#] #] ********************* #] "$d_web"'References/Mathematics/0_math theorem list, wikipedia.txt' # www.BillHowell.ca 01Feb2024 initial # view in text editor, using constant-width font (eg courier), tabWidth = 3 #48************************************************48 #24************************24 # Table of Contents, generate with : # $ grep "^#]" "$d_web"'References/Mathematics/0_math theorem list, wikipedia.txt' | sed "s/^#\]/ /" # #24************************24 # Setup, ToDos, to "clean out" formatting commands from text : see: str_chrP_rmBetween() - rm chrs between chr[start, end] = chrP, chrP example '{}' in "$d_bin"'strings.sh' !!!!************!!!! EXTREMELY IMPORTANT! https://en.wikipedia.org/wiki/List_of_theorems This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities List of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. 08******** Inequalities in pure mathematics Analysis Agmon In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,[1] consist of two closely related interpolation inequalities between the Lebesgue space L∞ and the Sobolev spaces Hs. It is useful in the study of partial differential equations. For the n-dimensional case, choose s1 and s2 such that s1 < n/2 < s2. Then, if 0 < θ < 1 and n/2 = θ*s1 + (1 - θ)*s2, the following inequality holds for any u ∈ H(s2(Ω) : ‖u‖subSup(L∞(Ω), ) ≤ C*‖u‖subSup(Hs1(Ω), θ) * ‖u‖(H(s2(Ω), (1-θ))) Askey–Gasper Babenko–Beckner Bernoulli Bernstein (mathematical analysis) Bessel Bihari–LaSalle Bohnenblust–Hille Borell–Brascamp–Lieb Brezis–Gallouet Carleman Chebyshev–Markov–Stieltjes inequalities Chebyshev sum Clarkson inequalities Eilenberg Fekete–Szegő Fenchel Friedrichs Gagliardo–Nirenberg interpolation In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the Lp-norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958. The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haim Brezis and Petru Mironescu in the late 2010s. Let 1 ≤ q ≤ + ∞ be a positive extended real quantity. Let j and m be non-negative integers such that j < m . Furthermore, let 1 ≤ r ≤ + ∞ be a positive extended real quantity, p ≥ 1 be real and θ ∈ [ 0 , 1 ] such that the relations 1 p = j n + θ ( 1 r − m n ) + 1 − θ q , j m ≤ θ ≤ 1 hold. Then, ‖ D j u ‖ L p ( R n ) ≤ C ‖ D m u ‖ L r ( R n ) θ ‖ u ‖ L q ( R n ) 1 − θ for any u ∈ L q ( R n ) such that D m u ∈ L r ( R n ) , with two exceptional cases: Gårding Grothendieck Grunsky inequalities Hanner inequalities Hardy Hardy–Littlewood Hardy–Littlewood–Sobolev Harnack Hausdorff–Young Hermite–Hadamard Hilbert Hölder Jackson Jensen Khabibullin conjecture on integral inequalities Kantorovich Karamata Korn Ladyzhenskaya In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data). Ladyzhenskaya's theorems are special cases of the Gagliardo–Nirenberg interpolation Landau–Kolmogorov Lebedev–Milin Lieb–Thirring Littlewood 4/3 Markov brothers' Mashreghi–Ransford Max–min Minkowski Poincaré Popoviciu Prékopa–Leindler Rayleigh–Faber–Krahn Remez Riesz rearrangement Schur test Shapiro Sobolev Steffensen Szegő Three spheres Trace inequalities Trudinger theorem Turán inequalities Von Neumann Wirtinger for functions Young convolution Young for products Inequalities relating to means Hardy–Littlewood maximal Inequality of arithmetic and geometric means Ky Fan Levinson Maclaurin Mahler Muirhead Newton inequalities Stein–Strömberg theorem Combinatorics Binomial coefficient bounds Factorial bounds XYZ Fisher Ingleton Lubell–Yamamoto–Meshalkin Nesbitt Rearrangement Schur Shapiro Stirling formula (bounds) Differential equations Grönwall Geometry See also: List of triangle inequalities Alexandrov–Fenchel Aristarchus Barrow Berger–Kazdan comparison theorem Blaschke–Lebesgue Blaschke–Santaló Bishop–Gromov Bogomolov–Miyaoka–Yau Bonnesen Brascamp–Lieb Brunn–Minkowski Castelnuovo–Severi Cheng eigenvalue comparison theorem Clifford theorem on special divisors Cohn-Vossen Erdős–Mordell Euler theorem in geometry Gromov for complex projective space Gromov systolic for essential manifolds Hadamard Hadwiger–Finsler Hinge theorem Hitchin–Thorpe Isoperimetric Jordan Jung theorem Loewner torus Łojasiewicz Loomis–Whitney Melchior Milman reverse Brunn–Minkowski Milnor–Wood Minkowski first for convex bodies Myers theorem Noether Ono Pedoe Ptolemy Pu Riemannian Penrose Toponogov theorem Triangle Weitzenböck Wirtinger (2-forms) Information theory Inequalities in information theory Kraft Log sum Welch bounds Algebra Abhyankar Pisier–Ringrose Linear algebra Abel Bregman–Minc Cauchy–Schwarz Golden–Thompson Hadamard Hoffman-Wielandt Peetre Sylvester rank Triangle Trace inequalities Eigenvalue inequalities Bendixson Weyl in matrix theory Cauchy interlacing theorem Poincaré separation theorem Number theory Bonse Large sieve Pólya–Vinogradov Turán–Kubilius Weyl Probability theory and statistics Azuma Bennett, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis, an upper bound on the variance of any bounded probability distribution Bernstein inequalities (probability theory) Boole Borell–TIS BRS-inequality Burkholder Burkholder–Davis–Gundy inequalities Cantelli Chebyshev Chernoff Chung–Erdős Concentration Cramér–Rao Doob martingale Dvoretzky–Kiefer–Wolfowitz Eaton, a bound on the largest absolute value of a linear combination of bounded random variables Emery Entropy power Etemadi Fannes–Audenaert Fano Fefferman Fréchet inequalities Gauss Gauss–Markov theorem, the statement that the least-squares estimators in certain linear models are the best linear unbiased estimators Gaussian correlation Gaussian isoperimetric Gibbs Hoeffding Hoeffding lemma Jensen Khintchine Kolmogorov Kunita–Watanabe Le Cam theorem Lenglart Marcinkiewicz–Zygmund Markov McDiarmid Paley–Zygmund Pinsker Popoviciu on variances Prophet Rao–Blackwell theorem Ross conjecture, a lower bound on the average waiting time in certain queues Samuelson Shearer Stochastic Gronwall Talagrand concentration Vitale random Brunn–Minkowski Vysochanskiï–Petunin Topology Berger for Einstein manifolds Inequalities particular to physics Ahlswede–Daykin Bell – see Bell theorem Bell original CHSH Clausius–Duhem Correlation – any of several inequalities FKG Ginibre inequal ity Griffiths Heisenberg Holley Leggett–Garg Riemannian Penrose Rushbrooke Tsirelson See also Comparison theorem List of mathematical identities Lists of mathematics topics List of set identities and relations +-----+ https://en.wikipedia.org/wiki/Schur_test In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version.[1] Let X , Y {\displaystyle X,\,Y} be two measurable spaces (such as R n {\displaystyle \mathbb {R} ^{n}}). Let T {\displaystyle \,T} be an integral operator with the non-negative Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)}, x ∈ X {\displaystyle x\in X}, y ∈ Y {\displaystyle y\in Y}: T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y . {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.} If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that ( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)} for almost all x {\displaystyle \,x} and ( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) {\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)} for almost all y {\displaystyle \,y}, then T {\displaystyle \,T} extends to a continuous operator T : L 2 → L 2 {\displaystyle T:L^{2}\to L^{2}} with the operator norm ‖ T ‖ L 2 → L 2 ≤ α β . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.} Such functions p ( x ) {\displaystyle \,p(x)}, q ( y ) {\displaystyle \,q(y)} are called the Schur test functions. In the original version, T {\displaystyle \,T} is a matrix and α = β = 1 {\displaystyle \,\alpha =\beta =1}.[2] Common usage and Young's inequality A common usage of the Schur test is to take p ( x ) = q ( y ) = 1. {\displaystyle \,p(x)=q(y)=1.} Then we get: ‖ T ‖ L 2 → L 2 2 ≤ sup x ∈ X ∫ Y | K ( x , y ) | d y ⋅ sup y ∈ Y ∫ X | K ( x , y ) | d x . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}^{2}\leq \sup _{x\in X}\int _{Y}|K(x,y)|\,dy\cdot \sup _{y\in Y}\int _{X}|K(x,y)|\,dx.} This inequality is valid no matter whether the Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} is non-negative or not. A similar statement about L p → L q {\displaystyle L^{p}\to L^{q}} operator norms is known as Young's inequality for integral operators:[3] if sup x ( ∫ Y | K ( x , y ) | r d y ) 1 / r + sup y ( ∫ X | K ( x , y ) | r d x ) 1 / r ≤ C , {\displaystyle \sup _{x}{\Big (}\int _{Y}|K(x,y)|^{r}\,dy{\Big )}^{1/r}+\sup _{y}{\Big (}\int _{X}|K(x,y)|^{r}\,dx{\Big )}^{1/r}\leq C,} where r {\displaystyle r} satisfies 1 r = 1 - ( 1 p - 1 q ) {\displaystyle {\frac {1}{r}}=1-{\Big (}{\frac {1}{p}}-{\frac {1}{q}}{\Big )}}, for some 1 ≤ p ≤ q ≤ ∞ {\displaystyle 1\leq p\leq q\leq \infty }, then the operator T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy} extends to a continuous operator T : L p ( Y ) → L q ( X ) {\displaystyle T:L^{p}(Y)\to L^{q}(X)}, with ‖ T ‖ L p → L q ≤ C . {\displaystyle \Vert T\Vert _{L^{p}\to L^{q}}\leq C.} Proof is provided. #08********08 #] ??Feb2024 #08********08 #] ??Feb2024 #08********08 #] ??Feb2024 #08********08 #] ??Feb2024 #08********08 #] ??Feb2024 #08********08 #] ??Feb2024 # enddoc