1,703 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Clones over Finite Sets and Minor Conditions

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    Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra. In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset. We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence

    University of Windsor Graduate Calendar 2023 Spring

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp

    Non-equilibrium universality: slow drives, measurements and dephasing

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    The behavior of quantum systems can be influenced by factors such as unitary evolution, measurements or decoherence. For large composite systems, these mechanisms can give rise to collective phenomena like phase transitions and universality. One example are quantum phase transitions in the ground states of a Hamiltonian. Close to the transition scale invariant behavior emerges, characterized by a set of universal critical exponents. If the system is driven in the vicinity of the transition, the drive scale can lead to a breakdown of the equilibrium scaling behavior. Nevertheless, the breakdown inherits universal properties and gives access to the leading critical exponents (Kibble-Zurek mechanism). However, the whole hierarchy of critical exponents, relevant and irrelevant, is accessible by a slow drive. We establish this generalized mechanism and its observable consequences at the level of elementary, but experimentally relevant, spin and fermion models. We construct drives that turn equilibrium irrelevant couplings into relevant drive couplings with an observable scaling in the excitation density. Criticality and universality also arise from competing unitary evolution and measurements, allowing for measurement-induced transitions. An example are (free) fermion models featuring a transition between an extended `critical' phase and a `pinned', weakly entangled phase. We investigate the role of dephasing/imperfect measurements onto the transition based on (i) numerical approaches (stochastic quantum trajectories), (ii) an effective bosonic replica field theory, and (iii) a perturbative treatment of the fermion dynamics. On the one hand, weak dephasing leaves the `critical' phase and measurement-induced transition in tact. On the other hand, we observe the emergence of a new, temperature-like scale for strong dephasing and weak measurements, enabled by the interplay of all three mechanisms. Despite the presence of the finite scale, observables like density-dependent correlations still feature scale invariant behavior. Paired with a perturbative treatment for strong dephasing, this behavior hints at a diffusion-like dynamics on the diagonal of the density matrix in the occupation number basis

    University of Windsor Graduate Calendar 2023 Winter

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Languages, groups and equations

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    The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in the area has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature groups. In virtually abelian groups the solutions sets are in fact rational, and one can obtain them as mm-regular sets. In the Heisenberg group producing the solutions to a single equation reduces to understanding the solutions to quadratic Diophantine equations and uses number theoretic techniques. In the Baumslag-Solitar groups the methods are combinatorial, and focus on the interplay of normal forms to solve particular classes of equations. In conclusion, EDT0L languages give an effective and simple combinatorial characterisation of sets of seemingly high complexity in many important classes of groups.Comment: 26 page

    Undergraduate and Graduate Course Descriptions, 2023 Spring

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    Wright State University undergraduate and graduate course descriptions from Spring 2023
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