Isomorphism Types Research Articles

SEPARABLE MODELS OF RANDOMIZATIONS

AbstractEvery complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of TR is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T. This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.

Definability in the Substructure Ordering of Simple Graphs

For simple graphs, we investigate and seek to characterize the properties first-order definable by the induced subgraph relation. Let \({\mathcal{P}\mathcal{G}}\) denote the set of finite isomorphism types of simple graphs ordered by the induced subgraph relation. We prove this poset has only one non-identity automorphism co, and for each finite isomorphism type G, the set {G, Gco} is definable. Furthermore, we show first-order definability in \({\mathcal{P}\mathcal{G}}\) captures, up to isomorphism, full second-order satisfiability among finite simple graphs. These results can be utilized to explore first-order definability in the closely associated lattice of universal classes. We show that for simple graphs, the lattice of universal classes has only one non-trivial automorphism, the set of finitely generated and finitely axiomatizable universal classes are separately definable, and each such universal subclass is definable up to the unique non-trivial automorphism.

Defining totality in the enumeration degrees

We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a longstanding question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in” relation for total degrees in the enumeration degrees. Two of the most fundamental concepts in computability theory are effective computation and effective enumeration, i.e., computation and enumeration of sets of natural numbers as achieved by computer programs. Each notion induces a natural measure of the complexity of sets of natural numbers. Turing reducibility was defined by Turing in the late 1930’s to capture the relative complexity of computing sets of natural numbers: A ≤T B means that a set A is computable from another set B. We use the word “reducibility” because the problem of computing A is being reduced to the problem of computing B. In the same way, Enumeration reducibility was defined by Friedberg and Rogers in the late 1950’s to capture the relative complexity of enumerating sets: A ≤e B means that there is a way to effectively produce an enumeration of a set A from any enumeration of a set B. Turing reducibility has been studied extensively, but enumeration reducibility also arises naturally in various contexts. Enumeration reducibility restricted to partial functions is equivalent to Kleene’s [12] notion of relative effective computability of partial functions. Scott [20] used enumeration reducibility to give a countable graph model of λ-calculus. C. F. Miller (unpublished manuscript) and M. Ziegler [28] applied enumeration reducibility in group theory, to state and prove an extension of Higman’s embedding theorem for finitely generated groups. J. Miller [15] used enumeration reducibility in his work on computable analysis; he answered a question of Pour El and Lempp by showing that Turing reducibility is not sufficient to measure the complexity of continuous functions on R (but enumeration reducibility is). Similarly, Richter [16] constructed a wide variety of countable structures for which enumeration reducibility, but not Turing reducibility, is sufficient to measure the complexity of their isomorphism type. There are several other occasions in 2010 Mathematics Subject Classification. 03D30.

Torsion of rational elliptic curves over quartic Galois number fields

Let E be an elliptic curve defined over Q, and let K be a number field of degree four that is Galois over Q. The goal of this article is to classify the different isomorphism types of E(K)tors.

Prosocial Activists in SNS: The Impact of Isomorphism and Social Presence on Prosocial Behaviors

The advent of information and communication technology has made people practice prosocial behavior in social networking services (SNSs) more easily. For this reason, the aim of the study was to identify the social and individual factors that induce prosociality in SNS. The concept of isomorphism for categorizing the characteristics of each social networks was adopted. The study also considered the concept of social presence for representing each individual. The experiment manipulated types of isomorphism (Mimetic, Normative, and Coercive) and degrees of social presence in an experimental SNS context. The study also measured individuals’ intention and activity of prosocial behavior. The experiment results indicate that mimetic and normative isomorphic conditions induce higher levels of prosocial intention and activity than coercive isomorphic condition. Also, a higher degree of social presence induces a higher level of prosocial intention. More interesting, the impact of mimetic condition is stronger when the social presence is higher.

A necessary condition for two string links to have the same closure up to concordance

To a link (obtained by closing a string link) we associate a certain diagram of groups. If two links are concordant, we show that there exists a certain type of isomorphism between the group diagrams.

Type-changing rewriting and semantics-preserving transformation

We have identified a class of whole-program transformations that are regular in structure and require changing the types of terms throughout a program while simultaneously preserving the initial semantics after transformation. This class of transformations cannot be safely performed with typical term rewriting techniques, which do not allow for changing the types of terms.In this paper, we present a formalization of type-and-transform systems, an automated approach to the whole-program transformation of terms of one type to terms of a different, isomorphic type using type-changing rewrite rules. A type-and-transform system defines typing and semantics relations between all corresponding source and target subprograms such that a complete transformation guarantees that the whole programs have equivalent types and semantics. We describe the type-and-transform system for the lambda calculus with let-polymorphism and general recursion, including several examples from the literature and properties of the system.

Isomorphism of intersection and union types

This paper gives a complete characterisation of type isomorphism definable by terms of a λ-calculus with intersection and union types. Unfortunately, when union is considered the Subject Reduction property does not hold in general. However, it is well known that in the λ-calculus, independently of the considered type system, the isomorphism between two types can be realised only by invertible terms. Notably, all invertible terms are linear terms. In this paper, the isomorphism of intersection and union types is investigated using a relevant type system for linear terms enjoying the Subject Reduction property. To characterise type isomorphism, a similarity between types and a type reduction are introduced. Types have a unique normal form with respect to the reduction rules and two types are isomorphic if and only if their normal forms are similar.

Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result which implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles' jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas which characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions which follows from the corresponding q-Hahn statement implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, q-TASEP and ASEP.

A Note on Point Stabilizers in Sharp Permutation Groups of Type {0, k}

We study sharp permutation groups of type {0, k} and observe that, once the isomorphism type of a point stabilizer is fixed, there are only finitely many possibilities for such a permutation group. We then show that a sharp permutation group of type {0, k} in which a point stabilizer is isomorphic to the alternating group on 5 letters must be a geometric group. There is, up to permutation isomorphism, one such permutation group.

Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X1,…, Xn] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.

Stochastic Mappings and Random Distribution Fields II: Stationarity

This paper treats the stationarity and stationarily cross correlatedness of infinite variate stochastic mappings. Moreover, for the case of infinite variate random distribution fields, a particular form for the operator cross covariance distribution is given, from which a Kolmogorov type isomorphism theorem and spectral representations are derived. Finally, it is proved that the stationarity of an infinite variate random distribution field is passed onto the summands in the general Wold decomposition.

Banach algebras generated by an invertible isometry of an Lp-space

We provide a complete description of those Banach algebras that are generated by an invertible isometry of an L p -space together with its inverse. Examples include the algebra P F p ( Z ) of p -pseudofunctions on Z , the commutative C ⁎ -algebra C ( S 1 ) and all of its quotients, as well as uncountably many ‘exotic’ Banach algebras. We associate to each isometry of an L p -space a spectral invariant called ‘spectral configuration’, which contains considerably more information than its spectrum as an operator. It is shown that the spectral configuration describes the isometric isomorphism type of the Banach algebra that the isometry generates together with its inverse. It follows from our analysis that these algebras are semisimple. With the exception of P F p ( Z ) , they are all closed under continuous functional calculus, and their Gelfand transform is an isomorphism. As an application of our results, we show that Banach algebras that act on L 1 -spaces are not closed under quotients. This answers the case p = 1 of a question asked by Le Merdy 20 years ago.

Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups

In recent years several new restrictions on integral partial augmentations for torsion units of \(\mathbb {Z} G\) have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups G. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 – the non-split covering group of S4, and one of order 72 of isomorphism type (C3 × C3) ⋊ D8. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of \(\mathbb {Q} G\) that are not conjugate to elements of \(\mathbb {Z} G\).

Infinite primitive and distance transitive directed graphs of finite out-valency

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.

Obésité masculine et incestualité

ObjectiveThis article proposes to shed light on the understanding of obesity in a familial perspective. Indeed, even if it is an individual symptom, the subject concerned is integrated into numerous groups and particularly into his primary membership group, namely his family. The clinical case-study of an obese teenager provides better understanding of the unconscious issues arising on the intergenerational familial scene. MethodWe propose an analysis of a symbiotic link woven between a mother and her son that is caused by a trauma experienced by the mother, and locked into an unconscious pact and the familial narcissistic contract. It is by using two projective family mediation tools, namely the drawing of the genealogical tree, or projective “genography”, and the drawing of the subject's dream house, or projective “spatiography”, that we were able to decipher the unconscious image of the family body in its synchronic and diachronic dimensions, and its evolution in the course of a process of therapeutic care. ResultsWe noted a familial functioning built on ties of an isomorphic type, demonstrating unconscious alliances and leaving no room for one member, in this case the teenager, to individualize, thus risking endangering the group. We therefore understood this functioning to be an adaptive answer to the lack of psychic elaboration in relation to the trauma. DiscussionFor want of symbolization, the bodily sphere, like a semaphore, then takes center stage to signal familial suffering that is unspeakable because it is unrepresentable. ConclusionThis article can thus contribute to our understanding of the challenges of global therapeutic care that is at once individual and familial, so as to consolidate results in terms of weight loss.

On Isomorphism of "Functional" Intersection and Union Types

Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the right type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott's inverse limit lambda-model. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction.

On Varieties of Lie Algebras of Maximal Class

AbstractWe study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ℂ, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ℕ-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of ℕ-graded Lie algebras of maximal class generated by L1 and L2, L = 〈L1; L2〉. Vergne described the structure of these algebras with the property L = 〈L1〉. In this paper we study those generated by the first and q-th components where q > 2, L = 〈L1; Lq〉. Under some technical condition, there can only be one isomorphism type of such algebras. For q = 3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

A corpus-based study of self-repairs in Russian spoken monologues

The paper investigates self-initiated self-repairs in the Prosodically Annotated Corpus of Spoken Russian (PrACS-Russ), a unique resource where spontaneous speech phenomena (speech errors, repairs, filled pauses etc.) in Russian natural discourse are systematically registered in transcripts synchronized to audio recordings. Based on a qualitative and a preliminary quantitative analysis of more than 800 repairs, two main strategies of self-repairing are identified and described: the ‘on-line strategy’ associated with speech disfluencies that break the lexical, grammatical or prosodic coherence of the produced discourse, e.g. with truncating the segment under repair, and the ‘off-line strategy’ that does not entail disfluency integrating repairs into fluent planned speech production. Our analysis shows that self-repairs appear with overall rates of 1.8–2.9 instances per 100 words. On-line repairs vastly outnumber off-line repairs: they comprise 82 % to 90 % of the total number of repairs in the analyzed subcorpora of PrACS-Russ. In the absolute majority (83–91 %) of repairs, the segment under repair and its corrected counterpart are structurally and semantically isomorphic, i.e. show systematic parallelism in their meaning, grammatical form, syntactic function or even in their phonetic shell (as, e.g. in ‘slips of the tongue’ cases). Main types of isomorphism between the segment under repair and its corrected counterpart are identified and investigated in the paper.

Non-commutative Frobenius characteristic of generalized parking functions : Application to enumeration

We give a recursive definition of generalized parking functions that allows them to be viewed as a species. From there we compute a non-commutative characteristic of the generalized parking function module and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan. Nous donnons une définition récursive des fonctions de parking généralisées nous permettant de munir ces dernières d’une structure d’espèce. Nous utilisons ce point de vu pour donner une caractéristique de Frobenius non-commutative du module des fonctions de parking généralisées que nous appliquons afin de donner de nombreuses formules d’énumération de structures et de type d’isomorphismes, ainsi qu’une interprétation dans plusieurs bases des fonctions symétriques non commutatives. Enfin, nousétudions une formule d’inclusion-exclusion provenant de Kung et Yan.