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Weierstrass Elliptic Functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script ''p''. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass \wp-function Motivation A cubic of the form C_^\mathbb=\ , where g_2,g_3\in\mathbb are complex numbers with g_2^3-27g_3^2\neq0, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it. For the quadric K=\left\; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Painlevé Property
Painlevé is a surname. Notable people with the surname include: * Jean Painlevé (1902–1989), French film director, actor, translator, animator, son of Paul * Paul Painlevé (1863–1933), French mathematician and politician, twice Prime Minister of France ** Painlevé conjecture, a conjecture about singularities in the n-body problem by Paul Painlevé ** Painlevé paradox, a paradox in rigid-body dynamics by Paul Painlevé ** Painlevé transcendents In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable ..., ordinary differential equation solutions discovered by Paul Painlevé Other * French aircraft carrier Painlevé, a planned ship named in honor of Paul Painlevé {{DEFAULTSORT:Painleve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Periodic Function
In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers ''u'' and ''v'' that are linearly independent as vectors over the field of real numbers. That ''u'' and ''v'' are periods of a function ''ƒ'' means that :f(z + u) = f(z + v) = f(z)\, for all values of the complex number ''z''. The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine, In the complex plane the exponential function ''e''''z'' is a singly periodic function, with period 2''πi''. Examples As an arbitrary mapping from pairs of reals (or complex numbers) to reals, a doubly periodic function can be constructed with little effort. For example, assume that the periods are 1 and ''i'', so that the repe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the ''degree''. That is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. This is also referred to a ''th-degree'' or ''th-order'' homogeneous function. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain of a function, domain and codomain are vector spaces over a Field (mathematics), field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces that contain S. It is often denoted pp. 29-30, §§ 2.5, 2.8 or \langle S \rangle. For example, in geometry, two linearly independent vectors span a plane. To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of . Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S is generally called the substructure ''generated'' by S. Definition Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', where ''n'' is the complex dimension of ''M''. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C''n'' considered as real vector space; then the quotient group V/\Lambda is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For ''n'' = 1 this is the classical period lattice construction of elliptic curves. For ''n'' > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when ''n'' > 1, and are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Absolute-convergence
In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. Motivation A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities. Definition Given a set ''X'' and functions f_n : X \to \mathbb (or to any normed vector space), the series :\sum_^ f_n(x) is called uniformly absolutely-convergent if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Period Lattice
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Definition A fundamental pair of periods is a pair of complex numbers \omega_1,\omega_2 \in \Complex such that their ratio \omega_2 / \omega_1 is not real. If considered as vectors in \R^2, the two are linearly independent. The lattice generated by \omega_1 and \omega_2 is :\Lambda = \left\. This lattice is also sometimes denoted as \Lambda(\omega_1, \omega_2) to make clear that it depends on \omega_1 and \omega_2. It is also sometimes denoted by \Omega\vphantom or \Omega(\omega_1, \omega_2), or simply by (\omega_1, \omega_2). The two generators \omega_1 and \omega_2 are called the ''lattice basis''. The parallelogram with vertices (0, \omega_1, \omega_1+\omega_2, \omega_2) is called the ''fundamental parallelogram''. While a fundamental pair gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Independence
In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of Dimension (vector space), dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist Scalar (mathematics), scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes ... [...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 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 have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in 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 complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
