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Ramanujan–Sato Series
In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only recently that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup \Gamma_0(n), while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators. Levels ''1–4A'' were given by Ramanujan (1914), level ''5'' by H. H. Chan and S. Cooper (2012), ''6A'' by Chan, Tanigawa, Yang, and Zudilin, ''6B'' by Sato (2002), ''6C'' by H. Chan, S. Chan, and Z. Liu (2004), ''6D'' by H. Chan and H. Verr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weber Modular Function
In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''1, and ''f''2,''f'', ''f''1 and ''f''2 are not Modular form#Modular functions, modular functions (per the Wikipedia definition), but every modular function is a rational function in ''f'', ''f''1 and ''f''2. Some authors use a non-equivalent definition of "modular functions". studied by Heinrich Martin Weber. Definition Let q = e^ where ''τ'' is an element of the upper half-plane. Then the Weber functions are :\begin \mathfrak(\tau) &= q^\prod_(1+q^) = \frac = e^\frac,\\ \mathfrak_1(\tau) &= q^\prod_(1-q^) = \frac,\\ \mathfrak_2(\tau) &= \sqrt2\, q^\prod_(1+q^)= \frac. \end These are also the definitions in Duke's paper ''"Continued Fractions and Modular Functions"''.https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ''Continued Fractions and Modular Functions'', W. Duke, pp 22-23 The function \eta(\tau) is the Dedekind eta function and (e^)^ should be interpreted as e^. The descriptions as ... [...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] |
Monstrous Moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. History In 1978, John McKay found that the first few ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube Root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of , denoted \sqrt[3]8, is , because , while the other cube roots of are -1+i\sqrt 3 and -1-i\sqrt 3. The three cube roots of are :3i, \quad \frac-\fraci, \quad \text \quad -\frac-\fraci. In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the ''principal cube root'', denoted with the radical sign \sqrt[3]. The cube root is the inverse function of the cube (algebra), cube function if considering only real numbers, but not if considering also complex numbers: although one has always \left(\sqrt[3]x\right)^3 =x, the cube of a nonzero number has more than one complex cube root and its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the ''radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sqrt. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Cubic
The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the semiconductors silicon and germanium, and silicon–germanium alloys in any proportion. There are also crystals, such as the high-temperature form of cristobalite, which have a similar structure, with one kind of atom (such as silicon in cristobalite) at the positions of carbon atoms in diamond but with another kind of atom (such as oxygen) halfway between those (see :Minerals in space group 227). Although often called the diamond lattice, this structure is not a lattice in the technical sense of this word used in mathematics. Crystallographic structure Diamond's cubic structure is in the Fdm space group (space group 227), which follows the face-centered cubic Bravais lattice. The lattice describes the repeat pattern; for diamond cubic cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apéry's Theorem
In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number :\zeta(3) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots = 1.2020569\ldots cannot be written as a fraction p/q where ''p'' and ''q'' are integers. The theorem is named after Roger Apéry. The special values of the Riemann zeta function at even integers 2n (n > 0) can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function's values are in general rational or not at the odd integers 2n+1 (n > 1) (though they are conjectured to be irrational). History Leonhard Euler proved that if ''n'' is a positive integer then :\frac + \frac + \frac + \frac + \cdots = \frac\pi^ for some rational number p/q. Specifically, writing the infinite series on the left as \zeta(2n), he showed :\zeta(2n) = (-1)^\frac where the B_n are the rational Bernoulli numbers. Once it was proved that \pi^n is always irrational, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Binomial Coefficient
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at ''n'' = 0 are: :, , , , , , 924, 3432, 12870, 48620, ...; Properties The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects. For example, n=2 represents ''AABB, ABAB, ABBA, BAAB, BABA, BBAA''. They also represent the number of combinations of ''A'' and ''B'' where there are never more ''B'' 's than ''A'' 's. For example, n=2 represents ''AAAA, AAAB, AABA, AABB, ABAA, ABAB''. The number of factors of ''2'' in \binom is equal to the number of ones in the binary representation of ''n'', so ''1'' is the only odd central binomial coefficient. Generating function The ordinary generating fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fischer Group Fi23
In the area of modern algebra known as group theory, the Fischer group ''Fi23'' is a sporadic simple group of order : 21831352711131723 : = 4089470473293004800 : ≈ 4. History ''Fi23'' is one of the 26 sporadic groups and is one of the three Fischer groups introduced by while investigating 3-transposition groups. The Schur multiplier and the outer automorphism group are both trivial. Representations The Fischer group Fi23 has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points The smallest faithful complex representation has dimension 782. The group has an irreducible representation of dimension 253 over the field with 3 elements. Generalized Monstrous Moonshine Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baby Monster Group
In the area of modern algebra known as group theory, the baby monster group ''B'' (or, more simply, the baby monster) is a sporadic simple group of order : 241313567211131719233147 : = 4154781481226426191177580544000000 : = 4,154,781,481,226,426,191,177,580,544,000,000 : ≈ 4. ''B'' is one of the 26 sporadic groups and has the second highest order of these, with the highest order being that of the monster group. The double cover of the baby monster is the centralizer of an element of order 2 in the monster group. The outer automorphism group is trivial and the Schur multiplier has order 2. History The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of -transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4. He investigated its properties and computed its character table. The first construction of the baby monst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
