Free Games over Coloured Automata

1985 ◽  
Vol 8 (2) ◽  
pp. 199-224
Author(s):  
Antoni Wiweger
Keyword(s):  
Category Theory ◽  
General Notion ◽  
Additional Structure ◽  
Explicit Constructions ◽  
Mealy Automaton ◽  
Special Cases ◽  
Abstract Games

Concepts of category theory are applied to the investigation of some relations between automata and abstract games. The notion of a coloured automaton introduced in this paper provides a framework for a unified treatment of automata and abstract games. Both games and automata can be viewed as special cases of this general notion. A coloured automaton is defined to be a Mealy automaton with the additional structure of a coloured graph on the set of inputs. Various categories of coloured automata, automata, and games are described. It is shown that some forgetful functors between these categories have left adjoints, and explicit constructions of these adjoints are given. The main result is Theorem 5.5 which describes a construction of a free abstract game over a coloured automaton satisfying some additional conditions.

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

scholarly journals Generalizations of 3-Sasakian manifolds and skew torsion

2020 ◽  
Vol 20 (3) ◽  
pp. 331-374 ◽  
Author(s):  
Ilka Agricola ◽  
Giulia Dileo
Keyword(s):  
Ricci Curvature ◽  
General Notion ◽  
Heisenberg Groups ◽  
Canonical Connection ◽  
Contact Metric Manifolds ◽  
Special Cases ◽  
Commutator Function ◽  
Killing Function ◽  
Generalized Killing Spinors ◽  
Sasaki Manifolds

AbstractIn the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α, δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α, δ)-Sasaki manifold is Einstein either if α = δ (the 3-α-Sasaki case) or if δ = (2n + 3)α, where dim M = 4n + 3.In the second part we find these adapted connections. We start with a very general notion of φ-compatible connections, where φ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α, δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

scholarly journals LIFTING THEOREMS FOR TENSOR FUNCTORS ON MODULE CATEGORIES

2011 ◽  
Vol 10 (01) ◽  
pp. 129-155 ◽  
Author(s):  
ROBERT WISBAUER
Keyword(s):  
General Theory ◽  
Tensor Product ◽  
Category Theory ◽  
Systematic Approach ◽  
Wreath Products ◽  
Baxter Equation ◽  
General Category ◽  
Module Theory ◽  
Elementary Module ◽  
Special Cases

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalized to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focusing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang–Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

Understanding the Biases of Generalised Recombination: Part II

2007 ◽  
Vol 15 (1) ◽  
pp. 95-131 ◽  
Author(s):  
Riccardo Poli ◽  
Christopher R. Stephens
Keyword(s):  
Population Model ◽  
Gene Deletion ◽  
Evolution Equations ◽  
Coarse Grained ◽  
General Notion ◽  
Schema Evolution ◽  
Infinite Population ◽  
Special Cases ◽  
Model Predictions ◽  
Hierarchical Nature

This is the second part of a two-part paper where we propose, model theoretically and study a general notion of recombination for fixed-length strings where homologous recombination, inversion, gene duplication, gene deletion, diploidy and more are just special cases. In Part I, we derived both microscopic and coarse-grained evolution equations for strings and schemata for a selecto-recombinative GA using generalised recombination, and we explained the hierarchical nature of the schema evolution equations. In this part, we provide a variety of fixed points for evolution in the case where recombination is used alone, thereby generalising Geiringer's theorem. In addition, we numerically integrate the infinite-population schema equations for some interesting problems, where selection and recombination are used together to illustrate how these operators interact. Finally, to assess by how much genetic drift can make a system deviate from the infinite-population-model predictions we discuss the results of real GA runs for the same model problems with generalised recombination, selection and finite populations of different sizes.

scholarly journals Promise Theory and the Alignment of Context, Processes, Types, and Transforms

2021 ◽  
Author(s):  
Jan Aldert Bergstra ◽  
Mark Burgess
Keyword(s):  
Category Theory ◽  
Structural Similarity ◽  
Additional Structure ◽  
Scaling Properties ◽  
Agent Based ◽  
Process Outcomes ◽  
Wide Range ◽  
Structural Details ◽  
Non Local ◽  
The Relationship

Promise Theory concerns the 'alignment', i.e. the degree of functional compatibility and the 'scaling' properties of process outcomes in agent-based models, with causality and intentional semantics. It serves as an umbrella for other theories of interaction, from physics to socio-economics, integrating dynamical and semantic concerns into a single framework. It derives its measures from sets, and can therefore incorporate a wide range of descriptive techniques, giving additional structure with predictive constraints. We review some structural details of Promise Theory, applied to Promises of the First Kind, to assist in the comparison of Promise Theory with other forms of physical and mathematical modelling, including Category Theory and Dynamical Systems.  We explain how Promise Theory is distinct from other kinds of model, but has a natural structural similarity to statistical mechanics and quantum theory, albeit with different goals; it respects and clarifies the bounds of locality, while incorporating non-local communication. We derive the relationship between promises and morphisms to the extent that this would be a useful comparison.

scholarly journals GENERALIZED CALABI–YAU STRUCTURES, K3 SURFACES, AND B-FIELDS

2005 ◽  
Vol 16 (01) ◽  
pp. 13-36 ◽  
Author(s):  
DANIEL HUYBRECHTS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Weak Form ◽  
K3 Surfaces ◽  
Standard Theory ◽  
General Notion ◽  
Symplectic Structures ◽  
Special Cases ◽  
Torelli Theorem

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

Understanding the Biases of Generalised Recombination: Part I

2006 ◽  
Vol 14 (4) ◽  
pp. 411-432 ◽  
Author(s):  
Riccardo Poli ◽  
Christopher R. Stephens
Keyword(s):  
Population Model ◽  
Gene Deletion ◽  
Evolution Equations ◽  
Coarse Grained ◽  
General Notion ◽  
Schema Evolution ◽  
Infinite Population ◽  
Special Cases ◽  
Model Predictions ◽  
Hierarchical Nature

This is the first part of a two-part paper where we propose, model theoretically and study a general notion of recombination for fixed-length strings, where homologous recombination, inversion, gene duplication, gene deletion, diploidy and more are just special cases. The analysis of the model reveals that the notion of schema emerges naturally from the model's equations. In Part I, after describing and characterising the notion of generalised recombination, we derive both microscopic and coarse-grained evolution equations for strings and schemata and illustrate their features with simple examples. Also, we explain the hierarchical nature of the schema evolution equations and show how the theory presented here generalises past work in evolutionary computation. In Part II, the study provides a variety of fixed points for evolution in the case where recombination is used alone, which generalise Geiringer's theorem. In addition, we numerically integrate the infinite-population schema equations for some interesting problems, where selection and recombination are used together to illustrate how these operators interact. Finally, to assess by how much genetic drift can make a system deviate from the infinite-population-model predictions we discuss the results of real GA runs for the same model problems with generalised recombination, selection and finite populations of different sizes.

Groupoid Enriched Categories and Homotopy Theory

1983 ◽  
Vol 35 (3) ◽  
pp. 385-416 ◽  
Author(s):  
P. H. H. Fantham ◽  
E. J. Moore
Keyword(s):  
Algebraic Topology ◽  
Category Theory ◽  
Homotopy Theory ◽  
Adjoint Functors ◽  
Homotopy Classes ◽  
Enriched Categories ◽  
Special Cases ◽  
Hopf Invariants

We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.

scholarly journals Categorical Nonstandard Analysis

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1573
Author(s):  
Hayato Saigo ◽  
Juzo Nohmi
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Degrees Of Freedom ◽  
Natural Transformation ◽  
Nonstandard Analysis ◽  
Axiomatic Approach ◽  
General Notion ◽  
Spatial Structures ◽  
Internal Set ◽  
Cartesian Closed

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor U on a topos of sets S together with a natural transformation υ, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called U-space, and the category USpace whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category USpace, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

Electron Beam Damage of Biomolecules Assessed by Energy Loss Spectroscopy

1978 ◽  
Vol 36 (3) ◽  
pp. 61-69
Author(s):  
M. Isaacson ◽  
M.L. Collins ◽  
M. Listvan
Keyword(s):  
Electron Microscopy ◽  
Radiation Damage ◽  
Structural Integrity ◽  
Analytical Electron Microscopy ◽  
High Dose ◽  
Dose Rates ◽  
Special Cases ◽  
Analytical Electron ◽  
Atom Migration ◽  
Beam Damage

Over the past five years it has become evident that radiation damage provides the fundamental limit to the study of blomolecular structure by electron microscopy. In some special cases structural determinations at very low doses can be achieved through superposition techniques to study periodic (Unwin & Henderson, 1975) and nonperiodic (Saxton & Frank, 1977) specimens. In addition, protection methods such as glucose embedding (Unwin & Henderson, 1975) and maintenance of specimen hydration at low temperatures (Taylor & Glaeser, 1976) have also shown promise. Despite these successes, the basic nature of radiation damage in the electron microscope is far from clear. In general we cannot predict exactly how different structures will behave during electron Irradiation at high dose rates. Moreover, with the rapid rise of analytical electron microscopy over the last few years, nvicroscopists are becoming concerned with questions of compositional as well as structural integrity. It is important to measure changes in elemental composition arising from atom migration in or loss from the specimen as a result of electron bombardment.

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