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Concept in topology
In mathematics, topological complexity of a topological space X (also denoted by TC(X)) is a topological invariant closely connected to the motion planning
Topological_complexity
Topological complexity in mathematics
norm) is a measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes. Given
Simplicial_volume
Number of "holes" of a surface
along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules. Arithmetic genus Geometric
Genus_(mathematics)
Node ordering for directed acyclic graphs
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed
Topological_sorting
Topological operation of turning a sphere inside-out without creasing
eversion can be described by a set of closed equations giving minimal topological complexity A six-dimensional sphere S 6 {\displaystyle S^{6}} in seven-dimensional
Sphere_eversion
Number representing system complexity
mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system
Topological_entropy
Soviet and American mathematician (born 1934)
first example of a topological quantum field theory. The Schwarz genus, one of the fundamental notions of topological complexity, is named after him
Albert_Schwarz
Research field in deep learning
graphs, or general topological spaces like simplicial complexes and CW complexes. TDL addresses this by incorporating topological concepts to process
Topological_deep_learning
introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces. Let M {\displaystyle
Thurston_norm
Mathematical method for optimizing material layout under given conditions
while 0 indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large
Topology_optimization
Concept in topology
functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct
Topological_conjugacy
Analysis of datasets using techniques from topology
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information
Topological_data_analysis
Measure of a graph's centrality, based on shortest paths
introduction by ego). Betweenness centrality has been used to analyze the topological complexity of river networks as well as their use in maritime trade. Betweenness
Betweenness_centrality
Concerned with the notion of stability in model theory
by restricting the topological complexity of the type spaces. However, Morley showed that (for countable theories) this topological restriction is equivalent
Stable_theory
of topological index. To increase the discrimination capability a few topological indices may be combined to superindex. Computational complexity is another
Topological_index
Observation in computer circuit design
the interconnection complexity of a circuit. Higher (intrinsic) Rent exponent values correspond to a higher topological complexity. One extreme example
Rent's_rule
Concept in mathematics
(physics) State space (physics) Farber, Michael; Grant, Mark (2009). "Topological complexity of configuration spaces". Proceedings of the American Mathematical
Configuration space (mathematics)
Configuration_space_(mathematics)
Feature of systems that defy description
Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity
Complexity
Mathematical subject
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics
Topological_combinatorics
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
is that passing to infinitely many dimensions causes much of the topological complexity of the unitary groups to vanish; but see the section on Bott's unitary
Kuiper's_theorem
Type of quantum computer
processors, the first used a toric code with twist defects as a topological degeneracy (or topological defect) while the second used a different but related protocol
Topological_quantum_computer
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
Field of mathematics and science based on non-linear systems and initial conditions
f^{k}(U)\cap V\neq \emptyset } . Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given
Chaos_theory
1950 short story by Armin Joseph Deutsch
himself, he believes that the new shuttle connection caused the topological complexity of the system to increase to the point where the connectivity became
A_Subway_Named_Mobius
Russian-American physicist (born 1963)
contributed to the classification of topological phases. He related two-dimensional lattice models of topological order to algebraic data describing the
Alexei_Kitaev
Mathematical theory
onto. Similar to stochastic closure, the full picture of a curve's topological complexity must be determined by sampling the knotoids of many projections
Open_knot_theory
Topological quantum error correcting code
quantum double models. It is also the simplest example of topological order—Z2 topological order (first studied in the context of Z2 spin liquid in 1991)
Surface_code
Computational complexity of quantum algorithms
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational
Quantum_complexity_theory
Study of discrete mathematical structures
metric spaces, there are more general discrete topological spaces, finite metric spaces, finite topological spaces. The time scale calculus is a unification
Discrete_mathematics
Type of spatial relationship
POLYVRT (Harvard University, 1976). The strategy of the topological data model is to store topological relationships (primarily adjacency) between features
Geospatial_topology
Topics referred to by the same term
algebraic structure Simplicial complex, a kind of topological space CW complex, a kind of topological space Line complex, a 3-dimensional family of lines
Complex
Italian-British applied mathematician
in particular) and structural complexity. He is known for his contributions to the field of geometric and topological fluid dynamics and, in particular
Renzo_L._Ricca
Mathematical description of mixing substances
weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator. Topological mixing
Mixing_(mathematics)
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
Computational complexity class of problems
In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial
BQP
Concept in math
other categories in geometry and algebra. The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between
Homotopy_category
Unsolved problem in computational complexity theory
time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem
Graph_isomorphism_problem
Computer hardware technology that uses quantum mechanics
using superconducting electronic circuits. Topological quantum computer — proposed approach using topological states such as anyons. Trapped-ion quantum
Quantum_computing
Algorithm to search the nodes of a graph
postorderings are A C B D and A B C D. Reverse postordering produces a topological sorting of any directed acyclic graph. This ordering is also useful in
Depth-first_search
Function that counts distinct factors of a string
exists as the logarithm of the complexity function is subadditive. Every real number between 0 and 1 occurs as the topological entropy of some sequence is
Complexity_function
Mathematical set with some added structure
linear and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is
Space_(mathematics)
Data structure used for indexing spatial information
leaf page. Leaf pages overlap little, but directory pages do. R*-tree topological split. The pages overlap very little since the R*-tree tries to minimize
R*-tree
Algorithm to be run on quantum computers
quantum Fourier transform, quantum walks, amplitude amplification and topological quantum field theory. Quantum algorithms may also be grouped by the type
Quantum_algorithm
Complexity class
computational complexity theory, the class QIP (which stands for Quantum Interactive Proof) is the quantum computing analogue of the classical complexity class
QIP_(complexity)
Tool to track locally defined data attached to the open sets of a topological space
{\displaystyle F} arises from a natural topological situation, E {\displaystyle E} may not have any clear topological interpretation. For example, if F {\displaystyle
Sheaf_(mathematics)
Real-valued number of spatial dimensions
dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. Unlike topological dimensions, the fractal index
Fractal_dimension
Mathematical structure
topological manifold with some additional mathematical structure that allows for differential calculus on the manifold. If M is already a topological
Differential_structure
peptide's N-terminus to form the knot. (Some lasso peptides also have topological complexity conferred by disulfide bonds.) Isopeptidase enzymes linearize the
Isopeptidase
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
Fewest graph edges whose removal breaks all cycles
of a topological space derived from the graph. It counts the ears in an ear decomposition of the graph, forms the basis of parameterized complexity on almost-trees
Cyclomatic_number
Graph algorithm
the strongly connected components are identified constitutes a reverse topological sort of the DAG formed by the strongly connected components. Donald Knuth
Tarjan's strongly connected components algorithm
Tarjan's_strongly_connected_components_algorithm
Pathological embedding of the sphere in 3D space
3-dimensional Euclidean space. The topological object was discovered by J. W. Alexander (1924). It is a particular topological embedding of a two-dimensional
Alexander_horned_sphere
Maximal subgraph whose vertices can reach each other
sets. Just as the number of connected components of a topological space is an important topological invariant, the zeroth Betti number, the number of components
Component_(graph_theory)
Computational Complexity, 8 (4): 316–329, doi:10.1007/s000370050002, S2CID 10641238. Ben-Amram, Amir M.; Galil, Zvi (2001), "Topological Lower Bounds on
Element_distinctness_problem
Theory of stochastic partial differential equations
inherent topological supersymmetry (TS) enabling the generalization of certain concepts from deterministic to stochastic models. Using tools of topological field
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Computational benchmark
engineering task of building a powerful quantum computer and the computational-complexity-theoretic task of finding a problem that can be solved by that quantum
Quantum_supremacy
Quantum Computing Division/Software
chip powered by a topological core architecture. The work created a new class of materials called topoconductors, which use topological superconductivity
Microsoft_Azure_Quantum
Quantum analog of probabilistic automata
The behaviour of topological automata is studied in the field of topological dynamics. The quantum automaton differs from the topological automaton in that
Quantum_finite_automaton
Subfield of mathematical topology
Computable topology (the study of the topological nature of computation) Computational geometry Digital topology Topological data analysis Spatial-temporal reasoning
Computational_topology
Topics referred to by the same term
number of lighthouse Fresnel lenses, defining size and focal length Topological order in quantum mechanics, an organized quantum state First-order hold
Order
Complexity class
number of satisfying assignments. Topologically sorting is easy in contrast to counting the number of topological sortings. A single perfect matching
♯P-complete
the mean (topological) dimension of a topological dynamical system is a non-negative extended real number that is a measure of the complexity of the system
Mean_dimension
Method for solving one problem using another
nontrivial problem, see p. 48. Schaefer, Marcus (2010), "Complexity of some geometric and topological problems" (PDF), Graph Drawing, 17th International Symposium
Polynomial-time_reduction
Flat-sided three-dimensional shape
notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks (the faces) whose pairwise
Polyhedron
Structural complexity is a science of applied mathematics that aims to relate fundamental physical or biological aspects of a complex system with the mathematical
Structural complexity (applied mathematics)
Structural_complexity_(applied_mathematics)
Copy of a directed graph with redundant edges removed
Garey & Ullman (1972), who provided tight bounds on the computational complexity of constructing them. More technically, the reduction is a directed graph
Transitive_reduction
Quantum algorithm
Bernstein–Vazirani algorithm was designed to prove an oracle separation between complexity classes BQP and BPP. Given an oracle that implements a function f : {
Bernstein–Vazirani_algorithm
Overview of and topical guide to algorithms
medians Order statistic tree Depth-first search Breadth-first search Topological sorting Flood fill Dijkstra's algorithm Bellman–Ford algorithm Floyd–Warshall
Outline_of_algorithms
Stage of electronic circuit design
gEDA suite) TopRouter (the topological pre-router in CadSoft/Autodesk's EAGLE 7.0 and higher) SimplifyPCB (a topological router with a focus on bundle
Routing (electronic design automation)
Routing_(electronic_design_automation)
Particle
entirely on braiding and performing topological charge measurements, and hence form a natural setting for topological quantum computing. This is in contrast
Fibonacci_anyons
Embedding a graph in a topological space, often Euclidean
cellular embeddings include the ribbon graph, a topological space formed by gluing together topological disks for the vertices and edges of an embedded
Graph_embedding
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)
Computational problem
reductions. Note that, in terms of time complexity, it can be solved in linear time simply by a topological sort. The Boolean formula value problem (or
Circuit_value_problem
Attribute of machine learning models
The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function
Sample_complexity
Computer science
In computational complexity theory, exact quantum polynomial time (EQP or sometimes QP) is the class of decision problems that can be solved by a quantum
Exact_quantum_polynomial_time
Basic unit of quantum information
up gapped topological system non-abelian anyons braiding of excitations depends on specific topological system depends on specific topological system vibrational
Qubit
Branch of game theory about two-player sequential games with perfect information
of taking objects from one or two piles Topological game, a type of mathematical game played in a topological space Zugzwang, being obliged to play when
Combinatorial_game_theory
Graph layout on multiple half-planes
include abstract algebra and knot theory. The notion of a book, as a topological space, was defined by C. A. Persinger and Gail Atneosen in the 1960s
Book_embedding
Swarming behaviour of birds when flying or foraging
where it was shown that Starlings typically interact with at most seven topological neighbors. Improvements: Spatial subdivision. The entire area/volume
Flocking
Study of sudden qualitative behavior changes caused by small parameter changes
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of
Bifurcation_theory
Type of shift space studied in ergodic theory
shifts of finite type. Subshifts of finite type are also sometimes called topological Markov shifts. Many chaotic dynamical systems are isomorphic to subshifts
Subshift_of_finite_type
Class of algorithms which use a moving line to solve geometrical problems
algorithm) and the Delaunay triangulation or boolean operations on polygons. Topological sweeping is a form of plane sweep with a simple ordering of processing
Sweep_line_algorithm
Process in quantum computing
requires at least five physical qubits. Beyond coding-theoretic designs, topological QECCs are particularly intuitive to visualize and can provide a clear
Quantum_error_correction
Hungarian and American mathematician and physicist (1903–1957)
defining locally convex spaces and topological vector spaces for the first time. In addition several other topological properties he defined at the time
John_von_Neumann
Integral expressing the amount of overlap of one function as it is shifted over another
general. In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless G is unimodular
Convolution
Types of quantum information
of up to 1,000 physical qubits. The approach of topological qubits, which takes advantage of topological effects in quantum mechanics, has been proposed
Physical_and_logical_qubits
Topological model
The Dimensionally Extended 9-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two
DE-9IM
Property of functions which is weaker than continuity
arbitrary topological space X {\displaystyle X} is locally constant on some dense open subset of X . {\displaystyle X.} If the topological space X {\displaystyle
Semi-continuity
Branch of mathematics
distinction when needed. Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds
Algebraic_geometry
Complex living system in the soil
fumigation. There are three different types of food web representations: topological (or traditional) food webs, flow webs and interaction webs. These webs
Soil_food_web
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
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)
Principle in computer science
similar thesis was later stated by Michael Freedman in an early review of topological quantum computing with Alexei Kitaev, Michael J. Larsen, and Zhenghan
Church–Turing–Deutsch principle
Church–Turing–Deutsch_principle
System capable of producing itself
unity in space in which they (the components) exist by specifying the topological domain of its realization as such a network." They describe the "space
Autopoiesis
Invariant in graph theory
In the mathematical field of graph theory, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using
Queue_number
Principle in statistical learning theory
minimization given a fixed function class can be derived using bounds on the VC complexity of the function class. For simplicity, considering the case of binary
Empirical_risk_minimization
Finding an optimal algorithm for playing chess
solved at least weakly. Calculated estimates of game-tree complexity and state-space complexity of chess exist which provide a bird's eye view of the computational
Solving_chess
Numerical method that reduces the complexity of computationally intensive simulations
orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics
Proper orthogonal decomposition
Proper_orthogonal_decomposition
Fewest edge crossings in drawing of a graph
1137/120872310. S2CID 6535755. Schaefer, Marcus (2010). Complexity of some geometric and topological problems (PDF). Graph Drawing, 17th International Symposium
Crossing number (graph theory)
Crossing_number_(graph_theory)
Problem in computer science
In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on
Simon's_problem
TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
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 COMPLEXITY
TOPOLOGICAL COMPLEXITY
Boy/Male
Tamil
Sun rays of God
Girl/Female
Hindu, Indian
Beautiful
Boy/Male
Hindu
Boy/Male
Latin
God of cattle.
Boy/Male
Hindu, Indian, Kannada, Sanskrit, Telugu
Lord of the Gods
Girl/Female
Hindu
Pleasing metrical composition
Boy/Male
Hindu, Indian, Sanskrit
Servant of Shiva
Male
Hawaiian
 Hawaiian unisex name KAILA means "style." Compare with strictly feminine Kaila.
Boy/Male
Tamil
Satisfied, Loved, Joyful
Surname or Lastname
English and Irish
English and Irish : variant spelling of Hamlin.Respelling of French Hamelin.
TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY
a.
Of or pertaining to zoology, or the science of animals.
v. i.
To introduce innovations in doctrine, esp. in theological doctrine.
a.
Of or pertaining to orology.
a.
Theological.
a.
Of or pertaining to nosology.
a.
Of or pertaining to noology.
a.
Of or pertaining to theology, or the science of God and of divine things; as, a theological treatise.
a.
Of or pertaining tootology.
a.
Relating to a horologe, or to horology.
a.
Pertaining to doxology; giving praise to God.
a.
Pertaining to posology.
a.
Of or pertaining to pomology.
a.
Alt. of Tropological
adv.
In a zoological manner; according to the principles of zoology.
n.
A student in a theological seminary.
a.
Alt. of Posological
a.
Pertaining to homology; having a structural affinity proceeding from, or base upon, that kind of relation termed homology.
a.
Characterized by tropes; varied by tropes; tropical.
a.
Of or pertaining to oology.
v. t.
To use in a tropological sense, as a word; to make a trope of.