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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
{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
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
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
Minkowksi 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
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
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
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)
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
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
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
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
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)
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
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
{\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
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
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
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
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)
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
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
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
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
Late 2026 cargo resupply mission to the International Space Station
Arrays. The mission also carried a payload designed by Langston University. LambdaVision-1 investigates the production of artificial retinal tissue in microgravity
SpaceX_CRS-35
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)
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
Equal-area pseudocylindrical global map projection
3 + A 1 θ , {\displaystyle {\begin{aligned}x&={\frac {2{\sqrt {3}}}{3}}\lambda \cdot {\frac {\cos {\theta }}{9A_{4}\theta ^{8}+7A_{3}\theta ^{6}+3A_{2}\theta
Equal_Earth_projection
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
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
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
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)
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
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
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
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
Construction 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
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
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
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
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
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
Last letter of the Greek alphabet
including 0 (sometimes written ω0) A primitive root of unity, like the complex cube roots of 1 The Wright Omega function A generic differential form In number
Omega
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
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
Forms which matter can take
helium-4, the most common isotope of helium, forms a superfluid below the lambda temperature of 2.17 K (−270.98 °C; −455.76 °F). The state is described as
State_of_matter
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
Modular function in mathematics
\left\lbrace {\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda }\right\rbrace
J-invariant
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
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
Mathematical function in general imaging
/\lambda )W(\theta )]} W ( θ ) = − z θ 2 / 2 + C s θ 4 / 4 {\displaystyle W(\theta )=-z\theta ^{2}/2+C_{s}\theta ^{4}/4} λ {\displaystyle \lambda } =
Contrast_transfer_function
Overview of and topical guide to algorithms
of effective computation Turing machine — model formalizing computation Lambda calculus — formal system used in the study of computation Von Neumann architecture
Outline_of_algorithms
Fourier transform of a real-space lattice, important in solid-state physics
wave is defined by its wavelength λ {\displaystyle \lambda } , where k λ = 2 π {\displaystyle k\lambda =2\pi } ; hence the corresponding wavenumber in reciprocal
Reciprocal_lattice
Mode of failure for which a material fractures into layers
factor λ {\displaystyle \lambda } , such that P Q = S 2 S 1 = λ {\displaystyle {\frac {P}{Q}}={\frac {S_{2}}{S_{1}}}=\lambda } , where S 1 {\displaystyle
Delamination
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
Voice assistants developed by Amazon
Most skills run code almost entirely in the cloud, using Amazon's AWS Lambda service. In April 2018, Amazon launched Blueprints, a tool for individuals
Amazon_Alexa
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
Pseudoazimuthal compromise map projection
x=2\operatorname {azeq} _{x}\left({\frac {\lambda }{2}},\varphi \right),\qquad y=\operatorname {azeq} _{y}\left({\frac {\lambda }{2}},\varphi \right)} where azeqx
Aitoff_projection
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)
Full-size luxury sedan (2015–present)
applied, and Layered Architecture has been applied to the Crest Grille. The cube-shaped projection light incorporates a white light, a daytime driving light
Genesis_G90
Observation in physical cosmology
− 1 = 1 + v c 1 − v c − 1 ≈ v c . {\displaystyle z={\frac {\lambda _{\text{o}}}{\lambda _{\text{e}}}}-1={\sqrt {\frac {1+{\frac {v}{c}}}{1-{\frac {v}{c}}}}}-1\approx
Hubble's_law
Real-valued function
computing it over cubes close to each other in position and scale. Precisely, if Q {\displaystyle Q} and R {\displaystyle R} are dyadic cubes such that their
Bounded_mean_oscillation
Topologically stable solution of a partial differential equation
solitons[clarification needed] of the Wess–Zumino–Witten model. Topological defects in lambda transition universality class[clarification needed] systems including:
Topological_defect
Particle detector for Cherenkov radiation
{\frac {dN_{\text{Cherenkov}}}{d\lambda }}\approx {\frac {2\pi \alpha }{\lambda ^{2}}}{\bigg (}1-{\frac {1}{\beta ^{2}n(\lambda )^{2}}}{\bigg )}(\Delta x)}
Cherenkov_detector
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
Theoretical subdimensional particle
Majorana checkerboard model, the stacked Kagome X-cube model, the hyperkagome X-cube model, and more. The X-cube model is constructed on a cubic lattice, with
Fracton (subdimensional particle)
Fracton_(subdimensional_particle)
{\displaystyle \lambda } there exists a cardinal μ ( λ ) {\displaystyle \mu (\lambda )} such that if an AEC K with LS(K) ≤ λ {\displaystyle {}\leq \lambda } is categorical
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
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