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In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and mer... | Wikipedia/Sorting_algorithm |
Algorithmic entities refer to autonomous algorithms that operate without human control or interference. Recently, attention is being given to the idea of algorithmic entities being granted (partial or full) legal personhood. Professor Shawn Bayern and Professor Lynn M. LoPucki popularized through their papers the idea ... | Wikipedia/Algorithmic_entities |
In mathematics, a system of bilinear equations is a special sort of system of polynomial equations, where each equation equates a bilinear form with a constant (possibly zero). More precisely, given two sets of variables represented as coordinate vectors x and y, then each equation of the system can be written
... | Wikipedia/System_of_bilinear_equations |
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the va... | Wikipedia/Matrix_difference_equation |
In number theory, the Erdős–Moser equation is
1
k
+
2
k
+
⋯
+
(
m
−
1
)
k
... | Wikipedia/Erdős–Moser_equation |
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation
has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are... | Wikipedia/Fermat–Catalan_conjecture |
In number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Sriniva... | Wikipedia/Ramanujan–Nagell_equation |
Diophantus and Diophantine Equations is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by Nauka in 1972 under the title Диофант и диофантовы уравнения. It was transla... | Wikipedia/Diophantus_and_Diophantine_Equations |
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural nu... | Wikipedia/Catalan's_conjecture |
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half s... | Wikipedia/Theta_functions |
The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer
n
{\displaystyle n}
that is greater than or equal to 2, there exist positive integers
x
{\displaystyle x}
,
... | Wikipedia/Erdős–Straus_conjecture |
The Beal conjecture is the following conjecture in number theory:
If
A
x
+
B
y
=
C
z
... | Wikipedia/Beal's_conjecture |
"The Equation" is the eighth episode of the first season of the American science fiction drama television series Fringe. The episode follows the Fringe team's investigation into the kidnapping of a young musical prodigy (Charlie Tahan) who has become obsessed with finishing one piece of music. Dr. Walter Bishop (John N... | Wikipedia/The_Equation |
The equation of time describes the discrepancy between two kinds of solar time. The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion along the celestial equator. Apparent solar time can be... | Wikipedia/Equation_of_time |
Equation were a British, young Devon-based folk supergroup formed in 1995, which combined the core talents of the Lakeman Brothers with Kathryn Roberts and Kate Rusby, later replaced for a spell by Cara Dillon. The name of the band refers to the initials of the band members' names, KR2 + SL3.
Their first single "He Lo... | Wikipedia/Equation_(band) |
The Equation Group, also known in China as APT-C-40, is a highly sophisticated threat actor suspected of being tied to the Tailored Access Operations (TAO) unit of the United States National Security Agency (NSA). Kaspersky Labs describes them as one of the most sophisticated advanced persistent threats in the world an... | Wikipedia/Equation_Group |
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow tha... | Wikipedia/Chemical_equation |
An equation clock is a mechanical clock which includes a mechanism that simulates the equation of time, so that the user can read or calculate solar time, as would be shown by a sundial. The first accurate clocks, controlled by pendulums, were patented by Christiaan Huyghens in 1657. For the next few decades, people we... | Wikipedia/Equation_clock |
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ... | Wikipedia/Numerical_solution |
In field theory, a branch of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient ... | Wikipedia/Minimal_polynomial_(field_theory) |
In mathematics, specifically algebraic geometry, a period or algebraic period is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the... | Wikipedia/Period_(algebraic_geometry) |
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exac... | Wikipedia/Fundamental_unit_(number_theory) |
In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry... | Wikipedia/Semialgebraic_set |
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the origin... | Wikipedia/End_(topology) |
In topology, puncturing a manifold is removing a finite set of points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point, it becomes twice-punctured, and so on.
Examples of punctured manifolds include the open d... | Wikipedia/Puncturing_(topology) |
In mathematics, the Teichmüller space
T
(
S
)
{\displaystyle T(S)}
of a (real) topological (or differential) surface
S
{\displaystyle S}
is a space that parametrizes complex structures on
... | Wikipedia/Teichmüller_theory |
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on
n
{\displaystyle n}
-dimensional Euclidean space
R
n
... | Wikipedia/Euclidean_topology |
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.
A subset of a topological space
... | Wikipedia/Connected_component_(topology) |
In topology, especially algebraic topology, the cone of a topological space
X
{\displaystyle X}
is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by
C
X
... | Wikipedia/Cone_(topology) |
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that ... | Wikipedia/Monoidal_transformation |
In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though so... | Wikipedia/Classification_of_algebraic_surfaces |
In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
The conjectures concern the generating functions (known as l... | Wikipedia/Weil_conjectures |
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate e... | Wikipedia/Algebraic_equivalence |
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
{
3
x
+
... | Wikipedia/Linear_simultaneous_equations |
A successive-approximation ADC (or SAR ADC) is a type of analog-to-digital converter (ADC) that digitizes each sample from a continuous analog waveform using a binary search through all possible quantization levels.
== History ==
The SAR ADC was first used for experimental pulse-code modulation (PCM) by Bell Labs in... | Wikipedia/Successive-approximation_ADC |
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
sin
θ
... | Wikipedia/Small-angle_approximation |
In philosophy of science, idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false but make models easier to understand or solve. That is, it is determined whether the phenomenon approximates an "ideal case," then the model is applied to make a predi... | Wikipedia/Idealization_(philosophy_of_science) |
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics... | Wikipedia/Circular_function |
A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in
Z
{\displaystyle \mathbb {Z} }
(the integers) and with alg... | Wikipedia/Hypertranscendental_function |
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of differential operators, in which the coeff... | Wikipedia/Algebraic_differential_equation |
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers
Q
{\displaystyle \mathbb {Q} }
, which would establish the transcendence of a... | Wikipedia/Schanuel's_conjecture |
Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.
== Introduction ==
A general approach to the least squares problem
m
i
n
‖
... | Wikipedia/Numerical_methods_for_linear_least_squares |
In computing, a cache-oblivious algorithm (or cache-transcendent algorithm) is an algorithm designed to take advantage of a processor cache without having the size of the cache (or the length of the cache lines, etc.) as an explicit parameter. An optimal cache-oblivious algorithm is a cache-oblivious algorithm that use... | Wikipedia/Cache-oblivious_algorithm |
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.
Th... | Wikipedia/Generalized_minimal_residual_method |
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in... | Wikipedia/Matrix_multiplication_algorithm |
Automatically Tuned Linear Algebra Software (ATLAS) is a software library for linear algebra. It provides a mature open source implementation of BLAS APIs for C and FORTRAN 77.
ATLAS is often recommended as a way to automatically generate an optimized BLAS library. While its performance often trails that of specialized... | Wikipedia/Automatically_Tuned_Linear_Algebra_Software |
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by < and >, respectively the less-than... | Wikipedia/Systems_of_polynomial_inequalities |
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, i... | Wikipedia/Wu's_method_of_characteristic_set |
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
== Introduction ==
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. Th... | Wikipedia/Regular_semi-algebraic_system |
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... | Wikipedia/Inconsistent_equations |
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a real coordinate space,... | Wikipedia/Transformation_(geometry) |
A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, message switching, or packet switching, to pass messages and signals.
Multiple n... | Wikipedia/Telecommunications_network |
A pantograph (or "pan" or "panto") is an apparatus mounted on the roof of an electric train, tram or trolley buses to collect power through contact with an overhead line. The term stems from the resemblance of some styles to the mechanical pantographs used for copying handwriting and drawings.
The pantograph is a commo... | Wikipedia/Pantograph_(rail) |
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:
f
(
a
b
)
=... | Wikipedia/Additive_function |
In number theory, a multiplicative function is an arithmetic function
f
{\displaystyle f}
of a positive integer
n
{\displaystyle n}
with the property that
f
(
1
)
=
1
... | Wikipedia/Multiplicative_function |
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.
== Introduction ==
A prototypical example, the Ri... | Wikipedia/Functional_equation_(L-function) |
In systems engineering, software engineering, and computer science, a function model or functional model is a structured representation of the functions (activities, actions, processes, operations) within the modeled system or subject area.
A function model, similar with the activity model or process model, is a graph... | Wikipedia/Functional_model |
Böttcher's equation, named after Lucjan Böttcher, is the functional equation
F
(
h
(
z
)
)
=
(
F
(
z
)
)
n
{\displaystyle... | Wikipedia/Böttcher's_equation |
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, ... | Wikipedia/Logarithm_function |
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
f
(
h
(
x
)
)
=
h
(
x
+
1
)
{\displaystyle f(h(x))=h(x+1)}
or
... | Wikipedia/Abel_equation |
Cauchy's functional equation is the functional equation:
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
.
{\displaystyle f(x+y)=f(x)+f(y).\ }
A function ... | Wikipedia/Cauchy's_functional_equation |
In mathematics, an even function is a real function such that
f
(
−
x
)
=
f
(
x
)
{\displaystyle f(-x)=f(x)}
for every
x
{\displaystyle x}
in its domain. Simil... | Wikipedia/Even_function |
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwel... | Wikipedia/Maxwell's_equations_in_curved_spacetime |
In electrical engineering, a transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer's core, which induces a varying electromotive force ... | Wikipedia/Transformer |
The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance (see magnetic capacitance section) rather than el... | Wikipedia/Gyrator–capacitor_model |
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may be said to have electric potential energy by virtue of either its own electric charge or... | Wikipedia/Electric_Potential_Energy |
In physics, the magnetomotive force (abbreviated mmf or MMF, symbol
F
{\displaystyle {\mathcal {F}}}
) is a quantity appearing in the equation for the magnetic flux in a magnetic circuit, Hopkinson's law. It is the property of certai... | Wikipedia/Magnetomotive_force |
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell str... | Wikipedia/Electromagnetic_stress–energy_tensor |
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, current sources, resistances, inductances, capacitances). An electrical circuit... | Wikipedia/Electrical_network |
Physics of Fluids is a monthly peer-reviewed scientific journal covering fluid dynamics, established by the American Institute of Physics in 1958, and is published by AIP Publishing. The journal focus is the dynamics of gases, liquids, and complex or multiphase fluids—and the journal contains original research resultin... | Wikipedia/Physics_of_Fluids |
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay (retarded time) of the fields due to the finite speed of light and relativistic eff... | Wikipedia/Jefimenko's_equations |
The London equations, developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the sim... | Wikipedia/London_equations |
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may be said to have electric potential energy by virtue of either its own electric charge or... | Wikipedia/Electrostatic_energy |
Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gl... | Wikipedia/topology |
In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology.
When some object X is said to cover another object Y, the cover is given by some surjective and... | Wikipedia/Cover_(algebra) |
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of eithe... | Wikipedia/Bijective_function |
In mathematics, the graph of a function
f
{\displaystyle f}
is the set of ordered pairs
(
x
,
y
)
{\displaystyle (x,y)}
, where
f
(
x
)
=
... | Wikipedia/Function_graph |
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finit... | Wikipedia/Simplicial_approximation_theorem |
The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics.
The elements of the n-dimensional grid cell topology (n ≥ 1) are all n-dimensional grid cubes and their k-dimensional faces ( for 0 ≤ k ≤ n−1); between thes... | Wikipedia/Grid_cell_topology |
A COVID‑19 vaccine is a vaccine intended to provide acquired immunity against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the virus that causes coronavirus disease 2019 (COVID‑19).
Knowledge about the structure and function of previous coronaviruses causing diseases like severe acute respiratory syndr... | Wikipedia/COVID-19_vaccine |
In mathematics, the Bing–Borsuk conjecture states that every
n
{\displaystyle n}
-dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the... | Wikipedia/Bing–Borsuk_conjecture |
In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhelm Waldhausen in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are ... | Wikipedia/Graph_manifold |
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Ri... | Wikipedia/Thurston's_geometrization_conjecture |
Cancer immunotherapy (immuno-oncotherapy) is the stimulation of the immune system to treat cancer, improving the immune system's natural ability to fight the disease. It is an application of the fundamental research of cancer immunology (immuno-oncology) and a growing subspecialty of oncology.
Cancer immunotherapy expl... | Wikipedia/Cancer_immunotherapy |
In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex
K
{\displaystyle K}
, the space
K
×
[
0
,
1
]
{... | Wikipedia/Zeeman_conjecture |
Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation.
It was found... | Wikipedia/Geometry_and_Topology |
In mathematics, the surgery structure set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which he... | Wikipedia/Surgery_structure_set |
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory
... | Wikipedia/Spectrum_(homotopy_theory) |
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory t... | Wikipedia/Obstruction_theory |
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem.
== Definition ==
Given a connected an... | Wikipedia/Signature_(topology) |
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension
>
4
{\displaystyle >4}
. The surgery structure set
S
... | Wikipedia/Surgery_exact_sequence |
In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topolo... | Wikipedia/Borel_conjecture |
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling.
== Definitions ==
Given a 3-manifold
M
... | Wikipedia/Dehn_surgery |
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse t... | Wikipedia/Morse_function |
In mathematics, specifically in surgery theory, the surgery obstructions define a map
θ
:
N
(
X
)
→
L
n
(
π
... | Wikipedia/Surgery_obstruction |
In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative
f
′
(
a
)
{\textstyle f'(a)}
of a function
f
... | Wikipedia/Fundamental_increment_lemma |
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of emp... | Wikipedia/Mathematical_theory |
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.
== Relation to other conjectures ==
A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered b... | Wikipedia/Elliptization_conjecture |
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space
H
3
/
Γ
{\displaystyle \mathbb {H} ^{3}/\Gamma }
wh... | Wikipedia/Kleinian_model |
In mathematics, the braid group on n strands (denoted
B
n
{\displaystyle B_{n}}
), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose... | Wikipedia/Braid_theory |
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