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In mathematics, an exponential field is a field with a further unary operation that is a homomorphism from the field's additive group to its multiplicative group. This generalizes the usual idea of exponentiation on the real numbers, where the base is a chosen positive real number.
## Definition
A field is an algebrai... | https://en.wikipedia.org/wiki/Exponential_field |
Optical character recognition or optical character reader (OCR) is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a scene photo (for example the text on signs and billboards in a landscape photo) ... | https://en.wikipedia.org/wiki/Optical_character_recognition |
In signal processing and related disciplines, aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the original signal, there are components at frequency exceeding a certain frequency ... | https://en.wikipedia.org/wiki/Aliasing |
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see ... | https://en.wikipedia.org/wiki/Duality_%28order_theory%29 |
Decision analysis (DA) is the discipline comprising the philosophy, methodology, and professional practice necessary to address important decisions in a formal manner. Decision analysis includes many procedures, methods, and tools for identifying, clearly representing, and formally assessing important aspects of a deci... | https://en.wikipedia.org/wiki/Decision_analysis |
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard e... | https://en.wikipedia.org/wiki/Curve_of_constant_width |
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold
$$
M
$$
is a manifold
$$
TM
$$
which assembles all the tangent vectors in
$$
M
$... | https://en.wikipedia.org/wiki/Tangent_bundle |
In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric stan... | https://en.wikipedia.org/wiki/Geometric_standard_deviation |
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is .
Triangulations may be viewed as special cases of planar straight-line graphs. When there are no hole... | https://en.wikipedia.org/wiki/Polygon_triangulation |
In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the sta... | https://en.wikipedia.org/wiki/Coefficient_of_variation |
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal gr... | https://en.wikipedia.org/wiki/Superconformal_algebra |
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.)
Formally, a cardinal κ i... | https://en.wikipedia.org/wiki/Weakly_compact_cardinal |
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely. The theory is computably axioma... | https://en.wikipedia.org/wiki/Presburger_arithmetic |
The helicoid, also known as helical surface, is a smooth surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its fixed axis of rotation. It is the third minimal surface to be known, after the plane and the catenoid.
## Descripti... | https://en.wikipedia.org/wiki/Helicoid |
Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two very first Fields Medals for this theory in 1936.
It can be considered as a generalization of the basic properties of covering maps to the
maps which are "almost cove... | https://en.wikipedia.org/wiki/Ahlfors_theory |
In geometry, a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points), and contains every point on the line that is between its endpoints. It is a special case of an arc, with zero curvature. The length of a line segment is given by the Euclidean distance between its en... | https://en.wikipedia.org/wiki/Line_segment |
## In physics
and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a numb... | https://en.wikipedia.org/wiki/Dimension |
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
$$
a_i x_{i-1} + b_i x_i + c_... | https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm |
A Lagged Fibonacci generator (LFG or sometimes LFib) is an example of a pseudorandom number generator. This class of random number generator is aimed at being an improvement on the 'standard' linear congruential generator. These are based on a generalisation of the Fibonacci sequence.
The Fibonacci sequence may be desc... | https://en.wikipedia.org/wiki/Lagged_Fibonacci_generator |
In mathematics, logic and computer science, a recursive (or decidable) language is a recursive subset of the Kleene closure of an alphabet. Equivalently, a formal language is recursive if there exists a Turing machine that decides the formal language. In theoretical computer science, such always-halting Turing machines... | https://en.wikipedia.org/wiki/Recursive_language |
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the... | https://en.wikipedia.org/wiki/Knot_invariant |
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of (integer) congruent numbers starts with
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29,... | https://en.wikipedia.org/wiki/Congruent_number |
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time fr... | https://en.wikipedia.org/wiki/Explicit_and_implicit_methods |
###
###
In mathematics,
### and
more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence
$$
0 \longrightarrow A \mathrel{\overset{q}{\longrightarrow}} B \mathrel{\overset{r}{\longrightarrow}} C \longrightar... | https://en.wikipedia.org/wiki/Splitting_lemma |
In computer science, k-approximation of k-hitting set is an approximation algorithm for weighted hitting set. The input is a collection S of subsets of some universe T and a mapping W from T to non-negative numbers called the weights of the elements of T. In k-hitting set the size of the sets in S cannot be larger than... | https://en.wikipedia.org/wiki/K-approximation_of_k-hitting_set |
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
The lemma was proved (assuming the axiom of choice) by Kazimier... | https://en.wikipedia.org/wiki/Zorn%27s_lemma |
__NOTOC__
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix
$$
A
$$
is skew-Hermitian if it satisfies the relation
where
$$
A^\textsf{H}
$$
denotes the conjugate transpose of t... | https://en.wikipedia.org/wiki/Skew-Hermitian_matrix |
In number theory, the Mertens function is defined for all positive integers n as
$$
M(n) = \sum_{k=1}^n \mu(k),
$$
where
$$
\mu(k)
$$
is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real numbers as follows:
$$
M(x) = M(\lfloor x \rfloor).
$$
Less f... | https://en.wikipedia.org/wiki/Mertens_function |
In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical statistics.
The motivation for studying empirical measures is that it is ofte... | https://en.wikipedia.org/wiki/Empirical_measure |
Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions.
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sententia... | https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined |
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
= 2463205976112133171923293141475971
≈ 8.
The finite simple groups have been completely classified. Every such group belongs ... | https://en.wikipedia.org/wiki/Monster_group |
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enri... | https://en.wikipedia.org/wiki/Internal_set_theory |
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance.
## In mathematics
, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two... | https://en.wikipedia.org/wiki/Symmetry |
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the corank of .
Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokerne... | https://en.wikipedia.org/wiki/Cokernel |
The quadratic residuosity problem (QRP) in computational number theory is to decide, given integers
$$
a
$$
and
$$
N
$$
, whether
$$
a
$$
is a quadratic residue modulo
$$
N
$$
or not.
Here
$$
N = p_1 p_2
$$
for two unknown primes
$$
p_1
$$
and
$$
p_2
$$
, and
$$
a
$$
is among the numbers which are not obv... | https://en.wikipedia.org/wiki/Quadratic_residuosity_problem |
In geometry, a conical surface is an unbounded three-dimensional surface formed from the union of infinite lines that pass through a fixed point and a space curve.
## Definitions
A (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex... | https://en.wikipedia.org/wiki/Conical_surface |
The Jefferson disk, also called the Bazeries cylinder or wheel cypher, is a cipher system commonly attributed to Thomas Jefferson that uses a set of wheels or disks, each with letters of the alphabet arranged around their edge in an order, which is different for each disk and is usually ordered randomly.
Each disk is m... | https://en.wikipedia.org/wiki/Jefferson_disk |
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a type of second-order ordinary differential equation named after the French physicist Alfred-Marie Liénard.
During the development of radio and vacuum tube technology, Liénard equations were intensely ... | https://en.wikipedia.org/wiki/Li%C3%A9nard_equation |
Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the effects of a system's actions (its outputs) return as inputs to that system, influencing subsequent action. It is concerned with general principles that are relevant across multiple contexts, including in e... | https://en.wikipedia.org/wiki/Cybernetics |
In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.
## Definition
For a Lie algebra
$$
\mathfrak{g}
$$
over a field
$$
K
$$
, if
$$
K[t,t^{-1}]
$$
is the space of Laurent polynomials, then
$$
L\mathfrak{g} := \mathfrak{g}\otimes K[t,t^{-1}],
$$
with the i... | https://en.wikipedia.org/wiki/Loop_algebra |
Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein estimation and empirical Bayes methods — and its applications to portfolio choice theory. The theorem gives a formula for ... | https://en.wikipedia.org/wiki/Stein%27s_lemma |
In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form M.
Manifold decomposition works in two directions: one can start with the smaller pieces and bui... | https://en.wikipedia.org/wiki/Manifold_decomposition |
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any... | https://en.wikipedia.org/wiki/Unique_factorization_domain |
In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line
$$
(-\infty, +\infty... | https://en.wikipedia.org/wiki/Square-integrable_function |
The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (... | https://en.wikipedia.org/wiki/Turn_%28angle%29 |
The Khintchine inequality, is a result in probability also frequently used in analysis bounding the expectation a weighted sum of Rademacher random variables with square-summable weights. It is named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet.
It states that for each
$$
p\in (0,\infty)... | https://en.wikipedia.org/wiki/Khintchine_inequality |
In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal
$$
x[n]
$$
and a finite impulse response (FIR) filter
$$
h[n]
$$
:
where for m outside the region .
This article uses common abstract notations, such as
$$
y(t) = x(t) * h... | https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method |
Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional
### NURBS
-based CAD design tools. Currently, it is necessary to convert data between CAD and FEA packages to analyse new designs during development, a difficult task since the t... | https://en.wikipedia.org/wiki/Isogeometric_analysis |
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group
$$
H_2(G, \Z)
$$
of a group G. It was introduced by in his work on projective representations.
## Examples and properties
The Schur multiplier
$$
\operatorname{M}(G)
$$
of a finite group G is a finite abelian gro... | https://en.wikipedia.org/wiki/Schur_multiplier |
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply be viewed as a function on superspace.... | https://en.wikipedia.org/wiki/Supermultiplet |
Motion blur is the apparent streaking of moving objects in a photograph or a sequence of frames, such as a film or animation. It results when the image being recorded changes during the recording of a single exposure, due to rapid movement or long exposure.
## Usages / Effects of motion blur
### Photography
When a came... | https://en.wikipedia.org/wiki/Motion_blur_%28media%29 |
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by .
The statement concerns the Taylor coe... | https://en.wikipedia.org/wiki/De_Branges%27s_theorem |
In computer graphics, the rendering equation is an integral equation that expresses the amount of light leaving a point on a surface as the sum of emitted light and reflected light. It was independently introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The equation is important in the th... | https://en.wikipedia.org/wiki/Rendering_equation |
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieti... | https://en.wikipedia.org/wiki/Fiber_product_of_schemes |
In mathematics and computer science, the syntactic monoid
$$
M(L)
$$
of a formal language
$$
L
$$
is the minimal monoid that recognizes the language
$$
L
$$
. By the Myhill–Nerode theorem, the syntactic monoid is unique up to unique isomorphism.
## Syntactic quotient
An alphabet is a finite set.
The free monoid on... | https://en.wikipedia.org/wiki/Syntactic_monoid |
A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in order to account for the presence of distortionary market instruments (e.g... | https://en.wikipedia.org/wiki/Shadow_price |
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid (liquids and... | https://en.wikipedia.org/wiki/Computational_fluid_dynamics |
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.
It typically involves using computer programs to compute approximate solutions to Maxwell's equa... | https://en.wikipedia.org/wiki/Computational_electromagnetics |
In differential geometry, the four-gradient (or 4-gradient)
$$
\boldsymbol{\partial}
$$
is the four-vector analogue of the gradient
$$
\vec{\boldsymbol{\nabla}}
$$
from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the vario... | https://en.wikipedia.org/wiki/Four-gradient |
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.... | https://en.wikipedia.org/wiki/Akaike_information_criterion |
In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
- In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space
$$
V
$$
into its field of scalars (that is, it is an el... | https://en.wikipedia.org/wiki/Functional_%28mathematics%29 |
In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.
## Definition
Let
$$
(M, d)
$$
be a complete separable metric space. Let
$$
\mathcal{K}
$$
denote the set of all compact subsets of
$$
M
... | https://en.wikipedia.org/wiki/Random_compact_set |
In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.. Equality between and is written , and read " equals ". In this equality, and are distinguished by calling them left-hand side (LHS), and right-hand sid... | https://en.wikipedia.org/wiki/Equality_%28mathematics%29 |
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutati... | https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups |
In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter
$$
x_0
$$
, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in o... | https://en.wikipedia.org/wiki/Location_parameter |
In mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934.
## Algorithm
The sieve starts with a list of the integers from 1 to . From this list,... | https://en.wikipedia.org/wiki/Sieve_of_Sundaram |
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal
$$
x
$$
, the unique real
$$
x_0
$$
infinitely close to i... | https://en.wikipedia.org/wiki/Standard_part_function |
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly eq... | https://en.wikipedia.org/wiki/Sparse_matrix |
A display list, also called a command list in Direct3D 12 and a command buffer in Vulkan, is a series of graphics commands or instructions that are run when the list is executed. Systems that make use of display list functionality are called retained mode systems, while systems that do not are as opposed to immediate m... | https://en.wikipedia.org/wiki/Display_list |
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each mapping from X to Y. The entry in row x and column y... | https://en.wikipedia.org/wiki/Incidence_matrix |
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of res... | https://en.wikipedia.org/wiki/Arithmetic_of_abelian_varieties |
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measu... | https://en.wikipedia.org/wiki/Ellipse |
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography. They published a list of semiprimes (numbers with exactly two prime fa... | https://en.wikipedia.org/wiki/RSA_Factoring_Challenge |
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see Fig. 1), the law of cosines states:
$$
\begin{align}
c^2 &= a^2 + b^2 - 2a... | https://en.wikipedia.org/wiki/Law_of_cosines |
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact ... | https://en.wikipedia.org/wiki/Markov%27s_inequality |
In mathematics, Sharkovskii's theorem (also spelled Sharkovsky, Sharkovskiy, Šarkovskii or Sarkovskii), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line ha... | https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem |
An exploded-view drawing is a diagram, picture, schematic or technical drawing of an object, that shows the relationship or order of assembly of various parts.
It shows the components of an object slightly separated by distance, or suspended in surrounding space in the case of a three-dimensional exploded diagram. An o... | https://en.wikipedia.org/wiki/Exploded-view_drawing |
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:
$$
p = x^2 + y^2,
$$
with x and y integers, if and only if
$$
p \equiv 1 \pmod{4}.
$$
The prime numbers for which this is true are called Pythagorean primes.
For example, the primes 5, 13, 17, 29, 37 and 4... | https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares |
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who... | https://en.wikipedia.org/wiki/Kleene_star |
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.
## Ramification theory of valuations
In mathematics, the ramification theory of valua... | https://en.wikipedia.org/wiki/Ramification_group |
Engineering is the practice of using natural science, mathematics, and the engineering design process to solve problems within technology, increase efficiency and productivity, and improve systems. Modern engineering comprises many subfields which include designing and improving infrastructure, machinery, vehicles, ele... | https://en.wikipedia.org/wiki/Engineering |
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive grou... | https://en.wikipedia.org/wiki/Pontryagin_duality |
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963.
Hoeffding's inequality is a special case of the Az... | https://en.wikipedia.org/wiki/Hoeffding%27s_inequality |
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data.
## History
Modern analytic celestial mechanics starte... | https://en.wikipedia.org/wiki/Celestial_mechanics |
Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions. For instance... | https://en.wikipedia.org/wiki/Probability_bounds_analysis |
Micro black holes, also known as mini black holes and quantum mechanical black holes, are hypothetical tiny (<1 ) black holes, for which quantum mechanical effects play an important role. The concept that black holes may exist that are smaller than stellar mass was introduced in 1971 by Stephen Hawking.
It is possible ... | https://en.wikipedia.org/wiki/Micro_black_hole |
Simplex noise is the result of an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with fewer directional artifacts, in higher dimensions, and a lower computational overhead. Ken Perlin designed the algorithm in 2001 to address the limitations of his classic noise function, especially in h... | https://en.wikipedia.org/wiki/Simplex_noise |
An attractor network is a type of recurrent dynamical network, that evolves toward a stable pattern over time. Nodes in the attractor network converge toward a pattern that may either be fixed-point (a single state), cyclic (with regularly recurring states), chaotic (locally but not globally unstable) or random (stocha... | https://en.wikipedia.org/wiki/Attractor_network |
A polyalphabetic cipher is a substitution, using multiple substitution alphabets. The Vigenère cipher is probably the best-known example of a polyalphabetic cipher, though it is a simplified special case. The Enigma machine is more complex but is still fundamentally a polyalphabetic substitution cipher.
## History
The... | https://en.wikipedia.org/wiki/Polyalphabetic_cipher |
In mathematics, the Weil conjectures were highly influential proposals by . 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 local zeta functio... | https://en.wikipedia.org/wiki/Weil_conjectures |
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".
Examples are 11, 666, 4444, and 999999. All repdigits are palindromic number... | https://en.wikipedia.org/wiki/Repdigit |
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple poly... | https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization |
A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. Tutorial review based on online MIT course notes. The key property of a PML that di... | https://en.wikipedia.org/wiki/Perfectly_matched_layer |
A palindrome (/ˈpæl.ɪn.droʊm/) is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as madam or racecar, the date "02/02/2020" and the sentence: "A man, a plan, a canal – Panama". The 19-letter Finnish word saippuakivikauppias (a soapstone vendor) is the longest singl... | https://en.wikipedia.org/wiki/Palindrome |
Parrondo's paradox, a paradox in game theory, describes how a combination of losing strategies can become a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996.
A simple example involves two coin flip games: Game A uses a biased coin that loses 50.5% of the time, while Gam... | https://en.wikipedia.org/wiki/Parrondo%27s_paradox |
In statistics, the algebra of random variables provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treatment of sums, products, ratios and general functions of random variables, as... | https://en.wikipedia.org/wiki/Algebra_of_random_variables |
In mathematics, a subset
$$
A
$$
of a Polish space
$$
X
$$
is universally measurable if it is measurable with respect to every complete probability measure on
$$
X
$$
that measures all Borel subsets of
$$
X
$$
. In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see below)... | https://en.wikipedia.org/wiki/Universally_measurable_set |
A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
## Formulation
### Basics
In decimal, unit fractions and have no repeating decimal, while repeats
$$
0.3333\dots
$$
indefinitely. The remainder of , on the other hand, repeats over six digits as,
$$
0.\b... | https://en.wikipedia.org/wiki/Prime_reciprocal_magic_square |
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
- A... | https://en.wikipedia.org/wiki/C%2A-algebra |
In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invaria... | https://en.wikipedia.org/wiki/Directed_algebraic_topology |
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