Notion Of Isomorphism Research Articles

The Hole Argument without the notion of isomorphism

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the analaysis of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure.

Geometries of topological groups

The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.

A Theoretical Investigation Based on the Rough Approximations of Hypergraphs

Rough sets are a key tool to model uncertainty and vagueness using upper and lower approximations without predefined functions and additional suppositions. Rough graphs cannot be studied more effectively when the inexact and approximate relations among more than two objects are to be discussed. In this research paper, the notion of a rough set is applied to hypergraphs to introduce the novel concept of rough hypergraphs based on rough relations. The notions of isomorphism, conformality, linearity, duality, associativity, commutativity, distributivity, Helly property, and intersecting families are illustrated in rough hypergraphs. The formulae of 2‐section, L2‐section, covering, coloring, rank, and antirank are established for certain types of rough hypergraphs. The relations among certain types of products of rough hypergraphs are studied in detail.

Weak containment of measure-preserving group actions

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Invariant Vector Fields and Groupoids

We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage from the space of invariant vector fields in a tube around a group orbit to the space invariant vector fields on a slice to the orbit. The notion comes from Hepworth's study of vector fields on stacks.

Construction and Comparison of Authentication Codes

One of the main problems we want to address is the fact that in the construction theory of authentication codes, the expression “new code” is used all the time, while it is almost never shown that the obtained codes indeed are new, or even what the notion “new” means to begin with. In this paper, we formally define authentication codes and introduce a comparison method for them, using the notion of isomorphism and the invariant isomorphism group. We then define operations in this framework which enable us to “multiply” and “add” arbitrary authentication codes, regardless of how they are initially constructed. We obtain (nonisomorphic) codes with easily calculated parameters, a large number of which are new.

Improved Enumeration of Simple Topological Graphs

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.

Exact fluid lumpability in Markovian process algebra

Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of agents of the system under scrutiny. When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one. For the analysis of massively parallel systems, however, lumping techniques may not be sufficient to yield a computationally tractable problem. Recently, much work has been directed towards forms of fluid techniques that provide a set of ordinary differential equations (ODEs) approximating the expected path of the stochastic process. Unfortunately, even fluid models of realistic systems may be too large for feasible analysis. This paper studies a behavioural relation for process algebra with fluid semantics, called projected label equivalence, which is shown to yield an exactly fluid lumpable model, i.e., an aggregated ODE system which can be related to the original one without any loss of information. Project label equivalence relates sequential components of a process term. In general, for any two sequential components that are related in the fluid sense, nothing can be said about their relationship from the stochastic viewpoint. We define and study a notion of well-posedness which allows us to relate fluid lumpability to the stochastic notion of semi-isomorphism, which is a weaker version of the common notion of isomorphism between the doubly labelled transition systems at the basis of the Markovian interpretation.

On isomorphisms of intersection types

The study of type isomorphisms for different λ-calculi started over twenty years ago, and a very wide body of knowledge has been established, both in terms of results and in terms of techniques. A notable missing piece of the puzzle was the characterization of type isomorphisms in the presence of intersection types. While, at first thought, this may seem to be a simple exercise, it turns out that not only finding the right characterization is not simple, but that the very notion of isomorphism in intersection types is an unexpectedly original element in the previously known landscape, breaking most of the known properties of isomorphisms of the typed λ-calculus. In particular, isomorphism is not a congruence and types that are equal in the standard models of intersection types may be nonisomorphic.

Adoption of ISO‐oriented quality management system in Greek universities

PurposeQuality management in Greek higher education at least until 2006 was in an early and debated stage. The intent of this paper is to present the extent of use of the ISO standards in Greek universities till 2006 and simultaneously to evaluate whether adoption of ISO‐oriented quality management tools is consistent with DiMaggio and Powell's notions of isomorphism (coercive, normative, and mimetic).Design/methodology/approachThe authors employed a mixed‐methods approach with sequential data collection with several alternations between quantitative and qualitative methods.FindingsIt is found that ISO‐oriented quality management system is fruitfully adopted in units only if all three types of neo‐institutional pressures (coercive, normative and mimetic) are present. These results and the high response rate suggest that there is a quality movement at the micro level in Greek higher education.Research limitations/implicationsQuality management (QM) research within the relatively uncharted Greek universities poses multiple challenges, e.g. in handling politically sensitive subjects, which may benefit readers in overcoming theirs.Originality/valueThe significance of the paper lies in the fact that no existing studies have investigated the adoption of ISO‐oriented quality management system in Greek universities, utilizing neo‐institutional theory and a mixed method research design. Especially relevant is that the study focuses on quality management at the micro level of units within higher education institutions. The study demonstrates how to distinguish different isomorphic pressures empirically.

The complexity of classifying separable Banach spaces up to isomorphism

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.

Multilevel Methods

The study of multilevel phenomena in organizations involves a complex interplay between methods and statistics on one hand and theory development on the other. In this introduction, the authors provide a short summary of the five articles in this feature topic and use them as a platform to discuss the broad need for work in the two areas of (a) multilevel construct validation and measurement and (b) statistical advances in variance decomposition. Within these two broad frameworks, the authors specifically discuss, first, the need to continue moving beyond notions of isomorphism in developing and testing aggregate-level constructs. Second, they discuss the potential value of using discontinuous growth models to understand transitions in longitudinal studies. Finally, they discuss some of the issues surrounding the ability to decompose variance in multilevel modeling of dichotomous and other nonnormal outcome data.

Assessing international accounting harmonization using Hegelian dialectic, isomorphism and Foucault

This paper applies an innovative approach to explore the processes, effects and likely future progress of the convergence of national accounting standards with International Financial Reporting Standards (IFRS). Particular use is made of the Hegelian dialectic concepts of thesis, antithesis and synthesis; notions of isomorphism and decoupling; and, to a lesser extent, Foucault's concept of power–knowledge. Factors that are likely to explain the extent, rhythm and pattern of convergence of national accounting standards with IFRS are discussed. The paper concludes by highlighting four matters that are thought likely to enhance understanding of the process of international accounting standards harmonization; and by averring that the international harmonization process is a dialectical process influenced by belief systems that are constructed in a context of isomorphism and decoupling.

Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups

The notion of isomorphism on stable AF-C^{\ast}-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e. is associated with a single square primitive incidence matrix. A C^{\ast}-isomorphism induces an equivalence relation on these matrices, called C^{\ast}-equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e. there is an algorithm that can be used to check in a finite number of steps whether two given primitive matrices are C^{\ast}-equivalent or not. Special cases of this problem will be considered in a forthcoming paper.

Measurement theory is a poor model of the relation of kinematic geometry and perception of motion

The Kubovy-Epstein proposal for the formalization of the relation between kinematic geometry and perception of motion has formal problems in itself. Motion phenomena are inadequately captured by the relational structures and the notion of isomorphism taken over from measurement theory. [Kubovy &amp; Epstein]

EXAMINING THE MULTIFACETED NOTION OF ISOMORPHISM IN MARRIAGE AND FAMILY THERAPY SUPERVISION: A QUEST FOR CONCEPTUAL CLARITY

The notion of isomorphism has been recommended as a conceptual framework to guide the practice of marriage and family therapy (MFT) supervision. The term is frequently cited in the MFT training literature but is often used in different ways. A panel of MFT supervisors rated the importance and relevance to both therapy and supervision of a large pool of variables. The majority of variables were found to be equally relevant or isomorphic to the domains of MFT and MFT supervision. A qualitative interview with a small subset of the panelists suggested that the concept, to varying degrees, has influenced their work as supervisors. The implications of the results for theory development, research and supervisory practice are discussed.

Abelian p-groups with no invariants

We prove for abelian p-groups a non-structure theorem relative to approximations of Ehrenfeucht-Fraïssé games of length ω 1 in terms of linear orderings with no uncountable descending sequences. Our result shows that there is a group which is too complicated to be characterized up to isomorphism by the Ehrenfeucht-Fraïssé game approximated by a fixed ordering. This means that such a group cannot have any complete invariants which are bounded in the sense of these approximations of the Ehrenfeucht-Fraïssé game. On the other hand, all the approximations characterize together the notion of isomorphism among groups of cardinality at most ω 1. From the point of view of Stability Theory, our result concerns certain stable theories with NDOP and NOTOP.

Classifying immersed curves

Let a collection γ \gamma of generically immersed curves be given in an oriented surface G G . To each component circle, associate a Gauss word by traveling once around the circle and recording the crossing points with signs. The set of these words forms a Gauss paragraph. If γ 1 {\gamma _1} and γ 2 {\gamma _2} fill the surface G G in the sense that the complementary regions are disks, then there is a homeomorphism of G G taking one to the other if and only if γ 1 {\gamma _1} and γ 2 {\gamma _2} have isomorphic Gauss paragraphs. This notion of isomorphism is defined here; it ignores the choices made in defining the Gauss words.

Classifying Immersed Curves

Let a collection $\gamma$ of generically immersed curves be given in an oriented surface $G$. To each component circle, associate a Gauss word by traveling once around the circle and recording the crossing points with signs. The set of these words forms a Gauss paragraph. If ${\gamma _1}$ and ${\gamma _2}$ fill the surface $G$ in the sense that the complementary regions are disks, then there is a homeomorphism of $G$ taking one to the other if and only if ${\gamma _1}$ and ${\gamma _2}$ have isomorphic Gauss paragraphs. This notion of isomorphism is defined here; it ignores the choices made in defining the Gauss words.

Normalization of fuzzy algebras

In this work the notion of fuzzy algebra is investigated. Given an algebra A and a poset S, a fuzzy algebra over A is an S-set f over the base set A of A such that every cut {x @? A | f(x) >= a}, a @? S, is a subalgebra of A. Some consequences are derived. A particular notion of equivalence between two fuzzy algebras is studied, which is more perspicuous than the notion of isomorphism. A set of properties which appear to be desirable for a fuzzy algebra is settled. The main theorem of the paper says that every fuzzy algebra is 'normalizable', i.e. another fuzzy algebra may be constructed which is equivalent to the former and which satisfies all the desired properties.