Partial Isomorphisms Research Articles

On the Endomorphism Monoid of Certain Ultrametric Speces

In this work we investigate the endomorphism monoid of certain ultrametric spaces. According to our main result, if X =<X,e> is an ultrametric space such that the range of e is finite, then the set of locally finite endomorphisms is dense in the endomorphism monoid of X and the endomorphism monoid of X has a dense, locally finite submonoid. This can be regarded as a homomorphism oriented counterpart of some ecently obtained results about the existence of dense, locally finite subgroups of the automorphism group of certain homogeneous structures. Further, as a byproduct, we obtain Hrushovski style extension theorems for the ages of certain ultrametric spaces, but here, instead of partial isomorphisms we extend partial homomorphisms.

A sharper definition of alignment for Panoptic Quality

The Panoptic Quality metric, developed by Kirillov et al. in 2019, makes object-level precision, recall and F1 measures available for evaluating image segmentation, and more generally any partitioning task, against a gold standard. Panoptic Quality is based on partial isomorphisms between hypothesized and true segmentations. Kirillov et al. desire that functions defining these one-to-one matchings should be simple, interpretable and effectively computable. They show that for t and h, true and hypothesized segments, the condition stating that there are more correct than wrongly predicted pixels, formalized as IoU(t,h)>.5 or equivalently as |t∩h|>.5|t∪h| has these properties. We show that a weaker function, requiring that more than half of the pixels in the hypothesized segment are in the true segment and vice-versa, formalized as |t∩h|>.5|t| and |t∩h|>.5|h|, is not only sufficient but also necessary. With a small proviso, every function defining a partial isomorphism satisfies this condition. We theoretically and empirically compare the two conditions.

On dense, locally finite subgroups of the automorphism group of certain homogeneous structures

AbstractLet be a countable structure such that each finite partial isomorphism of it can be extended to an automorphism. Evans asked if the age (set of finite substructures) of satisfies Hrushovski's extension property, then is it true that the automorphism group of contains a dense, locally finite subgroup? In order to investigate this question, in the previous decades a coherent variant of Hrushovski's extension property has been introduced and studied. Among other results, we provide equivalent conditions for the existence of a dense, locally finite subgroup of in terms of a (new) variant of the coherent extension property. We also compare our notion with other coherent extension properties.

Inapproximability of Unique Games in Fixed-Point Logic with Counting

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

On the Polishness of the inverse semigroup Γ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (X)$$\\end{document} on a compact metric space x

Let Γ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (X)$$\\end{document} be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space X. There is a topology, denoted τhco\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _\ extrm{hco}$$\\end{document}, that makes Γ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (X)$$\\end{document} a topological inverse semigroup. We address the question of whether τhco\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _\ extrm{hco}$$\\end{document} is Polish. For a 0-dimensional compact metric space X, we prove that (Γ(X),τhco)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Gamma (X), \ au _\ extrm{hco})$$\\end{document} is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup I(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I({\\mathbb N})$$\\end{document}. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open sets.

Wasserstein Distances, Geodesics and Barycenters of Merge Trees.

This paper presents a unified computational framework for the estimation of distances, geodesics and barycenters of merge trees. We extend recent work on the edit distance [104] and introduce a new metric, called the Wasserstein distance between merge trees, which is purposely designed to enable efficient computations of geodesics and barycenters. Specifically, our new distance is strictly equivalent to the $L$2-Wasserstein distance between extremum persistence diagrams, but it is restricted to a smaller solution space, namely, the space of rooted partial isomorphisms between branch decomposition trees. This enables a simple extension of existing optimization frameworks [110] for geodesics and barycenters from persistence diagrams to merge trees. We introduce a task-based algorithm which can be generically applied to distance, geodesic, barycenter or cluster computation. The task-based nature of our approach enables further accelerations with shared-memory parallelism. Extensive experiments on public ensembles and SciVis contest benchmarks demonstrate the efficiency of our approach - with barycenter computations in the orders of minutes for the largest examples - as well as its qualitative ability to generate representative barycenter merge trees, visually summarizing the features of interest found in the ensemble. We show the utility of our contributions with dedicated visualization applications: feature tracking, temporal reduction and ensemble clustering. We provide a lightweight C++ implementation that can be used to reproduce our results.

Finite Model Property and Bisimulation for LFD

Recently, Baltag and van Benthem introduced a decidable logic of functional dependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras (CRS) with atomic local dependence statements. Its semantics can be given in terms of generalised assignment models or their modal counterparts, hence the logic is both a first-order and a modal logic. We show that LFD has the finite model property (FMP) using Herwig's theorem on extending partial isomorphisms, and prove a bisimulation invariance theorem characterizing LFD as a fragment of first-order logic.

The Notions of Center, Commutator and Inner Isomorphism for Groupoids

In this paper we introduce some algebraic properties of subgroupoids and normal subgroupoids. we define other things, we define the normalizer of a wide subgroupoid H of a groupoid G and show that, as in the case of groups, this normalizer is the greatest wide subgroupoid of G in which H is normal. Furthermore, we provide definitions of the center Z(G) and the commutator G' of the groupoid G and prove that both of them are normal subgroupoids. We give the notions of inner and partial isomorphism of G and show that the groupoid I(G) given by the set of all the inner isomorphisms of G is a normal subgroupoid of A(G), the set of all the partial isomorphisms of G. Moreover, we prove that I(G) is isomorphic to the quotient groupoid G/Z(G), which extends to groupoids the corresponding well-known result for groups.

Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories

In flowchart languages, predicates play an interesting double role. In the textual representation, they are often presented as conditions, i.e., expressions which are easily combined with other conditions (often via Boolean combinators) to form new conditions, though they only play a supporting role in aiding branching statements choose a branch to follow. On the other hand, in the graphical representation they are typically presented as decisions, intrinsically capable of directing control flow yet mostly oblivious to Boolean combination.While categorical treatments of flowchart languages are abundant, none of them provide a treatment of this dual nature of predicates. In the present paper, we argue that extensive restriction categories are precisely categories that capture such a condition/decision duality, by means of morphisms which, coincidentally, are also called decisions. Further, we show that having these categorical decisions amounts to having an internal logic: Analogous to how subobjects of an object in a topos form a Heyting algebra, we show that decisions on an object in an extensive restriction category form a De Morgan quasilattice, the algebraic structure associated with the (three-valued) weak Kleene logicK3w. Full classical propositional logic can be recovered by restricting to total decisions, yielding extensive categories in the usual sense, and confirming (from a different direction) a result from effectus theory that predicates on objects in extensive categories form Boolean algebras.As an application, since (categorical) decisions are partial isomorphisms, this approach provides naturally reversible models of classical propositional logic and weak Kleene logic.

Colored Khovanov–Rozansky homology for infinite braids

We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp.

A New Method for Measuring Topological Structure Similarity between Complex Trajectories

We proposed a new framework to measure the similarity of topological structure between complex trajectories. There are three steps in the framework. A complex trajectory is first represented by a graph structure which consists of nodes and edges. Secondly, we developed a Comprehensive Structure Matching (CSM) algorithm to identify all common structures between the complex trajectories of interest. Thirdly, we used the Jaccard similarity coefficient to evaluate the similarity between two complex trajectories. We used synthetic graph data to evaluate the CSM method and examine its performance by comparing against that of the VF2 and the exact graph edit distance (EGED) algorithms. Results show that the CSM algorithm outperforms the EGED in terms of the computation efficiency. The CSM is more comprehensive than the VF2 algorithm as it further considers the partial isomorphism. We used the CSM algorithm to examine the 1993 to 2012 complex trajectories of anticyclonic eddies in the South China Sea (SCS). The CSM successfully found the complex trajectories that are similar to a thoroughly-studied ACE3 trajectory in the SCS. From the similar trajectories, we identified a dominant migrating path of the ocean eddies in the northern SCS. The CSM also successfully identified some new complex trajectories that propagated across the 18°N parallel in the SCS, which were not reported before. It also further identified multiple common structure models of the complex trajectories. These findings help us better understand the behaviors and the evolution of the mesoscale eddies in the SCS.

The Hrushovski property for hypertournaments and profinite topologies

We study the problem of extending partial isomorphisms for hypertournaments, which are relational structures generalizing tournaments. This is a generalized version of an old question of Herwig and Lascar. We show that the generalized problem has a negative answer, and we provide a positive answer in a special case. As a corollary, we show that the extension property holds for tournaments in case the partial isomorphisms have pairwise disjoint ranges and pairwise disjoint domains.

The Category TOF

We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds -- and follows the proof of Cockett et al.-- which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in which all of the identities for the cnot gate hold; and then a normal form for the restriction idempotents is constructed which corresponds precisely to subobjects of the total points of TOF. This is then used to show that TOF is equivalent to FPinj2, the full subcategory of sets and partial isomorphisms in which objects have cardinality a power of 2.

The Category CNOT

We exhibit a complete set of identities for CNOT, the symmetric monoidal category generated by the controlled-not gate, the swap gate, and the computational ancillae. We prove that CNOT is a discrete inverse category. Moreover, we prove that CNOT is equivalent to the category of partial isomorphisms of finitely-generated non-empty commutative torsors of characteristic 2. Equivalently this is the category of affine partial isomorphisms between finite-dimensional Z2 vector spaces.

The classification of infinite abelian groups with partial decomposition bases in $L_\infty \omega $

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language $L_{\infty \omega }$, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in $L_{\infty \omega }$.

A generalisation to Birkhoff–von Neumann theorem

The classic Birkhoff–von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this paper, we study a generalisation of this theorem in the type II1 setting. Namely, we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms, of the unit interval, with similar properties. We show that a weaker version of this theorem still holds.

Representación y comprensión de la fotografia: uma epistemologia Del mundo de Cartier-Bresson

http://dx.doi.org/10.5007/1808-1711.2016v20n2p215 The article aims to conceptualize both representation and understanding in photography, an activity whose main goal consist in elucidating the process through which the photographic image is constructed on a partial isomorphism relationship, as well as in enabling to understand a meaningful message. We appeal to Nelson Goodman’s account, according to which such a construction is based on data provided by the image, on the one hand, and by viewer’s knowledge, on the other. Given that those sources give viewer a new knowledge about the world and that the inferential processes depend upon a general theory of symbols, we both show and account for the inferential procedure that raises from the photographic ‘information’ in several case-studies taken from Henry Cartier-Bresson’s work.

To imitate or differentiate: Cross-level identity work in an innovation network

Survival in global high-tech industries requires many organizations to participate in specialized innovation networks. However, sustained participation in these networks often proves more challenging than expected for organizations and their representatives, due to complex cross-level identity tensions that are indiscernible when only one level of analysis is considered. The purpose of this study is to analyze cross-level identity tensions at the interface of personal and organizational identities in an innovation network. We identify three key cross-level identity tensions related to intellectual property, communication and market definition, which together contribute to an overall organizational-personal identity tension opposing differentiation and imitation. These tensions are indicative of a complex process of “partial isomorphism” in identity work, which can facilitate collaboration while simultaneously fostering innovation among personal and organizational network members.

Privacy, CCTV, and School Surveillance in the Shadow of Imagined Law

How do managers make decisions that affect human rights of other people? The article examines one such case: the decision-making process of Israeli school principals in installing Closed Circuit TV (CCTV) systems in schools. One unexpected source that principals relied on was that of imagined law: they wrongly assumed that there was a law that guided them in the matter. The deployment of CCTV systems in Israeli schools is relatively new and takes place at an accelerated pace. School principals are the ones that make the decision about introducing such systems into their schools. The study traced and explored this process. Based on semi-structured interviews, the findings portray a picture of partial isomorphism among schools. We frame the findings within institutional theory, which differentiates between exogenous and endogenous sources of decision-making. Most school principals relied on endogenous sources that were shaped by practical considerations and their own perceptions as to security, privacy, and education. Yet, the interviews indicate an additional and surprising source of organizational decision-making: imagined law. Some of the principals assumed the existence of specific legal rules. The principals did not search for professional guidance, and did not consult others. Instead, they filled the imagined law with endogenous sources, namely, their own perceptions.

The affine structure of gravitational theories: Symplectic groups and geometry

We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due to the partial isomorphism between CA(3, 1) and the centrally extended Sp( 8), we perform a nonlinear realization of the centrally extended ( CE )Sp( 8) in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie algebra and then, by the nonlinear realization, we define the coset field transformations, the Cartan forms and the inverse Higgs constraints. Finally, we discuss the geometrical Lagrangians where all the information on matter fields and their interactions can be contained.