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Type of algebraic equation
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problems. That is, given a number of functions
Modular_equation
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Symmetric holomorphic function
(\tau )=m.} The modular equation of degree p {\displaystyle p} (where p {\displaystyle p} is a prime number) is an algebraic equation in λ ( p τ ) {\displaystyle
Modular_lambda_function
17th-century conjecture proved by Andrew Wiles in 1994
solutions existed to Fermat's equation; or the modularity theorem was false. This meant that a proof of the modularity theorem would automatically prove
Fermat's_Last_Theorem
Monster and modular connection
moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular the j function. The initial numerical observation
Monstrous_moonshine
Plane algebraic curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ))
Classical_modular_curve
Analytic function on the upper half-plane with a certain behavior under the modular group
More precisely, a modular form is a holomorphic function on the complex upper half-plane that roughly satisfies a functional equation with respect to the
Modular_form
Type of lattice in mathematical order theory
condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally
Modular_lattice
Real root of the polynomial x^5+x+a
modular equation with n = 5 {\displaystyle n=5} may be related to the Bring–Jerrard quintic by the following function of the six roots of the modular
Bring_radical
Continued fraction closely related to the Rogers–Ramanujan identities
denominator are polynomial invariants of the icosahedron. Using the modular equation between R ( q ) {\displaystyle R(q)} and R ( q 5 ) {\displaystyle R(q^{5})}
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
Orientation-preserving mapping class group of the torus
solutions to Pell's equation. In both cases, the numbers can be arranged to form a semigroup subset of the modular group. The modular group can be shown
Modular_group
Computation modulo a fixed integer
non-linear modular arithmetic equations is NP-complete. Boolean ring Circular buffer Division (mathematics) Finite field Legendre symbol Modular exponentiation
Modular_arithmetic
Modular function in mathematics
In mathematics, the j-invariant or j function is a modular function of weight zero for the special linear group SL ( 2 , Z ) {\displaystyle \operatorname
J-invariant
Relates rational elliptic curves to modular forms
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way
Modularity_theorem
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
Algebraic variety
X0(N) can be defined over Q. The equations defining modular curves are the best-known examples of modular equations. The "best models" can be very different
Modular_curve
Series related to Ramanujan's pi formulas
and Moonshine". arXiv:1211.6563 [math.NT]. Ramanujan, S. (1914). "Modular equations and approximations to π". Quart. J. Math. 45. Oxford: 350–372. Chan;
Ramanujan–Sato_series
Mathematical equation
upper half-plane and Γ {\displaystyle \Gamma } is the modular group. The Picard–Fuchs equation is then d 2 y d j 2 + 1 j d y d j + 31 j − 4 144 j 2 (
Picard–Fuchs_equation
Weil pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function Modular group Supersingular primes Fermat curve Bézout's
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Indian mathematician (1887–1920)
as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions
Srinivasa_Ramanujan
Otherwise, Euler's equation may refer to a non-differential equation, as in these three cases: Euler–Lotka equation, a characteristic equation employed in mathematical
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Factorisation algorithm
Coppersmith's attack. Coppersmith's approach is a reduction of solving modular polynomial equations to solving polynomials over the integers. Let F ( x ) = x n +
Coppersmith_method
On unit fractions adding to 4/n
get a positive integer solution to the equation. Nevertheless, modular arithmetic, and identities based on modular arithmetic, have proven a very important
Erdős–Straus_conjecture
Plane curve
Ramanujan gave two close approximations for the circumference in §16 of "Modular Equations and Approximations to π {\displaystyle \pi } "; they are C π ≈ 3 (
Ellipse
Concept in modular arithmetic
linear congruence is a modular congruence of the form a x ≡ b ( mod m ) . {\displaystyle ax\equiv b{\pmod {m}}.} Unlike linear equations over the reals, linear
Modular multiplicative inverse
Modular_multiplicative_inverse
Collection of Srinivasa Ramanujan's discoveries in mathematics
about q-series and mock theta functions, about a third are about modular equations and singular moduli, and the remaining formulas are mainly about integrals
Ramanujan's_lost_notebook
English mathematician (1886–1965)
many more modular equations than all of his mathematical predecessors combined. Watson provided proofs for most of Ramanujan's modular equations. Bruce C
G._N._Watson
1995 publication in mathematics
constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x {\displaystyle x} and
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to
Timeline of numerals and arithmetic
Timeline_of_numerals_and_arithmetic
Special modular forms
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function
Hilbert_modular_form
Varying methods used to calculate pi
Archived from the original (PDF) on 6 July 2011. Ramanujan, S. (1914). "Modular equations and approximations to π". Quarterly Journal of Mathematics. 45: 350–372
Approximations_of_pi
Relates theta constants to the branch points of a hyperelliptic curve
general method. Camille Jordan showed that any algebraic equation may be solved by use of modular functions. This was accomplished by Thomae in 1870. Thomae
Thomae's_formula
Number, approximately 3.14
mathematical depth and rapid convergence. One of his formulae, based on modular equations, is 1 π = 2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) k ! 4 396
Pi
Exponentation in modular arithmetic
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography
Modular_exponentiation
strip. 1858 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions. 1859 – Bernhard Riemann formulates the Riemann
Timeline_of_mathematics
Mathematician
O.L. Atkin and Swinnerton-Dyer: if a modular form f(τ) is not modular for some congruence subgroup of the modular group, then the Fourier coefficients
Yunqing_Tang
Finding values for variables that make an equation true
success. If the solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic, for example), or can be
Equation_solving
Type of Diophantine equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x 2 − n y 2 = 1 , {\displaystyle x^{2}-ny^{2}=1,} where
Pell's_equation
Algorithm for fast modular multiplication
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing
Montgomery modular multiplication
Montgomery_modular_multiplication
N-th root of the product of n numbers
Archived from the original on 2011-03-02. Ramanujan, S. (1914). "Modular equations and approximations to π" (PDF). Quarterly Journal of Mathematics.
Geometric_mean
Class of mathematical functions
The modular discriminant Δ {\displaystyle \Delta } is defined as the discriminant of the characteristic polynomial of the differential equation ℘ ′ 2
Weierstrass_elliptic_function
Mathematical conjecture
the modular setting special points are the singular moduli and special varieties are irreducible components of varieties defined by modular equations. Given
Zilber–Pink_conjecture
Mathematical concept
A modular elliptic curve is an elliptic curve E that admits a parametrization X0(N) → E by a modular curve. This is not the same as a modular curve that
Modular_elliptic_curve
Problem of constructing equal-area shapes
ISBN 0-387-96568-8. Reprinted as The Trisectors. Ramanujan, S. (1914). "Modular equations and approximations to π" (PDF). Quarterly Journal of Mathematics.
Squaring_the_circle
Trail in which only the first and last vertices are equal
1177/0263276412451161. S2CID 146875675. Veblen, Oswald (1912), "An Application of Modular Equations in Analysis Situs", Annals of Mathematics, Second Series, 14 (1):
Cycle_(graph_theory)
History of a branch of mathematics
now called Galois theory. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group
History_of_group_theory
Result concerning properties of Galois representations associated with modular forms
of the equation ap + bp = cp, makes it clear that one of a, b, c is even and hence so is N. By the Taniyama–Shimura conjecture, E is a modular elliptic
Ribet's_theorem
Equation whose unknown is a function
and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates
Functional_equation
mathematics, almost holomorphic modular forms, also called nearly holomorphic modular forms, are a generalization of modular forms that are polynomials in
Almost holomorphic modular form
Almost_holomorphic_modular_form
Function defined by a hypergeometric series
distribution Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Morita, Tohru (1996). "Use of the Gauss contiguous
Hypergeometric_function
Computational operation
determines which of the two consecutive quotients must be used to satisfy equation (1). In number theory, the positive remainder is always chosen, but in
Modulo
Branch of pure mathematics
{\displaystyle n} . Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra
Number_theory
Generalization of a magic square
into the hypercube: nHm = k=0Σn-1 LPk mk J.R.Hendricks often uses modular equation, conditions to make hypercubes of various quality can be found on http://www
Magic_hypercube
Polynomial function of degree 5
their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric
Quintic_function
Method for increasing reliability
In computing, triple modular redundancy, sometimes called triple-mode redundancy, (TMR) is a fault-tolerant form of N-modular redundancy, in which three
Triple_modular_redundancy
Algebraic curve in mathematics
simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough
Elliptic_curve
following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg
Frobenius solution to the hypergeometric equation
Frobenius_solution_to_the_hypergeometric_equation
Unsolved problem in mathematics
conjecture is a conjecture concerning the growth rate of coefficients of modular forms and more generally, automorphic forms. The name of the conjecture
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Indian mathematician and professor (born 1972)
of his Ph.D. thesis was Contributions to Ramanujan's Schlafli-type Modular Equations, Class Invariants, Theta-functions, and Continued Fractions. Following
Nayandeep_Deka_Baruah
Canadian mathematician (1953–2020)
Borwein, J. M.; Borwein, P. B.; Bailey, D. H. (1989). "Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of
Peter_Borwein
Listing all imaginary quadratic fields with a given class number
the case n = 1 was first discussed by Kurt Heegner, using modular forms and modular equations to show that no further such field could exist. This work
Class_number_problem
Formulation of classical mechanics
This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Newton's laws and the concept of forces are
Lagrangian_mechanics
Complex-differentiable part of a Maass wave function
mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight
Mock_modular_form
British mathematician who proved Fermat's Last Theorem
satisfy the equation an + bn = cn for any integer value of n greater than 2) could be proven as a corollary of a limited form of the modularity theorem (unproven
Andrew_Wiles
Cycles in a graph that generate all cycles
doi:10.4064/fm-28-1-22-32. Veblen, Oswald (1912), "An application of modular equations in analysis situs", Annals of Mathematics, Second Series, 14 (1):
Cycle_basis
conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular group Congruence subgroup
List_of_number_theory_topics
Method for computing the relation of two integers with their greatest common divisor
are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly
Extended_Euclidean_algorithm
Nonlinear differential operator used to study conformal mappings
theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory
Schwarzian_derivative
Elliptic curve associated with a Fermat triple
{\displaystyle \alpha +\beta =\gamma } . This relates properties of solutions of equations to elliptic curves. This curve was popularized in its application to Fermat’s
Frey_curve
Type of Diophantine equation in number theory
Siksek, S. (2006). "Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation". Compositio Mathematica. 142:
Ramanujan–Nagell_equation
Mathematical function
-{\frac {1}{2}}\right).} Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character
Dedekind_eta_function
All even-degree subgraphs of a graph
27–30, ISBN 9780486419756. Veblen, Oswald (1912), "An application of modular equations in analysis situs", Annals of Mathematics, Second Series, 14 (1):
Cycle_space
Roots of multiple multivariate polynomials
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials
System of polynomial equations
System_of_polynomial_equations
Scottish mathematician (1951–2016)
Borwein, J. M.; Borwein, P. B.; Bailey, D. H. (1989). "Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of
Jonathan_Borwein
proofs to G. H. Hardy. 1914 — Srinivasa Aaiyangar Ramanujan publishes Modular Equations and Approximations to π. 1910s — Srinivasa Aaiyangar Ramanujan develops
Timeline_of_number_theory
Special mathematical function
the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used
Nome_(mathematics)
Measure of network community structure
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters
Modularity_(networks)
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer
Topological_modular_forms
German mathematician
Modulargleichungen der elliptischen Functionen (On the transformation of the modular equations of the elliptic functions) was supervised by Leo Königsberger. In
Martin_Krause_(mathematician)
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
Analytic function in mathematics
a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime
Riemann_zeta_function
French mathematician (1856–1941)
his introduction of a kind of symmetry group for a linear differential equation. He also introduced the Picard group in the theory of algebraic surfaces
Émile_Picard
Function studied by Ramanujan
is the Dedekind eta function, Δ ( z ) {\displaystyle \Delta (z)} is the modular discriminant, and q = e 2 π i z {\displaystyle q=e^{2\pi iz}} with I m
Ramanujan_tau_function
Clustering and community detection algorithm
cluster}}\\0&{\text{otherwise}}\end{cases}}\end{aligned}}} Based on the above equation, the modularity of a community c can be calculated as: Q c = 1 2 m ∑ i ∑ j A i
Louvain_method
Mathematical conjecture
equations in several variables involving addition, multiplication, and some special meromorphic transcendental functions (e.g. exponential or modular
Existential closedness conjecture
Existential_closedness_conjecture
Conformal field theory on a 2D spacetime
conformal bootstrap equations. While the Ward identities are linear equations for correlation functions, the conformal bootstrap equations depend non-linearly
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Mathematical map for transforming elliptic modular forms
mathematics, the Doi–Naganuma lifting is a map from elliptic modular forms to Hilbert modular forms of a real quadratic field, introduced by Doi & Naganuma
Doi–Naganuma_lifting
Relating two numbers and their greatest common divisor
Bézout's equation and was used by Bachet to solve the problems appearing on pages 199 ff. See also: Maarten Bullynck (February 2009). "Modular arithmetic
Bézout's_identity
Algebraic curve
by the Fermat equation: X n + Y n = Z n . {\displaystyle X^{n}+Y^{n}=Z^{n}.\ } Therefore, in terms of the affine plane its equation is: x n + y n =
Fermat_curve
Generalization of Fermat's Last Theorem and of Catalan's conjecture,
Bennett (2006). "The equation x2n + y2n = z5" (PDF). Journal de Théorie des Nombres de Bordeaux. 18: 315–321. Andrew Wiles (1995). "Modular Elliptic Curves
Fermat–Catalan_conjecture
In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber. Let q = e 2 π i τ {\displaystyle
Weber_modular_function
function to prove the functional equation. Erich Hecke, and later Hans Maass, applied the same Mellin transform method to modular forms on the upper half-plane
Rankin–Selberg_method
Solving integer equations from all modular solutions
Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo
Hasse_principle
Integer side lengths of a right triangle
the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. There
Pythagorean_triple
Unproved conjecture in mathematics
Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Quadratic imaginary number fields with unique factorisation
various similar proofs using modular functions. (Heegner's paper dealt mainly with the congruent number problem, also using modular functions.) Alan Baker's
Stark–Heegner_theorem
Branch of algebraic geometry
geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields,
Arithmetic_geometry
American mathematician
conceptually significant in the number theory of elliptic curves. The equation describes a modular curve whose solutions characterize the one remaining unsolved
Jennifer_Balakrishnan
Type of mathematical expression
mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems
Polynomial
Conjecture in number theory
ISSN 0025-5718. Dahmen, Sander (2011). "A Refined Modular Approach to the Diophantie Equation x2 + y2n = z3". International Journal of Number Theory
Beal_conjecture
MODULAR EQUATION
MODULAR EQUATION
Boy/Male
Hindu, Indian
Popular
Girl/Female
Bengali, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Famous; Popular
Boy/Male
Muslim
Accepted, Popular
Boy/Male
Muslim/Islamic
Popular
Boy/Male
Arabic
Popular; Famous
Boy/Male
Tamil
Parishrut | பரீஷà¯à®°à¯à®¤
Popular, Renown
Parishrut | பரீஷà¯à®°à¯à®¤
Boy/Male
Indian
Accepted, Popular
Girl/Female
Hindu, Indian
Popular Around
Boy/Male
Muslim
Familiar, Popular
Boy/Male
Muslim
Accepted, Popular
Boy/Male
Arabic, Muslim
Familiar; Popular
Girl/Female
Biblical Greek
Popular.
Girl/Female
Indian
Popular
Boy/Male
Arabic, Muslim
Famous; Popular
Boy/Male
Indian
Love
Boy/Male
Arabic, Hindu, Indian, Muslim
Accepted; Popular
Girl/Female
Indian
Popular
Girl/Female
Tamil
Popular
Girl/Female
Greek
Popular.
Boy/Male
Hindu
Popular, Renown
MODULAR EQUATION
MODULAR EQUATION
Biblical
standard; miracle
Girl/Female
Assamese, Hindu, Indian, Traditional
Blue Lotus Eyed
Boy/Male
Tamil
Light, Lustrous, Power
Female
English
Variant spelling of Middle English Gytha, GITHA means "strife, war."
Boy/Male
English
Strong; gifted ruler. Blend of Jer- and Derrick.
Girl/Female
Irish
Joy.
Boy/Male
American, Australian, French, German, Greek, Latin, Shakespearean, Spanish
Light; Illumination; From Lucanus; A Region of Southern Italy; Spanish Form of Luke Light
Girl/Female
Hindu
Related to shy
Boy/Male
Tamil
Unerring
Surname or Lastname
English (mainly Yorkshire)
English (mainly Yorkshire) : occupational name for an archer, Middle English schut(te), schit(te) (from Old English scytta, a primary derivative of scēotan ‘to shoot’).Americanized spelling of German Schutt.
MODULAR EQUATION
MODULAR EQUATION
MODULAR EQUATION
MODULAR EQUATION
MODULAR EQUATION
a.
Prevailing among the people; epidemic; as, a popular disease.
n.
A popular or jocular name for a drinking vessel.
pl.
of Morula
a.
Popular; famous.
a.
Of, pertaining to, or in the form of, a nodule or knot.
v. t.
To vary or inflect in a natural, customary, or musical manner; as, the organs of speech modulate the voice in reading or speaking.
imp. & p. p.
of Modulate
a.
Given to jesting; jocose; as, a jocular person.
a.
Of or pertaining to mode, modulation, module, or modius; as, modular arrangement; modular accent; modular measure.
a.
Relating or belonging to an ovule; as, an ovular growth.
a.
Depending on, or perceived by, the eye; received by actual sight; personally seeing or having seen; as, ocular proof.
p. pr. & vb. n.
of Modulate
a.
Having power to grind; grinding; as, the molar teeth; also, of or pertaining to the molar teeth.
n.
Any one of the teeth back of the incisors and canines. The molar which replace the deciduous or milk teeth are designated as premolars, and those which are not preceded by deciduous teeth are sometimes called true molars. See Tooth.
pl.
of Modulus
a.
Adapted to the means of the common people; possessed or obtainable by the many; hence, cheap; common; ordinary; inferior; as, popular prices; popular amusements.
a.
Of or pertaining to the common people, or to the whole body of the people, as distinguished from a select portion; as, the popular voice; popular elections.
n.
The size of some one part, as the diameter of semi-diameter of the base of a shaft, taken as a unit of measure by which the proportions of the other parts of the composition are regulated. Generally, for columns, the semi-diameter is taken, and divided into a certain number of parts, called minutes (see Minute), though often the diameter is taken, and any dimension is said to be so many modules and minutes in height, breadth, or projection.
n.
To model; also, to modulate.
a.
Beloved or approved by the people; pleasing to people in general, or to many people; as, a popular preacher; a popular law; a popular administration.