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Framework for studying stochastic partial differential equations
Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential
Regularity_structure
Axiom of set theory
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty
Axiom_of_regularity
Topics referred to by the same term
Look up regular or regularity in Wiktionary, the free dictionary. Regular and regularity may refer to: "Regular" (Badfinger song) Regular tunings of stringed
Regular
Mathematics award
differential equations, and in particular for the creation of a theory of regularity structures for such equations." Maryam Mirzakhani Stanford University, US Stanford
Fields_Medal
General three-dimensional form of local segments of proteins
glycine are sometimes known as "helix breakers" because they disrupt the regularity of the α helical backbone conformation; However, these preferences are
Protein_secondary_structure
Regularity in sensory qualia or abstract ideas
Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical
Pattern
Austrian-British mathematician
sampling and theory of rough paths and, in 2014, on his theory of regularity structures. Under the name HairerSoft, he develops Macintosh software. Regius
Martin_Hairer
Graph partition into regular subgraphs
In extremal graph theory, Szemerédi's regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between
Szemerédi_regularity_lemma
Objects that generalize functions
a consistent way of multiplying distributions with certain structure (regularity structures), available in many examples from stochastic analysis, notably
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Concept in stochastic analysis
the KPZ equation, and later developed the more general theory of regularity structures, for which he was awarded the Fields Medal in 2014. Rough path theory
Rough_path
Causal relationships between points in a manifold
vector is timelike or null at all points in the curve. The requirements of regularity and nondegeneracy of Σ {\displaystyle \Sigma } ensure that closed causal
Causal_structure
Non-linear stochastic partial differential equation
this work on the KPZ equation, along with rough paths theory and regularity structures. There were 6 different analytic self-similar solutions found for
Kardar–Parisi–Zhang_equation
Concept in probability theory
relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure. The result was first stated and
Dudley's_theorem
Public university in Lausanne, Switzerland
Martin Hairer (Professor of Mathematics, developed the theory of regularity structures, awarded a Fields Medal in 2014) Martin Vetterli (Professor in the
École Polytechnique Fédérale de Lausanne
École_Polytechnique_Fédérale_de_Lausanne
Set of daily solar adorations
the cosmic forces represented by these deities. These structured rituals emphasize regularity and discipline in the practitioner's spiritual practice
Liber_Resh_vel_Helios
Hydrated amorphous form of silica
diffraction of light passing through the microstructure of the opal. The regularity of the sizes and the packing of these spheres is a prime determinant of
Opal
American mathematician and Nobel Laureate (1928–2015)
solutions of such equations resolved Hilbert's nineteenth problem on regularity in the calculus of variations, which had been a well-known open problem
John_Forbes_Nash_Jr.
Mathematics award
with Matthew Strom Borman and Yakov Eliashberg, overtwisted contact structures in higher dimensions." Xinwen Zhu – "For work in arithmetic algebraic
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Abstract mathematical system of two types of objects and a relation between them
Normally a block design is required to satisfy numerical regularity conditions. As an incidence structure, X is the set of points and F is the set of lines,
Incidence_structure
2013 video game
According to them, the "grind of just doing PvP until co-op Jobs arrive with regularity" was lost, and newcomers would likely find multiplayer enjoyable and balanced
Grand_Theft_Auto_V
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set X {\displaystyle X} ,
Partition_regularity
Form of figurative language
did not share necessarily a relation, but also an idea, a pattern, a regularity, an attribute, an effect or a philosophy. These authors also accepted
Analogy
Differential equations involving stochastic processes
Hairer, M. (2020). A Course on Rough Paths with an Introduction to Regularity Structures, 2nd ed., Springer-Verlag, Heidelberg, DOI https://doi.org/10
Stochastic differential equation
Stochastic_differential_equation
Totality of existing entities
electrical conductivity. Laws of nature are uniformities or regularities in the structure of reality: they describe patterns of how entities interact
Reality
Field of mathematics which studies incidence structures
Neither regularity condition implies the other, so it has to be assumed that r > 1. A finite partial linear space satisfying both regularity conditions
Incidence_geometry
Area of discrete mathematics
structure, or equivalently, a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Szemerédi's regularity
Graph_theory
Technique used for determining crystal structures and identifying mineral compounds
cases new structures are deposited in databases available to the international community. Crystals, though long admired for their regularity and symmetry
X-ray_crystallography
Australian and American mathematician (born 1975)
Global regularity of wave maps I. Small critical Sobolev norm in high dimensions. Internat. Math. Res. Notices (2001), no. 6, 299–328. Global regularity of
Terence_Tao
Interpretation of sensory information
processing over half a dozen visual cues, each of which is based on a regularity of the physical world. Vision evolved to respond to the narrow range of
Perception
Thermoplastic polymer
e-modulus and hardness. Atactic polypropylene, on the other hand, lacks any regularity, which prevents it from crystallization, thereby creating an amorphous
Polypropylene
West Germanic language
for learners of English reading in learning the specific sound-symbol regularities that occur in the standard English spellings of commonly used words.
English_language
Global Masonic organisation
The International Order of Freemasonry for Men and Women, LE DROIT HUMAIN is a global Masonic order, membership of which is available to men and women
Le_Droit_Humain
Structured system of communication
affecting only one particular word or a few words, without any seeming regularity. Sometimes a simple change triggers a chain shift in which the entire
Language
Function type in graph theory
give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs. A graphon is a symmetric
Graphon
Italian mathematician (born 1985)
was awarded the EMS Prize, "for his outstanding contributions to the regularity of solutions of Monge–Ampère equation and optimal maps and for his deep
Guido_De_Philippis
Italian mathematician (born 1984)
worked in the theory of optimal transport, with particular emphasis on the regularity theory of optimal transport maps and its connections to Monge–Ampère equations
Alessio_Figalli
Association of one output to each input
were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century
Function_(mathematics)
Convention of symbols representing language
such, the use of writing by a community presupposes an analysis of the structure of language at some level. The symbols used in writing correspond systematically
Writing_system
Notion in statistics
then the regularity conditions must hold for every component of θ. It is easy to find an example of a density that does not satisfy the regularity conditions:
Fisher_information
Tabular arrangement of the chemical elements
it, as it clarifies helium's nature as an s-block element, increases regularity by having all period lengths repeated, faithfully follows Madelung's rule
Periodic_table
Scientific field of study
usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world, which may explain the
Physics
International governing body for association football in North America
together as a unit with Warner acting as a party whip. It happened with such regularity that sports political commentators referred to the CFU votes as the "Caribbean
CONCACAF
Form of motorsport
navigate and follow the itinerary is tested. Most non-regularity rally itineraries follow this base structure even where driving tests or special stages are
Rallying
Type of differential equation
various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness
Partial_differential_equation
15th-century motet by Josquin des Prez
vocalists join in climbing melodic lines and dense syncopation. While the regularity of imitation initially articulates the phrases, the middle verses exemplify
Ave_Maria_..._virgo_serena
Visible regularity of form found in the natural world
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled
Patterns_in_nature
Graph where every edge is in one triangle
MR 0980587, S2CID 120911900 Brouwer, A. E.; Haemers, W. H. (1992), "Structure and uniqueness of the (81,20,1,6) strongly regular graph", A collection
Locally_linear_graph
Founder of Islam (c. 570–632)
bringing forth springs of water, yet he declined, reasoning that the regularities of nature already served as sufficient proof of God's majesty. Some satirized
Muhammad
Scientific study of human society and relationships
statements that generalize from segregated hypotheses and empirical regularities rather than starting with an abstract idea of a social whole. The German
Sociology
Polygon shape with eight sides
The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's
Octagon
Unpredictable phenomenon in complex systems
syntactical structures of the text and the author style (Slautina, Marusenko, 2014). It has also been argued that the structure and regularity of language
Emergence
Type of secondary structure of proteins
arrangement in the secondary structure of proteins. It is also the most extreme type of local structure, and it is the local structure that is most easily predicted
Alpha_helix
Differential operator in mathematics
Dyatlov, Semyon. "Lecture notes for 18.155: distributions, elliptic regularity, Fourier integral operators, and wave propagation" (PDF). Retrieved 2026-03-21
Laplace_operator
Standard system of axiomatic set theory
{\displaystyle \mathbb {N} } does not require the axiom of regularity; it follows naturally from the structure of the construction.) By definition, a set z {\displaystyle
Zermelo–Fraenkel_set_theory
Hierarchical outline list of articles related to the physical sciences
techniques, abstractions, apparatus, and organization for exhibiting the regularities and securing their law-like descriptions." —p.vii, J. L. Heilbron, (2003
Outline_of_physical_science
Fundamental unit of cognition
processes. Concept formation happens as people recognize similarities and regularities in the world, identify common features, and group instances into mental
Concept
Form of literature
other patterns, such as dactylic hexameter, tend to be highly irregular. Regularity can vary between language. In addition, different patterns often develop
Poetry
Smooth manifold with an inner product on each tangent space
g_{p}:T_{p}M\times T_{p}M\to \mathbb {R} } in a smooth way (see the section on regularity below). This induces a norm ‖ ⋅ ‖ p : T p M → R {\displaystyle \|\cdot
Riemannian_manifold
Subset of artificial intelligence
Imieliński and Arun Swami introduced association rules for discovering regularities between products in large-scale transaction data recorded by point-of-sale
Machine_learning
Function related to statistics and probability theory
likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are assumed in various proofs involving likelihood
Likelihood_function
Branch of mathematics
structures play a role in mathematical logic and especially Boolean algebras have major applications in computer science. Finally, various structures
Order_theory
Physiological and psychological changes in preparation for sexual intercourse
regularity observed in frogs is applicable to humans remains unknown. Unambiguous experimental evidence for the existence of the Tarkhanov regularity
Sexual_arousal
Mathematical modelling alogorithm
experimental data. Period 1968–1971 is characterized by the application of only regularity criterion for solving of the problems of identification, pattern recognition
Group_method_of_data_handling
1999 natural disaster in Venezuela
deltas here show that geologically similar catastrophes have occurred with regularity since prehistoric times. Since the 17th century, at least two large-magnitude
Vargas_tragedy
Municipal-level Turkish law enforcement body
tradition and Islamic culture, based on which the first office for public regularity was created. During the time of the Ottoman Empire, services related to
Municipal_police_(Turkey)
Topological space with a notion of uniform properties
topology (hence coincides with it) is a simple consequence of complete regularity: for any x ∈ X {\displaystyle x\in X} and a neighbourhood X {\displaystyle
Uniform_space
Danish tennis player (born 1990)
that she prefers hard and grass courts; she likes hard courts due to the regularity of the bounce, whereas the fast pace of grass courts allows her to defend
Caroline_Wozniacki
Country in Central America
From the early 19th century to the mid-1950s, Izalco erupted with a regularity that earned it the name "Lighthouse of the Pacific". Its brilliant flares
El_Salvador
French philosopher (1930–2004)
on an acute awareness of level-distinctions, a marked thoroughness and regularity... Deconstruction must be understood, we contend, as the attempt to "account
Jacques_Derrida
Refractory compound of boron and nitrogen with formula BN
of boron nitride (a-BN) is non-crystalline, lacking any long-distance regularity in the arrangement of its atoms. It is analogous to amorphous carbon.
Boron_nitride
Restriction enzyme
"How Enzymes, Proteins, and Antibodies Recognize Extended DNAs; General Regularities". International Journal of Molecular Sciences. 22 (3): 1369. doi:10.3390/ijms22031369
EcoRI
Generalization of "n-th" to infinite cases
paradox. Assuming the axiom of regularity, "strictly well-ordered" can be weakened to "strictly totally ordered", as regularity prevents infinite descending
Ordinal_number
Story in the Book of Daniel in the Hebrew Bible
Gesetzmäßigkeit frühchristlicher Ikonographie [Daniel in the lions' den. On the regularity of early Christian iconography]. Wiesbaden: Reichert. ISBN 3-89500-469-3
Daniel_in_the_lions'_den
Globally accessible seed bank on Spitsbergen, Svalbard, Norway
failures, funding cuts, and natural disasters. These events occur with some regularity. War and civil strife have a history of destroying some genebanks. The
Svalbard_Global_Seed_Vault
Stable natural languages that have developed from a pidgin
neogrammarian "tree model" for the evolution of languages, and its postulated regularity of sound changes (these critics including the earliest advocates of the
Creole_language
Most populous city in Illinois, US
direction and one street per 100 meters in the other direction. The grid's regularity provided an efficient means of developing new real estate property. A
Chicago
Diagram used in various mystical traditions
interdependence and dynamism rather than a fixed spatial order or diagrammatic regularity. Scholarly analysis indicates that early Kabbalistic literature employed
Tree_of_life_(Kabbalah)
Infection caused by Bacillus anthracis bacteria
2001. Anthrax outbreaks occur in some wild animal populations with some regularity. Russian researchers estimate arctic permafrost contains around 1.5 million
Anthrax
American mathematician
Aaron Naber on Aug16Sat". YouTube. Seoul ICM VOD. 18 August 2014. "Structure and Regularity of Nonlinear Harmonic Maps - Aaron Naber". YouTube. Università
Aaron_Naber
Smooth manifold
as the regularity hypothesis weakens). Suppose M is equipped with a symplectic form ω, a Riemannian metric g, and an almost complex structure J. Since
Almost_complex_manifold
Structure or mechanism of social order
each individual whose behavior it influences that generates behavioral regularities." Additionally, they specify that organizations "are institutional elements
Institution
Poetic meter of four anapestic feet per line
used in comic form, anapestic tetrameter is often highly regular, as the regularity emphasizes the breezy, melodic feel of the meter, though the initial unstressed
Anapestic_tetrameter
decision making but "the rule of law", which was based on procedural regularity in the exercise of power. The CCP continues to operate on the Leninist
Democracy_in_China
Second-order partial differential equation
and is positive elsewhere near the boundary. For Laplace's equation, regularity of boundary points is characterized by the classical Wiener criterion
Laplace's_equation
Interplay between observation, experiment, and theory in science
Newton; Berkeley emphasizes that scientist should seek 'reduction to regularity'. Atherton (ed.) 1999 selects Locke, Berkeley, and Hume as part of the
Scientific_method
Problem-solving method
is the process of discovering general laws ... Induction tries to find regularity and coherence ... Its most conspicuous instruments are generalization
Heuristic
Family of DNA sequences found in prokaryotic organisms
other end of the spacer cut using a ruler mechanism, thus maintaining the regularity of the spacer size in the CRISPR array. The conservation of the PAM sequence
CRISPR
Cosmological theory
mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start
Mathematical universe hypothesis
Mathematical_universe_hypothesis
Kneecap, bone covering knee joint
stabilize it during exercise. Patellar dislocations occur with significant regularity, particularly in young female athletes. It involves the patella sliding
Patella
Collection of random variables
as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined. The set T {\displaystyle T} is called
Stochastic_process
Dictator of Italy from 1922 to 1943
his mental calmness and lack of concern for discomfort, his zeal and regularity in carrying out his assignments. Mussolini's military experience is told
Benito_Mussolini
When analysing the regularities and structure of music as well as the processing of music in the brain, certain findings lead to the question of whether
Musical_syntax
Phenomenon observed in the study of Artificial Neural Networks
of overparametrization. A key mechanism of the F-Principle is that the regularity of the activation function translates into the decay rate of the loss
Frequency principle/spectral bias
Frequency_principle/spectral_bias
Family of languages and dialects Indigenous to North Africa
languages are fairly regular. Some examples, of varying importance and regularity, include [g/ž/y]; [k/š]; [l/ř/r]; [l/ž, ll/ddž]; [trill/ vocalized r];
Berber_languages
Basic organisational unit of Freemasonry
third degree). He is first usually required to check, and certify, the regularity of the relationship of the Lodge – and be able to satisfy that Lodge of
Masonic_lodge
Tool for analyzing divide-and-conquer algorithms
, and therefore, yes, c > log b a {\displaystyle c>\log _{b}a} The regularity condition also holds: 2 ( n 2 4 ) ≤ k n 2 {\displaystyle 2\left({\frac
Master theorem (analysis of algorithms)
Master_theorem_(analysis_of_algorithms)
Type of recessed barrier
lawn within; and the garden in its turn was to be set free from its prim regularity, that it might assort with the wilder country without. — Horace Walpole
Ha-ha
System where only the inputs and outputs can be viewed, and not its implementation
probes or experiments, and constructs explanatory patterns from observed regularities, knowledge gained from black box analysis is shaped by the investigation
Black_box
Book by Barbara Herrnstein Smith
Terminal Features Problems of Closure Herrnstein Smith observes that regularity — such as the regular repetition of lines of iambic tetrameter — builds
Poetic_Closure
Branch of mathematical combinatorics
has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order n; that is, there are
Ramsey_theory
REGULARITY STRUCTURE
REGULARITY STRUCTURE
Surname or Lastname
English
English : nickname from Middle English flo(u)r ‘flower’, ‘blossom’ (Old French flur, from Latin flos, genitive floris). This was a conventional term of endearment in medieval romantic poetry, and as early as the 13th century it is also regularly found as a female personal name.English : metonymic occupational name for a miller or flour merchant, or perhaps a nickname for a pasty-faced person, from Middle English flo(u)r ‘flour’. This is in origin the same word as in 1, with the transferred sense ‘flower, pick of the meal’. Although the two words are now felt to be accidental homophones, they were not distinguished in spelling before the 18th century.English : occupational name for an arrowsmith, from an agent derivative of Middle English flŠ‘arrow’ (Old English flÄ).Welsh : Anglicized form of the Welsh personal name Llywarch, of unexplained origin.Translation of French Lafleur.
Surname or Lastname
Scottish
Scottish : habitational name from Berwick-on-Tweed, on the Northumbrian coast at the mouth of the Tweed river, a border town that regularly changed hands between the Scots and the English.English : variant of Barwick.
Girl/Female
Indian
One who remembers Allah regularly
Girl/Female
Indian
Shape, Structure
Girl/Female
Indian
Structure
Girl/Female
Tamil
Shape, Structure
Girl/Female
Tamil
Shape, Structure
Surname or Lastname
English (West Midlands)
English (West Midlands) : from Middle English schovel ‘shovel’, hence a metonymic occupational name for a maker or seller of shovels, or for someone who regularly used a shovel in his work.
Girl/Female
Indian, Kashmiri
Body Structure
Boy/Male
Indian
Good Structure
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.
Boy/Male
Muslim
Solid structure
Boy/Male
Indian
Solid structure
Girl/Female
Muslim
One who remembers Allah regularly
Girl/Female
Indian
Shape, Structure
Girl/Female
Hindu, Indian, Telugu
The Structure of God
Boy/Male
Afghan, Arabic, Gujarati, Indian, Muslim
Solid Structure; Lifetime
Girl/Female
Muslim/Islamic
One who remembers Allah regularly
REGULARITY STRUCTURE
REGULARITY STRUCTURE
Female
English
English name derived from Greek aster, ASTRA means "star."
Boy/Male
English American Latin
Contemporary phonetic'enduring.
Boy/Male
Hindu, Indian, Tamil
Art Maker; Lord of Krishna
Girl/Female
Hindu, Indian
Heaven
Male
English
Short form of English Adolph, DOLPH means "noble wolf."
Boy/Male
Hindu, Indian, Telugu
Brightness
Boy/Male
Latin
West wind.
Boy/Male
Tamil
Vajrajit | வஜà¯à®°à®¾à®œà®¿à®¤
Lord Indra
Boy/Male
Assamese, Belgium, Indian, Russian
Lord Krishna
Boy/Male
Tamil
Trivendra | தà¯à®°à®¿à®µà¯‡à®¨à¯à®¤à¯à®°
The meaning of the name Trivendra is the master of the three super power like Shiv bharma and Vishnu
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
adv.
With irregularity.
a.
Being without order or regularity; disorderly; out of rule.
n.
Irregularity; disorder.
a.
Coming without regularity; occasional; incidental; as, casual expenses.
adv.
At stated times; regularly.
n.
Abnormal regularity; the state of certain flowers, which, being naturally irregular, have become regular through a symmetrical repetition of the special irregularity.
n. pl.
A division of Echini which includes the circular, or regular, sea urchins.
n.
Regularity.
a.
Shapeless; without a determinate form; wanting regularity of shape.
n.
The condition or quality of being regular; as, regularity of outline; the regularity of motion.
n.
The quality or state of being angular; angularness.
n.
Exact proportion; justness of workmanship and conduct; regularity; rule.
n.
the practice of doing things with undiscriminating, mechanical regularity.
adv.
In a regular manner; in uniform order; methodically; in due order or time.
pl.
of Irregularity
n.
Supreme attention to the things of the present life; worldliness.
a.
Being without rhythm or regularity, as the pulse.
n.
Disorder; irregularity.
n.
The state or quality of being irregular; that which is irregular.
v. t.
To cause to become regular; to regulate.