3,229 research outputs found

    Elementary abelian subgroups: from algebraic groups to finite groups

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    We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p-subgroups for torsion primes p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p-local subgroups and p-radical subgroups, both of which play a crucial role in modular representation theory

    A matrix unified framework for deriving various impulse responses in Markov switching VAR: Evidence from oil and gas markets

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    We propose a new method to compute various impulse response functions (IRF) for a Markov switching VAR model in terms of neat matrix expressions in closed form. The key is to derive a suitable closed form representation for Markov switching VAR models using a state-space representation. By this representation, the IRF analysis can be processed with respect to either an asymmetric discrete or a symmetric continuous shocks. A simulation study demonstrates the actual advantages of the proposed matrix methodology. To illustrate the feasibility and the usefulness of our approach, we present empirical applications to oil and natural gas markets showing the relevance of accommodating asymmetries in the relationship between their price shocks and economic activities

    Contract-Based Design of Dataflow Programs

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    Quality and correctness are becoming increasingly important aspects of software development, as our reliance on software systems in everyday life continues to increase. Highly complex software systems are today found in critical appliances such as medical equipment, cars, and telecommunication infrastructure. Failures in these kinds of systems may have disastrous consequences. At the same time, modern computer platforms are increasingly concurrent, as the computational capacity of modern CPUs is improved mainly by increasing the number of processor cores. Computer platforms are also becoming increasingly parallel, distributed and heterogeneous, often involving special processing units, such as graphics processing units (GPU) or digital signal processors (DSP) for performing specific tasks more efficiently than possible on general-purpose CPUs. These modern platforms allow implementing increasingly complex functionality in software. Cost efficient development of software that efficiently exploits the power of this type of platforms and at the same time ensures correctness is, however, a challenging task. Dataflow programming has become popular in development of safetycritical software in many domains in the embedded community. For instance, in the automotive domain, the dataflow language Simulink has become widely used in model-based design of control software. However, for more complex functionality, this model of computation may not be expressive enough. In the signal processing domain, more expressive, dynamic models of computation have attracted much attention. These models of computation have, however, not gained as significant uptake in safety-critical domains due to a great extent to that it is challenging to provide guarantees regarding e.g. timing or determinism under these more expressive models of computation. Contract-based design has become widespread to specify and verify correctness properties of software components. A contract consists of assumptions (preconditions) regarding the input data and guarantees (postconditions) regarding the output data. By verifying a component with respect to its contract, it is ensured that the output fulfils the guarantees, assuming that the input fulfils the assumptions. While contract-based verification of traditional object-oriented programs has been researched extensively, verification of asynchronous dataflow programs has not been researched to the same extent. In this thesis, a contract-based design framework tailored specifically to dataflow programs is proposed. The proposed framework supports both an extensive subset of the discrete-time Simulink synchronous language, as well as a more general, asynchronous and dynamic, dataflow language. The proposed contract-based verification techniques are automatic, only guided by user-provided invariants, and based on encoding dataflow programs in existing, mature verification tools for sequential programs, such as the Boogie guarded command language and its associated verifier. It is shown how dataflow programs, with components implemented in an expressive programming language with support for matrix computations, can be efficiently encoded in such a verifier. Furthermore, it is also shown that contract-based design can be used to improve runtime performance of dataflow programs by allowing more scheduling decisions to be made at compile-time. All the proposed techniques have been implemented in prototype tools and evaluated on a large number of different programs. Based on the evaluation, the methods were proven to work in practice and to scale to real-world programs.Kvalitet och korrekthet blir idag allt viktigare aspekter inom mjukvaruutveckling, dÄ vi i allt högre grad förlitar oss pÄ mjukvarusystem i vÄra vardagliga sysslor. Mycket komplicerade mjukvarusystem finns idag i kritiska tillÀmpningar sÄ som medicinsk utrustning, bilar och infrastruktur för telekommunikation. Fel som uppstÄr i de hÀr typerna av system kan ha katastrofala följder. Samtidigt utvecklas kapaciteten hos moderna datorplattformar idag frÀmst genom att öka antalet processorkÀrnor. DÀrtill blir datorplattformar allt mer parallella, distribuerade och heterogena, och innefattar ofta specialla processorer sÄ som grafikprocessorer (GPU) eller signalprocessorer (DSP) för att utföra specifika berÀkningar snabbare Àn vad som Àr möjligt pÄ vanliga processorer. Den hÀr typen av plattformar möjligör implementering av allt mer komplicerade berÀkningar i mjukvara. Kostnadseffektiv utveckling av mjukvara som effektivt utnyttjar kapaciteten i den hÀr typen av plattformar och samtidigt sÀkerstÀller korrekthet Àr emellertid en mycket utmanande uppgift. Dataflödesprogrammering har blivit ett populÀrt sÀtt att utveckla mjukvara inom flera omrÄden som innefattar sÀkerhetskritiska inbyggda datorsystem. Till exempel inom fordonsindustrin har dataflödessprÄket Simulink kommit att anvÀndas i bred utstrÀckning för modellbaserad design av kontrollsystem. För mer komplicerad funktionalitet kan dock den hÀr modellen för berÀkning vara för begrÀnsad betrÀffande vad som kan beksrivas. Inom signalbehandling har mera expressiva och dynamiska modeller för berÀkning attraherat stort intresse. De hÀr modellerna för berÀkning har ÀndÄ inte tagits i bruk i samma utstrÀckning inom sÀkerhetskritiska tillÀmpningar. Det hÀr beror till en stor del pÄ att det Àr betydligt svÄrare att garantera egenskaper gÀllande till exempel timing och determinism under sÄdana hÀr modeller för berÀkning. Kontraktbaserad design har blivit ett vanligt sÀtt att specifiera och verifiera korrekthetsegenskaper hos mjukvarukomponeneter. Ett kontrakt bestÄr av antaganden (förvillkor) gÀllande indata och garantier (eftervillkor) gÀllande utdata. Genom att verifiera en komponent gentemot sitt konktrakt kan man bevisa att utdatan uppfyller garantierna, givet att indatan uppfyller antagandena. Trots att kontraktbaserad verifiering i sig Àr ett mycket beforskat omrÄde, sÄ har inte verifiering av asynkrona dataflödesprogram beforskats i samma utstrÀckning. I den hÀr avhandlingen presenteras ett ramverk för kontraktbaserad design skrÀddarsytt för dataflödesprogram. Det föreslagna ramverket stödjer sÄ vÀl en stor del av det synkrona sprÄket. Simulink med diskret tid som ett mera generellt asynkront och dynamiskt dataflödessprÄk. De föreslagna kontraktbaserade verifieringsteknikerna Àr automatiska. Utöver kontraktets för- och eftervillkor ger anvÀndaren endast de invarianter som krÀvs för att möjliggöra verifieringen. Verifieringsteknikerna grundar sig pÄ att omkoda dataflödesprogram till input för existerande och beprövade verifieringsverktyg för sekventiella program sÄ som Boogie. Avhandlingen visar hur dataflödesprogram implementerade i ett expressivt programmeringssprÄk med inbyggt stöd för matrisoperationer effektivt kan omkodas till input för ett verifieringsverktyg som Boogie. Utöver detta visar avhandlingen ocksÄ att kontraktbaserad design ocksÄ kan förbÀttra prestandan hos dataflödesprogram i körningsskedet genom att möjliggöra flera schemalÀggningsbeslut redan i kompileringsskedet. Alla tekniker som presenteras i avhandlingen har implementerats i prototypverktyg och utvÀrderats pÄ en stor mÀngd olika program. UtvÀrderingen bevisar att teknikerna fungerar i praktiken och Àr tillrÀckligt skalbara för att ocksÄ fungera pÄ program av realistisk storlek

    Algebraic solutions of linear differential equations: an arithmetic approach

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    Given a linear differential equation with coefficients in Q(x)\mathbb{Q}(x), an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz pp-curvature conjecture.Comment: 47 page

    A Generalised abc Conjecture and Quantitative Diophantine Approximation

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    The abc Conjecture and its number field variant have huge implications across a wide range of mathematics. While the conjecture is still unproven, there are a number of partial results, both for the integer and the number field setting. Notably, Stewart and Yu have exponential abc bounds for integers, using tools from linear forms in logarithms, while GyƑry has exponential abc bounds in the number field case, using methods from S-unit equations [20]. In this thesis, we aim to combine these methods to give improved results in the number field case. These results are then applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc conjecture. The smooth abc conjecture concerns counting the number of solutions to a+b = c with restrictions on the values of a, b and c. this leads us to more general methods of counting solutions to Diophantine problems. Many of these results are asymptotic in nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these lemmas effective rather than asymptotic other than on a set of size ή > 0, where ή is arbitrary. From there, we apply these tools to give an effective Schmidt’s Theorem, a quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin- Schaeffer Theorem), and to give effective results on inhomogeneous Diophantine Approximation on M0-sets, normal numbers and give an effective Strong Law of Large Numbers. We conclude this thesis by giving general versions of Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory"

    Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras

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    We calculate the fusion rules among Z2\mathbb{Z}_2-twisted modules Lsl2(ℓ,0)L_{\mathfrak{sl}_2}(\ell,0) at admissible levels. We derive a series MLDEs for normalized characters of ordinary twisted modules of quasi-lisse vertex algebras. Examples include affine VOAs of type A1(1)A_1^{(1)} at boundary admissible level, admissible level k=−1/2k=-1/2, A2(1)A^{(1)}_{2} at boundary admissible level k=−3/2k=-3/2, and BPk\mathrm{BP}^{k}-algebra with special value k=−9/4k=-9/4. We also derive characters of some non-vacuum modules for affine VOA of type D4D_4 at non-admissible level −2-2 from spectral flow automorphism

    Shuffle approach towards quantum affine and toroidal algebras

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    These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of sln\mathfrak{sl}_n (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of gl1\mathfrak{gl}_1, and the quantum toroidal algebras of sln\mathfrak{sl}_n. We provide the shuffle realization of the corresponding ``positive'' subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of ``specialization maps''. Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type AA quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant KK-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of sln\mathfrak{sl}_n using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras.Comment: v2: 124 pages, significantly improved version, typos fixed. v1: 126 page

    Locality and Exceptional Points in Pseudo-Hermitian Physics

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    Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iÎł, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when Îł increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states. Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case

    Doubly Efficient Batched Private Information Retrieval

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    Private information retrieval (PIR) allows a client to read data from a server, without revealing which information they are interested in. A PIR is doubly efficient if the server runtime is, after a one-time pre-processing, sublinear in the database size. A recent breakthrough result from Lin, Mook, and Wichs [STOC’23] proposed the first-doubly efficient PIR with (online) server computation poly-logarithmic in the size of the database, assuming the hardness of the standard Ring-LWE problem. In this work, we consider the problem of doubly efficient batched PIR (DEBPIR), where the client wishes to download multiple entries. This problem arises naturally in many practical applications of PIR, or when the database contains large entries. Our main result is a construction of DEBPIR where the amortized communication and server computation overhead is O~(1)\tilde{O}(1), from the Ring-LWE problem. This represents an exponential improvement compared with known constructions, and it is optimal up to poly-logarithmic factors in the security parameter. Interestingly, the server’s online operations are entirely combinatorial and all algebraic computations are done in the pre-processing or delegated to the client

    Modular curves and N\'eron models of generalized Jacobians

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    Let XX be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring RR, and m\mathfrak{m} a modulus on XX, given by a closed subscheme of XX which is geometrically reduced. The generalized Jacobian JmJ_\mathfrak{m} of XX with respect to m\mathfrak{m} is then an extension of the Jacobian of XX by a torus. We describe its N\'eron model, together with the character and component groups of the special fibre, in terms of a regular model of XX over RR. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves X0(N)X_0(N) with moduli supported on the cusps.Comment: 36 pages, minor corrections and references added. Accepted version, to appear in Compositio Mat
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