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Coarse Structure
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties. Properly, a coarse structure is not the large-scale analog o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
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] |
Binary Relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number 2 is related to numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Compact Subspace
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsets
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Metric
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly. If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the . In particular, if is some vector such that x = \left(x_1, x_2, \ldots, x_n\right) in finite dimensional coordinate space, it takes the form: :\, x\, _\infty := \max \left(\left, x_1\ , \ldots , \left, x_n\\right). Metric a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negligible Set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible. For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If and are both ideals of subsets of the same set , then one may speak of ''-negligible'' and ''-negligible'' subsets. The opposite of a negligible set is a generic property, which has various forms. Examples Let ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (relational Algebra)
In relational algebra, a projection is a unary operation written as \Pi_( R ), where R is a relation and a_1,...,a_n are attribute names. Its result is defined as the set obtained when the components of the tuples in R are restricted to the set \ – it ''discards'' (or ''excludes'') the other attributes. In practical terms, if a relation is thought of as a table, then projection can be thought of as picking a subset of its columns. For example, if the attributes are (name, age), then projection of the relation onto attribute list (age) yields – we have discarded the names, and only know what ages are present. Projections may also modify attribute values. For example, if R has attributes a, b, c, where the values of b are numbers, then \Pi_( R ) is like R, but with all b-values halved.http://www.csee.umbc.edu/~pmundur/courses/CMSC661-02/rel-alg.pdf ''See Problem 3.8.B on page 3'' Related concepts The closely related concept in set theory (see: projection (set theory)) differs ... [...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 . Multiple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
