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Mathematical theorem
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of
Mercer's_theorem
In functional analysis, a Hilbert space
existence of a unique reproducing kernel by the Riesz Representation theorem. Mercer's theorem can also be extended to address the vector-valued setting and
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Compact operator for which a finite trace can be defined
\operatorname {Tr} (T^{2})=\sum _{i=1}^{\infty }\lambda _{i}(T^{2})<\infty .} Mercer's theorem provides another example of a trace class operator. That is, suppose
Trace_class
proved Mercer's theorem, which states that positive-definite kernels can be expressed as a dot product in a high-dimensional space. This theorem is the
James_Mercer_(mathematician)
Class of algorithms for pattern analysis
measure ensuring the function k {\displaystyle k} satisfies Mercer's condition. Mercer's theorem is similar to a generalization of the result from linear
Kernel_method
Concept in regression analysis mathematics
\mathbb {R} ^{m}} , and can be infinite dimensional. This follows from Mercer's theorem, which states that a continuous, symmetric, positive definite kernel
Regularized_least_squares
Theory of stochastic processes
processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem states that a stochastic
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Spatial statistical signal analysis
kernel matrix are thus non-linear in the location of the data (see Mercer's theorem and the kernel trick for more information). Blind signal separation
Empirical orthogonal functions
Empirical_orthogonal_functions
Parameterization of a rotation into a unit vector and angle
associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient construction of the Cartesian representation of the
Axis–angle_representation
Statistical learning theory
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle
Representer_theorem
Serbian mathematician (1902–1967)
discovered a new class of theorems of Tauberian type, today known as Karamata's tauberian theorems. He also worked on Mercer's theorems, Frullani integral,
Jovan_Karamata
Fundamental theorem of Hilbert spaces Gram–Schmidt process Hellinger–Toeplitz theorem Hilbert space Inner product space Legendre polynomials Matrices Mercer's theorem
List of functional analysis topics
List_of_functional_analysis_topics
Matrix of inner products of vectors
{\displaystyle M} . The infinite-dimensional analog of this statement is Mercer's theorem. If M {\displaystyle M} is the Gram matrix of vectors v 1 , … , v n
Gram_matrix
Approximations used in machine learning
the Moore–Penrose pseudoinverse of K 11 {\displaystyle K_{11}} . By Mercer's theorem, we can decompose the kernel matrix as a Gram matrix: K = X T X {\textstyle
Low-rank matrix approximations
Low-rank_matrix_approximations
Branch of statistics mathematics
f\in H} , all the φ j {\displaystyle \varphi _{j}} are continuous. Mercer's theorem then states that sup s , t ∈ [ 0 , 1 ] | Σ ( s , t ) − ∑ j = 1 K λ
Functional_data_analysis
Functional analysis concept
stated above, the techniques used to prove results, e.g., the spectral theorem, in the non-compact case are typically different, involving operator-valued
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
arguments, continuous and positive semidefinite in a certain sense). Mercer's theorem asserts that K {\displaystyle K} can then be expressed as K ( x , y
String_kernel
valued reproducing kernel hilbert spaces of integrable functions and mercer theorem," Anal. Appl. (Singap.), 4 Mauricio A. Álvarez, Lorenzo Rosasco, and
Kernel methods for vector output
Kernel_methods_for_vector_output
McKay, known for his discovery of monstrous moonshine James Mercer, proved Mercer's theorem. Edward Milne, the study of radiative equilibrium, the structure
List of University of Manchester people
List_of_University_of_Manchester_people
Mathematical theorem about the Fourier series
In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: Fejér's Theorem—Let f : R → C {\displaystyle f:\mathbb
Fejér's_theorem
Mathematical logician and philosopher
theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two
Kurt_Gödel
British mathematician (1885–1977)
Littlewood showed that if the Riemann hypothesis is true, then the prime number theorem follows and obtained the error term. This work won him his Trinity fellowship
John_Edensor_Littlewood
Operation in differential calculus
that point, if the latter two both exist. Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker
Symmetric_derivative
German mathematician (1805–1859)
mathematician. In number theory, he proved special cases of Fermat's Last Theorem and created analytic number theory. In analysis, he advanced the theory
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Generalization of a positive-definite matrix
between positive definite kernels and RKHS is given by the following theorem Theorem— Every reproducing kernel is positive-definite, and every positive
Positive-definite_kernel
Nyquist–Shannon sampling theorem. The name Nyquist–Shannon sampling theorem honours Harry Nyquist and Claude Shannon, but the theorem was also previously discovered
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
American actor (born 1969)
The Joe Rogan Experience, where he attempted to debunk the Pythagorean theorem, claimed he can kill gravity, said he does not believe in the number zero
Terrence_Howard
American mathematician and billionaire (1938–2024)
California, Berkeley alumni Plateau's problem Simons' formula Simons' theorem Coy, Peter (April 11, 2019). "Meet Marilyn Simons, the Bricklayer's Daughter
Jim_Simons
Mathematical concept in algebra
last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities) This theorem has
Nilpotent_matrix
English mathematician (1937–2020)
function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line
John_Horton_Conway
Hungarian-American priest-theologian and theoretical physicist (1924–2009)
that Gödel's incompleteness theorem is relevant for theories of everything (TOE) in theoretical physics. Gödel's theorem states that any theory that includes
Stanley_Jaki
American songwriter (born 1940)
Records / 2016 "When Love Is Near" The Original Caste 1969 Tartaglian Theorem / Capitol Records / 2012 "You and Me Against the World" Helen Reddy 1974
Paul_Williams_(songwriter)
Prize from University of Cambridge in mathematics and theoretical physics
examination question on a particular theorem that William Thomson had written to him about, which is now known as Stokes' theorem. T. W. Körner notes Only a small
Smith's_Prize
Climatic cooling after the Medieval Warm Period (16th–19th centuries)
field of the solar surface and solar atmosphere (corona). Because Alfvén's theorem applies, the coronal magnetic field is dragged out into the heliosphere
Little_Ice_Age
Domain of convergence of power series
the distance exceeds that number; this statement is the Cauchy–Hadamard theorem. Note that r = 1/0 is interpreted as an infinite radius, meaning that f
Radius_of_convergence
Statistical measure of how far values spread from their average
standard error of the sample mean, which is used in the central limit theorem. To prove the initial statement, it suffices to show that Var ( X + Y
Variance
basketball player Andrew Wiles, British mathematician, proved Fermat's Last Theorem Andrew Wilson (swimmer), American swimmer Andrew Wingard (born 1996), American
List of people with given name Andrew
List_of_people_with_given_name_Andrew
Topics referred to by the same term
Navier–Stokes equations Reynolds decomposition Reynolds number Reynolds transport theorem Reynolds v. Sims, a 1964 U.S. Supreme Court case concerning State legislature
Reynolds
2012), "UPDATE: Toronto: Terry Gilliam Confirms Christoph Waltz For Zero Theorem", DeadLine. "Android Cop (2014) - Mark Atkins | Synopsis, Movie Info, Moods
List of science fiction films of the 2010s
List_of_science_fiction_films_of_the_2010s
Stone–von Neumann theorem, Stone–Čech compactification, Stone's representation theorem for Boolean algebras, Stone space, Stone–Weierstrass theorem, Stone's representation
List_of_eponyms_(L–Z)
Property of a mathematical function
derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity
Semi-differentiability
Postgraduate center in New Jersey, US
Social Science. ISBN 978-0-691-08841-9 Villani, Cédric (2015). Birth of a Theorem : A Mathematical Adventure, Faber and Faber. ISBN 978-0-86547-767-4 Wittrock
Institute_for_Advanced_Study
Topics referred to by the same term
probability space TREE, an extremely fast-growing function; see Kruskal's tree theorem Tree (automata theory) Tree (command), a recursive directory listing program
Tree_(disambiguation)
Kleroterion Liberalism Libertarianism Majoritarianism Majority rule Median voter theorem Motion Ochlocracy Nomination rules Peaceful transition of power Political
List_of_political_ideologies
Community school in Penistone, South Yorkshire, England
alumni including Nicholas Saunderson, the probable inventor of Bayes' theorem, in the 18th century. At various times in its history it has been single-sex
Penistone_Grammar_School
German polymath (1646–1716)
relation of integration and differentiation, later called the fundamental theorem of calculus, by means of a figure in his 1693 paper Supplementum geometriae
Gottfried_Wilhelm_Leibniz
Subjective attitude that something is true
Subjective validation Suggestibility Suggestion Theory of justification Thomas theorem Tinkerbell effect Trust Unintended consequence Validity Value (personal
Belief
Audio or video represented as a stream of discrete numbers
reconstructed, within the constraints of the Nyquist–Shannon sampling theorem, which dictates the sampling rate and quantization error dependent on the
Digital_recording
German mathematician (1885–1955)
Brose) Raum - Zeit - Materie (German, 1918) Majorana–Weyl spinor Peter–Weyl theorem Schur–Weyl duality Weyl algebra Weyl basis of the gamma matrices Weyl chamber
Hermann_Weyl
Stone, 85, American mathematician at Harvard (Stone–von Neumann uniqueness theorem), stroke. Bill Terry, 90, American Major League baseball player (New York
Deaths_in_January_1989
Algorithm in mathematics
} We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ )
Baum–Welch_algorithm
Optimization technique
finding the global optimum. Also worth mentioning are the no-free-lunch theorems, which state that there can be no metaheuristic that is better than all
Metaheuristic
Solid material with highly ordered microscopic structure
impossible for an ordinary periodic crystal (see crystallographic restriction theorem). The International Union of Crystallography has redefined the term "crystal"
Crystal
American philosopher, classicist, and academician (1930–1992)
the canon was to be re-read and re-defined so that it fitted the latest theorem of gender or race. Bloom would have none of it. He loved people who were
Allan_Bloom
Borough in New Jersey, United States
Princeton Andrew Wiles (born 1953), mathematician who proved Fermat's Last Theorem, Princeton University professor Woodrow Wilson (1856–1924), 28th President
Princeton,_New_Jersey
Humor about mathematics or mathematicians
by Alfréd Rényi: "A mathematician is a device for turning coffee into theorems." If the dual is taken, the quote becomes A comathematician is a device
Mathematical_joke
Scottish-born American blacksmith, mechanic and inventor
drawing illustrating the Pythagorean theorem. Neagle signed and dated the 1829 portrait above the Pythagorean theorem illustration. The original portrait
Patrick_Lyon_(blacksmith)
Argument for the existence of God
used still today, such as the unseen watchmaker, and the infinite monkey theorem, have their roots in this period. While the Stoics became the most well-known
Teleological_argument
Property of a statement that can be logically contradicted
example, Bayesian inductive logic is justified by theorems that make explicit assumptions. These theorems are obtained with deductive logic. They are sometimes
Falsifiability
Machine translation paradigm
which lends itself well to computer implementation is to apply Bayes' theorem, that is p ( e | f ) ∝ p ( f | e ) p ( e ) {\displaystyle p(e|f)\propto
Statistical machine translation
Statistical_machine_translation
American paleoconservative
explores such topics as complex numbers, field theory, the prime number theorem, the zeta function, the harmonic series, and others. The biographical sections
John_Derbyshire
Polish mathematician
conjecture of what is now called the Bregman–Minc inequality (or Bregman's theorem), proved in 1973 by Lev M. Bregman. Henryk Minc was born in Łódź, Poland
Henryk_Minc
Philosophical system originating from Thomas Aquinas
understands nothing without a phantasm." Hence the peripatetic axiom. (Another theorem to be drawn from this is that error is a result of drawing false conclusions
Thomism
American stage magician (1869–1950)
power. Popular Mechanics claimed she exercised the "pivot-and-fulcrum theorem of physics." Joe Nickell investigated Hurst and found her use of force
Lulu_Hurst
List of concepts in artificial intelligence
colloquially as working backward from the goal. It is used in automated theorem provers, inference engines, proof assistants, and other artificial intelligence
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
University in Brandon, Canada
Manitoba Neil Robertson, mathematician known for the Robertson–Seymour theorem John W. M. Thompson, Manitoba MLA and provincial cabinet minister Higher
Brandon_University
Berkeley; known for the Morse–Kelley set theory, Morse–Sard theorem and the Federer–Morse theorem John Mylopoulos (Sc.B. 1966) – professor emeritus of Computer
List of Brown University alumni
List_of_Brown_University_alumni
glassmaking Walther Nernst (1864–1941), German physical chemist whose heat theorem led the way to the third law of thermodynamics, 1920 Nobel Prize in Chemistry
List_of_chemists
Month of 1943
David Hilbert, 81, leading German mathematician known for Hilbert's basis theorem, Hilbert space, and many other contributions. The Battle of Demyansk began
February_1943
woman and a younger male lover; for example, in Pasolini's novel and movie Theorem, Harold Prince's film Something For Everyone (1970) and Bill Condon's film
Male sex work in popular culture
Male_sex_work_in_popular_culture
Month of 1937
Born: Vadim G. Vizing, Soviet Ukrainian mathematician known for Vizing's theorem; in Kiev (d. 2017) Died: Georges Valmier, 51, French abstract and impressionist
March_1937
American science documentary television series
battling against the Frost giants with his hammer. Gödel's incompleteness theorems state that there is no system that can proves every assertion within itself
Through_the_Wormhole
important for his work in pure mathematics, having authored a number of theorems Yakov Zeldovich – Soviet physicist born in Belarus; played an important
List of Jewish atheists and agnostics
List_of_Jewish_atheists_and_agnostics
Month of 1926
French telegraphy engineer known for his postulation in 1883 of Thévenin's theorem to calculate currents in complex electrical circuits to reduce the current
September_1926
Chemistry Jeffrey Mandula (1962), physicist known for the Coleman–Mandula theorem Allen Neuringer (1962), psychologist, prominent in the field of the experimental
List of Columbia College people
List_of_Columbia_College_people
important for his work in pure mathematics, having authored a number of theorems. Frank Wilczek (born 1951): American theoretical physicist. Along with
List_of_agnostics
areas in the study of X-rays Thomas Bayes, mathematician, known for Bayes' theorem and Bayesian Statistics Max Born, principal founder of Quantum mechanics
List of University of Edinburgh people
List_of_University_of_Edinburgh_people
Series of books published by Springer-Verlag
Mathematical Economics: Linear and Nonlinear Programming. Fixed-Point Theorems. ISBN 978-0-387-90481-8. Macki, Jack; Strauss, Aaron (1981). Introduction
Undergraduate Texts in Mathematics
Undergraduate_Texts_in_Mathematics
Istanbul David Blackwell, Ph.D. 1941 – mathematician; 2010 Rao–Blackwell theorem; first African-American to be inducted into the National Academy of Sciences
List of University of Illinois Urbana-Champaign people
List_of_University_of_Illinois_Urbana-Champaign_people
noted by GLAAD and Out Films: Sex in Chains (1928), Les Biches (1968), Theorem (1968), Score (1974), The Comedian (2012), and Appropriate Behavior (2014)
Media portrayals of bisexuality
Media_portrayals_of_bisexuality
member of National Academy of Sciences; known for Wigner-Eckart theorem, Eckart-Young theorem Thomas F. Frist Jr. (MD 1965): co-founder of HCA Healthcare
List of Washington University alumni
List_of_Washington_University_alumni
Branch of nonparametric statistics
{\displaystyle \{\lambda _{j}\}_{j=1}^{\infty }} . By the Karhunen-Loève theorem, f ( t ) = μ ( t ) + ∑ j = 1 ∞ ξ j ϕ j ( t ) {\displaystyle f(t)=\mu (t)+\sum
Distributional_data_analysis
Class of enzymes
1959 and is referred to as the Călugăreanu, or Călugăreanu–White–Fuller, theorem. Lk cannot be altered without breaking one or both strands of the helix;
Topoisomerase
Month of 1976
a 40-hour journey to Athens on the Mediterranean Sea. The "four color theorem", a mathematical conjecture that no more than four colors are required
June_1976
Irish educator, clergyman and author
(also rightly enough) brief. The pamphlet proves nothing, of course; its theorem is not to be proved. Gilbert Austin (1806). Chironomia, or a Treatise on
Gilbert_Austin
Month of 1926
Ryll-Nardzewski, Polish mathematician known for the Ryll-Nardzewski fixed point theorem; Wilno, Second Polish Republic (now Vilnius, Lithuania) (d. 2015) Opal
October_1926
Month of 1923
Polish theoretical physicist and co-author of the Haag–Lopuszanski–Sohnius theorem; in Lwow, Poland (present-day Lviv, Ukraine) (d. 2008) Two pro-Royalist
October_1923
(born 1902) 1 October John Stewart Bell, physicist, discoverer of Bell's theorem (born 1928) Sir Edmund George Irving, Royal Navy admiral and hydrographer
1990_in_the_United_Kingdom
American ecologist (1924–2002)
efficiency maximum power principle" (Odum 1970), and "maximum power output theorem" (Golley 1993, p. 87). (Bocking 1997, p.73) Cevolatti and Maud 2004. This
Howard_T._Odum
Month of 1925
J. Bussgang, Polish-born American mathematician known for the Bussgang theorem, founder of the signal processing firm Signatron Inc.; in Lwow, Poland
March_1925
Month of 1926
partial differential equations, and known for the Babuška–Lax–Milgram theorem and for the Ladyzhenskaya–Babuška–Brezzi condition; in Prague, Czechoslovakia
March_1926
Month of 1932
Italian mathematician whose name is associated with the Vitali convergence theorem and the Vitali set, from a heart attack after completing a lecture at the
February_1932
Month of 1937
William Lawvere, American mathematician known for Lawvere's fixed-point theorem and for the category of Lawvere theory sets; in Muncie, Indiana (d. 2023)
February_1937
– Fort Collins Talbott's Cider Company – Palisade The Infinite Monkey Theorem – Denver The Old Mine Cidery & Brewpub – Erie Wild Cider – Firestone Averill
List of cideries in the United States
List_of_cideries_in_the_United_States
Month in 1925
[page needed] Born: Klaus Roth, German-born British mathematician known for Roth's theorem on the diophantine approximation of algebraic numbers; in Breslau, Niedersachsen
October_1925
COVID-19. M. S. Narasimhan, 88, Indian mathematician (Narasimhan–Seshadri theorem). Raili Pietilä, 94, Finnish architect (Mäntyniemi). Richard L. Rubenstein
Deaths_in_May_2021
Month of 1926
German-born U.S. mathematician known for the Hochschild–Kostant–Rosenberg theorem; in Berlin (d.2007) Died: Claude Monet, 86, French Impressionist painter
December_1926
Month in 1909
Wolfskehl Prize and 1,000 marks for a partial proof of Fermat's last theorem. The theorem, first postulated by Pierre de Fermat, would finally be proven in
November_1909
Month of 1937
Ukrainian mathematician known for the Kolmogorov–Arnold representation theorem, the Arnold conjecture and for Arnold invariants; in Odessa, Ukrainian
June_1937
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