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the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism of schemes can be factorized as
Stein_factorization
German mathematician
for complex analysis and cryptography. Stein manifolds and Stein factorization are named after him. Karl Stein received his doctorate with his dissertation
Karl_Stein_(mathematician)
American mathematician (1931–2018)
maximal function), Stein complementary series representations, Nikishin–Pisier–Stein factorization in operator theory, the Tomas–Stein restriction theorem
Elias_M._Stein
Term in algebraic geometry
from a compact space to a Hausdorff space is a closed subset. The Stein factorization theorem states that any proper morphism to a locally noetherian scheme
Proper_morphism
Algorithms for matrix decomposition
non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more
Non-negative matrix factorization
Non-negative_matrix_factorization
Type of matrix factorization
an LDU factorization (with all diagonal entries of L and U equal to 1), then the factorization is unique. In that case, the LU factorization is also
LU_decomposition
Theorem of algebraic geometry and commutative algebra
Fulton–Hansen connectedness theorem Grothendieck's connectedness theorem Stein factorization Theorem on formal functions Danilov, V.I. (2001) [1994], "Zariski
Zariski's_main_theorem
Block design in combinatorial mathematics
matching with the factorization labels in turn. Similarly add three more blocks 12CDEF, 34CDEF, and 56CDEF, replacing the factorization labels by the corresponding
Steiner_system
Number divisible only by 1 and itself
although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes
Prime_number
as it is an analog of a fiber space in algebraic topology. By the Stein factorization, any surjective projective morphism is a contraction morphism followed
Contraction_morphism
Integer factorization algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and
Pollard's_rho_algorithm
variety. A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
limit. The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational
Theorem_on_formal_functions
Algorithm for public-key cryptography
proven that none exists; see integer factorization for a discussion of this problem. The first RSA-512 factorization in 1999 used hundreds of computers
RSA_cryptosystem
Complex number whose real and imaginary parts are both integers
every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up
Gaussian_integer
Large number defined as ten to the 100th power
duotrigintillion (short scale) or ten sexdecilliard (long scale). Its prime factorization is 2100 × 5100. The term was coined in 1920 by nine-year-old Milton
Googol
Branch of number theory
arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors
Algebraic_number_theory
Algorithm for computing greatest common divisors
essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic
Euclidean_algorithm
Discrete Fourier transform algorithm
factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but
Fast_Fourier_transform
Visualization of the prime numbers formed by arranging the integers into a spiral
few hundred points". Shortly afterwards, Ulam, with collaborators Myron Stein and Mark Wells, used MANIAC II at Los Alamos Scientific Laboratory to extend
Ulam_spiral
Algorithm for determining whether a number is prime
integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought
Primality_test
Number theory library written in C
various ring arithmetics as well as derived functionality such as integer factorization using a quadratic sieve. The library is designed to be compiled with
Fast Library for Number Theory
Fast_Library_for_Number_Theory
Concept within complex analysis
function, as defined below (Rudin 1987, Thm 17.17). This "Beurling factorization" allows the Hardy space to be completely characterized by the spaces
Hardy_space
Complexity class used to classify decision problems
problem in polynomial time. The decision problem version of the integer factorization problem: given integers n and k, is there a factor f with 1 < f < k
NP_(complexity)
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Algorithm for solving the discrete logarithm problem
number, a theory of factorization and genera. In Proc. Symp. Pure Math. 20, pages 415—440. AMS, Providence, R.I., 1971. A. Stein and E. Teske, Optimized
Baby-step_giant-step
Field (mathematics) generated by the square root of an integer
rings, the ideal class number, which measures the failure of unique factorization, is given in OEIS A003649; for the imaginary case, they are given in
Quadratic_field
Polynomial equation of degree 3
straightforward computation allows verifying that the existence of this factorization is equivalent with Δ 0 = Δ 1 = 0. {\displaystyle \Delta _{0}=\Delta
Cubic_equation
Computation modulo a fixed integer
coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic
Modular_arithmetic
Largest integer that divides given integers
not assured in arbitrary integral domains. However, if R is a unique factorization domain or any other GCD domain, then any two elements have a GCD. If
Greatest_common_divisor
Regularization technique for ill-posed problems
classification with logistic regression or support vector machines, and matrix factorization. Since Tikhonov Regularization simply adds a quadratic term to the objective
Ridge_regression
Type of mathematical functions
from a given and principal parts (Cousin I problem), and Weierstrass factorization theorem was able to create a global meromorphic function from a given
Function of several complex variables
Function_of_several_complex_variables
Dutch mathematician (born 1949)
number of variables is fixed (in 1983); Discovering the elliptic curve factorization method (in 1987); Computing all solutions to the inverse Fermat equation
Hendrik_Lenstra
calculus) Lovász local lemma Stein's lemma Wald's lemma Glivenko–Cantelli lemma Neyman–Pearson lemma Robbins lemma Factorization lemma Fatou's lemma Frostman's
List_of_lemmas
Submodule of fractions in abstract algebra
integers O K {\displaystyle {\mathcal {O}}_{K}} is from being a unique factorization domain (UFD). This is because h K = 1 {\displaystyle h_{K}=1} if and
Fractional_ideal
Natural number
(definition 2): numbers n with an even number of digits which have a factorization n = i*j where i and j have the same number of digits and the multiset
6000_(number)
Equation from stability analysis
systems. In particular, the discrete-time Lyapunov equation (also known as Stein equation) for X {\displaystyle X} is A X A H − X + Q = 0 {\displaystyle
Lyapunov_equation
Area of discrete mathematics
genus. Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Dénes Kőnig. The works
Graph_theory
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
1000_(number)
Coincidence in mathematics
whether the digits of a composite number could be the same as its prime factorization. A similar example (in fact the smallest) in binary is 255987 = 3 3
Mathematical_coincidence
Algebraic curve in mathematics
find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective conic
Elliptic_curve
Probabilistic primality test
return “composite” return “probably prime” This is not a probabilistic factorization algorithm because it is only able to find factors for numbers n which
Miller–Rabin_primality_test
1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. Cereceda's conjecture
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
List of unsolved computational problems
possible? Log-rank conjecture Hartmanis–Stearns conjecture Can integer factorization be done in polynomial time on a classical (non-quantum) computer? Can
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Natural number
super-prime, completes the ninth prime quadruplet set 3264 – solution to Steiner's conic problem: number of smooth conics tangent to 5 given conics in general
3000_(number)
Discrete probability distribution
and sufficient statistic for λ. To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the
Poisson_distribution
Fundamental theorem in probability theory and statistics
doi:10.56021/9780801868665. ISBN 9780801876387. Retrieved 2016-08-11. Stein, C. (1972). "A bound for the error in the normal approximation to the distribution
Central_limit_theorem
German polymath (1646–1716)
characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic
Gottfried_Wilhelm_Leibniz
{\displaystyle b\geq 2} for the base 2. Woodall first announced his work on factorization in a 1911 publication, acknowledging in it his communication on the
H._J._Woodall
Type of function in mathematics
Mathematics". encyclopediaofmath.org. Retrieved 2020-08-30. Krantz & Parks 2002. Stein & Shakarchi 2003, p. 15 harvnb error: no target: CITEREFSteinShakarchi2003
Analytic_function
Conjecture on zeros of the zeta function
50 (3): 221–242, doi:10.4064/aa-50-3-221-242, MR 0960551 Mazur, Barry; Stein, William (2015), Prime Numbers and the Riemann Hypothesis Montgomery, Hugh
Riemann_hypothesis
Fast Fourier Transform algorithm
to be an optimal cache-oblivious algorithm. The general Cooley–Tukey factorization rewrites the indices k and n as k = N 2 k 1 + k 2 {\displaystyle k=N_{2}k_{1}+k_{2}}
Cooley–Tukey_FFT_algorithm
Describes approximate behavior of a function
subexponential; examples of this include the fastest known algorithms for integer factorization and the function n log n {\displaystyle n^{\log n}} . We may ignore
Big_O_notation
Mathematics timeline
countable manifolds, survived into the late 1950s. differentiable stack factorization homology Kuranishi theory Floer homology Glossary of algebraic topology
Timeline_of_manifolds
Natural number
graph with six vertices (and fifteen edges) K6, which yields 132 blocks in Steiner system S(5,6,12). Refers to the Yo Soy 132 movement to vote in 2012 Mexican
132_(number)
Aspect of algebraic number theory
factorisation, and there aren't many imaginary quadratic fields with unique factorization — it exhibits many of the features of the theory. Writing G for the
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
7000_(number)
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
9000_(number)
Estimate of time taken for running an algorithm
sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the general number field sieve, which runs in time about 2 O ( n 1
Time_complexity
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
2000_(number)
Natural number
8]} . The supports of its codewords of weight 8, called octads, form the Steiner system S ( 5 , 8 , 24 ) {\displaystyle S(5,8,24)} , also known as the Witt
24_(number)
Probabilistic primality test
CR]. Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2001). "Section 31.8: Primality testing". Introduction to Algorithms (Second ed
Fermat_primality_test
x^{2}\equiv A{\bmod {B}}} . The problem remains NP-complete even if a prime factorization of B {\displaystyle B} is provided. Serializability of database histories
List_of_NP-complete_problems
Basic unit of quantum information
ISBN 978-1-107-00217-3. Shor, Peter (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer∗". SIAM Journal on Computing
Qubit
Overview of and topical guide to algorithms
algorithm Extended Euclidean algorithm Sieve of Eratosthenes Integer factorization Primality test AKS primality test Modular exponentiation Fast Fourier
Outline_of_algorithms
Function in discrete mathematics
Cormen, Thomas H.; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2001). "Chapter 30: Polynomials and the FFT". Introduction to Algorithms
Discrete_Fourier_transform
Weierstrass factorization theorem (complex analysis) Appell–Humbert theorem (complex manifold) Baily–Borel theorem (algebraic geometry) Behnke–Stein theorem
List_of_theorems
Model for the potential energy of a diatomic molecule
be found using operator methods. One approach involves applying the factorization method to the Hamiltonian. To write the stationary states on the Morse
Morse_potential
Branch of mathematics studying functions of a complex variable
1966). Shaw, W. T., Complex Analysis with Mathematica (Cambridge, 2006). Stein, E. & R. Shakarchi, Complex Analysis. (Princeton, 2003). Sveshnikov, A.
Complex_analysis
Type of linear error-correcting code
) As a cyclic code: The perfect G23 code can be constructed via the factorization of x 23 + 1 {\displaystyle x^{23}+1} over the binary field GF(2): x
Binary_Golay_code
Matrix with a multiplicative inverse
above two block matrix inverses can be combined to provide the simple factorization By the Weinstein–Aronszajn identity, one of the two matrices in the
Invertible_matrix
English mathematician
1090/cbms/079, ISBN 0-8218-0731-5, MR 1107300 Trakhtman, V. A. (1973), "Factorization of matrices of the Walsh function ordered according to Paley and repetition
Raymond_Paley
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
5000_(number)
Special mathematical functions defined on the surface of a sphere
to remove the factor of 4π. Whittaker & Watson 1927, p. 395 Unsöld 1927 Stein & Weiss 1971, §IV.2 Brink, D. M.; Satchler, G. R. Angular Momentum. Oxford
Spherical_harmonics
Machine learning model for vision processing
Josip; Mustafa, Basil; Padlewski, Piotr; Heek, Jonathan; Gilmer, Justin; Steiner, Andreas; Caron, Mathilde; Geirhos, Robert (2023-02-10), Scaling Vision
Vision_transformer
Notion in statistics
the same as that of the sample X. This may be seen by using Neyman's factorization criterion for a sufficient statistic. If T(X) is sufficient for θ, then
Fisher_information
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
8000_(number)
Method for computing the relation of two integers with their greatest common divisor
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill,
Extended_Euclidean_algorithm
1090/S0002-9947-1972-0293384-6. Stein, Elias (1993). "5". Harmonic Analysis. Princeton University Press. Jones, Peter W. (1980). "Factorization of Ap weights". Ann
Muckenhoupt_weights
graph with a 1-factor. factorization A graph factorization is a partition of the edges of the graph into factors; a k-factorization is a partition into k-factors
Glossary_of_graph_theory
Natural number
(n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and
4000_(number)
Algorithm characteristic in computations
candidate one-way functions are based on hard problems such as integer factorization or computing the discrete log. Note that it is not desirable for the
Average-case_complexity
Divergent sum of positive unit fractions
MR 1329545. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. "Chapter 7: Quicksort". Introduction to Algorithms
Harmonic_series_(mathematics)
Equation for radii of tangent circles
common divisor. Every primitive root quadruple can be found from a factorization of a sum of two squares, n 2 + m 2 = d e {\displaystyle n^{2}+m^{2}=de}
Descartes'_theorem
Variant of heap data structure
binary representation of n and e2(n) is the exponent of 2 in the prime factorization of n. The average case is more complex to analyze, but it can be shown
Binary_heap
Exponent of a power of two
for instance, they appear in Euclid's Elements, Props. IX.32 (on the factorization of powers of two) and IX.36 (half of the Euclid–Euler theorem, on the
Binary_logarithm
introduces the main ideas which were then developed in Peter Shor's factorization algorithm. Peter Shor, at AT&T's Bell Labs in New Jersey, publishes
Timeline of quantum computing and communication
Timeline_of_quantum_computing_and_communication
Plane curve: conic section
introduced in this section. The tangent vector can be rewritten by factorization: p → ′ ( t ) = 1 t ( f → 1 t − f → 2 1 t ) . {\displaystyle {\vec
Hyperbola
Theorem in probability theory
X 3 X 4 ] {\displaystyle E[X_{1}X_{2}X_{3}X_{4}]} . Using quadratic factorization − x T Σ − 1 x / 2 + v T x − v T Σ v / 2 = − ( x − Σ v ) T Σ − 1 ( x
Isserlis's_theorem
Algorithm to multiply matrices
"Communication-optimal parallel 2.5D matrix multiplication and LU factorization algorithms" (PDF). Proceedings of the 17th International Conference
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Computer-based method for summarizing a text
by latent semantic analysis (LSA) combined with non-negative matrix factorization (NMF). Although they did not replace other approaches and are often
Automatic_summarization
Greek-American computer scientist
(2006), "Learning from incomplete ratings using non-negative matrix factorization", in Ghosh, Joydeep; Lambert, Diane; Skillicorn, David B.; Srivastava
Fillia_Makedon
Argentine mathematician
Unión Matemática Argentina. 19: 3–41. 1960. Convolution Operators and Factorization, McGill Analysis Seminar, McGill University, Montreal, 1972 "Moment
Mischa_Cotlar
Generalization of graph theory
arrangement of finite sets Factor graph – Function graph representing factorization Greedoid – Set system used in greedy optimization Incidence structure –
Hypergraph
Soviet and Russian mathematician (1933–2019)
theorem describes homogeneous spaces of complex reductive groups that are Stein manifolds. In addition to Lie groups and algebras, Onishchik also worked
Arkady_Onishchik
mathematical objects including quasigroups, Latin squares, graph factorizations, and Steiner triple systems. Combinatorial design Magic square Square matrices
Room_square
Chinese anthology text (202–186 BC)
problems cover elementary arithmetic; fractions; inverse proportion; factorization of numbers; geometric progressions, in particular interest rate calculations
Book on Numbers and Computation
Book_on_Numbers_and_Computation
Symmetric arrangement of finite sets
example of a BTD(3) is given by The columns of a BTD(n) provide a 1-factorization of the complete graph on 2n vertices, K2n. BTD(n)s can be used to schedule
Combinatorial_design
142. Lemmermeyer, F. (2013). "Václav Šimerka: quadratic forms and factorization". LMS Journal of Computation and Mathematics. 16: 118–129. doi:10
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
Combinatorics problem proposed by Thomas Penyngton Kirkman
solution to this problem is an example of a Kirkman triple system, which is a Steiner triple system having a parallelism, that is, a partition of the blocks
Kirkman's_schoolgirl_problem
STEIN FACTORIZATION
STEIN FACTORIZATION
Boy/Male
German
Star
Male
Swedish
Swedish form of Old Norse Steinn, STEN means "stone."
Boy/Male
Anglo, British, Danish, English, Norse, Norwegian, Scandinavian
Young; Youth; Boy
Male
Norwegian
Norwegian form of Old Norse Steinn, STEIN means "stone."
Boy/Male
Australian, Spanish
Innocent
Boy/Male
Hindu
Stern
Boy/Male
Norse
Son of Stein.
Boy/Male
Tamil
Stern
Male
Danish
, stone.
Boy/Male
Norse
Young.
Surname or Lastname
German and Jewish (Ashkenazic)
German and Jewish (Ashkenazic) : from Middle High German stern, German Stern ‘star’, a habitational name for someone living at a house distinguished by the sign of a star, or a Jewish ornamental name.English : nickname for a severe person, from Middle English stern(e) ‘strict’, ‘austere’.
Boy/Male
Australian, German, Norwegian, Swedish
Stone
Boy/Male
Norse
Happy.
Surname or Lastname
English
English : habitational name from Stain in Lincolnshire, named with Old Norse steinn ‘stone’, ‘rock’.
Male
Norwegian
Norwegian form of Old Norse Eysteinn, ØYSTEIN means "island stone."
Male
Norwegian
Norwegian form of Old Norse Sveinn, SVEIN means "boy."
Male
Norse
Old Norse name derived from the word steinn, STEINN means "stone."
Boy/Male
Anglo Saxon
Stem.
Boy/Male
Australian, Chinese, Danish, Finnish, German, Swedish, Teutonic
Stone
Boy/Male
Australian, Danish, Finnish, French, German, Swedish, Teutonic
Stone
STEIN FACTORIZATION
STEIN FACTORIZATION
Girl/Female
Tamil
Abinaya | அபீநாயாÂ
Abinaya means expressions
Surname or Lastname
English
English : from a pet form of the personal name Hodge.
Boy/Male
Hindu
Boy/Male
Tamil
Boy/Male
Tamil
Son
Male
Egyptian
, the name of an Egyptian mummy in the Leyden Museum.
Boy/Male
Scottish
From the winding valley.
Boy/Male
Indian, Tamil
Genuine
Girl/Female
Muslim
Wild horse, Born with feet first (1)
Boy/Male
Hindu, Indian, Tamil
God's Name; Son of Goddess of Victory; Lord Vishnu.
STEIN FACTORIZATION
STEIN FACTORIZATION
STEIN FACTORIZATION
STEIN FACTORIZATION
STEIN FACTORIZATION
v. t.
To color, as wood, glass, paper, cloth, or the like, by processess affecting, chemically or otherwise, the material itself; to tinge with a color or colors combining with, or penetrating, the substance; to dye; as, to stain wood with acids, colored washes, paint rubbed in, etc.; to stain glass.
a.
Having a paddle wheel at the stern; as, a stern-wheel steamer.
a. & adv.
Rough; stern; angry.
n.
A steamboat having a stern wheel instead of side wheels.
imp. & p. p.
of Stain
v. t.
To discolor by the application of foreign matter; to make foul; to spot; as, to stain the hand with dye; armor stained with blood.
n.
See Skein.
v. t.
To stain.
v. i.
To give or receive a stain; to grow dim.
p. pr. & vb. n.
of Stain
n.
Stain; brand.
n. & v.
See Steen.
superl.
Having a certain hardness or severity of nature, manner, or aspect; hard; severe; rigid; rigorous; austere; fixed; unchanging; unrelenting; hence, serious; resolute; harsh; as, a sternresolve; a stern necessity; a stern heart; a stern gaze; a stern decree.
n. & v.
See Steen.
a.
Tall; strong; stern.
n.
Stain; foulness.
n.
Stain; pollution.
n.
Stern.
a.
Being in the stern, or being astern; as, the stern davits.
n.
A discoloration by foreign matter; a spot; as, a stain on a garment or cloth.