Quotient Construction Research Articles

Isolated Suborders and their Application to Counting Closure Operators

In this paper we investigate the interplay between isolated suborders and closures. Isolated suborders are a special kind of suborders and can be used to diminish the number of elements of an ordered set by means of a quotient construction. The decisive point is that there are simple formulae establishing relationships between the number of closures in the original ordered set and the quotient thereof induced by isolated suborders. We show how these connections can be used to derive a recursive algorithm for counting closures, provided the ordered set under consideration contains suitable isolated suborders.

A hyper-Kähler metric on the moduli spaces of monopoles with arbitrary symmetry breaking

We construct the hyper-Kähler moduli space of framed monopoles over R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence arbitrary symmetry breaking. In order to do so, we define a configuration space of pairs with appropriate asymptotic conditions and perform an infinite-dimensional quotient construction. We make use of the b and scattering calculuses to study the relevant differential operators.

Moduli spaces of instantons in flag manifold sigma models. Vortices in quiver gauge theories

In this paper, we discuss lumps (sigma model instantons) in flag manifold sigma models. In particular, we focus on the moduli space of BPS lumps in general Kähler flag manifold sigma models. Such a Kähler flag manifold, which takes the form Un1+⋯+nL+1Un1×⋯×UnL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\ extrm{U}\\left({n}_1+\\cdots +{n}_{L+1}\\right)}{\ extrm{U}\\left({n}_1\\right)\ imes \\cdots \ imes \ extrm{U}\\left({n}_{L+1}\\right)} $$\\end{document}, can be realized as a vacuum moduli space of a U(N1) × ··· × U(NL) quiver gauged linear sigma model. When the gauge coupling constants are finite, the gauged linear sigma model admits BPS vortex configurations, which reduce to BPS lumps in the low energy effective sigma model in the large gauge coupling limit. We derive an ADHM-like quotient construction of the moduli space of BPS vortices and lumps by generalizing the quotient construction in U(N) gauge theories by Hanany and Tong. As an application, we check the dualities of the 2d models by computing the vortex partition functions using the quotient construction.

Characteristic, C-Characteristic and Positive Cones in Hyperfields

We study the notions of the positive cone, characteristic and C-characteristic in (Krasner) hyperfields. We demonstrate how these interact in order to produce interesting results in the theory of hyperfields. For instance, we provide a criterion for deciding whether certain hyperfields cannot be obtained via Krasner’s quotient construction. We prove that any positive integer (larger than 1) can be realized as the characteristic of some infinite hyperfield and an analogous result for the C-characteristic. Finally, we study the (directed) graph associated with the strict partial order induced by a positive cone in a hyperfield in various examples.

Whole-grain Petri Nets and Processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P -processes is shown to be the free coloured prop-in-groupoids on P . There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best–Devillers, involving taking π 0 of the groupoids of states.)

Quantitative controller synthesis for consumption Markov decision processes

Consumption Markov decision processes (CMDPs) are a kind of resource-constrained probabilistic decision-making systems allowing the consumption and replenishment to capacity of resource. Making each decision in CMDPs consumes some amount of resource, which can be reloaded to the full capacity in a specified set of states. We consider here the quantitative reachability probability controller synthesis problems for CMDPs that extract a policy preventing resource exhaustion and achieve (repeated) reachability goals with the maximal probability, in contrast to the original work focusing on the qualitative controller synthesis for CMDPs, e.g., fulfilling the (repeated) reachability goals with positive probability or probability one. We first prove that the exact quantitative reachability probability problems are NP-hard by a reduction from the 0–1 Knapsack problem. We solve them by converting a given CMDP to a flattened Markov decision process (MDP) and then constructing an optimal policy for that MDP. Unfortunately, the naive flattened MDPs are prohibitively large and thus make the subsequent synthesis not scale well. To overcome this drawback, we propose a pruned construction and a quotient construction of the flattened MDP to reduce the sizes of MDPs. The empirical evaluation shows that our optimizations can significantly improve the scalability of the purely flattening-based synthesis method for CMDPs on benchmarks from real-world scenarios — autonomous electric vehicle routing and helicopter planning in Mars Perseverance Rover project.

Finite Generation of Cox Rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

Partial actions on convergence spaces

Abstract Continuous partial actions and continuous enveloping actions are investigated in the category of convergence spaces. Product and quotient constructions are considered. Further, it is shown that a continuous partial action on a convergence space can be extended to a continuous partial action on a compactification of the convergence space.

A note on the Severi problem for toric surfaces

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

Integrability of quotients in Poisson and Dirac geometry

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids integrating two interesting families of geometric structures: (i) a special class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson groups and (ii) Dirac homogeneous spaces.

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

A quotient of the Lubin\u2013Tate tower II

In this article we construct the quotient mathcal {M}_mathbf {1}/P(K) of the infinite-level Lubin–Tate space mathcal {M}_mathbf {1} by the parabolic subgroup P(K) subset mathrm {GL} _n(K) of block form (n-1,1) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and K/{mathbb {Q}} _p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of mathcal {M}_mathbf {1}/P(K) when n=2, and shows that mathcal {M}_mathbf {1}/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.

Quotient theorems in polyfold theory and S1‐equivariant transversality

We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in [42]. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of $S^1$-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbations. We outline the application to Hamiltonian-Floer cohomology and a proof of the weak Arnold conjecture for general symplectic manifolds, assuming the existence of Hamiltonian-Floer cohomology polyfolds.

Moment maps, strict linear precision, and maximum likelihood degree one

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

Fraction, restriction, and range categories from stable systems of morphisms

For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s,f) in C with first “leg” s lying in S, and give an alternative construction of its quotient category C[S−1] of S-fractions. Instead of trying to turn S-morphisms “directly” into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr(C,S) and Sect(C,S). The fraction category C[S−1] is their largest joint quotient category.Without confining S to be a class of monomorphisms of C, we show that Sect(C,S) admits a quotient category, Par(C,S), whose name is justified by two facts. On one hand, for S a class of monomorphisms in C, it returns the category of S-spans in C, also called S-partial maps in this case; on the other hand, we prove that Par(C,S) is a split restriction category (in the sense of Cockett and Lack). A further quotient construction produces even a range category (in the sense of Cockett, Guo and Hofstra), RaPar(C,S), which is still large enough to admit C[S−1] as its quotient.Both, Par and RaPar, are the left adjoints of global 2-adjunctions. When restricting these to their “fixed objects”, one obtains precisely the 2-equivalences by which their name givers characterized restriction and range categories. Hence, both Par(C,S) and RaPar(C,S) may be naturally presented as Par(D,T) and RaPar(D,T), respectively, where now T is a class of monomorphisms in D. In summary, while there is no a priori need for the exclusive consideration of classes of monomorphisms, one may resort to them naturally.

Cluster Structures and Subfans in Scattering Diagrams

We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing ${\rm SL}_{2}/{\rm PGL}_{2}$-local systems on the once punctured torus. Then we prove these statements. Along the way, using distinct subfans in the scattering diagram, we produce an example of a cluster variety with two non-equivalent cluster structures. To overcome the technical difficulty of infinite non-cluster wall-crossing in the scattering diagram, we introduce quiver folding into the machinery of scattering diagrams and give a quotient construction of scattering diagrams.

Monoids, their boundaries, fractals and C*-algebras

AbstractIn this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C*-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives rise to examples of C*-algebras (1.9) generalizing the boundary quotients \partial C_\lambda ^*(\mathcal{M}) discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients, while the cone topology can be used to define canonical measures on the attractor of an ℳ-fractal for a finitely 1-generated monoid ℳ.

Opacity of Nondeterministic Transition Systems: A (Bi)Simulation Relation Approach

In this paper, we propose several opacity-preserving (bi)simulation relations for general nondeterministic transition systems (NTS) in terms of initial-state opacity, current-state opacity, K-step opacity, and infinite-step opacity. We also show how one can leverage quotient construction to compute such relations. In addition, we use a two-way observer method to verify opacity of nondeterministic finite transition systems (NFTSs). As a result, although the verification of opacity for infinite NTSs is generally undecidable, if one can find such an opacity-preserving relation from an infinite NTS to an NFTS, the (lack of) opacity of the NTS can be easily verified over the NFTS which is decidable.

A Graphical Calculus for Semi-Groupal Categories

Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street's graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street's construction of a free monoidal category on a tensor scheme.

Modular Interpretation of a Non-Reductive Chow Quotient

AbstractThe space of n distinct points and adisjoint parametrized hyperplane in projective d-space up to projectivity – equivalently, configurations of n distinct points in affine d-space up to translation and homothety – has a beautiful compactification introduced by Chen, Gibney and Krashen. This variety, constructed inductively using the apparatus of Fulton–MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline M _{0,n}$ and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of (ℙd)n by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen–Gibney–Krashen space Td, n. This is a non-reductive analogue of Kapranov's famous quotient construction of $\overline M _{0,n}$, and indeed as a special case we show that $\overline M _{0,n}$ is the Chow quotient of (ℙ1)n−1 by an action of 𝔾m ⋊ 𝔾a.