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Moduli Space For Abelian Varieties
Modulus is the diminutive from the Latin word ''modus'' meaning measure or manner. It, or its plural moduli, may refer to the following: Physics, engineering and computing * Moduli (physics), scalar fields for which the potential energy function has continuous families of global minima * The measurement of standard pitch in the teeth of a rotating gear * Bulk modulus, a measure of compression resistance * Elastic modulus, a measure of stiffness * Shear modulus, a measure of elastic stiffness * Young's modulus, a specific elastic modulus * Modulo operation (a % b, mod(a, b), etc.), in both math and programming languages; results in remainder of a division * Casting modulus used in Chvorinov's rule. Mathematics * Modulus (modular arithmetic), base of modular arithmetic * Modulus, the absolute value of a real or complex number ( ) * Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects * Conformal modulus, a measure of the size of a cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminutive
A diminutive is a root word that has been modified to convey a slighter degree of its root meaning, either to convey the smallness of the object or quality named, or to convey a sense of intimacy or endearment. A (abbreviated ) is a word-formation device used to express such meanings. In many languages, such forms can be translated as "little" and diminutives can also be formed as multi-word constructions such as " Tiny Tim". Diminutives are often employed as nicknames and pet names when speaking to small children and when expressing extreme tenderness and intimacy to an adult. The opposite of the diminutive form is the augmentative. Beyond the ''diminutive form'' of a single word, a ''diminutive'' can be a multi-word name, such as "Tiny Tim" or "Little Dorrit". In many languages, formation of diminutives by adding suffixes is a productive part of the language. For example, in Spanish can be a nickname for someone who is overweight, and by adding an suffix, it becomes which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus Robot
The household robot Modulus, described by the manufacturer as "the friend of ''Homo sapiens''", was made by Sirius, a company Massimo Giuliana set up in 1982 for marketing home and personal computers, and which decided to start building its own domestic robot back in 1984. When the first "Modulus" prototype had been realized, the company asked Isao Hosoe, a Japanese designer who has been living and working in Milan for many years, to study its "body-work". Hosoe's work, however, went well beyond this, and was followed by a complete technological reprocessing of the robot. Data Process was responsible for the design and manufacture of the electronic and mechanical parts, while Sirius used the expertise of an American company, the RB Robot Corporation, for the software (its founder, Joseph H. Bosworth, is known by some as "the father of personal robotics"). Development Two million dollars were invested in developing this particular piece of equipment. Research carried out in the Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus Guitars
Modulus Graphite (formerly, ''Modulus Guitars'') is an American manufacturer of musical instruments best known for building bass guitars with carbon fiber necks. The company, originally called Modulus Graphite, was founded in part by Geoff Gould, a bassist who also worked for an aerospace company in Palo Alto, California, and coworker Jerry Dorsch. When they split, Jerry started Graphite Guitar Systems in Washington state. History The name is a reference to Young's modulus, a measure of the stiffness of an elastic material, used in the field of solid mechanics. Carbon fiber has an exceptionally high modulus. Traditionally, electric guitar and bass necks are made from hardwoods (such as maple or mahogany) reinforced with an adjustable steel " truss rod." Wood, being a naturally occurring material, is prone to variations in density and flexibility. This, coupled with the high stresses created by stretching steel strings across them lengthwise, makes wood necks prone to certain u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus (gastropod)
''Modulus'' is a genus of small sea snails, marine gastropod molluscs in the family Modulidae. Species Species within the genus ''Modulus'' include: * '' Modulus ambiguus'' Dautzenberg, 1910 * '' Modulus bayeri'' Petuch, 2001 * '' Modulus bermontianus'' Petuch, 1994 * '' Modulus cerodes'' A. Adams, 1851 * '' Modulus disculus'' (Philippi, 1846) * '' Modulus guernei'' Dautzenberg, 1900 * '' Modulus hennequini'' Petuch, 2013 * '' Modulus honkerorum'' Petuch, 2013 * '' Modulus kaicherae'' Petuch, 1987 * '' Modulus lindae'' Petuch, 1987 * '' Modulus modulus'' (Linnaeus, 1758) - the type species, known as the "button snail" * '' Modulus nodosus'' Macsotay & Campos, 2001 * '' Modulus pacei'' Petuch, 1987 * '' Modulus turbinoides'' (Locard, 1897) ;Taxa inquirenda: * ''Modulus duplicatus'' A. Adams, 1851 * ''Modulus morleti'' P. Fischer, 1882 * ''Modulus obliquus'' A. Adams, 1851 * ''Modulus obtusatus'' (Philippi, 1847) ;Species brought into synonymy: * ''Modulus calusa'' Petuch, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus (algebraic Number Theory)
In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. Definition Let ''K'' be a global field with ring of integers ''R''. A modulus is a formal product :\mathbf = \prod_ \mathbf^,\,\,\nu(\mathbf)\geq0 where p runs over all places of ''K'', finite or infinite, the exponents ν(p) are zero except for finitely many p. If ''K'' is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If ''K'' is a function field, ν(p) = 0 for all infinite places. In the function field case, a modulus is the same thing as an effective divisor, and in the number field case, a modulus can be considered as special form of Arakelov divisor. The notion of congruence can be extended to the setting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1) \chi(ab) = \chi(a)\chi(b); i.e. \chi is completely multiplicative. :2) \chi(a) \begin =0 &\text\; \gcd(a,m)>1\\ \ne 0&\text\;\gcd(a,m)=1. \end (gcd is the greatest common divisor) :3) \chi(a + m) = \chi(a); i.e. \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text\; \gcd(a,m)>1\\ 1 &\text\;\gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus Of Continuity
In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f(y), \leq\omega(, x-y, ), for all ''x'' and ''y'' in the domain of ''f''. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families. For instance, the modulus ω(''t'') := ''kt'' describes the k-Lipschitz functions, the moduli ω(''t'') := ''kt''α describe the Hölder continuity, the modulus ω(''t'') := ''kt''(, log ''t'', +1) describes the almost Lipschitz class, and so on. In general, the role of ω is to fix some explicit functional dependence of ε on δ in the (ε, δ) definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Modulus
In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves \Gamma is a measure of the size of \Gamma that is invariant under conformal mappings. More specifically, suppose that D is an open set in the complex plane and \Gamma is a collection of paths in D and f:D\to D' is a conformal mapping. Then the extremal length of \Gamma is equal to the extremal length of the image of \Gamma under f. One also works with the conformal modulus of \Gamma, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of \Gamma makes them useful tools in the study of conformal and quasi-conformal mappings. One also works with extremal length in dimensions greater than two and certain other metric spaces, but the following deals primarily with the two dimensional setting. Definition of extremal length To define extremal length, we need to first introduce several related quantities. Let ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and For example, the absolute value of 3 and the absolute value of −3 is The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. Terminology and notation In 1806, Jean-Robert Argand introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,Oxford English Dictionary, Draft Revision, June 2008 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli (physics)
In quantum field theory, the term moduli (or more properly moduli fields) is sometimes used to refer to scalar fields whose potential energy function has continuous families of global minima. Such potential functions frequently occur in supersymmetric systems. The term "modulus" is borrowed from mathematics, where it is used synonymously with "parameter". The word moduli (''Moduln'' in German) first appeared in 1857 in Bernhard Riemann's celebrated paper "Theorie der Abel'schen Functionen".Bernhard Riemann, Journal für die reine und angewandte Mathematik, vol. 54 (1857), pp. 101-155 Moduli spaces in quantum field theories In quantum field theories, the possible vacua are usually labeled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish. These vacuum expectation values can take any value for which the potential function is a minimum. Consequently, when the potential function ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
