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Function in mathematical analysis
In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R {\displaystyle
Schur-convex_function
Preorder on vectors of real numbers
of a Schur-convex function is the max function, max ( x ) = x 1 ↓ {\displaystyle \max(\mathbf {x} )=x_{1}^{\downarrow }} . Schur convex functions are necessarily
Majorization
Algebra theorem about convex functions
turn to the concept of Schur-convex functions. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. If x1
Karamata's_inequality
complement method Schur complement Schur-convex function Schur decomposition Schur functor Schur index Schur's inequality Schur's lemma (from Riemannian
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
German mathematician (1875–1941)
Schur's inequality Schur's theorem Schur-convex function Schur–Weyl duality Lehmer–Schur algorithm Schur's property for normed spaces. Jordan–Schur theorem
Issai_Schur
Tool in linear algebra and matrix analysis
The Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics. It is defined for a block
Schur_complement
Classical averages studied in ancient Greece
majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while
Pythagorean_means
Whenever certain curvatures are pointwise constant then they must be globally constant
\operatorname {R} _{p}} The Schur lemma states the following: Suppose that n {\displaystyle n} is not equal to two. If there is a function κ {\displaystyle \kappa
Schur's lemma (Riemannian geometry)
Schur's_lemma_(Riemannian_geometry)
Concept in Hlibert spaces mathematics
in fact, not operator monotone! A function f : I → R {\displaystyle f:I\to \mathbb {R} } is said to be operator convex if for all n {\displaystyle n} and
Trace_inequality
Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Normed vector space that is complete
reflexive spaces to certain optimization problems. For example, every convex continuous function on the unit ball B {\displaystyle B} of a reflexive space attains
Banach_space
Property of a mathematical matrix
nonnegative. Covariance matrix M-matrix Positive-definite function Positive-definite kernel Schur complement Sylvester's criterion Numerical range Williamson
Definite_matrix
inequalities can be found based on the notion of Schur-convexity. Related to the above, Bernstein functions are defined as those that are non-negative and
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Robertson–Seymour theorem (graph theory) Schnyder's theorem (graph theory) Schur's theorem (Ramsey theory) Schwenk's theorem (graph theory) Sensitivity theorem
List_of_theorems
method — simple and robust; linear convergence Lehmer–Schur algorithm — variant for complex functions Fixed-point iteration Newton's method — based on linear
List of numerical analysis topics
List_of_numerical_analysis_topics
Subfield of convex optimization
efficiently solved by interior point methods. All linear programs and (convex) quadratic programs can be expressed as SDPs, and the sum of squares hierarchy
Semidefinite_programming
Theorem about projections of coadjoint orbits of a connected compact Lie group
is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of Schur (1923)
Kostant's_convexity_theorem
Rayleigh–Faber–Krahn inequality Remez inequality Riesz rearrangement inequality Schur test Shapiro inequality Sobolev inequality Steffensen's inequality Szegő
List_of_inequalities
Theorem in functional analysis
+\lambda _{k}(A)} is a convex function, and ξ 1 ( A ) + ⋯ + ξ k ( A ) {\textstyle \xi _{1}(A)+\dots +\xi _{k}(A)} is concave. (Schur-Horn inequality) ξ 1
Min-max_theorem
1966 mathematics textbook by Serge Lang
ends with a proof of Schur's lemma and an explanation of the Jordan normal form. The final chapter covers basic concepts of convex sets, culminating in
Linear_Algebra_(book)
Vector space of infinite sequences
entry. The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent (Schur 1921). However, since the weak
Sequence_space
Maps whose domain and codomain are acted on by the same group, and the map commutes
the image of K[G] is a simple algebra, with centre K (by what is called Schur's lemma: see simple module). As a consequence, in important cases the construction
Equivariant_map
Mathematical object
irreducible representations in characteristic zero. The centralizer algebra (Schur algebra) of the permutation representation is commutative. (G/N, K/N) is
Gelfand_pair
American-Canadian mathematician (1930-1995)
mathematician, specializing in harmonic analysis. His name is attached to the Herz–Schur multiplier. He held professorships at Cornell University and McGill University
Carl_S._Herz
Periodic set of points
\mathrm {d} (\Lambda )} , or more generally the volume of a symmetric convex set S {\displaystyle S} , to the number of lattice points contained in
Lattice_(group)
Inverse of the average of the inverses of a set of numbers
x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.} The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for
Harmonic_mean
one adds the assumption that the category has a set of generators. The Schur multiplier of the Mathieu group M22 is particularly notorious as it was
List_of_incomplete_proofs
Description in Riemannian geometry
p\in M} . The Schur lemma states that if (M, g) is a connected Riemannian manifold with dimension at least three, and if there is a function f : M → R {\displaystyle
Sectional_curvature
Group of symmetries of an n-dimensional hypercube
form the 4-group. The second homology groups, known classically as the Schur multipliers, were computed in (Ihara & Yokonuma 1965). They are: H 2 ( C
Hyperoctahedral_group
Concept in mathematics
submodule (the socle) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the
Reductive_group
Gartenkunst und die mathematischen Wissenschaften 2011 Reinhard Siegmund-Schultze Schur und Landau: eine Freundschaft in unmenschlicher Zeit 2010 Hans Föllmer Von
Euler_Lecture
Branch of mathematics
considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra
Abstract_algebra
Group that is also a differentiable manifold with group operations that are smooth
Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra
Lie_group
British-Lebanese mathematician (1929–2019)
result due to Schur and Horn states that the set of possible diagonal vectors of an Hermitian matrix with given eigenvalues is the convex hull of all the
Michael_Atiyah
Emilie Virginia Haynsworth (1916–1985), American linear algebraist known for Schur complements and Haynsworth inertia additivity formula Olive Hazlett (1890–1974)
List_of_women_in_mathematics
sabbatical year at Harvard and MIT (1970-1971), he wrote “Manifestations of the Schur Complement’’, one of his most cited papers. In 1974, he started working
Richard_W._Cottle
Representation theory of the symplectic group
∗ = W ( g ( z ) ) . {\displaystyle \pi (g)W(z)\pi (g)^{*}=W(g(z)).} By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with
Oscillator_representation
theory and combinatorics Otto Schreier (1901–1929), group theory Issai Schur (1875–1941), group representations, combinatorics and number theory Arthur
List_of_Jewish_mathematicians
special linear Lie algebra s l {\displaystyle {\mathfrak {sl}}} n: Issai Schur's result in his 1901 dissertation that the weight multiplicities could be
Littelmann_path_model
Mathematical group of loops in a Lie group
MR 2435422 Neeb, Karl-Hermann (2006), "Towards a Lie theory of locally convex groups", Japanese Journal of Mathematics, 1 (2): 291–468, arXiv:1501.06269
Loop_group
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
Boy/Male
Irish
Hero.
Surname or Lastname
English
English : unexplained.
Boy/Male
Anglo Saxon
Storm.
Boy/Male
Irish American
Hound lover. Full of desire; much desire.
Surname or Lastname
Italian
Italian : from the title of rank conte ‘count’ (from Latin comes, genitive comitis ‘companion’). Probably in this sense (and the Late Latin sense of ‘traveling companion’), it was a medieval personal name; as a title it was no doubt applied ironically as a nickname for someone with airs and graces or simply for someone who worked in the service of a count.English : variant of Count, cognate with 1.French : nickname for someone in the service of a count or for someone who behaved pretentiously, from Old French conte, cunte ‘count’ (of the same derivation as 1).French (Conté) : variant of Comté (see Comte).
Surname or Lastname
English
English : habitational name from a place named Cove, examples of which are found in Devon, Hampshire, and Suffolk, from Old English cofa ‘cove’, ‘bay’, ‘inlet’, also ‘shelter’, ‘hut’, or a topographic name with the same meaning.
Male
English
Variant spelling of English Connor, CONNER means "hound-lover."
Surname or Lastname
English
English : from Old French covine ‘fraud’, ‘deceit’, hence a derogatory nickname for a trickster.English : habitational name from a place in Staffordshire named Coven ‘(place) at the huts or shelters (Old English cofa, dative plural cofum)’.
Male
English
Anglicized form of Irish Gaelic Conláed, CONLEY means "purifying fire."
Surname or Lastname
English
English : metathesized form of the occupational name Coyner.English : possibly an occupational name for a dealer in rabbits or rabbit skins, from an agent derivative of Middle English cony ‘rabbit’ (see Coney).
Boy/Male
Irish American
Strong willed or wise. Also a : Hero.
Boy/Male
British, Christian, English
Wagoner; To Convey
Surname or Lastname
English (Leicestershire)
English (Leicestershire) : variant of Culver.
Biblical
wall; ox; that beholds
Boy/Male
British, English
Peasant
Surname or Lastname
Spanish and Portuguese
Spanish and Portuguese : nickname from the title of rank conde ‘count’, a derivative of Latin comes, comitis ‘companion’.English : unexplained.
Surname or Lastname
English
English : from Middle English cony ‘rabbit’ (a back-formation from conies, from Old French conis, plural of conil), a nickname for someone thought to resemble a rabbit in some way or a metonymic occupational name for a dealer in rabbits or rabbit skins.
Surname or Lastname
Irish
Irish : variant spelling of Connor, now common in Scotland.English : occupational name for an inspector of weights and measures, Middle English connere, cunnere ‘inspector’, an agent derivative of cun(nen) ‘to examine’.
Boy/Male
American, British, English
Dove
Boy/Male
Hindu
Wall, Ox, That beholds
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
Surname or Lastname
English
English : plural variant of Hollen.
Boy/Male
Hindu
King of all Era
Girl/Female
Indian
Valley of Flowers
Girl/Female
Indian
Beautiful
Girl/Female
Arabic, Muslim
Light
Girl/Female
Tamil
Grace, Beauty
Girl/Female
Indian, Traditional
Fairy; Power
Male
Egyptian
, the great lord and ruler Cheres (Ares).
Boy/Male
Indian, Telugu
Lord Sai Baba; Great Friend
Girl/Female
Hindu, Indian, Traditional
Wife of Shiva
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
SCHUR CONVEX-FUNCTION
a.
Convex on both sides; double convex. See under Convex, a.
adv.
In a convex form; as, a body convexly shaped.
a.
Concave on one side and convex on the other, as an eggshell or a crescent.
p. pr. & vb. n.
of Scour
a.
Plane or flat on one side, and convex on the other; as, a plano-convex lens. See Convex, and Lens.
a.
Convex on one side, and concave on the other. The curves of the convex and concave sides may be alike or may be different. See Meniscus.
a.
Made convex; protuberant in a spherical form.
v. t.
To accompany; to convoy.
a.
Convex on one side, and flat on the other; plano-convex.
v. t.
To pass swiftly over; to brush along; to traverse or search thoroughly; as, to scour the coast.
v. t.
To context.
n.
A convex body or surface.
v. i.
To move hastily; to scour.
imp. & p. p.
of Scour
v. t.
To impart or communicate; as, to convey an impression; to convey information.
a.
Convex on both sides; as, a biconvex lens.
n. & v.
See Conge, Conge.
v. t.
To cause to pass from one place or person to another; to serve as a medium in carrying (anything) from one place or person to another; to transmit; as, air conveys sound; words convey ideas.
dv.
In a convex form; convexly.
v. t.
To purge; as, to scour a horse.