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In algebraic number theory, a quadratic field is an algebraic number field of degree two over
$$
\mathbf{Q}
$$
, the rational numbers.
Every such quadratic field is some
$$
\mathbf{Q}(\sqrt{d})
$$
where
$$
d
$$
is a (uniquely defined) square-free integer different from
$$
0
$$
and
$$
1
$$
. If
$$
d>0
$$
, the ... | https://en.wikipedia.org/wiki/Quadratic_field |
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold for all integers less than but remains unproven despite considerable effort... | https://en.wikipedia.org/wiki/Goldbach%27s_conjecture |
In set theory, the intersection of two sets
$$
A
$$
and
$$
B,
$$
denoted by
$$
A \cap B,
$$
is the set containing all elements of
$$
A
$$
that also belong to
$$
B
$$
or equivalently, all elements of
$$
B
$$
that also belong to
$$
A.
$$
## Notation and terminology
Intersection is written using the symbol "
... | https://en.wikipedia.org/wiki/Intersection_%28set_theory%29 |
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra
$$
\mathfrak{g}
$$
over the real numbers is associated to a Lie group . The theorem is part of the Lie group–Lie algebra correspondence.
Historically, the third theorem referred to a different but related result. Th... | https://en.wikipedia.org/wiki/Lie%27s_third_theorem |
In physics, the Schrödinger picture or Schrödinger representation is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential
$$
V
$$
changes). Th... | https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture |
In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but ... | https://en.wikipedia.org/wiki/Infinitesimal_transformation |
In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighborin... | https://en.wikipedia.org/wiki/Leverage_%28statistics%29 |
Parareal is a parallel algorithm from numerical analysis and used for the solution of initial value problems.
It was introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of the most widely studied parallel-in-time integration methods.
## Parallel-in-time integration methods
In contrast to e.g.... | https://en.wikipedia.org/wiki/Parareal |
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative ri... | https://en.wikipedia.org/wiki/Krull_dimension |
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the alg... | https://en.wikipedia.org/wiki/Free_object |
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex o... | https://en.wikipedia.org/wiki/Legendre_transformation |
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that... | https://en.wikipedia.org/wiki/Irrational_number |
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respecti... | https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior |
In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.
##
### Definition
The joint Shannon entropy (in bits) of two discrete random variables
$$
X
$$
and
$$
Y
$$
with images
$$
\mathcal X
$$
and
$$
\mathcal Y
$$
is defined as
$$
\Eta(X,Y) = -\sum_{x\in\mathcal ... | https://en.wikipedia.org/wiki/Joint_entropy |
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space
$$
X
$$
is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point
$$
x
$$
in
$$
X
$$
there exists a s... | https://en.wikipedia.org/wiki/First-countable_space |
In mathematical logic, a tautology (from ) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of what a ball is and regard... | https://en.wikipedia.org/wiki/Tautology_%28logic%29 |
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group
$$
G
$$
is the algorithmic problem of deciding whether two words in the generators represent the same element of
$$
G
$$
. The word problem is a well-known example of an un... | https://en.wikipedia.org/wiki/Word_problem_for_groups |
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function
$$
f
$$
is upper (respectively, lower) semicontinuous at a point
$$
x_0
$$
if, roughly speaking, the function values for arguments near
$$
x... | https://en.wikipedia.org/wiki/Semi-continuity |
In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.
For "pure" vector spaces, the concept has been introduced in homological algebra, and it ... | https://en.wikipedia.org/wiki/Graded_vector_space |
In trigonometry, Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points , and two unknown points . From and an observer measures the angles made by the lines of sight to each of the other ... | https://en.wikipedia.org/wiki/Hansen%27s_problem |
Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations.
## Lemma statement
Let
$$
V
$$
be a real Hilbert space with the norm
$$
\|\cdot\|.
$$
Let
$$... | https://en.wikipedia.org/wiki/C%C3%A9a%27s_lemma |
In mathematics, a read-once function is a special type of Boolean function that can be described by a Boolean expression in which each variable appears only once.
More precisely, the expression is required to use only the operations of logical conjunction, logical disjunction, and negation. By applying De Morgan's laws... | https://en.wikipedia.org/wiki/Read-once_function |
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of
#### Euclidean space
. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interv... | https://en.wikipedia.org/wiki/Compact_space |
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this ... | https://en.wikipedia.org/wiki/Lipschitz_continuity |
Bit blit (also written BITBLT, BIT BLT, BitBLT, Bit BLT, Bit Blt etc., which stands for bit block transfer) is a data operation commonly used in computer graphics in which several bitmaps are combined into one using a boolean function.
The operation involves at least two bitmaps: a "source" (or "foreground") and a "des... | https://en.wikipedia.org/wiki/Bit_blit |
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorph... | https://en.wikipedia.org/wiki/Bialgebra |
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
The most basic of these rules, called
## Simpson's 1/3 rule
, or just Simpson's rule, reads
$$
\int_a^b f(x) \, dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f... | https://en.wikipedia.org/wiki/Simpson%27s_rule |
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set
$$
\{0\}
$$
is a singleton whose single element is
$$
0
$$
.
## Properties
Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element ... | https://en.wikipedia.org/wiki/Singleton_%28mathematics%29 |
In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric
$$
q_{ab} (x)
$$
on the spatial slice and the metric's conjugate momentum
$$
K^{ab} (x)
$$
, which is related to the extrinsic curvature and is a measure of how... | https://en.wikipedia.org/wiki/Ashtekar_variables |
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σan, if its Euler transform converges to a sum, then that sum is called the Eul... | https://en.wikipedia.org/wiki/Euler_summation |
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation for a given polynomial . One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "... | https://en.wikipedia.org/wiki/Laguerre%27s_method |
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family ... | https://en.wikipedia.org/wiki/Product_%28category_theory%29 |
An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.
... | https://en.wikipedia.org/wiki/RLC_circuit |
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably ... | https://en.wikipedia.org/wiki/Core_model |
The information bottleneck method is a technique in information theory introduced by Naftali Tishby, Fernando C. Pereira, and William Bialek. It is designed for finding the best tradeoff between accuracy and complexity (compression) when summarizing (e.g. clustering) a random variable X, given a joint probability distr... | https://en.wikipedia.org/wiki/Information_bottleneck_method |
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory.
The historical roots of the theory lie in the idea of the adjoint ... | https://en.wikipedia.org/wiki/Coherent_duality |
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by .
## Definition
### Matrices
L... | https://en.wikipedia.org/wiki/Spectral_radius |
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices... | https://en.wikipedia.org/wiki/Matrix_calculus |
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust co... | https://en.wikipedia.org/wiki/General_frame |
In computational geometry, a Pitteway triangulation is a point set triangulation in which the nearest neighbor of any point p within the triangulation is one of the vertices of the triangle containing p.
Alternatively, it is a Delaunay triangulation in which each internal edge crosses its dual Voronoi diagram edge. Pit... | https://en.wikipedia.org/wiki/Pitteway_triangulation |
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space
$$
(X, \tau)
$$
is said to be metrizable if there is a metric
$$
d : X \times X \to [0, \infty)
$$
such that the topology induced by
$$
d
$$
is
$$
\tau.
$$
... | https://en.wikipedia.org/wiki/Metrizable_space |
Brane cosmology refers to several theories in particle physics and cosmology related to string theory, superstring theory and M-theory.
## Brane and bulk
The central idea is that the visible, four-dimensional spacetime is restricted to a brane inside a higher-dimensional space, called the "bulk" (also known as "hypersp... | https://en.wikipedia.org/wiki/Brane_cosmology |
In computer graphics, a digital differential analyzer (DDA) is hardware or software used for interpolation of variables over an interval between start and end point. DDAs are used for rasterization of lines, triangles and polygons. They can be extended to non linear functions, such as perspective correct texture mappin... | https://en.wikipedia.org/wiki/Digital_differential_analyzer_%28graphics_algorithm%29 |
In mathematics, more specifically in point-set topology, the derived set of a subset
$$
S
$$
of a topological space is the set of all limit points of
$$
S.
$$
It is usually denoted by
$$
S'.
$$
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets ... | https://en.wikipedia.org/wiki/Derived_set_%28mathematics%29 |
In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given per... | https://en.wikipedia.org/wiki/Trigonometric_interpolation |
Digital art, or the digital arts, is artistic work that uses digital technology as part of the creative or presentational process. It can also refer to computational art that uses and engages with digital media. Since the 1960s, various names have been used to describe digital art, including computer art, electronic ar... | https://en.wikipedia.org/wiki/Digital_art |
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective
$$
\lor
$$
can be used to join the two atomic formulas
$$... | https://en.wikipedia.org/wiki/Logical_connective |
Competitive analysis is a method invented for analyzing online algorithms, in which the performance of an online algorithm (which must satisfy an unpredictable sequence of requests, completing each request without being able to see the future) is compared to the performance of an optimal offline algorithm that can view... | https://en.wikipedia.org/wiki/Competitive_analysis_%28online_algorithm%29 |
In theoretical physics, the matrix theory is a quantum mechanical model proposed in 1997 by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind; it is also known as BFSS matrix model, after the authors' initials.
## Overview
This theory describes the behavior of a set of nine large matrices. In their origi... | https://en.wikipedia.org/wiki/Matrix_theory_%28physics%29 |
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number
$$
0 \leq k < 1
$$
such that for all x and y in M,
$$
d(f(x),f(y)) \leq k\,d(x,y).
$$
The smallest such value of k is called the Lipschitz c... | https://en.wikipedia.org/wiki/Contraction_mapping |
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order n – that is, a magic square which contains the numbers 1, 2, ..., n2 – the magic constant ... | https://en.wikipedia.org/wiki/Magic_constant |
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar natu... | https://en.wikipedia.org/wiki/Min-max_theorem |
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population... | https://en.wikipedia.org/wiki/Branching_process |
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as
$$
(u \cdot v)' = u ' \cdot v + u \cdot v'
$$
or in Leibniz's notation as
$$
\frac{d}{dx} (u\cdot v) = ... | https://en.wikipedia.org/wiki/Product_rule |
Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design optimization (MSDO), and multidisciplinary design analysis and optimization (MDAO).
MDO allows designe... | https://en.wikipedia.org/wiki/Multidisciplinary_design_optimization |
##
In 1931, the International Commission on Illumination (CIE) published the CIE 1931 color spaces which define the relationship between the visible spectrum and human color vision. The CIE color spaces are mathematical models that comprise a "standard observer", which is a static idealization of the color vision of a... | https://en.wikipedia.org/wiki/CIE_1931_color_space |
In computer graphics, a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel-based displays and printers. On such media, line drawing requires an approximation (in nontrivial cases). Basic algorithms rasterize lines in one color. A better representation with... | https://en.wikipedia.org/wiki/Line_drawing_algorithm |
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation a... | https://en.wikipedia.org/wiki/Pyramid_%28image_processing%29 |
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all positive integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides and area .
Heron's formula i... | https://en.wikipedia.org/wiki/Heronian_triangle |
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument.
## Mathematics
### Functions
For example, the function
$$
f(x,y) = x^2 + y^2
$$
is invariant under rotations of the plane around the orig... | https://en.wikipedia.org/wiki/Rotational_invariance |
In medical research, epidemiology, social science, and biology, a cross-sectional study (also known as a cross-sectional analysis, transverse study, prevalence study) is a type of observational study that analyzes data from a population, or a representative subset, at a specific point in time—that is, cross-sectional d... | https://en.wikipedia.org/wiki/Cross-sectional_study |
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth ... | https://en.wikipedia.org/wiki/Optional_stopping_theorem |
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th c... | https://en.wikipedia.org/wiki/Computer_algebra_system |
In computer graphics, a T-spline is a mathematical model for defining freeform surfaces. A T-spline surface is a type of surface defined by a network of control points where a row of control points is allowed to terminate without traversing the entire surface. The control net at a terminated row resembles the letter "T... | https://en.wikipedia.org/wiki/T-spline |
In computer graphics, slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius great circle arc, given the ends and an interpolation parameter between 0 an... | https://en.wikipedia.org/wiki/Slerp |
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
The existence of electron spin an... | https://en.wikipedia.org/wiki/Spin_%28physics%29 |
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or takin... | https://en.wikipedia.org/wiki/Noncommutative_algebraic_geometry |
In mathematics, an element of a -algebra is called positive if it is the sum of elements of the form
## Definition
Let
$$
\mathcal{A}
$$
be a *-algebra. An element
$$
a \in \mathcal{A}
$$
is called positive if there are finitely many elements
$$
a_k \in \mathcal{A} \; (k = 1,2,\ldots,n)
$$
, so that
$$
a = \sum_... | https://en.wikipedia.org/wiki/Positive_element |
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.
## Definition
In a triangle with sides , where the vertices are labeled in counterclockwise order, there is exactly one point such that the line segments form the same angle, , w... | https://en.wikipedia.org/wiki/Brocard_points |
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually ... | https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions |
In mathematics, the irregularity of a complex surface X is the Hodge number
$$
h^{0,1}= \dim H^1(\mathcal{O}_X)
$$
, usually denoted by q. The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in charact... | https://en.wikipedia.org/wiki/Irregularity_of_a_surface |
In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underl... | https://en.wikipedia.org/wiki/A%C2%B9_homotopy_theory |
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.
## Definition
They are defin... | https://en.wikipedia.org/wiki/Pythagorean_means |
Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processing, signal processing, machine learning, medical imaging, and more.
## Sparse dec... | https://en.wikipedia.org/wiki/Sparse_approximation |
In computer science, a lookup table (LUT) is an array that replaces runtime computation of a mathematical function with a simpler array indexing operation, in a process termed as direct addressing. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying ou... | https://en.wikipedia.org/wiki/Lookup_table |
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a spacetime interval between any two events is independent of the inertial frame... | https://en.wikipedia.org/wiki/Minkowski_space |
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
## Name
Integral curves are known by various other names, depending on the nature and interpretation of the differential equation or vector field. In physics, integral... | https://en.wikipedia.org/wiki/Integral_curve |
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely ... | https://en.wikipedia.org/wiki/Automorphism |
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to its... | https://en.wikipedia.org/wiki/Affine_group |
+ Legendre symbol ()for various a (along top) and p (along left side). 0 1 2 3 4 5 6 7 8 9 10 3 0 1 −1 5 0 1 −1 −1 1 7 0 1 1 −1 1 −1 −1 11 0 1 −1 1 1 1 −1 −1 −1 1 −1Only 0 ≤ a < p are shown, since due to the first property below any other a can be reduced modulo p. Quadratic... | https://en.wikipedia.org/wiki/Legendre_symbol |
Partial least squares (PLS) regression is a statistical method that bears some relation to principal components regression and is a reduced rank regression; instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted v... | https://en.wikipedia.org/wiki/Partial_least_squares_regression |
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector quantity, meaning that both magnitude and direction are needed to define it. The scalar absolute value (ma... | https://en.wikipedia.org/wiki/Velocity |
In number theory, a Liouville number is a real number
$$
x
$$
with the property that, for every positive integer
$$
n
$$
, there exists a pair of integers
$$
(p,q)
$$
with
$$
q>1
$$
such that
$$
0<\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.
$$
The inequality implies that Liouville numbers possess an excellent sequ... | https://en.wikipedia.org/wiki/Liouville_number |
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
## Solution as a Pell equation
Write
$$
N_k
$$
for the
$$
k
$$
th square triangular number, and write
$$... | https://en.wikipedia.org/wiki/Square_triangular_number |
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
## Abstract commensurability
Two groups G1 and G2 are said to be (abstractly) commensurable if there are ... | https://en.wikipedia.org/wiki/Commensurability_%28group_theory%29 |
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main stat... | https://en.wikipedia.org/wiki/Kummer_theory |
In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
$$
\frac{a}{\sin{\alpha}} \,=\, \frac{b}{\sin{\beta}} \,=\, \frac{c}{\sin{\gamma}} \,=\, 2R,
$$
where , and a... | https://en.wikipedia.org/wiki/Law_of_sines |
Digital sculpting, also known as sculpt modeling or 3D sculpting, is the use of software that offers tools to push, pull, smooth, grab, pinch or otherwise manipulate a digital object as if it were made of a real-life substance such as clay.
## Sculpting technology
The geometry used in digital sculpting programs to repr... | https://en.wikipedia.org/wiki/Digital_sculpting |
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is ... | https://en.wikipedia.org/wiki/Online_algorithm |
A vanishing point is a point on the image plane of a perspective rendering where the two-dimensional perspective projections of parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their... | https://en.wikipedia.org/wiki/Vanishing_point |
Fourier-transform spectroscopy (FTS) is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the radiation, electromagnetic or not. It can be applied to a variety of types of spectroscopy including optical ... | https://en.wikipedia.org/wiki/Fourier-transform_spectroscopy |
In computer science, Cannon's algorithm is a distributed algorithm for matrix multiplication for two-dimensional meshes first described in 1969 by Lynn Elliot Cannon.
It is especially suitable for computers laid out in an N × N mesh. While Cannon's algorithm works well in homogeneous 2D grids, extending it to heteroge... | https://en.wikipedia.org/wiki/Cannon%27s_algorithm |
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R3, the Gauss map is a map N: X → S2 (where S2 is the unit sphere) such that for e... | https://en.wikipedia.org/wiki/Gauss_map |
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin const... | https://en.wikipedia.org/wiki/L-system |
In abstract algebra, a cyclic group or monogenous group is a group, denoted
$$
C_n
$$
(also frequently
$$
\Z_n
$$
or
$$
Z_n
$$
, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operat... | https://en.wikipedia.org/wiki/Cyclic_group |
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable
$$
t^{1/2}
$$
with integer coefficients.
## Definitio... | https://en.wikipedia.org/wiki/Jones_polynomial |
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multi... | https://en.wikipedia.org/wiki/Outer_product |
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exi... | https://en.wikipedia.org/wiki/Recursively_enumerable_language |
In mathematics a radial basis function (RBF) is a real-valued function
$$
\varphi
$$
whose value depends only on the distance between the input and some fixed point, either the origin, so that
$$
\varphi(\mathbf{x}) = \hat\varphi(\left\|\mathbf{x}\right\|)
$$
, or some other fixed point
$$
\mathbf{c}
$$
, called a ... | https://en.wikipedia.org/wiki/Radial_basis_function |
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Different conventions concerning the value are in use. It is an example of the gene... | https://en.wikipedia.org/wiki/Heaviside_step_function |
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