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Vector Measure
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions and first consequences Given a field of sets (\Omega, \mathcal F) and a Banach space X, a finitely additive vector measure (or measure, for short) is a function \mu:\mathcal \to X such that for any two disjoint sets A and B in \mathcal one has \mu(A\cup B) =\mu(A) + \mu (B). A vector measure \mu is called countably additive if for any sequence (A_i)_^ of disjoint sets in \mathcal F such that their union is in \mathcal F it holds that \mu = \sum_^\mu(A_i) with the series on the right-hand side convergent in the norm of the Banach space X. It can be proved that an additive vector measure \mu is countably additive if and only if for any sequence (A_i)_^ as above one has where \, \cdot\, is the norm on X. Countably additive vector measur ... [...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] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bang–bang Control
In control theory, a bang–bang controller (2 step or on–off controller), is a feedback controller that switches abruptly between two states. These controllers may be realized in terms of any element that provides hysteresis. They are often used to control a plant that accepts a binary input, for example a furnace that is either completely on or completely off. Most common residential thermostats are bang–bang controllers. The Heaviside step function in its discrete form is an example of a bang–bang control signal. Due to the discontinuous control signal, systems that include bang–bang controllers are variable structure systems, and bang–bang controllers are thus variable structure controllers. Bang–bang solutions in optimal control In optimal control problems, it is sometimes the case that a control is restricted to be between a lower and an upper bound. If the optimal control switches from one extreme to the other (i.e., is strictly never in between the bounds), t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonotope
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. Zonohedra that tile space The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Kluvánek
Igor Kluvánek (27 January 1931 – 24 July 1993) was a Slovak-Australian mathematician. Academic career Igor Kluvánek obtained his first degree in electrical engineering from the Slovak Polytechnic University, Bratislava, in 1953. His first appointment was in the Department of Mathematics of the same institution. At the same time he worked for his C.Sc. degree obtained from the Slovak Academy of Sciences. In the early 60's he joined the Department of Mathematical Analysis of the University of Pavol Jozef Šafárik in Košice. During 1967–68 he held a visiting position at The Flinders University of South Australia. The events of 1968 in Czechoslovakia made it impossible for him and his family to return to their homeland. The Flinders University of South Australia was able to create a chair in applied mathematics to which he was appointed in January 1969 and occupied until his resignation in 1986. Early years Kluvánek graduated in 1953 from the Slovak Polytechnic Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (measure Theory)
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. Definition Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subset A with \mu(B) of A are atoms, and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes. Examples * Consider the set ''X'' = and let the sigma-algebra \Sigma be the power set of ''X''. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider the Lebesgue measure on the real line. This measure has no atoms. Atomic measures A \sigma-finite measure \mu on a measurable space (X, \Sigma) is called atomic or pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexey Lyapunov
Alexey Andreevich Lyapunov (russian: Алексе́й Андре́евич Ляпуно́в; 25 September 1911 – 23 June 1973) was a Soviet mathematician and an early pioneer of computer science. One of the founders of Soviet cybernetics, Lyapunov was member of the Soviet Academy of Sciences and a specialist in the fields of real function theory, mathematical problems of cybernetics, set theory, programming theory, mathematical linguistics, and mathematical biology. Biography Composer Sergei Lyapunov, mathematician Aleksandr Lyapunov, and philologist Boris Lyapunov were close relatives of Alexey Lyapunov. In 1928, Lyapunov enrolled at Moscow State University to study mathematics, and in 1932 he became a student of Nikolai Luzin. Under his mentorship, Lyapunov began his research in descriptive set theory. He became world-wide known for his theorem on the range of an atomless vector-measure in finite dimensions, now called the Lyapunov Convexity Theorem. From 1934 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively exhaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
