id stringlengths 2 8 | title stringlengths 1 130 | text stringlengths 0 252k | formulas listlengths 1 823 | url stringlengths 38 44 |
|---|---|---|---|---|
1010522 | Disjunction and existence properties | In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
Definitions.
Related properties.
Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (... | [
{
"math_id": 0,
"text": "(\\exists x \\in \\mathbb{N})\\varphi(x)"
},
{
"math_id": 1,
"text": "\\varphi(\\bar{n})"
},
{
"math_id": 2,
"text": "n \\in \\mathbb{N}\\text{.}"
},
{
"math_id": 3,
"text": "\\bar{n}"
},
{
"math_id": 4,
"text": "T"
},
{
"math_... | https://en.wikipedia.org/wiki?curid=1010522 |
10105237 | Sylvester equation | In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:
formula_0
It is named after English mathematician James Joseph Sylvester. Then given matrices "A", "B", and "C", the problem is to find the possible matrices "X" that obey this equation. All matrices are assumed to ha... | [
{
"math_id": 0,
"text": "A X + X B = C."
},
{
"math_id": 1,
"text": "\\operatorname{vec}"
},
{
"math_id": 2,
"text": " (I_m \\otimes A + B^T \\otimes I_n) \\operatorname{vec}X = \\operatorname{vec}C,"
},
{
"math_id": 3,
"text": "A"
},
{
"math_id": 4,
"text": ... | https://en.wikipedia.org/wiki?curid=10105237 |
10105571 | Fort space | Examples of topological spaces
In mathematics, there are a few topological spaces named after M. K. Fort, Jr.
Fort space.
Fort space is defined by taking an infinite set "X", with a particular point "p" in "X", and declaring open the subsets "A" of "X" such that:
The subspace formula_0 has the discrete topology and is ... | [
{
"math_id": 0,
"text": "X\\setminus\\{p\\}"
}
] | https://en.wikipedia.org/wiki?curid=10105571 |
10106425 | Orr–Sommerfeld equation | The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Somme... | [
{
"math_id": 0,
"text": "\\mathbf{u} = \\left(U(z)+u'(x,z,t), 0 ,w'(x,z,t)\\right)"
},
{
"math_id": 1,
"text": "(U(z), 0, 0)"
},
{
"math_id": 2,
"text": "\\mathbf{u}' \\propto \\exp(i \\alpha (x - c t))"
},
{
"math_id": 3,
"text": "\\frac{\\mu}{i\\alpha\\rho} \\left({d^2 ... | https://en.wikipedia.org/wiki?curid=10106425 |
1010712 | Closed convex function | Terms in Maths
In mathematics, a function formula_0 is said to be closed if for each formula_1, the sublevel set
formula_2
is a closed set.
Equivalently, if the epigraph defined by
formula_3
is closed, then the function formula_4 is closed.
This definition is valid for any function, but most used for convex functions. ... | [
{
"math_id": 0,
"text": "f: \\mathbb{R}^n \\rightarrow \\mathbb{R} "
},
{
"math_id": 1,
"text": " \\alpha \\in \\mathbb{R}"
},
{
"math_id": 2,
"text": " \\{ x \\in \\mbox{dom} f \\vert f(x) \\leq \\alpha \\} "
},
{
"math_id": 3,
"text": " \\mbox{epi} f = \\{ (x,t) \\in \\... | https://en.wikipedia.org/wiki?curid=1010712 |
10109430 | Reynolds analogy | The Reynolds Analogy is popularly known to relate turbulent momentum and heat transfer. That is because in a turbulent flow (in a pipe or in a boundary layer) the transport of momentum and the transport of heat largely depends on the same turbulent eddies: the velocity and the temperature profiles have the same shape. ... | [
{
"math_id": 0,
"text": " \\frac{f}{2} = \\frac{h}{C_p\\times G} = \\frac{k'_c}{V_{av}} "
}
] | https://en.wikipedia.org/wiki?curid=10109430 |
10109649 | Courant algebroid | In a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototyp... | [
{
"math_id": 0,
"text": "TM\\oplus T^*M"
},
{
"math_id": 1,
"text": "E\\to M"
},
{
"math_id": 2,
"text": "[\\cdot,\\cdot]:\\Gamma E \\times \\Gamma E \\to \\Gamma E"
},
{
"math_id": 3,
"text": "\\langle \\cdot, \\cdot \\rangle: E\\times E\\to M\\times\\R"
},
{
"ma... | https://en.wikipedia.org/wiki?curid=10109649 |
10109665 | Chilton and Colburn J-factor analogy | Chilton–Colburn J-factor analogy (also known as the "modified Reynolds analogy") is a successful and widely used analogy between heat, momentum, and mass transfer. The basic mechanisms and mathematics of heat, mass, and momentum transport are essentially the same. Among many analogies (like Reynolds analogy, Prandtl–Ta... | [
{
"math_id": 0,
"text": "J_M=\\frac{f}{2} = J_H = \\frac{h}{c_p\\, G}\\,{Pr}^{\\frac{2}{3}}= J_D = \\frac{k'_c}{\\overline{v}} \\cdot {Sc}^{\\frac{2}{3}}"
},
{
"math_id": 1,
"text": "J_M = \\frac{f}{2} = \\frac{Sh}{Re\\, Sc^{\\frac{1}{3}}} = J_H = \\frac{f}{2} = \\frac{Nu}{Re\\, Pr^{\\frac{1}{3}... | https://en.wikipedia.org/wiki?curid=10109665 |
101107 | Dentition | Development and arrangement of teeth
Dentition pertains to the development of teeth and their arrangement in the mouth. In particular, it is the characteristic arrangement, kind, and number of teeth in a given species at a given age. That is, the number, type, and morpho-physiology (that is, the relationship between th... | [
{
"math_id": 0,
"text": "(di^2\\text{-}dc^1\\text{-}dm^2) / (di_2\\text{-}dc_1\\text{-}dm_2) \\times 2 =20."
},
{
"math_id": 1,
"text": "(I^2\\text{-}C^1\\text{-}P^2\\text{-}M^3) / (I_2\\text{-}C_1\\text{-}P_2\\text{-}M_3) \\times 2 =32."
}
] | https://en.wikipedia.org/wiki?curid=101107 |
1011270 | Bourbaki–Witt theorem | Fixed-point theorem
In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets. It states that if "X" is a non-empty chain complete poset, and
formula_0
such that
formula_1 for all formula_2
then "f" has a fixed po... | [
{
"math_id": 0,
"text": "f : X \\to X"
},
{
"math_id": 1,
"text": "f (x) \\geq x"
},
{
"math_id": 2,
"text": "x,"
},
{
"math_id": 3,
"text": " x_{n+1}=f(x_n), n=0,1,2,\\ldots, "
},
{
"math_id": 4,
"text": " x_n=x_{\\infty},"
},
{
"math_id": 5,
"tex... | https://en.wikipedia.org/wiki?curid=1011270 |
1011332 | Predicate variable | In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capital roman letters such as formula_0, formula... | [
{
"math_id": 0,
"text": "P"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "R"
},
{
"math_id": 3,
"text": "x"
},
{
"math_id": 4,
"text": " =, \\ \\in , \\ \\le,\\ <, \\ \\sub,... "
},
{
"math_id": 5,
"text": " =, \\ \\in , \\ \\le,\\ <, ... | https://en.wikipedia.org/wiki?curid=1011332 |
10113455 | Cophenetic correlation | In statistics, and especially in biostatistics, cophenetic correlation (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the field of biostatistics ... | [
{
"math_id": 0,
"text": "x(i,j) = |X_i-X_j|"
},
{
"math_id": 1,
"text": "t(i,j)"
},
{
"math_id": 2,
"text": "T_i"
},
{
"math_id": 3,
"text": "T_j"
},
{
"math_id": 4,
"text": "\\bar{x}"
},
{
"math_id": 5,
"text": "\\bar{t}"
},
{
"math_id": 6... | https://en.wikipedia.org/wiki?curid=10113455 |
10119238 | Essential extension | Concept in mathematics
In mathematics, specifically module theory, given a ring "R" and an "R"-module "M" with a submodule "N", the module "M" is said to be an essential extension of "N" (or "N" is said to be an essential submodule or large submodule of "M") if for every submodule "H" of "M",
formula_0 implies that for... | [
{
"math_id": 0,
"text": "H\\cap N=\\{0\\}\\,"
},
{
"math_id": 1,
"text": "H=\\{0\\}\\,"
},
{
"math_id": 2,
"text": "N\\subseteq_e M\\,"
},
{
"math_id": 3,
"text": "N\\trianglelefteq M"
},
{
"math_id": 4,
"text": "N+H=M\\,"
},
{
"math_id": 5,
"text"... | https://en.wikipedia.org/wiki?curid=10119238 |
10121045 | Pore space in soil | Volume occupied by liquid and gas phases in a soil
The pore space of soil contains the liquid and gas phases of soil, i.e., everything but the solid phase that contains mainly minerals of varying sizes as well as organic compounds.
In order to understand porosity better a series of equations have not been used to expre... | [
{
"math_id": 0,
"text": "\\rho_{dry} = \\frac{M_{solid}}{V_{total}}"
},
{
"math_id": 1,
"text": "\\rm{Dry \\ bulk \\ density} = \\frac{\\rm{(mass \\ of \\ oven \\ dry \\ soil)}}{\\rm{(total \\ sample \\ volume)}}"
},
{
"math_id": 2,
"text": "\\eta = \\frac{V_{pore}}{V_{total}} = \\fr... | https://en.wikipedia.org/wiki?curid=10121045 |
10121788 | Fuel fraction | In aerospace engineering, an aircraft's fuel fraction, fuel weight fraction, or a spacecraft's propellant fraction, is the weight of the fuel or propellant divided by the gross take-off weight of the craft (including propellant):
formula_0
The fractional result of this mathematical division is often expressed as a perc... | [
{
"math_id": 0,
"text": "\\ \\zeta = \\frac{\\Delta W}{W_1} "
},
{
"math_id": 1,
"text": "-\\ln(1-\\ \\zeta) "
}
] | https://en.wikipedia.org/wiki?curid=10121788 |
10122951 | Succinct data structure | Data structure which is efficient to both store in memory and query
In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but (unlike other compressed representations) still allows for efficient query operations. The co... | [
{
"math_id": 0,
"text": "Z"
},
{
"math_id": 1,
"text": "Z + O(1)"
},
{
"math_id": 2,
"text": "Z + o(Z)"
},
{
"math_id": 3,
"text": "O(Z)"
},
{
"math_id": 4,
"text": "2Z"
},
{
"math_id": 5,
"text": "Z + \\sqrt{Z}"
},
{
"math_id": 6,
"tex... | https://en.wikipedia.org/wiki?curid=10122951 |
10125391 | Mapping cone (homological algebra) | Tool in homological algebra
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so... | [
{
"math_id": 0,
"text": "A, B"
},
{
"math_id": 1,
"text": "d_A, d_B;"
},
{
"math_id": 2,
"text": "A = \\dots \\to A^{n - 1} \\xrightarrow{d_A^{n - 1}} A^n \\xrightarrow{d_A^n} A^{n + 1} \\to \\cdots"
},
{
"math_id": 3,
"text": "B."
},
{
"math_id": 4,
"text": "... | https://en.wikipedia.org/wiki?curid=10125391 |
10125619 | Fundamental matrix (linear differential equation) | Matrix consisting of linearly independent solutions to a linear differential equation
In mathematics, a fundamental matrix of a system of "n" homogeneous linear ordinary differential equationsformula_0is a matrix-valued function formula_1 whose columns are linearly independent solutions of the system.
Then every soluti... | [
{
"math_id": 0,
"text": " \\dot{\\mathbf{x}}(t) = A(t) \\mathbf{x}(t) "
},
{
"math_id": 1,
"text": " \\Psi(t) "
},
{
"math_id": 2,
"text": "\\mathbf{x}(t) = \\Psi(t) \\mathbf{c}"
},
{
"math_id": 3,
"text": "\\mathbf{c}"
},
{
"math_id": 4,
"text": " \\Psi "
}... | https://en.wikipedia.org/wiki?curid=10125619 |
10125731 | Rupture field | In abstract algebra, a rupture field of a polynomial formula_0 over a given field formula_1 is a field extension of formula_1 generated by a root formula_2 of formula_0.
For instance, if formula_3 and formula_4 then formula_5 is a rupture field for formula_0.
The notion is interesting mainly if formula_0 is irreducible... | [
{
"math_id": 0,
"text": "P(X)"
},
{
"math_id": 1,
"text": "K"
},
{
"math_id": 2,
"text": "a"
},
{
"math_id": 3,
"text": "K=\\mathbb Q"
},
{
"math_id": 4,
"text": "P(X)=X^3-2"
},
{
"math_id": 5,
"text": "\\mathbb Q[\\sqrt[3]2]"
},
{
"math_id... | https://en.wikipedia.org/wiki?curid=10125731 |
1012633 | Price dispersion | In economics, price dispersion is variation in prices across sellers of the same item, holding fixed the item's characteristics. Price dispersion can be viewed as a measure of trading frictions (or, tautologically, as a violation of the law of one price). It is often attributed to consumer search costs or unmeasured at... | [
{
"math_id": 0,
"text": "\nF\\left(x\\right) =\\begin{cases}\n 0, & \\text{if } p < \\underline{p} \\left( q \\right)\\\\\n 1 - \\left( \\frac{p^{*} - p}{p - r}\\right)\\left( \\frac{q}{2\\left( 1 - q \\right)}\\right), & \\text{if } \\underline{p} \\left( q... | https://en.wikipedia.org/wiki?curid=1012633 |
1012687 | Coefficient of variation | Statistical parameter
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 a... | [
{
"math_id": 0,
"text": " \\sigma "
},
{
"math_id": 1,
"text": " \\mu "
},
{
"math_id": 2,
"text": "| \\mu |"
},
{
"math_id": 3,
"text": "\\sigma"
},
{
"math_id": 4,
"text": "\\mu"
},
{
"math_id": 5,
"text": "CV = \\frac{\\sigma}{\\mu}."
},
{
... | https://en.wikipedia.org/wiki?curid=1012687 |
10129659 | Ducci sequence | Sequence of n-tuples of integers
A Ducci sequence is a sequence of "n"-tuples of integers, sometimes known as "the Diffy game", because it is based on sequences.
Given an "n"-tuple of integers formula_0, the next "n"-tuple in the sequence is formed by taking the absolute differences of neighbouring integers:
formula_1
... | [
{
"math_id": 0,
"text": "(a_1,a_2,...,a_n)"
},
{
"math_id": 1,
"text": "(a_1,a_2,...,a_n) \\rightarrow (|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\\, ."
},
{
"math_id": 2,
"text": "k(b_1, b_2, ... b_n)"
},
{
"math_id": 3,
"text": "k"
},
{
"math_id": 4,
"text": "b_i... | https://en.wikipedia.org/wiki?curid=10129659 |
1013089 | Solovay–Strassen primality test | The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967
(see Theorem E in the paper). This test has been largely superseded by the ... | [
{
"math_id": 0,
"text": "a^{(p-1)/2} \\equiv \\left(\\frac{a}{p}\\right) \\pmod p "
},
{
"math_id": 1,
"text": "\\left(\\tfrac{a}{p}\\right)"
},
{
"math_id": 2,
"text": "\\left(\\tfrac{a}{n}\\right)"
},
{
"math_id": 3,
"text": " a^{(n-1)/2} \\equiv \\left(\\frac{a}{n}\\ri... | https://en.wikipedia.org/wiki?curid=1013089 |
10131478 | PDIFF | Category of piecewise-smooth manifolds
In geometric topology, PDIFF, for "p"iecewise "diff"erentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between them) and PL (the category of piecewise li... | [
{
"math_id": 0,
"text": "\\text{Diff} \\to \\text{PDiff} \\to \\text{PL}."
}
] | https://en.wikipedia.org/wiki?curid=10131478 |
10132968 | Fuel mass fraction | In combustion physics, fuel mass fraction is the ratio of fuel mass flow to the total mass flow of a fuel mixture. If an air flow is fuel free, the fuel mass fraction is zero; in pure fuel without trapped gases, the ratio is unity. As fuel is burned in a combustion process, the fuel mass fraction is reduced. The defini... | [
{
"math_id": 0,
"text": "Y_F = \\frac{m_F}{m_{\\rm{tot}}}"
},
{
"math_id": 1,
"text": "m_F"
},
{
"math_id": 2,
"text": "m_{\\rm{tot}}"
}
] | https://en.wikipedia.org/wiki?curid=10132968 |
10133123 | Splitting storm | Meteorological process associated with developing thunderstorms
A splitting storm is a phenomenon when a convective thunderstorm will separate into two supercells, with one propagating towards the left (the left mover) and the other to the right (the right mover) of the mean wind shear direction across a deep layer of ... | [
{
"math_id": 0,
"text": "p'_d"
},
{
"math_id": 1,
"text": "z"
},
{
"math_id": 2,
"text": "\\frac{\\partial p_d'}{\\partial z} \\propto 2 \\frac{\\partial}{\\partial z}\\mathbf{S} \\cdot \\nabla_h w' - \\frac{1}{2}\\frac{\\partial \\zeta'^2}{\\partial z}"
},
{
"math_id": 3,
... | https://en.wikipedia.org/wiki?curid=10133123 |
101334 | Euler–Jacobi pseudoprime | In number theory, an odd integer "n" is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base "a", if "a" and "n" are coprime, and
formula_0
where formula_1 is the Jacobi symbol.
If "n" is an odd composite integer that satisfies the above congruence, then "n" is called an Euler–Jac... | [
{
"math_id": 0,
"text": "a^{(n-1)/2} \\equiv \\left(\\frac{a}{n}\\right)\\pmod{n}"
},
{
"math_id": 1,
"text": "\\left(\\frac{a}{n}\\right)"
}
] | https://en.wikipedia.org/wiki?curid=101334 |
10134 | Electromagnetic spectrum | Range of frequencies or wavelengths of electromagnetic radiation
The electromagnetic spectrum is the full range of electromagnetic radiation, organized by frequency or wavelength. The spectrum is divided into separate bands, with different names for the electromagnetic waves within each band. From low to high frequency... | [
{
"math_id": 0,
"text": "f = \\frac{c}{\\lambda}, \\quad\\text{or}\\quad f = \\frac{E}{h}, \\quad\\text{or}\\quad E=\\frac{hc}{\\lambda},"
}
] | https://en.wikipedia.org/wiki?curid=10134 |
1013550 | Quantum channel | Foundational object in quantum communication theory
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the general dynamics of a qubit. An example of classical information is a text do... | [
{
"math_id": 0,
"text": "H_A"
},
{
"math_id": 1,
"text": "H_B"
},
{
"math_id": 2,
"text": "L(H_A)"
},
{
"math_id": 3,
"text": "H_A."
},
{
"math_id": 4,
"text": " \\Phi"
},
{
"math_id": 5,
"text": "I_n \\otimes \\Phi,"
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=1013550 |
1013588 | Newmark-beta method | The Newmark-beta method is a method of numerical integration used to solve certain differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark, former Professor... | [
{
"math_id": 0,
"text": "M\\ddot{u} + C\\dot{u} + f^{\\textrm{int}}(u) = f^{\\textrm{ext}} \\,"
},
{
"math_id": 1,
"text": "M"
},
{
"math_id": 2,
"text": "C"
},
{
"math_id": 3,
"text": "f^{\\textrm{int}}"
},
{
"math_id": 4,
"text": "f^{\\textrm{ext}}"
},
{... | https://en.wikipedia.org/wiki?curid=1013588 |
10136 | Expert system | Computer system emulating the decision-making ability of a human expert
In artificial intelligence (AI), an expert system is a computer system emulating the decision-making ability of a human expert.
Expert systems are designed to solve complex problems by reasoning through bodies of knowledge, represented mainly as if... | [
{
"math_id": 0,
"text": "R1: \\mathit{Man}(x) \\implies \\mathit{Mortal}(x)"
},
{
"math_id": 1,
"text": "^{n}"
}
] | https://en.wikipedia.org/wiki?curid=10136 |
10137513 | Drag-divergence Mach number | The drag-divergence Mach number (not to be confused with critical Mach number) is the Mach number at which the aerodynamic drag on an airfoil or airframe begins to increase rapidly as the Mach number continues to increase. This increase can cause the drag coefficient to rise to more than ten times its low-speed value.
... | [
{
"math_id": 0,
"text": "M_\\text{dd} + \\frac{1}{10}c_{l,\\text{design}} + \\frac{t}{c} = K,"
},
{
"math_id": 1,
"text": "M_\\text{dd}"
},
{
"math_id": 2,
"text": "c_{l,\\text{design}}"
},
{
"math_id": 3,
"text": "K"
}
] | https://en.wikipedia.org/wiki?curid=10137513 |
1013769 | Systemic risk | Risk of collapse of an entire financial system or entire market
In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming the entire sys... | [
{
"math_id": 0,
"text": "i = 1, 2"
},
{
"math_id": 1,
"text": "a_i \\geq 0"
},
{
"math_id": 2,
"text": "T \\geq 0"
},
{
"math_id": 3,
"text": "d_i \\geq 0"
},
{
"math_id": 4,
"text": "T"
},
{
"math_id": 5,
"text": "a_i"
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=1013769 |
10137896 | Tensor product of quadratic forms | In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as "quadratic spaces". If "R" is a commutative ring where 2 is invertible, and if formula_0 and formula_1 are two quadratic spaces over "R", then their tensor product formula_2 is the quadratic space whose... | [
{
"math_id": 0,
"text": "(V_1, q_1)"
},
{
"math_id": 1,
"text": "(V_2,q_2)"
},
{
"math_id": 2,
"text": "(V_1 \\otimes V_2, q_1 \\otimes q_2)"
},
{
"math_id": 3,
"text": "V_1 \\otimes V_2"
},
{
"math_id": 4,
"text": "q_1"
},
{
"math_id": 5,
"text": ... | https://en.wikipedia.org/wiki?curid=10137896 |
10138003 | Fourier integral operator | Class of differential and integral operators
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases.
A Fourier int... | [
{
"math_id": 0,
"text": "T"
},
{
"math_id": 1,
"text": "(Tf)(x)=\\int_{\\mathbb{R}^n} e^{2\\pi i \\Phi(x,\\xi)}a(x,\\xi)\\hat{f}(\\xi) \\, d\\xi "
},
{
"math_id": 2,
"text": "\\hat f"
},
{
"math_id": 3,
"text": "f"
},
{
"math_id": 4,
"text": "a(x,\\xi)"
},
... | https://en.wikipedia.org/wiki?curid=10138003 |
1013834 | Beverage antenna | Type of radio antenna
The Beverage antenna or "wave antenna" is a long-wire receiving antenna mainly used in the low frequency and medium frequency radio bands, invented by Harold H. Beverage in 1921. It is used by amateur radio operators, shortwave listeners, longwave radio DXers and for military applications.
A Bever... | [
{
"math_id": 0,
"text": "\\theta_\\text{max} = \\arccos \\biggl(1 - \\frac{\\lambda}{2 L} \\biggr),"
},
{
"math_id": 1,
"text": "L"
},
{
"math_id": 2,
"text": "\\lambda"
}
] | https://en.wikipedia.org/wiki?curid=1013834 |
10138549 | Morton number |
In fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number or Bond number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase, "c".
It is named after Rose Morton, who described it with W. L. Haberman in 1953.
Definition.
The Mort... | [
{
"math_id": 0,
"text": "\\mathrm{Mo} = \\frac{g \\mu_c^4 \\, \\Delta \\rho}{\\rho_c^2 \\sigma^3}, "
},
{
"math_id": 1,
"text": "\\mu_c"
},
{
"math_id": 2,
"text": "\\rho_c"
},
{
"math_id": 3,
"text": " \\Delta \\rho"
},
{
"math_id": 4,
"text": "\\sigma"
},
... | https://en.wikipedia.org/wiki?curid=10138549 |
1013950 | Heilbronn triangle problem | On point sets with no small-area triangles
<templatestyles src="Unsolved/styles.css" />
Unsolved problem in mathematics:
What is the asymptotic growth rate of the area of the smallest triangle determined by three out of formula_0 points in a square, when the points are chosen to maximize this area?
In discrete ge... | [
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "D"
},
{
"math_id": 2,
"text": "\\Delta_D(n)"
},
{
"math_id": 3,
"text": "\\tbinom63=20"
},
{
"math_id": 4,
"text": "\\Delta_D(6)=\\tfrac18"
},
{
"math_id": 5,
"text": "D'"
},
{
"math_i... | https://en.wikipedia.org/wiki?curid=1013950 |
10144353 | Fraser filter | A Fraser filter, named after Douglas Fraser, is typically used in geophysics when displaying VLF data. It is effectively the first derivative of the data.
If formula_0 represents the collected data then formula_1 is the average of two values. Consider this value to be plotted between point 1 and point 2 and do the same... | [
{
"math_id": 0,
"text": "f(i) = f_i"
},
{
"math_id": 1,
"text": "average_{12}=\\frac{f_1 + f_2}{2}"
},
{
"math_id": 2,
"text": "average_{34}=\\frac{f_3 + f_4}{2}"
},
{
"math_id": 3,
"text": "\\Delta x"
},
{
"math_id": 4,
"text": "\\frac{average_{12}-average_{3... | https://en.wikipedia.org/wiki?curid=10144353 |
10144855 | Kneser's theorem (differential equations) | Mathematical theorem
In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations:
Statement of the theorem due to A. Kneser.
Consider an ordinary linear homogeneous differential equation of the form
formula_0
with
formula_1
continuous.
We say this equation is "o... | [
{
"math_id": 0,
"text": "y'' + q(x)y = 0"
},
{
"math_id": 1,
"text": "q: [0,+\\infty) \\to \\mathbb{R}"
},
{
"math_id": 2,
"text": "\\limsup_{x \\to +\\infty} x^2 q(x) < \\tfrac{1}{4}"
},
{
"math_id": 3,
"text": "\\liminf_{x \\to +\\infty} x^2 q(x) > \\tfrac{1}{4}."
},
... | https://en.wikipedia.org/wiki?curid=10144855 |
10144971 | Oscillation theory | In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation
formula_0
is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution.
T... | [
{
"math_id": 0,
"text": "F(x,y,y',\\ \\dots,\\ y^{(n-1)})=y^{(n)} \\quad x \\in [0,+\\infty)"
},
{
"math_id": 1,
"text": "y'' + y = 0"
}
] | https://en.wikipedia.org/wiki?curid=10144971 |
101453 | Dirichlet's theorem on arithmetic progressions | Theorem on the number of primes in arithmetic sequences
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers "a" and "d", there are infinitely many primes of the form "a" + "nd", where "n" is also a positive integer. In other words, the... | [
{
"math_id": 0,
"text": "a,\\ a+d,\\ a+2d,\\ a+3d,\\ \\dots,\\ "
},
{
"math_id": 1,
"text": "\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{11}+\\frac{1}{19}+\\frac{1}{23}+\\frac{1}{31}+\\frac{1}{43}+\\frac{1}{47}+\\frac{1}{59}+\\frac{1}{67}+\\cdots"
},
{
"math_id": 2,
"text": "\\varphi(d).\\ "... | https://en.wikipedia.org/wiki?curid=101453 |
1014534 | Projectively extended real line | Real numbers with an added point at infinity
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, formula_0, by a point denoted ∞. It is thus the set formula_1 with the standard arithmetic operations extende... | [
{
"math_id": 0,
"text": "\\mathbb{R}"
},
{
"math_id": 1,
"text": "\\mathbb{R}\\cup\\{\\infty\\}"
},
{
"math_id": 2,
"text": "\\mathbb{R}^*"
},
{
"math_id": 3,
"text": "\\widehat{\\mathbb{R}}."
},
{
"math_id": 4,
"text": "\\frac{a}{0} = \\infty"
},
{
"m... | https://en.wikipedia.org/wiki?curid=1014534 |
10145406 | Difference-map algorithm | The difference-map algorithm is a search algorithm for general constraint satisfaction problems. It is a meta-algorithm in the sense that it is built from more basic algorithms that perform projections onto constraint sets. From a mathematical perspective, the difference-map algorithm is a dynamical system based on a m... | [
{
"math_id": 0,
"text": "x"
},
{
"math_id": 1,
"text": "A"
},
{
"math_id": 2,
"text": "B"
},
{
"math_id": 3,
"text": "P_A"
},
{
"math_id": 4,
"text": "P_B"
},
{
"math_id": 5,
"text": "\n\\begin{align}\nx \\mapsto D(x) &= x + \\beta \\left[ P_A \\le... | https://en.wikipedia.org/wiki?curid=10145406 |
1014694 | Real projective space | Type of topological space
In mathematics, real projective space, denoted &NoBreak;&NoBreak; or &NoBreak;&NoBreak; is the topological space of lines passing through the origin 0 in the real space &NoBreak;&NoBreak; It is a compact, smooth manifold of dimension n, and is a special case &NoBrea... | [
{
"math_id": 0,
"text": "(-1)^p"
},
{
"math_id": 1,
"text": "\\pi_1(\\mathbf{RP}^n)"
},
{
"math_id": 2,
"text": "(-1)^{n+1}"
},
{
"math_id": 3,
"text": "\\mathbf{RP}^n = \\mathbf{RP}^{n-1} \\cup_f D^n."
},
{
"math_id": 4,
"text": "\n\\begin{array}{c}\n[*:0:0:\... | https://en.wikipedia.org/wiki?curid=1014694 |
10148835 | Membrane analogy | The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist Ludwig Prandtl in 1903.
It describes the stress distribution on a long bar in torsion. The cross section of the bar is constant along its length, and need not be circular. The differential equation that g... | [
{
"math_id": 0,
"text": "3T/bt^2"
}
] | https://en.wikipedia.org/wiki?curid=10148835 |
1014906 | Cyclomatic complexity | Measure of the structural complexity of a software program
Cyclomatic complexity is a software metric used to indicate the complexity of a program. It is a quantitative measure of the number of linearly independent paths through a program's source code. It was developed by Thomas J. McCabe, Sr. in 1976.
Cyclomatic comp... | [
{
"math_id": 0,
"text": "S"
},
{
"math_id": 1,
"text": "P"
},
{
"math_id": 2,
"text": "S/P"
},
{
"math_id": 3,
"text": "M = E - N + 2P,"
},
{
"math_id": 4,
"text": "M = E - N + P."
},
{
"math_id": 5,
"text": "M = E - N + 2."
},
{
"math_id":... | https://en.wikipedia.org/wiki?curid=1014906 |
10151043 | Moni Naor | Israeli computer scientist (born 1961)
Moni Naor () is an Israeli computer scientist, currently a professor at the Weizmann Institute of Science. Naor received his Ph.D. in 1989 at the University of California, Berkeley. His advisor was Manuel Blum.
He works in various fields of computer science, mainly the foundations... | [
{
"math_id": 0,
"text": "\\epsilon"
},
{
"math_id": 1,
"text": "\\delta"
}
] | https://en.wikipedia.org/wiki?curid=10151043 |
10151726 | Atmospheric tide | Global-scale periodic oscillations of the atmosphere
Atmospheric tides are global-scale periodic oscillations of the atmosphere. In many ways they are analogous to ocean tides. They can be excited by:
General characteristics.
The largest-amplitude atmospheric tides are mostly generated in the troposphere and stratosphe... | [
{
"math_id": 0,
"text": "u"
},
{
"math_id": 1,
"text": "v"
},
{
"math_id": 2,
"text": "w"
},
{
"math_id": 3,
"text": "\\Phi"
},
{
"math_id": 4,
"text": "\\int g(z,\\varphi) \\, dz"
},
{
"math_id": 5,
"text": "N^2"
},
{
"math_id": 6,
"t... | https://en.wikipedia.org/wiki?curid=10151726 |
10151936 | Random regular graph | A random "r"-regular graph is a graph selected from formula_0, which denotes the probability space of all "r"-regular graphs on formula_1 vertices, where formula_2 and formula_3 is even. It is therefore a particular kind of random graph, but the regularity restriction significantly alters the properties that will hold,... | [
{
"math_id": 0,
"text": "\\mathcal{G}_{n,r}"
},
{
"math_id": 1,
"text": "n"
},
{
"math_id": 2,
"text": "3 \\le r < n"
},
{
"math_id": 3,
"text": "nr"
},
{
"math_id": 4,
"text": "m"
},
{
"math_id": 5,
"text": " r \\ge 3 "
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=10151936 |
1015276 | Shoe size | Measurement scale indicating the fitting size of a shoe
A shoe size is an indication of the fitting size of a shoe for a person.
There are a number of different shoe-size systems used worldwide. While all shoe sizes use a number to indicate the length of the shoe, they differ in exactly what they measure, what unit of ... | [
{
"math_id": 0,
"text": "\\begin{align}\n \\text{EUR shoe size} &= \\frac{L + 2\\times{6.66\\bar{6}} } {6.6\\bar{6}} = \\frac{3}{20}\\times{L} + 2 \\\\[3pt]\n \\text{UK shoe size} &= \\frac{L + 2\\times{8.4\\bar{6}} } {8.4\\bar{6}} - 25= \\frac{3}{25.4}\\times{L} - 23\n\\end{align}"
},
{
"math_i... | https://en.wikipedia.org/wiki?curid=1015276 |
1015425 | Stolper–Samuelson theorem | Macroeconomic trade theorem
The Stolper–Samuelson theorem is a theorem in Heckscher–Ohlin trade theory. It describes the relationship between relative prices of output and relative factor returns—specifically, real wages and real returns to capital.
The theorem states that—under specific economic assumptions (constant ... | [
{
"math_id": 0,
"text": "P(C)= ar + bw, \\, "
},
{
"math_id": 1,
"text": "P(W) = cr + dw \\, "
}
] | https://en.wikipedia.org/wiki?curid=1015425 |
10159772 | Percus–Yevick approximation | In statistical mechanics the Percus–Yevick approximation is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. The approximation is named after Jerome K. Per... | [
{
"math_id": 0,
"text": " c(r)=g_{\\rm total}(r) - g_{\\rm indirect}(r) \\, "
},
{
"math_id": 1,
"text": "g_{\\rm total}(r)"
},
{
"math_id": 2,
"text": "g(r)=\\exp[-\\beta w(r)]"
},
{
"math_id": 3,
"text": "g_{\\rm indirect}(r)"
},
{
"math_id": 4,
"text": "u(r... | https://en.wikipedia.org/wiki?curid=10159772 |
10159868 | Spinors in three dimensions | Spin representations of the SO(3) group
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
Formulation.
The association of a spinor with ... | [
{
"math_id": 0,
"text": "\\vec{x} \\rightarrow X \\ =\\left(\\begin{matrix}x_3&x_1-ix_2\\\\x_1+ix_2&-x_3\\end{matrix}\\right)."
},
{
"math_id": 1,
"text": " X\\equiv {\\vec \\sigma}\\cdot{\\vec x} "
},
{
"math_id": 2,
"text": " {\\vec \\sigma}\\equiv (\\sigma_1, \\sigma_2, \\sigma_3... | https://en.wikipedia.org/wiki?curid=10159868 |
10160091 | Spin representation | Particular projective representations of the orthogonal or special orthogonal groups
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they... | [
{
"math_id": 0,
"text": "Q(z_1,\\ldots, z_n) = z_1^2+ z_2^2+\\cdots+z_n^2."
},
{
"math_id": 1,
"text": "Q(x_1,\\ldots, x_n) = x_1^2+ x_2^2+\\cdots+x_p^2-(x_{p+1}^2+\\cdots +x_{p+q}^2)."
},
{
"math_id": 2,
"text": "\\mathfrak{so}(V,Q) = \\mathfrak{so}(n,\\mathbb C)."
},
{
"mat... | https://en.wikipedia.org/wiki?curid=10160091 |
1016017 | Thiocyanate | Ion (S=C=N, charge –1)
<templatestyles src="Chembox/styles.css"/>
Chemical compound
Thiocyanates are salts containing the thiocyanate anion (also known as rhodanide or rhodanate). is the conjugate base of thiocyanic acid. Common salts include the colourless salts potassium thiocyanate and sodium thiocyanate. Merc... | [
{
"math_id": 0,
"text": "\\ce{S=C=N^\\ominus <-> {^{\\ominus}S}-C}\\ce{#N}"
}
] | https://en.wikipedia.org/wiki?curid=1016017 |
10160606 | Overdetermined system | More equations than unknowns (mathematics)
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solu... | [
{
"math_id": 0,
"text": "\\begin{align}\nY&=-2X-1\\\\\nY&=3X-2\\\\\nY&=X+1.\n\\end{align}"
},
{
"math_id": 1,
"text": "\n\\begin{bmatrix}\n 2 & 1 \\\\\n-3 & 1 \\\\\n-1 & 1 \\\\\n\\end{bmatrix}\n\\begin{bmatrix} X \\\\ Y \\end{bmatrix}\n = \n\\begin{bmatrix} -1 \\\\ -2 \\\\ 1 \\end{bmatrix}\n"
... | https://en.wikipedia.org/wiki?curid=10160606 |
10161645 | Kazhdan–Lusztig polynomial | In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial formula_0 is a member of a family of integral polynomials introduced by David Kazhdan and George Lusztig (1979). They are indexed by pairs of elements "y", "w" of a Coxeter group "W", which can in particular be the Weyl group of a Lie grou... | [
{
"math_id": 0,
"text": "P_{y,w}(q)"
},
{
"math_id": 1,
"text": "\\ell"
},
{
"math_id": 2,
"text": "\\ell(w)"
},
{
"math_id": 3,
"text": "T_w"
},
{
"math_id": 4,
"text": "w\\in W"
},
{
"math_id": 5,
"text": "\\mathbb{Z}[q^{1/2}, q^{-1/2}]"
},
{... | https://en.wikipedia.org/wiki?curid=10161645 |
10162277 | Sturm–Picone comparison theorem | In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real doma... | [
{
"math_id": 0,
"text": "(p_1(x) y^\\prime)^\\prime + q_1(x) y = 0 "
},
{
"math_id": 1,
"text": "(p_2(x) y^\\prime)^\\prime + q_2(x) y = 0 "
},
{
"math_id": 2,
"text": "0 < p_2(x) \\le p_1(x)"
},
{
"math_id": 3,
"text": "q_1(x) \\le q_2(x)."
}
] | https://en.wikipedia.org/wiki?curid=10162277 |
10162448 | Volatility swap | Financial derivative instrument
In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to
formula_0
where:... | [
{
"math_id": 0,
"text": "(\\sigma_{\\text{realised}}-K_{\\text{vol}})N_{\\text{vol}}"
},
{
"math_id": 1,
"text": "\\sigma_{\\text{realised}}"
},
{
"math_id": 2,
"text": "K_{\\text{vol}}"
},
{
"math_id": 3,
"text": "N_{\\text{vol}}"
},
{
"math_id": 4,
"text": "... | https://en.wikipedia.org/wiki?curid=10162448 |
10163003 | Bruhat order | Partial order on a Coxeter group
In mathematics, the Bruhat order (also called the strong order, strong Bruhat order, Chevalley order, Bruhat–Chevalley order, or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.
History.
The Bru... | [
{
"math_id": 0,
"text": "\\mu(\\pi,\\sigma)=(-1)^{\\ell(\\sigma)-\\ell(\\pi)}"
}
] | https://en.wikipedia.org/wiki?curid=10163003 |
10163132 | Heston model | Model in finance
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a ra... | [
{
"math_id": 0,
"text": "\ndS_t = \\mu S_t\\,dt + \\sqrt{\\nu_t} S_t\\,dW^S_t,\n"
},
{
"math_id": 1,
"text": "\n\\sqrt{\\nu_t}\n"
},
{
"math_id": 2,
"text": "\nd \\sqrt{\\nu_t} = -\\theta \\sqrt{\\nu_t} \\,dt + \\delta\\,dW^\\nu_t.\n"
},
{
"math_id": 3,
"text": "\\nu_t"
... | https://en.wikipedia.org/wiki?curid=10163132 |
10163390 | Picone identity | In the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. Since its inception in 1910 it has been used with tremendous success in association with an almost immediate proof of the Sturm comparison t... | [
{
"math_id": 0,
"text": "(p_1(x) u')' + q_1(x) u = 0 "
},
{
"math_id": 1,
"text": "(p_2(x) v')' + q_2(x) v = 0. "
},
{
"math_id": 2,
"text": "\\left(\\frac{u}{v}(p_1 u' v - p_2 u v')\\right)' = \\left(q_2 - q_1\\right) u^2 + \\left(p_1 - p_2\\right)u'^2 + p_2\\left(u'-v'\\frac{u}{v}\... | https://en.wikipedia.org/wiki?curid=10163390 |
1016345 | Cayley–Purser algorithm | 1999 public-key cryptography algorithm
The Cayley–Purser algorithm was a public-key cryptography algorithm published in early 1999 by 16-year-old Irishwoman Sarah Flannery, based on an unpublished work by Michael Purser, founder of Baltimore Technologies, a Dublin data security company. Flannery named it for mathematic... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}0 & 1 \\\\ 2 & 3\\end{bmatrix} +\n\\begin{bmatrix}1 & 2 \\\\ 3 & 4\\end{bmatrix} =\n\\begin{bmatrix}1 & 3 \\\\ 5 & 7\\end{bmatrix} \\equiv\n\\begin{bmatrix}1 & 3 \\\\ 0 & 2\\end{bmatrix}"
},
{
"math_id": 1,
"text": "\\begin{bmatrix}0 & 1 \\\\ 2 & 3 \\end... | https://en.wikipedia.org/wiki?curid=1016345 |
1016556 | Induced seismicity | Minor earthquakes and tremors caused by human activity
Induced seismicity is typically earthquakes and tremors that are caused by human activity that alters the stresses and strains on Earth's crust. Most induced seismicity is of a low magnitude. A few sites regularly have larger quakes, such as The Geysers geothermal ... | [
{
"math_id": 0,
"text": "\\tau_c =\\tau_0 +\\mu(\\sigma_n -P)"
},
{
"math_id": 1,
"text": "\\tau_c "
},
{
"math_id": 2,
"text": "\\tau_0 "
},
{
"math_id": 3,
"text": "\\sigma_n"
},
{
"math_id": 4,
"text": "\\mu"
},
{
"math_id": 5,
"text": "P"
},
... | https://en.wikipedia.org/wiki?curid=1016556 |
10165595 | Sturm separation theorem | In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such ... | [
{
"math_id": 0,
"text": "\\displaystyle u"
},
{
"math_id": 1,
"text": "\\displaystyle v"
},
{
"math_id": 2,
"text": "\\displaystyle W[u,v]"
},
{
"math_id": 3,
"text": "W[u,v](x)\\equiv W(x)\\neq 0"
},
{
"math_id": 4,
"text": "\\displaystyle x"
},
{
"ma... | https://en.wikipedia.org/wiki?curid=10165595 |
10167616 | Quantile function | Statistical function that defines the quantiles of a probability distribution
In probability and statistics, the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value. Intuitively, the quantile function associates with a range at and below... | [
{
"math_id": 0,
"text": " F_X\\colon \\mathbb{R} \\to [0,1]"
},
{
"math_id": 1,
"text": "Q\\colon [0, 1] \\to \\mathbb{R}"
},
{
"math_id": 2,
"text": "F_X(x) := \\Pr(X \\le x) = p\\,,"
},
{
"math_id": 3,
"text": "Q(p) =F_X^{-1}(p)\\,."
},
{
"math_id": 4,
"text... | https://en.wikipedia.org/wiki?curid=10167616 |
10167815 | Shape moiré | Type of moiré patterns
Shape moiré is one type of moiré patterns demonstrating the phenomenon of moiré magnification. 1D shape moiré is the particular simplified case of 2D shape moiré. One-dimensional patterns may appear when superimposing an opaque layer containing tiny horizontal transparent lines on top of a layer ... | [
{
"math_id": 0,
"text": "p_m=-\\frac{p_b \\cdot p_r}{p_b-p_r}"
},
{
"math_id": 1,
"text": "\\frac{v_m}{v_r}=-\\frac{p_b}{p_b-p_r}"
}
] | https://en.wikipedia.org/wiki?curid=10167815 |
101700 | Diophantine set | Solution of some Diophantine equation
In mathematics, a Diophantine equation is an equation of the form "P"("x"1, ..., "x""j", "y"1, ..., "y""k") = 0 (usually abbreviated "P"("x", "y") = 0) where "P"("x", "y") is a polynomial with integer coefficients, where "x"1, ..., "x""j" indicate parameters and "y"1, ..., "y""k" i... | [
{
"math_id": 0,
"text": "\\mathbb{N}^j"
},
{
"math_id": 1,
"text": "\\bar{x} \\in S \\iff (\\exists \\bar{y} \\in \\mathbb{N}^{k})(P(\\bar{x},\\bar{y})=0) ."
},
{
"math_id": 2,
"text": "\\mathbb{Q}"
},
{
"math_id": 3,
"text": "x = (y_1 + 1)(y_2 + 1)"
},
{
"math_id... | https://en.wikipedia.org/wiki?curid=101700 |
1017002 | Upper topology | In mathematics, the upper topology on a partially ordered set "X" is the coarsest topology in which the closure of a singleton formula_0 is the order section formula_1 for each formula_2 If formula_3 is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets. Howeve... | [
{
"math_id": 0,
"text": "\\{a\\}"
},
{
"math_id": 1,
"text": "a] = \\{x \\leq a\\}"
},
{
"math_id": 2,
"text": "a\\in X."
},
{
"math_id": 3,
"text": "\\leq"
},
{
"math_id": 4,
"text": "(-\\infty, +\\infty] = \\R \\cup \\{+\\infty\\}"
},
{
"math_id": 5,... | https://en.wikipedia.org/wiki?curid=1017002 |
10171509 | Hirschberg's algorithm | Algorithm for aligning two sequences
In computer science, Hirschberg's algorithm, named after its inventor, Dan Hirschberg, is a dynamic programming algorithm that finds the optimal sequence alignment between two strings. Optimality is measured with the Levenshtein distance, defined to be the sum of the costs of insert... | [
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "Y"
},
{
"math_id": 2,
"text": "\\operatorname{length}(X) = n"
},
{
"math_id": 3,
"text": "\\operatorname{length}(Y) = m"
},
{
"math_id": 4,
"text": "O(nm)"
},
{
"math_id": 5,
"text": "O(\\min\... | https://en.wikipedia.org/wiki?curid=10171509 |
10171965 | Joel Bowman | American chemist
Joel Mark Bowman is an American physical chemist and educator. He is the Samuel Candler Dobbs Professor of Theoretical Chemistry at Emory University.
Publications, honors and awards.
Bowman is the author or co-author of more than 600 publications and is a member of the International Academy of Quantum ... | [
{
"math_id": 0,
"text": "y_{ij}=exp(-r_{ij}/a)"
},
{
"math_id": 1,
"text": "r_{ij}"
},
{
"math_id": 2,
"text": "i"
},
{
"math_id": 3,
"text": "j"
},
{
"math_id": 4,
"text": "a"
}
] | https://en.wikipedia.org/wiki?curid=10171965 |
10172238 | Nanofluidics | Dynamics of fluids confined in nanoscale structures
Nanofluidics is the study of the behavior, manipulation, and control of fluids that are confined to structures of nanometer (typically 1–100 nm) characteristic dimensions (1 nm = 10−9 m). Fluids confined in these structures exhibit physical behaviors not observed in l... | [
{
"math_id": 0,
"text": "\\frac{1}{r}\\frac{d}{dr}\\left (r \\frac{d\\phi}{dr} \\right )= \\kappa^2 \\phi,"
},
{
"math_id": 1,
"text": "\\kappa = \\sqrt{\\frac{8\\pi n e^2}{\\epsilon k T}}, "
},
{
"math_id": 2,
"text": "\\frac{1}{r} \\frac{d}{dr} \\left (r \\frac{d v_z}{dr} \\right )... | https://en.wikipedia.org/wiki?curid=10172238 |
10172878 | Classical group | In mathematics, the classical groups are defined as the special linear groups over the reals formula_0, the complex numbers formula_1 and the quaternions formula_2 together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, ... | [
{
"math_id": 0,
"text": "\\mathbb{R}"
},
{
"math_id": 1,
"text": "\\mathbb{C}"
},
{
"math_id": 2,
"text": "\\mathbb{H}"
},
{
"math_id": 3,
"text": "\\varphi(x\\alpha, y\\beta) = \\alpha\\varphi(x, y)\\beta, \\quad \\forall x,y \\in V, \\forall \\alpha,\\beta \\in F."
},... | https://en.wikipedia.org/wiki?curid=10172878 |
10175953 | Zero dagger | In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be "no" set of natural numbers ... | [
{
"math_id": 0,
"text": "(L,\\in,U)"
}
] | https://en.wikipedia.org/wiki?curid=10175953 |
10176565 | Complete homogeneous symmetric polynomial | In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials.
Definition.
The complete hom... | [
{
"math_id": 0,
"text": "h_k (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq i_1 \\leq i_2 \\leq \\cdots \\leq i_k \\leq n} X_{i_1} X_{i_2} \\cdots X_{i_k}."
},
{
"math_id": 1,
"text": "h_k (X_1, X_2, \\dots,X_n) = \\sum_{l_1+l_2+ \\cdots + l_n=k \\atop l_i \\geq 0 } X_{1}^{l_1} X_{2}^{l_2} \\cdots X_{n... | https://en.wikipedia.org/wiki?curid=10176565 |
1018 | Algebraically closed field | Algebraic structure where all polynomials have roots
In mathematics, a field "F" is algebraically closed if every non-constant polynomial in "F"["x"] (the univariate polynomial ring with coefficients in "F") has a root in "F".
Examples.
As an example, the field of real numbers is not algebraically closed, because the p... | [
{
"math_id": 0,
"text": "x^2+1=0"
},
{
"math_id": 1,
"text": "\\mathbb F_p"
},
{
"math_id": 2,
"text": "\\begin{pmatrix}\n 0 & 0 & \\cdots & 0 & -a_0\\\\\n 1 & 0 & \\cdots & 0 & -a_1\\\\\n 0 & 1 & \\cdots & 0 & -a_2\\\\\n \\vdots & \\vdots & \\ddots & \\vdots & \\vdots\\\\\n 0 &... | https://en.wikipedia.org/wiki?curid=1018 |
1018020 | Evaporative cooling (atomic physics) | Atomic physics technique to achieve high phase space densities
Evaporative cooling is an atomic physics technique to achieve high phase space densities which optical cooling techniques alone typically can not reach.
Atoms trapped in optical or magnetic traps can be evaporatively cooled via two primary mechanisms, usual... | [
{
"math_id": 0,
"text": "\\Delta E\\propto-m_{F}B_{Z}"
}
] | https://en.wikipedia.org/wiki?curid=1018020 |
10180397 | Hapke parameters | The Hapke parameters are a set of parameters for an empirical model that is commonly used to describe the directional reflectance properties of the airless regolith surfaces of bodies in the Solar System. The model has been developed by astronomer Bruce Hapke at the University of Pittsburgh.
The parameters are:
The Hap... | [
{
"math_id": 0,
"text": "\\bar{\\omega}_0"
},
{
"math_id": 1,
"text": "K_s/(K_s+K_a)"
},
{
"math_id": 2,
"text": "K_s"
},
{
"math_id": 3,
"text": "K_a"
},
{
"math_id": 4,
"text": "h"
},
{
"math_id": 5,
"text": "B_0"
},
{
"math_id": 6,
"... | https://en.wikipedia.org/wiki?curid=10180397 |
1018257 | 3-manifold | Mathematical space
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like ou... | [
{
"math_id": 0,
"text": "M"
},
{
"math_id": 1,
"text": "\\pi = \\pi_1(M)"
},
{
"math_id": 2,
"text": "\\begin{align}\nH_0(M) &= H^3(M) =& \\mathbb{Z} \\\\\nH_1(M) &= H^2(M) =& \\pi/[\\pi,\\pi] \\\\\nH_2(M) &= H^1(M) =& \\text{Hom}(\\pi,\\mathbb{Z}) \\\\\nH_3(M) &= H^0(M) = & \\mathbb... | https://en.wikipedia.org/wiki?curid=1018257 |
10182648 | Monetary inflation | Sustained increase in a state's money supply (not prices)
Monetary inflation is a sustained increase in the money supply of a country (or currency area). Depending on many factors, especially public expectations, the fundamental state and development of the economy, and the transmission mechanism, it is likely to resul... | [
{
"math_id": 0,
"text": "MV = PT"
}
] | https://en.wikipedia.org/wiki?curid=10182648 |
1018336 | Lawson criterion | Criterion for igniting a nuclear fusion chain reaction
The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy being generated by fusion reactions within the fusion fuel to the rate of energy losses to the environment. When the rate of production is higher than the rate... | [
{
"math_id": 0,
"text": "\\tau_E"
},
{
"math_id": 1,
"text": "P_B = 1.4 \\cdot 10^{-34} \\cdot N^2 \\cdot T^{1/2} \\frac{\\mathrm{W}}{\\mathrm{cm}^3}"
},
{
"math_id": 2,
"text": "^2_1\\mathrm{D} +\\, ^3_1\\mathrm{T} \\rightarrow\\, ^4_2\\mathrm{He} \\left(3.5\\, \\mathrm{MeV}\\right... | https://en.wikipedia.org/wiki?curid=1018336 |
1018347 | Utility maximization problem | Problem of allocation of money by consumers in order to most benefit themselves
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize m... | [
{
"math_id": 0,
"text": "\\succcurlyeq"
},
{
"math_id": 1,
"text": "(x+\\epsilon, y)\\succcurlyeq(x,y)"
},
{
"math_id": 2,
"text": "(x,y+\\epsilon)\\succcurlyeq(x,y)"
},
{
"math_id": 3,
"text": "(x+\\epsilon, y+\\epsilon)\\succ(x,y)"
},
{
"math_id": 4,
"text":... | https://en.wikipedia.org/wiki?curid=1018347 |
101843 | Degenerate distribution | The probability distribution of a random variable which only takes a single value
In mathematics, a degenerate distribution (sometimes also Dirac distribution) is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with sup... | [
{
"math_id": 0,
"text": "F_{k_0}(x)=\\left\\{\\begin{matrix} 1, & \\mbox{if }x\\ge k_0 \\\\ 0, & \\mbox{if }x<k_0 \\end{matrix}\\right."
},
{
"math_id": 1,
"text": " k_0 \\in \\mathbb{R} "
},
{
"math_id": 2,
"text": "\\Pr(X = k_0) = 1,"
},
{
"math_id": 3,
"text": "X(\\ome... | https://en.wikipedia.org/wiki?curid=101843 |
10184674 | Large deviations of Gaussian random functions | A random function – of either one variable (a random process), or two or more variables
(a random field) – is called Gaussian if every finite-dimensional distribution is a multivariate normal distribution. Gaussian random fields on the sphere are useful (for example) when analysing
Sometimes, a value of a Gaussian rand... | [
{
"math_id": 0,
"text": "M"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "0"
},
{
"math_id": 3,
"text": "1"
},
{
"math_id": 4,
"text": "a>0"
},
{
"math_id": 5,
"text": "P(M>a)"
},
{
"math_id": 6,
"text": "C a \\exp(-a^2/2) ... | https://en.wikipedia.org/wiki?curid=10184674 |
101848 | ALOHAnet | Computer networking system
ALOHAnet, also known as the ALOHA System, or simply ALOHA, was a pioneering computer networking system developed at the University of Hawaii. ALOHAnet became operational in June 1971, providing the first public demonstration of a wireless packet data network.
The ALOHAnet used a new method o... | [
{
"math_id": 0,
"text": "\\frac{G^k e^{-G}}{k!}"
},
{
"math_id": 1,
"text": "\\frac{(2G)^k e^{-2G}}{k!}"
},
{
"math_id": 2,
"text": "Prob_{pure}"
},
{
"math_id": 3,
"text": "Prob_{pure}=e^{-2G}"
},
{
"math_id": 4,
"text": "S_{pure}"
},
{
"math_id": 5,
... | https://en.wikipedia.org/wiki?curid=101848 |
101851 | Hilbert's tenth problem | On solvability of Diophantine equations
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a ... | [
{
"math_id": 0,
"text": "3x^2-2xy-y^2z-7=0"
},
{
"math_id": 1,
"text": "x=1,\\ y=2,\\ z=-2"
},
{
"math_id": 2,
"text": "x^2+y^2+1=0"
},
{
"math_id": 3,
"text": "\\overline{HSI}."
},
{
"math_id": 4,
"text": "a_1x + a_2y = a_3"
},
{
"math_id": 5,
"te... | https://en.wikipedia.org/wiki?curid=101851 |
101863 | Linear independence | Vectors whose linear combinations are nonzero
In the theory of vector spaces, a set of vectors is said to be <templatestyles src="Template:Visible anchor/styles.css" />linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exi... | [
{
"math_id": 0,
"text": "\\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_k"
},
{
"math_id": 1,
"text": "a_1, a_2, \\dots, a_k,"
},
{
"math_id": 2,
"text": "a_1\\mathbf{v}_1 + a_2\\mathbf{v}_2 + \\cdots + a_k\\mathbf{v}_k = \\mathbf{0},"
},
{
"math_id": 3,
"text": "\\mat... | https://en.wikipedia.org/wiki?curid=101863 |
10186385 | Quadrature domains | In the branch of mathematics called potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D (an open connected set) together with
a finite subset {"z"1, …, z"k"} of D such that, for every function "u" harmonic and integrable over D with respect to area measure, the integral of "u" wi... | [
{
"math_id": 0,
"text": "\n\\iint_D u\\, dx dy = \\sum_{j=1}^k c_j u(z_j),\n"
}
] | https://en.wikipedia.org/wiki?curid=10186385 |
1018676 | Field of sets | Algebraic concept in measure theory, also referred to as an algebra of sets
In mathematics, a field of sets is a mathematical structure consisting of a pair formula_0 consisting of a set formula_1 and a family formula_2 of subsets of formula_1 called an algebra over formula_1 that contains the empty set as an element, ... | [
{
"math_id": 0,
"text": "( X, \\mathcal{F} )"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "\\mathcal{F}"
},
{
"math_id": 3,
"text": "X,"
},
{
"math_id": 4,
"text": "X "
},
{
"math_id": 5,
"text": "X \\in \\mathcal{F}"
},
{
"ma... | https://en.wikipedia.org/wiki?curid=1018676 |
1018783 | Marshallian demand function | Microeconomic function
In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the ... | [
{
"math_id": 0,
"text": " L "
},
{
"math_id": 1,
"text": " p "
},
{
"math_id": 2,
"text": " x "
},
{
"math_id": 3,
"text": " I "
},
{
"math_id": 4,
"text": "B(p, I) = \\{x : p \\cdot x \\leq I\\},"
},
{
"math_id": 5,
"text": " p \\cdot x = \\sum_i^... | https://en.wikipedia.org/wiki?curid=1018783 |
1018951 | List of convexity topics | This is a list of convexity topics, by Wikipedia page. | [
{
"math_id": 0,
"text": "\\mathbb{R}^n"
}
] | https://en.wikipedia.org/wiki?curid=1018951 |
1019002 | Dirac measure | Measure that is 1 if and only if a specified element is in the set
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element "x" or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
Definitio... | [
{
"math_id": 0,
"text": "\\delta_x (A) = 1_A(x)= \\begin{cases} 0, & x \\not \\in A; \\\\ 1, & x \\in A. \\end{cases}"
},
{
"math_id": 1,
"text": "\\int_{X} f(y) \\, \\mathrm{d} \\delta_x (y) = f(x),"
},
{
"math_id": 2,
"text": "\\int_X f(y) \\delta_x (y) \\, \\mathrm{d} y = f(x),"
... | https://en.wikipedia.org/wiki?curid=1019002 |
1019142 | Expenditure function | In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods.
Formally, if there is a utility function formula_0 that describes preferences over "n " commodities, the expenditur... | [
{
"math_id": 0,
"text": "u"
},
{
"math_id": 1,
"text": "e(p, u^*) : \\textbf R^n_+ \\times \\textbf R\n \\rightarrow \\textbf R"
},
{
"math_id": 2,
"text": "u^*"
},
{
"math_id": 3,
"text": "p"
},
{
"math_id": 4,
"text": "e(p, u^*) = \\min_{x \\in \\geq(u^*)} p... | https://en.wikipedia.org/wiki?curid=1019142 |
1019195 | Expenditure minimization problem | In microeconomics, the expenditure minimization problem is the dual of the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a consumer's utility function, prices, and a utility target,
Expenditure function.
Formally, the expenditure... | [
{
"math_id": 0,
"text": "u"
},
{
"math_id": 1,
"text": "L"
},
{
"math_id": 2,
"text": "p"
},
{
"math_id": 3,
"text": "u^*"
},
{
"math_id": 4,
"text": "e(p, u^*) = \\min_{x \\in \\geq{u^*}} p \\cdot x"
},
{
"math_id": 5,
"text": "\\geq{u^*} = \\{x \... | https://en.wikipedia.org/wiki?curid=1019195 |
1019406 | Cuthill–McKee algorithm | In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George and Joseph Liu is th... | [
{
"math_id": 0,
"text": "R_i"
},
{
"math_id": 1,
"text": "i=1, 2,.."
},
{
"math_id": 2,
"text": " R_{i+1} "
},
{
"math_id": 3,
"text": " R_i"
},
{
"math_id": 4,
"text": " R_i "
},
{
"math_id": 5,
"text": "n\\times n"
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=1019406 |
10195749 | Power sum symmetric polynomial | In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rati... | [
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": " p_k (x_1, x_2, \\dots,x_n) = \\sum_{i=1}^n x_i^k \\, ."
},
{
"math_id": 2,
"text": "p_0 (x_1, x_2, \\dots,x_n) = 1 + 1 + \\cdots + 1 = n \\, ,"
},
{
"math_id": 3,
"text": "p_1 (x_1, x_2, \\dots,x_n) = x_1 + x_2 + \\... | https://en.wikipedia.org/wiki?curid=10195749 |
1019627 | Howard T. Odum | American ecologist (1924–2002)
Howard Thomas Odum (September 1, 1924 – September 11, 2002), usually cited as H. T. Odum, was an American ecologist. He is known for his pioneering work on ecosystem ecology, and for his provocative proposals for additional laws of thermodynamics, informed by his work on general systems t... | [
{
"math_id": 0,
"text": "J"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "C"
},
{
"math_id": 3,
"text": "J = CX"
}
] | https://en.wikipedia.org/wiki?curid=1019627 |
10196392 | Gaisser–Hillas function | The Gaisser–Hillas function is used in astroparticle physics. It parameterizes the longitudinal particle density in a cosmic ray air shower. The function was proposed in 1977 by Thomas K. Gaisser and Anthony Michael Hillas.
The number of particles formula_0 as a function of traversed atmospheric depth formula_1 is expr... | [
{
"math_id": 0,
"text": "N(X)"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "N(X)= N_\\text{max}\\left(\\frac{X-X_0}{X_\\text{max}-X_0}\\right)^{\\frac{X_\\text{max}-X_{0}}{\\lambda}}\\exp\\left(\\frac{X_\\text{max}-X}{\\lambda}\\right),"
},
{
"math_id": 3,
"... | https://en.wikipedia.org/wiki?curid=10196392 |
1019760 | Weight transfer | Change in wheel load or center of mass in a vehicle
Weight transfer and load transfer are two expressions used somewhat confusingly to describe two distinct effects:
In the automobile industry, weight transfer customarily refers to the change in load borne by different wheels during acceleration. This would be more pr... | [
{
"math_id": 0,
"text": "\\Delta \\mathrm{Weight}_\\mathrm{front} = a\\frac{h}{b}m"
},
{
"math_id": 1,
"text": "\\Delta \\mathrm{Weight}_\\mathrm{front} = \\frac{a}{g} \\frac{h}{b}w"
},
{
"math_id": 2,
"text": "\\Delta \\mathrm{Weight}_\\mathrm{front}"
},
{
"math_id": 3,
... | https://en.wikipedia.org/wiki?curid=1019760 |
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