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Mathematical sequence
In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl: The sequence of all multiples of an irrational
Weyl_sequence
Type of number sequence
sequence p(n) is uniformly distributed modulo 1. This was proven by Weyl and is an application of van der Corput's difference theorem. The sequence log(n)
Equidistributed_sequence
Algorithm that generates an approximation of a random number sequence
generators. A recent innovation is to combine the middle square with a Weyl sequence. This method produces high-quality output through a long period (see
Pseudorandom_number_generator
(rcb_generator)". Technical Report. Widynski, Bernard (2017). "Middle-Square Weyl Sequence RNG". arXiv:1704.00358 [cs.CR]. Kneusel, Ron (2018). Random Numbers
List of random number generators
List_of_random_number_generators
Pseudorandom number generator
Modifying the middle-square algorithm with a Weyl sequence improves period and randomness. To generate a sequence of n-digit pseudorandom numbers, an n-digit
Middle-square_method
Non-tensorial representation of the spin group
3-dimensional Euclidean space are quaternionic, Weyl spinors in 4-dimensional Euclidean space are quaternionic, Weyl spinors in Lorentzian signature ( 3 , 1 )
Spinor
Weyl group Weyl integral Weyl integration formula Weyl law Weyl metrics Weyl module Weyl notation Weyl quantization Weyl relations Weyl scalar Weyl semimetal
List of things named after Hermann Weyl
List_of_things_named_after_Hermann_Weyl
Aspect of mathematical spectrum theory
Such a sequence is called a singular sequence or Weyl sequence. By sparsifying the sequence and applying Gram–Schmidt process, the sequence can be made
Essential_spectrum
Class of pseudorandom number generators
with a simple additive counter modulo 232 (which he calls a "Weyl sequence" after Weyl's equidistribution theorem). This also increases the period by
Xorshift
Algorithm for generating pseudo-randomized numbers
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
Finite sum formed using the exponential function
sequence xn, to show a degree of randomness. The techniques involved are ingenious and subtle. A variant of 'Weyl differencing' investigated by Weyl involving
Exponential_sum
Topics referred to by the same term
Weyl's criterion may refer to: Equidistributed sequence#Weyl's criterion in uniform distribution theory Essential spectrum#Weyl's criterion in spectral
Weyl's_criterion
Integer multiples of any irrational mod 1 are uniformly distributed on the circle
Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence a, 22a
Equidistribution_theorem
Rotation of a circle by an angle of π times an irrational number
arithmetic Siegel disc Toeplitz algebra Phase locking (circle map) Weyl sequence Fisher, Todd (2007). "Circle Homomorphisms" (PDF). Archived from the
Irrational_rotation
Type of vector space in math
unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great
Hilbert_space
Natural number
made by another famous mathematician, Hermann Weyl, in a published article. Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products
57_(number)
Set of eigenvalues of a matrix
contains no convergent subsequence. Such a sequence is called a singular sequence (or a singular Weyl sequence). Example: λ = 0 ∈ σ e s s , 2 ( B ) {\displaystyle
Spectrum (functional analysis)
Spectrum_(functional_analysis)
pair of chambers a minimal sequence of reflections (in the Weyl group) to go from one chamber to the other. An adjacent sequence of chambers in a building
Weyl_distance_function
Continuous progression from past to future
causality, being a component quantity of various measurements used to sequence events, to compare the duration of events (or the intervals between them)
Time
Function that "converges" to periodicity
studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion
Almost_periodic_function
Ranges of numbers contained in each other
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I n {\displaystyle I_{n}} on the real
Nested_intervals
German mathematician (1882–1935)
described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics
Emmy_Noether
Natural number
A. (ed.). "Sequence A162328 (Number of reduced words of length n in the Weyl group D_17)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
900_(number)
Vector bundle associated with conformal manifolds
the local twistor bundle. Local twistors can be represented as pairs of Weyl spinors on M (in general from different spin representations, determined
Local_twistor
Natural number
(March 2010). "Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits" (PDF). Journal of Physics A: Mathematical and Theoretical.
238_(number)
Gamma matrices for arbitrary Clifford algebras
theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors. Gamma matrices also
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
52-dimensional exceptional simple Lie group
groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations of F4 are (sequence A121738 in the OEIS): 1
F4_(mathematics)
Exact solution for the Einstein field equations
across the equatorial plane the odd order Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by a 0 = M , a 1 = 0
Kerr_metric
Group of symmetries of an n-dimensional hypercube
in geometry, the hyperoctahedral groups also appear in Lie theory, as the Weyl group associated to the symplectic groups and the orthogonal groups and their
Hyperoctahedral_group
Simple Lie group; the automorphism group of the octonions
groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A104599 in the OEIS): 1, 7
G2_(mathematics)
Set of philosophical problems
points, hence there is no infinite sequence of movements, and the paradox is resolved. According to Hermann Weyl, the assumption that space is made of
Zeno's_paradoxes
2. Weierstrass M-test. Weitzenböck Weitzenböck formula. Weyl 1. Weyl calculus. 2. Weyl quantization. Whitney 1. The Whitney extension theorem gives
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
248-dimensional exceptional simple Lie group
groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121732 in the OEIS): 1, 248
E8_(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
78-dimensional exceptional simple Lie group
groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121737 in the OEIS): 1, 27
E6_(mathematics)
Number with all digits equally frequent
the sequence ( b k x ) k = 0 ∞ {\displaystyle {\left(b^{k}x\right)}_{k=0}^{\infty }} is equidistributed modulo 1, or equivalently, using Weyl's criterion
Normal_number
Mathematical logic concept
interpretability, Gentzen's theory is stronger than Peano arithmetic. Hermann Weyl made the following comment in 1946 regarding the significance of Gentzen's
Gentzen's_consistency_proof
Group in group theory and physics
x_{n}^{\ell _{n}}~.} This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra Wn is a quotient of U ( h n ) {\displaystyle
Heisenberg_group
On when a family of real, continuous functions has a uniformly convergent subsequence
differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of
Arzelà–Ascoli_theorem
133-dimensional exceptional simple Lie group
groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121736 in the OEIS): 1, 56
E7_(mathematics)
Group of unitary complex matrices with determinant of 1
rank n − 1 is given by the set of diagonal matrices with determinant 1. The Weyl group of SU(n) is the symmetric group Sn, which is represented by signed
Special_unitary_group
Algebraic operation on coordinate vectors
the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number.
Dot_product
Type of group in mathematics
for their images by g (for details, see Affine space § Subtraction and Weyl's axioms). The kernel of p is the vector space of the translations. So, the
Orthogonal_group
Indian mathematician (1933-2012)
Verma, Daya-Nand (1971), Mobius inversion for the Bruhat ordering on a Weyl Group, Ann. Sci. Ecole Norm. Sup. 4e Serie, t.4, pp. 393–398. J.E.Humphreys
Daya-Nand_Verma
Number representing a continuous quantity
Series in Mathematics. 2015-01-05. Wheeler, John Archibald (1986). "Hermann Weyl and the Unity of Knowledge: In the linkage of four mysteries—the "how come"
Real_number
Direct sum of irreducible modules
semisimple. Classic examples of simple, but not semisimple, rings are the Weyl algebras, such as the Q-algebra A = Q ⟨ x , y ⟩ / ⟨ x y − y x − 1 ⟩ , {\displaystyle
Semisimple_module
subalgebra Unitarian trick Verma module Weyl 1. Hermann Weyl (1885 – 1955), a German mathematician 2. A Weyl chamber is one of the connected components
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
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
Theory of irregularities of distribution
history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. Discrepancy theory is based on
Discrepancy_theory
Heterocyclic aromatic organic compound
Hetero-Rings with Maximum Unsaturation) — Part 2b". In Schaumann E (ed.). Houben-Weyl Methods of Organic Chemistry. Vol. E 9b/2 (4th Supplement ed.). Thieme. p
Purine
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
Mathematical operation on vector spaces
include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general. The exterior algebra
Tensor_product
Cosmological model
other aspects of cosmology. First, the boundary between aeons satisfies the Weyl curvature hypothesis, thus providing a certain kind of low-entropy past as
Conformal_cyclic_cosmology
Matrices similar to diagonal matrices
p-Schatten class is the set of all operators with finite p-Schatten norm. Weyl, von Neumann, and Kuroda, showed the following: For any p > 1 {\displaystyle
Diagonalizable_matrix
Compact astronomical body
stars. In 1926, Ralph Fowler showed that these stars are not like main-sequence stars, where thermal pressure balances gravity. Instead, a type of quantum-mechanical
Black_hole
Part of spectral theory
with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Foundational principle in quantum physics
momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928: σ x σ p ≥ ℏ 2 {\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar
Uncertainty_principle
Combinatorial object in representation theory
occurring exactly once in the sum. Robinson–Schensted correspondence Schur–Weyl duality Knuth, Donald E. (1973), The Art of Computer Programming, Vol. III:
Young_tableau
View that mathematics does not necessarily represent reality, but is more akin to a game
Magazine. 52 (4): 207–216. doi:10.1080/0025570X.1979.11976784. Reid, Constance; Weyl, Hermann (1970). Hilbert. Springer-Verlag. p. 198. ISBN 9783662286159. Gödel
Formalism (philosophy of mathematics)
Formalism_(philosophy_of_mathematics)
Objects in representation theory of Lie algebras
Bruhat ordering of the Weyl group. Let 0 ⊂ A ⊂ B ⊂ W λ {\displaystyle 0\subset A\subset B\subset W_{\lambda }} be a sequence of g {\displaystyle {\mathfrak
Verma_module
on Hochschild cochain complex. Dixmier conjecture: any endomorphism of a Weyl algebra is an automorphism. Fröberg conjecture on the Hilbert functions of
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Set of quantum operations
It has 2 1.5 n 2 + O ( n ) {\displaystyle 2^{1.5n^{2}+O(n)}} elements. Weyl group, which is generated by the SWAP and Hadamard gates. It has 2 n log
Clifford_group
Algebra associated to any vector space
Multilinear algebra Symmetric algebra, the symmetric analog Tensor algebra Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
Exterior_algebra
Vector operator in vector calculus
variable i is used to refer to an arbitrary component, such as xi. The Voss-Weyl formula, which allows the divergence to be determined using simply partial
Divergence
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
Hypothesis about sapient life and the universe
positive whole number, then wave impulses become distorted. In 1922, Hermann Weyl claimed that Maxwell's theory of electromagnetism can be expressed in terms
Anthropic_principle
Summatory function of the Möbius function
_{n=1}^{\infty }{\frac {(-1)^{n-1}(2\pi )^{2n}}{(2n)!n\zeta (2n+1)x^{2n}}}.} Weyl conjectured that the Mertens function satisfied the approximate functional-differential
Mertens_function
Orthogonal symmetric polynomial family
positive Weyl chamber. ρ is the Weyl vector: half the sum of the positive roots; this is a special element of P+ in the interior of the positive Weyl chamber
Macdonald_polynomials
Integral expressing the amount of overlap of one function as it is shifted over another
finite-dimensional unitary representations form an orthonormal basis in L2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along
Convolution
Four-dimensional analog of the dodecahedron
Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits O ( Λ ) = W ( H 4 ) = I {\displaystyle O(\Lambda )=W(H_{4})=I}
120-cell
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
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
Set of vectors used to define coordinates
ordered basis, which is therefore not simply an unstructured set, but a sequence, an indexed family, or similar; see § Ordered bases and coordinates below
Basis_(linear_algebra)
1938 film by Michael Curtiz and William Keighley
received four nominations, winning three—Best Art Direction (Carl Jules Weyl), Best Film Editing (Ralph Dawson) and Best Original Score (Erich Wolfgang
The_Adventures_of_Robin_Hood
Concept in mathematics
variation. The Weyl group of a reductive group G means the quotient group of the normalizer of a maximal torus by the torus, W = NG(T)/T. The Weyl group is
Reductive_group
Universal construction in multilinear algebra
include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two
Tensor_algebra
24-dimensional repeating pattern of points
then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using L and this Weyl vector. Conway
Leech_lattice
Canadian geometer (1907–2003)
University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, and Solomon Lefschetz. Returning to Trinity for a year, he
Harold Scott MacDonald Coxeter
Harold_Scott_MacDonald_Coxeter
Mathematical manifold theory
proof appeared in 1933, but he considered it "crude in the extreme". Hermann Weyl, one of the most brilliant mathematicians of the era, found himself unable
Hodge_theory
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
the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums ∑ k = 0 m − 1 f ( x + k ω ) {\displaystyle \sum _{k=0}^{m-1}f(x+k\omega
Denjoy–Koksma_inequality
Rationality-focused community blog
and voted on them using the quadratic voting system, popularized by Glen Weyl and Vitalik Buterin. From the 2000+ posts published that year, the Review
LessWrong
Schoenflies conjecture (disproved 1910) Tait's conjecture Von Neumann conjecture Weyl–Berry conjecture Williamson conjecture In contemporary mathematics, ideas
List_of_conjectures
Stochastic matrix representing links between entities
12,13,15]. In a number of cases the spectrum is described by the fractal Weyl law [10,12]. The Google matrix can be constructed also for other directed
Google_matrix
Monster and modular connection
one finds that the two Lie algebras are isomorphic, and in particular, the Weyl denominator formula for m {\displaystyle {\mathfrak {m}}} is precisely the
Monstrous_moonshine
Mathematician (1845–1918)
contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections;
Georg_Cantor
Algebraic structure
symmetric group and the partition algebra generalizes the original Schur-Weyl duality between the general linear group and the symmetric group. There are
Partition_algebra
Method of determining fractal dimension
dimension. Correlation dimension Packing dimension Uncertainty exponent Weyl–Berry conjecture Lacunarity Wagon, Stan (2010). Mathematica in Action: Problem
Minkowski–Bouligand_dimension
Decomposition of periodic functions
the [−π,π] case. An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous
Fourier_series
Self-intersecting compact surface, an immersion of the real projective plane
conformal geometry (Robert Bryant)". The Mathematical Heritage of Hermann Weyl (May 12–16, 1987, Duke University, Durham, North Carolina). Proc. Sympos
Boy's_surface
Straight path on a curved surface or a Riemannian manifold
length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic. The metric
Geodesic
Given a right module M over the Weyl algebra A n {\displaystyle A_{n}} , the Gelfand–Kirillov dimension of M over the Weyl algebra coincides with the dimension
Gelfand–Kirillov_dimension
Catalan solid with 60 faces
construction method uses quaternions and the icosahedral symmetry of the Weyl group orbits O ( Λ ) = W ( H 3 ) / C 2 ≈ A 5 = I {\displaystyle O(\Lambda
Pentagonal_hexecontahedron
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
Decomposition in multilinear algebra
approximated arbitrarily well by a sequence of tensor of rank at most r {\displaystyle r} , even though the limit of the sequence converges to a tensor of rank
Tensor_rank_decomposition
Area of mathematics
(1973). "Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities". Functional Analysis and
Catastrophe_theory
computation of the Riemann curvature tensor and associated objects (Ricci tensor, Weyl tensor). SageManifolds can also deal with generic affine connections, not
Sage_Manifolds
Group that is a topological space with continuous group operations
representations, which are all finite-dimensional; this is part of the Peter–Weyl theorem. For example, the theory of Fourier series describes the decomposition
Topological_group
Models of computation
been independently developed by many different authors, including Hermann Weyl (1927). Philosophie der Mathematik und Naturwissenschaft.; the model is discussed
Hypercomputation
Rational-number approximation of a real number
'expected' value. Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence. This showed that
Diophantine_approximation
WEYL SEQUENCE
WEYL SEQUENCE
Girl/Female
Indian
Well-established, Well-found
Boy/Male
Indian
Well-established, Well-found
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Girl/Female
Tamil
Well wisher
Boy/Male
Tamil
Hitakrit | ஹிதாகà¯à®°à®¿à®¤Â
Well wisher, Well to do
Hitakrit | ஹிதாகà¯à®°à®¿à®¤Â
Girl/Female
Tamil
Hitishini | ஹிதீஷீநீ
Well-wisher
Hitishini | ஹிதீஷீநீ
Boy/Male
Hungarian
Well.
Girl/Female
American, Australian, Christian, Danish, Finnish, French, German, Greek, Portuguese, Swedish
Eloquent; Well-spoken; To Talk Well
Boy/Male
Irish
Well.
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
Hindu
Well wisher, Well to do
Girl/Female
Muslim
Well-established, Well-found
Girl/Female
Biblical
Well educated, well brought up.
Girl/Female
Gujarati, Hindu, Indian
Well Wisher; Friend; Well-wisher
Boy/Male
Tamil
Well born
Girl/Female
Indian
Well-arranged, Well-ordered
Biblical
well educated; well brought up
Boy/Male
Muslim
Well-established, Well-found
Surname or Lastname
English
English : variant spelling of Way.Dutch : variant of Wei.
Girl/Female
Muslim
Well-arranged, Well-ordered
WEYL SEQUENCE
WEYL SEQUENCE
Surname or Lastname
English
English : variant of Brissenden.
Boy/Male
Bengali, Hindu, Indian
Entertainer
Girl/Female
Gujarati, Indian, Kannada, Punjabi, Sikh
Lovely Flower
Boy/Male
Norse
Son of Raud.
Girl/Female
Norse
Mother of Tongue-Stein.
Surname or Lastname
English
English : patronymic from Kipp.
Girl/Female
Latin American Greek
Girl/Female
Indian
Cute
Boy/Male
Celtic
Song.
Surname or Lastname
English
English : from a pet form of the female personal name Isabel (see Hibbs 2).English : from a pet form of the personal name Hilbert.
WEYL SEQUENCE
WEYL SEQUENCE
WEYL SEQUENCE
WEYL SEQUENCE
WEYL SEQUENCE
a.
Prosperous; well.
a.
Balanced or considered with reference to public weal.
a.
Common weal.
a.
Safe; as, a chip warranted well at a certain day and place.
imp. & p. p.
of Well
v. t.
To pour forth, as from a well.
p. pr. & vb. n.
of Well
a. & adv.
Well.
a.
Spoken with propriety; as, well-spoken words.
v. t.
To promote the weal of; to cause to be prosperous.
a.
Being well folded.
a.
Correctly informed; provided with information; well furnished with authentic knowledge; intelligent.
a.
Being in health; sound in body; not ailing, diseased, or sick; healthy; as, a well man; the patient is perfectly well.
n.
Prosperity; happiness; well-being; weal.
a.
Good in condition or circumstances; desirable, either in a natural or moral sense; fortunate; convenient; advantageous; happy; as, it is well for the country that the crops did not fail; it is well that the mistake was discovered.
n.
The state or condition of being well; welfare; happiness; prosperity; as, virtue is essential to the well-being of men or of society.
a.
Speaking well; speaking with fitness or grace; speaking kindly.
n.
One who wishes well, or means kindly.
a.
Polite; well-bred; complaisant; courteous.
a.
Well put together; having symmetry of parts.