Q4815938 P2534 "\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbf{Z}) c \equiv 0 \pmod{N} \right\}" S143 Q328
Q22907477 P2534 " n = n_p + n_s\text{ and } P = P_p \oplus P_s" S143 Q328
Q7598372 P2534 " \displaystyle m \big( ( A \cap Z ) \cup ( B \setminus Z ) \big) = p \, \operatorname{mes} (A) + (1-p) \operatorname{mes} (B) " S143 Q328
Q5198174 P2534 "r(G) = 1 + \min_{v\in V} r(G-v),\,{{pad|4em}}where G - v is the digraph resulting from deletion of vertex v and all edges beginning or ending at v." S143 Q328
Q6731660 P2534 "E/V = K_1 \left(\alpha^2+\beta^2\right) = K_1\left(1-\gamma^2\right). <ref>The lowest-order term in the energy can be written in more than one way because, by definition, {{math| <var>&alpha;<sup>2</sup>+&beta;<sup>2</sup>+&gamma;<sup>2</sup><var> {{=}} 1}}.</ref>" S143 Q328
Q6888417 P2534 " P(t) = \big(P_{ij}(t)\big) where each individual entry, P_{ij}(t)\ refers to the probability that state E_i\ will change to state E_j\ in time t\ ." S143 Q328
Q134574 P2534 " P = \frac{\Delta p Q}{\eta}" S143 Q328
Q917713 P2534 " \operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}." S143 Q328
Q3554820 P2534 "\phi_{\alpha\beta} = \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta(U_\beta\cap U_\alpha)} is smooth for all pairs of indices &alpha;, &beta;." S143 Q328
Q2634828 P2534 "A = \varprojlim_{i\in I} A_i = \Big\{\vec a \in \prod_{i\in I}A_i \;\Big|\; a_i = f_{ij}(a_j) \text{ for all } i \leq j \text{ in } I\Big\}." S143 Q328
Q4724022 P2534 " \left\{ \begin{pmatrix} t & du \\ u & t \end{pmatrix} t,u \in \mathbb F_q, t^2 - du^2=1 \right\} \subset \mathrm{SL}_2(\mathbb F_q) ." S143 Q328
Q8062835 P2534 "G(m,n,r) = \langle a,b | a^n = b^m = 1, a^b = a^r \rangle , where ''mn'' is the order of ''G''(''m'',''n'',''r''), the [[greatest common divisor]], gcd((''r''-1)''n'', ''m'') = 1, and ''r''<sup>''n''</sup> ≡ 1 (mod ''m'')." S143 Q328
Q1378504 P2534 " s \xrightarrow[R]{} t if and only if there exist x, y, u, v \in \Sigma^* such that s = xuy, t = xvy, and u \rightarrow v." S143 Q328
Q1737729 P2534 "F(a,b,c,d,e) = d \oplus e \oplus ac \oplus ae \oplus bc \oplus be \oplus cd \oplus de \oplus ade \oplus ace \oplus abd \oplus abc" S143 Q328
Q7254653 P2534 "\left\{\{a,b,c,d\},\{a,b,c\},\{a,b,d\},\{a,b\},\{a\},\{b\},\emptyset\right\}. This topology corresponds to the partial order a<c,b<c,a<d,b<d where open sets are downward closed sets. ''X'' is highly [[pathological (mathematics)|pathological]] from the usual viewpoint of [[general topology]] as it fails to satisfy any [[separation axiom]] besides [[Kolmogorov space|T<sub>0</sub>]]. However, from the viewpoint of [[algebraic topology]] ''X'' has the remarkable property that it is indistinguishable from the [[circle]] ''S''<sup>1</sup>." S143 Q328
Q898323 P2534 " \epsilon H = p(t)(q(t+\epsilon) - q(t)) - \epsilon L \," S143 Q328
Q7784630 P2534 "C_1=\{(0,0),(-1,-1),(0,-1),(1,-1)\} and D_1=\{(-1,1),(0,1),(1,1)\}," S143 Q328
Q1765138 P2534 " \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \text{ for } |x|<1 <ref name =singh>{{cite journal | last1 = Singh | first1 = A. N. | date = 1936 | title = On the Use of Series in Hindu Mathematics | url = | journal = Osiris | volume = 1 | issue = | pages = 606–628 |doi = 10.1086/368443 }}</ref>" S143 Q328
Q7278838 P2534 "angle = 2\pi \frac{part}{total}, where ''angle'' is given in [[radian]]s (change 2&pi; to 360 for degrees), ''part'' is the partial amount represented by the slice and ''total'' is the total amount represented by all slices. &mdash; [[UserKieff|Kieff]] 0219, 12 January 2007 (UTC)" S143 Q328
Q7269444 P2534 " |f(x) - f(y)| < \epsilon \;\;\;\; \forall y \in G " S143 Q328
Q71746 P2534 " a \text{ (digit at } i\text{ )} \times b \text{ (digit at } (n-i)\text{)}." S143 Q328
Q1052034 P2534 " z^{\mathrm{T}}I z = \begin{bmatrix} a & b\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix} a \\ b\end{bmatrix}= a^2 + b^2." S143 Q328
Q1155211 P2534 "\!\,r=\cos(k\theta)<ref>''Mathematical Models'' by [[Martyn Cundy|H. Martyn Cundy]] and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73.</ref>" S143 Q328
Q5500215 P2534 " s = 2 \sinh \left( \frac{1}{2} d \right) = \sqrt{2 (\cosh d -1) } where ''d'' is the distance between the two points, and sinh and cosh are [[hyperbolic functions]].<ref>{{cite book|last1=Smogorzhevsky|title=Lobachevskian Geometry|date=1976|publisher=Mir|location=Moscow|page=65}}</ref>" S143 Q328
Q5159240 P2534 "Payout = Claim \times \frac {Sum\ Insured} {Current\ Value} \!<ref name="NCA">{{cite web |url=http//www.nca.ie/eng/Research_Zone/Reports/Home_Construction/NCA-Home-construction-Volume-4.pdf |title=The Home Construction Industry and the Consumer in Ireland, Volume 4 |author= Grant Thornton (Ireland) |date=2008-11 |page=27 |work=Review of insurance issues |publisher=National Consumer Agency |accessdate=2010-02-23}}</ref>" S143 Q328
Q7114196 P2534 "\mathcal O=\{(x,y,z)\in K^3 \; |\; z=xy+x^2x^\sigma+y^\sigma \} \; \cup \; \{\text{point of infinity of the } z\text{-axis}\} is an ovoid in the 3-dimensional projective space over {{mvar|K}} (represented in inhomogeneous coordinates)." S143 Q328
Q3115556 P2534 "\left\{\begin{matrix} - & c_1 & \cdots & c_{d-1} & c_d \\ a_0 & a_1 & \cdots & a_{d-1} & a_d \\ b_0 & b_1 & \cdots & b_{d-1} & - \end{matrix}\right\}, " S143 Q328
Q215313 P2534 "\bar{D_T} is the mass-averaged absorbed dose of the entire item T" S143 Q328
Q2292874 P2534 "\begin{matrix}L & = & \{uvwxy u,y \in \{0,1,2,3\}^*; v,w,x \in \{0,1,2,3\} \and (v=w \or v=x \or x=w)\} \\ & & \cup \ \{w w \in \{0,1,2,3\}^*\and \text {precisely } \tfrac 1 7 \text{ of the characters in }w \text{ are 3's}\}\end{matrix}." S143 Q328
Q284089 P2534 "\rm CO_2 + H_2O \xrightarrow{Carbonic\ anhydrase} H_2CO_3 (in [[Biological tissue|tissue]]s - high CO<sub>2</sub> concentration)<ref>Carbonic acid has a pK<sub>a</sub> of around 6.36 (the exact value depends on the medium) so at pH 7 a small percentage of the bicarbonate is protonated. See [[carbonic acid]] for details concerning the equilibria HCO<sub>3</sub><sup>-</sup> + H<sup>+</sup>\rightleftharpoons H<sub>2</sub>CO<sub>3</sub> and H<sub>2</sub>CO<sub>3</sub>\rightleftharpoons CO<sub>2</sub> + H<sub>2</sub>O</ref>" S143 Q328
Q5062085 P2534 " P(A\mid B) = P(A) \text{ or } P(B\mid A) = P(B)<ref>Conditional Independence in Statistical theory [http//edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf "Conditional Independence in Statistical Theory", A. P. Dawid"]</ref><ref>Probabilistic independence on Britannica [http//www.britannica.com/EBchecked/topic/477530/probability-theory/32768/Applications-of-conditional-probability#toc32769 "Probability->Applications of conditional probability->independence (equation 7) "]</ref> (''A'' and ''B'' are independent)" S143 Q328
Q6423024 P2534 "D \subset \cup \mathfrak{p}_i Let ''x'' be in ''D''. Then ''xy'' = 0 for some nonzero ''y''. Since ''R'' is reduced, (0) is the intersection of all \mathfrak{p}_i and thus ''y'' is not in some \mathfrak{p}_i. Since ''xy'' is in all \mathfrak{p}_j; in particular, in \mathfrak{p}_i, ''x'' is in \mathfrak{p}_i." S143 Q328
Q1876526 P2534 "\lim_{z \to a}f(z) &nbsp; and &nbsp; \lim_{z \to a}\frac{1}{f(z)} &nbsp; exist, then ''a'' is a [[removable singularity]] of both ''f'' and 1/''f''. " S143 Q328
Q5532460 P2534 "ppp means "play very very softly". p would mean "piano" or softly, pp would mean "pianissimo" or very softly. ppp is thus pianississimo, meaning play very very softly. See [[Dynamics (music)]] for a thorough explanation. --[[UserJayron32|<font style="color#000099">Jayron</font>]]'''''[[User talkJayron32|<font style="color#009900">32</font>]]''''' 1829, 15 May 2010 (UTC)" S143 Q328
Q7662826 P2534 "x+\sqrt{b^2-y^2}= a \ln \frac{b+\sqrt{b^2-y^2}}{y}.<ref>George Salmon (1879). ''A Treatise on the Higher Plane Curves Intended as a Sequel to A Treatise on Conic Sections''. Published by Hodges, Foster, and Figgis. Page 290. [http//books.google.com/books?id=Vv43AAAAMAAJ&pg=PA289]</ref>" S143 Q328
Q837551 P2534 " \left \{ f_k^i \mathbf{R}^n \to \mathbf{R} \ \ 1 \leq i \leq n, 1 \leq k \leq r \right \}" S143 Q328
Q1359990 P2534 " f({{v}_{1}}^{\prime },{{v}_{2}}^{\prime })f({{v}_{1}},{{v}_{2}})\ge f({{v}_{1}},{{v}_{2}}^{\prime })f({{v}_{1}}^{\prime },{{v}_{2}}), for all v and all {v}'<v." S143 Q328
Q2409122 P2534 " R_{\zeta} = \left( \zeta I - T \right)^{-1}." S143 Q328
Q1541210 P2534 "<math alt="L = \int_{S_0}^S n ds">L=\int_{S_o}^{S}n\,ds, where ''n'' is the refractive index and ''S'' is the arc length of the curve. If [[Cartesian coordinate]]s are used, this equation is modified to incorporate the change in arc length for a spherical gradient, to each physical dimension" S143 Q328
Q2737027 P2534 "\lambda(t) = \frac{f(t)}{R(t)}, where f(t) is the time to (first) failure distribution (i.e. the failure density function) and R(t)=1-F(t)." S143 Q328
Q2829907 P2534 "\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} " S143 Q328
Q1361825 P2534 "\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \and \varphi(x, w_1, \ldots, w_n , A) ] )" S143 Q328
Q15146798 P2534 "\{x\} = \begin{cases} 0, & x < 0 \\ x, & x \ge 0. \end{cases}" S143 Q328
Q1252988 P2534 "S(-f) = S(f)^*, &nbsp; which is the [[complex conjugate]] of S(f)." S143 Q328
Q5638612 P2534 " \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right)= \frac{1}{\mu} \frac{\partial p}{\partial z} where \mu is the dynamic viscosity of the fluid." S143 Q328
Q1781358 P2534 "X\vee Y = (X\amalg Y)\;/{\sim},\, <!-- NOTE In expressions like "a\sim b", the symbols \sim, being a binary relation symbol, has a certain amount of space before it and after it. In this present expression, that spacing is not appropriate, and so it is avoided by enclosing "\sim" in curly braces, thus "{\sim}". -->" S143 Q328
Q4859826 P2534 "\forall x \Box Fx \rightarrow \Box \forall x Fx." S143 Q328
Q1921708 P2534 "\sum_{k=0}^n (-1)^k {n \choose k} (x - k)^n = n! <!-- feel free to unmathify this, if possible -->" S143 Q328
Q4698465 P2534 "\|\boldsymbol{v}\| = \sqrt{v_1^2+v_2^2+v_3^2} is the Euclidean norm of the vector, analogous to the calculation of the [[Euclidean distance]] between two points in [[Euclidean space]]." S143 Q328
Q4116368 P2534 "1/Db = w/Dw + f/Df + p/Dp + m/Dm<ref name=Siri>{{cite journal |author=Siri William E |title=The gross composition of the body |journal=Advances in Biological and Medical Physics |volume=4|pages=239–280|year=1956|pmid=13354513}}</ref>{{rp|262}}" S143 Q328
Q865746 P2534 "d X \times X \to [0,\infty)," S143 Q328
Q5203038 P2534 " d' = \frac{\mu_S - \mu_N}{\sqrt{\frac{1}{2}(\sigma_S^2 + \sigma_N^2)}}<ref>Samuel Gale and David Perkel. A Basal Ganglia Pathway Drives Selective Auditory Responses in Songbird Dopaminergic Neurons via Disinhibition. The Journal of Neuroscience (2010). 30(3)1027–1037</ref>" S143 Q328
Q298521 P2534 " X\;R\;Y if and only if for every formula A, if \Box A\in X then A\in Y," S143 Q328
Q895997 P2534 "T_b = \frac{a}{Rb}" S143 Q328
Q6472739 P2534 "\sum_n\frac{1}{|z_n|^2} converges, with zeros counted according to their [[multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]])" S143 Q328
Q958871 P2534 "e^{\hat{M}}_N (x,\overline{\theta},\theta)^* = e^{\hat{M}^*}_{N^*}(x,\theta,\overline{\theta}) where \mu^*=\mu, \alpha^*=\dot{\alpha}, and \dot{\alpha}^*=\alpha and \omega(x,\overline{\theta},\theta)^*=\omega(x,\theta,\overline{\theta})." S143 Q328
Q685892 P2534 "\Delta _G U=\Delta _G H -p\Delta V_m, where \Delta _G U is the molar lattice energy, \Delta _G H the molar lattice enthalpy and \Delta V_m the change of the volume per mol. Therefore the lattice enthalpy further takes into account that work has to be performed against an outer pressure p. Lattice Energy of an ionic compound depends upon charge of the ion and size of the ions.Moreover factors such as packing of ions doesn't matter efficiently" S143 Q328
Q1121 P2534 "\forall X \left[ \emptyset \notin X \implies \exists f \colon X \rightarrow \bigcup X \quad \forall A \in X \, ( f(A) \in A ) \right] \,." S143 Q328
Q348793 P2534 " \Pi_A (\Pi_B) is the [[Peltier coefficient]]<ref name="YAS-AIK">Yu. A. Skripnik, A. I. Khimicheva. Methods and devices for measuring the Peltier coefficient of an [[inhomogeneous]] electric circuit. Measurement Techniques July 1997, Volume 40, Issue 7, pp 673-677</ref><ref name="SA-AB-TC">See also [[Current source#Constant current source with thermal compensation|Constant current source with thermal compensation]]</ref> of [[primary circuit|conductor A]] ([[secondary circuit|conductor B]]), and" S143 Q328
Q3064205 P2534 "B_{ij} = \frac{\mbox{Energy absorbed at }A_{j}\mbox{ originating as emission at } A_{i}}{\mbox{Total radiation emitted from }A_{i}}" S143 Q328
Q207643 P2534 " f(a_1 \mathbf{x}_1+\cdots+a_m \mathbf{x}_m) = a_1 f(\mathbf{x}_1)+\cdots+a_m f(\mathbf{x}_m). \!<ref>{{harvnb|Rudin|1991|page=14}}<br>Suppose now that {{mvar|X}} and {{mvar|Y}} are vector spaces ''over the same scalar field''. A mapping \Lambda X \to Y is said to be ''linear'' if \Lambda(\alpha x + \beta y) = \alpha\Lambda x + \beta \Lambda y for all {{mvar|x}} and {{mvar|y}} in {{mvar|X}} and all scalars \alpha and \beta. Note that one often writes \Lambda x, rather than \Lambda(x), when \Lambda is linear.</ref><ref>{{harvnb|Rudin|1976|page=206}}<br>A mapping {{mvar|A}} of a vector space {{mvar|X}} into a vector space {{mvar|Y}} is said to be a ''linear transformation'' if<br>A({\bf x}_1 + {\bf x}_2) = A{\bf x}_1 + A{\bf x}_2,\;\;\;\;\;A(c{\bf x})=cA{\bf x}<br>for all {\bf x},{\bf x}_1,{\bf x}_2 \in X and all scalars {{mvar|c}}. Note that one often writes A{\bf x} instead of A({\bf x}) if {{mvar|A}} is linear.</ref>" S143 Q328
Q7642959 P2534 " \Delta r \approx \frac{\Delta}{\sqrt{1+I_\max/I_s}}&nbsp;&nbsp;&nbsp;({{EquationRef|1}})" S143 Q328
Q7169283 P2534 "A = \begin{bmatrix}2 & -2 \\ -2 & 1\end{bmatrix}" S143 Q328
Q2123192 P2534 "\tan \left( \frac{p}{2} \right) e^{i \theta} = \mathrm{cn} \left( z, \tfrac{1}{\sqrt{2}} \right), \text{ where } w = p e^{i \theta} \text{ and } z = x + i y." S143 Q328
Q5645461 P2534 " H_3 \quad \longrightarrow \quad H_3^+ \ + \ e^- <ref>Helm H. et al. [http//books.google.de/books?id=njAjmdxOH9oC&pg=PA275|Coupling of Bound States to Continuum States in Neutral Triatomic Hydrogen. in ''Dissociative Recombination''], ed. S. Guberman, Kluwer Academic, Plenum Publishers, USA, 275-288 (2003) ISBN 0-306-47765-3</ref>" S143 Q328
Q223722 P2534 " (1-\epsilon)(k\ln k)^{2-\epsilon}<\ln A(k)<(1+\epsilon)(k\ln k)^{2+\epsilon} for sufficiently large ''k''." S143 Q328
Q7306402 P2534 " \epsilon = h \sin \theta" S143 Q328
Q60354 P2534 "\mathrm{VO_2\; max} = Q \times\ (\mathrm{C_aO_2} - \mathrm{C_vO_2}), when these values are obtained during an exertion at a maximal effort." S143 Q328
Q2590607 P2534 "\begin{pmatrix}h & 0\\0 & -h\end{pmatrix}\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix} = \begin{pmatrix}0 & h\\h & 0\end{pmatrix}." S143 Q328
Q6981 P2534 "(\rho v^2)_{sw}\approx \left( \frac{4 B(r)^2}{2\mu_0} \right) _m<ref group="note">The reason for the factor of 4 is because the magnetic field strength just inside the magnetopause is twice the dipole value for a planar magnetopause</ref> where \rho and v are the [[density]] and [[velocity]] of the [[solar wind]], and" S143 Q328
Q870797 P2534 "\text{SL}(2, \mathbf Z) = \left \{ \left. \begin{pmatrix}a & b \\ c & d \end{pmatrix} \right| a, b, c, d \in \mathbf Z,\ ad-bc = 1 \right \}" S143 Q328
Q7801630 P2534 "z \in I^* if and only if there exists a c \in R, where c is not contained in any minimal prime ideal of R, such that c z^{p^e} \in I^{[p^e]} for all e \gg 0. If R is reduced, then one can instead consider all e > 0." S143 Q328
Q5161000 P2534 " \Gamma(n) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb Z) a, d \equiv 1 \pmod n, b, c \equiv 0 \pmod n \right\} " S143 Q328
Q5475803 P2534 "\sum_{n=1}^\infty n = -\frac{1}{12}? Does that have any deeper meaning? In similar cases (1+2+4+8+...=-1, for example), the strange answer is actually because you're working in a strange metric without knowing it (2-adic in the case of that example), but I can't think of a metric in which 1+2+3+4+... would converge. --[[UserTango|Tango]] ([[User talkTango|talk]]) 0401, 19 July 2008 (UTC)" S143 Q328
Q5159256 P2534 " P(A\mid B) = P(A) \text{ or } P(B\mid A) = P(B)<ref>Conditional Independence in Statistical theory [http//edlab-www.cs.umass.edu/cs589/2010-lectures/conditional%20independence%20in%20statistical%20theory.pdf "Conditional Independence in Statistical Theory", A. P. Dawid"]</ref><ref>Probabilistic independence on Britannica [http//www.britannica.com/EBchecked/topic/477530/probability-theory/32768/Applications-of-conditional-probability#toc32769 "Probability->Applications of conditional probability->independence (equation 7) "]</ref> (''A'' and ''B'' are independent)" S143 Q328
Q1353947 P2534 "\Psi_B(\cdots,\bold{r}_i,\cdots,\bold{r}_j,\cdots)=+\Psi_B(\cdots,\bold{r}_j,\cdots,\bold{r}_i,\cdots) if the particles are [[bosons]]," S143 Q328
Q2601117 P2534 "\left\{ {n \atop n} \right\} = 1 and for n \geq 1, \left\{ {n \atop 1}\right\} = 1 the only way to partition an ''n''-element set into ''n'' parts is to put each element of the set into its own part, and the only way to partition a nonempty set into one part is to put all of the elements in the same part." S143 Q328
Q332683 P2534 "f_\mathrm{img} = \begin{cases} f + 2f_\mathrm{IF} , & \text{if } f_\mathrm{LO} > f \text{ (high side injection)}\\ f- 2f_\mathrm{IF}, & \text{if } f_\mathrm{LO} < f \text{ (low side injection)} \end{cases} " S143 Q328
Q387743 P2534 "E/V = K_1 \left(\alpha^2+\beta^2\right) = K_1\left(1-\gamma^2\right). <ref>The lowest-order term in the energy can be written in more than one way because, by definition, {{math| <var>&alpha;<sup>2</sup>+&beta;<sup>2</sup>+&gamma;<sup>2</sup><var> {{=}} 1}}.</ref>" S143 Q328
Q846912 P2534 " \forall x,y\in U, ~ x\ne y, when h is drawn randomly from the family H, the difference h(x)-h(y) ~\bmod~ m is uniformly distributed in [m]." S143 Q328
Q909693 P2534 "\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}, where ''M''<sub>o</sub> is the molecular mass of the repeating unit." S143 Q328
Q1386472 P2534 "KIE=\frac{k_L}{k_H}" S143 Q328
Q5135329 P2534 "\left[-\ -\right] V^{op} \times V \to V ," S143 Q328
Q869338 P2534 "\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}" S143 Q328
Q4747168 P2534 "A = \begin{bmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \end{bmatrix}." S143 Q328
Q7100784 P2534 "\psi(\alpha) is the smallest ordinal which cannot be expressed from 0, 1, \omega and \Omega using sums, products, exponentials, and the \psi function itself (to previously constructed ordinals less than \alpha)." S143 Q328
Q2717459 P2534 "A_{C} = 2 \pi r^2 + 2 \pi r h = 2 \pi r ( r + h ).\," S143 Q328
Q7999139 P2534 "W_\delta(t)({ b}) is the set of points within a distance δ of some point '''b'''(''x'') of the path '''b''' with 0≤''x''≤''t''." S143 Q328
Q486921 P2534 " p = {n R T \over V} = {\text{constant} \over V} " S143 Q328
Q902120 P2534 "P(\theta)\propto{}\exp\left(-U(\theta)/kT\right), where U(\theta) is the potential determining the probability of each value of \theta." S143 Q328
Q3893192 P2534 "\bold{A} = (A_{ij})" S143 Q328
Q10286397 P2534 "A \cup \operatorname{cl}(A) \cup \operatorname{cl}(\operatorname{cl}(B)) = \operatorname{cl}(A \cup B) \setminus \operatorname{cl}(\varnothing) \text{ for all subsets }A, B \subseteq X." S143 Q328
Q2606520 P2534 "angle = 2\pi \frac{part}{total}, where ''angle'' is given in [[radian]]s (change 2&pi; to 360 for degrees), ''part'' is the partial amount represented by the slice and ''total'' is the total amount represented by all slices. &mdash; [[UserKieff|Kieff]] 0219, 12 January 2007 (UTC)" S143 Q328
Q780487 P2534 "\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \and \varphi(x, w_1, \ldots, w_n , A) ] )" S143 Q328
Q7160302 P2534 "C_{abcd} = \frac{C^{(1)}_{abcd}}{\lambda}+\frac{C^{(2)}_{abcd}}{\lambda^2}+\frac{C^{(3)}_{abcd}}{\lambda^3}+\frac{C^{(4)}_{abcd}}{\lambda^4}+O\left(\frac{1}{\lambda^5}\right)" S143 Q328Q5310229 P2534 "\langle , \rangle X \times Y \to F" S143 Q328
Q7410129 P2534 "\begin{cases} X_{t} \sim \mathrm{Unif} (\{X_{t-1} - 1, X_{t-1} + 1\}), & t \mbox{ an integer;} \\ X_{t} = X_{\lfloor t \rfloor}, & t \mbox{ not an integer;} \end{cases}" S143 Q328
Q5382669 P2534 " \kappa^{2} \equiv \frac{2 \Omega}{R}\frac{d}{dR}(R^2 \Omega), where R is the radial co-ordinate.<ref>p161, Astrophysical Flows, Pringle and King 2007</ref>" S143 Q328
Q460184 P2534 "\begin{alignat}{2}\text{Net Cash Flows from Financing Activities} = & \left[\text{Dividends received from }3^{{\rm rd}}\text{ parties}\right]\\ & -\left[\text{Dividends paid to }3^{{\rm rd}}\text{ parties}\right]\\ & - [\text{Dividends paid to NCI but not} \\ & \text{intracompany dividend payments}] \end{alignat}" S143 Q328
Q124656 P2534 "2\sqrt{3}\approx 3.464 (for the equatorial geodesic), \sqrt{13}\approx 3.606, 4 (for the geodesic through the midpoints of opposite edges), 2\sqrt{7}\approx 5.292, and \sqrt{19}\approx 4.359." S143 Q328
Q2344309 P2534 " q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right)&nbsp;which is called the ''magnitude'' of the vector {{math|''x''}}." S143 Q328
Q470531 P2534 "(a\ b\ c) for ax^2 + bx + c." S143 Q328
Q909313 P2534 " Ar= \frac{Gr}{Re^2} <ref>{{cite book | author=[[Frank P. Incropera|Incropera, F. P.]]| title=Fundamentals of Heat and Mass Transfer, 5th Ed.| publisher=Wiley| year=2001 | isbn=978-0471386506 }}</ref>" S143 Q328
Q5462733 P2534 "\frac{\mathrm{d}X(t)}{\mathrm{d}t} = \begin{cases} r_i & \text{ if } X(t)>0 \\ \max(r_i,0) & \text{ if } X(t)=0.\end{cases}" S143 Q328
Q5067368 P2534 " \exists W_1 \subseteq V \exists W_2 \subseteq V \exists W_3 \subseteq V \forall v \in V (v \in W_1 \vee v \in W_2 \vee v \in W_3) \wedge " S143 Q328
Q740060 P2534 "(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in K^3;\\-1,&\mbox{ if not.}\end{cases}" S143 Q328
Q2908602 P2534 "t' = t - \frac{|\mathbf{r}-\mathbf{r}'|}{c}" S143 Q328
Q751048 P2534 "\varepsilon=\begin{pmatrix}0 & 1 \\0 & 0 \end{pmatrix}\quad\text{and}\quad a + b\varepsilon = \begin{pmatrix}a & b \\ 0 & a \end{pmatrix}." S143 Q328
Q3694453 P2534 " M = m - 5 (\log_{10}{D_L} - 1) - K_{Corr}\!\," S143 Q328
Q3895032 P2534 "\begin{pmatrix}1 & 1 \\ 1 & -\end{pmatrix}" S143 Q328
Q898680 P2534 " C = \frac{3 \times \mbox{number of triangles}}{\mbox{number of connected triplets of vertices}} = \frac{\mbox{number of closed triplets}}{\mbox{number of connected triplets of vertices}}." S143 Q328
Q4916502 P2534 " \frac{dL}{dx} = S \frac{\frac{dE}{dx}}{1+k_B\frac{dE}{dx}}." S143 Q328
Q2153925 P2534 "\mathbf{B} = \nabla \times \mathbf{A}.<ref name="Feynman1515">{{Harvtxt|Feynman|1964|p=15_15}}</ref> and \mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t } <ref name="Feynman1515"/> and so even if the region outside the windings is devoid of '''B''' field, it is filled with non-zero '''E''' field." S143 Q328
Q1238125 P2534 "\mathbf{A}\otimes\mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B} \end{bmatrix}, " S143 Q328
Q230913 P2534 "\Gamma = V \left(\frac{dp}{de}\right)_V" S143 Q328
Q193835 P2534 "\begin{pmatrix} 0 & 1 & 0& 1 \\ 1 & 0 & 2 & 1 \\ 0 & 2 & 0 & 0 \\ 1 & 1 & 0 & 0 \end{pmatrix} and its cube" S143 Q328
Q2462410 P2534 " \mathbf{A} \cdot \mathbf{B} is a 4-vector style, which is typically more compact and can use [[Vector (mathematics and physics)|vector]] [[Vector notation|notation]], (such as the inner product "dot"), always using bold uppercase to represent the 4-vector, and bold lowercase to represent 3-vectors, eg. \vec{\mathbf{a}} \cdot \vec{\mathbf{b}}. Most of the 3-vector rules have analogues in 4-vector mathematics." S143 Q328
Q5326898 P2534 " f_{\rho}(x) = \int_{Y} y d\rho(y|x), x \in X," S143 Q328
Q827230 P2534 "\lambda^*(E) = \operatorname{inf} \left\{\sum_{k=1}^\infty l(I_k) {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subseteq \bigcup_{k=1}^\infty I_k\right\}." S143 Q328
Q6042601 P2534 "\left(\begin{array}{cccr} 5 & 2 & 6 & 0\\ 4 & 7 & 3 & 8\\ 5 & 9 & 0 & 4\\ 3 & 1 & 0 & -3\\ 9 & 0 & 2 & 1\end{array}\right)&nbsp; &nbsp; and &nbsp; &nbsp; \left(\begin{array}{ccc} 1 & 5 & 0\\ 0 & 9 & 2\\ 1 & 7 & 3\end{array}\right)" S143 Q328
Q6420194 P2534 " U = \left( \begin{array}{cccc} 1 & \cdots & 0 & c_1 \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 1 & c_{n-1} \\ 0 & \cdots & 0 & c_n \end{array} \right)" S143 Q328
Q3044461 P2534 "Loan = {X}\frac{1-(1+i)^{-n}}{i^{(12)}} \," S143 Q328Q1268618 P2534 "insert(u, j) is u with "[]" inserted into the jth position" S143 Q328
Q210729 P2534 "\Phi in this relationship does not necessarily represent anything physically meaningful. In the case of the current generator, Q, the time integral of current, represents the quantity of electric charge physically delivered by the generator. Here \Phi is the time integral of voltage but whether or not that represents a physical quantity depends on the nature of the voltage source. For a voltage generated by magnetic induction it is meaningful, but for an electrochemical source, or a voltage that is the output of another circuit, no physical meaning is attached to it." S143 Q328
Q2607586 P2534 "C - D = \frac{\nu_c - \nu_d}{\nu_a - \nu_b} (A - B) + k, " S143 Q328
Q282327 P2534 "\Lambda(n) = \begin{cases} \log p & \text{if }n=p^k \text{ for some prime } p \text{ and integer } k \ge 1, \\ 0 & \text{otherwise.} \end{cases}" S143 Q328
Q6552939 P2534 " M = \begin{pmatrix} e(a_1,b_1) & e(a_1,b_2) & \cdots & e(a_1,b_n) \\ e(a_2,b_1) & e(a_2,b_2) & \cdots & e(a_2,b_n) \\ \vdots & \vdots & \ddots & \vdots \\ e(a_n,b_1) & e(a_n,b_2) & \cdots & e(a_n,b_n) \end{pmatrix} ." S143 Q328
Q17008711 P2534 " Z = \pi^{-1}(Y' \oplus R) = \pi^{-1} (Y \oplus N \oplus R), " S143 Q328
Q17099544 P2534 "{n-1\choose k} + {n-1\choose k-1} = {n\choose k}\quad\text{for }1 \le k \le n " S143 Q328
Q161172 P2534 "\mathcal{M}_2(\mathbb{R}) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\ a,b,c,d \in \mathbb{R} \right\}. " S143 Q328
Q6708764 P2534 "\left ( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right ), \ a, b, c \in \mathbb Z." S143 Q328
Q15846555 P2534 " \models_{\mathcal S} \varphi\ \to\ \vdash_{\mathcal S} \varphi.<ref name="metalogic">Hunter, Geoffrey, Metalogic An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971</ref>" S143 Q328
Q6901647 P2534 "x*y\le z if and only if x\le (y\Rightarrow z)." S143 Q328
Q3042809 P2534 " \begin{array}{ccc} O_K & \hookrightarrow & O_L \\ \downarrow & & \downarrow \\ K & \hookrightarrow & L \end{array} " S143 Q328
Q5194340 P2534 "\displaystyle \omega_{m,n}(x) = \frac{e^{-x+\pi i (m/2-n)}}{\Gamma(1+n-m/2)}U(m/2-n,1+m,x)." S143 Q328
Q3743693 P2534 "\left(\frac{\partial S}{\partial X}\right)_U=0 &nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; \left(\frac{\partial ^2S}{\partial X^2}\right)_U < 0&nbsp;&nbsp;&nbsp;&nbsp; at equilibrium." S143 Q328
Q7606831 P2534 "(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in F^3;\\-1,&\mbox{ if not.}\end{cases}" S143 Q328
Q5347247 P2534 "M=\frac{\max x + \min x}{2}." S143 Q328
Q5436946 P2534 "\begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\ -(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1} \end{bmatrix}" S143 Q328
Q25304378 P2534 " \operatorname{St}(\Omega)=\{x+ye_nx+ze_n\in \Omega \text{ for some } z \text{ and } |y|\leq\frac{1}{2} |\Omega\cap L_x|\}." S143 Q328
Q6602900 P2534 " \mbox{absolute margin of victory} = \begin{cases}0; & w \le \frac{c}{2} \\ w - \max\{r, \frac{c}{2}\}; & w > \frac{c}{2} \end{cases}" S143 Q328
Q692689 P2534 " X = \coprod_{j\in J}X_j" S143 Q328
Q2140269 P2534 " \Delta = \frac{2\pi t }{\lambda} C ( \sigma_{1} - \sigma_{2}) " S143 Q328
Q1150815 P2534 "\sqrt{3}." S143 Q328
Q913012 P2534 " L(x) = \begin{cases} \operatorname{sinc}(x)\, \operatorname{sinc}(x/a) & \text{if}\;\; -a < x < a\\ 0 & \text{otherwise} \end{cases} " S143 Q328
Q4735170 P2534 " K \subseteq A \leftrightarrow \exists H \exists J[\langle H,J,K \rangle \in R_0 \wedge H \subseteq B \wedge J \subseteq \alpha / B ]," S143 Q328
Q6958465 P2534 " B(R) = \left\lbrace{ \left({\begin{array}{*{20}c} a & b \\ 0 & d \end{array}}\right) a,d \in R^*, ~ b \in R }\right\rbrace. " S143 Q328
Q7884465 P2534 " 10\log(|H_{NEXT}(f)|^2)=\begin{cases} -66 + 6\log(f) dB & f < 20 KHz \\ -50.5 + 15\log(f) dB & f >= 20 KHz \end{cases}" S143 Q328
Q3433864 P2534 " L_{R, n} \equiv \frac{NH}{n \pi f_0}, where \,N is the [[Brunt–Väisälä frequency]], \,H is the [[scale height]], and ''n'' = 1, 2, ...." S143 Q328
Q5701030 P2534 "{QP - PQ = \frac{ih}{2\pi}} <br />[The symbol '''Q''' is the matrix for displacement, '''P''' is the matrix for momentum, '''i''' stands for the square root of negative one, and '''h''' is Planck's constant.<ref>See ''Introduction to quantum mechanics''. by Henrik Smith, p. 58 for a readable introduction. See Ian J. R. Aitchison, et al., "Understanding Heisenberg's 'magical' paper of July 1925," Appendix A, for a mathematical derivation of this relationship.</ref>]" S143 Q328
Q1455709 P2534 "r_n = \sqrt{n \lambda f + \frac{n^2\lambda^2}{4}}<ref>{{cite book|chapterurl=http//xdb.lbl.gov/Section4/Sec_4-4.html|publisher=Center for X-ray Optics and Advanced Light Source, Lawrence Berkeley National Laboratory|accessdate=13 January 2015|title=X-Ray Data Booklet|chapter=Zone Plates}}</ref>" S143 Q328
Q5316014 P2534 " \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} a \\ b \\ d \end{bmatrix} = \begin{bmatrix} a \\ b \\ b \\ d \end{bmatrix}" S143 Q328
Q6818052 P2534 "\mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1} where s_i represents a value 0 or 1 depending on the state of the ''i<sup>th</sup>'' switch." S143 Q328
Q5282315 P2534 "\dot m = \rho \, A \, v." S143 Q328
Q4493413 P2534 "J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ),\,<!-- Do not delete "\," it improves display of formula on certain browsers. --->" S143 Q328
Q7818354 P2534 "V_c=kT_c/q" S143 Q328
Q1162676 P2534 " g S = \{g.s\,\,s \in S\}." S143 Q328
Q899029 P2534 "\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T \times \Delta S_{\text{fus}} < 0, where \Delta H_{\text{fus}} is the enthalpy or [[heat of fusion]]." S143 Q328
Q5152373 P2534 " \mathrm{comm}_G(H)=\{g\in G gHg^{-1} \cap H \text{ has finite index in both } H \text{ and } gHg^{-1}\}.<ref>Geoghegan (2008), p. 348</ref>" S143 Q328
Q5348913 P2534 " H(k-1) = \begin{bmatrix}Y(k) & Y(k+1) & \cdots & Y(k+p) \\ Y(k+1) & \ddots & & \vdots \\ \vdots & & & \\ Y(k+r) & \cdots & & Y(k+p+r) \end{bmatrix}" S143 Q328
Q17082552 P2534 "X^Tr = \mu" S143 Q328
Q3117686 P2534 " (X Y Z 0) \equiv (a X a Y a Z 0) ," S143 Q328
Q138224 P2534 "\cosh s = (\cosh b -1) \cosh^2 l + 1 = \cosh b \cdot \cosh^2 l - \sinh^2 l<ref>P. Buser and H. Karcher. Gromov's almost flat manifolds. Asterisque 81 (1981), page 104.</ref>" S143 Q328
Q899628 P2534 "\Delta T = Q \times R_{\theta}\,." S143 Q328
Q6935403 P2534 "H^i(X,L^{-1})=0\text{ for }i = 0,1.\ " S143 Q328
Q17105936 P2534 "\eta \left\{\eta_bE + \frac{Px}{v}\right\} = \left\{W C_{rr1} + N C_{rr2} v + \frac{1}{2}\rho C_d A v^2\right\}x +Wh + \frac{N_a W v^2}{2g}<ref name="Equation">Solar Vehicle Performance, Dr. Eric Slimko, December 1, 1991</ref>" S143 Q328
Q1473659 P2534 "\textbf{x}_{i} = \textbf{x}_{1} + \sigma(\textbf{x}_{i} - \textbf{x}_{1}) \text{ for all i } \in\{2,\dots,n+1\}. go to step 1." S143 Q328
Q1430460 P2534 "\frac{\pi^{d/2}}{d2^{d-1}\Gamma(d/2)}\rightarrow 0 as d \rightarrow \infty. Furthermore, the distance between the center and the corners is r\sqrt{d}, which increases without bound for fixed r." S143 Q328
Q1253278 P2534 "\textbf{x}_{i} = \textbf{x}_{1} + \sigma(\textbf{x}_{i} - \textbf{x}_{1}) \text{ for all i } \in\{2,\dots,n+1\}. go to step 1." S143 Q328
Q17085832 P2534 "h=\sqrt{\frac{3}{4}-\left(\!{\color{Blue}R}-\frac{1}{2}\cot\!\left(\frac{\pi}{n}\right)\!\right)^{2}} + \sqrt{{1}-\left(\!{\color{Blue}R}-\frac{1}{2}\csc\!\left(\frac{\pi}{n}\right)\!\right)^{2}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; if order n is &nbsp;&nbsp;&nbsp;&nbsp; 3\leq n\leq9" S143 Q328
Q5404299 P2534 "P_0^s (S) = \lim_{\delta \downarrow 0} \sup \left\{ \left. \sum_{i \in I} \mathrm{diam} (B_i)^s \right| \begin{matrix} \{ B_i \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint closed balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}." S143 Q328
Q1799665 P2534 "r = a - n \operatorname{trunc}\left(\frac{a}{n}\right)" S143 Q328
Q2091879 P2534 "\log y = k \log x + \log a." S143 Q328
Q182571 P2534 " \left[\begin{array}{c} R \\ G \\ B \end{array}\right]=\left[\begin{array}{ccc}255/R'_w & 0 & 0 \\ 0 & 255/G'_w & 0 \\ 0 & 0 & 255/B'_w\end{array}\right]\left[\begin{array}{c}R' \\ G' \\ B' \end{array}\right]" S143 Q328
Q1622788 P2534 "N_g = 0.04\,{T_d}^{1.25} <ref>Anderson R.B., Eriksson A.J., Kroninger H., Meal D.V. and Smith M.A. "Lightning and thunderstorm parameters" IEE Conference Publication No. 236, "Lightning and Power Systems", London, June 1984</ref> " S143 Q328
Q8021980 P2534 "u(w_0 + WTA , 1) = u(w_0 , 0). <ref>Horowitz, John Keith and Mcconnell, Kenneth, (2003), [http//www.ag.unr.edu/moeltner/APEC%20464_664/Spring_07/pdf/Horowitz_McConnell_2003.pdf "Willingness to accept, willingness to pay and the income effect"], ''Journal of Economic Behavior & Organization'', Vol. 51, No. 4, p.&nbsp;537–545. {{wayback|url=http//www.ag.unr.edu/moeltner/APEC%20464_664/Spring_07/pdf/Horowitz_McConnell_2003.pdf |date=20110720095243 }}</ref>" S143 Q328
Q7705272 P2534 "t(I) = \{r \in R \mbox{ }|\mbox{ } \forall s \notin I, \mbox{ }\exists x \in (s)\mbox{ } x \notin I \text{ and } (x)(r) \subset I \}. \, " S143 Q328
Q32980 P2534 "\|\boldsymbol{v}\| = \sqrt{v_1^2+v_2^2+v_3^2} is the Euclidean norm of the vector, analogous to the calculation of the [[Euclidean distance]] between two points in [[Euclidean space]]." S143 Q328
Q3595339 P2534 "\text{Hypothesis }H_0\text{ No leak}" S143 Q328
Q5884046 P2534 "R = \frac {\text{Isentropic enthalpy change in rotor}}{\text{Isentropic enthalpy change in stage}} .<ref>Peng, William W., Fundamentals of turbomachinery, John Wiley, 2008</ref>" S143 Q328
Q17098676 P2534 "P_\lambda=\sum_{\mu\le \lambda}u_{\lambda\mu}m_\mu where ''u''<sub>&lambda;&mu;</sub> is a rational function of ''q'' and ''t'' with ''u''<sub>&lambda;&lambda;</sub> = 1;" S143 Q328
Q7623975 P2534 " \sin x \cos x = \frac{\sin(2x)}{2}?" S143 Q328
Q5121662 P2534 " U^R = \left( \begin{array}{ccccccc} 0 & -1 & & & & & \\ 1 & 0 & & & & & \\ & & 0 & -1 & & & \\ & & 1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0& -1\\ & & & & & 1 & 0 \end{array} \right) U^T \left( \begin{array}{ccccccc} 0 & 1 & & & & & \\ -1 & 0 & & & & & \\ & & 0 & 1 & & & \\ & & -1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0& 1\\ & & & & & -1 & 0 \end{array} \right)~. " S143 Q328
Q955475 P2534 "\begin{smallmatrix}2\pi\left(c^2+b\sqrt{a^2-c^2}E(\alpha,m)+\frac{bc^2}{\sqrt{a^2-c^2}}F(\alpha,m)\right),\,\!\end{smallmatrix} where \begin{smallmatrix}\alpha=\arccos\left(\frac{c}{a}\right)\,\,\!\end{smallmatrix} is the modular angle, or '''[[angular eccentricity]]'''; \begin{smallmatrix}m=\frac{b^2-c^2}{b^2\sin(\alpha)^2}\,\!\end{smallmatrix} and \begin{smallmatrix}F(\alpha,m)\,\!\end{smallmatrix}, \begin{smallmatrix}E(\alpha,m)\,\!\end{smallmatrix} are the incomplete [[elliptic integral]]s of the first and second kind, respectively. The values 980&nbsp;km, 759&nbsp;km, and 498&nbsp;km were used for a, b, and c respectively." S143 Q328
Q1530275 P2534 "E(X) = \int_\Omega X dP" S143 Q328
Q743364 P2534 "A \to B \quad C" S143 Q328
Q909339 P2534 "\Delta\lambda = \lambda^{\mathrm{state 2}}_{\mathrm{observed}} - \lambda^{\mathrm{state 1}}_{\mathrm{observed}} where \lambda is the wavelength of the spectral peak of interest and \lambda^{\mathrm{state 2}}_{\mathrm{observed}} > \lambda^{\mathrm{state 1}}_{\mathrm{observed}}" S143 Q328
Q1361106 P2534 " EROEI = \frac{\hbox{Energy Delivered}}{\hbox{Energy Required to Deliver that Energy}} <ref>Hall CA, Lambert JG, Balogh SB. 2013. EROI of different fuels and the implications for society. Energy Policy. 141–52 & Atlason R, Unnthorsson R. 2014. Ideal EROI (energy return on investment) deepens the understanding of energy systems. Energy. 241–45</ref>" S143 Q328
Q4774357 P2534 " \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 5 & 0 & 0 \\ 0 & 7 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \end{bmatrix}. " S143 Q328
Q806352 P2534 "E_g(T)=E_g(0)-\frac{\alpha T^2}{T+\beta}, where ''E<sub>g</sub>''(0), α and β are material constants.<ref>[http//ece-www.colorado.edu/~bart/book/eband5.htm Temperature dependence of the energy bandgap]. Ece-www.colorado.edu. Retrieved on 2013-04-03.</ref>" S143 Q328
Q1751970 P2534 "sgn is the [[sign function]], \text{abs} is the [[absolute value]], and R_i is the [[Ranking|rank]]. Notice that pairs 3 and 9 are tied in absolute value. They would be ranked 1 and 2, so each gets the average of those ranks, 1.5." S143 Q328
Q10965882 P2534 " \forall t \exists ! s (\bigvee_{q\in Q} H_q (s,t))\land \neg\exists s \exists t (\bigvee_{q,q'\in Q, q\neq q} H_q(s,t)\land H_{q'} (s,t))" S143 Q328
Q1747683 P2534 " p \overset{\alpha}{\rightarrow} p' " S143 Q328
Q3417792 P2534 "S(-f) = S(f)^*, &nbsp; which is the [[complex conjugate]] of S(f)." S143 Q328
Q3775923 P2534 "O\left( 2^{n_1} \sum_{i>1} n_i\right)." S143 Q328Q2126650 P2534 "M(B) = \inf\{ m(V) \mid V \text{ is an open set with } B \subseteq V \subseteq X \} ." S143 Q328
Q845727 P2534 "(L/D)_{max} = \frac{1}{2} \sqrt{\frac{\pi \epsilon AR}{C_{D,0}}},<ref>{{cite web|author=Loftin, LK, Jr.|title=Quest for performance The evolution of modern aircraft. NASA SP-468|url=http//www.hq.nasa.gov/pao/History/SP-468/cover.htm|accessdate=2006-04-22}}</ref>" S143 Q328
Q7524238 P2534 " u(t) = \begin{cases} b, & \phi(x,\lambda,t)<0 \\ ?, & \phi(x,\lambda,t)=0 \\ a, & \phi(x,\lambda,t)>0.\end{cases}" S143 Q328
Q7449314 P2534 "A^T A = I \text{ or } A A^T = I. \, <ref > Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press.</ref>" S143 Q328
Q3773126 P2534 "\lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h}.<ref name="Mercer2014">{{cite book|author=Peter R. Mercer|title=More Calculus of a Single Variable|year=2014|publisher=Springer|isbn=978-1-4939-1926-0|page=173}}</ref><ref name="tp1">Thomson, p. 1</ref>" S143 Q328
Q18012454 P2534 "<math alt="rho-hat">\mathbf{\hat{\rho}}<sub>n</sub> is the respective unit vector in the direction of the position vector ρ (from observation point to orbiting body in Topocentric Equatorial Coordinate System)" S143 Q328
Q6047819 P2534 "\alpha = \frac{\mbox{number of de-excitations via electron emission}}{\mbox{number of de-excitations via gamma-ray emission}}" S143 Q328
Q90587 P2534 "\lambda = \frac{1}{(1.74+2 log(\frac{r}{k}))^2} <ref>[http//www.upi.com/Science_News/2006/01/31/A-73-year-old-experiment-yields-secrets/UPI-44631138742579/ A 73-year-old experiment yields secrets]. ''[[United Press International]]''. January 31, 2006. Accessed on January 13, 2011. This press release refers to this scientific article [http//web.mechse.illinois.edu/research/gioia/Art/gioia_Chakraborty_pipes_PRL.pdf ]</ref>" S143 Q328
Q388886 P2534 "\theta_0=\frac{1}{6\pi\epsilon_0}\frac{e^2}{m_0c^3}\ <ref>Farias & Recami, p.11. Caldirola's original paper has a different formula due to not working in standard units.</ref>" S143 Q328Q1332432 P2534 "\mathcal L = -\frac14F_{\mu\nu}^a F_{\mu\nu}^a" S143 Q328
Q1607213 P2534 " f( x; a,b ) = \frac{ 1 }{ x [ \log_e( b ) - \log_e( a ) ]} \quad \text{ for } a \le x \le b \text{ and } a > 0." S143 Q328
Q3010607 P2534 "\operatorname{recc}(A) = \{y \in X \forall x \in A, \forall \lambda \geq 0 x + \lambda y \in A\}.<ref>{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1}}</ref>" S143 Q328
Q2269249 P2534 "\sqrt{B}=\bigcap\{ P\subseteq R \mid B \subseteq P, P \mbox{ a prime ideal} \}\subseteq\{x\in R\mid x^n\in B \mbox{ for some }n\in\mathbb{N}^+ \} \," S143 Q328
Q597183 P2534 "\begin{alignat}{2}\text{Net Cash Flows from Financing Activities} = & \left[\text{Dividends received from }3^{{\rm rd}}\text{ parties}\right]\\ & -\left[\text{Dividends paid to }3^{{\rm rd}}\text{ parties}\right]\\ & - [\text{Dividends paid to NCI but not} \\ & \text{intracompany dividend payments}] \end{alignat}" S143 Q328
Q7269080 P2534 " \begin{bmatrix} S_{11} & \cdots & S_{1 n}\\ \vdots & \ddots & \vdots \\ S_{n 1} & \cdots & S_{n n}\end{bmatrix} " S143 Q328
Q4751110 P2534 "\beta_{E}(Q)=\frac{1}{\ell(Q)}\inf\{\delta\text{ there is a line }L\text{ so that for every }x\in E\cap Q, \; \text{dist}(x,L)<\delta\}," S143 Q328
Q5421524 P2534 "\mathcal{E}_{q}^{\epsilon}(d)=\,\! Choose r\,\! with probability proportional to e^{\epsilon q(d,r)}\times\mu(r)\,\!, where d\in\mathcal{D}^n,r\in R\,\!." S143 Q328
Q17086746 P2534 "H^{-i}(j_x^*C)\ne 0 or H^{i}(j_x^!C)\ne 0" S143 Q328
Q900231 P2534 "dU = \delta Q - \delta W.\; <ref>Freedman, Roger A., and Young, Hugh D. (2008). 12th Edition. Chapter 19 First Law of Thermodynamics, page 656. Pearson Addison-Wesley, San Francisco.</ref>" S143 Q328
Q17004900 P2534 " \text{Risk} = (\text{expected loss in case of the accident}) \times (\text{probability of the accident occurring})" S143 Q328
Q2088081 P2534 "\Delta\lambda = \lambda^{\mathrm{state 2}}_{\mathrm{observed}} - \lambda^{\mathrm{state 1}}_{\mathrm{observed}} where \lambda is the wavelength of the spectral peak of interest and \lambda^{\mathrm{state 2}}_{\mathrm{observed}} > \lambda^{\mathrm{state 1}}_{\mathrm{observed}}" S143 Q328
Q4086855 P2534 " \left( y' \right)^m = f(x,y),\, when m is a natural number (i.e., a [[positive integer]]), and f(x,y) is a [[polynomial]] in two variables (i.e., a bivariate polynomial)." S143 Q328
Q9300786 P2534 "F_D is the drag [[force]], which is by definition the force component in the direction of the flow velocity,<ref>See [[lift force]] and [[vortex induced vibration]] for a possible force components transverse to the flow direction.</ref>" S143 Q328
Q6667312 P2534 "\mathrm{CodeValue} = \sum_{i=1}^N s_i \cdot 2^{i-1} where s_i represents a value 0 or 1 depending on the state of the ''i<sup>th</sup>'' switch." S143 Q328
Q1058762 P2534 "M_1=MB*m \, definitional relationship between monetary base ''MB'' (bank reserves plus currency held by the non-bank public) the narrowly defined [[money supply]], M_1," S143 Q328
Q17080460 P2534 "\frac{R_{\text{ac}}}{\mu L} = a B_{\text{max}} f + c f + e f^2" S143 Q328
Q836475 P2534 "\forall C \biggl( \lnot \exist W \left( C \in W \right) \iff \exist F \Bigl( \forall x \bigl( \exist W \left( x \in W \right) \Rightarrow \exist s \left( s \in C \and \langle s, x \rangle \in F \right) \bigr) \and \forall x \forall y \forall s \bigl( \left( \langle s, x \rangle \in F \and \langle s, y \rangle \in F \right) \Rightarrow x = y \bigr) \Bigr) \biggr)." S143 Q328
Q5666472 P2534 " A \leftrightarrow \neg \neg A." S143 Q328
Q6889727 P2534 "\begin{vmatrix} x_1 & x_1^q & x_1^{q^2}\\x_2 & x_2^q & x_2^{q^2}\\x_3 & x_3^q & x_3^{q^2} \end{vmatrix}" S143 Q328
Q2383161 P2534 "\alpha = \frac{\mbox{number of de-excitations via electron emission}}{\mbox{number of de-excitations via gamma-ray emission}}" S143 Q328
Q247184 P2534 "\text{Surplus Transfer Value} = \left( {{\text{Total value of Candidate's votes} - \text{Quota}} \over \text{Total value of Candidate's votes}} \right)\times \text{Value of each vote}" S143 Q328
Q13440 P2534 "\frac{m_A}{m_P}=2.512^{(5.05-3.62)}=3.73</ref> Beginning in October of each year, Pleione along with the rest of the cluster can be seen rising in the east in the early morning before [[dawn]].<ref name="ESOPLEIADES">" S143 Q328
Q282453 P2534 " \vec{w}\cdot\vec{x} + b=0,\, [[FileSvm_max_sep_hyperplane_with_margin.png|right|thumb|Maximum-margin hyperplane and margins for an SVM trained with samples from two classes. Samples on the margin are called the support vectors.]]" S143 Q328
Q17006927 P2534 " C = \dfrac{\varepsilon_0 K A}{d} <ref name="Physics">{{cite book|isbn=0-87901-135-1|title=Physics Second Edition|author=Paul Allen Tipler|pages=653–660|publisher=Worth Publishers|year=1982}}</ref>" S143 Q328
Q1344747 P2534 " \text{gear inches} = \text{drive wheel diameter in inches}\times\frac{\text{number of teeth in front chainring}}{\text{number of teeth in rear sprocket}}" S143 Q328
Q5953313 P2534 " S_d [i_r, i_z] = \frac{S[i_r, i_z]}{N\Delta V(i_r)} where \Delta V is the grid volume and N is the number of photon packets." S143 Q328
Q214604 P2534 "f(x)=x^2-5x+6" S143 Q328
Q185969 P2534 "P(\theta)\propto{}\exp\left(-U(\theta)/kT\right), where U(\theta) is the potential determining the probability of each value of \theta." S143 Q328
Q900625 P2534 " \text{EQE} = \frac{\text{electrons/sec}}{\text{photons/sec}}= \frac{\text{current}/\text{(charge of one electron)}}{(\text{total power of photons})/(\text{energy of one photon})}" S143 Q328
Q5637579 P2534 "\begin{bmatrix}1 & 1 & 1 & 1\\1 & -1 & -1 & 1\\1 & 1 & -1 & -1\\1 & -1 & 1 & -1\end{bmatrix}\rightarrow\left[\begin{array}{c|ccc}1 & 1 & 1 & 1\\\hline0 & -2 & -2 & 0\\0 & 0 & -2 & -2\\0 & -2 & 0 & -2\end{array}\right]\rightarrow\begin{bmatrix}-2 & -2 & 0\\0 & -2 & -2\\-2 & 0 & -2\end{bmatrix}\rightarrow\begin{bmatrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{bmatrix}" S143 Q328
Q734439 P2534 "\text{LHA}_{\text{object}} = {\text{LST}} - \alpha_{\text{object}} (If result is negative, add 360 degrees. If result is greater than 360, subtract 360 degrees.)" S143 Q328
Q1915551 P2534 " \begin{bmatrix} V \\ F \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I \\ v \end{bmatrix} " S143 Q328
Q5514013 P2534 "Payout = Claim \times \frac {Sum\ Insured} {Current\ Value} \!<ref name="NCA">{{cite web |url=http//www.nca.ie/eng/Research_Zone/Reports/Home_Construction/NCA-Home-construction-Volume-4.pdf |title=The Home Construction Industry and the Consumer in Ireland, Volume 4 |author= Grant Thornton (Ireland) |date=2008-11 |page=27 |work=Review of insurance issues |publisher=National Consumer Agency |accessdate=2010-02-23}}</ref>" S143 Q328
Q7239587 P2534 "f(x,t)=\frac{1}{2\pi} \int \hat{\zeta}_0(\omega) \exp \left[-i\left(k(\omega)x-\omega t \right)\right] d\omega " S143 Q328Q1496380 P2534 "2p &nbsp; as the parameter (i.e., the latus rectum) of a body's orbit," S143 Q328
Q913837 P2534 "M=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}, " S143 Q328
Q185359 P2534 "A \cup B = \{ x x \in A \text{ or } x \in B\}.<ref name="0">{{Cite book|url=https//books.google.com/books?id=LBvpfEMhurwC|title=Basic Set Theory|last=Vereshchagin|first=Nikolai Konstantinovich|last2=Shen|first2=Alexander|date=2002-01-01|publisher=American Mathematical Soc.|isbn=9780821827314|language=en}}</ref>" S143 Q328
Q7300313 P2534 "I'_R = I_R x 10^{-at} (Hull, 1951, p. 74).</blockquote>Hull's decay formula is somewhat awkward and might give rise to confusion. For example, ''I'''<sub>''R''</sub> does not refer to the derivative of ''I''<sub>''R''</sub>. A more convenient way of writing the formula would be as follows" S143 Q328
Q2246795 P2534 " s = 2 \sinh \left( \frac{1}{2} d \right) = \sqrt{2 (\cosh d -1) } where ''d'' is the distance between the two points, and sinh and cosh are [[hyperbolic functions]].<ref>{{cite book|last1=Smogorzhevsky|title=Lobachevskian Geometry|date=1976|publisher=Mir|location=Moscow|page=65}}</ref>" S143 Q328
Q176645 P2534 "\Pr(X_{n+1}=x\mid X_1=x_1, X_2=x_2, \ldots, X_n=x_n) = \Pr(X_{n+1}=x\mid X_n=x_n), if both [[conditional probability|conditional probabilities]] are well defined, i.e. if \Pr(X_1=x_1,...,X_n=x_n)>0." S143 Q328
Q165474 P2534 "x * y = y * x\qquad\mbox{for all }x,y\in S" S143 Q328
Q7100757 P2534 "x=[0,1,2]\," S143 Q328
Q215579 P2534 " \ddot{a}_{\overline{n|}i}=a_{\overline{n}|i}(1 + i)=a_{\overline{n-1|}i}+1 (value at the time of the first of ''n'' payments of 1)" S143 Q328
Q6520159 P2534 "\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x)\ e^{- 2\pi i x \xi}\,dx, &nbsp; for any [[real number]] ''[[Xi (letter)|ξ]]''." S143 Q328
Q1163221 P2534 "\mathrm{ROA} = \frac{\mbox{Net Income}}{\mbox{Average Total Assets}}<ref name="isbn0-618-73661-1">{{cite book |author1=Susan V. Crosson |author2=Belverd E., Jr Needles |author3=Needles, Belverd E. |author4=Powers, Marian |title=Principles of accounting |publisher=Houghton Mifflin |location=Boston |year=2008 |page=209 |isbn=0-618-73661-1}}</ref>" S143 Q328
Q17083884 P2534 "g_m = {\mu \over r_p}<ref>van der Bijl, H. J. (1919). Theory and Operating Characteristics of the Thermionic" S143 Q328Q17093018 P2534 "\hat{y} = \sgn(\mathbf{w}^\top \mathbf{x})" S143 Q328
Q12013681 P2534 "(A,\mathfrak{m}_A) dominates (B,\mathfrak{m}_B) if A \supset B and \mathfrak{m}_A \cap B = \mathfrak{m}_B.<ref>Efrat (2006) p.55</ref>" S143 Q328
Q7138412 P2534 "P_k = \frac{k}{|G|} \sum_{g \in G} c_k(g) where ''c''<sub>''k''</sub>(''g'') is the number of ''k''-cycles in the cycle decomposition of ''g''." S143 Q328
Q6824327 P2534 "\frac{\text{new tempo}}{\text{old tempo}} = \frac{\text{number of pivot note values in new measure}}{\text{number of pivot note values in old measure}}" S143 Q328
Q471120 P2534 "\Delta\lambda = \lambda^{\mathrm{state 1}}_{\mathrm{observed}} - \lambda^{\mathrm{state 2}}_{\mathrm{observed}} where \lambda is the wavelength of the spectral peak of interest and \lambda^{\mathrm{state 1}}_{\mathrm{observed}} > \lambda^{\mathrm{state 2}}_{\mathrm{observed}}" S143 Q328
Q25109770 P2534 "y_{it} = X_{it}\mathbf{\beta}+\alpha_i+u_{it} for t=1,\ldots,T and i=1,\ldots,N" S143 Q328
Q6365582 P2534 "R = \frac {\text{Isentropic enthalpy change in rotor}}{\text{Isentropic enthalpy change in stage}} .<ref>Peng, William W., Fundamentals of turbomachinery, John Wiley, 2008</ref>" S143 Q328
Q1128678 P2534 "i \colon M \rightarrow K" S143 Q328
Q327408 P2534 "\begin{bmatrix} R \\ G \\ B \end{bmatrix} = \begin{bmatrix} 3.1956 & 2.4478 & -0.1434 \\ -2.5455 & 7.0492 & 0.9963 \\ 0.0000 & 0.0000 & 1.0000 \end{bmatrix} \begin{bmatrix} X \\ Y \\ Z \end{bmatrix}." S143 Q328
Q2845279 P2534 "d\phi = \frac{\part \phi}{\part p} dp + \frac{\part \phi}{\part q} dq . The slow uptake of the continental methods in calculus led to the formation of the Analytical Society by [[Charles Babbage]], [[John Herschel]] and [[George Peacock]].<ref name=Boyer/>" S143 Q328
Q536400 P2534 " u_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<ref>{{cite web |url=http//mathworld.wolfram.com/BinetsFibonacciNumberFormula.html |title=Binet's Fibonacci Number Formula |author=Weisstein, Eric W |publisher=From MathWorld—A Wolfram Web Resource |accessdate=10 January 2011}}</ref>" S143 Q328
Q7767516 P2534 "{F}=\frac{Q_1Q_2}{4\pi\mathrm{D}^2\varepsilon_0\varepsilon_r}<ref name="Butt">Butt, Hans, Kh. Graf, and Michael Kappl. Physics and chemistry of interfaces. 2nd., rev. and enl. ed. Weinheim Wiley-VCH, 2006. Print.</ref>" S143 Q328
Q84052 P2534 " \sqrt{a^2+r} \approx a + \frac{r}{2a} - \frac{(r/2a)^2}{2(a+\frac{r}{2a})}, with a = 4/3 and r = 2/9 <ref name="cooke 200">{{cite book |last=Cooke |authorlink=Roger Cooke |title= |year=1997 |chapter=The Mathematics of the Hindus |pages=200 | quote = The Hindus had a very good system of approximating irrational square roots. Three of the ''Sulva Sutras'' contain the approximation 1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot4 \cdot 34} for the diagonal of a square of side 1 (that is \sqrt{2}). [...] We can only conjecture how such an approximation was obtained. One guess is the approximation \sqrt{a^2+r} = a + \frac{r}{2a} - \frac{(r/2a)^2}{2(a+\frac{r}{2a})} with a = 4/3 and r = 2/9. This approximation follows a rule given by the twelfth century Muslim mathematician Al-Hassar.}}</ref>" S143 Q328
Q25303870 P2534 "(\forall x)(x\in A \to \mathcal{P}(x) \subseteq A).<ref>See Definition 9.8 of {{cite book|last1=Zaring W.M.|first1= G. Takeuti|title=Introduction to axiomatic set theory|date=1971|publisher=Springer-Verlag|location=New York|isbn=0387900241|edition=2nd, rev.}}</ref>" S143 Q328
Q592634 P2534 "m_\text{P}=\sqrt{\frac{\hbar c}{G}} ≈ {{val|1.220910|e=19|u=[[GeV]]/c<sup>2</sup>}} = {{val|2.176470|(51)|e=-8|u=kg}} = {{val|21.76470|u=[[Microgram|μg]]}} = {{val|1.3107|e=19|u=[[atomic mass unit|amu]]}},<ref>CODATA 2016 [http//physics.nist.gov/cgi-bin/cuu/Value?plkmc2gev value in GeV], [http//physics.nist.gov/cgi-bin/cuu/Value?plkm value in kg]</ref>" S143 Q328
Q2995290 P2534 "\frac{L}{L_{\odot}} \sim {\left ( \frac{M}{M_{\odot}} \right )}^{3.9} What is the ~ mean in this case? This is a formula taken in [[luminosity]] article.[[UserPendragon5|Pendragon5]] ([[User talkPendragon5|talk]]) 1909, 12 February 2012 (UTC)" S143 Q328
Q223325 P2534 " \vec F_g = - \hat r ~ G ~ \frac{M m}{R^2}" S143 Q328
Q6038052 P2534 "\beta_j(p,q) = P(w_{pq}|N^j_{pq}, G)" S143 Q328Q6322827 P2534 "i=\text{number of invites sent by each customer } (e.g. if each new customer invites five friends, ''i'' = 5)" S143 Q328
Q1224527 P2534 "J = \frac{q D n e^{-\Phi / V_t}\big|_0^{x_d}}{\int_0^{x_d} e^{-\Phi / V_t}dx}" S143 Q328
Q5439524 P2534 "xp_x+yp_y \leq I" S143 Q328
Q7661853 P2534 "\mathcal{L}_X\omega=0.<ref name="CdS">{{Citation|title=Lectures on Symplectic Geometry|series=Lecture Notes in Mathematics|volume=1764|first=Ana|last=Cannas da Silva|publisher=Springer-Verlag|year=2001|isbn=978-3-540-42195-5|page=106}}.</ref>" S143 Q328
Q2914865 P2534 " \text{PPV} = \frac{\text{number of true positives}}{\text{number of true positives}+\text{number of false positives}} = \frac{\text{number of true positives}}{\text{number of positive calls}}" S143 Q328
Q179692 P2534 "\forall X \left[ \emptyset \notin X \implies \exists f \colon X \rightarrow \bigcup X \quad \forall A \in X \, ( f(A) \in A ) \right] \,." S143 Q328
Q18393178 P2534 "P^{(n)} = P^n R_P \cap R = \{ f \in R \mid sf \in P^n \text{ for some }s \in R - P \}.<ref>Here, by abuse of notation, we write I \cap R to mean the pre-image of ''I'' along the localization map R \to R_P.</ref>" S143 Q328