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Homomorphic Equivalence
In the mathematics of graph theory, two graphs, ''G'' and ''H'', are called homomorphically equivalent if there exists a graph homomorphism f\colon G\to H and a graph homomorphism g\colon H\to G. An example usage of this notion is that any two cores of a graph are homomorphically equivalent. Homomorphic equivalence also comes up in the theory of databases. Given a database schema, two instances I and J on it are called homomorphically equivalent if there exists an instance homomorphism f\colon I\to J and an instance homomorphism g\colon J\to I. In fact for any category ''C'', one can define homomorphic equivalence. It is used in the theory of accessible categories The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates i ..., where "weak universality" is the best one can hope for in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core (graph Theory)
In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms. Definition Graph C is a core if every homomorphism f:C \to C is an isomorphism, that is it is a bijection of vertices of C. A core of a graph G is a graph C such that # There exists a homomorphism from G to C, # there exists a homomorphism from C to G, and # C is minimal with this property. Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores. Examples * Any complete graph is a core. * A cycle of odd length is a core. * A graph G is a core if and only if the core of G is equal to G. * Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph ''K''2. * By the Beckman–Quarles theorem, the infinite unit distance graph on all points of the Euclidean plane or of any higher-dimensional Euclidean s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Databases
In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases spans formal techniques and practical considerations, including data modeling, efficient data representation and storage, query languages, security and privacy of sensitive data, and distributed computing issues, including supporting concurrent access and fault tolerance. A database management system (DBMS) is the software that interacts with end users, applications, and the database itself to capture and analyze the data. The DBMS software additionally encompasses the core facilities provided to administer the database. The sum total of the database, the DBMS and the associated applications can be referred to as a database system. Often the term "database" is also used loosely to refer to any of the DBMS, the database system or an applicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Database Schema
The database schema is the structure of a database described in a formal language supported by the database management system (DBMS). The term "schema" refers to the organization of data as a blueprint of how the database is constructed (divided into database tables in the case of relational databases). The formal definition of a database schema is a set of formulas (sentences) called integrity constraints imposed on a database. These integrity constraints ensure compatibility between parts of the schema. All constraints are expressible in the same language. A database can be considered a structure in realization of the database language. The states of a created conceptual schema are transformed into an explicit mapping, the database schema. This describes how real-world entities are modeled in the database. "A database schema specifies, based on the database administrator's knowledge of possible applications, the facts that can enter the database, or those of interest to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instance (database)
Instantiation or instance may refer to: Philosophy * A modern concept similar to ''participation'' in classical Platonism; see the Theory of Forms * The instantiation principle, the idea that in order for a property to exist, it must be had by some object or substance; the instance being a specific object rather than the idea of it * Universal instantiation * An instance (predicate logic), a statement produced by applying universal instantiation to a universal statement * Existential fallacy, also called existential instantiation * A substitution instance, a formula of mathematical logic that can be produced by substituting certain strings of symbols for others in formula, also can be used as the mathematical order to represent the data in an algorithm Computing * Instance (computer science), referring to any running process, or specifically to an object as an instance of a class * Table instance (or database instance), a concept in database design; see Row (database) * Creatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accessible Categories
The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic. A standard text book by Adámek and Rosický appeared in 1994. Accessible categories also have applications in homotopy theory.J. RosickÃ"On combinatorial model categories" ''arXiv'', 16 August 2007. Retrieved on 19 January 2008.Rosický, J. "Injectivity and accessible categories." ''Cubo Matem. Educ'' 4 (2002): 201-211. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript ''Les dérivateurs''. Some properties of accessible categories depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
