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Pascal's Simplex
In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem. Generic Pascal's ''m''-simplex Let ''m'' (''m'' > 0) be a number of terms of a polynomial and ''n'' (''n'' ≥ 0) be a power the polynomial is raised to. Let \wedge^m denote a Pascal's ''m''-simplex. Each Pascal's ''m''-simplex is a semi-infinite object, which consists of an infinite series of its components. Let \wedge^m_n denote its ''n''th component, itself a finite (''m − 1'')-simplex with the edge length ''n'', with a notational equivalent \vartriangle^_n. ''n''th component \wedge^m_n = \vartriangle^_n consists of the coefficients of multinomial expansion of a polynomial with ''m'' terms raised to the power of ''n'': :, x, ^n=\sum_;\ \ x\in\mathbb^m,\ k\in\mathbb^m_0,\ n\in\mathbb_0,\ m\in\mathbb where \textstyle, x, =\sum_^m,\ , k, =\sum_^m,\ x^k=\prod_^m. Example for \wedge^4 Pascal's 4-simplex , sliced along the ''k4'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Pyramid 3d
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education school in Zaandam, the Netherlands * Pasca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Matrix
In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are: L_5 = \begin 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 1 & 3 & 3 & 1 & 0 \\ 1 & 4 & 6 & 4 & 1 \end\,\,\,U_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 \end\,\,\,S_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 70 \end=L_5 \times U_5 There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity. Definition The non-zero elements of a Pascal matrix are given by the bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinomial Expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by :(a+b+c)^n = \sum_ \, a^i \, b^ \;\! c^k, where is a nonnegative integer and the sum is taken over all combinations of nonnegative indices and such that . The trinomial coefficients are given by : = \frac \,. This formula is a special case of the multinomial formula for . The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron. Properties The number of terms of an expanded trinomial is the triangular number : t_ = \frac, where is the exponent to which the trinomial is raised.. Example An example of a trinomial expansion with n=2 is : (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca See also * Binomial expansion * Pascal's pyramid * Multinomial coefficient * Trinomial triangle The trinomial triangle is a variation of Pascal's triangle. The difference between the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Tetrahedron
In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names. Structure of the tetrahedron Because the tetrahedron is a three-dimensional object, displaying it on a piece of paper, a computer screen or other two-dimensional medium is difficult. Assume the tetrahedron is divided into a number of levels, or floors, or slices, or layers. The top layer (the apex) is labelled "Layer 0". Other layers can be thought of as overhead views of the tetrahedron with the previous layers removed. The first six layers are as follows: The layers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Expansion
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example, for , (x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. The coefficient in the term of is known as the binomial coefficient \tbinom or \tbinom (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where \tbinom gives the number of different combinations of elements that can be chosen from an -element set. Therefore \tbinom is often pronounced as " choose ". History Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid ment ... [...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] |
Semi-infinite
In mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways. In ordered structures and Euclidean spaces Generally, a semi-infinite set is bounded in one direction, and unbounded in another. For instance, the natural numbers are semi-infinite considered as a subset of the integers; similarly, the intervals (c,\infty) and (-\infty,c) and their closed counterparts are semi-infinite subsets of \R. Half-spaces are sometimes described as semi-infinite regions. Semi-infinite regions occur frequently in the study of differential equations. For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar. A semi-infinite integral is an improper integral over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite. Most forms of semi-infiniteness are boundedness properties, not cardinality or measure properties: semi-i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
