scholarly journals Entropy in Thermodynamics: from Foliation to Categorization

2021 ◽  
Vol 29 (1) ◽  
pp. 49-66
Author(s):  
Radosław A. Kycia
Keyword(s):  
Natural Language ◽  
Category Theory ◽  
Differential Forms ◽  
Ordered Sets ◽  
Smooth Case ◽  
Axiomatic Definition ◽  
Adiabatic Processes ◽  
Definition Of

Abstract We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

scholarly journals Finding Global Minima with a Filled Function Approach for Non-Smooth Global Optimization

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Weixiang Wang ◽  
Youlin Shang ◽  
Ying Zhang
Keyword(s):  
Global Optimization ◽  
Optimization Problem ◽  
Numerical Test ◽  
Function Approach ◽  
Global Optimization Problem ◽  
Computational Results ◽  
Filled Function ◽  
Global Minima ◽  
Smooth Case ◽  
Definition Of

A filled function approach is proposed for solving a non-smooth unconstrained global optimization problem. First, the definition of filled function in Zhang (2009) for smooth global optimization is extended to non-smooth case and a new one is put forwarded. Then, a novel filled function is proposed for non-smooth the global optimization and a corresponding non-smooth algorithm based on the filled function is designed. At last, a numerical test is made. The computational results demonstrate that the proposed approach is effcient and reliable.

Q-orders with no unit on Q

2021 ◽  
pp. 1-8
Author(s):  
Cheng-Jie Zhou ◽  
Wei Yao
Keyword(s):  
Ordered Sets ◽  
Complete Lattices ◽  
Definition Of ◽  
Self Adaptive

For a usual commutative quantale Q (does not necessarily have a unit), we propose a definition of Q-ordered sets by introducing a kind of self-adaptive self-reflexivity. We study their completeness and the related Q-modules of complete lattices. The main result is that, the complete Q-ordered sets and the Q-modules of complete lattices are categorical isomorphic.

scholarly journals Object-Free Definition of Categories

2013 ◽  
Vol 21 (3) ◽  
pp. 193-205
Author(s):  
Marco Riccardi
Keyword(s):  
Binary Relation ◽  
Category Theory ◽  
Final Part ◽  
One To One ◽  
Definition Of

Summary Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

scholarly journals Categories as Mathematical Models

2018 ◽  
Author(s):  
David I. Spivak
Keyword(s):  
Dynamical Systems ◽  
Mathematical Modelling ◽  
Mathematical Models ◽  
Category Theory ◽  
Vector Spaces ◽  
Mathematical Objects ◽  
Working Definition ◽  
Modelling Framework ◽  
Categorical Models ◽  
Definition Of

Category theory is presented as a mathematical modelling framework that highlights the relationships between objects, rather than the objects in themselves. A working definition of model is given, and several examples of mathematical objects, such as vector spaces, groups, and dynamical systems, are considered as categorical models.

Category theory

2020 ◽  
pp. 19-38
Author(s):  
Ash Asudeh ◽  
Gianluca Giorgolo
Keyword(s):  
Natural Language ◽  
Category Theory ◽  
Categorial Grammar ◽  
Natural Language Semantics ◽  
Fundamental Notion ◽  
Language Semantics ◽  
Natural Transformations ◽  
Glue Semantics

This chapter aims to introduce sufficient category theory to enable a formal understanding of the rest of the book. It first introduces the fundamental notion of a category. It then introduces functors, which are maps between categories. Next it introduces natural transformations, which are natural ways of mapping between functors. The stage is then set to at last introduces monads, which are defined in terms of functors and natural transformations. The last part of the chapter provides a compositional calculus with monads for natural language semantics (in other words, a logic for working with monads) and then relates the compositional calculus to Glue Semantics and to a very simple categorial grammar for parsing. The chapter ends with some exercises to aid understanding.

Migrating from ISO/IEC 9126 to SQUARE

2017 ◽  
pp. 160-178
Author(s):  
Paula Estrella ◽  
Nikos Tsourakis
Keyword(s):  
Natural Language ◽  
Potential User ◽  
Use Case ◽  
Relative Importance ◽  
Quality Model ◽  
Evaluation Methodologies ◽  
Model Based ◽  
Fair Comparison ◽  
Definition Of ◽  
Current Standards

When it comes to the evaluation of natural language systems, it is well acknowledged that there is a lack of common evaluation methodologies, making the fair comparison of such systems a difficult task. Many attempts to standardize this process have used a quality model based on the ISO/IEC 9126 standards. The authors have also used these standards for the definition of a weighted quality model for the evaluation of a medical speech translator, showing the relative importance of the system's features depending on the potential user (patient or doctor, developer). More recently, ISO/IEC 9126 has been replaced by a new series of standards, the 25000 or SQuaRE series, indicating that the model should be migrated to the new series in order to maintain compliance adherence to current standards. This chapter demonstrates how to migrate from ISO/IEC 9126 to ISO 25000 by using the authors' previous work as a use case.

scholarly journals Contralog: a Prolog conform forward-chaining environment and its application for dynamic programming and natural language parsing

2016 ◽  
Vol 8 (1) ◽  
pp. 41-62
Author(s):  
Imre Kilián
Keyword(s):  
Dynamic Programming ◽  
Natural Language ◽  
Inference Engine ◽  
Recursive Definition ◽  
Target Model ◽  
Forward Chaining ◽  
Natural Language Parsing ◽  
Multiplication Problem ◽  
Inference Strategy ◽  
Definition Of

Abstract The backward-chaining inference strategy of Prolog is inefficient for a number of problems. The article proposes Contralog: a Prolog-conform, forward-chaining language and an inference engine that is implemented as a preprocessor-compiler to Prolog. The target model is Prolog, which ensures mutual switching from Contralog to Prolog and back. The Contralog compiler is implemented using Prolog's de facto standardized macro expansion capability. The article goes into details regarding the target model. We introduce first a simple application example for Contralog. Then the next section shows how a recursive definition of some problems is executed by their Contralog definition automatically in a dynamic programming way. Two examples, the well-known matrix chain multiplication problem and the Warshall algorithm are shown here. After this, the inferential target model of Prolog/Contralog programs is introduced, and the possibility for implementing the ReALIS natural language parsing technology is described relying heavily on Contralog's forward chaining inference engine. Finally the article also discusses some practical questions of Contralog program development.

Constructing Corpora for the Development and Evaluation of Paraphrase Systems

2008 ◽  
Vol 34 (4) ◽  
pp. 597-614 ◽  
Author(s):  
Trevor Cohn ◽  
Chris Callison-Burch ◽  
Mirella Lapata
Keyword(s):  
Natural Language Processing ◽  
Natural Language ◽  
Language Processing ◽  
Syntactic Structure ◽  
Nominal Data ◽  
Parallel Corpus ◽  
Word Alignments ◽  
Definition Of

Automatic paraphrasing is an important component in many natural language processing tasks. In this article we present a new parallel corpus with paraphrase annotations. We adopt a definition of paraphrase based on word alignments and show that it yields high inter-annotator agreement. As Kappa is suited to nominal data, we employ an alternative agreement statistic which is appropriate for structured alignment tasks. We discuss how the corpus can be usefully employed in evaluating paraphrase systems automatically (e.g., by measuring precision, recall, and F1) and also in developing linguistically rich paraphrase models based on syntactic structure.

scholarly journals Quasi-Perfect Stackelberg Equilibrium

2019 ◽  
Vol 33 ◽  
pp. 2117-2124 ◽  
Author(s):  
Alberto Marchesi ◽  
Gabriele Farina ◽  
Christian Kroer ◽  
Nicola Gatti ◽  
Tuomas Sandholm
Keyword(s):  
Broad Class ◽  
Stackelberg Equilibrium ◽  
Correlated Equilibrium ◽  
Equilibrium Concept ◽  
Extensive Form ◽  
Tree Form ◽  
Axiomatic Definition ◽  
Security Games ◽  
Definition Of ◽  
Perfect Equilibria

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasiperfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings—that is, settings where a leader can commit to a strategy—which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes.

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