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Partial order on matrices
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize
Loewner_order
American mathematician (1893–1968)
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Karl Loewner was
Charles_Loewner
Property of a mathematical matrix
partial ordering M > N . {\displaystyle M>N.} The ordering is called the Loewner order. Every positive definite matrix is invertible and its inverse is also
Definite_matrix
Mathematical concept
complex matrices ordered by Loewner ordering. It states that for any pair A , B {\displaystyle A,B} of complex matrices of order n {\displaystyle n} there
Young's inequality for products
Young's_inequality_for_products
Technique in mathematical modeling
interpolation Transfer function interpolation Piecewise tangential interpolation Loewner framework (Empirical) cross Gramian Krylov subspace methods Nonlinear and
Model_order_reduction
Notion in statistics
symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix addition and inversion, as well
Fisher_information
Norm on a vector space of matrices
\|\cdot \|} is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if A ≼ B ⇒ ‖ A ‖ ≤ ‖ B ‖ . {\displaystyle
Matrix_norm
Mathematical set closed under positive linear combinations
{\displaystyle \mathbb {R} ^{n},} and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in semidefinite programming
Convex_cone
Function that maps matrices to matrices
is mapped to itself. Algebraic Riccati equation Sylvester's formula Loewner order Matrix calculus Trace inequalities Trigonometric functions of matrices
Analytic_function_of_a_matrix
Vector space with a partial order
symmetric n × n {\displaystyle n\times n} matrices with real entries. The Loewner order ≼ {\displaystyle \preccurlyeq } on two symmetric matrices A , B ∈ Sym
Ordered_vector_space
Experimental design that is optimal with respect to some statistical criterion
symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix-matrix addition, under matrix-inversion
Optimal_experimental_design
Form of differential geometry
invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail
Systolic_geometry
Features that do not change if length or energy scales are multiplied by a common factor
dimensions of random walks, where the random walks are defined via Schramm–Loewner evolution (SLE). As we have seen above, CFTs describe the physics of phase
Scale_invariance
systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given
Systoles_of_surfaces
Mathematics award
Zurich, Switzerland "For his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal
Fields_Medal
Second-order partial differential equation
3702449. S2CID 118707082. Berestycki, Nathanaël. "Lectures on Schramm–Loewner Evolution" (PDF). Retrieved 2026-03-21. Bishop, Christopher J. (2018).
Laplace's_equation
Mathematical theory on behavior of connected clusters in a random graph
predictions are backed up by arguments from conformal field theory and Schramm–Loewner evolution, and include predicted numerical values for the exponents. Most
Percolation_theory
Canadian-American mathematician (1925–2020)
the metric of hyperbolic space to the unit ball. Nirenberg and Charles Loewner studied the more general means of naturally assigning a complete Riemannian
Louis_Nirenberg
(1897). "XXIV.—On the p-discriminant of a Differential Equation of the First Order, and on Certain Points in the General Theory of Envelopes connected therewith"
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Random motion of particles suspended in a fluid
with reflecting spatial boundaries Rotational Brownian motion Schramm–Loewner evolution – Concept in probability theory Single particle tracking Single
Brownian_motion
Sequence of moves on a lattice
limit of the self-avoiding walk is conjectured to be described by Schramm–Loewner evolution with parameter κ = 8/3. Critical phenomena – Physics associated
Self-avoiding_walk
Threshold of percolation theory models
ISSN 1539-3755. PMID 17994964. Javerzat, Nina (January 2, 2024). "Schramm-Loewner Evolution in 2D Rigidity Percolation". Physical Review Letters. 132 (1)
Percolation_threshold
Statement in complex analysis; formerly the Bieberbach conjecture
independently proved the conjecture for starlike functions. Then Charles Loewner (Löwner (1923)) proved | a 3 | ≤ 3 {\displaystyle |a_{3}|\leq 3} , using
De_Branges's_theorem
Collection of random variables
winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic
Stochastic_process
Reasoning about equations with free variables
for totality is I ⊆ R R T . {\displaystyle I\subseteq RR^{T}.} Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation
Algebraic_logic
Physics of many interacting particles
function (mathematics) Quantum probability Percolation theory Schramm–Loewner evolution List of textbooks in thermodynamics and statistical mechanics
Statistical_mechanics
can then be understood as the consistency condition between the set of N Loewner equations describing the growth of each slit: ∂ p ∂ λ i = − ∂ A 0 ∂ λ j
Gibbons–Tsarev_equation
(1797–1866), mathematician Michel Loève (1907–1979), probability theory Charles Loewner (1893–1968), mathematician Alfred Loewy (1873–1935), representation theory
List_of_Jewish_mathematicians
cluster. Can also be generated by a hull-generating walk, or by Schramm-Loewner Evolution. 91 48 {\displaystyle {\frac {91}{48}}} 1.8958 2D percolation
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
smoothing Scheffé's method Scheirer–Ray–Hare test Schilder's theorem Schramm–Loewner evolution Schuette–Nesbitt formula Schwarz criterion Score (statistics)
List_of_statistics_articles
Mathematics of smooth surfaces
smallest length of any non-contractible closed curve on the surface. In 1949 Loewner proved a torus inequality for metrics on the torus, namely that the area
Differential geometry of surfaces
Differential_geometry_of_surfaces
Representation of a type of random process
notation A R ( p ) {\displaystyle AR(p)} indicates an autoregressive model of order p. The AR(p) model is defined as X t = ∑ i = 1 p φ i X t − i + ε t {\displaystyle
Autoregressive_model
Mathematical theorem
conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation. The following is known about numerically approximating
Riemann_mapping_theorem
American mathematician
Complex Analysis. vol. 3: Harmonic Analysis. vol. 4: Operator Theory. Loewner's theorem on monotone matrix functions Springer, 2019, ISBN 978-3-030-22421-9
Barry_Simon
Problem book in mathematical analysis
sources. The section on geometry (IX) contains many problems contributed by Loewner (in differential geometry) and Hirsch (in algebraic geometry). The book
Problems and Theorems in Analysis
Problems_and_Theorems_in_Analysis
etc. (see also). A non-probabilistic proof is found 18 years later. The Loewner's torus inequality relates the area of a compact surface (topologically
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Reversible diffusion Runge–Kutta method Russo–Vallois integral Schramm–Loewner evolution Semimartingale Stochastic calculus Stochastic differential equation
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Roma. 1942: 201–208. Titus, C. J. (1973). "A proof of a conjecture of Loewner and of the conjecture of Carathéodory on umbilic points". Acta Math. 131
Carathéodory_conjecture
Mathematical model for neuron networks
or baseline potential V i B {\displaystyle V_{i}^{\mathsf {B}}} , on the order of −40 to −80 millivolts. However, this apparent discrepancy exists only
Galves–Löcherbach_model
LOEWNER ORDER
LOEWNER ORDER
Surname or Lastname
English
English : variant of Lanier 1.Dutch : variant of Leonard.Jewish (western Ashkenazic) : name taken by someone who was good at chanting the Pentateuch at public worship in the synagogue or who regularly did so, from West Yiddish layner ‘reader’ (a derivative of West Yiddish laynen ‘to read’, which comes ultimately from Latin legere ‘to read’).Jewish (Ashkenazic) : occupational name for a flax grower or merchant, from German Lein ‘flax’ + agent suffix -er.
Surname or Lastname
English
English : patronymic from Lown 1.
Boy/Male
Afghan, African, Arabic, Australian, French, Iranian
Owner
Boy/Male
Muslim
Master. Owner.
Boy/Male
Christian & English(British/American/Australian)
Owner
Surname or Lastname
English
English : occupational name for a scholar or schoolmaster, from an agent derivative of Middle English lern(en), which meant both ‘to learn’ and ‘to teach’ (Old English leornian).South German : habitational name for someone from Lern near Freising.South German : nickname from Middle High German lerner ‘pupil’, ‘schoolboy’.Jewish (Ashkenazic) : occupational name from Yiddish lerner ‘Talmudic student or scholar’.
Boy/Male
Arabic
Master; Owner
Surname or Lastname
English (Sussex)
English (Sussex) : topographic name for one who lived in a township or village, Middle English toun, + -er, a characteristic topographic ending of Sussex surnames.English (Sussex) : occupational name for a toll taker or tax collector, from tolnere, an agent derivative of Middle English toll ‘tax’, ‘payment’. Compare Toller.
Boy/Male
Hindu
Land owner
Girl/Female
British, English
Controller
Surname or Lastname
English
English : unexplained. It may be of the same origin as 2.Possibly an Americanized spelling of German Lohner.
Surname or Lastname
English
English : variant of Lower.
Boy/Male
Muslim
Owner
Boy/Male
Indian
Master, Owner
Girl/Female
Hindu
Water owner
Boy/Male
Indian
Owner
Boy/Male
Muslim
Master, Owner
Girl/Female
Hindu, Indian, Kannada, Telugu
Refreshing; Owner
Girl/Female
Tamil
Sarawathi | ஸாரவாதீÂ
Water owner
Sarawathi | ஸாரவாதீÂ
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : occupational name denoting a servant who carried the ewer to guests at table so that they could wash their hands, Anglo-Norman French and Middle English ewerer (related to ewere ‘jug’), with the French definite article l’.Cornish : variant of Flower 4.
LOEWNER ORDER
LOEWNER ORDER
Girl/Female
Indian
Beautiful Angel of God
Boy/Male
Hindu, Indian, Traditional
Lord of the Universe
Boy/Male
Arabic, Australian, Muslim
Generous
Surname or Lastname
English (Cheshire)
English (Cheshire) : perhaps a habitational name from Cromwell in Nottinghamshire or Cromwell Bottom in West Yorkshire, both named from Old English crumb ‘crooked’ + wella ‘stream’, ‘spring’. The latter is recorded as Crumbel (1251) and Crumble (1566).Probably an altered spelling of German Krumpel or Krümpel, a nickname for someone with a deformity, from Middle High German krum(p) ‘deformed’, ‘crooked’; skeletal deformities were common in the Middle Ages, often as a result of rickets.
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Full Moon; Moonlight
Girl/Female
Arabic, Swahili
Woman; Life
Boy/Male
British, English
Spearman
Girl/Female
Muslim/Islamic
Wish
Boy/Male
Muslim
One who beats, Sticker
Boy/Male
Bengali, Indian
Calm; Silent
LOEWNER ORDER
LOEWNER ORDER
LOEWNER ORDER
LOEWNER ORDER
LOEWNER ORDER
a.
Compar. of Low, a.
a.
Placed in a lower order, class, or rank; holding a lower or inferior position.
a.
To let descend by its own weight, as something suspended; to let down; as, to lower a bucket into a well; to lower a sail or a boat; sometimes, to pull down; as, to lower a flag.
n.
One who owns; a rightful proprietor; one who has the legal or rightful title, whether he is the possessor or not.
p. pr. & vb. n.
of Lower
n.
Cloudiness; gloominess.
n.
A frowning; sullenness.
n.
A maker of bits, spurs, and metal mounting for bridles and saddles; hence, a saddler.
v. i.
To frown; to look sullen.
a.
To reduce the height of; as, to lower a fence or wall; to lower a chimney or turret.
n.
Alt. of Loriner
a.
To reduce in value, amount, etc. ; as, to lower the price of goods, the rate of interest, etc.
a.
To bring down; to humble; as, to lower one's pride.
v. i.
To be dark, gloomy, and threatening, as clouds; to be covered with dark and threatening clouds, as the sky; to show threatening signs of approach, as a tempest.
a.
To reduce the degree, intensity, strength, etc., of; as, to lower the temperature of anything; to lower one's vitality; to lower distilled liquors.
a.
To depress as to direction; as, to lower the aim of a gun; to make less elevated as to object; as, to lower one's ambition, aspirations, or hopes.
v. i.
To fall; to sink; to grow less; to diminish; to decrease; as, the river lowered as rapidly as it rose.
imp. & p. p.
of Lower
a.
Pertaining to, or kept in, the lower case; -- used to denote the small letters, in distinction from capitals and small capitals. See the Note under 1st Case, n., 3.
p. pr. & vb. n.
of Lower