id stringlengths 2 8 | title stringlengths 1 130 | text stringlengths 0 252k | formulas listlengths 1 823 | url stringlengths 38 44 |
|---|---|---|---|---|
10198492 | Nilpotent orbit | In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role
in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras.
Definition.
An element "X" of a semisimple Lie algebra "g" is called nilpotent if its adjoint endomorphism
"ad X": "g" →... | [
{
"math_id": 0,
"text": "n\\times n"
},
{
"math_id": 1,
"text": "\\lambda_1\\geq \\lambda_2\\geq\\ldots\\geq\\lambda_r,"
},
{
"math_id": 2,
"text": "\\lambda"
},
{
"math_id": 3,
"text": " A=\\begin{bmatrix}x & y\\\\ z & -x \\end{bmatrix}, \\quad (x,y,z)\\ne (0,0,0)\\quad{... | https://en.wikipedia.org/wiki?curid=10198492 |
1020021 | Distance (graph theory) | Length of shortest path between two nodes of a graph
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there ... | [
{
"math_id": 0,
"text": "\\epsilon(v) = \\max_{u \\in V}d(v,u)."
},
{
"math_id": 1,
"text": "r = \\min_{v \\in V} \\epsilon(v) = \\min_{v \\in V}\\max_{u \\in V}d(v,u)."
},
{
"math_id": 2,
"text": "d = \\max_{v \\in V}\\epsilon(v) = \\max_{v \\in V}\\max_{u \\in V}d(v,u)."
},
{
... | https://en.wikipedia.org/wiki?curid=1020021 |
10200558 | Hot band | In molecular vibrational spectroscopy, a hot band is a band centred on a hot transition, which is a transition between two excited vibrational states, i.e. neither is the overall ground state. In infrared or Raman spectroscopy, hot bands refer to those transitions for a particular vibrational mode which arise from a st... | [
{
"math_id": 0,
"text": "\\nu_1"
},
{
"math_id": 1,
"text": "\\nu_2"
},
{
"math_id": 2,
"text": "\\nu_3"
},
{
"math_id": 3,
"text": "101"
},
{
"math_id": 4,
"text": " 001"
},
{
"math_id": 5,
"text": "{{N}\\over{N_0}} = {{e^{-E/k_BT}}}"
},
{
... | https://en.wikipedia.org/wiki?curid=10200558 |
10201 | Exothermic process | Thermodynamic process that releases energy to its surroundings
In thermodynamics, an exothermic process (from grc " ' ()" 'outward' and " ' ()" 'thermal') is a thermodynamic process or reaction that releases energy from the system to its surroundings, usually in the form of heat, but also in a form of light (e.g. a spa... | [
{
"math_id": 0,
"text": "Q > 0."
},
{
"math_id": 1,
"text": "\\Delta H < 0,"
},
{
"math_id": 2,
"text": "\\Delta U = Q + 0 > 0."
}
] | https://en.wikipedia.org/wiki?curid=10201 |
10201642 | Polar alignment | Method of orienting telescopes and other celestial observation devices
Polar alignment is the act of aligning the rotational axis of a telescope's equatorial mount or a sundial's gnomon with a celestial pole to parallel Earth's axis.
Alignment methods.
The method to use differs depending on whether the alignment is tak... | [
{
"math_id": 0,
"text": "{\\displaystyle \\Delta\\alpha=\\Delta e\\cdot \\tan{(\\delta)}\\cdot \\sin(h)+\\Delta a\\cdot\\left(\\sin(\\Phi)-\\cos(\\Phi)\\cdot\\tan(\\delta)\\cdot\\cos(h)\\right)}"
},
{
"math_id": 1,
"text": "\\Delta\\delta=\\Delta e\\cdot\\cos(h)+\\Delta a\\cdot\\cos(\\Phi)\\cdot... | https://en.wikipedia.org/wiki?curid=10201642 |
10202429 | Relative change | Comparisons in quantitative sciences
In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a "standard" or "reference" or "starting" value. The comparison is expressed as a ra... | [
{
"math_id": 0,
"text": " \\text{Relative change}(v_\\text{ref}, v) = \\frac{\\text{Actual change}}{v_\\text{ref}} = \\frac{v - v_\\text{ref}}{v_\\text{ref}}."
},
{
"math_id": 1,
"text": " \\text{Relative change}(v_\\text{ref}, v) = \\frac{v - v_\\text{ref}}{|v_\\text{ref}|}."
},
{
"mat... | https://en.wikipedia.org/wiki?curid=10202429 |
1020537 | Bow shock | Shock wave caused by blowing stellar wind
In astrophysics, bow shocks are shock waves in regions where the conditions of density and pressure change dramatically due to blowing stellar wind. Bow shock occurs when the magnetosphere of an astrophysical object interacts with the nearby flowing ambient plasma such as the s... | [
{
"math_id": 0,
"text": "c_s^2 = \\gamma p/ \\rho "
},
{
"math_id": 1,
"text": " \\gamma "
},
{
"math_id": 2,
"text": " p "
},
{
"math_id": 3,
"text": " \\rho "
},
{
"math_id": 4,
"text": " v"
},
{
"math_id": 5,
"text": "V_A"
},
{
"math_id"... | https://en.wikipedia.org/wiki?curid=1020537 |
1020661 | Lie–Kolchin theorem | Theorem in the representation theory of linear algebraic groups
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
It states that if "G" is a connected and solvable linear algebraic group defined over an alge... | [
{
"math_id": 0,
"text": "\\rho\\colon G \\to GL(V)"
},
{
"math_id": 1,
"text": "\\rho(G)(L) = L."
},
{
"math_id": 2,
"text": " \\rho(g), \\,\\, g \\in G "
},
{
"math_id": 3,
"text": "\\rho(G)"
},
{
"math_id": 4,
"text": " \\{ x+iy \\in \\mathbb{C} \\mid x^2+y^... | https://en.wikipedia.org/wiki?curid=1020661 |
10208822 | Möbius aromaticity | In organic chemistry, Möbius aromaticity is a special type of aromaticity believed to exist in a number of organic molecules. In terms of molecular orbital theory these compounds have in common a monocyclic array of molecular orbitals in which there is an odd number of out-of-phase overlaps, the opposite pattern compar... | [
{
"math_id": 0,
"text": "p_z"
},
{
"math_id": 1,
"text": "\\cos\\omega"
},
{
"math_id": 2,
"text": "\\omega=\\pi/N"
},
{
"math_id": 3,
"text": "\\beta^\\prime"
},
{
"math_id": 4,
"text": "\\beta^\\prime=\\beta\\cos(\\pi/N)"
},
{
"math_id": 5,
"text... | https://en.wikipedia.org/wiki?curid=10208822 |
1020980 | Starling equation | Mathematical description of fluid movements
The Starling principle holds that extracellular fluid movements between blood and tissues are determined by differences in hydrostatic pressure and colloid osmotic pressure (oncotic pressure) between plasma inside microvessels and interstitial fluid outside them. The Starling... | [
{
"math_id": 0,
"text": " P_c "
},
{
"math_id": 1,
"text": " \\pi_i "
},
{
"math_id": 2,
"text": " \\pi_p "
},
{
"math_id": 3,
"text": " P_i "
},
{
"math_id": 4,
"text": " J_s "
},
{
"math_id": 5,
"text": "\\ J_v = L_\\mathrm{p} S ( [P_\\mathrm{c} ... | https://en.wikipedia.org/wiki?curid=1020980 |
1021 | Aspect ratio | Attribute of a geometric shape
The aspect ratio of a geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangle is oriented as a "landscape".
The aspect ratio is most... | [
{
"math_id": 0,
"text": "\\sqrt{2}:1 = 1.414..."
},
{
"math_id": 1,
"text": "\\sqrt{2}"
},
{
"math_id": 2,
"text": "\\sqrt{W^2/WH} = \\sqrt{W/H}"
}
] | https://en.wikipedia.org/wiki?curid=1021 |
1021099 | Hodrick–Prescott filter | Mathematical tool used in macroeconomics
The Hodrick–Prescott filter (also known as Hodrick–Prescott decomposition) is a mathematical tool used in macroeconomics, especially in real business cycle theory, to remove the cyclical component of a time series from raw data. It is used to obtain a smoothed-curve representati... | [
{
"math_id": 0,
"text": "\\lambda"
},
{
"math_id": 1,
"text": "y_t\\,"
},
{
"math_id": 2,
"text": "t = 1, 2, ..., T\\,"
},
{
"math_id": 3,
"text": "\\tau_t"
},
{
"math_id": 4,
"text": "c_t"
},
{
"math_id": 5,
"text": "y_t\\ = \\tau_t\\ + c_t\\,"
... | https://en.wikipedia.org/wiki?curid=1021099 |
10211794 | Exner function | Parameter in atmospheric modeling
The Exner function is an important parameter in atmospheric modeling. The Exner function can be viewed as non-dimensionalized pressure and can be defined as:
formula_0
where formula_1 is a standard reference surface pressure, usually taken as 1000 hPa; formula_2 is the gas constant fo... | [
{
"math_id": 0,
"text": "\\Pi = \\left( \\frac{p}{p_0} \\right)^{R_d/c_p} = \\frac{T}{\\theta} "
},
{
"math_id": 1,
"text": "p_0"
},
{
"math_id": 2,
"text": "R_d"
},
{
"math_id": 3,
"text": "c_p"
},
{
"math_id": 4,
"text": "T"
},
{
"math_id": 5,
"t... | https://en.wikipedia.org/wiki?curid=10211794 |
1021312 | Gerhard Frey | German mathematician (born 1944)
Gerhard Frey (; born 1 June 1944) is a German mathematician, known for his work in number theory. Following an original idea of , he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to... | [
{
"math_id": 0,
"text": "\\varepsilon"
}
] | https://en.wikipedia.org/wiki?curid=1021312 |
102140 | Perturbation theory | In math and applied mathematics, methods for finding an approximate solution to a problem
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique... | [
{
"math_id": 0,
"text": "\\varepsilon"
},
{
"math_id": 1,
"text": "\\ A\\ ,"
},
{
"math_id": 2,
"text": " A \\equiv A_0 + \\varepsilon^1 A_1 + \\varepsilon^2 A_2 + \\varepsilon^3 A_3 + \\cdots "
},
{
"math_id": 3,
"text": "\\ A_0\\ "
},
{
"math_id": 4,
"text":... | https://en.wikipedia.org/wiki?curid=102140 |
1021510 | Dilution of precision (navigation) | Propagation of error with varying topology
Dilution of precision (DOP), or geometric dilution of precision (GDOP), is a term used in satellite navigation and geomatics engineering to specify the error propagation as a mathematical effect of navigation satellite geometry on positional measurement precision.
Introduction... | [
{
"math_id": 0,
"text": "\\operatorname{GDOP} = \\frac{\\Delta ( \\text{output location} )}{\\Delta ( \\text{measured data} )}"
},
{
"math_id": 1,
"text": "\\Delta ( \\text{measured data} )"
},
{
"math_id": 2,
"text": "x"
},
{
"math_id": 3,
"text": "y"
},
{
"math_... | https://en.wikipedia.org/wiki?curid=1021510 |
1021521 | Equity premium puzzle | Economics concept
The equity premium puzzle refers to the inability of an important class of economic models to explain the average equity risk premium (ERP) provided by a diversified portfolio of equities over that of government bonds, which has been observed for more than 100 years. There is a significant disparity b... | [
{
"math_id": 0,
"text": "E_0 \\left[\\sum_{t=0}^\\infty \\beta^t U(c_t)\\right]"
},
{
"math_id": 1,
"text": "0<\\beta<1"
},
{
"math_id": 2,
"text": "c_t"
},
{
"math_id": 3,
"text": "t"
},
{
"math_id": 4,
"text": "U(c, \\alpha) = \\frac{c^{(1-\\alpha)}}{1-\\alp... | https://en.wikipedia.org/wiki?curid=1021521 |
1021753 | Variety (universal algebra) | Class of algebraic structures
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhof... | [
{
"math_id": 0,
"text": "f(o_A(a_1, \\dots, a_n)) = o_B(f(a_1), \\dots, f(a_n))"
},
{
"math_id": 1,
"text": "x(yz) = (xy)z."
},
{
"math_id": 2,
"text": "x(yz) = (xy)z"
},
{
"math_id": 3,
"text": "1 x = x 1 = x"
},
{
"math_id": 4,
"text": "x x^{-1} = x^{-1} x =... | https://en.wikipedia.org/wiki?curid=1021753 |
1021754 | Sound localization | Biological sound detection process
Sound localization is a listener's ability to identify the location or origin of a detected sound in direction and distance.
The sound localization mechanisms of the mammalian auditory system have been extensively studied. The auditory system uses several cues for sound source localiz... | [
{
"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "\\theta"
},
{
"math_id": 2,
"text": "r"
},
{
"math_id": 3,
"text": "c"
},
{
"math_id": 4,
"text": "ITD= \\begin{cases} 3\\times\\frac{r}{c}\\times\\sin\\theta, & \\text{if }f\\leq\\text{4000Hz } \\\\ 2\\t... | https://en.wikipedia.org/wiki?curid=1021754 |
102182 | Celestial mechanics | Branch of astronomy
<templatestyles src="Hlist/styles.css"/>
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ep... | [
{
"math_id": 0,
"text": "n=2"
},
{
"math_id": 1,
"text": "n>2"
}
] | https://en.wikipedia.org/wiki?curid=102182 |
10218909 | Bootstrapping (finance) | Method for constructing a fixed-income yield curve
In finance, bootstrapping is a method for constructing a (zero-coupon) fixed-income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps.
A "bootstrapped curve", correspondingly, is one where the prices of the instruments used as an "in... | [
{
"math_id": 0,
"text": "1 = C_{n} \\cdot \\Delta_1 \\cdot df_{1} + C_{n} \\cdot \\Delta_2 \\cdot df_{2} + C_{n} \\cdot \\Delta_3 \\cdot df_{3} + \\cdots + (1+ C_{n} \\cdot \\Delta_n ) \\cdot df_n "
},
{
"math_id": 1,
"text": "df_{n} = {(1 - \\sum_{i=1}^{n-1} C_{n} \\cdot \\Delta_i \\cdot df... | https://en.wikipedia.org/wiki?curid=10218909 |
10220067 | Crystalline cohomology | In mathematics, crystalline cohomology is a Weil cohomology theory for schemes "X" over a base field "k". Its values "H""n"("X"/"W") are modules over the ring "W" of Witt vectors over "k". It was introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974).
Crystalline cohomology is partly... | [
{
"math_id": 0,
"text": "\\ell"
},
{
"math_id": 1,
"text": "\\mathbf{Z}_\\ell"
},
{
"math_id": 2,
"text": "H^i(X/W)=\\varprojlim H^i(X/W_n)"
},
{
"math_id": 3,
"text": "H^i(X/W_n)= H^i(\\operatorname{Cris}(X/W_n),O)"
},
{
"math_id": 4,
"text": "H^i(X/W) = H^i_... | https://en.wikipedia.org/wiki?curid=10220067 |
10220473 | Eilenberg–Zilber theorem | Links the homology groups of a product space with those of the individual spaces
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space formula_0 and those of the spaces formula_1 and formula_2. The ... | [
{
"math_id": 0,
"text": "X \\times Y"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "Y"
},
{
"math_id": 3,
"text": "C_*(X)"
},
{
"math_id": 4,
"text": "C_*(Y)"
},
{
"math_id": 5,
"text": "C_*(X \\times Y) "
},
{
"math_id": 6,
... | https://en.wikipedia.org/wiki?curid=10220473 |
10220713 | Zero field splitting | Zero field splitting (ZFS) describes various interactions of the energy levels of a molecule or ion resulting from the presence of more than one unpaired electron. In quantum mechanics, an energy level is called degenerate if it corresponds to two or more different measurable states of a quantum system. In the presence... | [
{
"math_id": 0,
"text": "\\hat{\\mathcal{H}}=D\\left(S_z^2-\\frac{1}{3}S(S+1)\\right)+E(S_x^2-S_y^2) "
},
{
"math_id": 1,
"text": "S_{x,y,z}"
},
{
"math_id": 2,
"text": "\\hat{\\mathcal{H}}_D=\\mathbf{SDS}"
},
{
"math_id": 3,
"text": "\\mathbf{D}"
},
{
"math_id": ... | https://en.wikipedia.org/wiki?curid=10220713 |
10220853 | Eilenberg–Moore spectral sequence | In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's origina... | [
{
"math_id": 0,
"text": "k"
},
{
"math_id": 1,
"text": "H_\\ast(-)=H_\\ast(-,k)"
},
{
"math_id": 2,
"text": "H^\\ast(-)=H^\\ast(-,k)"
},
{
"math_id": 3,
"text": "E_f"
},
{
"math_id": 4,
"text": " \\begin{array}{c c c} E_f &\\rightarrow & E \\\\ \\downarrow & ... | https://en.wikipedia.org/wiki?curid=10220853 |
10221371 | Mechanical watch | Type of watch which uses a clockwork mechanism to measure the passage of time
A mechanical watch is a watch that uses a clockwork mechanism to measure the passage of time, as opposed to quartz watches which function using the vibration modes of a piezoelectric quartz tuning fork, or radio watches, which are quartz watc... | [
{
"math_id": 0,
"text": "n_2 = \\frac{n_1 \\cdot z_1}{z_2}"
},
{
"math_id": 1,
"text": "z_1"
},
{
"math_id": 2,
"text": "z_2"
},
{
"math_id": 3,
"text": "n_1"
},
{
"math_id": 4,
"text": "n_2"
},
{
"math_id": 5,
"text": "T = 2 \\pi \\sqrt{ \\frac {I... | https://en.wikipedia.org/wiki?curid=10221371 |
102221 | Orthogonality | Various meanings of the terms
In mathematics, orthogonality is the generalization of the geometric notion of "perpendicularity". Whereas "perpendicular" is typically followed by "to" when relating two lines to one another (e.g., "line A is perpendicular to line B"), "orthogonal" is commonly used without "to" (e.g., "or... | [
{
"math_id": 0,
"text": " \\psi_m "
},
{
"math_id": 1,
"text": " \\psi_n "
},
{
"math_id": 2,
"text": " \\langle \\psi_m | \\psi_n \\rangle = 0 "
}
] | https://en.wikipedia.org/wiki?curid=102221 |
10222175 | Wythoff's game | Two-player mathematical subtraction game
Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal. The game ends when ... | [
{
"math_id": 0,
"text": "n_k = \\lfloor k \\phi \\rfloor = \\lfloor m_k \\phi \\rfloor -m_k \\,"
},
{
"math_id": 1,
"text": "m_k = \\lfloor k \\phi^2 \\rfloor = \\lceil n_k \\phi \\rceil = n_k + k \\,"
},
{
"math_id": 2,
"text": "\\frac{1}{\\phi} + \\frac{1}{\\phi^2} = 1 \\,."
}
] | https://en.wikipedia.org/wiki?curid=10222175 |
10223066 | Spacetime algebra | Setting of relativistic physics in geometric algebra
In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the... | [
{
"math_id": 0,
"text": "a,b"
},
{
"math_id": 1,
"text": "ab"
},
{
"math_id": 2,
"text": "a \\cdot b"
},
{
"math_id": 3,
"text": "a \\wedge b"
},
{
"math_id": 4,
"text": " a \\cdot b = \\frac{ab +ba}{2} = b \\cdot a, \\quad a \\wedge b= \\frac{ab-ba}{2} = - b ... | https://en.wikipedia.org/wiki?curid=10223066 |
10224194 | Kai (conjunction) | Kai ( "and"; ; ; sometimes abbreviated "k") is a letter that is a conjunction in Greek, Coptic (ⲕⲁⲓ) and Esperanto ("kaj"; ).
"Kai" is the most frequent word in any Greek text and thus used by statisticians to assess authorship of ancient manuscripts based on the number of times it is used.
Ligature.
Because of its fre... | [
{
"math_id": 0,
"text": "\\chi^2"
}
] | https://en.wikipedia.org/wiki?curid=10224194 |
10225 | Elliptic curve | Algebraic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in "K"2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then ... | [
{
"math_id": 0,
"text": "y^2 = x^3 + ax + b"
},
{
"math_id": 1,
"text": "\\mathbb{H}^2"
},
{
"math_id": 2,
"text": "\\Delta"
},
{
"math_id": 3,
"text": "\\Delta = -16\\left(4a^3 + 27b^2\\right) \\neq 0"
},
{
"math_id": 4,
"text": "\\frac{Y^2}{Z^2} =\n\\frac{X^... | https://en.wikipedia.org/wiki?curid=10225 |
1022661 | Symbolic Cholesky decomposition | In the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used to determine the non-zero pattern for the formula_0 factors of a symmetric sparse matrix when applying the Cholesky decomposition or variants.
Algorithm.
Let
formula_1
be a sparse symmetric positive definite matr... | [
{
"math_id": 0,
"text": "L"
},
{
"math_id": 1,
"text": "A=(a_{ij}) \\in \\mathbb{K}^{n \\times n}"
},
{
"math_id": 2,
"text": "\\mathbb{K}"
},
{
"math_id": 3,
"text": "A = LL^T\\,"
},
{
"math_id": 4,
"text": "\\mathcal{A}_i"
},
{
"math_id": 5,
"tex... | https://en.wikipedia.org/wiki?curid=1022661 |
10226740 | Free recoil | Term for recoil energy of a firearm not supported from behind
Free recoil / Frecoil is a vernacular term or jargon for recoil energy of a firearm not supported from behind. Free recoil denotes the translational kinetic energy ("Et") imparted to the shooter of a small arm when discharged and is expressed in joules (J), ... | [
{
"math_id": 0,
"text": "E_{tgu} = 0.5 \\cdot [\\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} { 1000 }]^2 / m_{gu}"
},
{
"math_id": 1,
"text": "v_{gu} = \\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} {1000 \\cdot m_{gu}}"
},
{
"math_id": 2,
"text": "E_{tgu} = 0.5 \\cdot m_{gu} \\cdot ... | https://en.wikipedia.org/wiki?curid=10226740 |
10231968 | $ (disambiguation) | $ is the dollar or peso currency sign (36 in ASCII), primarily used to represent currencies.
$ may also refer to:
<templatestyles src="Template:TOC_right/styles.css" />
Currency.
The sign is used for:
See also.
Topics referred to by the same term
<templatestyles src="Dmbox/styles.css" />
This page lists as... | [
{
"math_id": 0,
"text": "\\mathrm{S}\\!\\!\\!\\Vert"
}
] | https://en.wikipedia.org/wiki?curid=10231968 |
1023353 | Burgers' equation | Partial differential equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced ... | [
{
"math_id": 0,
"text": "u(x,t)"
},
{
"math_id": 1,
"text": "\\nu"
},
{
"math_id": 2,
"text": "\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = \\nu\\frac{\\partial^2 u}{\\partial x^2}."
},
{
"math_id": 3,
"text": "u\\partial u/\\partial x"
},
{... | https://en.wikipedia.org/wiki?curid=1023353 |
1023390 | Elastance | Mechanical stiffness, or inverse of electrical capacitance or fluid flow compliance
Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units ... | [
{
"math_id": 0,
"text": " C = {Q \\over V}"
},
{
"math_id": 1,
"text": " S = {V \\over Q} \\ . "
},
{
"math_id": 2,
"text": "\\mathbf{A}= s^2 \\mathbf{L} + s \\mathbf{R} + \\mathbf{S} = s \\mathbf{Z}"
}
] | https://en.wikipedia.org/wiki?curid=1023390 |
10237 | Exponentiation by squaring | Algorithm for fast exponentiation
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as squ... | [
{
"math_id": 0,
"text": "n > 0"
},
{
"math_id": 1,
"text": " x^n=\n \\begin{cases}\n x \\, ( x^{2})^{(n - 1)/2}, & \\mbox{if } n \\mbox{ is odd} \\\\\n (x^{2})^{n/2} , & \\mbox{if } n \\mbox{ is even}\n \\end{cases}\n"
},
{
"math_id": 2,
"text": "... | https://en.wikipedia.org/wiki?curid=10237 |
1023857 | Positive set theory | Class of alternative set theories
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas formula_0 (the smallest class of formulas containing atomic membership and equality formulas and closed under conj... | [
{
"math_id": 0,
"text": "\\phi"
},
{
"math_id": 1,
"text": "\\mathrm{GPK}^+_\\infty"
},
{
"math_id": 2,
"text": "\\forall x \\forall y (\\forall z (z \\in x \\leftrightarrow z \\in y) \\to x = y)"
},
{
"math_id": 3,
"text": "\\exists x \\forall y (y \\in x \\leftrightarro... | https://en.wikipedia.org/wiki?curid=1023857 |
1023920 | NTSC-J | Japanese variation of the NTSC analog television standard
NTSC-J or "System J" is the informal designation for the analogue television standard used in Japan. The system is based on the US NTSC (NTSC-M) standard with minor differences. While NTSC-M is an official CCIR and FCC standard, NTSC-J or "System J" are a colloq... | [
{
"math_id": 0,
"text": "I"
},
{
"math_id": 1,
"text": "Q"
},
{
"math_id": 2,
"text": "C = (Cb-512)*(0.545)*(\\sin\\omega t) + (Cr-512)*(0.769)*(\\cos\\omega t)"
},
{
"math_id": 3,
"text": "C = (Cb-512)*(0.504)*(\\sin\\omega t) + (Cr-512)*(0.711)*(\\cos\\omega t)"
}
] | https://en.wikipedia.org/wiki?curid=1023920 |
10240442 | CM-field | Complex multiplication field
In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.
The abbreviation "CM" was introduced by .
Formal definition.
A number field "K" is a CM-field if it is a quadratic extensio... | [
{
"math_id": 0,
"text": "\\mathbb C "
},
{
"math_id": 1,
"text": "\\mathbb R "
},
{
"math_id": 2,
"text": "\\sqrt{\\alpha} "
},
{
"math_id": 3,
"text": " \\mathbb Q"
},
{
"math_id": 4,
"text": "F"
},
{
"math_id": 5,
"text": "\\mathbb C"
},
{
... | https://en.wikipedia.org/wiki?curid=10240442 |
10240807 | Unbiased estimation of standard deviation | Procedure to estimate standard deviation from a sampleIn statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a wa... | [
{
"math_id": 0,
"text": "s = \\sqrt{\\frac{\\sum_{i=1}^n (x_i - \\overline{x})^2}{n-1}},"
},
{
"math_id": 1,
"text": "\\{x_1,x_2,\\ldots,x_n\\}"
},
{
"math_id": 2,
"text": "\\overline{x}"
},
{
"math_id": 3,
"text": "(n-1) s^2/\\sigma^2"
},
{
"math_id": 4,
"tex... | https://en.wikipedia.org/wiki?curid=10240807 |
1024131 | Jean Bourgain | Belgian mathematician (1954–2018)
Jean Louis, baron Bourgain (; (1954--)28 1954 – (2018--)22 2018) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and ... | [
{
"math_id": 0,
"text": "l_p"
},
{
"math_id": 1,
"text": "O(\\log^2 (n))"
},
{
"math_id": 2,
"text": "O(\\log(n))"
}
] | https://en.wikipedia.org/wiki?curid=1024131 |
10242885 | Algebraic character | Mathematical concept
An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.
Definition.
... | [
{
"math_id": 0,
"text": "\\mathfrak{g}"
},
{
"math_id": 1,
"text": "\\mathfrak{h},"
},
{
"math_id": 2,
"text": "A=\\mathbb{Z}[[\\mathfrak{h}^*]]"
},
{
"math_id": 3,
"text": "e^{\\mu}"
},
{
"math_id": 4,
"text": "\\mu\\in\\mathfrak{h}^*"
},
{
"math_id":... | https://en.wikipedia.org/wiki?curid=10242885 |
1024314 | Mental accounting | Mental accounting (or psychological accounting) is a model of consumer behaviour developed by Richard Thaler that attempts to describe the process whereby people code, categorize and evaluate economic outcomes. Mental accounting incorporates the economic concepts of prospect theory and transactional utility theory to e... | [
{
"math_id": 0,
"text": "(x, y)"
},
{
"math_id": 1,
"text": "Value(x+y)"
},
{
"math_id": 2,
"text": "Value(x) + Value(y)"
},
{
"math_id": 3,
"text": "x"
},
{
"math_id": 4,
"text": "y"
},
{
"math_id": 5,
"text": "Value(x) + Value(y) > Value(x+y)"
... | https://en.wikipedia.org/wiki?curid=1024314 |
1024323 | Pyrgeometer | Device which measures infra-red radiation
A pyrgeometer is a device that measures near-surface infra-red (IR) radiation, approximately from 4.5 μm to 100 μm on the electromagnetic spectrum (thereby excluding solar radiation).
It measures the resistance/voltage changes in a material that is sensitive to the net energy t... | [
{
"math_id": 0,
"text": "E_\\mathrm{net} = E_\\mathrm{in} - E_\\mathrm{out}"
},
{
"math_id": 1,
"text": " E_\\mathrm{net} = \\frac{U_\\mathrm{emf}}{S}"
},
{
"math_id": 2,
"text": "E_\\mathrm{out} = \\sigma T^4"
},
{
"math_id": 3,
"text": "E_\\mathrm{in} = \\frac{U_\\math... | https://en.wikipedia.org/wiki?curid=1024323 |
1024614 | Taylor–Proudman theorem | In fluid mechanics, the Taylor–Proudman theorem (after Geoffrey Ingram Taylor and Joseph Proudman) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity formula_0, the fluid velocity will be uniform along any line parallel to the axis of rotation. formula_0 m... | [
{
"math_id": 0,
"text": "\\Omega"
},
{
"math_id": 1,
"text": "\n\\rho({\\mathbf u}\\cdot\\nabla){\\mathbf u}={\\mathbf F}-\\nabla p,"
},
{
"math_id": 2,
"text": "{\\mathbf u}"
},
{
"math_id": 3,
"text": "\\rho"
},
{
"math_id": 4,
"text": "p"
},
{
"math... | https://en.wikipedia.org/wiki?curid=1024614 |
10246759 | Poincaré–Lindstedt method | Technique used in perturbation theory
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bou... | [
{
"math_id": 0,
"text": "\\ddot{x} + x + \\varepsilon\\, x^3 = 0\\,"
},
{
"math_id": 1,
"text": "x(0) = 1,\\,"
},
{
"math_id": 2,
"text": " \\dot x(0) = 0.\\,"
},
{
"math_id": 3,
"text": "x(t) = \\cos(t) + \\varepsilon \\left[ \\tfrac{1}{32}\\, \\left( \\cos(3t) - \\cos(t... | https://en.wikipedia.org/wiki?curid=10246759 |
102476 | Log-normal distribution | Probability distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then has a normal distribution. Equivalently, if Y has a normal dist... | [
{
"math_id": 0,
"text": "\\sigma"
},
{
"math_id": 1,
"text": "\\ Z\\ "
},
{
"math_id": 2,
"text": "\\mu"
},
{
"math_id": 3,
"text": "\\sigma > 0"
},
{
"math_id": 4,
"text": " X = e^{\\mu + \\sigma Z} "
},
{
"math_id": 5,
"text": "\\ X\\ "
},
{
... | https://en.wikipedia.org/wiki?curid=102476 |
10251864 | Multi-objective optimization | Mathematical concept
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more t... | [
{
"math_id": 0,
"text": "\n\\min_{x \\in X} (f_1(x), f_2(x),\\ldots, f_k(x))\n"
},
{
"math_id": 1,
"text": "k\\geq 2"
},
{
"math_id": 2,
"text": "X"
},
{
"math_id": 3,
"text": " X \\subseteq \\mathbb R^n "
},
{
"math_id": 4,
"text": "n"
},
{
"math_id":... | https://en.wikipedia.org/wiki?curid=10251864 |
10252066 | Choquet integral | A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, where it is used as a way of measuring the expected utility of an uncertain... | [
{
"math_id": 0,
"text": "S"
},
{
"math_id": 1,
"text": "\\mathcal{F}"
},
{
"math_id": 2,
"text": "f : S\\to \\mathbb{R}"
},
{
"math_id": 3,
"text": "\\nu : \\mathcal{F}\\to \\mathbb{R}^+"
},
{
"math_id": 4,
"text": "f"
},
{
"math_id": 5,
"text": "\... | https://en.wikipedia.org/wiki?curid=10252066 |
1025272 | Bohr–Einstein debates | Series of public disputes between physicists Niels Bohr and Albert Einstein
The Bohr–Einstein debates were a series of public disputes about quantum mechanics between Albert Einstein and Niels Bohr. Their debates are remembered because of their importance to the philosophy of science, insofar as the disagreements—and t... | [
{
"math_id": 0,
"text": " \\Delta t "
},
{
"math_id": 1,
"text": " \\nu_0 "
},
{
"math_id": 2,
"text": " \\Delta x "
},
{
"math_id": 3,
"text": " \\Delta t = \\Delta x/v "
},
{
"math_id": 4,
"text": " \\Delta \\nu "
},
{
"math_id": 5,
"text": " \\D... | https://en.wikipedia.org/wiki?curid=1025272 |
1025455 | Killing vector field | Vector field on a Riemannian manifold that preserves the metric
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of iso... | [
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "g"
},
{
"math_id": 2,
"text": "\\mathcal{L}_{X} g = 0 \\,."
},
{
"math_id": 3,
"text": "g\\left(\\nabla_Y X, Z\\right) + g\\left(Y, \\nabla_Z X\\right) = 0 \\,"
},
{
"math_id": 4,
"text": "Y"
},
{
... | https://en.wikipedia.org/wiki?curid=1025455 |
1025655 | Balassa–Samuelson effect | Tendency for consumer prices to be systematically higher in more developed countries
The Balassa–Samuelson effect, also known as Harrod–Balassa–Samuelson effect (Kravis and Lipsey 1983), the Ricardo–Viner–Harrod–Balassa–Samuelson–Penn–Bhagwati effect (Samuelson 1994, p. 201), or productivity biased purchasing power par... | [
{
"math_id": 0,
"text": "MPL_{nt,1}=MPL_{nt,2}=1"
},
{
"math_id": 1,
"text": "w_1=p_{nt,1}*MPL_{nt,1}=p_{t}*MPL_{t,1}"
},
{
"math_id": 2,
"text": "w_2=p_{nt,2}*MPL_{nt,2}=p_{t}*MPL_{t,2}"
},
{
"math_id": 3,
"text": "MPL_{t,1}<MPL_{t,2}"
},
{
"math_id": 4,
"tex... | https://en.wikipedia.org/wiki?curid=1025655 |
1025748 | Lévy process | Stochastic process in probability theory
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time... | [
{
"math_id": 0,
"text": "X=\\{X_t:t \\geq 0\\}"
},
{
"math_id": 1,
"text": "X_0=0 \\,"
},
{
"math_id": 2,
"text": "0 \\leq t_1 < t_2<\\cdots <t_n <\\infty"
},
{
"math_id": 3,
"text": "X_{t_2}-X_{t_1}, X_{t_3}-X_{t_2},\\dots,X_{t_n}-X_{t_{n-1}}"
},
{
"math_id": 4,
... | https://en.wikipedia.org/wiki?curid=1025748 |
1025794 | Minimum degree algorithm | Matrix manipulation algorithm
In numerical analysis, the minimum degree algorithm is an algorithm used to permute the rows and columns of a symmetric sparse matrix before applying the Cholesky decomposition, to reduce the number of non-zeros in the Cholesky factor.
This results in reduced storage requirements and means... | [
{
"math_id": 0,
"text": " \\mathbf{A}\\mathbf{x} = \\mathbf{b}"
},
{
"math_id": 1,
"text": "n \\times n"
},
{
"math_id": 2,
"text": "\\mathbf{P}^T\\mathbf{A}\\mathbf{P}"
},
{
"math_id": 3,
"text": " \\left(\\mathbf{P}^T\\mathbf{A}\\mathbf{P}\\right) \\left(\\mathbf{P}^T\\... | https://en.wikipedia.org/wiki?curid=1025794 |
1025901 | Reynolds decomposition | In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition.
For example, for a quantity formula_0 the decomposition would be
formula_1
where formula_2 denotes the expectation value of formula_0, (o... | [
{
"math_id": 0,
"text": "u"
},
{
"math_id": 1,
"text": "u(x,y,z,t) = \\overline{u(x,y,z)} + u'(x,y,z,t) "
},
{
"math_id": 2,
"text": "\\overline{u}"
},
{
"math_id": 3,
"text": "u'"
},
{
"math_id": 4,
"text": "\\langle u\\rangle"
},
{
"math_id": 5,
... | https://en.wikipedia.org/wiki?curid=1025901 |
10261414 | Hot chocolate effect | Phenomenon of wave mechanics
The hot chocolate effect is a phenomenon of wave mechanics in which the pitch heard from tapping a cup of hot liquid rises after the addition of a soluble powder. The effect is thought to happen because upon initial stirring, entrained gas bubbles reduce the speed of sound in the liquid, lo... | [
{
"math_id": 0,
"text": "\nf = \\frac{1}{4}\\frac{v}{h}\n"
},
{
"math_id": 1,
"text": "\\rho"
},
{
"math_id": 2,
"text": "K"
},
{
"math_id": 3,
"text": "\nv = \\sqrt{\\frac{K}{\\rho}}\n"
}
] | https://en.wikipedia.org/wiki?curid=10261414 |
10261692 | Ground track | Path on the surface of the Earth or another body directly below an aircraft or satellite
A ground track or ground trace is the path on the surface of a planet directly below an aircraft's or satellite's trajectory. In the case of satellites, it is also known as a suborbital track or subsatellite track, and is the verti... | [
{
"math_id": 0,
"text": "\\Delta L_1 = -2 \\pi \\frac{T}{T_E}"
},
{
"math_id": 1,
"text": "T"
},
{
"math_id": 2,
"text": "T_E"
},
{
"math_id": 3,
"text": "\\Delta L_2 = - \\frac{3 \\pi J_2 R_e^2 cos(i)}{a^2(1-e^2)^2}"
},
{
"math_id": 4,
"text": "J_2"
},
{
... | https://en.wikipedia.org/wiki?curid=10261692 |
1026409 | Opel Monza | The Opel Monza is an executive fastback coupe produced by the German automaker Opel from 1977 to 1986. It was marketed in the United Kingdom as the Vauxhall Royale Coupé by Vauxhall.
Monza A1 (1977–1982).
The Monza was planned as a successor for the Commodore Coupé. In the late 1970s the Commodore C model was made as a... | [
{
"math_id": 0,
"text": "\\scriptstyle C_\\mathrm x\\,"
}
] | https://en.wikipedia.org/wiki?curid=1026409 |
102651 | Bernoulli process | Random process of binary (boolean) random variables
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables... | [
{
"math_id": 0,
"text": "i"
},
{
"math_id": 1,
"text": "2=\\{H,T\\} ."
},
{
"math_id": 2,
"text": "2=\\{H,T\\}"
},
{
"math_id": 3,
"text": "\\Omega=2^\\mathbb{N}=\\{H,T\\}^\\mathbb{N}"
},
{
"math_id": 4,
"text": "\\Omega=2^\\mathbb{Z}"
},
{
"math_id": ... | https://en.wikipedia.org/wiki?curid=102651 |
1026522 | Boltzmann equation | Equation of statistical mechanics
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.
The classic example of such a system is a fluid with temperature gradients in space caus... | [
{
"math_id": 0,
"text": "d^3 \\mathbf{r}"
},
{
"math_id": 1,
"text": "\\mathbf{r}"
},
{
"math_id": 2,
"text": " \\mathbf{p}"
},
{
"math_id": 3,
"text": "d^3 \\mathbf{p}"
},
{
"math_id": 4,
"text": " d^3\\mathbf{r} \\, d^3\\mathbf{p} = dx \\, dy \\, dz \\, dp_x... | https://en.wikipedia.org/wiki?curid=1026522 |
102653 | Bernoulli trial | Any experiment with two possible random outcomes
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named afte... | [
{
"math_id": 0,
"text": "p"
},
{
"math_id": 1,
"text": "q"
},
{
"math_id": 2,
"text": "\np = 1 - q, \\quad \\quad q = 1 - p, \\quad \\quad p + q = 1."
},
{
"math_id": 3,
"text": "p:q"
},
{
"math_id": 4,
"text": "q:p."
},
{
"math_id": 5,
"text": "o_... | https://en.wikipedia.org/wiki?curid=102653 |
10265555 | Notation for differentiation | Notation of differential calculus
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantag... | [
{
"math_id": 0,
"text": "\\frac{dy}{dx}."
},
{
"math_id": 1,
"text": "\\frac{df}{dx}(x)\\text{ or }\\frac{d f(x)}{dx}\\text{ or }\\frac{d}{dx} f(x)."
},
{
"math_id": 2,
"text": "\\frac{d^2y}{dx^2}, \\frac{d^3y}{dx^3}, \\frac{d^4y}{dx^4}, \\ldots, \\frac{d^ny}{dx^n}."
},
{
"ma... | https://en.wikipedia.org/wiki?curid=10265555 |
1026848 | Weyl tensor | Measure of the curvature of a pseudo-Riemannian manifold
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body fe... | [
{
"math_id": 0,
"text": "C = R - \\frac{1}{n-2}\\left(\\mathrm{Ric} - \\frac{s}{n}g\\right) {~\\wedge\\!\\!\\!\\!\\!\\!\\!\\!\\;\\bigcirc~} g - \\frac{s}{2n(n - 1)}g {~\\wedge\\!\\!\\!\\!\\!\\!\\!\\!\\;\\bigcirc~} g"
},
{
"math_id": 1,
"text": "h {~\\wedge\\!\\!\\!\\!\\!\\!\\!\\!\\;\\bigcirc~} k... | https://en.wikipedia.org/wiki?curid=1026848 |
1027229 | Method of analytic tableaux | Tool for proving a logical formula
In proof theory, the semantic tableau (; plural: tableaux), also called an analytic tableau, truth tree, or simply tree, is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed... | [
{
"math_id": 0,
"text": "A_1,\\ldots,A_n"
},
{
"math_id": 1,
"text": "B"
},
{
"math_id": 2,
"text": "\\{A_1,\\ldots,A_n,\\neg B\\}"
},
{
"math_id": 3,
"text": "\\neg A"
},
{
"math_id": 4,
"text": "A"
},
{
"math_id": 5,
"text": "A \\wedge B"
},
... | https://en.wikipedia.org/wiki?curid=1027229 |
10273855 | Positive-definite kernel | Generalization of a positive-definite matrix
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations... | [
{
"math_id": 0,
"text": " \\mathcal X "
},
{
"math_id": 1,
"text": " K: \\mathcal X \\times \\mathcal X \\to \\mathbb{R}"
},
{
"math_id": 2,
"text": "\\mathcal X"
},
{
"math_id": 3,
"text": "x_1, \\dots, x_n\\in \\mathcal X"
},
{
"math_id": 4,
"text": "n\\in \... | https://en.wikipedia.org/wiki?curid=10273855 |
10273917 | Projective cover | In the branch of abstract mathematics called category theory, a projective cover of an object "X" is in a sense the best approximation of "X" by a projective object "P". Projective covers are the dual of injective envelopes.
Definition.
Let formula_0 be a category and "X" an object in formula_0. A projective cover is a... | [
{
"math_id": 0,
"text": "\\mathcal{C}"
},
{
"math_id": 1,
"text": "p : P \\to X"
},
{
"math_id": 2,
"text": "p(N) \\ne M"
},
{
"math_id": 3,
"text": "p':P'\\rightarrow M"
},
{
"math_id": 4,
"text": "p\\alpha=p'"
}
] | https://en.wikipedia.org/wiki?curid=10273917 |
10274 | Enthalpy | Measure of energy in a thermodynamic system
Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant external pressure, which is convenient... | [
{
"math_id": 0,
"text": "W"
},
{
"math_id": 1,
"text": "P_{ext}"
},
{
"math_id": 2,
"text": "V_{system, initial}=0"
},
{
"math_id": 3,
"text": "V_{system, final}"
},
{
"math_id": 4,
"text": "W=P_{ext}\\Delta V"
},
{
"math_id": 5,
"text": " H = \\su... | https://en.wikipedia.org/wiki?curid=10274 |
10274436 | Topological modular forms | In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer "n" there is a topological space formula_0, and these spaces are equipped with certain maps between them, so that for any topological space "X", one obtains an abe... | [
{
"math_id": 0,
"text": "\\operatorname{tmf}^{n}"
},
{
"math_id": 1,
"text": "\\operatorname{tmf}^{n}(X)"
},
{
"math_id": 2,
"text": "\\operatorname{tmf}^{0}"
}
] | https://en.wikipedia.org/wiki?curid=10274436 |
10274608 | Stress majorization | Geometric placement based on ideal distances
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of "formula_0" "formula_1"-dimensional data items, a configuration "formula_2" of formula_0 points in "formula_3 formula_4"-dimensional space is sought that minimizes the ... | [
{
"math_id": 0,
"text": "n"
},
{
"math_id": 1,
"text": "m"
},
{
"math_id": 2,
"text": "X"
},
{
"math_id": 3,
"text": "r"
},
{
"math_id": 4,
"text": "(\\ll m)"
},
{
"math_id": 5,
"text": "\\sigma(X)"
},
{
"math_id": 6,
"text": "2"
},
... | https://en.wikipedia.org/wiki?curid=10274608 |
10275945 | 6-demicube | Uniform 6-polytope
In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a "6-cube" (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it... | [
{
"math_id": 0,
"text": "\\left\\{3 \\begin{array}{l}3, 3, 3\\\\3\\end{array}\\right\\}"
}
] | https://en.wikipedia.org/wiki?curid=10275945 |
10275954 | 7-cube | 7-dimensional hypercube
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.
It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be c... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}\n128 & 7 & 21 & 35 & 35 & 21 & 7 \n\\\\ 2 & 448 & 6 & 15 & 20 & 15 & 6 \n\\\\ 4 & 4 & 672 & 5 & 10 & 10 & 5 \n\\\\ 8 & 12 & 6 & 560 & 4 & 6 & 4 \n\\\\ 16 & 32 & 24 & 8 & 280 & 3 & 3 \n\\\\ 32 & 80 & 80 & 40 & 10 & 84 & 2 \n\\\\ 64 & 192 & 240 & 160 & ... | https://en.wikipedia.org/wiki?curid=10275954 |
10275958 | 7-demicube | Uniform 7-polytope
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labe... | [
{
"math_id": 0,
"text": "\\left\\{3 \\begin{array}{l}3, 3, 3, 3\\\\3\\end{array}\\right\\}"
}
] | https://en.wikipedia.org/wiki?curid=10275958 |
10275959 | 8-demicube | In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM8 for an... | [
{
"math_id": 0,
"text": "\\left\\{3 \\begin{array}{l}3, 3, 3, 3, 3\\\\3\\end{array}\\right\\}"
}
] | https://en.wikipedia.org/wiki?curid=10275959 |
10275960 | 9-demicube | Uniform 9-polytope
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM9 fo... | [
{
"math_id": 0,
"text": "\\left\\{3 \\begin{array}{l}3, 3, 3, 3, 3, 3\\\\3\\end{array}\\right\\}"
}
] | https://en.wikipedia.org/wiki?curid=10275960 |
10275967 | 8-cube | 8-dimensional hypercube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around e... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}\n256 & 8 & 28 & 56 & 70 & 56 & 28 & 8\n\\\\ 2 & 1024 & 7 & 21 & 35 & 35 & 21 & 7\n\\\\ 4 & 4 & 1792 & 6 & 15 & 20 & 15 & 6\n\\\\ 8 & 12 & 6 & 1792 & 5 & 10 & 10 & 5\n\\\\ 16 & 32 & 24 & 8 & 1120 & 4 & 6 & 4\n\\\\ 32 & 80 & 80 & 40 & 10 & 448 & 3 & 3... | https://en.wikipedia.org/wiki?curid=10275967 |
10275976 | 7-orthoplex | Regular 7- polytope
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells "4-faces", 448 "5-faces", and 128 "6-faces".
It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second w... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}\n14 & 12 & 60 & 160 & 240 & 192 & 64 \n\\\\ 2 & 84 & 10 & 40 & 80 & 80 & 32 \n\\\\ 3 & 3 & 280 & 8 & 24 & 32 & 16 \n\\\\ 4 & 6 & 4 & 560 & 6 & 12 & 8 \n\\\\ 5 & 10 & 10 & 5 & 672 & 4 & 4 \n\\\\ 6 & 15 & 20 & 15 & 6 & 448 & 2 \n\\\\ 7 & 21 & 35 & 35 & ... | https://en.wikipedia.org/wiki?curid=10275976 |
10275985 | 6-simplex | Uniform 6-polytope
</math>0.654654|
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alternate names.
It can also be called a hept... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}7 & 6 & 15 & 20 & 15 & 6 \\\\ 2 & 21 & 5 & 10 & 10 & 5 \\\\ 3 & 3 & 35 & 4 & 6 & 4 \\\\ 4 & 6 & 4 & 35 & 3 & 3 \\\\ 5 & 10 & 10 & 5 & 21 & 2 \\\\ 6 & 15 & 20 & 15 & 6 & 7 \\end{matrix}\\end{bmatrix}"
},
{
"math_id": 1,
"text": "\\left(\\sqr... | https://en.wikipedia.org/wiki?curid=10275985 |
10276020 | 7-simplex | Type of 7-polytope
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.
Alternate names.
It can also be ... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}8 & 7 & 21 & 35 & 35 & 21 & 7 \\\\ 2 & 28 & 6 & 15 & 20 & 15 & 6 \\\\ 3 & 3 & 56 & 5 & 10 & 10 & 5 \\\\ 4 & 6 & 4 & 70 & 4 & 6 & 4 \\\\ 5 & 10 & 10 & 5 & 56 & 3 & 3 \\\\ 6 & 15 & 20 & 15 & 6 & 28 & 2 \\\\ 7 & 21 & 35 & 35 & 21 & 7 & 8 \\end{matrix}\\en... | https://en.wikipedia.org/wiki?curid=10276020 |
10276044 | 8-orthoplex | In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells "4-faces", 1792 "5-faces", 1024 "6-faces", and 256 "7-faces".
It has two constructive forms, the first being regular with Schläfli symbol {36,4}, and the second w... | [
{
"math_id": 0,
"text": "\\begin{bmatrix}\\begin{matrix}\n 16 & 14 & 84 & 280 & 560 & 672 & 448 & 128\n\\\\ 2 & 112 & 12 & 60 & 160 & 240 & 192 & 64\n\\\\ 3 & 3 & 448 & 10 & 40 & 80 & 80 & 32\n\\\\ 4 & 6 & 4 & 1120 & 8 & 24 & 32 & 16\n\\\\ 5 & 10 & 10 & 5 & 1792 & 6 & 12 & 8\n\\\\ 6 & 15 & 20 & 15 & 6 & 179... | https://en.wikipedia.org/wiki?curid=10276044 |
10276223 | Proper equilibrium | Proper equilibrium is a refinement of Nash Equilibrium by Roger B. Myerson.
Proper equilibrium further refines Reinhard Selten's notion of a
trembling hand perfect equilibrium by assuming that more costly trembles are made with
significantly smaller probability than less
costly ones.
Definition.
Given a normal form ... | [
{
"math_id": 0,
"text": "\\epsilon > 0"
},
{
"math_id": 1,
"text": "\\sigma"
},
{
"math_id": 2,
"text": "\\epsilon"
},
{
"math_id": 3,
"text": " u(s,\\sigma_{-i})<u(s',\\sigma_{-i})"
}
] | https://en.wikipedia.org/wiki?curid=10276223 |
10277277 | Octonion algebra | In mathematics, an octonion algebra or Cayley algebra over a field "F" is a composition algebra over "F" that has dimension 8 over "F". In other words, it is a 8-dimensional unital non-associative algebra "A" over "F" with a non-degenerate quadratic form "N" (called the "norm form") such that
formula_0
for all "x" and ... | [
{
"math_id": 0,
"text": "N(xy) = N(x)N(y)"
},
{
"math_id": 1,
"text": "(q + Qe)(r + Re) = (qr + \\gamma R^* Q) + (Rq + Q r^* )e ."
},
{
"math_id": 2,
"text": "H^1(F, G_2)"
}
] | https://en.wikipedia.org/wiki?curid=10277277 |
1027784 | Simplicial set | Mathematical construction used in homotopy theory
In mathematics, a simplicial set is an object composed of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functo... | [
{
"math_id": 0,
"text": "\\delta^{n,0},\\dotsc,\\delta^{n,n}\\colon[n-1]\\to[n]"
},
{
"math_id": 1,
"text": "\\delta^{n,i}"
},
{
"math_id": 2,
"text": "[n-1]\\to[n]"
},
{
"math_id": 3,
"text": "i"
},
{
"math_id": 4,
"text": "d_{n,0},\\dotsc,d_{n,n}"
},
{
... | https://en.wikipedia.org/wiki?curid=1027784 |
10279126 | Aristotelian physics | Natural sciences as described by Aristotle
Aristotelian physics is the form of natural philosophy described in the works of the Greek philosopher Aristotle (384–322 BC). In his work "Physics", Aristotle intended to establish general principles of change that govern all natural bodies, both living and inanimate, celesti... | [
{
"math_id": 0,
"text": "\n{dy\\over dt} \\propto y\n"
},
{
"math_id": 1,
"text": "y(0)=0"
},
{
"math_id": 2,
"text": "y=0"
}
] | https://en.wikipedia.org/wiki?curid=10279126 |
10280254 | Isothermal coordinates | In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form
formula_0
where formula_1 is a positive smooth functio... | [
{
"math_id": 0,
"text": " g = \\varphi (dx_1^2 + \\cdots + dx_n^2),"
},
{
"math_id": 1,
"text": "\\varphi"
},
{
"math_id": 2,
"text": " ds^2 = E \\, dx^2 + 2F \\, dx \\, dy + G \\, dy^2,"
},
{
"math_id": 3,
"text": " z = x + iy"
},
{
"math_id": 4,
"text": " ds... | https://en.wikipedia.org/wiki?curid=10280254 |
10280692 | Digital biquad filter | Second order recursive digital linear filter
In signal processing, a digital biquad filter is a second order recursive linear filter, containing two poles and two zeros. "Biquad" is an abbreviation of "biquadratic", which refers to the fact that in the Z domain, its transfer function is the ratio of two quadratic funct... | [
{
"math_id": 0,
"text": "\\ H(z)=\\frac{b_0+b_1z^{-1}+b_2z^{-2}} {a_0+a_1z^{-1}+a_2z^{-2} }"
},
{
"math_id": 1,
"text": "\\ H(z)=\\frac{b_0+b_1z^{-1}+b_2z^{-2}} {1+a_1z^{-1}+a_2z^{-2} }"
},
{
"math_id": 2,
"text": "\\ y[n] = \\frac{1}{a_0} \\left ( b_0x[n] + b_1x[n-1] + b_2x[n-2] - a... | https://en.wikipedia.org/wiki?curid=10280692 |
1028158 | Canonical basis | Basis of a type of algebraic structure
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
Representation theory.
The canonical basis for the irreducible representations of a quantized enveloping algebra of
type formula_2 and also for ... | [
{
"math_id": 0,
"text": "(X^i)_i"
},
{
"math_id": 1,
"text": "A"
},
{
"math_id": 2,
"text": "ADE"
},
{
"math_id": 3,
"text": "q"
},
{
"math_id": 4,
"text": "q=1"
},
{
"math_id": 5,
"text": "q=0"
},
{
"math_id": 6,
"text": "\\mathcal{Z}:... | https://en.wikipedia.org/wiki?curid=1028158 |
10282799 | Lie bracket of vector fields | Operator in differential topology
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields "X" and "Y" on a smooth manifold "M" a third vector field denoted ["X", "Y... | [
{
"math_id": 0,
"text": "\\mathcal{L}_X Y"
},
{
"math_id": 1,
"text": "X : M \\rightarrow TM"
},
{
"math_id": 2,
"text": "f(p)"
},
{
"math_id": 3,
"text": "p \\in M"
},
{
"math_id": 4,
"text": "f"
},
{
"math_id": 5,
"text": "C^\\infty(M)"
},
{
... | https://en.wikipedia.org/wiki?curid=10282799 |
10283 | Erlang (unit) | Load measure in telecommunications
<templatestyles src="Template:Infobox/styles-images.css" />
The erlang (symbol E) is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single co... | [
{
"math_id": 0,
"text": " E = \\lambda h "
},
{
"math_id": 1,
"text": "P_b = B(E,m) = \\frac{\\frac{E^m}{m!}} { \\sum_{i=0}^m \\frac{E^i}{i!}} "
},
{
"math_id": 2,
"text": "P_b"
},
{
"math_id": 3,
"text": "B(E,0) = 1. \\,"
},
{
"math_id": 4,
"text": "B(E,j) = ... | https://en.wikipedia.org/wiki?curid=10283 |
1028314 | Penman equation | The Penman equation describes evaporation ("E") from an open water surface, and was developed by Howard Penman in 1948. Penman's equation requires daily mean temperature, wind speed, air pressure, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is im... | [
{
"math_id": 0,
"text": "E_{\\mathrm{mass}}=\\frac{m R_n + \\rho_a c_p \\left(\\delta e \\right) g_a }{\\lambda_v \\left(m + \\gamma \\right) }\n"
},
{
"math_id": 1,
"text": "E_{\\mathrm{mass}}=\\frac{m R_n + \\gamma * 6.43\\left(1+0.536 * U_2 \\right)\\delta e}{\\lambda_v \\left(m + \\gamma \\r... | https://en.wikipedia.org/wiki?curid=1028314 |
1028321 | Wigner semicircle distribution | Probability distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−"R", "R"] whose probability density function "f" is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):
formula_0
for −"R" ≤ "x" ≤ "R", and "f"("x") = 0 if "|x|" > "R"... | [
{
"math_id": 0,
"text": "f(x)={2 \\over \\pi R^2}\\sqrt{R^2-x^2\\,}\\, "
},
{
"math_id": 1,
"text": "\\frac{1}{n+1}\\left({R \\over 2}\\right)^{2n} {2n\\choose n}\\, "
},
{
"math_id": 2,
"text": "s(z)=-\\frac{2}{R^2}(z-\\sqrt{z^2-R^2})"
},
{
"math_id": 3,
"text": "\\varph... | https://en.wikipedia.org/wiki?curid=1028321 |
1028589 | Normal basis | In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number ... | [
{
"math_id": 0,
"text": "F\\subset K"
},
{
"math_id": 1,
"text": "G"
},
{
"math_id": 2,
"text": "\\beta\\in K"
},
{
"math_id": 3,
"text": "\\{g(\\beta) : g\\in G\\}"
},
{
"math_id": 4,
"text": "\\alpha \\in K"
},
{
"math_id": 5,
"text": "\\alpha = ... | https://en.wikipedia.org/wiki?curid=1028589 |
102883 | Belief | Mental state of holding a proposition or premise to be true
A belief is a subjective attitude that a proposition is true or a state of affairs is the case. A subjective attitude is a mental state of having some stance, take, or opinion about something. In epistemology, philosophers use the term "belief" to refer to att... | [
{
"math_id": 0,
"text": "S"
},
{
"math_id": 1,
"text": "P"
}
] | https://en.wikipedia.org/wiki?curid=102883 |
1028841 | Simplex category | Category of non-empty finite ordinals and order-preserving maps
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects.
Formal definition.
The simpl... | [
{
"math_id": 0,
"text": "\\Delta"
},
{
"math_id": 1,
"text": " [n] = \\{0, 1, \\dots, n\\} "
},
{
"math_id": 2,
"text": " [n] "
},
{
"math_id": 3,
"text": " n+1 "
},
{
"math_id": 4,
"text": "\\Delta_+"
},
{
"math_id": 5,
"text": "\\Delta_+=\\Delta\... | https://en.wikipedia.org/wiki?curid=1028841 |
1028978 | Combs method | The Combs method is a rule base reduction method of writing fuzzy logic rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion in fuzzy logic rules.
The Combs method takes advantage of the logical equality formula_0.
Equality proof.
The simplest proof of given equality involves u... | [
{
"math_id": 0,
"text": "((p \\land q) \\Rightarrow r) \\iff ((p \\Rightarrow r) \\lor (q \\Rightarrow r))"
},
{
"math_id": 1,
"text": "S^N"
},
{
"math_id": 2,
"text": "S \\times N"
}
] | https://en.wikipedia.org/wiki?curid=1028978 |
10290343 | Dirac algebra | In mathematical physics, the Dirac algebra is the Clifford algebra formula_0. This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.
The gamma matrice... | [
{
"math_id": 0,
"text": "\\text{Cl}_{1,3}(\\mathbb{C})"
},
{
"math_id": 1,
"text": "4\\times 4"
},
{
"math_id": 2,
"text": "\\{\\gamma^\\mu\\} = \\{\\gamma^0,\\gamma^1, \\gamma^2, \\gamma^3\\}"
},
{
"math_id": 3,
"text": "\\mathbb{C}"
},
{
"math_id": 4,
"text"... | https://en.wikipedia.org/wiki?curid=10290343 |
1029137 | Eigenplane | In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term "eigenvector" for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term eigenplane can be used to describe a two-dimensional plane (a "... | [
{
"math_id": 0,
"text": "M \\; [ \\mathbf{s} \\; \\mathbf{t} ] \\; = \\; [ \\mathbf{s} \\; \\mathbf{t} ] \\Lambda_\\theta "
}
] | https://en.wikipedia.org/wiki?curid=1029137 |
1029177 | Primitive polynomial (field theory) | Minimal polynomial of a primitive element in a finite field
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF("p""m"). This means that a polynomial "F"("X") of degree m with coefficients in GF("p") = Z/"p"Z is a "primitive pol... | [
{
"math_id": 0,
"text": "\\{0,1,\\alpha, \\alpha^2,\\alpha^3, \\ldots \\alpha^{p^m-2}\\}"
},
{
"math_id": 1,
"text": "\\begin{align}x^3+2x+1 & = (x-\\gamma)(x-\\gamma^3)(x-\\gamma^9)\\\\\nx^3+2x^2+x+1 &= (x-\\gamma^5)(x-\\gamma^{5\\cdot3})(x-\\gamma^{5\\cdot9}) = (x-\\gamma^5)(x-\\gamma^{15})(x... | https://en.wikipedia.org/wiki?curid=1029177 |
10294 | Encryption | Process of converting plaintext to ciphertext
In cryptography, encryption is the process of transforming (more specifically, encoding) information in a way that, ideally, only authorized parties can decode. This process converts the original representation of the information, known as plaintext, into an alternative for... | [
{
"math_id": 0,
"text": "O(\\log\\log M)"
}
] | https://en.wikipedia.org/wiki?curid=10294 |
10295395 | Elliptic cohomology | Algebraic invariant of topological spaces
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
History and motivation.
Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirze... | [
{
"math_id": 0,
"text": "S^1"
},
{
"math_id": 1,
"text": "A^*"
},
{
"math_id": 2,
"text": "A^i = 0"
},
{
"math_id": 3,
"text": "u\\in A^2"
},
{
"math_id": 4,
"text": "A"
},
{
"math_id": 5,
"text": "A^0 = R"
},
{
"math_id": 6,
"text": "E... | https://en.wikipedia.org/wiki?curid=10295395 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.