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Algebraic Addition Theorem
In mathematics, an addition theorem is a formula such as that for the exponential function: :''e''''x'' + ''y'' = ''e''''x'' · ''e''''y'', that expresses, for a particular function ''f'', ''f''(''x'' + ''y'') in terms of ''f''(''x'') and ''f''(''y''). Slightly more generally, as is the case with the trigonometric functions and , several functions may be involved; this is more apparent than real, in that case, since there is an algebraic function of (in other words, we usually take their functions both as defined on the unit circle). The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To "classify" addition theorems it is necessary to put some restriction on the type of function ''G'' admitted, such that :''F''(''x'' + ''y'') = ''G''(''F''(''x''), ''F''(''y'')). In this identity one can assume that ''F'' an ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mordell–Weil Theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of k-rational point, ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elliptic curve E and K the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties. History The ''tangent-chord process'' (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(\mathbb)/2E(\mathbb) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(\mathbb) to be finitely generated; and it shows that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timeline Of Abelian Varieties
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves. Early history * c. 1000 Al-Karaji writes on congruent numbers Seventeenth century * Fermat studies descent for elliptic curves * 1643 Fermat poses an elliptic curve Diophantine equation * 1670 Fermat's son published his ''Diophantus'' with notes Eighteenth century * 1718 Giulio Carlo Fagnano dei Toschi, studies the rectification of the lemniscate, addition results for elliptic integrals. * 1736 Leonhard Euler writes on the pendulum equation without the small-angle approximation. * 1738 Euler writes on curves of genus 1 considered by Fermat and Frenicle * 1750 Euler writes on elliptic integrals * 23 December 1751 – 27 January 1752: Birth of the theory of elliptic functions, according to later remarks of Jacobi, as Euler writes on Fagnano's work. * 1775 John Landen publishes Landen's transformation, an isogeny formula. * 1786 Adrien-Marie Legendre begins to write on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Group
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only if it can be written in the form : f(x) = \frac where P\, and Q\, are polynomial functions of x\, and Q\, is not the zero function. The domain of f\, is the set of all valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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