can make human living environment become more comfortable [1]. Although the "interior space design" theory is not a long time, but has been constantly being human practice. Since ancient times, human beings have been in the pursuit of the security of living space. Full, comfortable and beautiful. People in the primitive society in the cave laying leaves, weeds and skins and in the walls of the cave with color drawing lines and patterns, and so on for living space transformation, which all belong to the indoor space design. Interior space design is not only focused on solving the relationship between people and nature, but also more and more focus on solving the problem of the relationship between people and society, so that people in the community better work and life. -
ADELIC VERSION of MARGULIS ARITHMETICITY THEOREM Hee Oh 1. Introduction Let R Denote the Set of All Prime Numbers Including
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis’s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one ∞ and set Q∞ = R. Let S be any subset of R. For each p ∈ S, let Gp be a connected semisimple adjoint Qp-group without any Qp-anisotropic factors and Dp ⊂ Gp(Qp) be a compact open subgroup for almost all finite prime p ∈ S. Let (GS , Dp) denote the restricted topological product of Gp(Qp)’s, p ∈ S with respect to Dp’s. Note that if S is finite, (GS , Dp) = Qp∈S Gp(Qp). We show that if Pp∈S rank Qp (Gp) ≥ 2, any irreducible lattice in (GS , Dp) is a rational lattice. We also present a criterion on the collections Gp and Dp for (GS , Dp) to admit an irreducible lattice. In addition, we describe discrete subgroups of (GA, Dp) generated by lattices in a pair of opposite horospherical subgroups. 1. Introduction Let R denote the set of all prime numbers including the infinite prime ∞ and Rf the set of finite prime numbers, i.e., Rf = R−{∞}. We set Q∞ = R. For each p ∈ R, let Gp be a non-trivial connected semisimple algebraic Qp-group and for each p ∈ Rf , let Dp be a compact open subgroup of Gp(Qp). The adele group of Gp, p ∈ R with respect to Dp, p ∈ Rf is defined to be the restricted topological product of the groups Gp(Qp) with respect to the distinguished subgroups Dp. -
A Quasideterminantal Approach to Quantized Flag Varieties
A QUASIDETERMINANTAL APPROACH TO QUANTIZED FLAG VARIETIES BY AARON LAUVE A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Vladimir Retakh & Robert L. Wilson and approved by New Brunswick, New Jersey May, 2005 ABSTRACT OF THE DISSERTATION A Quasideterminantal Approach to Quantized Flag Varieties by Aaron Lauve Dissertation Director: Vladimir Retakh & Robert L. Wilson We provide an efficient, uniform means to attach flag varieties, and coordinate rings of flag varieties, to numerous noncommutative settings. Our approach is to use the quasideterminant to define a generic noncommutative flag, then specialize this flag to any specific noncommutative setting wherein an amenable determinant exists. ii Acknowledgements For finding interesting problems and worrying about my future, I extend a warm thank you to my advisor, Vladimir Retakh. For a willingness to work through even the most boring of details if it would make me feel better, I extend a warm thank you to my advisor, Robert L. Wilson. For helpful mathematical discussions during my time at Rutgers, I would like to acknowledge Earl Taft, Jacob Towber, Kia Dalili, Sasa Radomirovic, Michael Richter, and the David Nacin Memorial Lecture Series—Nacin, Weingart, Schutzer. A most heartfelt thank you is extended to 326 Wayne ST, Maria, Kia, Saˇsa,Laura, and Ray. Without your steadying influence and constant comraderie, my time at Rut- gers may have been shorter, but certainly would have been darker. Thank you. Before there was Maria and 326 Wayne ST, there were others who supported me. -
Geometric Modeling in Shape Space
Geometric Modeling in Shape Space Martin Kilian Niloy J. Mitra Helmut Pottmann Vienna University of Technology Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an as-isometric- as-possible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. Abstract space, line geometry interprets straight lines as points on a quadratic surface [Berger 1987], and the various types of sphere geometries We present a novel framework to treat shapes in the setting of Rie- model spheres as points in higher dimensional space [Cecil 1992]. mannian geometry. Shapes – triangular meshes or more generally Other examples concern kinematic spaces and Lie groups which straight line graphs in Euclidean space – are treated as points in are convenient for handling congruent shapes and motion design. a shape space. We introduce useful Riemannian metrics in this These examples show that it is often beneficial and insightful to space to aid the user in design and modeling tasks, especially to endow the set of objects under consideration with additional struc- explore the space of (approximately) isometric deformations of a ture and to work in more abstract spaces. We will show that many given shape. Much of the work relies on an efficient algorithm to geometry processing tasks can be solved by endowing the set of compute geodesics in shape spaces; to this end, we present a multi- closed orientable surfaces – called shapes henceforth – with a Rie- resolution framework to solve the interpolation problem – which mannian structure. -
Analyzing Space Dimensions in Video Games
SBC { Proceedings of SBGames 2019 | ISSN: 2179-2259 Art & Design Track { Full Papers Analyzing Space Dimensions in Video Games Leandro Ouriques Geraldo Xexéo Eduardo Mangeli Programa de Engenharia de Sistemas e Programa de Engenharia de Sistemas e Programa de Engenharia de Sistemas e Computação, COPPE/UFRJ Computação, COPPE/UFRJ Computação, COPPE/UFRJ Center for Naval Systems Analyses Departamento de Ciência da Rio de Janeiro, Brazil (CASNAV), Brazilian Nay Computação, IM/UFRJ E-mail: [email protected] Rio de Janeiro, Brazil Rio de Janeiro, Brazil E-mail: [email protected] E-mail: [email protected] Abstract—The objective of this work is to analyze the role of types of game space without any visualization other than space in video games, in order to understand how game textual. However, in 2018, Matsuoka et al. presented a space affects the player in achieving game goals. Game paper on SBGames that analyzes the meaning of the game spaces have evolved from a single two-dimensional screen to space for the player, explaining how space commonly has huge, detailed and complex three-dimensional environments. significance in video games, being bound to the fictional Those changes transformed the player’s experience and have world and, consequently, to its rules and narrative [5]. This encouraged the exploration and manipulation of the space. led us to search for a better understanding of what is game Our studies review the functions that space plays, describe space. the possibilities that it offers to the player and explore its characteristics. We also analyze location-based games where This paper is divided in seven parts. -
Chapter 6: Design and Design Frameworks: Investing in KBC and Economic Performance
323 | DESIGN AND DESIGN FRAMEWORKS: INVESTMENT IN KBC AND ECONOMIC PERFORMANCE CHAPTER 6. DESIGN AND DESIGN FRAMEWORKS: INVESTMENT IN KBC AND ECONOMIC PERFORMANCE This chapter addresses the nature and the economic impact of design by looking at design-related intellectual property and how businesses protect their knowledge based capital. The chapter reviews the nature and various definitions of design and how design-related IP, specifically registered designs, relates to other formal IP mechanisms such as patents, trademarks, and copyright. It looks at the primary areas of design activity in a subset of OECD countries and investigates the similarities and differences of the constituent design IP regimes as well as the various treaties governing international design IP regulation. The review continues with an examination of how design-related IP functions in comparison to and in conjunction with other formal and informal IP protection mechanisms and what factors motivate firms to choose and appropriate combinations of protection mechanisms. By examining historical patterns of design registrations in a variety of ways, this chapter identifies trends, at the national level, of how firms perceive the importance of design-related IP. Analysis of national origins of registrations in both the European Community and the United States provides an indicator of the activity of those countries’ businesses relative to their proximities to the markets. It explores the existence of possible alternative indicators for design activity and of industry-specific variations across the sample set. The chapter concludes with a review of input and output measures as stated in the limited set of studies that have endeavoured to establish or quantify the value and/or benefit of design and design-related IP. -
Intelligent Design Creationism and the Constitution
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Washington University St. Louis: Open Scholarship Washington University Law Review Volume 83 Issue 1 2005 Is It Science Yet?: Intelligent Design Creationism and the Constitution Matthew J. Brauer Princeton University Barbara Forrest Southeastern Louisiana University Steven G. Gey Florida State University Follow this and additional works at: https://openscholarship.wustl.edu/law_lawreview Part of the Constitutional Law Commons, Education Law Commons, First Amendment Commons, Religion Law Commons, and the Science and Technology Law Commons Recommended Citation Matthew J. Brauer, Barbara Forrest, and Steven G. Gey, Is It Science Yet?: Intelligent Design Creationism and the Constitution, 83 WASH. U. L. Q. 1 (2005). Available at: https://openscholarship.wustl.edu/law_lawreview/vol83/iss1/1 This Article is brought to you for free and open access by the Law School at Washington University Open Scholarship. It has been accepted for inclusion in Washington University Law Review by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. Washington University Law Quarterly VOLUME 83 NUMBER 1 2005 IS IT SCIENCE YET?: INTELLIGENT DESIGN CREATIONISM AND THE CONSTITUTION MATTHEW J. BRAUER BARBARA FORREST STEVEN G. GEY* TABLE OF CONTENTS ABSTRACT ................................................................................................... 3 INTRODUCTION.................................................................................................. -
Twisted Loop Groups and Their Affine Flag Varieties
TWISTED LOOP GROUPS AND THEIR AFFINE FLAG VARIETIES G. PAPPAS* AND M. RAPOPORT Introduction Loop groups are familiar objects in several branches of mathematics. Let us mention here three variants. The first variant is differential-geometric in nature. One starts with a Lie group G (e.g., a compact Lie group or its complexification). The associated loop group is then the group of (C0-, or C1-, or C∞-) maps of S1 into G, cf. [P-S] and the literature cited there. A twisted version arises from an automorphism α of G. The associated twisted loop group is the group of maps γ : R → G such that γ(θ + 2π) = α(γ(θ)) . The second variant is algebraic and arises in the context of Kac-Moody algebras. Here one constructs an infinite-dimensional algebraic group variety with Lie algebra equal or closely related to a given Kac-Moody algebra. (This statement is an oversimplification and the situation is in fact more complicated: there exist various constructions at a formal, a minimal, and a maximal level which produce infinite-dimensional groups with Lie algebras closely related to the given Kac-Moody Lie algebra, see [Ma2], also [T2], [T3] and the literature cited there). The third variant is algebraic-geometric in nature and is our main concern in this paper. Let us recall the basic definitions in the untwisted case. Let k be a field and let G0 be an algebraic group over Spec (k). We consider the functor LG0 on the category of k-algebras, R 7→ LG0(R) = G0(R((t))). -
An Overview of the Building Delivery Process
An Overview of the Building Delivery CHAPTER Process 1 (How Buildings Come into Being) CHAPTER OUTLINE 1.1 PROJECT DELIVERY PHASES 1.11 CONSTRUCTION PHASE: CONTRACT ADMINISTRATION 1.2 PREDESIGN PHASE 1.12 POSTCONSTRUCTION PHASE: 1.3 DESIGN PHASE PROJECT CLOSEOUT 1.4 THREE SEQUENTIAL STAGES IN DESIGN PHASE 1.13 PROJECT DELIVERY METHOD: DESIGN- BID-BUILD METHOD 1.5 CSI MASTERFORMAT AND SPECIFICATIONS 1.14 PROJECT DELIVERY METHOD: 1.6 THE CONSTRUCTION TEAM DESIGN-NEGOTIATE-BUILD METHOD 1.7 PRECONSTRUCTION PHASE: THE BIDDING 1.15 PROJECT DELIVERY METHOD: CONSTRUCTION DOCUMENTS MANAGEMENT-RELATED METHODS 1.8 PRECONSTRUCTION PHASE: THE SURETY BONDS 1.16 PROJECT DELIVERY METHOD: DESIGN-BUILD METHOD 1.9 PRECONSTRUCTION PHASE: SELECTING THE GENERAL CONTRACTOR AND PROJECT 1.17 INTEGRATED PROJECT DELIVERY METHOD DELIVERY 1.18 FAST-TRACK PROJECT SCHEDULING 1.10 CONSTRUCTION PHASE: SUBMITTALS AND CONSTRUCTION PROGRESS DOCUMENTATION Building construction is a complex, significant, and rewarding process. It begins with an idea and culminates in a structure that may serve its occupants for several decades, even centuries. Like the manufacturing of products, building construction requires an ordered and planned assembly of materials. It is, however, far more complicated than product manufacturing. Buildings are assembled outdoors by a large number of diverse constructors and artisans on all types of sites and are subject to all kinds of weather conditions. Additionally, even a modest-sized building must satisfy many performance criteria and legal constraints, requires an immense variety of materials, and involves a large network of design and production firms. Building construction is further complicated by the fact that no two buildings are identical; each one must be custom built to serve a unique function and respond to its specific context and the preferences of its owner, user, and occupant. -
On Some Recent Developments in the Theory of Buildings
On some recent developments in the theory of buildings Bertrand REMY∗ Abstract. Buildings are cell complexes with so remarkable symmetry properties that many groups from important families act on them. We present some examples of results in Lie theory and geometric group theory obtained thanks to these highly transitive actions. The chosen examples are related to classical and less classical (often non-linear) group-theoretic situations. Mathematics Subject Classification (2010). 51E24, 20E42, 20E32, 20F65, 22E65, 14G22, 20F20. Keywords. Algebraic, discrete, profinite group, rigidity, linearity, simplicity, building, Bruhat-Tits' theory, Kac-Moody theory. Introduction Buildings are cell complexes with distinguished subcomplexes, called apartments, requested to satisfy strong incidence properties. The notion was invented by J. Tits about 50 years ago and quickly became useful in many group-theoretic situations [75]. By their very definition, buildings are expected to have many symmetries, and this is indeed the case quite often. Buildings are relevant to Lie theory since the geometry of apartments is described by means of Coxeter groups: apartments are so to speak generalized tilings, where a usual (spherical, Euclidean or hyper- bolic) reflection group may be replaced by a more general Coxeter group. One consequence of the existence of sufficiently large automorphism groups is the fact that many buildings admit group actions with very strong transitivity properties, leading to a better understanding of the groups under consideration. The beginning of the development of the theory is closely related to the theory of algebraic groups, more precisely to Borel-Tits' theory of isotropic reductive groups over arbitrary fields and to Bruhat-Tits' theory of reductive groups over non-archimedean valued fields.