1,096 research outputs found

    A qualitative approach to the identification, visualisation and interpretation of repetitive motion patterns in groups of moving point objects

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    Discovering repetitive patterns is important in a wide range of research areas, such as bioinformatics and human movement analysis. This study puts forward a new methodology to identify, visualise and interpret repetitive motion patterns in groups of Moving Point Objects (MPOs). The methodology consists of three steps. First, motion patterns are qualitatively described using the Qualitative Trajectory Calculus (QTC). Second, a similarity analysis is conducted to compare motion patterns and identify repetitive patterns. Third, repetitive motion patterns are represented and interpreted in a continuous triangular model. As an illustration of the usefulness of combining these hitherto separated methods, a specific movement case is examined: Samba dance, a rhythmical dance will? many repetitive movements. The results show that the presented methodology is able to successfully identify, visualize and interpret the contained repetitive motions

    Soergel calculus

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    On the "principle of the quantumness", the quantumness of Relativity, and the computational grand-unification

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    After reviewing recently suggested operational "principles of the quantumness", I address the problem on whether Quantum Theory (QT) and Special Relativity (SR) are unrelated theories, or instead, if the one implies the other. I show how SR can be indeed derived from causality of QT, within the computational paradigm "the universe is a huge quantum computer", reformulating QFT as a Quantum-Computational Field Theory (QCFT). In QCFT SR emerges from the fabric of the computational network, which also naturally embeds gauge invariance. In this scheme even the quantization rule and the Planck constant can in principle be derived as emergent from the underlying causal tapestry of space-time. In this way QT remains the only theory operating the huge computer of the universe. Is QCFT only a speculative tautology (theory as simulation of reality), or does it have a scientific value? The answer will come from Occam's razor, depending on the mathematical simplicity of QCFT. Here I will just start scratching the surface of QCFT, analyzing simple field theories, including Dirac's. The number of problems and unmotivated recipes that plague QFT strongly motivates us to undertake the QCFT project, since QCFT makes all such problems manifest, and forces a re-foundation of QFT.Comment: To be published on AIP Proceedings of Vaxjo conference. The ideas on Quantum-Circuit Field Theory are more recent. V4 Largely improved, with new interesting results and concepts. Dirac equation solve

    Thick Soergel calculus in type A

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    Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we extend it to a presentation of the category of generalized Bott-Samelson bimodules. We also diagrammatically categorify the representations of the Hecke algebra which are induced from trivial representations of parabolic subgroups. The main tool is an explicit description of the idempotent which picks out a generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson bimodule. This description uses a detailed analysis of the reduced expression graph of the longest element of S_n, and the semi-orientation on this graph given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper relies extensively on color figure

    A Priori Error Analysis and Spring Arithmetic

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    WOSInternational audienceError analysis is defined by the following concern: bounding the output variation of a (nonlinear) function with respect to a given variation of the input variables. This paper investigates this issue in the framework of interval analysis. The classical way of analyzing the error is to linearize the function around the point corresponding to the actual input, but this method is local and not reliable. Both drawbacks can be easily circumvented by a combined use of interval arithmetic and domain splitting. However, because of the underlying linearization, a standard interval algorithm leads to a pessimistic bound, and even simply fails (i.e., returns an infinite error) in case of singularity. We propose an original nonlinear approach where intervals are used in a more sophisticated way through the so-called "springs". This new structure allows to represent an (infinite) set of intervals constrained by their midpoints and their radius. The output error is then calculated with a spring arithmetic in the same way as the image of a function is calculated with interval arithmetic. Our method is illustrated on two examples, including an application of geopositioning

    Diagrammatic Languages and Formal Verification : A Tool-Based Approach

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    The importance of software correctness has been accentuated as a growing number of safety-critical systems have been developed relying on software operating these systems. One of the more prominent methods targeting the construction of a correct program is formal verification. Formal verification identifies a correct program as a program that satisfies its specification and is free of defects. While in theory formal verification guarantees a correct implementation with respect to the specification, applying formal verification techniques in practice has shown to be difficult and expensive. In response to these challenges, various support methods and tools have been suggested for all phases from program specification to proving the derived verification conditions. This thesis concerns practical verification methods applied to diagrammatic modeling languages. While diagrammatic languages are widely used in communicating system design (e.g., UML) and behavior (e.g., state charts), most formal verification platforms require the specification to be written in a textual specification language or in the mathematical language of an underlying logical framework. One exception is invariant-based programming, in which programs together with their specifications are drawn as invariant diagrams, a type of state transition diagram annotated with intermediate assertions (preconditions, postconditions, invariants). Even though the allowed program states—called situations—are described diagrammatically, the intermediate assertions defining a situation’s meaning in the domain of the program are still written in conventional textual form. To explore the use of diagrams in expressing the intermediate assertions of invariant diagrams, we designed a pictorial language for expressing array properties. We further developed this notation into a diagrammatic domain-specific language (DSL) and implemented it as an extension to the Why3 platform. The DSL supports expression of array properties. The language is based on Reynolds’s interval and partition diagrams and includes a construct for mapping array intervals to logic predicates. Automated verification of a program is attained by generating the verification conditions and proving that they are true. In practice, full proof automation is not possible except for trivial programs and verifying even simple properties can require significant effort both in specification and proof stages. An animation tool which supports run-time evaluation of the program statements and intermediate assertions given any user-defined input can support this process. In particular, an execution trace leading up to a failed assertion constitutes a refutation of a verification condition that requires immediate attention. As an extension to Socos, a verificion tool for invariant diagrams built on top of the PVS proof system, we have developed an execution model where program statements and assertions can be evaluated in a given program state. A program is represented by an abstract datatype encoding the program state, together with a small-step state transition function encoding the evaluation of a single statement. This allows the program’s runtime behavior to be formally inspected during verification. We also implement animation and interactive debugging support for Socos. The thesis also explores visualization of system development in the context of model decomposition in Event-B. Decomposing a software system becomes increasingly critical as the system grows larger, since the workload on the theorem provers must be distributed effectively. Decomposition techniques have been suggested in several verification platforms to split the models into smaller units, each having fewer verification conditions and therefore imposing a lighter load on automatic theorem provers. In this work, we have investigated a refinement-based decomposition technique that makes the development process more resilient to change in specification and allows parallel development of sub-models by a team. As part of the research, we evaluated the technique on a small case study, a simplified version of a landing gear system verification presented by Boniol and Wiels, within the Event-B specification language.Vikten av programvaras korrekthet har accentuerats dĂ„ ett vĂ€xande antal sĂ€kerhetskritiska system, vilka Ă€r beroende av programvaran som styr dessa, har utvecklas. En av de mer framtrĂ€dande metoderna som riktar in sig pĂ„ utveckling av korrekt programvara Ă€r formell verifiering. Inom formell verifiering avses med ett korrekt program ett program som uppfyller sina specifikationer och som Ă€r fritt frĂ„n defekter. Medan formell verifiering teoretiskt sett kan garantera ett korrekt program med avseende pĂ„ specifikationerna, har tillĂ€mpligheten av formella verifieringsmetod visat sig i praktiken vara svĂ„r och dyr. Till svar pĂ„ dessa utmaningar har ett stort antal olika stödmetoder och automatiseringsverktyg föreslagits för samtliga faser frĂ„n specifikationen till bevisningen av de hĂ€rledda korrekthetsvillkoren. Denna avhandling behandlar praktiska verifieringsmetoder applicerade pĂ„ diagrambaserade modelleringssprĂ„k. Medan diagrambaserade sprĂ„k ofta anvĂ€nds för kommunikation av programvarudesign (t.ex. UML) samt beteende (t.ex. tillstĂ„ndsdiagram), krĂ€ver de flesta verifieringsplattformar att specifikationen kodas medelst ett textuellt specifikationsspĂ„k eller i sprĂ„ket hos det underliggande logiska ramverket. Ett undantag Ă€r invariantbaserad programmering, inom vilken ett program tillsammans med dess specifikation ritas upp som sk. invariantdiagram, en typ av tillstĂ„ndstransitionsdiagram annoterade med mellanliggande logiska villkor (förvillkor, eftervillkor, invarianter). Även om de tillĂ„tna programtillstĂ„nden—sk. situationer—beskrivs diagrammatiskt Ă€r de logiska predikaten som beskriver en situations betydelse i programmets domĂ€n fortfarande skriven pĂ„ konventionell textuell form. För att vidare undersöka anvĂ€ndningen av diagram vid beskrivningen av mellanliggande villkor inom invariantbaserad programming, har vi konstruerat ett bildbaserat sprĂ„k för villkor över arrayer. Vi har dĂ€refter vidareutvecklat detta sprĂ„k till ett diagrambaserat domĂ€n-specifikt sprĂ„k (domain-specific language, DSL) och implementerat stöd för det i verifieringsplattformen Why3. SprĂ„ket lĂ„ter anvĂ€ndaren uttrycka egenskaper hos arrayer, och Ă€r baserat pĂ„ Reynolds intevall- och partitionsdiagram samt inbegriper en konstruktion för mappning av array-intervall till logiska predikat. Automatisk verifiering av ett program uppnĂ„s genom generering av korrekthetsvillkor och Ă„tföljande bevisning av dessa. I praktiken kan full automatisering av bevis inte uppnĂ„s utom för trivial program, och Ă€ven bevisning av enkla egenskaper kan krĂ€va betydande anstrĂ€ngningar bĂ„de vid specifikations- och bevisfaserna. Ett animeringsverktyg som stöder exekvering av sĂ„vĂ€l programmets satser som mellanliggande villkor för godtycklig anvĂ€ndarinput kan vara till hjĂ€lp i denna process. SĂ€rskilt ett exekveringspĂ„r som leder upp till ett falskt mellanliggande villkor utgör ett direkt vederlĂ€ggande (refutation) av ett bevisvillkor, vilket krĂ€ver omedelbar uppmĂ€rksamhet frĂ„n programmeraren. Som ett tillĂ€gg till Socos, ett verifieringsverktyg för invariantdiagram baserat pĂ„ bevissystemet PVS, har vi utvecklat en exekveringsmodell dĂ€r programmets satser och villkor kan evalueras i ett givet programtillstĂ„nd. Ett program representeras av en abstrakt datatyp för programmets tillstĂ„nd tillsammans med en small-step transitionsfunktion för evalueringen av en enskild programsats. Detta möjliggör att ett programs exekvering formellt kan analyseras under verifieringen. Vi har ocksĂ„ implementerat animation och interaktiv felsökning i Socos. Avhandlingen undersöker ocksĂ„ visualisering av systemutveckling i samband med modelluppdelning inom Event-B. Uppdelning av en systemmodell blir allt mer kritisk dĂ„ ett systemet vĂ€xer sig större, emedan belastningen pĂ„ underliggande teorembe visare mĂ„ste fördelas effektivt. Uppdelningstekniker har föreslagits inom mĂ„nga olika verifieringsplattformar för att dela in modellerna i mindre enheter, sĂ„ att varje enhet har fĂ€rre verifieringsvillkor och dĂ€rmed innebĂ€r en mindre belastning pĂ„ de automatiska teorembevisarna. I detta arbete har vi undersökt en refinement-baserad uppdelningsteknik som gör utvecklingsprocessen mer kapabel att hantera förĂ€ndringar hos specifikationen och som tillĂ„ter parallell utveckling av delmodellerna inom ett team. Som en del av forskningen har vi utvĂ€rderat tekniken pĂ„ en liten fallstudie: en förenklad modell av automationen hos ett landningsstĂ€ll av Boniol and Wiels, uttryckt i Event-B-specifikationsprĂ„ket

    Rigid C^*-tensor categories of bimodules over interpolated free group factors

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    Given a countably generated rigid C^*-tensor category C, we construct a planar algebra P whose category of projections Pro is equivalent to C. From P, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C^*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F_infty).Comment: 50 pages, many figure

    Webs and quantum skew Howe duality

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    We give a diagrammatic presentation in terms of generators mod relations of the representation category of Uq(sln)U_q(\mathfrak{sl}_n). More precisely, we produce all the relations among SLn\rm{SL}_n-webs, thus describing the full subcategory tensor-generated by fundamental representations ⋀kCn\bigwedge^k \mathbb{C}^n (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim [arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an application of quantum skew Howe duality.Comment: 32 pages, added a missing relation which had been implicitly used; this version has the same content as the published versio
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