id stringlengths 2 8 | title stringlengths 1 130 | text stringlengths 0 252k | formulas listlengths 1 823 | url stringlengths 38 44 |
|---|---|---|---|---|
10002357 | Khintchine inequality | Theorem in probability
In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick formula_0 complex numbers formula_1, and add them together e... | [
{
"math_id": 0,
"text": " N "
},
{
"math_id": 1,
"text": " x_1,\\dots,x_N \\in\\mathbb{C}"
},
{
"math_id": 2,
"text": "\\pm 1 "
},
{
"math_id": 3,
"text": " \\sqrt{|x_1|^{2}+\\cdots + |x_N|^{2}}"
},
{
"math_id": 4,
"text": " \\{\\varepsilon_n\\}_{n=1}^N "
},... | https://en.wikipedia.org/wiki?curid=10002357 |
10004115 | Brushed DC electric motor | Internally commutated electric motor
A brushed DC electric motor is an internally commutated electric motor designed to be run from a direct current power source and utilizing an electric brush for contact.
Brushed motors were the first commercially important application of electric power to driving mechanical energy,... | [
{
"math_id": 0,
"text": "I = \\frac{V_\\text{applied} - V_\\text{cemf}}{R_\\text{armature}}"
},
{
"math_id": 1,
"text": "P = I \\cdot V_\\text{cemf}"
},
{
"math_id": 2,
"text": "\\begin{align}\n T &= \\frac{1}{2\\pi} k_b I_a \\Phi \\\\\n &= k_T I_a \\Phi\n\\end{align}"
},
{
... | https://en.wikipedia.org/wiki?curid=10004115 |
10004409 | Shear strength (soil) | Magnitude of the shear stress that a soil can sustain
Shear strength is a term used in soil mechanics to describe the magnitude of the shear stress that a soil can sustain. The shear resistance of soil is a result of friction and interlocking of particles, and possibly cementation or bonding of particle contacts. Due t... | [
{
"math_id": 0,
"text": "\\tau"
},
{
"math_id": 1,
"text": "\\mu"
}
] | https://en.wikipedia.org/wiki?curid=10004409 |
1000441 | Artificial chemistry | An artificial chemistry is a chemical-like system that usually consists of objects, called molecules, that interact according to rules resembling chemical reaction rules. Artificial chemistries are created and studied in order to understand fundamental properties of chemical systems, including prebiotic evolution, as w... | [
{
"math_id": 0,
"text": "\\subset"
}
] | https://en.wikipedia.org/wiki?curid=1000441 |
1000450 | Johann Heinrich von Thünen | German economist (1783–1850)
Johann Heinrich von Thünen (24 June 1783 – 22 September 1850), sometimes spelled Thuenen, was a prominent nineteenth-century economist and a native of Mecklenburg-Strelitz, now in northern Germany.
Even though he never held a professorial position, von Thunen had substantial influence on ec... | [
{
"math_id": 0,
"text": "R = Y(p - c) - YFm \\,"
}
] | https://en.wikipedia.org/wiki?curid=1000450 |
10005756 | Sample mean and covariance | Statistics computed from a sample of data
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.
The sample mean is the average value (or mean value) of a sample of numbers t... | [
{
"math_id": 0,
"text": " \\bar{X}=\\frac{1}{N}\\sum_{i=1}^{N}X_{i}."
},
{
"math_id": 1,
"text": "\\bar{x} = (1+4+1)/3 = 2"
},
{
"math_id": 2,
"text": "\\mu = (1+1+3+4+0+2+1+0) /8 = 12/8 = 1.5"
},
{
"math_id": 3,
"text": "x_{ij}"
},
{
"math_id": 4,
"text": "\\... | https://en.wikipedia.org/wiki?curid=10005756 |
10006830 | Disk loading | Characteristic of rotors/propellers
In fluid dynamics, disk loading or disc loading is the average pressure change across an actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including helicopter main rotors and tail rotors; propellers typically have a higher ... | [
{
"math_id": 0,
"text": "A"
},
{
"math_id": 1,
"text": "v"
},
{
"math_id": 2,
"text": "\\rho"
},
{
"math_id": 3,
"text": "\\dot{m}"
},
{
"math_id": 4,
"text": "\\dot m = \\rho \\, A \\, v."
},
{
"math_id": 5,
"text": "w"
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=10006830 |
10008 | Electrode | Electrical conductor used to make contact with nonmetallic parts of a circuit
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials ... | [
{
"math_id": 0,
"text": "\\qquad \\qquad"
},
{
"math_id": 1,
"text": "\\qquad"
},
{
"math_id": 2,
"text": "\\Delta G^{\\dagger}"
},
{
"math_id": 3,
"text": "\\Delta G^{0}"
},
{
"math_id": 4,
"text": "\\Delta G^{\\dagger} = \\frac{1}{4 \\lambda} (\\Delta G^{0} ... | https://en.wikipedia.org/wiki?curid=10008 |
1001293 | Irreducibility (mathematics) | In mathematics, the concept of irreducibility is used in several ways.
<templatestyles src="Dmbox/styles.css" />
Index of articles associated with the same name
This includes a list of related items that share the same name (or similar names). <br> If an [ internal link] incorrectly led you here, you may wi... | [
{
"math_id": 0,
"text": "\\mathbb RP^2"
}
] | https://en.wikipedia.org/wiki?curid=1001293 |
1001329 | Class function | In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group "G" that is constant on the conjugacy classes of "G". In other words, it is invariant under the conjugation map on "G". Such functions play a basic role in representation theory.
Chara... | [
{
"math_id": 0,
"text": " \\sum_{g \\in G} f(g) g"
},
{
"math_id": 1,
"text": " \\langle \\phi , \\psi \\rangle = \\frac{1}{|G|} \\sum_{g \\in G} \\phi(g) \\overline{\\psi(g)} "
},
{
"math_id": 2,
"text": " \\langle \\phi, \\psi \\rangle = \\int_G \\phi(t) \\overline{\\psi(t)}\\, dt.... | https://en.wikipedia.org/wiki?curid=1001329 |
1001361 | Semisimple module | Direct sum of irreducible modules
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring... | [
{
"math_id": 0,
"text": "0 \\to A \\xrightarrow{f} B \\xrightarrow{g} C \\to 0 "
},
{
"math_id": 1,
"text": "B \\cong A \\oplus C"
},
{
"math_id": 2,
"text": "B \\cong f(A) \\oplus s(C)."
},
{
"math_id": 3,
"text": " A=\\mathbf{Q}{\\left[x,y\\right]}/\\langle xy-yx-1\\ran... | https://en.wikipedia.org/wiki?curid=1001361 |
10013925 | Multiple inert gas elimination technique | Medical technique
The multiple inert gas elimination technique (MIGET) is a medical technique used mainly in pulmonology that involves measuring the concentrations of various infused, inert gases in mixed venous blood, arterial blood, and expired gas of a subject. The technique quantifies true shunt, physiological dead... | [
{
"math_id": 0,
"text": "V_A/Q=8.63 \\times \\frac{C_{c'}\\ce{O2} - C_v\\ce{O2}}{P_I\\ce{O2} - P_A\\ce{O2}}"
},
{
"math_id": 1,
"text": "V_A/Q=8.63 \\times \\frac{C_v\\ce{CO2} - C_{c'}\\ce{CO2}}{P_A\\ce{CO2}}"
},
{
"math_id": 2,
"text": "V_A/Q = 8.63 \\times \\ce{solubility} \\times ... | https://en.wikipedia.org/wiki?curid=10013925 |
10014466 | Copper cable certification | Cable testing regimen
In copper twisted pair wire networks, copper cable certification is achieved through a thorough series of tests in accordance with Telecommunications Industry Association (TIA) or International Organization for Standardization (ISO) standards. These tests are done using a certification-testing too... | [
{
"math_id": 0,
"text": "2"
},
{
"math_id": 1,
"text": "\\sqrt 2"
}
] | https://en.wikipedia.org/wiki?curid=10014466 |
1001490 | Convex conjugate | Generalization of the Legendre transformation
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (afte... | [
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "X^{*}"
},
{
"math_id": 2,
"text": "\\langle \\cdot , \\cdot \\rangle : X^{*} \\times X \\to \\mathbb{R}"
},
{
"math_id": 3,
"text": "\\left( x^*, x \\right) \\mapsto x^* (x)."
},
{
"math_id": 4,
"text": "... | https://en.wikipedia.org/wiki?curid=1001490 |
10016360 | Excellent ring | In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing m... | [
{
"math_id": 0,
"text": "\\mathbb{Z}"
},
{
"math_id": 1,
"text": "R"
},
{
"math_id": 2,
"text": "k"
},
{
"math_id": 3,
"text": "K"
},
{
"math_id": 4,
"text": "R\\otimes_kK"
},
{
"math_id": 5,
"text": "R \\to S"
},
{
"math_id": 6,
"text"... | https://en.wikipedia.org/wiki?curid=10016360 |
1002045 | Émilie du Châtelet | French mathematician, physicist, and author (1706–1749)
Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French natural philosopher and mathematician from the early 1730s until her death due to complications during childbirth in 1749.
Her most recognized ach... | [
{
"math_id": 0,
"text": "mv^2"
}
] | https://en.wikipedia.org/wiki?curid=1002045 |
1002128 | Giant magnetoresistance | Phenomenom involving the change of conductivity in metallic layers
Giant magnetoresistance (GMR) is a quantum mechanical magnetoresistance effect observed in multilayers composed of alternating ferromagnetic and non-magnetic conductive layers. The 2007 Nobel Prize in Physics was awarded to Albert Fert and Peter Grünber... | [
{
"math_id": 0,
"text": "\\delta_H = \\frac{R(H)-R(0)}{R(0)}"
},
{
"math_id": 1,
"text": "w = - J (\\mathbf M_1 \\cdot \\mathbf M_2). "
},
{
"math_id": 2,
"text": "R = R_0 + \\Delta R \\sin^2 \\frac{\\theta}{2},"
},
{
"math_id": 3,
"text": "\\rho_{F\\pm}=\\frac{2\\rho_F}{... | https://en.wikipedia.org/wiki?curid=1002128 |
10022123 | Vaccine efficacy | Reduction of disease among the vaccinated comparing to the unvaccinated
Vaccine efficacy or vaccine effectiveness is the percentage reduction of disease cases in a vaccinated group of people compared to an unvaccinated group. For example, a vaccine efficacy or effectiveness of 80% indicates an 80% decrease in the numbe... | [
{
"math_id": 0,
"text": "VE = \\frac{ARU - ARV}{ARU} \\times 100\\%,"
},
{
"math_id": 1,
"text": "VE"
},
{
"math_id": 2,
"text": "ARU"
},
{
"math_id": 3,
"text": "ARV"
},
{
"math_id": 4,
"text": "VE = (1 - RR) \\times 100\\%,"
},
{
"math_id": 5,
"t... | https://en.wikipedia.org/wiki?curid=10022123 |
10023038 | Benefit–cost ratio | Indicator of value-for-money of a project or proposal
A benefit–cost ratio (BCR) is an indicator, used in cost–benefit analysis, that attempts to summarize the overall value for money of a project or proposal. A BCR is the ratio of the benefits of a project or proposal, expressed in monetary terms, relative to its cost... | [
{
"math_id": 0,
"text": "BCR = \\frac{\\text{Discounted value of incremental benefits}}{\\text{Discounted value of incremental costs}}"
}
] | https://en.wikipedia.org/wiki?curid=10023038 |
10023138 | Chandrasekhar number | The Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent ratio of the Lorentz force to the viscosity. It is named after the Indian astrophysicist Subrahmanyan Chandrasekhar.
The number's main function is as a measure of the magnetic field, being proportional to the square of a chara... | [
{
"math_id": 0,
"text": "\\ Q"
},
{
"math_id": 1,
"text": "\\frac{1}{\\sigma}\\left(\\frac{\\partial^{}\\mathbf{u}}{\\partial t^{}}\\ +\\ (\\mathbf{u} \\cdot \\nabla) \\mathbf{u}\\right)\\ =\\ - {\\mathbf \\nabla }p\\ +\\ \\nabla^2 \\mathbf{u}\\ +\\frac {\\sigma}{\\zeta} {Q}\\ ({\\mathbf \\nabla... | https://en.wikipedia.org/wiki?curid=10023138 |
1002551 | Flap (aeronautics) | Anti-stalling high-lift device on aircraft
A flap is a high-lift device used to reduce the stalling speed of an aircraft wing at a given weight. Flaps are usually mounted on the wing trailing edges of a fixed-wing aircraft. Flaps are used to reduce the take-off distance and the landing distance. Flaps also cause an inc... | [
{
"math_id": 0,
"text": "L = \\tfrac12 \\rho V^2 S C_L"
},
{
"math_id": 1,
"text": "\\rho"
},
{
"math_id": 2,
"text": "C_L"
}
] | https://en.wikipedia.org/wiki?curid=1002551 |
100267 | Dihedral group | Group of symmetries of a regular polygon
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
The notation fo... | [
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "2n"
},
{
"math_id": 2,
"text": "n \\ge 3"
},
{
"math_id": 3,
"text": "\\mathrm{D}_n"
},
{
"math_id": 4,
"text": "n/2"
},
{
"math_id": 5,
"text": "\\mathrm{r}_i\\,\\mathrm{r}_j = \\mathrm{r}_{i... | https://en.wikipedia.org/wiki?curid=100267 |
1002779 | Azeotropic distillation | Any of a range of techniques used to break an azeotrope in distillation
In chemistry, azeotropic distillation is any of a range of techniques used to break an azeotrope in distillation. In chemical engineering, "azeotropic distillation" usually refers to the specific technique of adding another component to generate a ... | [
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] | https://en.wikipedia.org/wiki?curid=1002779 |
1003242 | Rodrigues' formula | Formula for the Legendre polynomials
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827). The name "Rodrigues formula" was introduced by Heine in... | [
{
"math_id": 0,
"text": "(P_n(x))_{n=0}^\\infty"
},
{
"math_id": 1,
"text": "[a, b]"
},
{
"math_id": 2,
"text": "\\int_a^b P_m(x) P_n(x) w(x) \\, dx = K_n \\delta_{m,n},"
},
{
"math_id": 3,
"text": "w(x)"
},
{
"math_id": 4,
"text": "K_n"
},
{
"math_id"... | https://en.wikipedia.org/wiki?curid=1003242 |
100337 | Standard ML | General-purpose functional programming language
Standard ML (SML) is a general-purpose, high-level, modular, functional programming language with compile-time type checking and type inference. It is popular for writing compilers, for programming language research, and for developing theorem provers.
Standard ML is a mo... | [
{
"math_id": 0,
"text": "f(x) = x^3-x-1"
},
{
"math_id": 1,
"text": "x=3"
},
{
"math_id": 2,
"text": "f'(3) = 27-1 = 26"
}
] | https://en.wikipedia.org/wiki?curid=100337 |
1003410 | S transform | S transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the "S" transform is a generalization of the short-time Fourier transform (STFT), extending the continuous wavelet transform and overcoming some of its disadvantages. For one, modulation sinusoids are fixed wi... | [
{
"math_id": 0,
"text": " S_x(t,f) = \\int_{-\\infty}^\\infty x(\\tau)|f|e^{- \\pi (t- \\tau)^2 f^2} e^{-j2 \\pi f \\tau} \\, d \\tau "
},
{
"math_id": 1,
"text": "x(\\tau) = \\int_{-\\infty}^\\infty \\left[\\int_{-\\infty}^{\\infty}S_x(t,f)\\, dt\\right]\\,e^{j2\\pi f\\tau}\\, df"
},
{
... | https://en.wikipedia.org/wiki?curid=1003410 |
10034460 | Catenary ring | In mathematics, a commutative ring "R" is catenary if for any pair of prime ideals "p", "q", any two strictly increasing chains
"p" = "p"0 ⊂ "p"1 ⊂ ... ⊂ "p""n" = "q"
of prime ideals are contained in maximal strictly increasing chains from "p" to "q" of the same (finite) length. In a geometric situation, in which the d... | [
{
"math_id": 0,
"text": "\\text{height}(P)\\le \\text{height}(p)+ \\text{tr.deg.}_A(B) - \\text{tr.deg.}_{\\kappa(p)}(\\kappa(P))."
},
{
"math_id": 1,
"text": "B=A[x_1,\\dots,x_n]"
}
] | https://en.wikipedia.org/wiki?curid=10034460 |
100349 | Legendre polynomials | System of complete and orthogonal polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlig... | [
{
"math_id": 0,
"text": "w(x) = 1"
},
{
"math_id": 1,
"text": " [-1,1]"
},
{
"math_id": 2,
"text": "P_n(x)"
},
{
"math_id": 3,
"text": "n"
},
{
"math_id": 4,
"text": "\\int_{-1}^1 P_m(x) P_n(x) \\,dx = 0 \\quad \\text{if } n \\ne m."
},
{
"math_id": 5,... | https://en.wikipedia.org/wiki?curid=100349 |
10039004 | Sergey Yablonsky | Soviet and Russian mathematician
Sergey Vsevolodovich Yablonsky (Russian: Серге́й Все́володович Ябло́нский, 6 December 1924 – 26 May 1998) was a Soviet and Russian mathematician, one of the founders of the Soviet school of mathematical cybernetics and discrete mathematics. He is the author of a number of classic result... | [
{
"math_id": 0,
"text": "c^n"
},
{
"math_id": 1,
"text": "c"
},
{
"math_id": 2,
"text": "n"
}
] | https://en.wikipedia.org/wiki?curid=10039004 |
10040846 | Random measure | In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.
Definition.
Random measures can be defined as transition kernels or as ran... | [
{
"math_id": 0,
"text": " E "
},
{
"math_id": 1,
"text": " \\mathcal E "
},
{
"math_id": 2,
"text": " \\sigma "
},
{
"math_id": 3,
"text": " \\R^n "
},
{
"math_id": 4,
"text": " \\zeta "
},
{
"math_id": 5,
"text": " (\\Omega, \\mathcal A, P) "
},... | https://en.wikipedia.org/wiki?curid=10040846 |
10042977 | Wind engineering | Study of the effects of wind on natural and built environments
Wind engineering is a subset of mechanical engineering, structural engineering, meteorology, and applied physics that analyzes the effects of wind in the natural and the built environment and studies the possible damage, inconvenience or benefits which may ... | [
{
"math_id": 0,
"text": "\\ v_z = v_g \\cdot \\left( \\frac {z} {z_g} \\right)^ \\frac {1} {\\alpha}, 0 < z < z_g\n"
},
{
"math_id": 1,
"text": "\\ v_z"
},
{
"math_id": 2,
"text": "\\ z"
},
{
"math_id": 3,
"text": "\\ v_g"
},
{
"math_id": 4,
"text": "\\ z_g "
... | https://en.wikipedia.org/wiki?curid=10042977 |
10043 | Estimator | Rule for calculating an estimate of a given quantity based on observed data
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, t... | [
{
"math_id": 0,
"text": " \\theta "
},
{
"math_id": 1,
"text": "\\widehat{\\theta}"
},
{
"math_id": 2,
"text": "\\theta"
},
{
"math_id": 3,
"text": "\\widehat{\\theta}(X)"
},
{
"math_id": 4,
"text": "x"
},
{
"math_id": 5,
"text": "X=x"
},
{
... | https://en.wikipedia.org/wiki?curid=10043 |
10043801 | Homotopy category of chain complexes | In homological algebra in mathematics, the homotopy category "K(A)" of chain complexes in an additive category "A" is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes "Kom(A)" of "A" and the derived category "D(A)" of "A" when "A" is a... | [
{
"math_id": 0,
"text": "h^n \\colon A^n \\to B^{n - 1}"
},
{
"math_id": 1,
"text": "f^n - g^n = d_B^{n - 1} h^n + h^{n + 1} d_A^n,"
},
{
"math_id": 2,
"text": " f - g = d_B h + h d_A."
},
{
"math_id": 3,
"text": "f - g"
},
{
"math_id": 4,
"text": "f \\sim g\\... | https://en.wikipedia.org/wiki?curid=10043801 |
1004401 | Nambu–Goto action | The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of zero-thickness (infinitely thin) string behavior, using the principles of Lagrangian mechani... | [
{
"math_id": 0,
"text": "L=K-U"
},
{
"math_id": 1,
"text": "S"
},
{
"math_id": 2,
"text": "S = \\int_{t_i}^{t_f} L \\, dt."
},
{
"math_id": 3,
"text": "-ds^2 = -(c \\, dt)^2 + dx^2 + dy^2 + dz^2, \\ "
},
{
"math_id": 4,
"text": "ds/c"
},
{
"math_id": 5... | https://en.wikipedia.org/wiki?curid=1004401 |
10046650 | Optimal matching | Sequence analysis in social science
Optimal matching is a sequence analysis method used in social science, to assess the dissimilarity of ordered arrays of tokens that usually represent a time-ordered sequence of socio-economic states two individuals have experienced. Once such distances have been calculated for a set ... | [
{
"math_id": 0,
"text": "S = (s_1, s_2, s_3, \\ldots s_T)"
},
{
"math_id": 1,
"text": "s_i"
},
{
"math_id": 2,
"text": "{\\mathbf S}"
},
{
"math_id": 3,
"text": "a_i: {\\mathbf S} \\rightarrow {\\mathbf S}"
},
{
"math_id": 4,
"text": "s"
},
{
"math_id"... | https://en.wikipedia.org/wiki?curid=10046650 |
10046651 | G-network | In queueing theory, a discipline within the mathematical theory of probability, a G-network (generalized queueing network, often called a Gelenbe network) is an open network of G-queues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic re-routing or traffi... | [
{
"math_id": 0,
"text": "\\scriptstyle{\\Lambda_i}"
},
{
"math_id": 1,
"text": "\\scriptstyle{\\lambda_i}"
},
{
"math_id": 2,
"text": "\\scriptstyle{p_{ij}^{+}}"
},
{
"math_id": 3,
"text": "\\scriptstyle{p_{ij}^{-}}"
},
{
"math_id": 4,
"text": "\\scriptstyle{d... | https://en.wikipedia.org/wiki?curid=10046651 |
1004679 | Needleman–Wunsch algorithm | Method for aligning biological sequences
The Needleman–Wunsch algorithm is an algorithm used in bioinformatics to align protein or nucleotide sequences. It was one of the first applications of dynamic programming to compare biological sequences. The algorithm was developed by Saul B. Needleman and Christian D. Wunsch a... | [
{
"math_id": 0,
"text": "O(mn)"
},
{
"math_id": 1,
"text": "F_{ij}"
},
{
"math_id": 2,
"text": "O(nm)"
},
{
"math_id": 3,
"text": "\\Theta(\\min \\{n,m\\})"
},
{
"math_id": 4,
"text": "i=0,\\dotsc,n"
},
{
"math_id": 5,
"text": "j=0,\\dotsc,m"
},
... | https://en.wikipedia.org/wiki?curid=1004679 |
1004743 | Similarity measure | Real-valued function that quantifies similarity between two objects
In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures... | [
{
"math_id": 0,
"text": " (x_1,y_1)"
},
{
"math_id": 1,
"text": " (x_2,y_2)"
},
{
"math_id": 2,
"text": " d = \\surd[(x_2-x_1)^2 + (y_2-y_1)^2]"
},
{
"math_id": 3,
"text": " J(A,B)={ A\\bigcap B\\over A\\bigcup B}"
},
{
"math_id": 4,
"text": " \\left\\vert x_1... | https://en.wikipedia.org/wiki?curid=1004743 |
1004764 | Gap penalty | A Gap penalty is a method of scoring alignments of two or more sequences. When aligning sequences, introducing gaps in the sequences can allow an alignment algorithm to match more terms than a gap-less alignment can. However, minimizing gaps in an alignment is important to create a useful alignment. Too many gaps can c... | [
{
"math_id": 0,
"text": "A+B\\cdot (L-1)"
},
{
"math_id": 1,
"text": "k"
},
{
"math_id": 2,
"text": "kA+kB (L-1) = k(A+B(L-1))"
},
{
"math_id": 3,
"text": "G(L)=A+C\\ln L"
}
] | https://en.wikipedia.org/wiki?curid=1004764 |
10048386 | Milner Baily Schaefer | Milner Baily ("Benny") Schaefer (1912 in Cheyenne, Wyoming – 1970 in San Diego, California), is notable for his work on the population dynamics of fisheries.
Career.
Schaefer worked as a biologist at the Washington State Fisheries Department. From 1937 to 1942 as a scientist for the International Pacific Salmon Fisheri... | [
{
"math_id": 0,
"text": "H(E,X)=q E X\\!"
},
{
"math_id": 1,
"text": "\\dot{X}=0"
},
{
"math_id": 2,
"text": "H(E)=q K E \\left(1-\\frac{qE}{r}\\right)"
}
] | https://en.wikipedia.org/wiki?curid=10048386 |
10049713 | Victorian Railways Dd class | Class of Australian 4-6-0 and 58 Australian 4-6-2T steam locomotives
The DD class (later reclassified into D1, D2 and D3 subclasses) was a passenger and mixed traffic steam locomotive that ran on Victorian Railways from 1902 to 1974. Originally introduced on mainline express passenger services, they were quickly supers... | [
{
"math_id": 0,
"text": "\\mathrm{D^D_E}"
}
] | https://en.wikipedia.org/wiki?curid=10049713 |
10049748 | Cophasing | Segmented mirror/telescope-related individual segment-controlling process in astronomy
In astronomy, the term cophasing or phasing describes the process of controlling the individual segments in a segmented mirror or a telescope so that the segments form a larger composite mirroring surface. Cophasing implies precise, ... | [
{
"math_id": 0,
"text": "\\lambda/40"
}
] | https://en.wikipedia.org/wiki?curid=10049748 |
10050297 | Maximal ergodic theorem | The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
Suppose that formula_0 is a probability space, that formula_1 is a (possibly noninvertible) measure-preserving transformation, and that formula_2. Define formula_3 by
formula_4
Then the maximal ergodic theorem states that
formu... | [
{
"math_id": 0,
"text": "(X, \\mathcal{B},\\mu)"
},
{
"math_id": 1,
"text": "T : X\\to X"
},
{
"math_id": 2,
"text": "f\\in L^1(\\mu,\\mathbb{R})"
},
{
"math_id": 3,
"text": "f^*"
},
{
"math_id": 4,
"text": "f^* = \\sup_{N\\geq 1} \\frac{1}{N} \\sum_{i=0}^{N-1... | https://en.wikipedia.org/wiki?curid=10050297 |
10050999 | Median absolute deviation | Statistical measure of variability
In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.
For a univariate data set "X"1, "X"2, ..., "Xn", ... | [
{
"math_id": 0,
"text": "\\tilde{X}=\\operatorname{median}(X) "
},
{
"math_id": 1,
"text": "\n\\operatorname{MAD} = \\operatorname{median}( |X_i - \\tilde{X}|)\n"
},
{
"math_id": 2,
"text": "\\sigma"
},
{
"math_id": 3,
"text": "\\hat{\\sigma} = k \\cdot \\operatorname{MA... | https://en.wikipedia.org/wiki?curid=10050999 |
10053499 | Oleg Lupanov | Russian mathematician (1932–2006)
Oleg Borisovich Lupanov (; 2 June 1932 – 3 May 2006) was a Soviet and Russian mathematician, dean of the Moscow State University's Faculty of Mechanics and Mathematics (1980–2006), head of the Chair of Discrete Mathematics of the Faculty of Mechanics and Mathematics (1981–2006).
Togeth... | [
{
"math_id": 0,
"text": "C(f)\\le \\frac{2^n}{n} + o\\left(\\frac{2^n}{n}\\right). "
}
] | https://en.wikipedia.org/wiki?curid=10053499 |
100558 | A* search algorithm | Algorithm used for pathfinding and graph traversal
A* (pronounced "A-star") is a graph traversal and pathfinding algorithm, which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. Given a weighted graph, a source node and a goal node, the algorithm finds the shortes... | [
{
"math_id": 0,
"text": "O(b^d)"
},
{
"math_id": 1,
"text": "f(n) = g(n) + h(n)"
},
{
"math_id": 2,
"text": "d(x,y)>\\varepsilon>0"
},
{
"math_id": 3,
"text": "\\varepsilon"
},
{
"math_id": 4,
"text": "w(n) = \\begin{cases} 1 - \\frac{d(n)}{N} & d(n) \\le N \\... | https://en.wikipedia.org/wiki?curid=100558 |
10056274 | Data transformation (statistics) | Application of a function to each point in a data set
In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point "zi" is replaced with the transformed value "yi" = "f"("zi"), where "f" is a function. Transforms are usually applied ... | [
{
"math_id": 0,
"text": "Y = a + bX"
},
{
"math_id": 1,
"text": "\\log(Y) = a + bX"
},
{
"math_id": 2,
"text": "Y = e^a e^{bX}"
},
{
"math_id": 3,
"text": "\\log(Y)"
},
{
"math_id": 4,
"text": "e^{b}\\!"
},
{
"math_id": 5,
"text": "10^{b}"
},
{... | https://en.wikipedia.org/wiki?curid=10056274 |
100563 | System on a chip | Micro-electronic component
A system on a chip or system-on-chip (SoC ; pl. "SoCs" ) is an integrated circuit that integrates most or all components of a computer or other electronic system. These components almost always include on-chip central processing unit (CPU), memory interfaces, input/output devices and interfac... | [
{
"math_id": 0,
"text": "P = IV = \\frac{V^2}{R} = {I^2}{R}"
}
] | https://en.wikipedia.org/wiki?curid=100563 |
1005746 | Supercritical flow | Flow velocity larger than wave velocity
A supercritical flow is a flow whose velocity is larger than the wave velocity. The analogous condition in gas dynamics is supersonic speed.
According to the website Civil Engineering Terms, supercritical flow is defined as follows:
The flow at which depth of the channel is less... | [
{
"math_id": 0,
"text": "Fr \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{U}{\\sqrt{gh}},"
},
{
"math_id": 1,
"text": " Fr < 1 "
},
{
"math_id": 2,
"text": " Fr > 1 "
},
{
"math_id": 3,
"text": " Fr \\approx 1 "
}
] | https://en.wikipedia.org/wiki?curid=1005746 |
10058495 | Admittance parameters | Properties of an electrical network in terms of a matrix of ratios of currents to voltages
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used t... | [
{
"math_id": 0,
"text": "I = Y V\\,"
},
{
"math_id": 1,
"text": "\\begin{pmatrix}I_1 \\\\ I_2\\end{pmatrix} = \\begin{pmatrix} Y_{11} & Y_{12} \\\\ Y_{21} & Y_{22} \\end{pmatrix}\\begin{pmatrix}V_1 \\\\ V_2\\end{pmatrix}"
},
{
"math_id": 2,
"text": "\\begin{align} \nY_{11} &= {I_1 \\... | https://en.wikipedia.org/wiki?curid=10058495 |
1005868 | Tachymeter (watch) | Scale sometimes inscribed around the rim of an analog watch
A tachymeter (pronounced ) is a scale sometimes inscribed around the rim of an analog watch with a chronograph. It can be used to conveniently compute the frequency in inverse-hours of an event of a known second-defined period, such as speed (distance over hou... | [
{
"math_id": 0,
"text": "T = \\frac{3600}{t}"
}
] | https://en.wikipedia.org/wiki?curid=1005868 |
10058792 | Backlash (engineering) | Clearance between mating components
In mechanical engineering, backlash, sometimes called lash, play, or slop, is a clearance or lost motion in a mechanism caused by gaps between the parts. It can be defined as "the maximum distance or angle through which any part of a mechanical system may be moved in one direction wi... | [
{
"math_id": 0,
"text": "b_t=t_i-t_a\\;"
},
{
"math_id": 1,
"text": "b_c = 2 \\left( \\Delta c \\right) \\tan\\phi"
},
{
"math_id": 2,
"text": "b_{avg}=0.04 * m"
}
] | https://en.wikipedia.org/wiki?curid=10058792 |
10059553 | Futile cycle | Metabolic process
A futile cycle, also known as a substrate cycle, occurs when two metabolic pathways run simultaneously in opposite directions and have no overall effect other than to dissipate energy in the form of heat. The reason this cycle was called "futile" cycle was because it appeared that this cycle operated ... | [
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] | https://en.wikipedia.org/wiki?curid=10059553 |
10059597 | GPS signals | Signals broadcast by GPS satellites
GPS signals are broadcast by Global Positioning System satellites to enable satellite navigation. Receivers on or near the Earth's surface can determine location, time, and velocity using this information. The GPS satellite constellation is operated by the 2nd Space Operations Squadr... | [
{
"math_id": 0,
"text": "i"
},
{
"math_id": 1,
"text": "\\begin{align}\n X_1(t) &= d(t) \\oplus d(t - 2) \\oplus d(t - 3) \\oplus d(t - 5) \\oplus d(t - 6) \\\\\n X_2(t) &= d(t) \\oplus d(t - 1) \\oplus d(t - 2) \\oplus d(t - 3) \\oplus d(t - 6) \\\\\n d'(t') &= \\begin{cases}\n X_1\\left(... | https://en.wikipedia.org/wiki?curid=10059597 |
10059981 | Category of manifolds | Category theory
In mathematics, the category of manifolds, often denoted Man"p", is the category whose objects are manifolds of smoothness class "C""p" and whose morphisms are "p"-times continuously differentiable maps. This is a category because the composition of two "C""p" maps is again continuous and of class "C""p... | [
{
"math_id": 0,
"text": "(M, p_0),"
},
{
"math_id": 1,
"text": "M"
},
{
"math_id": 2,
"text": "C^p"
},
{
"math_id": 3,
"text": "p_0 \\in M ,"
},
{
"math_id": 4,
"text": "F: (M,p_0) \\to (N,q_0),"
},
{
"math_id": 5,
"text": "F(p_0) = q_0."
},
{
... | https://en.wikipedia.org/wiki?curid=10059981 |
10061569 | Perpendicular axis theorem | The perpendicular axis theorem (or plane figure theorem) states that, "The moment of inertia ("Iz") of a laminar body about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent."
Define perpendicula... | [
{
"math_id": 0,
"text": "x"
},
{
"math_id": 1,
"text": "y"
},
{
"math_id": 2,
"text": "z"
},
{
"math_id": 3,
"text": "O"
},
{
"math_id": 4,
"text": "xy"
},
{
"math_id": 5,
"text": "I_z = I_x + I_y"
},
{
"math_id": 6,
"text": "I_x"
},
... | https://en.wikipedia.org/wiki?curid=10061569 |
100625 | Inverter (logic gate) | Logic gate implementing negation
In digital logic, an inverter or NOT gate is a logic gate which implements logical negation. It outputs a bit opposite of the bit that is put into it. The bits are typically implemented as two differing voltage levels.
Description.
The NOT gate outputs a zero when given a one, and a one... | [
{
"math_id": 0,
"text": "f(a)=1-a"
},
{
"math_id": 1,
"text": "f(0)=1-0=1"
},
{
"math_id": 2,
"text": "f(1)=1-1=0"
}
] | https://en.wikipedia.org/wiki?curid=100625 |
10063629 | Rank–size distribution | Rank–size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as the rank–frequency distribution, when the source data are from a frequenc... | [
{
"math_id": 0,
"text": "1 - p"
}
] | https://en.wikipedia.org/wiki?curid=10063629 |
10063692 | Constant-Q transform | Short-time Fourier transform with variable resolution
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform and very closely related to the complex Morlet wavelet tran... | [
{
"math_id": 0,
"text": "\\delta f_k = 2^{1/n} \\cdot \\delta f_{k-1}\n= \\left( 2^{1/n} \\right)^k \\cdot \\delta f_\\text{min},"
},
{
"math_id": 1,
"text": "X[k,m] = \\sum_{n=0}^{N-1} W[n-m] x[n] e^{-j 2 \\pi k n/N}. "
},
{
"math_id": 2,
"text": " Q = \\frac{f_k}{\\delta f_k}."
}... | https://en.wikipedia.org/wiki?curid=10063692 |
100638 | Controllability | Dynamic system property
Controllability is an important property of a control system and plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observability are dual aspects of the same problem.
Roughly, the concept of controllabili... | [
{
"math_id": 0,
"text": "\\mathbf{x_0}"
},
{
"math_id": 1,
"text": "\\mathbf{x_f}"
},
{
"math_id": 2,
"text": "\\dot{\\mathbf{x}} = \\mathbf{A}\\mathbf{x}(t) + \\mathbf{B}\\mathbf{u}(t)"
},
{
"math_id": 3,
"text": "\\mathbf{x}"
},
{
"math_id": 4,
"text": "\\ma... | https://en.wikipedia.org/wiki?curid=100638 |
10063937 | Dual cone and polar cone | Concepts in convex analysis
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.
Dual cone.
In a vector space.
The dual cone "C*" of a subset "C" in a linear space "X" over the reals, e.g. Euclidean space R"n", with dual space "X*" is the set
formula_0
where formula_1 is th... | [
{
"math_id": 0,
"text": "C^* = \\left \\{y\\in X^*: \\langle y , x \\rangle \\geq 0 \\quad \\forall x\\in C \\right \\},"
},
{
"math_id": 1,
"text": "\\langle y, x \\rangle"
},
{
"math_id": 2,
"text": "\\langle y, x\\rangle = y(x)"
},
{
"math_id": 3,
"text": "C^{\\prime}... | https://en.wikipedia.org/wiki?curid=10063937 |
10064136 | Separation principle | In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of the system, which feed... | [
{
"math_id": 0,
"text": "\n\\begin{align}\n\\dot{x}(t) & = A x(t) + B u(t) \\\\\ny(t) & = C x(t)\n\\end{align}\n"
},
{
"math_id": 1,
"text": "u(t)"
},
{
"math_id": 2,
"text": "y(t)"
},
{
"math_id": 3,
"text": "x(t)"
},
{
"math_id": 4,
"text": "\\dot{\\hat{x}} ... | https://en.wikipedia.org/wiki?curid=10064136 |
10065 | Empirical formula | Simplest whole number ratio of atoms present in a compound
In chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms present in a compound. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, would simply be SO, as is the empirical formu... | [
{
"math_id": 0,
"text": "\\left(\\frac{48.64 \\mbox{ g C}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }}{12.01 \\mbox{ g C}}\\right) = 4.049\\ \\text{mol}"
},
{
"math_id": 1,
"text": "\\left(\\frac{8.16 \\mbox{ g H}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }}{1.007 \\mbox{ g H}}\\right) = 8.095\\ \\tex... | https://en.wikipedia.org/wiki?curid=10065 |
1006597 | Nanorobotics | Emerging technology field
Nanoid robotics, or for short, nanorobotics or nanobotics, is an emerging technology field creating machines or robots, which are called nanorobots or simply nanobots, whose components are at or near the scale of a nanometer (10−9 meters). More specifically, nanorobotics (as opposed to microro... | [
{
"math_id": 0,
"text": "\\boldsymbol{F} = V \\cdot (\\boldsymbol{M} \\cdot \\nabla \\boldsymbol{B})"
},
{
"math_id": 1,
"text": "\\boldsymbol{\\tau} = V \\cdot (\\boldsymbol{M} \\boldsymbol{\\times} \\boldsymbol{B})"
},
{
"math_id": 2,
"text": "(\\boldsymbol{M} \\boldsymbol{\\times}... | https://en.wikipedia.org/wiki?curid=1006597 |
10066313 | Burgers vector | Vector representing lattice distortion due to dislocations in a crystal
In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as b, that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice.
Concepts.
T... | [
{
"math_id": 0,
"text": "\n\\|\\mathbf{b}\\|\\ = (a/2)\\sqrt{h^2+k^2+l^2}\n"
},
{
"math_id": 1,
"text": "\\|\\mathbf{b}\\|"
},
{
"math_id": 2,
"text": "\\mathbf b = \\tfrac{a}{2} \\langle h k l \\rangle ;"
},
{
"math_id": 3,
"text": "\\tfrac{a}{2} \\langle h k l \\rangle ... | https://en.wikipedia.org/wiki?curid=10066313 |
10066673 | Causal sets | Approach to quantum gravity using discrete spacetime
The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order... | [
{
"math_id": 0,
"text": "C"
},
{
"math_id": 1,
"text": "\\preceq"
},
{
"math_id": 2,
"text": "x \\in C"
},
{
"math_id": 3,
"text": " x \\preceq x "
},
{
"math_id": 4,
"text": "x, y \\in C"
},
{
"math_id": 5,
"text": " x \\preceq y"
},
{
"ma... | https://en.wikipedia.org/wiki?curid=10066673 |
10067215 | Quality engineering | Principles and practice of product and service quality assurance and control
Quality engineering is the discipline of engineering concerned with the principles and practice of product and service quality assurance and control. In software development, it is the management, development, operation and maintenance of IT s... | [
{
"math_id": 0,
"text": "\\text{Quality} = \\frac{\\text{Results of work efforts}}{\\text{Total costs}}"
}
] | https://en.wikipedia.org/wiki?curid=10067215 |
10067276 | Shifted Gompertz distribution | The shifted Gompertz distribution is the distribution of the larger of two independent random variables one of which has an exponential distribution with parameter formula_1 and the other has a Gumbel distribution with parameters formula_2 and formula_1. In its original formulation the distribution was expressed referr... | [
{
"math_id": 0,
"text": "b \\geq 0"
},
{
"math_id": 1,
"text": " b "
},
{
"math_id": 2,
"text": "\\eta"
},
{
"math_id": 3,
"text": " f(x;b,\\eta) = b e^{-bx} e^{-\\eta e^{-bx}}\\left[1 + \\eta\\left(1 - e^{-bx}\\right)\\right] \\text{ for }x \\geq 0. \\,"
},
{
"m... | https://en.wikipedia.org/wiki?curid=10067276 |
10070469 | Herbertsmithite | Halide mineral
Herbertsmithite is a mineral with chemical structure ZnCu3(OH)6Cl2. It is named after the mineralogist Herbert Smith (1872–1953) and was first found in 1972 in Chile. It is polymorphous with kapellasite and closely related to paratacamite. Herbertsmithite is generally found in and around Anarak, Iran, he... | [
{
"math_id": 0,
"text": "\\mathbb{Z}_2"
}
] | https://en.wikipedia.org/wiki?curid=10070469 |
10070867 | Bidirectional map | In computer science, a bidirectional map is an associative data structure in which the formula_0 pairs form a one-to-one correspondence. Thus the binary relation is functional in each direction: each formula_1 can also be mapped to a unique formula_2. A pair formula_3 thus provides a unique coupling between formula_4 a... | [
{
"math_id": 0,
"text": "(key, value)"
},
{
"math_id": 1,
"text": "value"
},
{
"math_id": 2,
"text": "key"
},
{
"math_id": 3,
"text": "(a, b)"
},
{
"math_id": 4,
"text": "a"
},
{
"math_id": 5,
"text": "b"
},
{
"math_id": 6,
"text": "f: ... | https://en.wikipedia.org/wiki?curid=10070867 |
10072717 | Categorical distribution | Discrete probability distribution
In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can take on one of "K" possible categories,... | [
{
"math_id": 0,
"text": "[x=i]"
},
{
"math_id": 1,
"text": "\nf(x=i\\mid \\boldsymbol{p} ) = p_i ,\n"
},
{
"math_id": 2,
"text": "\\boldsymbol{p} = (p_1,\\ldots,p_k)"
},
{
"math_id": 3,
"text": "p_i"
},
{
"math_id": 4,
"text": "\\textstyle{\\sum_{i=1}^k p_i = ... | https://en.wikipedia.org/wiki?curid=10072717 |
10073693 | 1/2 + 1/4 + 1/8 + 1/16 + ⋯ | Mathematical infinite series
In mathematics, the infinite series + + + + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
In summation notation, this may be expressed as
formula_0
The series is related to philosophical questions considered in antiquity, particula... | [
{
"math_id": 0,
"text": "\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots = \\sum_{n=1}^\\infty \\left({\\frac 12}\\right)^n = 1. "
},
{
"math_id": 1,
"text": "\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots"
},
{
"math_id": 2,
"text": "s_n=\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\... | https://en.wikipedia.org/wiki?curid=10073693 |
10073845 | 1/4 + 1/16 + 1/64 + 1/256 + ⋯ | Infinite series equal to 1/3 at its limit
In mathematics, the infinite series + + + + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. As it is a geometric series with first term and common ratio , its sum is
formula_0
Visual de... | [
{
"math_id": 0,
"text": " \\sum_{n=1}^\\infty \\frac{1}{4^n}=\\frac {\\frac 1 4} {1 - \\frac 1 4} = \\frac 1 3. "
},
{
"math_id": 1,
"text": " 3\\left(\\frac14+\\frac{1}{4^2}+\\frac{1}{4^3}+\\frac{1}{4^4}+\\cdots\\right) = 1."
},
{
"math_id": 2,
"text": " \\sum_{n=1}^\\infty \\frac{3... | https://en.wikipedia.org/wiki?curid=10073845 |
1007613 | Bell state | Quantum states of two qubits
In quantum information science, the Bell's states or EPR pairs25 are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell's states are a form of entangled and normalized basis vectors. This normalization implies that the overall probab... | [
{
"math_id": 0,
"text": "\\langle \\Phi|\\Phi \\rangle = 1"
},
{
"math_id": 1,
"text": "2\\sqrt{2}"
},
{
"math_id": 2,
"text": "|\\Phi^+\\rangle = \\frac{1}{\\sqrt{2}} \\big(|0\\rangle_A \\otimes |0\\rangle_B + |1\\rangle_A \\otimes |1\\rangle_B\\big) \\qquad (1)"
},
{
"math_... | https://en.wikipedia.org/wiki?curid=1007613 |
1007660 | Self-verifying theories | Systems capable of proving their own consistency
Self-verifying theories are consistent first-order systems of arithmetic, much weaker than Peano arithmetic, that are capable of proving their own consistency. Dan Willard was the first to investigate their properties, and he has described a family of such systems. Accor... | [
{
"math_id": 0,
"text": "\\Pi^0_2"
},
{
"math_id": 1,
"text": "(\\forall x,y)\\ (\\exists z)\\ {\\rm multiply}(x,y,z)."
},
{
"math_id": 2,
"text": "{\\rm multiply}"
},
{
"math_id": 3,
"text": "z/y=x."
},
{
"math_id": 4,
"text": "\\Pi^0_1"
}
] | https://en.wikipedia.org/wiki?curid=1007660 |
10077292 | Eventually (mathematics) | In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it does not have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for su... | [
{
"math_id": 0,
"text": "\\mathbb{R}"
},
{
"math_id": 1,
"text": "P"
},
{
"math_id": 2,
"text": "x"
},
{
"math_id": 3,
"text": "\\forall"
},
{
"math_id": 4,
"text": "\\exists"
},
{
"math_id": 5,
"text": "\\exists a \\in \\mathbb{R}"
},
{
"m... | https://en.wikipedia.org/wiki?curid=10077292 |
1007903 | Generalized singular value decomposition | Name of two different techniques based on the singular value decomposition
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions differ because one version decomposes two matrices (somewhat like th... | [
{
"math_id": 0,
"text": "\\mathbb{F} = \\mathbb{R}"
},
{
"math_id": 1,
"text": "\\mathbb{F} = \\mathbb{C}"
},
{
"math_id": 2,
"text": "A_1 \\in \\mathbb{F}^{m_1 \\times n}"
},
{
"math_id": 3,
"text": "A_2 \\in \\mathbb{F}^{m_2 \\times n}"
},
{
"math_id": 4,
"t... | https://en.wikipedia.org/wiki?curid=1007903 |
1007969 | Iterative reconstruction | Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques.
For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are usually a
better, but computationally more expensive... | [
{
"math_id": 0,
"text": "f(r)"
},
{
"math_id": 1,
"text": "\\mathbf{A}x+\\epsilon"
},
{
"math_id": 2,
"text": "\\epsilon"
}
] | https://en.wikipedia.org/wiki?curid=1007969 |
10083278 | 5-cubic honeycomb | Tiling of five-dimensional space
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an "order-4 penteractic honeycomb".
It is analogous to the square tiling... | [
{
"math_id": 0,
"text": "{\\tilde{C}}_5"
},
{
"math_id": 1,
"text": "{\\tilde{B}}_5"
},
{
"math_id": 2,
"text": "{\\tilde{D}}_5"
}
] | https://en.wikipedia.org/wiki?curid=10083278 |
10083462 | 6-cube | 6-dimensional hypercube
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}64 & 6 & 15 & 20 & 15 & 6 \\\\ 2 & 192 & 5 & 10 & 10 & 5 \\\\ 4 & 4 & 240 & 4 & 6 & 4 \\\\ 8 & 12 & 6 & 160 & 3 & 3 \\\\ 16 & 32 & 24 & 8 & 60 & 2 \\\\ 32 & 80 & 80 & 40 & 10 & 12 \\end{matrix}\\end{bmatrix}"
}
] | https://en.wikipedia.org/wiki?curid=10083462 |
10083518 | 6-orthoplex | In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell "4-faces", and 64 "5-faces".
It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboard... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}12 & 10 & 40 & 80 & 80 & 32 \\\\ 2 & 60 & 8 & 24 & 32 & 16 \\\\ 3 & 3 & 160 & 6 & 12 & 8 \\\\ 4 & 6 & 4 & 240 & 4 & 4 \\\\ 5 & 10 & 10 & 5 & 192 & 2 \\\\ 6 & 15 & 20 & 15 & 6 & 64 \\end{matrix}\\end{bmatrix}"
}
] | https://en.wikipedia.org/wiki?curid=10083518 |
1008471 | Wigner–Eckart theorem | Theorem used in quantum mechanics for angular momentum calculations
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one of whic... | [
{
"math_id": 0,
"text": "T^{(k)}"
},
{
"math_id": 1,
"text": "j"
},
{
"math_id": 2,
"text": "j'"
},
{
"math_id": 3,
"text": "\\langle j \\| T^{(k)} \\| j' \\rangle"
},
{
"math_id": 4,
"text": "m"
},
{
"math_id": 5,
"text": "m'"
},
{
"math_i... | https://en.wikipedia.org/wiki?curid=1008471 |
10084899 | Static synchronous compensator | Regulating device used on transmission networks
In Electrical Engineering , a static synchronous compensator (STATCOM) is a shunt-connected, reactive compensation device used on transmission networks. It uses power electronics to form a voltage-source converter that can act as either a source or sink of reactive AC pow... | [
{
"math_id": 0,
"text": "Q=\\frac{V_S*(\\Delta V)}{X}*\\cos(\\delta)"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "V_S"
},
{
"math_id": 3,
"text": "\\Delta V"
},
{
"math_id": 4,
"text": "V_R"
},
{
"math_id": 5,
"text": "X"
},
{
... | https://en.wikipedia.org/wiki?curid=10084899 |
10086335 | Complete theory | In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence formula_0 the theory formula_1 contains the sentence or its negation but not both (that is, either formula_2 or formula_3). ... | [
{
"math_id": 0,
"text": "\\varphi,"
},
{
"math_id": 1,
"text": "T"
},
{
"math_id": 2,
"text": "T \\vdash \\varphi"
},
{
"math_id": 3,
"text": "T \\vdash \\neg \\varphi"
},
{
"math_id": 4,
"text": "S"
},
{
"math_id": 5,
"text": "A \\land B \\in S"
... | https://en.wikipedia.org/wiki?curid=10086335 |
10087500 | Impedance parameters | Set of properties used in electrical engineering
Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are... | [
{
"math_id": 0,
"text": "I_n\\,"
},
{
"math_id": 1,
"text": "V_n\\,"
},
{
"math_id": 2,
"text": "V = Z I\\,"
},
{
"math_id": 3,
"text": "\\begin{pmatrix} V_1 \\\\ V_2\\end{pmatrix} = \\begin{pmatrix} Z_{11} & Z_{12} \\\\ Z_{21} & Z_{22} \\end{pmatrix}\\begin{pmatrix}I_1 \... | https://en.wikipedia.org/wiki?curid=10087500 |
10087606 | Symmetry operation | Geometric transformation which produces an identical image
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a <templatestyles src="Fraction/styles.css" />1⁄3 turn rotation of a regular triangle about... | [
{
"math_id": 0,
"text": "i^n=E"
},
{
"math_id": 1,
"text": "i^n=-E"
},
{
"math_id": 2,
"text": "S_n^n = E,"
},
{
"math_id": 3,
"text": "S_n^{2n} = E."
},
{
"math_id": 4,
"text": "S_4^2 = C_2."
}
] | https://en.wikipedia.org/wiki?curid=10087606 |
10088188 | Order of integration | Summary statistic
In statistics, the order of integration, denoted "I"("d"), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
Integration of order "d".
A time series is integrated of order "d" if
formula_0
is a stationary process... | [
{
"math_id": 0,
"text": "(1-L)^d X_t \\ "
},
{
"math_id": 1,
"text": "L"
},
{
"math_id": 2,
"text": "1-L "
},
{
"math_id": 3,
"text": "(1-L) X_t = X_t - X_{t-1} = \\Delta X. "
},
{
"math_id": 4,
"text": "(1-L)^0 X_t = X_t "
},
{
"math_id": 5,
"tex... | https://en.wikipedia.org/wiki?curid=10088188 |
10088265 | Papkovich–Neuber solution | The Papkovich–Neuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed to solve the equations of linear elasticity.
It can be shown that any Stokes flow with body force formula_0 can be written in the form:
formula_1
formula_2
... | [
{
"math_id": 0,
"text": "\\mathbf{f}=0"
},
{
"math_id": 1,
"text": "\\mathbf{u} = {1\\over{2 \\mu}} \\left[ \\nabla ( \\mathbf{x} \\cdot \\mathbf{\\Phi} + \\chi) - 2 \\mathbf{\\Phi} \\right]"
},
{
"math_id": 2,
"text": "p = \\nabla \\cdot \\mathbf{\\Phi}"
},
{
"math_id": 3,
... | https://en.wikipedia.org/wiki?curid=10088265 |
10090547 | Peano existence theorem | Theorem regarding the existence of a solution to a differential equation.
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the ... | [
{
"math_id": 0,
"text": "D"
},
{
"math_id": 1,
"text": "\\mathbb{R}\\times\\mathbb{R}"
},
{
"math_id": 2,
"text": "f\\colon D \\to \\mathbb{R}"
},
{
"math_id": 3,
"text": "y'(x) = f\\left(x,y(x)\\right)"
},
{
"math_id": 4,
"text": "y\\left(x_0\\right) = y_0"
... | https://en.wikipedia.org/wiki?curid=10090547 |
10092186 | Nagata ring | In commutative algebra, an N-1 ring is an integral domain formula_0 whose integral closure in its quotient field is a finitely generated formula_0-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension formula_1 of its quotient field formula_2, the integral closure of formula_0 in formula_1... | [
{
"math_id": 0,
"text": "A"
},
{
"math_id": 1,
"text": "L"
},
{
"math_id": 2,
"text": "K"
},
{
"math_id": 3,
"text": "p"
},
{
"math_id": 4,
"text": "k"
},
{
"math_id": 5,
"text": "K^p\\subseteq k"
},
{
"math_id": 6,
"text": "R"
},
{... | https://en.wikipedia.org/wiki?curid=10092186 |
10092550 | Body force | Force which acts throughout the volume of a body
In physics, a body force is a force that acts throughout the volume of a body. Forces due to gravity, electric fields and magnetic fields are examples of body forces. Body forces contrast with "contact forces" or "surface forces" which are exerted to the surface of an ob... | [
{
"math_id": 0,
"text": "\\mathbf{F}_{\\mathrm{body}} = \\int\\limits_{V}\\mathbf{f}(\\mathbf{r}) \\mathrm{d} V \\,,"
},
{
"math_id": 1,
"text": "\\mathbf{f} (\\mathbf{r})=\\rho (\\mathbf{r})\\mathbf{a} (\\mathbf{r})"
},
{
"math_id": 2,
"text": "g = 9.81 \\frac{\\mathrm m}{\\mathrm ... | https://en.wikipedia.org/wiki?curid=10092550 |
10094198 | Hardy–Littlewood maximal function | In mathematics, the Hardy–Littlewood maximal operator "M" is a significant non-linear operator used in real analysis and harmonic analysis.
Definition.
The operator takes a locally integrable function "f" : R"d" → C and returns another function "Mf".
For any point "x" ∈ R"d", the function "Mf" returns the maximum of ... | [
{
"math_id": 0,
"text": " Mf(x)=\\sup_{r>0} \\frac{1}{|B(x, r)|}\\int_{B(x, r)} |f(y)|\\, dy "
},
{
"math_id": 1,
"text": "\\left |\\{Mf > \\lambda\\} \\right |< \\frac{C_d}{\\lambda} \\Vert f\\Vert_{L^1 (\\mathbf{R}^d)}."
},
{
"math_id": 2,
"text": " \\Vert Mf\\Vert_{L^p (\\mathbf{R... | https://en.wikipedia.org/wiki?curid=10094198 |
10099552 | Zakai equation | In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows o... | [
{
"math_id": 0,
"text": "dx = f(x,t) dt + dw"
},
{
"math_id": 1,
"text": "dz = h(x,t) dt + dv"
},
{
"math_id": 2,
"text": "w, v"
},
{
"math_id": 3,
"text": "p(x,t)"
},
{
"math_id": 4,
"text": "dp = L[p] dt + p h^T dz"
},
{
"math_id": 5,
"text": "L[... | https://en.wikipedia.org/wiki?curid=10099552 |
1010127 | Carbon-13 | Rare isotope of carbon
Carbon-13 (13C) is a natural, stable isotope of carbon with a nucleus containing six protons and seven neutrons. As one of the environmental isotopes, it makes up about 1.1% of all natural carbon on Earth.
Detection by mass spectrometry.
A mass spectrum of an organic compound will usually contain... | [
{
"math_id": 0,
"text": "C = \\frac{100Y}{1.1X}"
}
] | https://en.wikipedia.org/wiki?curid=1010127 |
1010141 | Gilbreath's conjecture | Conjecture in number theory
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. T... | [
{
"math_id": 0,
"text": "(p_n)"
},
{
"math_id": 1,
"text": "(d^1_n)"
},
{
"math_id": 2,
"text": "d^1_n = p_{n+1} - p_n,"
},
{
"math_id": 3,
"text": "n"
},
{
"math_id": 4,
"text": "k"
},
{
"math_id": 5,
"text": "(d_n^k)"
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=1010141 |
10101991 | Newton's inequalities | In mathematics, the Newton inequalities are named after Isaac Newton. Suppose "a"1, "a"2, ..., "a""n" are non-negative real numbers and let formula_0 denote the "k"th elementary symmetric polynomial in "a"1, "a"2, ..., "a""n". Then the elementary symmetric means, given by
formula_1
satisfy the inequality
formula_2
Equa... | [
{
"math_id": 0,
"text": "e_k"
},
{
"math_id": 1,
"text": "S_k = \\frac{e_k}{\\binom{n}{k}},"
},
{
"math_id": 2,
"text": "S_{k-1}S_{k+1} \\le S_k^2."
}
] | https://en.wikipedia.org/wiki?curid=10101991 |
10102876 | Vitali covering lemma | Combinatorial and geometric result used in measure theory of Euclidean spaces
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. T... | [
{
"math_id": 0,
"text": "\\mathbb{R}^d"
},
{
"math_id": 1,
"text": "B = B(x,r)"
},
{
"math_id": 2,
"text": "c \\geq 0 "
},
{
"math_id": 3,
"text": "cB"
},
{
"math_id": 4,
"text": "B(x,cr)"
},
{
"math_id": 5,
"text": " B_{1}, \\dots, B_{n} "
},
... | https://en.wikipedia.org/wiki?curid=10102876 |
10103 | Electroweak interaction | Unified description of electromagnetism and the weak interaction
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two force... | [
{
"math_id": 0,
"text": "Q = T_3 + \\tfrac{1}{2}\\,Y_\\mathrm{W}"
},
{
"math_id": 1,
"text": " \\begin{pmatrix}\n\\gamma \\\\\nZ^0 \\end{pmatrix} = \\begin{pmatrix}\n\\cos \\theta_\\text{W} & \\sin \\theta_\\text{W} \\\\\n-\\sin \\theta_\\text{W} & \\cos \\theta_\\text{W} \\end{pmatrix} \\begin{... | https://en.wikipedia.org/wiki?curid=10103 |
10103794 | Slenderness ratio | Ratio of width and height in architecture
In architecture, the slenderness ratio, or simply slenderness, is an aspect ratio, the quotient between the height and the width of a building.
In structural engineering, slenderness is used to calculate the propensity of a column to buckle. It is defined as formula_0 where fo... | [
{
"math_id": 0,
"text": "l/k"
},
{
"math_id": 1,
"text": "l"
},
{
"math_id": 2,
"text": "k"
},
{
"math_id": 3,
"text": "k^2=I/A"
},
{
"math_id": 4,
"text": "A"
},
{
"math_id": 5,
"text": "I"
},
{
"math_id": 6,
"text": "E"
}
] | https://en.wikipedia.org/wiki?curid=10103794 |
10104622 | Kuratowski's closure-complement problem | In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922.... | [
{
"math_id": 0,
"text": "S"
},
{
"math_id": 1,
"text": "kS"
},
{
"math_id": 2,
"text": "cS"
},
{
"math_id": 3,
"text": "kkS=kS"
},
{
"math_id": 4,
"text": "ccS=S"
},
{
"math_id": 5,
"text": "kckckckcS=kckcS"
},
{
"math_id": 6,
"text": "... | https://en.wikipedia.org/wiki?curid=10104622 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.