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Symmetric holomorphic function
In mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the
Modular_lambda_function
Class of mathematical functions
{\displaystyle e_{3}} are related to the modular lambda function: λ ( τ ) = e 3 − e 2 e 1 − e 2 , τ = ω 2 ω 1 . {\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}
Weierstrass_elliptic_function
Topics referred to by the same term
zeta function Liouville function, λ(n) = (–1)Ω(n) Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p Modular lambda function, λ(τ)
Lambda_function
Mathematical functions
lemniscate sine can be used for the computation of values of the modular lambda function: ∏ k = 1 n sl ( 2 k − 1 2 n + 1 ϖ 2 ) = λ ( ( 2 n + 1 ) i ) 1 −
Lemniscate_elliptic_functions
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
Analytic function on the upper half-plane with a certain behavior under the modular group
In number theory and complex analysis, a modular form is a type of function of a complex number variable that possesses a high degree of symmetry, of a
Modular_form
functions Lemniscate elliptic functions Theta functions Neville theta functions Modular lambda function Closely related are the modular forms, which include J-invariant
List of mathematical functions
List_of_mathematical_functions
Function defined by a hypergeometric series
The j-invariant, a modular function, is a rational function in λ ( τ ) {\displaystyle \lambda (\tau )} . Incomplete beta functions Bx(p, q) are related
Hypergeometric_function
Zeta-like functions approximate arbitrary holomorphic functions
universality property has been shown for the Lerch zeta function L ( λ , α , s ) {\displaystyle L(\lambda ,\alpha ,s)} , at least when the parameter α is a
Zeta_function_universality
Function in mathematical number theory
3, 5, and 7. There are no primitive roots modulo 8. The Carmichael lambda function of a prime power can be expressed in terms of the Euler totient. Any
Carmichael_function
Class of periodic mathematical functions
this theory led to hyperelliptic functions and modular forms. A meromorphic function is called an elliptic function, if there are two R {\displaystyle
Elliptic_function
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
z = λ ( 1 + 5 i ) {\displaystyle z=\lambda (1+5i)} and λ {\displaystyle \lambda } is the modular lambda function. Khrushchev, Sergey (2008). Orthogonal
Lemniscate_constant
Type of algebraic equation
group) expressed in terms of complex analysis. Modular lambda function Ramanujan's lost notebook Weisstein, Eric W. "Modular Equation". MathWorld. v t e
Modular_equation
Special function defined by an integral
{Q} ^{+}} (where λ is the modular lambda function), then K(k) is expressible in closed form in terms of the gamma function. For example, r = 2, r = 3
Elliptic_integral
Special functions of several complex variables
derivation formulas see the articles Nome (mathematics) and Modular lambda function! For the theta functions these integrals are valid: ∫ 0 1 θ 2 ( x ) d x = ∑
Theta_function
Function whose domain is the positive integers
λ(n) be Liouville's function. Then | λ ( n ) | μ ( n ) = λ ( n ) | μ ( n ) | = μ ( n ) , {\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n)
Arithmetic_function
Meromorphic function on the complex plane
so-called complete L-function of f {\displaystyle \textstyle f} : Λ ( f , s ) = q ( f ) s / 2 γ ( f , s ) L ( f , s ) . {\displaystyle \Lambda (f,s)=q(f)^{s/2}\gamma
L-function
Mapping function
The term modular set function is equivalent to additive set function; see modularity below. Let μ {\displaystyle \mu } be a set function defined on
Sigma-additive_set_function
Algebraic curve in mathematics
{\left(\lambda ^{2}-\lambda +1\right)^{3}}{\lambda ^{2}\left(\lambda -1\right)^{2}}}} with j-invariant j(τ) and λ(τ) is sometimes called the modular lambda function
Elliptic_curve
Conformal mappings in complex analysis
i{\frac {K(1-z)}{K(z)}}} . This expression is the inverse of the modular lambda function. The Schwarz–Christoffel transformation gives the mapping from
Schwarz_triangle_function
Orientation-preserving mapping class group of the torus
reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry. The action of the modular group on the rational numbers
Modular_group
Linear operator acting on modular forms
{\textstyle \Lambda '} . Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous
Hecke_operator
Theorem about the range of an analytic function
original proof was based on properties of the modular lambda function, usually denoted by λ {\textstyle \lambda } , and which performs, using modern terminology
Picard_theorem
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
Correspondence between subfields and subgroups
F=\mathbb {Q} (j),} where j is the j-invariant written in terms of the modular lambda function: j = 256 ( 1 − λ ( 1 − λ ) ) 3 ( λ ( 1 − λ ) ) 2 = 256 ( 1 − λ
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
lattice Λ a theta function given by Θ Λ ( τ ) = ∑ x ∈ Λ e i π τ ‖ x ‖ 2 I m τ > 0. {\displaystyle \Theta _{\Lambda }(\tau )=\sum _{x\in \Lambda }e^{i\pi \tau
Theta_function_of_a_lattice
_{\lambda \in L^{g}}\exp(\pi iTr(\lambda T\lambda ^{t}))} where T is an element of the Siegel upper half plane of degree g. This is a Siegel modular form
Siegel_theta_series
Named function defined within a function
provide similar benefit. For example, a lambda function also allows for a function to be defined inside of a function (as well as elsewhere) and allows for
Nested_function
Type of mathematical function
{\displaystyle L(s,\chi )} and Λ ( s , χ ) {\displaystyle \Lambda (s,\chi )} are entire functions of s {\displaystyle s} . Again, this assumes that χ {\displaystyle
Dirichlet_L-function
Equation in Fourier analysis
{\displaystyle \mathbb {R} ^{n}/\Lambda } to an L 1 ( R n / Λ ) {\displaystyle L^{1}(\mathbb {R} ^{n}/\Lambda )} function having Fourier series f Λ ( x )
Poisson_summation_formula
Special function of two variables
analogue of a classical elliptic modular function. Note that E ( z , s ) {\displaystyle E(z,s)} is not a square-integrable function of z {\displaystyle z} with
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Number of integers coprime to and less than n
Pollack, P. (2023), "Two problems on the distribution of Carmichael's lambda function", Mathematika, 69 (4): 1195–1220, arXiv:2303.14043, doi:10.1112/mtk
Euler's_totient_function
36 mathematical problems stated in 1955
elliptic modular functions of level N {\displaystyle N} , and especially to decompose the Jacobian variety J {\displaystyle J} of this function field into
Taniyama's_problems
Uses of the constant
\infty }{\frac {1}{n}}\ln {\frac {16}{\lambda (ni)}}} (where λ {\displaystyle \lambda } is the modular lambda function) π = lim n → ∞ 24 n ln ( 2 1 / 4
List_of_formulae_involving_π
Matrix group
a congruence cover of the modular surface with eigenvalue λ {\displaystyle \lambda } then λ ⩾ 3 16 {\displaystyle \lambda \geqslant {\tfrac {3}{16}}}
Congruence_subgroup
Implementations Considered Harmful, or, Lambda: The Ultimate GOTO 1978: The Art of the Interpreter or, the Modularity Complex (Parts Zero, One, and Two) 1978:
History of the Scheme programming language
History_of_the_Scheme_programming_language
Concept in combinatorics
Ramanujan's congruences on the partition function and for representation theory of the symmetric group, especially modular representation theory. A partition
T-core_partition
Mathematical method in functional analysis
functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a
Tomita–Takesaki_theory
Theory of a class of elliptic curves
non-trivial endomorphisms rather than referring to a singular curve. The modular function j(τ) is algebraic on imaginary quadratic numbers τ: these are the only
Complex_multiplication
{\displaystyle \lambda (n)=\varphi (n),} where λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are respectively the Carmichael function and Euler's
Root_of_unity_modulo_n
Topics referred to by the same term
mock modular form of weight 1/2 Ramanujan theta function, f ( a , b ) {\displaystyle f(a,b)} Neville theta functions Riemann–Siegel theta function, θ (
Theta function (disambiguation)
Theta_function_(disambiguation)
Class of complex vector function
variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegel modular forms of genus 2. Examples with
Jacobi_form
Programming paradigm based on applying and composing functions
(since Java 8). The lambda calculus, developed in the 1930s by Alonzo Church, is a formal system of computation built from function application. In 1937
Functional_programming
L-function, Shimura showed that F ( z ) = ∑ n = 1 ∞ Λ ( n ) q n {\displaystyle F(z)=\sum _{n=1}^{\infty }\Lambda (n)q^{n}} is a holomorphic modular function
Shimura_correspondence
Algebraic stack in mathematics
Λ {\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda } into P 2 {\displaystyle \mathbb {P} ^{2}} from the Weierstrass P function pg 165. This isomorphic
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Type of symmetric polynomials in mathematics
{\displaystyle s_{\lambda }=\det(h_{\lambda _{i}+j-i})_{i,j=1}^{l(\lambda )}=\det \left[{\begin{matrix}h_{\lambda _{1}}&h_{\lambda _{1}+1}&\dots &h_{\lambda _{1}+n-1}\\h_{\lambda
Schur_polynomial
Differential operator
defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function. The eta invariant of self-adjoint
Eta_invariant
Theorem in number theory
Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms. The modulus of the Euler function (see
Pentagonal_number_theorem
Way of defining a lattice in the complex plane
This type of lattice is the underlying object with which elliptic functions and modular forms are defined. A fundamental pair of periods is a pair of complex
Fundamental_pair_of_periods
t(x))a = t(a)) with a modular equivalence relation based on a congruency. The λ-algebra describing the algebraic structure of the lambda-calculus is found
Computable_topology
Inconsistent pure type systems related to Girard's paradox
◻ . ( ( k → k ) → k → k ) . {\displaystyle \lambda k^{\square }.\ \lambda \alpha ^{k\to k}.\ \lambda \beta ^{k}.\ \alpha (\alpha \,\beta )\ :\ \Pi k:\square
System_U
Function from sets to numbers
{\mathcal {F}}.} Every finitely additive function on a field of sets is modular. In geometry, a set function valued in some abelian semigroup that possess
Set_function
Concept in mathematical group theory
with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful
Character_theory
Dialect of Lisp
Steele and Gerald Jay Sussman, via a series of memos now known as the Lambda Papers. It was the first dialect of Lisp to choose lexical scope and the
Scheme_(programming_language)
Infinite series in mathematical analysis
}a_{n}e^{-\lambda _{n}s},} where a n {\displaystyle a_{n}} , s {\displaystyle s} are complex numbers and { λ n } {\displaystyle \{\lambda _{n}\}} is a
General_Dirichlet_series
Periodic set of points
is a fundamental domain of the modular group, contain one complex number for each 2D lattice Λ {\displaystyle \Lambda } up to scaling and rotation. The
Lattice_(group)
Representation theory
( x ) {\displaystyle \lambda '(kx)=\Delta _{AN}(x)^{1/2}\lambda (x)} for k in K and x in AN, where ΔAN is the modular function of AN. Two different characters
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Mathematical theory of data types
New function terms may be constructed using lambda expressions, and are called lambda terms. These terms are also defined inductively: a lambda term
Type_theory
Programming language family
doing a function application: we execute the anonymous function by passing to it the value 5. Named functions are created by storing a lambda expression
Lisp_(programming_language)
Symbols for constants, special functions
mode in a waveguide the gamma function, a generalization of the factorial the upper incomplete gamma function the modular group, the group of fractional
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Two-dimensional conformal field theory
to the modular invariance of the torus one-point function. Due to remarkable identities of conformal blocks and structure constants, this modular invariance
Liouville_field_theory
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
\left|\operatorname {Re} (z)\right|<{\frac {1}{2}},|z|>1\right\}} (see Modular form). A function f : H → C {\displaystyle f:{\mathcal {H}}\to \mathbb {C} } is
Maass_wave_form
Type of Dirichlet series associated to number field extensions
, ρ , L / K ) {\displaystyle \Lambda (s,\rho ,L/K)=\prod _{v}L_{v}(s,\rho ,L/K)} Then the "completed" Artin L-function satisfies the following functional
Artin_L-function
Complex numbers with non-negative imaginary part
{\displaystyle y>0} . It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by y
Upper_half-plane
Set-to-real map with diminishing returns
f(T)\leq f(S)} . Examples of monotone submodular functions include: Linear (Modular) functions Any function of the form f ( S ) = ∑ i ∈ S w i {\displaystyle
Submodular_set_function
Ability of a computing system to simulate Turing machines
). Springer. ISBN 978-3-642-37801-0. "Announcing LAMBDA: Turn Excel formulas into custom functions". TECHCOMMUNITY.MICROSOFT.COM. 3 December 2020. Retrieved
Turing_completeness
Addison-Wesley; 2nd ed (1974) ISBN 0-201-00288-4. Apostol, T.M. (1997), Modular Functions and Dirichlet Series in Number Theory (2nd ed.), New York: Springer-Verlag
Laplace–Stieltjes_transform
Type of mathematical group
the modular surface, and it has been verified for some small groups. Selberg himself proved the lower bound λ 1 ⩾ 1 16 , {\displaystyle \lambda _{1}\geqslant
Arithmetic_Fuchsian_group
Data type in type theory
2023-09-13. Hofmann, Martin (1995). "A simple model for quotient types". Typed Lambda Calculi and Applications. Lecture Notes in Computer Science. Vol. 902. Berlin
Quotient_type
Programming language
including functions, which are first-class objects. Thus, the following constructs function max x y; if x > y then x else y close end; and vars max; lambda x
POP-2
Adaptive filter algorithm for digital signal processing
{\displaystyle C(\mathbf {w} _{n})=\sum _{i=0}^{n}\lambda ^{n-i}e^{2}(i)} where 0 < λ ≤ 1 {\displaystyle 0<\lambda \leq 1} is the "forgetting factor" which gives
Recursive least squares filter
Recursive_least_squares_filter
Algorithm for public key cryptography
existence of the following modular multiplicative inverse: μ = ( L ( g λ mod n 2 ) ) − 1 mod n {\displaystyle \mu =(L(g^{\lambda }{\bmod {n}}^{2}))^{-1}{\bmod
Paillier_cryptosystem
transcendental number theory concerning the exponential function have analogues involving the modular function j. Writing q = e2πiτ for the nome and j(τ) = J(q)
Four_exponentials_conjecture
Mathematical structure in differential geometry
{\displaystyle \textstyle {\rm {div}}_{\lambda }(X)={\frac {{\mathcal {L}}_{X}\lambda }{\lambda }}} . The modular vector field of an orientable Poisson
Poisson_manifold
Rational function of the form (az + b)/(cz + d)
obtain the modular group PSL(2, Z), a discrete subgroup of PSL(2, R) important in the study of lattices in the complex plane, elliptic functions and elliptic
Möbius_transformation
Λ d ρ ) {\displaystyle \Psi ^{k}(\rho )=N_{k}(\Lambda ^{1}\rho ,\Lambda ^{2}\rho ,\ldots ,\Lambda ^{d}\rho )\ } where the Λiρ are the exterior powers
Representation_ring
Methods of safely sharing general data
{\sqrt {2}}\lambda \,\!} ). Now in our case we define the output function of A {\displaystyle {\mathcal {A}}\,\!} as a real valued function (called as
Differential_privacy
Transformation of a mathematical sequence
binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined
Binomial_transform
Nonlinear differential operator used to study conformal mappings
particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and
Schwarzian_derivative
Mathematical function
modular inversion: The function λ {\displaystyle \lambda } , defined by λ ( τ ) = θ 2 ( τ ) 4 θ 3 ( τ ) 4 , {\displaystyle \lambda (\tau )={\frac {\theta
Jacobi_elliptic_functions
Third letter of the Greek alphabet
In Archaic Greece, the shape of gamma was closer to a classical lambda (Λ), while lambda retained the Phoenician L-shape (𐌋). Letters that arose from
Gamma
Natural number
programming languages. In lambda calculus and computability theory, natural numbers are represented by Church encoding as functions, where the Church numeral
1
2023 edition of the C++ programming language standard
nullary lambda expressions attributes on lambda expressions constexpr changes: non-literal variables, labels, and gotos in constexpr functions allowing
C++23
Cryptographic attack on the RSA system
encryption exponent e and λ(N) also must be relatively prime so that there is a modular inverse. The factorization of N and the private key d are kept secret,
Wiener's_attack
be a continuous function on G. If G and π {\displaystyle \pi } are algebraic, it would be a regular function on G. modular The modular representation theory
Glossary of representation theory
Glossary_of_representation_theory
Technique in functional programming
(define (make-weak-memoizer proc) (let ((cache (make-weak-table equal?))) (lambda args (let ((x (weak-table-ref cache args))) (if (bwp-object? x) (let ((r
Hash_consing
the lambda calculus were expressed using variable substitution in which the values of parameters were substituted into the body of an invoked lambda expression
History_of_the_Actor_model
Statistics models class
j λ j S j / ϕ {\displaystyle S_{\lambda }=\sum _{j}\lambda _{j}S_{j}/\phi } . Since the penalty allows some functions through unpenalized (straight lines
Generalized_additive_model
Matrix-valued random variable
{Z}}_{N}}}e^{-H_{N}(\lambda )}\mathrm {d} \lambda ,\qquad H_{N}(\lambda )=-\sum \limits _{j\neq k}\ln |\lambda _{j}-\lambda _{k}|+N\sum \limits _{j=1}^{N}Q(\lambda _{j})
Random_matrix
Representation theory
q}\right)\prod _{n,m=1}^{\infty }(1-p^{n}q^{m})^{c_{nm}}} for the elliptic modular function j. Peterson gave a recursion formula for the multiplicities mult(β)
Weyl_character_formula
Decomposition of an integer as a sum of positive integers
} Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For
Integer_partition
Theory in supersymmetric gauge theory
^{\dagger }]^{2}-i\lambda \sigma ^{\mu }D_{\mu }{\bar {\lambda }}-i{\bar {\psi }}{\bar {\sigma }}^{\mu }D_{\mu }\psi -i{\sqrt {2}}[\lambda ,\psi ]\phi ^{\dagger
Seiberg–Witten_theory
{1}{z^{2}}}+\sum _{\lambda \in \Lambda \smallsetminus \{0\}}\!\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)} The function ℘ ( z ) {\displaystyle
Dixon_elliptic_functions
Representation of the control state of a computer program
function. ; ; In this case, the function argument assigns that ; continuation to the variable the-continuation. ; (call/cc (lambda (k) (set! the-continuation
Continuation
(D)f=\lambda _{D}f,} i.e. f is a simultaneous eigenfunction of the operators π(D). If ψ is a zonal spherical function, then, regarded as a function on G/K
Zonal_spherical_function
software was modular based on the universal model of computation and the lambda calculus. Six Church instructions hide the details of a named function application
Plessey_System_250
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
Algorithm for public-key cryptography
one-way function, possibly because the difficulty of factoring was not well-studied at the time. Moreover, like Diffie-Hellman, RSA is based on modular exponentiation
RSA_cryptosystem
Family of hypersurfaces in algebraic geometry
context of local zeta-functions, such families have been shown to have relationships with mirror symmetry and extensions of the modularity theorem. The Dwork
Dwork_family
Highly viscous liquid exhibiting a kind of semi-solid behavior
f_{\text{gel}}(\lambda _{1},\lambda _{2},\lambda _{3})=f_{\text{net}}(\lambda _{1},\lambda _{2},\lambda _{3})+f_{\text{mix}}(\lambda _{1},\lambda _{2},\lambda _{3})
Gel
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION
MODULAR LAMBDA-FUNCTION