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Framework in lambda calculus
In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different
Lambda_cube
Type whose definition depends on a value
to types, for example). The lambda cube is generalized further by pure type systems. The system λ Π {\displaystyle \lambda \Pi } of pure first order dependent
Dependent_type
Formalism in computer science
(LF), a pure lambda calculus with dependent types. Based on work by Berardi on pure type systems, Henk Barendregt proposed the lambda cube to systematize
Typed_lambda_calculus
Mathematical-logic system
typed lambda calculus with types as first-class values These formal systems are extensions of lambda calculus that are not in the lambda cube: Binary
Lambda_calculus
Typed lambda calculus
(also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism
System_F
Form of typed lambda calculus
cube of constructive logics akin to the lambda cube (these specifications are non-dependent). A modification of this cube was later called the L-cube
Pure_type_system
Rocq and Lean. The lambda cube was not a new type theory but a categorization of existing type theories. The eight corners of the cube included some existing
History_of_type_theory
Concept in Aristotelian logic
to identify the allowed logical conversions from one type to another. Lambda cube Logical hexagon Square of opposition Triangle of opposition Hans Reichenbach
Logical_cube
Type theory created by Thierry Coquand
higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible
Calculus_of_constructions
Mathematical theory of data types
combinatory logic others defined in the lambda cube (also known as pure type systems) others under the name typed lambda calculus Homotopy type theory explores
Type_theory
Topics referred to by the same term
(physics), theory organizing subatomic baryons and mesons into octets Lambda cube Octal, base-8 number system Octant (solid geometry) Octave (poetry) Octetra
Octet
Concept in philosophical logic
Lambda cube Logical cube Square of opposition Triangle of opposition N-opposition theory logical hexagon Moretti, Alessio. "The oppositional cube (or
Logical_hexagon
Special forms of a typed lambda calculus
Sørensen, Morten Heine; Urzyczyn, Paweł (2006). "Pure type systems and the lambda cube". Lectures on the Curry–Howard isomorphism. Elsevier. doi:10.1016/S0049-237X(06)80015-7
System_U
Concept in Aristotelian logic
triangle of contraries and Sir William Hamilton’s subcontraries. Lambda cube Logical cube Logical hexagon Square of opposition Bazhanov, Valentin (January
Triangle_of_opposition
Curry–Howard isomorphism Calculus of constructions Constructivist analysis Lambda cube System F Introduction to topos theory LF (logical framework) Computability
List of mathematical logic topics
List_of_mathematical_logic_topics
Type of logic diagram
{\displaystyle s(A)=\emptyset } ). Boole's syllogistic Free logic Lambda cube Logical cube Logical hexagon Semiotic square Triangle of opposition Per The
Square_of_opposition
Basis of generic programming
frequently studied impredicative typed λ-calculi are based on those of the lambda cube, especially System F. Leivant's notion of rank can be generalized to
Parametric_polymorphism
Kind of proof calculus
polymorphism have been considered in the literature, the most famous being the lambda cube of Henk Barendregt. The intersection of logic and type theory is a vast
Natural_deduction
Branch of type theory
each variable in a lambda abstraction, turning them into Π types. And they extended the lambda cube to what they call the f-cube, which has with FSD-encoded
Intersection_type_discipline
{\displaystyle \lambda \leq \kappa } , the space I λ {\displaystyle I^{\lambda }} is embeddable in I κ {\displaystyle I^{\kappa }} . The Tychonoff cube I κ {\displaystyle
Tychonoff_cube
Way to represent data types in the lambda calculus
science, Scott encoding is a way to represent algebraic data types in the lambda calculus, following their syntactic definition without regard whether they
Mogensen–Scott_encoding
Hypercube partition of Euclidean space
dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale
Dyadic_cubes
Approximation of a black body's spectral radiance
B T λ 4 , {\displaystyle B_{\lambda }(T)={\frac {2ck_{\text{B}}T}{\lambda ^{4}}},} where B λ {\displaystyle B_{\lambda }} is the spectral radiance (the
Rayleigh–Jeans_law
Family of map projections
{S}}(\lambda -\lambda _{0})\\y&={\frac {\sin \varphi }{\sqrt {S}}}\end{aligned}}} x = λ − λ 0 y = sin φ {\displaystyle {\begin{aligned}x&=\lambda -\lambda
Cylindrical equal-area projection
Cylindrical_equal-area_projection
Computer program for the Boolean satisfiability problem
"cubes". A cube can also be seen as a conjunction of a subset of variables of the original formula. In conjunction with the formula, each of the cubes
SAT_solver
{3}{2}}{\frac {\lambda _{x}^{4}+\lambda _{y}^{4}+\lambda _{z}^{4}}{(\lambda _{x}^{2}+\lambda _{y}^{2}+\lambda _{z}^{2})^{2}}}-{\frac {1}{2}}}
Gyration_tensor
Square root: Yields a number whose square is the given one. Cube root: Yields a number whose cube is the given one. Transcendental functions are functions
List of mathematical functions
List_of_mathematical_functions
Feature of a typed formal language that builds new types from old ones
defined by recursively composing type constructors. For example, simply typed lambda calculus can be seen as a language with a single non-basic type constructor—the
Type_constructor
Conic conformal map projection
{\begin{aligned}x&=\rho \sin \left[n\left(\lambda -\lambda _{0}\right)\right]\\y&=\rho _{0}-\rho \cos \left[n\left(\lambda -\lambda _{0}\right)\right]\end{aligned}}}
Lambert conformal conic projection
Lambert_conformal_conic_projection
Cylindrical conformal map projection
{\displaystyle x(\lambda )=\int _{\lambda _{0}}^{\lambda }R\,du,\qquad y(\varphi )=\int _{0}^{\varphi }R\sec v\,dv.} The value λ 0 {\displaystyle \lambda _{0}}
Mercator_projection
and only if λ ( n ) = φ ( n ) , {\displaystyle \lambda (n)=\varphi (n),} where λ {\displaystyle \lambda } and φ {\displaystyle \varphi } are respectively
Root_of_unity_modulo_n
Mathematical folklore
measures are modified or omitted. The Lebesgue measure λ {\displaystyle \lambda } on the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is locally
Infinite-dimensional Lebesgue measure
Infinite-dimensional_Lebesgue_measure
Chebyshev center Chebyshev constants Chebyshev cube root Chebyshev distance Chebyshev equation Chebyshev's equioscillation theorem Chebyshev filter, a
List of things named after Pafnuty Chebyshev
List_of_things_named_after_Pafnuty_Chebyshev
Correspondence between subfields and subgroups
{\displaystyle G=\left\{\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda \right\}\subset
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
Description in spectral theory
π ) − d ω d v o l ( Ω ) {\displaystyle \lim _{\lambda \rightarrow \infty }{\frac {N(\lambda )}{\lambda ^{d/2}}}=(2\pi )^{-d}\omega _{d}\mathrm {vol} (\Omega
Weyl_law
Device to deploy CubeSats into orbit from the International Space Station
Nanoracks CubeSat Deployer (NRCSD) is a device to deploy CubeSats into orbit from the International Space Station (ISS). In 2014, two CubeSat deployers
Nanoracks_CubeSat_Deployer
Method in physics
dependence of the heat capacity of solids, which is proportional to the cube of temperature – the Debye T 3 law. Similarly to the Einstein photoelectron
Debye_model
Pseudoazimuthal compromise map projection
{\begin{aligned}x&={\frac {1}{2}}\left(\lambda \cos \varphi _{1}+{\frac {2\cos \varphi \sin {\frac {\lambda }{2}}}{\operatorname {sinc} \alpha }}\right)
Winkel_tripel_projection
Type of map projection
y}{\partial \varphi }}\cdot {\frac {\partial x}{\partial \lambda }}-{\frac {\partial y}{\partial \lambda }}\cdot {\frac {\partial x}{\partial \varphi }}=s\cdot
Equal-area_projection
Table that displays the frequency of variables
association). Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement
Contingency_table
Open-source distributed analytics engine
Spark Cube engine - completed (v2.5) Connect more data sources (MySQL, Oracle, SparkSQL, etc.) - completed (v2.6) Real-time analytics with Lambda Architecture
Apache_Kylin
Set of vectors used to define coordinates
, b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda } is any real number. A simple basis of this
Basis_(linear_algebra)
Spectral density of light emitted by a black body
{\displaystyle \lambda } instead of per unit frequency: B λ ( λ , T ) = 2 h c 2 λ 5 1 exp ( h c λ k B T ) − 1 {\displaystyle B_{\lambda }(\lambda ,T)={\frac
Planck's_law
Relative deformation of a physical body
{\displaystyle \lambda ={\frac {l}{L}}} The extension ratio λ is related to the engineering strain e by e = λ − 1 {\displaystyle e=\lambda -1} This equation
Strain_(mechanics)
Characterization of distortion in map projections
}}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\[4pt]\sin \theta '&={\frac
Tissot's_indicatrix
Symbols for constants, special functions
of the compensation for the risk borne in investment the α-conversion in lambda calculus the independence number of a graph a placeholder for ordinal numbers
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Disproved conjecture in number theory
lambda (1-(a-3b)(a^{2}+3b^{2}))\\[2pt]x_{2}&=\lambda ((a+3b)(a^{2}+3b^{2})-1)\\[2pt]x_{3}&=\lambda ((a+3b)-(a^{2}+3b^{2})^{2})\\[2pt]x_{4}&=\lambda
Euler's sum of powers conjecture
Euler's_sum_of_powers_conjecture
Mathematical version of an order change
5 ) − 1 λ 6 = ( 23 ) {\displaystyle \lambda _{2}(13)\lambda _{2}((15)\lambda _{4})^{4}(\lambda _{5})^{-1}\lambda _{6}=(23)} ( 14325 ) − 1 {\displaystyle
Permutation
Function that takes one or more functions as an input or that outputs a function
Functor (disambiguation). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming
Higher-order_function
Area of mathematical analysis
{\displaystyle \lambda >0} , one selects intervals or cubes on which the average size of f {\displaystyle f} is larger than λ {\displaystyle \lambda } . The function
Harmonic_analysis
Adaptation of the standard Mercator projection
{\begin{aligned}x(\lambda ,\varphi )&={\frac {1}{2}}k_{0}a\ln \left[{\frac {1+\sin \lambda \cos \varphi }{1-\sin \lambda \cos \varphi }}\right],\\[5px]y(\lambda ,\varphi
Transverse Mercator projection
Transverse_Mercator_projection
Cylindrical equal-area map projection
{\displaystyle {\begin{aligned}x&={\frac {R\pi \lambda \cos 45^{\circ }}{180^{\circ }}}={\frac {R\pi \lambda }{180^{\circ }{\sqrt {2}}}}\\y&={\frac {R\sin
Gall–Peters_projection
Change in the shape or size of an object
internal deformation, the dimensionless change in shape of an infinitesimal cube of material relative to a reference configuration. Mechanical strains are
Deformation_(engineering)
Pseudocylindrical equal-area map projection
{2}{\sqrt {4\pi +\pi ^{2}}}}R\,(\lambda -\lambda _{0})(1+\cos \theta )\approx 0.422\,2382\,R\,(\lambda -\lambda _{0})(1+\cos \theta ),\\[8pt]y&=2{\sqrt
Eckert_IV_projection
Minkowsi sum of line segments
\Lambda \subset \mathbb {R} ^{d}} such that the union of all translates Z + λ {\displaystyle Z+\lambda } ( λ ∈ Λ {\displaystyle \lambda \in \Lambda }
Zonotope
In computer programming, an anonymous function (function literal, lambda function, or block) is a function definition that is not bound to an identifier
Examples of anonymous functions
Examples_of_anonymous_functions
Conic equal-area map projection
{\displaystyle {R}} is the radius, λ {\displaystyle \lambda } is the longitude, λ 0 {\displaystyle \lambda _{0}} the reference longitude, φ {\displaystyle
Albers_projection
Geometric space with six dimensions
polytopes, of which there are only three in six dimensions: the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes, constructed
Six-dimensional_space
Energy driving the accelerated expansion of the universe
universe. It also slows the rate of structure formation. Assuming that the lambda-CDM model of cosmology is correct, dark energy dominates the universe, contributing
Dark_energy
Pseudocylindrical compromise map projection
) , y = 1.3523 R Y , {\displaystyle {\begin{aligned}x&=0.8487\,RX(\lambda -\lambda _{0}),\\y&=1.3523\,RY,\end{aligned}}} where R is the radius of the
Robinson_projection
One of two different regular graphs with 16 vertices
10-regular graph with 80 edges. The 80-edge graph is the dimension-5 halved cube graph; it was called the Clebsch graph by Seidel (1968) because of its relation
Clebsch_graph
Azimuthal equidistant map projection
_{0}\cos \varphi \cos \left(\lambda -\lambda _{0}\right)\\\tan \theta &={\frac {\cos \varphi \sin \left(\lambda -\lambda _{0}\right)}{\cos \varphi _{0}\sin
Azimuthal equidistant projection
Azimuthal_equidistant_projection
Electromagnetic radiation generated by the thermal motion of particles
λ {\displaystyle I_{\lambda }} as follows, E λ ( λ ) = π I λ ( λ ) {\displaystyle E_{\lambda }(\lambda )=\pi I_{\lambda }(\lambda )} where both spectral
Thermal_radiation
{\displaystyle x^{3}+y^{3}+z^{3}-\lambda xyz=0.} Each curve in the pencil is determined by the parameter λ {\displaystyle \lambda } and consists of the points
Hesse_pencil
General-purpose programming language
called a generator expression. Anonymous functions are implemented using lambda expressions; however, there may be only one expression in each body. Conditional
Python_(programming_language)
Equal-area pseudocylindrical global map projection
θ 3 + A 1 θ , {\displaystyle {\begin{aligned}x&={\frac {2{\sqrt {3}}\,\lambda \cos {\theta }}{3\,(9\,A_{4}\,\theta ^{8}+7\,A_{3}\,\theta ^{6}+3\,A_{2}\
Equal_Earth_projection
Vast empty spaces between filaments with few or no galaxies
the context of the standard general-relativistic cosmological model, the Lambda-CDM model, in which the Universe is on average expanding, the spatial curvature
Void_(astronomy)
Group of symmetries of an n-dimensional hypercube
mathematical groups that arise as the group of symmetries of the square, the cube, and their higher-dimensional counterparts (the hypercubes), as well as the
Hyperoctahedral_group
Generalization of volume to non-integer number of dimensions
Lebesgue measure λ d {\displaystyle \lambda _{d}} , which is normalized so that the Lebesgue measure of the unit cube [0,1]d is 1. In fact, for any Borel
Hausdorff_measure
Family of probability distributions
,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )[(1+s/\theta )^{\alpha }-1]&\quad p\neq 1,2,\\-\lambda \log(1+s/\theta )&\quad p=2,\\\lambda e^{\theta
Tweedie_distribution
Equations in physical cosmology
although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors
Friedmann_equations
Regions of an electromagnetic field
decreases by the inverse-distance squared, the reactive field by an inverse-cube law, resulting in a diminished power in the parts of the electric field by
Near_and_far_field
Measure of material deformation perpendicular to loading
^{\text{Hencky}}&=-{\frac {\ln \lambda _{\text{trans}}}{\ln \lambda _{\text{axial}}}}\\[6pt]\nu ^{\text{Biot}}&={\frac {1-\lambda _{\text{trans}}}{\lambda _{\text{axial}}-1}}\\[6pt]\nu
Poisson's_ratio
Computer backgammon program (1992)
net trained by a form of temporal-difference learning, specifically TD-Lambda. It explored strategies that humans had not pursued and led to advances
TD-Gammon
Unit of volume
no longer exact. A litre is a cubic decimetre, which is the volume of a cube 10 centimetres × 10 centimetres × 10 centimetres (1 L ≡ 1 dm3 ≡ 1000 cm3)
Litre
Curve from a cone intersecting a plane
{\displaystyle {\frac {{\tilde {x}}^{2}}{-S/(\lambda _{1}^{2}\lambda _{2})}}+{\frac {{\tilde {y}}^{2}}{-S/(\lambda _{1}\lambda _{2}^{2})}}=1,} or equivalently x ~
Conic_section
computable from the equation's data. The numbers λ , μ , ν {\displaystyle \lambda ,\mu ,\nu } are (up to permutations, sign changes and addition of ( ℓ ,
Schwarz's_list
Symbolic description of a mathematical object
the lambda expression, was introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. The lambda operators (lambda abstraction
Expression_(mathematics)
Cylindrical equidistant map projection
) cos φ 1 y = R ( φ − φ 0 ) {\displaystyle {\begin{aligned}x&=R(\lambda -\lambda _{0})\cos \varphi _{1}\\y&=R(\varphi -\varphi _{0})\end{aligned}}}
Equirectangular_projection
Geographic coordinate specifying north-south position
latitude ( ϕ {\displaystyle \phi } ) and longitude ( λ {\displaystyle \lambda } ) are defined on a spherical model. The graticule spacing is 10 degrees
Latitude
Unit of measure used in weather radar
D m a x N 0 e − Λ D D 6 d D {\displaystyle Z=\int _{0}^{Dmax}N_{0}e^{-\Lambda D}D^{6}\mathrm {d} D} As rain droplets have a diameter on the order of 1
DBZ_(meteorology)
Pseudocylindrical compromise map projection
{3\lambda }{2}}{\sqrt {{\frac {1}{3}}-\left({\frac {\varphi }{\pi }}\right)^{2}}}\\y&=\varphi \end{aligned}}} where λ {\displaystyle \lambda } is the
Kavrayskiy_VII_projection
Graph defined from a mathematical group
\Lambda _{i}(S)} . Then the set of eigenvalues of Γ ( G , S ) {\displaystyle \Gamma (G,S)} is exactly ⋃ i Λ i ( S ) , {\textstyle \bigcup _{i}\Lambda _{i}(S)
Cayley_graph
Theorem in geometry
{\textstyle \mu (\lambda A+(1-\lambda )B)\geq (\mu (\lambda A)^{1/n}+\mu ((1-\lambda )B)^{1/n})^{n}=(\lambda \mu (A)^{1/n}+(1-\lambda )\mu (B)^{1/n})^{n}
Brunn–Minkowski_theorem
Interferometric technique
interferometers) which consisted of lenses, beam splitter, mirrors, and corner cube, the possibility of creating a much simpler and more compact system was investigated
Self-mixing_interferometry
Optical filter in fluorescence microscopy
commonly packaged with an emission filter and a dichroic beam splitter in a cube so that the group is inserted together into the microscope. The dichroic
Excitation_filter
Pseudocylindrical equal-area map projection
{\displaystyle {\begin{aligned}x&=R{\frac {2{\sqrt {2}}}{\pi }}\left(\lambda -\lambda _{0}\right)\cos \theta ,\\[5px]y&=R{\sqrt {2}}\sin \theta ,\end{aligned}}}
Mollweide_projection
Lowest possible energy of a quantum system or field
{k} \lambda }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)\right]&=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}\\[10px]\left[a_{\mathbf
Zero-point_energy
Last letter of the Greek alphabet
{\displaystyle \omega _{0}} ) A primitive root of unity, like the complex cube roots of 1 The Wright Omega function A generic differential form In number
Omega
Mercator variant map projection
{\begin{aligned}x&=\left\lfloor {\frac {1}{2\pi }}\cdot 2^{\text{zoom level}}\left(\pi +\lambda \right)\right\rfloor {\text{ pixels}}\\[5pt]y&=\left\lfloor {\frac {1}{2\pi
Web_Mercator_projection
Topic in group theory
}),h)\cdot (\lambda ,\omega '):=(a_{h(\omega ')}\lambda ,h\omega ').} The primitive wreath product action on Λ Ω {\displaystyle \Lambda ^{\Omega }} :
Wreath_product
Topics referred to by the same term
formerly called Masala TV Masala (surname) Massachusetts Area South Asian Lambda Association, an LGBT group for people of South Asian ethnicity Marsala (disambiguation)
Masala
Frequency change of a wave for observer relative to its source
{mob}}}{\lambda _{\rm {c}}}}\cos \phi \cos \theta } where v mob {\displaystyle v_{\text{mob}}} is the speed of the mobile station, λ c {\displaystyle \lambda _{\rm
Doppler_effect
Retroazimuthal compromise map projection
{\begin{aligned}x&=\lambda -\lambda _{0}\\y&={\frac {\lambda -\lambda _{0}}{\sin \left(\lambda -\lambda _{0}\right)}}{\Big (}\sin \varphi \cos \left(\lambda -\lambda _{0}\right)-\tan
Craig retroazimuthal projection
Craig_retroazimuthal_projection
Theorem concerning ratios of line segments
\lambda \cdot ({\vec {a}}+{\vec {b}})=\lambda \cdot {\vec {a}}+\lambda \cdot {\vec {b}}} and ‖ λ a → ‖ = | λ | ⋅ ‖ a → ‖ {\displaystyle \|\lambda {\vec
Intercept_theorem
Precision-guided bomb
Tian Ge (Chinese: 天戈; pinyin: tiān gē; lit. 'Lambda Boötis'), abbreviated as TG or GB, is a series of precision-guided munitions (PGM) developed by Harbin
TG_PGB
Lines not in the same plane
not coplanar. If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three
Skew_lines
1887 investigation of the speed of light
\lambda _{1}-\Delta \lambda _{2}}{\lambda }}\approx {\frac {2Lv^{2}}{\lambda c^{2}}}.} Note the difference between Δ λ {\displaystyle \Delta \lambda }
Michelson–Morley_experiment
Nuclear weapon component
=\lambda _{f}^{core}/v_{n}} and d c o r e = λ f c o r e λ t c o r e 3 ( − α + ν − 1 ) {\displaystyle d_{core}={\sqrt {\frac {\lambda _{f}^{core}\lambda
Tamper_(nuclear_weapon)
Pseudocylindrical equal-area map projection
( λ − λ 0 ) cos φ y = φ {\displaystyle {\begin{aligned}x&=\left(\lambda -\lambda _{0}\right)\cos \varphi \\y&=\varphi \,\end{aligned}}} where φ {\displaystyle
Sinusoidal_projection
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