1,703 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
"Le present est plein de l’avenir, et chargé du passé" : Vorträge des XI. Internationalen Leibniz-Kongresses, 31. Juli – 4. August 2023, Leibniz Universität Hannover, Deutschland. Band 2
[No abstract available]Deutschen Forschungsgemeinschaft (DFG)/Projektnr. 517991912VGH VersicherungNiedersächsisches Ministerium für Wissenschaft und Kultur (MWK
Clones over Finite Sets and Minor Conditions
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
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp
Non-equilibrium universality: slow drives, measurements and dephasing
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
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Languages, groups and equations
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 -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
Wright State University undergraduate and graduate course descriptions from Spring 2023
Recommended from our members
Highly symmetric embeddings of graphs on surfaces
This thesis considers highly symmetric maps, that is embeddings of graphs in surfaces such that the automorphism group is “large”. This may be when the automorphism group of the map acts regularly on the flag-set of the map, as for the fully regular maps studied in Part I. In contrast, Part II focusses on a class of maps where the automorphism group has (up to) two orbits on the flag-set and may not be edge-transitive.
Part I is dedicated to advancing the understanding of fully regular maps with external symmetries. Chapter 2 proves that for arbitrary valency greater than three, a fully regular map with Trinity symmetry exists, extending the previously-known existence of such a map for every even valency. Chapter 3 addresses a group of operators which acts on fully regular maps whose automorphism group is isomorphic to SL(2, 2^α). The group of operators, which depends on the value of α and is defined more precisely in Chapter 3, includes the dual and Petrie operators as well as the allowable hole operators. One approach is by exploring the orbits of this group as it acts on the space of all maps with automorphism group isomorphic to SL(2, 2^α) for the given α. A detailed investigation is presented for the group of operators acting on the set consisting of all maps with automorphism group A5 which is isomorphic to SL(2, 4).
In Part II, the focus is on edge-biregular maps. These maps can be identified with group presentations which have a particular form, namely they are generated by four involutions which partition into two distinct sets each consisting of a pair of commuting involutions. Edge-biregular maps correspond to the most symmetric examples of maps with bipartite medial graph. By the definition, each edge-biregular map inherits a two-colouring on the edges, and so long as the map is not degenerate in some way, both the valency and the face length are even. In Chapter 4 these maps are introduced, foundations are laid and degeneracies are addressed. Chapter 5 is a partial classification covering edge-biregular maps whose colour-preserving automorphism group is dihedral, and/or whose surface has Euler characteristic which is either non-negative or negative and prime. The context for Chapter 6 is edge-biregular maps whose underlying group is symmetric or alternating. A genuinely edge-biregular map is an edge-biregular map which (when disregarding the colouring of edges) is not a fully regular map. The chapter includes a proof that, with the exception of some small cases, a genuinely edge-biregular map of every feasible type exists such that the colour preserving automorphism group is symmetric or alternating.</br
- …