12,062 research outputs found

    Graduate Catalog of Studies, 2023-2024

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    Cyclic proof systems for modal fixpoint logics

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    This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one

    Fragments and frame classes:Towards a uniform proof theory for modal fixed point logics

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    This thesis studies the proof theory of modal fixed point logics. In particular, we construct proof systems for various fragments of the modal mu-calculus, interpreted over various classes of frames. With an emphasis on uniform constructions and general results, we aim to bring the relatively underdeveloped proof theory of modal fixed point logics closer to the well-established proof theory of basic modal logic. We employ two main approaches. First, we seek to generalise existing methods for basic modal logic to accommodate fragments of the modal mu-calculus. We use this approach for obtaining Hilbert-style proof systems. Secondly, we adapt existing proof systems for the modal mu-calculus to various classes of frames. This approach yields proof systems which are non-well-founded, or cyclic.The thesis starts with an introduction and some mathematical preliminaries. In Chapter 3 we give hypersequent calculi for modal logic with the master modality, building on work by Ori Lahav. This is followed by an Intermezzo, where we present an abstract framework for cyclic proofs, in which we give sufficient conditions for establishing the bounded proof property. In Chapter 4 we generalise existing work on Hilbert-style proof systems for PDL to the level of the continuous modal mu-calculus. Chapter 5 contains a novel cyclic proof system for the alternation-free two-way modal mu-calculus. Finally, in Chapter 6, we present a cyclic proof system for Guarded Kleene Algebra with Tests and take a first step towards using it to establish the completeness of an algebraic counterpart

    UMSL Bulletin 2023-2024

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    The 2023-2024 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1088/thumbnail.jp

    Graduate Catalog of Studies, 2023-2024

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    Log-majorization type inequalities

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    Several inequalities have been established in the context of Hilbert spaces operators or operator algebras. Our discussion will be limited to matrices. Important inequalities in mathematics and other sciences, such as Golden-Thompson inequality or von Neumann trace inequality, and extensions, are revisited. Our main goal is to emphasize the link between majorization theory and other relevant inequalities.Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), Centre for Mathematics of the University of Coimbra (CMUC), Center for Research and Development in Mathematics and Applications (CIDMA) , CMAT-UTADpublishe

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    From a Doodle to a Theorem: A Case Study in Mathematical Discovery

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    We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

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    Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of dd-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.Comment: 46 pages mai
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