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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, both have unusual conformational
Protein_secondary_structure
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
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
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)
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
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
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
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
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
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
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
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
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
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
British mathematician
theorem. In 2003, Gowers established a regularity lemma for hypergraphs, analogous to the Szemerédi regularity lemma for graphs. In 2005, he introduced
Timothy_Gowers
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.
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
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
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
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
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
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
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
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
How one process influences another
relations directly. On this ground, the scholar distinguished between the regularity view of causality and the counterfactual notion. According to the counterfactual
Causality
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
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
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
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
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
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
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
French philosopher (1930–2019)
being solely a structure imposed by the mind, the transcendental exists in the real world itself. There are regularities, structures, codes inscribed
Michel_Serres
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
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
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
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
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
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
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
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
Hungarian and American mathematician and physicist (1903–1957)
the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA. During World War II, von Neumann
John_von_Neumann
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
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)
Philanthropy conception of meaning
meaning as use, a feature of a social convention and conventions as regularities of a specific sort. Lewis' work was an application of game theory in
Meaning_(philosophy)
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
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
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
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
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
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
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
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)
City in Peru
astonished the Spaniards by the beauty of its edifices, the length and regularity of its streets." The great square was surrounded by several palaces, since
Cusco
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
Systematic endeavour to gain knowledge
techniques, abstractions, apparatus, and organization for exhibiting the regularities and securing their law-like descriptions. Cohen, Eliel (2021). "The boundary
Science
American vlogger and entrepreneur (born 1980)
brothers announced that videos might not be released with their usual regularity, and for the first time abandoned their four-minute limit for non-educational
Hank_Green
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
Portuguese-Dutch philosopher (1632–1677)
Spinoza's ideas strongly appealed to them: the unity of all that exists, the regularity of all that happens, and the identity of spirit and nature. By 1879, Spinoza's
Baruch_Spinoza
Type of neutron star with beams of radiation
Experimental demonstrations have been reported in 2018. Generally, the regularity of pulsar emission does not rival the stability of atomic clocks. They
Pulsar
Church in Jerusalem
occasional disagreements, religious services take place in the Church with regularity and coexistence is generally peaceful. An example of concord between the
Church_of_the_Holy_Sepulchre
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
1994 software engineering book
program should reflect only the problem it needs to solve. Any other regularity in the code is a sign, to me at least, that I'm using abstractions that
Design_Patterns
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
Problem-solving method
is the process of discovering general laws ... Induction tries to find regularity and coherence ... Its most conspicuous instruments are generalization
Heuristic
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
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
Government of France from 1870 to 1940
center-right party after World War I. Governing coalitions collapsed with regularity, rarely lasting more than a few months, as radicals, socialists, liberals
French_Third_Republic
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
American Sport
than there being five home and away games for each team, has no clear regularity in which opponents play how many times. In a late August 2025 interview
2026_UFL_season
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
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
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
Rule system for formal languages
linguistics, some authors use the term phrase structure grammar to refer to context-free grammars, whereby phrase-structure grammars are distinct from dependency
Context-free_grammar
Preserved brain of the scientist
Albert Einstein's brain was removed shortly after his death. His apparent regularities or irregularities in the brain have been used to support various ideas
Brain_of_Albert_Einstein
Non-arbitrary attachment of sounds to object shapes
of possible words. He speculates that there must therefore be certain regularities "which the human soul follows in the process of name-giving". German
Bouba/kiki_effect
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
Informal understanding of acceptable conduct
behavior and may be held privately. "Prevalent behaviors" and behavioral regularities are not necessarily norms. Instinctual or biological reactions, personal
Social_norm
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
Constructed human language based on predicate logic
written in any orthographic system as long as it satisfies the required regularities and unambiguities. Some of the reasons for such elasticity would be as
Lojban
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
Generalized manifold
subdivision X ' of X always satisfies the following condition, weaker than regularity: whenever σ and g·σ are subsimplices of some simplex τ, they are equal
Orbifold
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
Dutch painter (1872–1944)
particularly obvious in the "lozenge" works that Mondrian began producing with regularity in the mid-1920s. The "lozenge" paintings are square canvases tilted 45
Piet_Mondrian
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
British mathematician (born 1977)
arithmetic regularity lemma for functions defined on the first N {\displaystyle N} natural numbers, somewhat analogous to the Szemerédi regularity lemma for
Ben_Green_(mathematician)
Irregular beating of the atria of the heart
300 beats per minute (bpm) around the atrium. In AF, there is no such regularity, except at the sources where the local activation rate can exceed 500
Atrial_fibrillation
American period drama television series (2007–2015)
bonus, etc.) unknown or little used at the time "creeps in with striking regularity." Mad Men is widely considered one of the most influential shows in the
Mad_Men
American lynching victim (1941–1955)
all time": "For almost a century, African Americans were lynched with regularity and impunity. Now, thanks to a mother's determination to expose the barbarousness
Emmett_Till
Structure at the rear of the vertebrate brain, beneath the cerebrum
intuition ("a ha") in working memory. The comparative simplicity and regularity of the cerebellar anatomy led to an early hope that it might imply a similar
Cerebellum
President of Indonesia from 1945 to 1967
nationalist militiamen and other Indonesians. Shootouts broke out with alarming regularity in Jakarta, including an attempted assassination of Prime Minister Sjahrir
Sukarno
Probabilistic graphical representation of causal relationships
assuming that previously computed expected values are correct. Under mild regularity conditions, this process converges on maximum likelihood (or maximum posterior)
Bayesian_network
Chemical element with atomic number 17 (Cl)
oxidation states are fractional. In addition, all the above chemical regularities are valid for "normal" or close to normal conditions, while at ultra-high
Chlorine
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
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)
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.
Boy/Male
Afghan, Arabic, Gujarati, Indian, Muslim
Solid Structure; Lifetime
Boy/Male
Indian
Solid structure
Girl/Female
Tamil
Shape, 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.
Girl/Female
Muslim/Islamic
One who remembers Allah regularly
Girl/Female
Indian
One who remembers Allah regularly
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.
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
Tamil
Shape, Structure
Girl/Female
Muslim
One who remembers Allah regularly
Girl/Female
Indian
Structure
Girl/Female
Indian
Shape, Structure
Girl/Female
Indian, Kashmiri
Body Structure
Boy/Male
Muslim
Solid structure
Girl/Female
Indian
Shape, Structure
Boy/Male
Indian
Good Structure
Girl/Female
Hindu, Indian, Telugu
The Structure of God
REGULARITY STRUCTURE
REGULARITY STRUCTURE
Boy/Male
Tamil
Great
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sikh, Telugu
Love of the Teacher; Love of Guru
Girl/Female
French English
Divine.
Girl/Female
Hindu
Lover of gold
Girl/Female
Hindu, Indian
Permanent; Who is Made Forever
Surname or Lastname
English
English : habitational name from places called Merton in London, Devon, Norfolk, and Oxfordshire, named in Old English with mere ‘lake’, ‘pool’ + tūn ‘enclosure’, ‘settlement’. Compare Marton, Martin 2.
Boy/Male
Anglo, British, Christian, Dutch, English, French, German
Brave Ruler
Girl/Female
Teutonic
Ruler of the home.
Male
Arthurian
, ("the foaming one"); a horse of Arthur's.
Boy/Male
Hindu
Learning ocean
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
REGULARITY STRUCTURE
a.
Coming without regularity; occasional; incidental; as, casual expenses.
n.
Abnormal regularity; the state of certain flowers, which, being naturally irregular, have become regular through a symmetrical repetition of the special irregularity.
n.
Irregularity; disorder.
a.
Shapeless; without a determinate form; wanting regularity of shape.
a.
Being without order or regularity; disorderly; out of rule.
n. pl.
A division of Echini which includes the circular, or regular, sea urchins.
n.
the practice of doing things with undiscriminating, mechanical regularity.
adv.
At stated times; regularly.
n.
The condition or quality of being regular; as, regularity of outline; the regularity of motion.
n.
The state or quality of being irregular; that which is irregular.
adv.
With irregularity.
n.
Exact proportion; justness of workmanship and conduct; regularity; rule.
v. t.
To cause to become regular; to regulate.
adv.
In a regular manner; in uniform order; methodically; in due order or time.
n.
Disorder; irregularity.
n.
Supreme attention to the things of the present life; worldliness.
n.
Regularity.
a.
Being without rhythm or regularity, as the pulse.
n.
The quality or state of being angular; angularness.
pl.
of Irregularity