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Star-convex Set
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s_0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set. Definition Given two points x and y in a vector space X (such as Euclidean space \R^n), the convex hull of \ is called the and it is denoted by \left , y\right~:=~ \left\ ~=~ x + (y - x) , 1 where z , 1:= \ for every vector z. A subset S of a vector space X is said to be s_0 \in S if for every s \in S, the closed interval \left _0, s\right\subseteq S. A set S is and is called a if there exists some point s_0 \in S such that S i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Domain
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s_0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set. Definition Given two points x and y in a vector space X (such as Euclidean space \R^n), the convex hull of \ is called the and it is denoted by \left , y\right~:=~ \left\ ~=~ x + (y - x) , 1 where z , 1:= \ for every vector z. A subset S of a vector space X is said to be s_0 \in S if for every s \in S, the closed interval \left _0, s\right\subseteq S. A set S is and is called a if there exists some point s_0 \in S such that S i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express this is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \leq 1. The balanced hull or balanced envelope of a set S is the smallest balanced set containing S. The balanced core of a subset S is the largest balanced set contained in S. Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Definition Let X be a vector space over the field \mathbb of real or complex numbers. Notation If S is a set, a is a scalar, and B \subseteq \mathbb then let a S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Length
Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album), 1997 album by the Australian band Regurgitator * The Units, a synthpunk band Television * ''The Unit'', an American television series * '' The Unit: Idol Rebooting Project'', South Korean reality TV survival show Business * Stock keeping unit, a discrete inventory management construct * Strategic business unit, a profit center which focuses on product offering and market segment * Unit of account, a monetary unit of measurement * Unit coin, a small coin or medallion (usually military), bearing an organization's insignia or emblem * Work unit, the name given to a place of employment in the People's Republic of China Science and technology Science and medicine * Unit, a vessel or section of a chemical plant * Blood unit, a measuremen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The number 9 is in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
