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In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random
Topological_recursion
Theory of computer sciences
In the formal language theory of computer science, left recursion is a special case of recursion where a string is recognized as part of a language by the
Left_recursion
Fractal analysis technique
dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity Iterated function system Barnsley fern Cantor set
Box_counting
Continuous function that is not absolutely continuous
dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity Iterated function system Barnsley fern Cantor set
Cantor_function
Type of binary relation
and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The
Well-founded_relation
Fractal related to the mandelbrot set
usually are not given by a formula (these are called topological polynomials): given a topological quadratic whose branch point is periodic with period
Douady_rabbit
Fractal creation method
dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity Iterated function system Barnsley fern Cantor set
Chaos_game
Fractal sets in complex dynamics of mathematics
dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity Iterated function system Barnsley fern Cantor set
Julia_set
American theoretical physicist
theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor
Edward_Witten
Infinitely detailed mathematical structure
curve map is not a homeomorphism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert
Fractal
Research institute in Saclay, France
Quantum field theory, conformal field theory, integrable systems, topological recursion, combinatorics, random geometries Condensed matter physics Statistical
Institute of Theoretical Physics, Saclay
Institute_of_Theoretical_Physics,_Saclay
Type of manifold in differential geometry
OCLC 22509804. Dunin-Barkowski, Petr (2024). "Symplectic duality for topological recursion". Transactions of the American Mathematical Society. arXiv:2206
Symplectic_manifold
American mathematician
Penner, R. C.; Reidys, Christian M.; Sułkowski, Piotr (2012). "Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces".
Robert_Penner
Rewriting system and type of formal grammar
above to the earlier recursion, one gets: Axiom First recursion Second recursion Third recursion Fourth recursion Seventh recursion, scaled down ten times
L-system
Finite or infinite ordered list of elements
the above theorems to spaces without metrics. The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped
Sequence
Open subset of the real–number line
Cantor's devil's staircase is not fractal because its Hausdorff and topological dimensions coincide. However, the Cantor staircase function possesses
Fractal_string
Set of points on a line segment with certain topological properties
that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace
Cantor_set
Topics referred to by the same term
Brightcove, an American company, stock ticker: BCOV BCOV equations in topological recursion This disambiguation page lists articles associated with the title
BCOV
Generalization of "n-th" to infinite cases
ordinals. This is known as transfinite recursion. Formally, a function F is defined by transfinite recursion on the ordinals if, for every ordinal α
Ordinal_number
Three-dimensional fractal
first described by Karl Menger in 1926, in his studies of the concept of topological dimension. It has similar properties as the Cantor set and the Cantor
Menger_sponge
Element mapped to itself by a mathematical function
imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked
Fixed_point_(mathematics)
American annual mathematics conference
Results on Ricci Flow Chiu-Chu Melissa Liu (Columbia University) - Topological Recursion and Crepant Transformation Conjecture Bing Wang (USTC) - Local entropy
Geometry_Festival
Geometric structure used in certain particle interactions
amplituhedron. Using twistor theory, Britto–Cachazo–Feng–Witten recursion (BCFW recursion) relations involved in the scattering process may be represented
Amplituhedron
Class of algorithms
self-referential (recursively defined) data structure, traversal can be defined by recursion or, more subtly, corecursion, in a natural and clear fashion; in these
Tree_traversal
Statistical model for 2D crystals
{\displaystyle T>0} . Melting of 2D crystals is mediated by the dissociation of topological defects, which destroy the order of the crystal. In 2016, Michael Kosterlitz
KTHNY_theory
Fractal named after mathematician Benoit Mandelbrot
increase in interest in complex dynamics and abstract mathematics, and the topological and geometric study of the Mandelbrot set remains a key topic in the
Mandelbrot_set
Value approached by a mathematical object
abstract space in which limits can be defined are topological spaces. If X {\displaystyle X} is a topological space with topology τ {\displaystyle \tau }
Limit_(mathematics)
Branch of mathematical logic
The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding
Reverse_mathematics
Set of all limit points of a set
Cantor–Bendixson derivative of a topological space X is defined by repeatedly applying the derived set operation using transfinite recursion as follows: X 0 = X {\displaystyle
Derived_set_(mathematics)
Aspect of theoretical physics
between N = 4 supersymmetric Yang–Mills theory and the perturbative topological B model string theory in twistor space. It was initially proposed by
Twistor_string_theory
Sphere with radius one, usually centered on the origin of the space
displayed precision. The A n {\displaystyle A_{n}} values satisfy the recursion: A 0 = 2 {\displaystyle A_{0}=2} A 1 = 2 π {\displaystyle A_{1}=2\pi }
Unit_sphere
Topological concept for collections of sets
is a topological space in which every open cover admits a point-finite open refinement. Every locally finite collection of subsets of a topological space
Point-finite_collection
Mathematical model of the time dependence of a point in space
needed] A topological dynamical system is a dynamical system (T, X, Φ) on a locally compact and/or Hausdorff topological space X. T is a topological isomorphism
Dynamical_system
Intersection of Set Theory and General Topology
topological space X are its cardinality and the cardinality of its topology, denoted respectively by |X| and o(X). The weight w(X ) of a topological space
Set-theoretic_topology
Three-holed sphere
construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms
Pair_of_pants_(mathematics)
Sierpiński–Menger type of fractal
corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one dimension
Mosely_snowflake
Technique in 3D computer graphics
described a technique for a direct evaluation of the limit surface without recursion. Catmull–Clark surfaces are defined recursively, using the following refinement
Catmull–Clark subdivision surface
Catmull–Clark_subdivision_surface
Swedish mathematician
is a Swedish mathematician/logician and expert on domain theory and recursion theory (also known as computability theory). Viggo received his PhD in
Viggo_Stoltenberg-Hansen
Infinite sum
Manfred P. (1999). Topological Vector Spaces (2nd ed.). New York: Springer. ISBN 978-1-4612-7155-0. Trèves, François (1967). Topological Vector Spaces, Distributions
Series_(mathematics)
Condition for a mathematical function to map some value to itself
function can be given, in computability theory, by applying Kleene's recursion theorem. These results are not equivalent theorems; the Knaster–Tarski
Fixed-point_theorem
Well-quasi-ordering of finite trees
second-order arithmetic theory with a form of arithmetical transfinite recursion). In 2004, the result was generalized from trees to graphs as the Robertson–Seymour
Kruskal's_tree_theorem
French mathematician (1920–1993)
differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis. Reeb was born in
Georges_Reeb
Mathematical proposition equivalent to the axiom of choice
directly using transfinite recursion, still assuming the axiom of choice. For that, see for example Transfinite recursion theorem § Example: a basis construction
Zorn's_lemma
Additional mathematical object
features are related in a certain way, then the structure becomes a topological group. A map between two similarly-structured sets that preserves their
Mathematical_structure
the properties of topological spaces and structures defined on them. It differs from other branches of topology as the topological spaces do not have
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Sequence of operations for a task
Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem
Algorithm
Whole of an object being mathematically similar to part of itself
Logarithmic spiral Long-range dependency Non-well-founded set theory Recursion Self-dissimilarity Self-reference Self-replication Self-similarity of
Self-similarity
Mathematical set containing no elements
turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique
Empty_set
Type of neural network which utilizes recursion
structures, or a scalar prediction on it, by traversing a given structure in topological order. These networks were first introduced to learn distributed representations
Recursive_neural_network
Class of mathematical orderings
below. Initial segments are also used in the statement of the transfinite recursion theorem. Properties of initial segments include: A well-ordered set is
Well-order
Every polynomial has a real or complex root
The proof that this statement results from the previous ones is done by recursion on n: when a root r 1 {\displaystyle r_{1}} has been found, the polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Turing. The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious. This article provides detail of that debate
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Monster and modular connection
Koike–Norton–Zagier identity is related to J. The twisted denominator identities imply recursion relations on the coefficients of Tg, and unpublished work of Koike showed
Monstrous_moonshine
Mathematical logic hierarchy
important in measure theory and analysis. The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains
Borel_hierarchy
Possible axiom for set theory
Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined;
Axiom_of_determinacy
Fractal composed of triangles
remaining smaller triangles infinitely. Each removed triangle (a trema) is topologically an open set. This process of recursively removing triangles is an example
Sierpiński_triangle
Functions of an angle
the language of topological groups. The set U {\displaystyle U} of complex numbers of unit modulus is a compact and connected topological group, which has
Trigonometric_functions
Subfield of mathematical logic
Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is
Descriptive_set_theory
Axiom of set theory
indexed family of compact topological spaces is compact. The closure of the product of any indexed family of subsets of a topological space is equal to the
Axiom_of_choice
Russian-American physicist (b. 1945)
confinement. In 1980, Migdal found that matrix models could be applied to topological quantum field theories such as quantum gravity. Initial results obtained
Alexander_Migdal
Function, homomorphism, or morphism
of mathematical functions Homeomorphism – Mapping which preserves all topological properties of a given space List of chaotic maps Maplet arrow (↦) – commonly
Map_(mathematics)
Book on philosophy of mathematics
effect ordered pair; relation; function; operation Proximity; connection Topological space; mereotopology Following Successive actions Function composition;
Mathematics, Form and Function
Mathematics,_Form_and_Function
supremum(X) ∈ O, then X ∩ O ≠ ∅. Using the Scott topological definition of open it is apparent that all topological properties are met. ⋅∅ and D, i.e. the empty
Computable_topology
Association of one output to each input
successor, and projection functions via the operators composition, primitive recursion, and minimization. Although defined only for functions from integers to
Function_(mathematics)
Mathematical technique in algebraic geometry
and nr. r is called the length of the dévissage. The last step of the recursion consists of a dévissage in dimension nr which includes a morphism αr :
Dévissage
Computational model used in machine learning
particle swarm optimization are other learning algorithms. Convergent recursion is a learning algorithm for cerebellar model articulation controller (CMAC)
Neural network (machine learning)
Neural_network_(machine_learning)
Set of all things that may be the input of a mathematical function
mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty
Domain_of_a_function
Mathematical concept for comparing objects
{\displaystyle X} is a topological space, there is a natural way of transforming X / ∼ {\displaystyle X/\sim } into a topological space; see Quotient space
Equivalence_relation
Branch of mathematics that studies sets
etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α {\displaystyle \alpha } , known as its rank. The
Set_theory
German mathematician (1868–1942)
sets"—the topological chapters—Hausdorff developed for the first time, based on the known neighborhood axioms, a systematic theory of topological spaces
Felix_Hausdorff
Theory proposed by Roger Penrose
in twistor space. Another key development was the introduction of BCFW recursion. This has a natural formulation in twistor space that in turn led to remarkable
Twistor_theory
Index of articles associated with the same name
different meaning, of a decomposition of a topological space X into disjoint subsets each of which is a topological manifold (so that in particular a stratification
Stratification_(mathematics)
Theorem in mathematical logic
the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty
Compactness_theorem
Maximally helicity violating amplitudes
Lorentz-violating counterterms. BCFW recursion, also known as the Britto–Cachazo–Feng–Witten (BCFW) on-shell recursion method, is a way of calculating scattering
MHV_amplitudes
Programming language for quantum algorithms
located in Santa Barbara and directed by Michael Freedman, that explored topological quantum computing. During a Microsoft Ignite Keynote on September 26
Q_Sharp
On solvability of Diophantine equations
non-member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intuitive notion of
Hilbert's_tenth_problem
Foundational controversy in twentieth-century mathematics
This is in fact the so-called "induction schema" used in the notion of "recursion" that was still in development at this time (van Heijenoort p. 493). This
Brouwer–Hilbert_controversy
Computer science algorithm
before exploring its breadth. A stack (often the program's call stack via recursion) is generally used when implementing the algorithm. The algorithm begins
Graph_traversal
Method of machine learning
\mathbb {R} ^{i}} and the sequence c i {\displaystyle c_{i}} satisfies the recursion: c 0 = 0 {\displaystyle c_{0}=0} ( c i ) j = ( c i − 1 ) j , j = 1 , 2
Online_machine_learning
(mathematical logic) Kirby–Paris theorem (proof theory) Kleene's recursion theorem (recursion theory) König's theorem (set theory, mathematical logic) Lindström's
List_of_theorems
Theorem in descriptive set theory
whether a subset of Aω will be determined depends to some extent on its topological structure. For the purposes of Gale–Stewart games, the set A is endowed
Borel_determinacy_theorem
Real number that can be computed within arbitrary precision
CS1 maint: multiple names: authors list (link) P. Odifreddi, Classical Recursion Theory (1989), p.8. North-Holland, 0-444-87295-7 Turing (1936). Minsky
Computable_number
Proposition in mathematical logic
sketch, but this was also incorrect, although it influenced later ideas in recursion theory. In 1906, Kőnig revised part of his attempted CH disproof and established
Continuum_hypothesis
isometry. For topological groups: group isomorphism which is also a homeomorphism of the underlying topological spaces. For topological vector spaces:
Equivalent definitions of mathematical structures
Equivalent_definitions_of_mathematical_structures
Set whose elements all belong to another set
of parts and the wholes they form Region – Connected open subset of a topological spacePages displaying short descriptions of redirect targets Subset sum
Subset
Mathematical transform that expresses a function of time as a function of frequency
This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional
Fourier_transform
algorithm: computes lowest common ancestors for pairs of nodes in a tree Topological sort: finds linear order of nodes (e.g. jobs) based on their dependencies
List_of_algorithms
subset of a topological space has the Baire property if it differs from an open set by a meager set 3. The Baire space is a topological space whose points
Glossary_of_set_theory
Programming paradigm
of programs that can be created easily (e.g. those involving loops or recursion), as well as making it harder for users to reason effectively about their
Differentiable_programming
Mathematical result or axiom on order relations
required to satisfy the above recursive condition, then the transfinite recursion theorem ensures this defines the function f {\displaystyle f} uniquely
Hausdorff_maximal_principle
the development of topological data analysis (TDA). This interdisciplinary field attempts to apply abstract geometric and topological concepts to identify
History_of_mathematics
Motifs in the works of Jorge Luis Borges
Jorge Luis Borges (1899–1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences, although Borges' strongest links
Jorge Luis Borges and mathematics
Jorge_Luis_Borges_and_mathematics
Size of a possibly infinite set
Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, Springer-Verlag. Eduard Čech, Topological Spaces,
Cardinal_number
Formula for the derivative of a product
of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f, denoted here Logder(f), is
Product_rule
Axiomatic set theories based on the principles of mathematical constructivism
{\displaystyle g(Sn)=f(g(n))} . This iteration- or recursion principle is akin to the transfinite recursion theorem, except it is restricted to set functions
Constructive_set_theory
Diagram that shows all possible logical relations between a collection of sets
a stained-glass window in memory of Venn. Edwards–Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum, which were based
Venn_diagram
History of maths
Čech cohomology, homotopy groups of a topological space. 1933 Solomon Lefschetz Singular homology of topological spaces. 1934 Reinhold Baer Ext groups
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Coefficients coupled with angular momentum
under even permutations. Compact groups form a wide class of groups with topological structure. They include the finite groups with added discrete topology
3-j_symbol
Problem in combinatorial optimization
35)=505,m(1,29)=505,m(1,23)=505\\\end{aligned}}} Besides, we can break the recursion and convert it into a tree. Then we can cut some leaves and use parallel
Knapsack_problem
Value indicating the relation of a proposition to truth
or computational content. For example, one may use the open sets of a topological space as intuitionistic truth values, in which case the truth value of
Truth_value
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
Surname or Lastname
English
English : variant of Sewell.Samuel Sewall (1652–1730) came with his parents from Bishop Stoke, Hampshire, England, to Newbury, MA, as a nine-year-old boy. In 1676 he married Hannah Hull, a wealthy heiress, and in 1681 he was appointed printer to the Council in Boston. He served as a judge in the infamous Salem witchcraft trials of 1692—the only one of the judges to admit publicly that he had been wrong. In 1700 he published The Selling of Joseph, which argues that all men are created equal and presents theological arguments against slavery.
Surname or Lastname
English
English : regional name from the district around Middlesbrough named Cleveland ‘the land of the cliffs’, from the genitive plural (clifa) of Old English clif ‘bank’, ‘slope’ + land ‘land’.Americanized spelling of Norwegian Kleiveland or Kleveland, habitational names from any of five farmsteads in Agder and Vestlandet named with Old Norse kleif ‘rocky ascent’ or klefi ‘closet’ (an allusion to a hollow land formation) + land ‘land’.Grover Cleveland (1837–1908), 22nd and 24th president of the U.S., was the fifth child of a country Presbyterian clergyman. His father, Richard Falley Cleveland, a graduate of Yale College and of the theological seminary at Princeton, was descended from a certain Moses Cleaveland who arrived in MA in 1635.
Surname or Lastname
English and French
English and French : from a medieval personal name, ultimately from Greek Basileios ‘royal’. The name was borne by a 4th-century bishop of Caesarea in Cappadocia, regarded as one of the four Fathers of the Eastern Church; he wrote important theological works and established a rule for religious orders of monks. Various other saints are also known under these and cognate names. The popularity of Vasili as a Russian personal name is largely due to the fact that this was the ecclesiastical name of St. Vladimir (956–1015), Prince of Kiev, who was chiefly responsible for the introduction of Christianity to Russia. As an American surname, this has also absorbed some Greek, Russian, and other derivatives of Greek Vasili.
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
Boy/Male
Muslim
The giver of life
Girl/Female
Hindu, Indian
Clean
Boy/Male
Hindu
Sarvaniki endrudu
Girl/Female
Tamil
Jennisha | ஜேநà¯à®¨à¯€à®·à®¾
Dispeller of ignorance
Boy/Male
Bengali, Indian
Good Luck; Good Light
Boy/Male
Indian, Sanskrit
Devotee of the Gods
Girl/Female
Greek Russian
God's gift.
Female
Turkish
Turkish name ELMAS means "diamond."
Girl/Female
Tamil
Amaya | ஆமய , அமயÂ
Night rain
Boy/Male
Norse
Half Dane.
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
TOPOLOGICAL RECURSION
a.
Alt. of Tropological
n.
A student in a theological seminary.
a.
Characterized by tropes; varied by tropes; tropical.
a.
Pertaining to doxology; giving praise to God.
a.
Of or pertaining tootology.
a.
Relating to a horologe, or to horology.
a.
Of or pertaining to theology, or the science of God and of divine things; as, a theological treatise.
a.
Of or pertaining to nosology.
a.
Of or pertaining to orology.
a.
Of or pertaining to noology.
a.
Theological.
a.
Alt. of Posological
v. t.
To use in a tropological sense, as a word; to make a trope of.
a.
Pertaining to posology.
adv.
In a zoological manner; according to the principles of zoology.
a.
Of or pertaining to oology.
a.
Of or pertaining to zoology, or the science of animals.
a.
Pertaining to homology; having a structural affinity proceeding from, or base upon, that kind of relation termed homology.
v. i.
To introduce innovations in doctrine, esp. in theological doctrine.
a.
Of or pertaining to pomology.