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Product Topology
Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer. In retailing, products are often referred to as ''Me ..., an item that serves as a solution to a specific consumer problem. * Product (project management) {{more citations needed, date=January 2021 In project management under the PRINCE2 methodology, a product breakdown structure (PBS) is a tool for analysing, documenting and communicating the outcomes of a project, and forms part of the product base ..., a deliverable or set of deliverables that contribute to a business solution Mathematics * Product (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
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Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer. In retailing, products are often referred to as ''Merchandising, merchandise'', and in manufacturing, products are bought as raw materials and then sold as finished goods. A Service (economics), service is also regarded to as a type of product. Commodity, Commodities are usually raw materials such as metals and agricultural products, but a commodity can also be anything widely available in the open market. In project management, products are the formal definition of the Product breakdown structure, project deliverables that make up or contribute to delivering the objectives of the project. A related concept is that of a sub-product, a secondary but useful result of a production (economics), production process. Dangerous products, particularly physical ones, that cause injuries to consume ...
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Multiplication Of Vectors
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., Vector multiplication refers to one of several techniques for the multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ... of two (or more) vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ... with themselves. It may concern any of the following articles: * Dot product In mathematics, the dot product or s ...
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Product (category Theory)
In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ..., the product of two (or more) objects Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ... in a category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ... is a notion designed to capture the essence behind constructions in other areas of mathematics Mathematics (from Greek: ) includes the study of such topics ...
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Monoidal Category
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a monoidal category (or tensor category) is a category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ... \mathbf C equipped with a bifunctor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... :\otimes : \mathbf \times \mathbf \to \mathbf that is associative In mathematics Mathematics (from Ancient Greek, Gree ...
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Wedge Sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the Quotient space (topology), quotient space of the Disjoint union (topology), disjoint union of ''X'' and ''Y'' by the identification x_0 \sim y_0: X \vee Y = (X \amalg Y)\;/, where \,\sim\, is the equivalence closure of the relation \left\. More generally, suppose \left(X_i\right)_ is a indexed family of pointed spaces with basepoints \left(p_i\right)_. The wedge sum of the family is given by: \bigvee_ X_i = \coprod_ X_i\;/, where \,\sim\, is the equivalence closure of the relation \left\. In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints \left(p_i\right)_, unless the spaces \left(X_i\right)_ are Homogeneous space, homogeneous. The wedge sum is agai ...
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Smash Product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the Quotient space (topology), quotient of the product space ''X'' × ''Y'' under the identifications (''x'', ''y''0) ∼ (''x''0, ''y'') for all ''x'' in ''X'' and ''y'' in ''Y''. The smash product is itself a pointed space, with basepoint being the equivalence class of (''x''0, ''y''0). The smash product is usually denoted ''X'' ∧ ''Y'' or ''X'' ⨳ ''Y''. The smash product depends on the choice of basepoints (unless both ''X'' and ''Y'' are homogeneous space, homogeneous). One can think of ''X'' and ''Y'' as sitting inside ''X'' × ''Y'' as the subspace (topology), subspaces ''X'' × and × ''Y''. These subspaces intersect at a single point: (''x''0, ''y''0), the basepoint of ''X'' × ''Y''. So the union of ...
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Slant Product
In algebraic topology the cap product is a method of adjoining a chain (algebraic topology), chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938. Definition Let ''X'' be a topological space and ''R'' a coefficient ring. The cap product is a bilinear map on singular homology and cohomology :\frown\;: H_p(X;R)\times H^q(X;R) \rightarrow H_(X;R). defined by contracting a singular chain \sigma : \Delta\ ^p \rightarrow\ X with a singular cochain \psi \in C^q(X;R), by the formula : : \sigma \frown \psi = \psi(\sigma, _) \sigma, _. Here, the notation \sigma, _ indicates the restriction of the simplicial map \sigma to its face spanned by the vectors of the base, see Simplex. Interpretation In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap ...
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Cup Product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''∗(''X''), called the cohomology ring. The cup product was introduced in work of James Waddell Alexander II, J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944. Definition In singular cohomology, the cup product is a construction giving a product on the graded ring, graded cohomology ring ''H''∗(''X'') of a topological space ''X''. The construction starts with a product of Cochain (algebraic topology), cochains: if \alpha^p is a ''p''-cochain and \beta^q is a ''q''-cochain, then :(\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_) \cdot \beta^q(\sigm ...
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Cap Product
In algebraic topology the cap product is a method of adjoining a chain (algebraic topology), chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938. Definition Let ''X'' be a topological space and ''R'' a coefficient ring. The cap product is a bilinear map on singular homology and cohomology :\frown\;: H_p(X;R)\times H^q(X;R) \rightarrow H_(X;R). defined by contracting a singular chain \sigma : \Delta\ ^p \rightarrow\ X with a singular cochain \psi \in C^q(X;R), by the formula : : \sigma \frown \psi = \psi(\sigma, _) \sigma, _. Here, the notation \sigma, _ indicates the restriction of the simplicial map \sigma to its face spanned by the vectors of the base, see Simplex. Interpretation In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap ...
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Product Topology
Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer. In retailing, products are often referred to as ''Me ..., an item that serves as a solution to a specific consumer problem. * Product (project management) {{more citations needed, date=January 2021 In project management under the PRINCE2 methodology, a product breakdown structure (PBS) is a tool for analysing, documenting and communicating the outcomes of a project, and forms part of the product base ..., a deliverable or set of deliverables that contribute to a business solution Mathematics * Product (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
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Tensor Product
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the tensor product V \otimes W of two vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...s and (over the same field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...) is a vector space that can be thought of as the ''space of all tensor In mathematics Mathematics (from Greek: ) includes the ...
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Triple Product
In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ... and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., the triple product is a product of three 3-dimensional File:Dimension levels.svg, thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum numb ... vectors, usually Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number ...
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