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Thomae's Formula
In mathematics, Thomae's formula is a formula introduced by relating theta constants to the branch points of a hyperelliptic curve . History In 1824, the Abel–Ruffini theorem established that polynomial equations of a degree of five or higher could have no solutions in radicals. It became clear to mathematicians since then that one needed to go beyond radicals in order to express the solutions to equations of the fifth and higher degrees. In 1858, Charles Hermite, Leopold Kronecker, and Francesco Brioschi independently discovered that the quintic equation could be solved with elliptic transcendents. This proved to be a generalization of the radical, which can be written as: \sqrt \exp \left(\right) = \exp \left(\frac\int^x_1\frac\right). With the restriction to only this exponential, as shown by Galois theory, only compositions of Abelian extensions may be constructed, which suffices only for equations of the fourth degree and below. Something more general is required for eq ... [...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] |
Comptes Rendus De L'Académie Des Sciences
(, ''Proceedings of the Academy of Sciences''), or simply ''Comptes rendus'', is a French scientific journal published since 1835. It is the proceedings of the French Academy of Sciences. It is currently split into seven sections, published on behalf of the Academy until 2020 by Elsevier: ''Mathématique, Mécanique, Physique, Géoscience, Palévol, Chimie, ''and'' Biologies.'' As of 2020, the ''Comptes Rendus'' journals are published by the Academy with a diamond open access model. Naming history The journal has had several name changes and splits over the years. 1835–1965 ''Comptes rendus'' was initially established in 1835 as ''Comptes rendus hebdomadaires des séances de l'Académie des Sciences''. It began as an alternative publication pathway for more prompt publication than the ''Mémoires de l'Académie des Sciences,'' which had been published since 1666. The ''Mémoires,'' which continued to be published alongside the ''Comptes rendus'' throughout the ninetee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Number Theory
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surfaces
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure (specifically a complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. Given this, the sphere and torus admit complex structures but the Möbius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bring–Jerrard Form
In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of the polynomial x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thus multi-valued), or a specific root, which is usually chosen such that the Bring radical is real-valued for real ''a'' and is an analytic function in a neighborhood of the real line. Because of the existence of four branch points, the Bring radical cannot be defined as a function that is continuous over the whole complex plane, and its domain of continuity must exclude four branch cuts. George Jerrard showed that some quintic equations can be solved in closed form using radicals and Bring radicals, which had been introduced by Erland Bring. In this article, the Bring radical of ''a'' is denoted \operatorname(a). For real argument, it is odd, monotonically decreasing, and unbounded, with asymptotic behavior \operatorname(a) \sim -a^ for larg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
