A set S of real numbers is called bounded above if there is a real number k such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded below and lower bound are similarly defined. A set S is bounded if it is bounded both above and below. Therefore, a set of real numbers is bounde ...

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Bounded set, Bounded set - Definition, Bounded set - Metric space, Bounded set - Boundedness in topological vector spaces, Bounded set - Boundedness in order theory

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Let C be a preadditive category. In particular, morphisms in C can be added. Given objects A1,...,An in C, suppose that we have: another object A1 ⊕ ··· ⊕ An in C (the biproduct); morphisms pk: A1 ⊕ ··· ⊕ An → Ak in C (the projection morphisms); and morphisms i< ...

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Biproduct, Biproduct - Definition, Biproduct - Examples, Biproduct - Properties

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For a formal definition, given three vector spaces V, W and X over the same base field F, a bilinear operator is a function B : V × W → X such that for any w in W the map is a linear operator from V to X, and for any v in V the map is a linear operator from W to X. In other words, if we hold the first entry of the bilinear operator fixed, while letting the second entry vary, the result is a linear operator, ...

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Bilinear operator, Bilinear operator - Definition, Bilinear operator - Properties, Bilinear operator - Examples

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An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (where the image of (x,y) is written as xy) such that the associative law holds: (x y) z = x (y z) for all x, y and z in A. The bilinearity of the multiplication can be expressed as (x + y) z = x z + y z  
...

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Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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For a graph G: = (V,E) with n vertices, the betweenness CB(v) for vertex v is: where σst is the number of shortest geodesic paths from s to t, and σst(v) the number of shortest geodesic paths from s to t that pass through a vertex v. This may be normalised by dividing through by See also:

Betweenness, Betweenness - Definition

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Given a homogeneous linear second-order ordinary differential equation Abel's identity can be written as where W(x) is the Wronskian of the two linearly independent solutions to the differential equation. ...

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Abel's identity, Abel's identity - Definition, Abel's identity - Derivation

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The analog hole (sometimes analog reconversion problem or analog reconversion issue) is a fundamental, and inevitable vulnerability in copy prevention schemes for noninteractive digital content which is intended to be played back using analog means. When the information is converted to a perceptible analog form, there are no restrictions on the resulting analog signal, and the content can be captured ba ...

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Analog hole, Analog hole - Definition, Analog hole - Engineering vs. Business and Political Views, Analog hole - Copyright law vs. particular techniques, Analog hole - Consumer vs. Professional equipment

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The most common choice is to take "small" to mean finite. In this case, two sets are almost disjoint if their intersection is finite, i.e. if (Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. However, the unit interval [0, 1] and the set of rational numbers Q are not almos ...

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Almost disjoint sets, Almost disjoint sets - Definition, Almost disjoint sets - Other meanings

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The admittance matrix of a graph G is defined as L: = D − A with D the degree matrix of G and A the adjacency matrix of G. More explicitly, given a graph G with n vertices the admittance matrix is defined as In the case of directed graphs, either the indegree or the outdeg ...

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Admittance matrix, Admittance matrix - Definition, Admittance matrix - Properties

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The algebraic connectivity of a graph G is the the second smallest eigenvalue of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. (This is a corollary to the fact that the multiplicity of 0 as an eigenvector in the Laplacian is the number of connected components in the graph.) The magnitude of this value reflects how well connected the overall graph is, and has implications for ...

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Algebraic connectivity, Algebraic connectivity - Definition, Algebraic connectivity - History

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Typically, character actors lack some of the stereotypical physical attributes associated with stars. A character actor may be very short or very tall, heavy or thin, balding, or simply unconventional-looking. Many older actors and actresses find their access to lead roles limited by age as well. Similarly, actors of colour were often barred from roles for which they were otherwise suited; some found work perfoming ethnic stereotypes. A character actor is usually playing a character who doesn't go through a major change in the course of the ...

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Character actor, Character actor - Definition, Character actor - Crossover actors, Character actor - Examples of character roles

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1 ångström (Å) = 10–10 metres = 0.1 nm For comparison, the average diameter of an atom, calculated from its empirical radius, ranges from approximately 0.5 Å for hydrogen (the smallest element) to 3.8 Å for uranium (one of the largest naturally occuring elements on Earth). ...

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Angstrom, Angstrom - Definition, Angstrom - Origin, Angstrom - Representation in Unicode

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The term "agent" describes a software abstraction, an idea, or a concept, similar to OOP terms such as methods, functions, and objects. The concept of an agent provides a convenient and powerful way to describe a complex software entity that is capable of acting with a certain degree of autonomy in order to accomplish tasks on behalf of its user. But unlike objects, which are defined in terms of methods and attributes, ...

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Software agent, Software agent - Definition, Software agent - Intelligent agents, Software agent - Autonomous agents, Software agent - Distributed agents, Software agent - Multi-agent systems, Software agent - Mobile agents, Software agent - What is not an agent ..., Software agent - History, Software agent - Examples, Software agent - Buyer agents shopping bots, Software agent - User agents personal agents, Software agent - Monitoring-and-surveillance predictive agents, Software agent - Data mining agents, Software agent - Other examples, Software agent - Design issues, Software agent - Footnotes

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An algebraic space X is comprised of an affine scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions: 1. R is an equivalence relation as a subset of U × U 2. The projections pi: R → U onto each factor are étale maps. If a third condition 3. R is the trivial equivalence relation ov ...

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Algebraic space, Algebraic space - Definition, Algebraic space - Facts about algebraic spaces, Algebraic space - Applications, Algebraic space - Reference

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In the subsequent, we use the notation na (where n is in the set N of natural numbers) for the sum of a with itself n times. An Archimedean group (G, +, ≤) is a linearly ordered group subject to the following condition: for any a and b in G which are greater than 0, the inequality na ≤ b fo ...

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Archimedean group, Archimedean group - Definition, Archimedean group - Examples of Archimedean groups, Archimedean group - Examples of non-Archimedean groups, Archimedean group - Theorems

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For a given cardinal number κ and a stationary set S ⊆ κ, ◊S is the statement that there is a sequence such that every Aδ is a subset of δ for every B ⊆ κ, the set is stationary is usually written as just ◊. ...

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Diamondsuit, Diamondsuit - Definition, Diamondsuit - Properties and use

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A subset C of a real vector space V is a (linear) cone iff λx belongs to C for any x in C and any positive scalar λ of V. The condition can be written more succintly as "λC = C for any positive scalar λ of V". The definition makes sense for any vector space V which allows the notion of "positive scalar", such as spaces over the rational, algebraic, or (more commonly) real numbe ...

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Cone linear algebra, Cone linear algebra - Definition, Cone linear algebra - Boolean additive and linear closure, Cone linear algebra - Pointed and blunt cones, Cone linear algebra - The cone of a set, Cone linear algebra - Salient cone, Cone linear algebra - Spherical section and projection, Cone linear algebra - Convex cone, Cone linear algebra - Affine cone, Cone linear algebra - Proper cone

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A partition of an interval [a,b] is a finite sequence Each [xi − 1,xi] is called a subinterval of the partition. A refinement of the partition is a partition such that for every i with there is an integer r(i) such that xi = y ...

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Darboux integral, Darboux integral - Definition, Darboux integral - Facts about the Darboux integral

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Let be a finite set, containing n objects or letters. The collection of all bi-infinite strings in these letters is denoted by where denotes the integers. This collection has a natural topology, the product topology. The basis of the topology are the cylinder sets Cylinder sets are clopen sets. ...

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Cylinder set, Cylinder set - Definition, Cylinder set - Applications

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In cohomology theory, the cup product is a construction giving a product on the graded cohomology ring H∗(X) of an object X. In the singular theory, this takes the form of a product of cocycle classes: if cp and dq are cocycle classes in Hp(X) and Hq(X), respectively, then the cup ...

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Cup product, Cup product - Definition, Cup product - Equations, Cup product - Examples, Cup product - Cup Product and Geometric Intersections

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A vertex covering for a graph G is a set of vertices V so that every edge of G is adjacent to at least one vertex in V. The minimum vertex covering the smallest vertex cover. We say V covers the edges of the graph. The vertex covering number ωV(G) for a graph GSee also:

Covering graph theory, Covering graph theory - Definition, Covering graph theory - Examples, Covering graph theory - Properties

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