IA Scholar Query: Automorphism Groups of Substructure Lattices of Vector Spaces in Computable Algebra.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 03 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Use and Abuse of Instance Parameters in the Lean Mathematical Library
https://scholar.archive.org/work/rof3xdk27rhihk7srigfkjzgo4
The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean's mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.Anne Baanen, June Andronick, Leonardo de Mourawork_rof3xdk27rhihk7srigfkjzgo4Wed, 03 Aug 2022 00:00:00 GMTCanonical Heights on Shimura Varieties and the André-Oort Conjecture
https://scholar.archive.org/work/najpsxgvizhbrpfaxoyzqawjqe
The main purpose of this work is to prove the Andr\'e-Oort conjecture in full generality.Jonathan Pila, Ananth N. Shankar, Jacob Tsimerman, Hélène Esnault, Michael Groechenigwork_najpsxgvizhbrpfaxoyzqawjqeSat, 23 Jul 2022 00:00:00 GMTThe Characterization of Substructures of γ -Anti Fuzzy Subgroups with Application in Genetics
https://scholar.archive.org/work/qxufguy3rndzje5dmedyws5tlm
Fuzzy and anti fuzzy normal subgroups are the current instrument for dealing with ambiguity in various decision-making challenges. This article discusses γ -anti fuzzy normal subgroups and γ -fuzzy normal subgroups. Set-theoretic properties of union and intersection are examined and it is observed that union and intersection of γ -anti fuzzy normal subgroups are γ -anti fuzzy normal subgroups. Employee selection impacts the input quality of employees and hence plays an important part in human resource management. The cost of a group is established in proportion to the fuzzy multisets of a fuzzy multigroup. It was a good idea to introduce anti-intuitionistic fuzzy sets and anti-intuitionistic fuzzy subgroups, as well as to demonstrate some of their algebraic features. Product of γ -anti fuzzy normal subgroups and γ -fuzzy normal subgroups is defined, the product's algebraic nature is analyzed, and the findings are supported by presenting γ -anti typical sections with blurring and γ -ordinary parts with the weirdness of well-defined and well-established groups of genetic codes.Kalaichelvan Kalaiarasi, P. Sudha, Nasreen Kausar, Sajida Kousar, Dragan Pamucar, Nasr Al Din Ide, Lele Qinwork_qxufguy3rndzje5dmedyws5tlmSat, 16 Jul 2022 00:00:00 GMTOn the connection between Hopf–Galois structures and skew braces
https://scholar.archive.org/work/y3i7dpw3wrbm7oljuudg2pqqya
We present a different point of view on the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results which involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.L. Stefanello, S. Trappenierswork_y3i7dpw3wrbm7oljuudg2pqqyaThu, 07 Jul 2022 00:00:00 GMTOn algebraically coisotropic submanifolds of holomorphic symplectic manifolds
https://scholar.archive.org/work/gjdrwlttenhlxbztpjvd7es4xe
We investigate algebraically coisotropic submanifolds X in a holomorphic symplectic projective manifold M. Motivated by our results in the hypersurface case, we raise the following question: when X is not uniruled, is it true that up to a finite étale cover, the pair (X,M) is a product (Z× Y, N× Y) where N, Y are holomorphic symplectic and Z⊂ N is Lagrangian? We prove that this is indeed the case when M is an abelian variety and give some partial answer when the canonical bundle K_X is semi-ample. In particular, when K_X is nef and big, X is Lagrangian in M. We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when M is irreducible hyperkähler.Ekaterina Amerik, Frédéric Campanawork_gjdrwlttenhlxbztpjvd7es4xeMon, 04 Jul 2022 00:00:00 GMTQuantum Gravity in 30 Questions
https://scholar.archive.org/work/tidlab4o7nhlddttjl4b3ublja
Quantum gravity is the missing piece in our understanding of the fundamental interactions today. Given recent observational breakthroughs in gravity, providing a quantum theory for what lies beyond general relativity is more urgent than ever. However, the complex history of quantum gravity and the multitude of available approaches can make it difficult to get a grasp of the topic and its main challenges and opportunities. We provide a guided tour of quantum gravity in the form of 30 questions, aimed at a mixed audience of learners and practitioners. The issues covered range from basic motivational and background material to a critical assessment of the status quo and future of the subject. The emphasis is on structural issues and our current understanding of quantum gravity as a quantum field theory of dynamical geometry beyond perturbation theory. We highlight the identification of quantum observables and the development of effective numerical tools as critical to future progress.R. Loll, G. Fabiano, D. Frattulillo, F. Wagnerwork_tidlab4o7nhlddttjl4b3ubljaTue, 14 Jun 2022 00:00:00 GMTExact Lagrangians from contracting ℂ^*-actions
https://scholar.archive.org/work/japxtedqobbvne5f7ltm5y5m2y
We obtain families of non-isotopic closed exact Lagrangian submanifolds in quasi-projective holomorphic symplectic manifolds that admit contracting ℂ^*-actions. We show that the Floer cohomologies of these Lagrangians are topological in nature, recovering the ordinary cohomologies of their intersection. Moreover, by using these Lagrangians and a version of Carrell-Goresky's integral decomposition theorem, we obtain degree-wise lower bounds on the symplectic cohomology of these spaces.Filip Živanovićwork_japxtedqobbvne5f7ltm5y5m2yMon, 13 Jun 2022 00:00:00 GMTTowards Hodge Theoretic Characterizations of 2d Rational SCFTs
https://scholar.archive.org/work/bjbk7ktphfhalovrq5k42syq6y
The study of rational conformal field theories in the moduli space is of particular interest since these theories correspond to points in moduli space where the algebraic and arithmetic structure are usually richer, while also being points where non–trivial physics occurs (such as in the study of attractor black holes and BPS states at rational points). This has led to various attempts to characterize and classify such rational points. In this paper, a conjectured characterization by Gukov–Vafa of rational conformal field theories whose target space is a Ricci flat Kähler manifold is analyzed carefully for the case of toroidal compactifications. We refine the conjectured statement as well as making an effort to verify it, using T^4 compactification as a test case. Seven common properties in terms of Hodge theory (including complex multiplication) have been identified for T^4-target rational conformal field theories. By imposing three properties out of the seven, however, there remain 𝒩 = (1,1) SCFTs that are not rational. Open questions, implications and future lines of work are discussed.Abhiram Kidambi, Masaki Okada, Taizan Watariwork_bjbk7ktphfhalovrq5k42syq6yFri, 10 Jun 2022 00:00:00 GMTCP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond
https://scholar.archive.org/work/45ksn43gh5evtkkemu4pehmime
These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While exisiting work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?Orr Shalit, Michael Skeidework_45ksn43gh5evtkkemu4pehmimeFri, 10 Jun 2022 00:00:00 GMTSuperposed Random Spin Tensor Networks and their Holographic Properties
https://scholar.archive.org/work/6vgaz3qhqjenhaooatvhdzf7xa
We study criteria for and properties of boundary-to-boundary holography in a class of spin network states defined by analogy to projected entangled pair states (PEPS). In particular, we consider superpositions of states corresponding to well-defined, discrete geometries on a graph. By applying random tensor averaging techniques, we map entropy calculations to a random Ising model on the same graph, with distribution of couplings determined by the relative sizes of the involved geometries. The superposition of tensor network states with variable bond dimension used here presents a picture of a genuine quantum sum over geometric backgrounds. We find that, whenever each individual geometry produces an isometric mapping of a fixed boundary region C to its complement, then their superposition does so iff the relative weight going into each geometry is inversely proportional to its size. Additionally, we calculate average and variance of the area of the given boundary region and find that the average is bounded from below and above by the mean and sum of the individual areas, respectively. Finally, we give an outlook on possible extensions to our program and highlight conceptual limitations to implementing these.Simon Langenscheidtwork_6vgaz3qhqjenhaooatvhdzf7xaThu, 19 May 2022 00:00:00 GMTProceedings of the 2022 Joint Workshop of the German Research Training Groups in Computer Science
https://scholar.archive.org/work/lvykkw5kcfhlvolc6paa2sxczu
Having spent two successive years running online to prevent the spread of the Corona virus, the traditional annual meeting of the German Research Training Groups (RTGs) funded by the Deutsche Forschungsgemeinschaft (DFG) in the field of computer science returns to Schloss Dagstuhl --– Leibniz Center for Informatics, one of the world's premier venues for computer science-related seminars. Returning to Dagstuhl and hosting this meeting as an in-person-only event was a deliberate decision to revive interaction modes that many of the funded researchers had yet to experience: fostering personal interchange of ideas and experiences in order to strengthen the connection within the German computer science community. This volume documents the abstracts of the research topics of funded researchers in the participating RTGs. The event was jointly organized by RTG 2475 (Cybercrime and Forensic Computing) and RTG 2428 (ConVeY --- Continuous Verification of Cyber-Physical Systems). It took place between Sunday, June 12 and Wednesday, June 15, 2022, as in-person only Dagstuhl Event 22243. The meeting featured the usual sequence of research presentations by funded researchers, networking meetings for PIs and RTG coordinators, as well as two invited talks, one by Professor Martina Seidl (JKU Linz, Austria) on "Competitions as Scientific Method" and another by Professor Jennifer Byrne (School of Medical Sciences, The University of Sydney, Australia) titled "An introduction to research paper mills". Because last year's event marked the 25th anniversary of the workshop series, it featured a live interview with Professor Otto Spaniol who had initiated the workshop series in 1996. We document the interview in this volume.Felix Freiling, Helmut Seidl, 2022 2022 Joint Workshop Of The German Research Training Groups In Computer Science June 12–June 15work_lvykkw5kcfhlvolc6paa2sxczuTue, 17 May 2022 00:00:00 GMTFourfolds of Weil type and the spinor map
https://scholar.archive.org/work/motoly5h2zc5fkprhtfzf465fe
Recent papers by Markman and O'Grady give, besides their main results on the Hodge conjecture and on hyperkaehler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six. In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.Bert van Geemenwork_motoly5h2zc5fkprhtfzf465feTue, 03 May 2022 00:00:00 GMTUse and abuse of instance parameters in the Lean mathematical library
https://scholar.archive.org/work/osj33n3sgrhbznyjrcoecnn33e
The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean's mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.Anne Baanenwork_osj33n3sgrhbznyjrcoecnn33eMon, 02 May 2022 00:00:00 GMTOn a class of semigroup products
https://scholar.archive.org/work/gjmdz7vegzb73gvxurcxvshpby
This paper concerns a class of semigroups that arise as products US, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U^1 that is suitably compatible with the product in the over-monoid. The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest. The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup US is a quotient of a semidirect product U⋊ S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups. The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for US out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.Scott Carson, Igor Dolinka, James East, Victoria Gould, Rida-e Zenabwork_gjmdz7vegzb73gvxurcxvshpbyFri, 29 Apr 2022 00:00:00 GMTTopological classification of symmetric quantum walks. Discrete symmetry types and chiral symmetric protocols
https://scholar.archive.org/work/u2io4zc7w5fdjpa2tcxzuzrtdq
In this thesis, we study the topological classification of symmetric quantum walks. These describe the discrete time evolution of single quantum particles on the lattice with additional locally acting symmetries. The thesis consists of three parts: In the first part, we discuss discrete symmetry types for self-adjoint and unitary operators from an abstract point of view, i.e. without assuming an underlying physical model. We reduce any abstract finite group of involutive symmetries and their projective representations to a smaller set of symmetry types, eliminating elements that are redundant for topological classifications. This reduction process leads to the well-known tenfold way for self-adjoint operators, and for unitary operators, we identify 38 non-redundant symmetry types. For these, we define a symmetry index, which labels equivalence classes of finite-dimensional representations up to trivial direct summands. We show that these equivalence classes naturally carry a group structure and finish the discussion by explicitly computing the corresponding index groups for all non-trivial symmetry types. Second, we develop a topological classification for symmetric quantum walks based on the symmetry index derived in the first part. We begin without a locality condition on the unitary time evolution operator but only assume an underlying discrete spatial structure. Unlike continuous-time systems, quantum walks exhibit non-gentle perturbations, i.e. local or compact perturbations that cannot be undone continuously. Using the symmetry index, we provide a complete topological classification of such perturbations of unitary operators on any lattice or graph. We add a locality condition on the one-dimensional lattice and detail the implications of such assumption on the classification. The spatial structure of the one-dimensional lattice allows us to define the left- and right symmetry index, which characterise a walks topological properties on the two half-chains. The sum of these two indices equals the overall symm [...]Tobias Geib, Technische Informationsbibliothek (TIB)work_u2io4zc7w5fdjpa2tcxzuzrtdqTue, 12 Apr 2022 00:00:00 GMTOligomorphic groups and tensor categories
https://scholar.archive.org/work/wttca64n2nfqji4uvb5zv4ihkq
Given an oligomorphic group G and a measure μ for G (in a sense that we introduce), we define a rigid tensor category Perm(G; μ) of "permutation modules," and, in certain cases, an abelian envelope Rep(G; μ) of this category. When G is the infinite symmetric group, this recovers Deligne's interpolation category, while other choices for G lead to fundamentally new tensor categories. In particular, we give the first example (so far as we know) of a 𝐂-linear rigid tensor category that bears no obvious relation to finite dimensional representations. One interesting aspect of our construction is that, unlike much previous work in this direction, our categories are concrete: the objects are modules over a ring, and the tensor product receives a universal bi-linear map. Central to our constructions is a novel theory of integration on oligomorphic groups, which could be of more general interest. Classifying the measures on an oligomorphic group appears to be a difficult problem, which we solve in only a few cases.Nate Harman, Andrew Snowdenwork_wttca64n2nfqji4uvb5zv4ihkqSat, 09 Apr 2022 00:00:00 GMTSuccessive Spectral Sequences
https://scholar.archive.org/work/7lqnmp6lfbdvphehhjtjlikbu4
In this paper, we develop a structure theory for generalized spectral sequences, which are derived from chain complexes that are filtered over arbitrary partially ordered sets. Also, a more general construction method reminiscent of exact couples is studied, together with examples where they arise naturally. As for ordinary spectral sequences we will see differentials and group extensions, however the real power comes from the appearance of natural isomorphims between pages of differing indices. The constructions reveal finer invariants than ordinary spectral sequences, and they connect to other fields such as Fary functors and perverse sheaves. They are based on a natural index scheme, which allows us to obtain new results even in the standard case of Z-filtered chain complexes, e.g. a useful criterion for a product structure for Grothendieck's spectral sequences, and news paths to connect the first or second page to the limit. This turns out to yield the right framework for unifying several spectral sequences that one would usually apply one after another. Examples that we work out are successive Leray--Serre spectral sequences, the Adams--Novikov spectral sequence following the chromatic spectral sequence, successive Grothendieck spectral sequences, and successive Eilenberg--Moore spectral sequences.Benjamin Matschkework_7lqnmp6lfbdvphehhjtjlikbu4Fri, 18 Mar 2022 00:00:00 GMTThe classical limit and spontaneous symmetry breaking in algebraic quantum theory
https://scholar.archive.org/work/6zrlewc365fxlglrcik466cdou
In this paper an overview of some recent developments on the classical limit and spontaneous symmetry breaking (SSB) in algebraic quantum theory is given. In such works, based on the theory of C^*-algebras, the concept of the classical limit has been formalized in a complete algebraic manner. Additionally, since this setting allows for commutative as well as non-commutative C^*-algebras, and hence for classical and quantum theories, it provides an excellent framework to study SBB as an emergent phenomenon when transitioning from the quantum to the classical world by turning off a semi-classical parameter. We summarize the main results and show that this algebraic approach sheds new light on the connection between the classical and the quantum realm, where particular emphasis is placed on the role of SSB in Theory versus Nature. To this end a detailed analysis is carried out and illustrated with three different physical models: Schrödinger operators, mean-field quantum spin systems and the Bose-Hubbard model.Christiaan J.F. van de Venwork_6zrlewc365fxlglrcik466cdouSun, 06 Mar 2022 00:00:00 GMTBeyond the Lascar Group
https://scholar.archive.org/work/7x6jrrwsljboti77hzs52j6tvq
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical core 𝒥 associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of 𝒥, modulo certain infinitesimal automorphisms, is a locally compact group 𝒢. The automorphism groups of models of the theory are related with 𝒢, not in general via a homomorphism, but by a quasi-homomorphism, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of SL_n(ℝ) or SL_n(ℚ_p).Ehud Hrushovskiwork_7x6jrrwsljboti77hzs52j6tvqMon, 21 Feb 2022 00:00:00 GMTSome Fundamental Theorems in Mathematics
https://scholar.archive.org/work/6lqit72adje3zlo54s5zpgviem
An expository hitchhikers guide to some theorems in mathematics.Oliver Knillwork_6lqit72adje3zlo54s5zpgviemFri, 04 Feb 2022 00:00:00 GMT