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Locally Finite Partially Ordered Set
In mathematics, a locally finite poset is a partially ordered set ''P'' such that for all ''x'', ''y'' ∈ ''P'', the interval 'x'', ''y''consists of finitely many elements. Given a locally finite poset ''P'' we can define its ''incidence algebra''. Elements of the incidence algebra are functions ''ƒ'' that assign to each interval 'x'', ''y''of ''P'' a real number ''ƒ''(''x'', ''y''). These functions form an associative algebra with a product defined by : (f * g)(x,y):=\sum_ f(x,z) g(z,y). There is also a definition of ''incidence coalgebra''. In theoretical physics a locally finite poset is also called a causal set and has been used as a model for spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen .... References Stanley, Richard P. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...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] |
Incidence Algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. Definition A locally finite poset is one in which every closed interval : 'a, b''= is finite. The members of the incidence algebra are the functions ''f'' assigning to each nonempty interval 'a, b''a scalar ''f''(''a'', ''b''), which is taken from the ''ring of scalars'', a commutative ring with unity. On this underlying set one defines addition and scalar multiplication pointwise, and "multiplication" in the incidence algebra is a convolution defined by :(f*g)(a, b)=\sum_f(a, x)g(x, b). An incidence algebra is finite-dimensional if and only if the underlying poset is finite. Related concepts An incidence algebra is analogous to a group algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates: * The laws of physics are invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard P
Richard is a male given name. It originates, via Old French, from Frankish language, Old Frankish and is a Compound (linguistics), compound of the words descending from Proto-Germanic language, Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick (nickname), Dick", "Dickon", "Dickie (name), Dickie", "Rich (given name), Rich", "Rick (given name), Rick", "Rico (name), Rico", "Ricky (given name), Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
