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A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria).
Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coo... | https://en.wikipedia.org/wiki/Heronian_tetrahedron |
In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.
Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation.
## Monotone cubic Hermite interpolation
Monotone ... | https://en.wikipedia.org/wiki/Monotone_cubic_interpolation |
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set
$$
x
$$
the existence of a set
$$
\mathcal{P}(x)
$$
, the power set of
$$
x
$$
, consisting precisely of the subsets of
$$
x
$$
. By the axiom of extensionality, the set
$$
\mathcal{P}... | https://en.wikipedia.org/wiki/Axiom_of_power_set |
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
## Motivation
Let X be a topological space, and let
$$
\mathcal{U}
$$
be an open cover of X. Let
$$
N(\mat... | https://en.wikipedia.org/wiki/%C4%8Cech_cohomology |
In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).
The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y.... | https://en.wikipedia.org/wiki/Tractor_bundle |
Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. This dimensionless quantity can also be rephrased as t... | https://en.wikipedia.org/wiki/Sinuosity |
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector b... | https://en.wikipedia.org/wiki/Serre%E2%80%93Swan_theorem |
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. ... | https://en.wikipedia.org/wiki/Dirichlet_problem |
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of... | https://en.wikipedia.org/wiki/Direct_multiple_shooting_method |
In mathematics, the Gudermannian function relates a hyperbolic angle measure
$$
\psi
$$
to a circular angle measure
$$
\phi
$$
called the gudermannian of
$$
\psi
$$
and denoted
$$
\operatorname{gd}\psi
$$
. The Gudermannian function reveals a close relationship between the circular functions and hyperbolic funct... | https://en.wikipedia.org/wiki/Gudermannian_function |
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphis... | https://en.wikipedia.org/wiki/Direct_limit |
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all ... | https://en.wikipedia.org/wiki/Dual_polyhedron |
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004, as a part of his PhD dissertation entitled Totally Antisymmetric Quasigroups.
##
### Strengths
and weaknesses
Strengths
The Damm al... | https://en.wikipedia.org/wiki/Damm_algorithm |
Scanline rendering (also scan line rendering and scan-line rendering) is an algorithm for visible surface determination, in 3D computer graphics, that works on a row-by-row basis rather than a polygon-by-polygon or pixel-by-pixel basis. All of the polygons to be rendered are first sorted by the top y coordinate at whi... | https://en.wikipedia.org/wiki/Scanline_rendering |
In number theory, Fermat's Last Theorem (sometimes called
### Fermat's conjecture
, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than . The cases and have been known since antiquity to have infinitely many solutions.
The propositio... | https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem |
Reyes rendering is a computer software architecture used in 3D computer graphics to render photo-realistic images. It was developed in the mid-1980s by Loren Carpenter and Robert L. Cook at Lucasfilm's Computer Graphics Research Group, which is now Pixar. It was first used in 1982 to render images for the Genesis effec... | https://en.wikipedia.org/wiki/Reyes_rendering |
In statistics, a studentized residual is the dimensionless ratio resulting from the division of a residual by an estimate of its standard deviation, both expressed in the same units. It is a form of a Student's t-statistic, with the estimate of error varying between points.
This is an important technique in the detecti... | https://en.wikipedia.org/wiki/Studentized_residual |
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all... | https://en.wikipedia.org/wiki/Incidence_geometry |
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. T... | https://en.wikipedia.org/wiki/Arithmetic_mean |
Smoothing splines are function estimates,
$$
\hat f(x)
$$
, obtained from a set of noisy observations
$$
y_i
$$
of the target
$$
f(x_i)
$$
, in order to balance a measure of goodness of fit of
$$
\hat f(x_i)
$$
to
$$
y_i
$$
with a derivative based measure of the smoothness of
$$
\hat f(x)
$$
. They provide a m... | https://en.wikipedia.org/wiki/Smoothing_spline |
Robotics is the interdisciplinary study and practice of the design, construction, operation, and use of robots.
Within mechanical engineering, robotics is the design and construction of the physical structures of robots, while in computer science, robotics focuses on robotic automation algorithms.
#### Other
discipli... | https://en.wikipedia.org/wiki/Robotics |
Cel shading or toon shading is a type of non-photorealistic rendering designed to make 3D computer graphics appear to be flat by using less shading color instead of a shade gradient or tints and shades. A cel shader is often used to mimic the style of a comic book or cartoon and/or give the render a characteristic pap... | https://en.wikipedia.org/wiki/Cel_shading |
In theoretical computer science and formal language theory, a regular grammar is a grammar that is right-regular or left-regular.
While their exact definition varies from textbook to textbook, they all require that
- all production rules have at most one non-terminal symbol;
- that symbol is either always at the end or... | https://en.wikipedia.org/wiki/Regular_grammar |
In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a trivial center, then is said to be complete.
An automorphism of a group... | https://en.wikipedia.org/wiki/Outer_automorphism_group |
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable is denoted or , with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a... | https://en.wikipedia.org/wiki/Exponential_function |
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.
One distinguishes the left regular representation λ given by left translation and the right regular representation ρ giv... | https://en.wikipedia.org/wiki/Regular_representation |
In control theory, a bang–bang controller (hysteresis, 2 step or on–off controller), is a feedback controller that switches abruptly between two states. These controllers may be realized in terms of any element that provides hysteresis. They are often used to control a plant that accepts a binary input, for example a f... | https://en.wikipedia.org/wiki/Bang%E2%80%93bang_control |
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution.
## Definition
The... | https://en.wikipedia.org/wiki/Shell_integration |
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is... | https://en.wikipedia.org/wiki/Clopen_set |
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.
Given an arithmetic function
$$
f
$$
and a prime
$$
p
$$
, define the formal power series
$$
f_p(x)
$$
, called the Bell series of
$$
f
$$
mod... | https://en.wikipedia.org/wiki/Bell_series |
In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindström in 1966. They were later studied for their appli... | https://en.wikipedia.org/wiki/Lindstr%C3%B6m_quantifier |
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number , one has
$$
e^{i x} = \cos x + i \sin x,
$$
where is the b... | https://en.wikipedia.org/wiki/Euler%27s_formula |
Texture mapping is a term used in computer graphics to describe how 2D images are projected onto 3D models. The most common variant is the UV unwrap, which can be described as an inverse paper cutout, where the surfaces of a 3D model is cut apart so that it can be unfolded into a 2D coordinate space (UV Space).
## Sema... | https://en.wikipedia.org/wiki/Texture_mapping |
In general relativity, a frame field (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by
$$
\vec{e}_0
$$
an... | https://en.wikipedia.org/wiki/Frame_fields_in_general_relativity |
In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems. For example, there are problems that can be solved with n2 time but not n time, where... | https://en.wikipedia.org/wiki/Time_hierarchy_theorem |
###
###
For other inequalities named after Wirtinger, see Wirtinger's inequality.
In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product. It has important consequences in com... | https://en.wikipedia.org/wiki/Wirtinger_inequality_%282-forms%29 |
A pangram or holoalphabetic sentence is a sentence using every letter of a given alphabet at least once. Pangrams have been used to display typefaces, test equipment, and develop skills in handwriting, calligraphy, and typing.
## Origins
The best-known English pangram is "The quick brown fox jumps over the lazy dog". I... | https://en.wikipedia.org/wiki/Pangram |
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function
$$
f
$$
defined on the real numbers with real values and given a point
$$
x_0
$$
in the domain of
$$
f
$$
, the fixed-point iteration is
$$
x_{n+1}=f(x_n), \, n=0, 1, 2, \dots
$$
wh... | https://en.wikipedia.org/wiki/Fixed-point_iteration |
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Ve... | https://en.wikipedia.org/wiki/Condition_number |
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The transpose of a matrix was introduced in 1858 by the British mathematician Arthu... | https://en.wikipedia.org/wiki/Transpose |
In econometrics, cointegration is a statistical property describing a long-term, stable relationship between two or more time series variables, even if those variables themselves are individually non-stationary (i.e., they have trends). This means that despite their individual fluctuations, the variables move together ... | https://en.wikipedia.org/wiki/Cointegration |
In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers, which is a subset of Rd.
The probl... | https://en.wikipedia.org/wiki/Klee%27s_measure_problem |
In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal conta... | https://en.wikipedia.org/wiki/Minimal_ideal |
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. The boundary of no escape is called the event horizon. A black hole has a great e... | https://en.wikipedia.org/wiki/Black_hole |
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form that gives it useful properties for use in logic programming, formal specification, universal algebra and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their signific... | https://en.wikipedia.org/wiki/Horn_clause |
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other ... | https://en.wikipedia.org/wiki/Pythagorean_theorem |
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.
A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M... | https://en.wikipedia.org/wiki/Loop_theorem |
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In mathematics, F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimension... | https://en.wikipedia.org/wiki/F4_%28mathematics%29 |
Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation rather than the treatment of g... | https://en.wikipedia.org/wiki/Loop_quantum_gravity |
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by , where they called these graphs "polar graphs" ().
A split graph may have more than one partition into a... | https://en.wikipedia.org/wiki/Split_graph |
An and-inverter graph (AIG) is a directed, acyclic graph that represents a structural implementation of the logical functionality of a circuit or network. An AIG consists of two-input nodes representing logical conjunction, terminal nodes labeled with variable names, and edges optionally containing markers indicating ... | https://en.wikipedia.org/wiki/And-inverter_graph |
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets A and B, such that each element of A is less th... | https://en.wikipedia.org/wiki/Dedekind_cut |
Malthusianism is a theory that population growth is potentially exponential, according to the Malthusian growth model, while the growth of the food supply or other resources is linear, which eventually reduces living standards to the point of triggering a population decline. This event, called a Malthusian catastrophe ... | https://en.wikipedia.org/wiki/Malthusianism |
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and .
No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate i... | https://en.wikipedia.org/wiki/Partition_function_%28number_theory%29 |
In mathematics, an annulus (: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annular (as in annular eclipse).
The open annulus... | https://en.wikipedia.org/wiki/Annulus_%28mathematics%29 |
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is al... | https://en.wikipedia.org/wiki/Discrete_Poisson_equation |
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of v... | https://en.wikipedia.org/wiki/Functional_calculus |
A render farm is a high-performance computer system, e.g. a computer cluster, built to render computer-generated imagery (CGI), typically for film and television visual effects.
A render farm is different from a render wall, which is a networked, tiled display used for real-time rendering. The rendering of images is a ... | https://en.wikipedia.org/wiki/Render_farm |
In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori Milstein who first published it in 1974.
## Description
Consider the autonomous Itō stochastic differential equation:
$$
\mathrm{d} X_t = a(X_t) \, \mathrm{d} t + b(... | https://en.wikipedia.org/wiki/Milstein_method |
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
The theory of algebraic surfaces is muc... | https://en.wikipedia.org/wiki/Algebraic_surface |
In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutativ... | https://en.wikipedia.org/wiki/Noncommutative_quantum_field_theory |
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equat... | https://en.wikipedia.org/wiki/Spectral_theory |
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt. These numbers are formed from a construction due to Gregory Chaitin.
... | https://en.wikipedia.org/wiki/Chaitin%27s_constant |
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use intermediate collocation points, as well as linear multistep methods that save a finite time history of the solution. John C. B... | https://en.wikipedia.org/wiki/General_linear_methods |
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set o... | https://en.wikipedia.org/wiki/Steiner_tree_problem |
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices
$$
A
$$
,
$$
B
$$
, and
$$
C
$$
, a general problem is to verify whether
$$
A \times B = C
$$
. A naïve algorithm would compute the product
$$
A \time... | https://en.wikipedia.org/wiki/Freivalds%27_algorithm |
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. The ter... | https://en.wikipedia.org/wiki/Lorenz_system |
In mathematics and functional analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in... | https://en.wikipedia.org/wiki/Direct_integral |
An engineering drawing is a type of technical drawing that is used to convey information about an object. A common use is to specify the geometry necessary for the construction of a component and is called a detail drawing. Usually, a number of drawings are necessary to completely specify even a simple component. These... | https://en.wikipedia.org/wiki/Engineering_drawing |
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or ... | https://en.wikipedia.org/wiki/Block_matrix |
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific prope... | https://en.wikipedia.org/wiki/Wavelet |
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized a... | https://en.wikipedia.org/wiki/Geometric_group_theory |
An electronic oscillator is an electronic circuit that produces a periodic, oscillating or alternating current (AC) signal, usually a sine wave, square wave or a triangle wave, powered by a direct current (DC) source. Oscillators are found in many electronic devices, such as radio receivers, television sets, radio and ... | https://en.wikipedia.org/wiki/Electronic_oscillator |
Resolution independence is where elements on a computer screen are rendered at sizes independent from the pixel grid, resulting in a graphical user interface that is displayed at a consistent physical size, regardless of the resolution of the screen.
## Concept
As early as 1978, the typesetting system TeX due to Donald... | https://en.wikipedia.org/wiki/Resolution_independence |
In mathematics, an almost periodic function is, loosely speaking, a function of a real variable that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Ab... | https://en.wikipedia.org/wiki/Almost_periodic_function |
Skeletal animation or rigging is a technique in computer animation in which a character (or other articulated object) is represented in two parts: a polygonal or parametric mesh representation of the surface of the object, and a hierarchical set of interconnected parts (called joints or bones, and collectively forming ... | https://en.wikipedia.org/wiki/Skeletal_animation |
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
## Linear interpolation between two known points
If the two known points are given by the coordinates
$$
(x_0,y_0)
$$
and the linear interpo... | https://en.wikipedia.org/wiki/Linear_interpolation |
Blend modes (alternatively blending modes or mixing modes) in digital image editing and computer graphics are used to determine how two layers are blended with each other. The default blend mode in most applications is simply to obscure the lower layer by covering it with whatever is present in the top layer (see alpha... | https://en.wikipedia.org/wiki/Blend_modes |
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Many groups of geometric transformations are algebraic groups, including orthog... | https://en.wikipedia.org/wiki/Algebraic_group |
In cryptography, the tabula recta (from Latin tabula rēcta) is a square table of alphabets, each row of which is made by shifting the previous one to the left. The term was invented by the German author and monk Johannes Trithemius in 1508, and used in his
## Trithemius cipher
.
Trithemius cipher
The Trithemius cipher... | https://en.wikipedia.org/wiki/Tabula_recta |
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras... | https://en.wikipedia.org/wiki/Ext_functor |
In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced in 1985 by the American mathematician Peter L. Montgomery.Martin Kochanski, "Montgomery Multiplication" a colloquial e... | https://en.wikipedia.org/wiki/Montgomery_modular_multiplication |
Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic, which is a formal theory of truth. It was introduced and so named by Giorgi Japaridze in 2003.
In classical logic, formulas represent true/fal... | https://en.wikipedia.org/wiki/Computability_logic |
In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by .
A cardinal number
$$
\kappa... | https://en.wikipedia.org/wiki/Indescribable_cardinal |
In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.
Such a multiplication of distributions has long ... | https://en.wikipedia.org/wiki/Colombeau_algebra |
Holography is a technique that enables a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interferometry. In principle, it is possible to make a hologram for any type of wave... | https://en.wikipedia.org/wiki/Holography |
In algebraic topology, a -chain
is a formal linear combination of the -cells in a cell complex. In simplicial complexes (respectively, cubical complexes), -chains are combinations of -simplices (respectively, -cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are equiv... | https://en.wikipedia.org/wiki/Chain_%28algebraic_topology%29 |
In class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K (in a fixed algebraic closure of K) and the generalized ideal class groups defined via a modulus of K.
It is called an existence the... | https://en.wikipedia.org/wiki/Takagi_existence_theorem |
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in
$$
\mathbb{R}^n
$$
. This number depends on the size and shape of the bodies, and their relative orientation to each other.
## Definition
Let
$$
K_1, K_2, \dots, K_r
$$
b... | https://en.wikipedia.org/wiki/Mixed_volume |
Artificial general intelligence (AGI)—sometimes called human‑level intelligence AI—is a type of artificial intelligence that would match or surpass human capabilities across virtually all cognitive tasks.
Some researchers argue that state‑of‑the‑art large language models already exhibit early signs of AGI‑level capabil... | https://en.wikipedia.org/wiki/Artificial_general_intelligence |
The Enigma machine is a cipher device developed and used in the early- to mid-20th century to protect commercial, diplomatic, and military communication. It was employed extensively by Nazi Germany during World War II, in all branches of the German military. The Enigma machine was considered so secure that it was used ... | https://en.wikipedia.org/wiki/Enigma_machine |
In applied mathematics, a DFT matrix is a square matrix as an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.
## Definition
An N-point DFT is expressed as the multiplication
$$
X = W x
$$
, where
$$
x
$$
is the original inpu... | https://en.wikipedia.org/wiki/DFT_matrix |
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.
One of several r... | https://en.wikipedia.org/wiki/Closed-subgroup_theorem |
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915Whitha... | https://en.wikipedia.org/wiki/Burgers%27_equation |
LAN Manager is a discontinued network operating system (NOS) available from multiple vendors and developed by Microsoft in cooperation with 3Com Corporation. It was designed to succeed 3Com's 3+Share network server software which ran atop a heavily modified version of MS-DOS.
## History
The LAN Manager OS/2 operating ... | https://en.wikipedia.org/wiki/LAN_Manager |
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precis... | https://en.wikipedia.org/wiki/Quantum_superposition |
In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy.
The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of in... | https://en.wikipedia.org/wiki/Arithmetical_set |
In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of imprope... | https://en.wikipedia.org/wiki/Improper_rotation |
In topology, especially algebraic topology, the cone of a topological space
$$
X
$$
is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by
$$
CX
$$
or by
$$
\operatorname{cone}(X)
$$
.
## Definitions
Formally, the cone of X is defi... | https://en.wikipedia.org/wiki/Cone_%28topology%29 |
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line
$$
\mathbb Z
$$
which starts at 0, and at ea... | https://en.wikipedia.org/wiki/Random_walk |
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