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In combinatorial geometry, the Weyl distance function is a function that behaves in some ways like the distance function of a metric space, but instead
Weyl_distance_function
formula Weyl distance function Weyl equation, a relativistic wave equation Weyl expansion Weyl fermion Weyl gauge Weyl gravity Weyl group Length of a Weyl group
List of things named after Hermann Weyl
List_of_things_named_after_Hermann_Weyl
Function that "converges" to periodicity
Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian
Almost_periodic_function
Mathematical structure
decomposition Generalized polygon Mostow rigidity Coxeter complex Weyl distance function Tits 1974. Tits 1981. Bruhat & Tits 1972. Ku 1998. Garrett 1997
Building_(mathematics)
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
Historical note. Weyl (1913) proved the existence of the harmonic function u by giving a direct proof of Dirichlet's principle. In Weyl (1940), he presented
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Philosophical argument
In philosophy, Weyl's tile argument, introduced by Hermann Weyl in 1949, is an argument against the notion that physical space is "discrete", as if composed
Weyl's_tile_argument
operators. The real functions correspond to the Hermitian operators. The function f ( ξ ) {\displaystyle f(\xi )} is called Weyl's symbol of operator f
Method of quantum characteristics
Method_of_quantum_characteristics
Representation theory
ideas of Hermann Weyl from the spectral theory of ordinary differential equations, Harish-Chandra introduced his celebrated c-function c(λ) to describe
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Type of vector space in math
y)} is a distance function meaning firstly that it is symmetric in x {\displaystyle x} and y , {\displaystyle y,} secondly that the distance between x
Hilbert_space
Mathematical equation
{\displaystyle \Delta } denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function u ∈ L l o c 1 ( Ω ) {\displaystyle u\in L_{\mathrm
Weyl's lemma (Laplace equation)
Weyl's_lemma_(Laplace_equation)
Integral expressing the amount of overlap of one function as it is shifted over another
a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle f*g} , as the
Convolution
Special mathematical functions defined on the surface of a sphere
University Press, 2012). Li, Feifei; Braun, Carol; Garg, Anupam (2013), "The Weyl-Wigner-Moyal Formalism for Spin", Europhysics Letters, 102 (6) 60006, arXiv:1210
Spherical_harmonics
Gravity theories that are invariant under Weyl transformations
{\displaystyle \Omega (x)} is a function on spacetime. The simplest theory in this category has the square of the Weyl tensor as the Lagrangian S = ∫ d
Conformal_gravity
Greek letter
(probably to better distinguish from elements w {\displaystyle w} of the Weyl group, than the usual notation ω {\displaystyle \omega } ). The lemniscate
Pi_(letter)
Lattice in 8-dimensional space with special properties
Weyl group contains a subgroup of order 128·8! consisting of all permutations of the coordinates and all even sign changes. This subgroup is the Weyl
E8_lattice
{\displaystyle \textstyle 1-{2GM}/{r}} . Weyl–Schouten theorem Conformal geometry Yamabe problem Ray D'Inverno. "6.13 The Weyl tensor". Introducing Einstein's
Conformally_flat_manifold
Mathematical function, in linear algebra
linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition
Linear_map
Method for specifying point positions
coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on
Coordinate_system
Function that encodes the dependence of a coupling parameter on the energy scale
theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, β(g), encodes the dependence of a coupling parameter, g,
Beta_function_(physics)
Process of calculating the causal factors that produced a set of observations
Hermann Weyl and published in 1911, describing the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. Today known as Weyl's law, it
Inverse_problem
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Object detection system using radio waves
signal for longer ranges (about 100 km (62 miles)). Distance may also be measured as a function of time. The radar mile is the time it takes for a radar
Radar
Type of mathematical functions
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space C n {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Type of wave propagating in 3 dimensions
free dictionary. Plane-wave expansion Rectilinear propagation Wave equation Weyl expansion Brekhovskikh, L. (1980). Waves in Layered Media (2 ed.). New York:
Plane_wave
Region in spacetime from which nothing can escape
comoving distance from which light emitted in the past could reach the observer at a given time. For events that occur beyond that distance, light has
Event_horizon
Invariance under a mathematical reflection
Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. Weyl, Hermann (1982) [1952]. Symmetry. Princeton: Princeton University Press.
Reflection_symmetry
Method of analysis applied to problems wave propagation
homogeneous spaces derived from the Coxeter group (so, for example, the Weyl groups of simple Lie algebras). The traditional statement of Huygens's principle
Huygens–Fresnel_principle
Set of philosophical problems
number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According to Hermann Weyl, the assumption
Zeno's_paradoxes
Measured time difference as explained by relativity theory
t'={\frac {2D}{c}}} The length of the half path can be calculated as a function of known quantities as: D = ( 1 2 v Δ t ′ ) 2 + L 2 {\displaystyle D={\sqrt
Time_dilation
Generalization of a positive-definite matrix
ingredients, namely kernels K {\displaystyle K} and distances d {\displaystyle d} . Here by a distance function between each pair of elements of some set X {\displaystyle
Positive-definite_kernel
electromagnetic field was proposed in 1918 by Weyl. In this work Weyl coins the term gauge theory. Weyl, in an attempt to generalize the geometrical ideas
History of classical field theory
History_of_classical_field_theory
Change in wavelength of light
Sitter in 1917 that redshift would be correlated with distance. In 1929 Hubble combined his distance estimates with redshift data from Slipher's reports
Redshift
German mathematician (1862–1943)
remained there for the rest of his life. Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John
David_Hilbert
Lie algebra with imaginary simple roots
finite-dimensional semisimple Lie algebras. In particular they have a Weyl group, Weyl character formula, Cartan subalgebra, roots, weights, and so on. A
Generalized_Kac–Moody_algebra
Procedure of coping with redundant degrees of freedom in physical field theories
be beneficial in specific situations have appeared in the literature. The Weyl gauge (also known as the Hamiltonian or temporal gauge) is an incomplete
Gauge_fixing
Mathematical model combining space and time
positive whole number, then wave impulses become distorted. In 1922, Hermann Weyl claimed that Maxwell's theory of electromagnetism can be expressed in terms
Spacetime
Algebraic operation on coordinate vectors
a weight function (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions u ( x ) {\displaystyle
Dot_product
Second-order partial differential equation
every test function φ ∈ C c ∞ ( Ω ) {\displaystyle \varphi \in C_{c}^{\infty }(\Omega )} . By Weyl's lemma, every weakly harmonic function is in fact
Laplace's_equation
Interpretation of quantum mechanics
universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum measurements
Many-worlds_interpretation
Differential Equations. New York: Springer Verlag. ISBN 978-0-387-95000-6. Weyl, Hermann (1989) [1952]. Symmetry. Princeton Science Library. Princeton University
Symmetry_in_mathematics
Inequality in mathematical physics
of large coupling, that is for potentials β V {\displaystyle \beta V} the Weyl asymptotics lim β → ∞ 1 β γ + n 2 t r ( − Δ + β V ) − γ = L γ , n c l ∫ R
Lieb–Thirring_inequality
Exact solution for the Einstein field equations
"monopole point source" of general relativity. Weyl multipole moments arise from treating a certain metric function (formally corresponding to Newtonian gravitational
Kerr_metric
Topological space that locally resembles Euclidean space
circle. By the implicit function theorem, every submanifold of Euclidean space is locally the graph of a function. Hermann Weyl gave an intrinsic definition
Manifold
Quantum mechanical statistic
density can be described by the Weyl properties of space. In Riemann flat space, the Bohm potential is shown to equal the Weyl curvature. According to Castro
Quantum_potential
Mathematical description of spacetime used in relativity
Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and
Minkowski_spacetime
Albert Einstein's hypothetical situations to argue scientific points
wave function for the pair links the positions of the particles as well as their linear momenta. The figure depicts the spreading of the wave function from
Einstein's thought experiments
Einstein's_thought_experiments
Dixmier conjecture: any endomorphism of a Weyl algebra is an automorphism. Fröberg conjecture on the Hilbert functions of a set of forms. Fujita conjecture
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
24-dimensional repeating pattern of points
elliptic functions. The vector ( 0 , 1 , 2 , 3 , … , 22 , 23 , 24 ) {\displaystyle (0,1,2,3,\dots ,22,23,24)} in this construction is really the Weyl vector
Leech_lattice
Integral transform and linear operator
lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation. Schur improved Hilbert's results about the discrete
Hilbert_transform
Smooth manifold with an inner product on each tangent space
Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel
Riemannian_manifold
Differential operator in mathematics
{\displaystyle H^{2}} . In particular, harmonic functions are smooth, and in fact real analytic. More generally, Weyl's lemma states that if u {\displaystyle u}
Laplace_operator
Discrete (i.e., incremental) version of infinitesimal calculus
axis and the piece-wise constant curve, which is the total distance traveled. Suppose a function is defined at the mid-points of n {\displaystyle n} intervals
Discrete_calculus
Theoretical framework in physics
additional generators Qα, called supercharges, which themselves transform as Weyl fermions. The symmetry group generated by all these generators is known as
Quantum_field_theory
Physics phenomenon
recognition of what would later be called entanglement. That same year, Hermann Weyl observed in his textbook on group theory and quantum mechanics that quantum
Quantum_entanglement
Tensor that describes the 4D geometry of spacetime
coordinates, Lemaître–Tolman metric, Peres metric, Rindler coordinates, Weyl–Lewis–Papapetrou coordinates, Gödel metric. Some of them are without the
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Force resulting from the quantisation of a field
the strength of the force falls off rapidly with distance, it is measurable only when the distance between the objects is small. This force becomes so
Casimir_effect
Thought experiment in special relativity
between two events, which was also pointed out in the textbooks of Hermann Weyl (1918) or Wolfgang Pauli (1921). Regarding the § Role of acceleration, Laue
Twin_paradox
Speed of electromagnetic waves in vacuum
299792458 m/s. This fixed value is used to define the metre as exactly the distance that light travels in vacuum in 1⁄299792458 of a second. The second is
Speed_of_light
Physical theory describing classical fields
mathematicians and physicists like Albert Einstein, Theodor Kaluza, Hermann Weyl, Arthur Eddington, Gustav Mie and Ernst Reichenbacher. Early attempts to
Classical_field_theory
Differential calculus on function spaces
control Direct method in calculus of variations Noether's theorem De Donder–Weyl theory Variational Bayesian methods Chaplygin problem Nehari manifold Hu–Washizu
Calculus_of_variations
Introductory article
Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. After the development of quantum mechanics, Weyl, Vladimir
Introduction_to_gauge_theory
Hypothetical topological feature of spacetime
1928, German mathematician, philosopher and theoretical physicist Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis
Wormhole
Description of physical properties at the atomic and subatomic scale
wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known
Quantum_mechanics
Structure defining distance on a manifold
space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).[citation needed] While the notion of
Metric_tensor
Fundamental mechanical principles
start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function between two states of
Action_principles
Periodic spatial graph
of the Laves graph has been studied, showing the existence of Dirac and Weyl points in this structure. The structure of the Laves graph, and of gyroid
Laves_graph
Set of mathematical concepts in quantum gravity
mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length. Each theory of quantum gravity
Quantum_geometry
Theorem about projections of coadjoint orbits of a connected compact Lie group
coordinates of Λ. Let K be a connected compact Lie group with maximal torus T and Weyl group W = NK(T)/T. Let their Lie algebras be k {\displaystyle {\mathfrak
Kostant's_convexity_theorem
Integral transform
Bruijn, N. G. (1973). "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence", Nieuw Arch. Wiskd., III. Ser
Linear canonical transformation
Linear_canonical_transformation
Euclidean space without distance and angles
a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms. An affine subspace
Affine_space
Theorem in differential geometry
each Busemann function is in fact (weakly) a harmonic function. Weyl's lemma implies the infinite differentiability of the Busemann functions. Then, the
Splitting_theorem
Attraction of masses and energy
product of their masses and inversely proportional to the square of the distance between them. Scientists are looking for a theory that describes gravity
Gravity
Mathematics of smooth surfaces
then given by a distance function d(p,q), namely the infimum of the lengths of piecewise smooth paths between p and q. This distance is realised locally
Differential geometry of surfaces
Differential_geometry_of_surfaces
Formulation of quantum mechanics
{\hat {q}}{\hat {p}}} , p ^ q ^ {\displaystyle {\hat {p}}{\hat {q}}} , or Weyl ordering prescription; conversely, q p {\displaystyle qp} can be translated
Path_integral_formulation
Construct allowing differentiation of tangent vector fields of manifolds
by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity)
Affine_connection
Branch of mathematics
particular importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight
Differential_geometry
German mathematician (1875–1941)
product Schur test Schur's inequality Schur's theorem Schur-convex function Schur–Weyl duality Lehmer–Schur algorithm Schur's property for normed spaces
Issai_Schur
Standard that defines a specific range of colors
linear space (vector space)... became widely known around 1920, when Hermann Weyl and others published formal definitions. In fact, such a definition had been
Color_space
Straight path on a curved surface or a Riemannian manifold
spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces;
Geodesic
Basic framework of mathematics
Birkhauser (1992). Weyl 1927 Comments on Hilbert's second lecture on the foundations of mathematics in van Heijenoort 1967:484. Although Weyl the intuitionist
Foundations_of_mathematics
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
with mass and distance from the axis. For a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis
Moment_of_inertia
Algebraic structure with addition and multiplication
| n | p {\displaystyle |m/n|_{p}=|m|_{p}/|n|_{p}} ). It defines a distance function on Q {\displaystyle \mathbb {Q} } and the completion of Q {\displaystyle
Ring_(mathematics)
Property of a thermodynamic system
scientist and engineer William Rankine in 1850 with the names thermodynamic function and heat-potential. In 1865, German physicist Rudolf Clausius, one of the
Entropy
American theoretical physicist
symmetry breaking near the Grand Unification Scale. However, recent work on Weyl invariant theories offers a better rationale for a natural inflation scenario
Christopher_T._Hill
Hungarian and American mathematician and physicist (1903–1957)
Rockefeller Foundation to study mathematics under David Hilbert. Hermann Weyl remembers how in the winter of 1926–1927 von Neumann, Emmy Noether, and he
John_von_Neumann
Array of numbers
function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, such as y, by "considering" the function for
Matrix_(mathematics)
Adages and sayings named after a person
description. Weyl law, in mathematics, describes the asymptotic behavior of eigenvalues of the Laplace-Beltrami operator. Named for Hermann Weyl. The Wiedemann–Franz
List_of_eponymous_laws
the Virasoro algebra Weyl 1. Named after Hermann Weyl 2. A Weyl transformation is a rescaling of the world-sheet metric. 3. Weyl spinor, an element of
Glossary_of_string_theory
Conservative political initiative in the United States
goals as aligned with their Agenda 47 program. Trump later attempted to distance himself from the plan. After he won the 2024 election, he nominated several
Project_2025
Algorithm for generating pseudo-randomized numbers
period, but is obviously non-random. Other values of c coprime to m produce a Weyl sequence, which is better distributed but still obviously non-random. Historically
Linear_congruential_generator
Algorithm that generates an approximation of a random number sequence
elaborate generators. A recent innovation is to combine the middle square with a Weyl sequence. This method produces high-quality output through a long period
Pseudorandom_number_generator
Lorentzian manifold where curvature vanishes at large distances
large families of solutions which are asymptotically flat, such as the AF Weyl metrics and their rotating generalizations, the AF Ernst vacuums (the family
Asymptotically_flat_spacetime
Expression that may be integrated over a region
(noncommutative) algebra of differential operators they generate is the Weyl algebra and is a noncommutative ("quantum") deformation of the symmetric
Differential_form
Quantum field theory
theory) to quantum mechanics. Weyl named the relevant symmetry in Noether's theorem the "gauge symmetry", by analogy to distance standardization in railroad
Yang–Mills_theory
Measure of relativistic velocity
motion, each frame being associated with distance and time coordinates. Using the inverse hyperbolic function artanh, the rapidity w corresponding to velocity
Rapidity
Theorem in physics
as developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl, the state of a quantum mechanical system is a vector | ψ ⟩ {\displaystyle
Bell's_theorem
French mathematician (1869–1951)
spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor Geometry and topology of Lie groups Riemannian geometry Symmetric
Élie_Cartan
Set of spacetime events, light-connected to a given event
cones so that they are all parallel is reflected in the non-vanishing of the Weyl tensor. Absolute future Absolute past Hyperbolic partial differential equation
Light_cone
Mathematical structures that allow quantum mechanics to be explained
Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Chinese-American mathematician (born 1949)
their analysis was an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem.[CY77b] Outside
Shing-Tung_Yau
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
Girl/Female
Muslim/Islamic
Some distance
Girl/Female
Muslim/Islamic
Some distance
Boy/Male
Indian, Modern
Full of Light
Girl/Female
Muslim
Some distance
Girl/Female
Arabic, Muslim, Sindhi
Some Distance
Girl/Female
Muslim
Some distance
Girl/Female
American, British, English, French, Greek
Fate; Certain Fortune; The Mythological Greek God of Fate
Female
French
French form of Latin Constantia, CUSTANCE means "steadfast."Â
Boy/Male
Hindu
Existance
Surname or Lastname
English
English : topographic name for someone who lived near a spring or stream, Middle English well(e) (Old English well(a)).German : from a short form of the personal names Wallo, Walilo.German : nickname from Middle High German wël ‘round’.
Boy/Male
Indian
Distance
Girl/Female
Muslim
Distinct
Girl/Female
Indian
Some distance
Girl/Female
Muslim
Some distance
Boy/Male
Arabic
Distance
Girl/Female
Arabic, Muslim, Sindhi
Some Distance
Girl/Female
Indian
Some distance
Girl/Female
Muslim
Some distance
Girl/Female
English French
Certain fortune; fate. The mythological Greek god of fate.
Surname or Lastname
English
English : variant spelling of Way.Dutch : variant of Wei.
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
Girl/Female
Arabic, Muslim, Zoroastrian
Ruby; Pearl
Boy/Male
Australian, French, German, Italian, Latin, Portuguese, Spanish, Swiss
Gift from God; Given; Given by God; Abbreviation of Donatello Gift from God
Boy/Male
English
Cut in two.
Girl/Female
Indian
Dear to the gods, Dear to the Goddess
Boy/Male
Muslim
Jupiter. Planet.
Girl/Female
Indian
Brave, Fearless, Intrepid
Boy/Male
Tamil
Ritambhara | ரீதாமà¯à®ªà®¾à®°à®¾
Religious
Boy/Male
British, Indian, Malaysian, Tamil
Rock; Strong
Girl/Female
Indian
Exalted
Boy/Male
Latin Arthurian Legend
Destroyer.
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
WEYL DISTANCE-FUNCTION
v. t.
To cause to appear as if at a distance; to make seem remote.
v. t.
To place at a distance or remotely.
n.
The interval between two notes; as, the distance of a fourth or seventh.
v. t.
To mention as a case or example; to refer to; to cite; as, to instance a fact.
imp. & p. p.
of Distance
v. t.
To promote the weal of; to cause to be prosperous.
a.
Separated; having an intervening space; at a distance; away.
a.
Reserved or repelling in manners; cold; not cordial; somewhat haughty; as, a distant manner.
a.
Being in health; sound in body; not ailing, diseased, or sick; healthy; as, a well man; the patient is perfectly well.
v. t.
To pour forth, as from a well.
a.
Being well folded.
a.
Indistinct; faint; obscure, as from distance.
v. t.
To outstrip by as much as a distance (see Distance, n., 3); to leave far behind; to surpass greatly.
a.
Far separated; far off; not near; remote; -- in place, time, consanguinity, or connection; as, distant times; distant relatives.
a. & adv.
Well.
a.
So separated as not to be confounded with any other thing; not liable to be misunderstood; not confused; well-defined; clear; as, we have a distinct or indistinct view of a prospect.
n.
Remoteness in succession or relation; as, the distance between a descendant and his ancestor.
n.
Prosperity; happiness; well-being; weal.
a.
Not conformable; discrepant; repugnant; as, a practice so widely distant from Christianity.
n.
Distance.