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searching for DFRAC 545 found (576 total)

alternate case: dFRAC

Jacobian matrix and determinant (3,549 words) [view diff] case mismatch in snippet view article find links to article

{\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial

_{f}={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial

F(x)=\operatorname {sgn}(x){\begin{cases}{\dfrac {A|x|}{1+\ln(A)}},&|x|<{\dfrac {1}{A}},\\[1ex]{\dfrac {1+\ln(A|x|)}{1+\ln(A)}},&{\dfrac {1}{A}}\leq |x|\leq 1,\end{cases}}}

_{x}={\dfrac {\partial \mathbf {r} }{\partial x}};\;\mathbf {e} _{y}={\dfrac {\partial \mathbf {r} }{\partial y}};\;\mathbf {e} _{z}={\dfrac {\partial

}{\text{kg}}}}={\dfrac {\text{C}}{\text{V}}}={\dfrac {{\text{A}}{\cdot }{\text{s}}}{\text{V}}}={\dfrac {{\text{W}}{\cdot }{\text{s}}}{{\text{V}}^{2}}}={\dfrac

_{i}={\begin{bmatrix}{\dfrac {\partial A_{11}}{\partial x_{1}}}+{\dfrac {\partial A_{12}}{\partial x_{2}}}+{\dfrac {\partial A_{13}}{\partial x_{3}}}\\{\dfrac {\partial

32 , ⋯ {\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color

}+K_{\mathrm {mA} }b+(K_{\mathrm {mB} }+b)a}}={\dfrac {{\dfrac {Vb}{K_{\mathrm {mB} }+b}}\cdot a}{{\dfrac {K_{\mathrm {iA} }K_{\mathrm {mB} }+K_{\mathrm

dfrac {\text{J}}{{\text{A}}^{2}}}={\dfrac {{\text{T}}{\cdot }{\text{m}}^{2}}{\text{A}}}={\dfrac {\text{Wb}}{\text{A}}}={\dfrac {{\text{V}}{\cdot

{\begin{bmatrix}{\dfrac {\mu }{\sigma ^{2}}}\\[10pt]-{\dfrac {1}{2\sigma ^{2}}}\end{bmatrix}}} [ − η 1 2 η 2 − 1 2 η 2 ] {\displaystyle {\begin{bmatrix}-{\dfrac {\eta

− p d ρ v 2 {\displaystyle \mathrm {Eu} ={\dfrac {\mbox{pressure forces}}{\mbox{inertial forces}}}={\dfrac {\mbox{(pressure)(area)}}{\mbox{(mass)(acc

≤ 1 , {\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2px-x^{2}\right),&0\leq x\leq p,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2px-x^{2}\right)

)\\[10pt]&={\dfrac {\kappa \lambda }{\kappa ^{2}+1}}{\begin{cases}{\dfrac {e^{-(\theta -m)\lambda \kappa }}{1-e^{-2\pi \lambda \kappa }}}-{\dfrac {e^{(\theta

{\displaystyle {\dfrac {{\dfrac {{\dfrac {{\dfrac {{\dfrac {\text{the}}{NP/N}}{\dfrac {\text{dog}}{N}}\qquad }{NP}}>}{S/(S\backslash NP)}}T_{>}\qquad {\dfrac

\Omega ={\dfrac {\text{V}}{\text{A}}}={\dfrac {1}{\text{S}}}={\dfrac {\text{W}}{{\text{A}}^{2}}}={\dfrac {{\text{V}}^{2}}{\text{W}}}={\dfrac

Lorentz force law. That is, T = N ⋅ s C ⋅ m . {\displaystyle \mathrm {T={\dfrac {N{\cdot }s}{C{\cdot }m}}} .} As an SI derived unit, the tesla can also

π 2 ≡ {\displaystyle \lambda ={\dfrac {\mu }{\rho }}{\sqrt {\dfrac {\pi }{2R\theta }}}={\dfrac {M}{Re}}{\sqrt {\dfrac {k\pi }{2}}}\equiv } mean free path

{\begin{array}{crclcl}&{\dfrac {H}{D/2}}&=&{\dfrac {x}{c/2}}\\\therefore &H&=&{\dfrac {Dx}{c}}&=&{\dfrac {D}{c}}{\Big (}f+{\dfrac {cf}{D}}{\Big )}\\&&=&{\dfrac {Df}{c}}+f&=&{\dfrac

={\dfrac {\partial Q/Q}{\partial P/P}}={\dfrac {\partial Q}{\partial P}}{\dfrac {P}{Q}}=-{\dfrac {1}{2}}\left(-2+{\dfrac {4}{Q}}\right)=1-{\dfrac {4}{2Q}}=1-{\dfrac

=-G\int \left[{\dfrac {[\rho ]}{r^{3}}}+{\dfrac {1}{r^{2}c}}\left[{\dfrac {\partial \rho }{\partial t}}\right]\right]\mathbf {(} r)dV'+{\dfrac {G}{c^{2}}}\int

{\mathcal {F}}}{\dfrac {1}{n}}\left|\sum _{i=1}^{n}f(X_{i})-\mathbb {E} f(Y_{i})\right|\leq \mathbb {E} _{Y}\sup _{f\in {\mathcal {F}}}{\dfrac {1}{n}}\left|\sum

{\begin{aligned}Q_{c}&={\dfrac {1}{2m}}\sum _{i}\sum _{j}A_{ij}\mathbf {1} \left\{c_{i}=c_{j}=c\right\}-\left(\sum _{i}{\dfrac {k_{i}}{2m}}\mathbf {1}

_{z}={\dfrac {F}{A}}={\dfrac {Pd^{2}}{(d+2t)^{2}-d^{2}}}\ } Though this may be approximated to σ z = P r 2 t   {\displaystyle \sigma _{z}={\dfrac {Pr}{2t}}\

{\displaystyle \operatorname {Ai} (x)={\dfrac {1}{\pi }}\int _{0}^{\infty }\cos \left({\dfrac {t^{3}}{3}}+xt\right)\,dt\equiv {\dfrac {1}{\pi }}\lim _{b\to \infty

Rebounds ) {\displaystyle {\text{Rebound Rate}}={\dfrac {100\times {\text{Rebounds}}\times {\dfrac {\text{Team Minutes Played}}{5}}}{{\text{Minutes Played}}\times

2 Γ 3 ⟨ ε ⟩ 2 {\displaystyle {\dfrac {d\left\langle \varepsilon ^{2}\right\rangle }{dt}}=lim_{dt\rightarrow 0}{\dfrac {\left\langle \varepsilon ^{2}(t+dt)\right\rangle

_{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac

diffusive transport rate, of source of gradient {\displaystyle \mathrm {Ma} ={\dfrac {\mbox{advective transport rate, due to surface tension gradient}}{\mbox{diffusive

= m U 2 k L 2 = U 2 ω 0 2 L 2 {\displaystyle {\dfrac {mU^{2}}{FL}}={\dfrac {mU^{2}}{kL^{2}}}={\dfrac {U^{2}}{\omega _{0}^{2}L^{2}}}} , where, m = mass

Lorenz gauge: ◻ φ = ρ ϵ 0 , ◻ A = μ 0 J {\displaystyle \Box \varphi ={\dfrac {\rho }{\epsilon _{0}}}\,,\quad \Box \mathbf {A} =\mu _{0}\mathbf {J} }

+ C {\displaystyle \int \sec \theta \,d\theta ={\begin{cases}{\dfrac {1}{2}}\ln {\dfrac {1+\sin \theta }{1-\sin \theta }}+C\\[15mu]\ln {{\bigl |}\sec \theta

{\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial

x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}} Imaginary part: y = Im ⁡ ( z ) = z − z ¯ 2 i {\displaystyle y=\operatorname {Im} (z)={\dfrac {z-{\overline

{\displaystyle MPK={\dfrac {\partial Y}{\partial K}}=\alpha AL^{\beta }K^{\alpha -1}=\alpha {\dfrac {AL^{\beta }K^{\alpha }}{K}}=\alpha {\dfrac {Y}{K}}} Because

= 72 30 = 12 5 {\displaystyle {\dfrac {\sigma (30)}{30}}={\dfrac {1+2+3+5+6+10+15+30}{30}}={\dfrac {72}{30}}={\dfrac {12}{5}}} σ ( 140 ) 140 = 1 + 2 +

u={\dfrac {\text{A}}{{\dfrac {\text{electron}}{{\text{m}}^{3}}}{\cdot }{\text{m}}^{2}\cdot {\dfrac {\text{C}}{\text{electron}}}}}={\dfrac {\dfrac

Euler-Bernoulli beam theory ± E I d 2 w d x 2 = M {\displaystyle \pm EI{\dfrac {d^{2}w}{dx^{2}}}=M} Where w {\displaystyle w} is the deflection and M {\displaystyle

dfrac {V_{\text{s3}}}{Z_{\text{s3}}}}\right)Z_{\text{s1Y}}\\[2ex]&V_{\text{s2Y}}=\left({\dfrac {V_{\text{s2}}}{Z_{\text{s2}}}}-{\dfrac

_{\text{c}}'}{i\omega }}\to {\dfrac {1}{Q_{1}\left({\dfrac {i\omega }{\omega _{01}}}+{\dfrac {\omega _{01}}{i\omega }}\right)}}+{\dfrac {1}{Q_{2}\left({\dfrac {i\omega }{\omega

product and γ = 1 1 − | v ( t ) | 2 c 2 {\displaystyle \gamma ={\dfrac {1}{\sqrt {1-{\dfrac {|\mathbf {v} (t)|^{2}}{c^{2}}}}}}} is the instantaneous Lorentz

{\displaystyle {\dfrac {1}{p}}={\dfrac {j}{n}}+\theta \left({\dfrac {1}{r}}-{\dfrac {m}{n}}\right)+{\dfrac {1-\theta }{q}},\qquad {\dfrac {j}{m}}\leq \theta

}} is defined as c d = 2 F d ρ u 2 A {\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho u^{2}A}}} where: F d {\displaystyle F_{\mathrm

{\displaystyle \left({\dfrac {p-1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{-3/2}\cdot

{\displaystyle a_{mn}={\begin{cases}1&{\text{if }}n=0,\\\\{\dfrac {1}{n!}}&{\text{if }}m=1,\\\\{\dfrac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1

square numbers behaves asymptotically as b x log ⁡ ( x ) . {\displaystyle {\dfrac {bx}{\sqrt {\log(x)}}}.} This constant b was rediscovered in 1913 by Srinivasa

_{m}}}&={\begin{bmatrix}{\dfrac {\partial \mu _{1}}{\partial \theta _{m}}}&{\dfrac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\dfrac {\partial \mu _{N}}{\partial

{H}}_{D}={}&2g_{\text{I}}\mu _{\text{N}}\mu _{\text{B}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {1}{L_{z}}}\sum _{i}{\dfrac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {I}

dfrac {d}{dx}}{\dfrac {1}{x}}\\={\dfrac {d}{d}}{\dfrac {1}{x^{2}}}\\={\dfrac {d\!\!\!\backslash }{d\!\!\!\backslash }}{\dfrac {1}{x^{2}}}\\=-{\dfrac

b a + b , {\displaystyle a\parallel b\mathrel {:=} {\frac {1}{{\dfrac {1}{a}}+{\dfrac {1}{b}}}}={\frac {ab}{a+b}},} where a, b, and a ∥ b {\displaystyle

{\displaystyle T(z)={\dfrac {z-1}{z}}Z[Y(s)]} T ( z ) = z − 1 z Z [ T ( s ) s ] {\displaystyle T(z)={\dfrac {z-1}{z}}Z[{\dfrac {T(s)}{s}}]} The bilinear

_{m}}}&={\begin{bmatrix}{\dfrac {\partial \mu _{1}}{\partial \theta _{m}}}&{\dfrac {\partial \mu _{2}}{\partial \theta _{m}}}&\cdots &{\dfrac {\partial \mu _{N}}{\partial

{\displaystyle T(z)={\dfrac {z-1}{z}}Z[Y(s)]} T ( z ) = z − 1 z Z [ T ( s ) s ] {\displaystyle T(z)={\dfrac {z-1}{z}}Z[{\dfrac {T(s)}{s}}]} The bilinear

{\;\cot {\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}-\cot ^{2\!}{\dfrac {\pi }{\sqrt {2\,}}}\cdot \coth {\dfrac {\pi }{\sqrt

the single thread performance, ( 1 1 − p = 20 ) {\displaystyle \left({\dfrac {1}{1-p}}=20\right)} . Amdahl's law can be formulated in the following way:

{\begin{bmatrix}{\dfrac {d\beta }{dt}}&{\dfrac {d^{2}\beta }{dt^{2}}}\end{bmatrix}}&=\det {\begin{bmatrix}{\dfrac {d\beta }{ds}}{\dfrac {ds}{dt}}&\left({\dfrac {d^{2}\beta

2 ) sin ⁡ ( A 2 ) {\displaystyle n_{21}={\dfrac {\sin \left({\dfrac {A+D_{m}}{2}}\right)}{\sin \left({\dfrac {A}{2}}\right)}}} This is useful to calculate

{\displaystyle {\dfrac {1}{r}}{\dfrac {\partial }{\partial r}}\left(r{\dfrac {\partial G}{\partial r}}\right)+{\dfrac {1}{\alpha }}\delta (t-\tau ){\dfrac {\delta

) ⋅ d X d s d s , {\displaystyle Tw={\dfrac {1}{2\pi }}\int \left(U\times {\dfrac {dU}{ds}}\right)\cdot {\dfrac {dX}{ds}}ds\;,} where d X / d s {\displaystyle

τ 2 2 {\displaystyle T(Y)={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}\leq {\dfrac {1}{2N}}\sum _{i,j}(N\tau )^{2}={\dfrac {N^{3}\tau ^{2}}{2}}\,\!} The

_{p+1}\cdots \alpha _{q}}={\dfrac {\partial }{\partial x^{\alpha _{q}}}}\cdots {\dfrac {\partial }{\partial x^{\alpha _{p+2}}}}{\dfrac {\partial }{\partial x^{\alpha

dfrac {\Delta t}{T_{i}}}+{\dfrac {T_{d}}{\Delta t}}\right)e(t_{k})+\left(-1-{\dfrac {2T_{d}}{\Delta t}}\right)e(t_{k-1})+{\dfrac {T_{d}}{\Delta

{\left({\dfrac {s_{1}^{2}}{n_{1}}}+{\dfrac {s_{2}^{2}}{n_{2}}}\right)^{2}}{{\dfrac {\left({\dfrac {s_{1}^{2}}{n_{1}}}\right)^{2}}{n_{1}-1}}+{\dfrac {\left({\dfrac

F'(x)\approx {\dfrac {F(x+h)-F(x)}{h}}={\dfrac {G(x)-F(x)}{h}}\,} so that h ≈ G ( x ) − F ( x ) F ′ ( x ) {\displaystyle h\approx {\dfrac {G(x)-F(x)}{F'(x)}}\

{\displaystyle ={\dfrac {5}{{\sqrt {3}}+4}}\times {\dfrac {{\sqrt {3}}-4}{{\sqrt {3}}-4}}} = 5 ( 3 − 4 ) ( 3 + 4 ) ( 3 − 4 ) {\displaystyle ={\dfrac {5({\sqrt

{\displaystyle {\dfrac {PR}{RB'}}={\dfrac {SQ}{B'S}},} From (1), (2), and (3), Δ B − Δ A + P A ′ L 1 = Δ C − Δ B − Q C ′ L 2 {\displaystyle {\dfrac {\Delta B-\Delta

dfrac {d}{dx}}{\dfrac {1}{x}}\\={\dfrac {d}{d}}{\dfrac {1}{x^{2}}}\\={\dfrac {d\!\!\!\backslash }{d\!\!\!\backslash }}{\dfrac {1}{x^{2}}}\\=-{\dfrac

( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}} G ( s ) {\displaystyle G(s)} is called the

{\displaystyle \mathrm {Wb} =\Omega {\cdot }{\text{C}}={\dfrac {\mathrm {J} }{\mathrm {A} }}={\dfrac {\mathrm {N} {\cdot }\mathrm {m} }{\mathrm {A} }},} where

function in t {\displaystyle t} with the period T = 2 π ω {\displaystyle T={\dfrac {2\pi }{\omega }}} . If ϵ = 0 {\displaystyle \epsilon =0} , then there is

dfrac {\partial u}{\partial u'}}&{\dfrac {\partial u}{\partial v'}}\\{\dfrac {\partial v}{\partial u'}}&{\dfrac {\partial v}{\partial

{\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial

{dx}{xS}}={\begin{cases}-{\dfrac {2}{\sqrt {b}}}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\-{\dfrac {2}{\sqrt

criterion is N W − P = R R p 2 C s D e f f ≤ 3 β {\displaystyle N_{W-P}={\dfrac {{\mathfrak {R}}R_{p}^{2}}{C_{s}D_{eff}}}\leq 3\beta } Where R {\displaystyle

{e_{1}}{x}}={\frac {-{\dfrac {\partial v_{1}}{\partial e_{1}}}{\dfrac {e_{1}}{v_{1}}}}{{\dfrac {\partial v_{2}}{\partial x}}{\dfrac {x}{v_{2}}}-{\dfrac {\partial v_{1}}{\partial

y ( γ − δ x ) , {\displaystyle {\begin{array}{rcl}{\dfrac {dx}{dt}}&=&x(\alpha -\beta y)\\{\dfrac {dy}{dt}}&=&-y(\gamma -\delta x),\end{array}}} where

I − 1 N ) 1 − 1 N {\textstyle HHI^{*}={\cfrac {\left(HHI-{\dfrac {1}{N}}\right)}{1-{\dfrac {1}{N}}}}} for N > 1 and H H I ∗ = 1 {\displaystyle HHI^{*}=1}

t)={\dfrac {1}{4\pi \varepsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\\&\mathbf {A} (\mathbf {r} ,t)={\dfrac {\mu

u s ¯ r − s | ≤ π ∑ r | u r | 2 . {\displaystyle \left|\sum _{r\neq s}{\dfrac {u_{r}{\overline {u_{s}}}}{r-s}}\right|\leq \pi \displaystyle \sum _{r}|u_{r}|^{2}

before skip also holds true afterwards. { P } skip { P } {\displaystyle {\dfrac {}{\{P\}{\texttt {skip}}\{P\}}}} The assignment axiom states that, after

\mathbf {v} =\left({\dfrac {\partial v_{z}}{\partial y}}-{\dfrac {\partial v_{y}}{\partial z}},{\dfrac {\partial v_{x}}{\partial z}}-{\dfrac {\partial v_{z}}{\partial

{\displaystyle u{\dfrac {\partial u}{\partial x}}+v{\dfrac {\partial u}{\partial y}}=-{\dfrac {1}{\rho }}{\dfrac {\partial p}{\partial x}}+{\nu }{\dfrac {\partial

{(-1)^{r-1}\pi a^{m+1-nr}\Gamma \left({\dfrac {m+1}{n}}\right)}{n\sin \left({\dfrac {m+1}{n}}\pi \right)(r-1)!\,\Gamma \left({\dfrac {m+1}{n}}-r+1\right)}}\quad {\text{for

x | C i | {\displaystyle \Delta _{i}={\dfrac {{\underset {x\in C_{i}}{\sum }}d(x,\mu )}{|C_{i}|}},\mu ={\dfrac {{\underset {x\in C_{i}}{\sum }}x}{|C_{i}|}}}

{\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)} where the step size h>0 and where the sinc function is

00 {\displaystyle {{\rm {SE}}={\sqrt {{\dfrac {1}{n_{11}}}+{\dfrac {1}{n_{10}}}+{\dfrac {1}{n_{01}}}+{\dfrac {1}{n_{00}}}}}}} . This is an asymptotic

=3\\&\#\mathbf {A} =0&\#\mathbf {P} =1\end{Bmatrix}}&={\dfrac {13-1}{9}}&={\dfrac {12}{9}}&={\dfrac {4}{3}}\\\\0.3789789\ldots &=0.3{\overline

dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial

-\left[{\dfrac {(\mathbf {y} -\mathbf {p} )\cdot \mathbf {a} }{\mathbf {a} \cdot \mathbf {a} }}\right]\mathbf {a} =\mathbf {y} -\left[{\dfrac {\mathbf

{\dfrac {(2\ell +1)\pi r}{m}}=-{\frac {\pi }{m}}\sum _{r=1}^{m-1}{\frac {r\cdot \sin {\dfrac {2\pi r}{m}}}{\cos {\dfrac {2\pi r}{m}}-\cos {\dfrac {(2\ell

1 2 ∫ R 2 J ( x ) K − 1 ( x ; y ) J ( y ) d x d y } . {\displaystyle {\dfrac {W[J]}{W[0]}}=\exp \left\lbrace {\frac {1}{2}}\int _{\mathbb {R} ^{2}}J(x)K^{-1}(x;y)J(y)\

)&={\dfrac {1}{4\pi ^{2}}}\iint {\dfrac {A_{h}(-\theta ,\tau )}{\phi (\theta ,\tau )}}e^{j\theta t+j\tau \omega }\,d\theta \,d\tau \\&={\dfrac {1}{4\pi

{\partial H}{\partial p_{i}}}\quad {\text{and}}\quad {\dfrac {{\text{d}}p_{i}}{{\text{d}}t}}=-{\dfrac {\partial H}{\partial x_{i}}}} where x i {\displaystyle

the relation ε 0 μ 0 = 1 c 2 {\displaystyle \varepsilon _{0}\mu _{0}={\dfrac {1}{c^{2}}}} is also used. Maxwell's equations can directly give inhomogeneous

_{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}} and

y)={\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\[5pt]{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial

{\displaystyle {\dfrac {dN}{dt}}={\dfrac {\sqrt {3}}{2\pi }}{\dfrac {q^{2}mc}{\hbar ^{2}}}{\dfrac {\chi }{\gamma }}\int _{0}^{\chi }{\dfrac {F(\chi ,\eta

dfrac {Z_{0}}{2\pi {\sqrt {\varepsilon _{\textrm {eff}}}}}}\ln \left(8{\dfrac {h}{w}}+{\dfrac {w}{4h}}\right),&{\text{when }}{\dfrac {w}{h}}\leq

\beta }{\Big (}y(t){\Big )}={\dfrac {M(\alpha )}{1-\alpha }}{\dfrac {d}{dt^{\beta }}}\int _{0}^{t}\exp {\Big (}-{\dfrac {\alpha }{1-\alpha }}(t-s){\Big

\left({\dfrac {\log n}{\epsilon ^{2}}}\right)} the probability that no bucket has more than ( 1 + ϵ ) ⋅ n p {\displaystyle (1+\epsilon )\cdot {\dfrac {n}{p}}}

dfrac {x}{r}}&{\dfrac {y}{r}}&{\dfrac {z}{r}}\\\\{\dfrac {xz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {yz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac

}}\\{\dfrac {1}{T_{s}}}{\dfrac {\sin \left[\pi {\dfrac {t}{T_{s}}}\left(1-\beta \right)\right]+4\beta {\dfrac {t}{T_{s}}}\cos \left[\pi {\dfrac {t}{T_{s}}}\left(1+\beta

_{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}} and

y y τ x y = 2 λ μ γ ˙ 2 μ γ ˙ = 2 γ ˙ λ . {\displaystyle {\text{Wi}}={\dfrac {\mbox{elastic forces}}{\mbox{viscous forces}}}={\frac {\tau _{xx}-\tau

{\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial u}}&{\dfrac {\partial f_{1}}{\partial v}}\\[6pt]{\dfrac {\partial f_{2}}{\partial u}}&{\dfrac {\partial f_{2}}{\partial

civilian population {\displaystyle {\text{Labor force participation rate}}={\dfrac {\text{Labor force}}{\text{Noninstitutional civilian population}}}} . Instead

y f ( f − 1 ( y ) ) = d d y y {\displaystyle {\dfrac {\mathrm {d} }{\mathrm {d} y}}f(f^{-1}(y))={\dfrac {\mathrm {d} }{\mathrm {d} y}}y} The right side

{\displaystyle u{\dfrac {\partial u}{\partial x}}+v{\dfrac {\partial u}{\partial y}}=-{\dfrac {1}{\rho }}{\dfrac {\partial p}{\partial x}}+{\nu }{\dfrac {\partial

\delta =a(a-1){\dfrac {\partial }{\partial a}}+(l+b){\dfrac {\partial }{\partial b}}+(2ac-c){\dfrac {\partial }{\partial c}}+(-aP+P){\dfrac {\partial }{\partial

y)&={\dfrac {1}{1-xyF(x,xy)}}&={\dfrac {1}{1-{\dfrac {xy}{1-x^{2}yF(x,x^{2}y)}}}}&=\cdots &={\dfrac {1}{1-{\dfrac {xy}{1-{\dfrac {x^{2}y}{1-{\dfrac {x^{3}y}{\cdots

{1}{2}}\left({\dfrac {v}{c}}\right)^{2}+{\dfrac {3}{8}}\left({\dfrac {v}{c}}\right)^{4}+{\dfrac {5}{16}}\left({\dfrac {v}{c}}\right)^{6}+\cdots

value of t is assumed of the order O ( 1 ε 2 ) {\displaystyle O\left({\dfrac {1}{\varepsilon _{2}}}\right)} (i.e. less approximation-value ε requires

\beta }{\Big (}y(t){\Big )}={\dfrac {M(\alpha )}{1-\alpha }}{\dfrac {d}{dt^{\beta }}}\int _{0}^{t}\exp {\Big (}-{\dfrac {\alpha }{1-\alpha }}(t-s){\Big

) 2 , {\displaystyle SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {1}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal

as, L = − u ∗ 3 κ g T Q ρ c p {\displaystyle L=-{\dfrac {u_{*}^{3}}{\kappa {\dfrac {g}{T}}{\dfrac {Q}{\rho c_{p}}}}}} where κ ≈ 0.40 {\displaystyle \kappa

w ) = 2 ∂ ∂ z J v ( z , w ) {\displaystyle J_{v-1}(z,w)-J_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}J_{v}(z,w)} Y v − 1 ( z , w ) − Y v + 1 ( z , w

{\displaystyle {\begin{pmatrix}{\dfrac {\partial \Phi }{\partial {x}}}\\[2ex]{\dfrac {\partial \Phi }{\partial {y}}}\\[2ex]{\dfrac {\partial \Phi }{\partial

a ) . {\displaystyle \quad N(r,a,f)=N\left(r,{\dfrac {1}{f-a}}\right),\quad m(r,a,f)=m\left(r,{\dfrac {1}{f-a}}\right).\,} For a = ∞, we set N(r,∞,f) = N(r

={\frac {{\dfrac {\partial u_{y}}{\partial x}}dx}{dx+{\dfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\dfrac {\partial u_{y}}{\partial x}}{1+{\dfrac {\partial

({\dfrac {1}{2}}).\alpha N\delta \left(({\dfrac {\delta }{2}})-({\dfrac {\lambda }{d}})\right)=\left(({\dfrac {1}{4}}).\alpha N(\delta ^{2}-O({\dfrac {\lambda

ln ⁡ y i , if  λ = 0 {\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if

\infty }=1-{\dfrac {r_{1}}{r_{2}}}+\ln {\dfrac {r_{1}}{r_{2}}}-{\dfrac {z}{2}}q_{1}\left(1-{\dfrac {r_{1}q_{2}}{r_{2}q_{1}}}+\ln {\dfrac

{\displaystyle {\dfrac {2mE}{\hbar ^{2}}}=-\nu ^{2}} , ν < 0 {\displaystyle \nu <0} and 2 m V 0 ℏ 2 a 2 = γ 2 {\displaystyle {\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma

≤ σ ≤ 1 n tan 2 ⁡ π 2 n {\displaystyle 0\leq \sigma \leq {\dfrac {1}{n}}\tan ^{2}{\dfrac {\pi }{2n}}} then Z n ( x , σ ) = ( 1 + σ ) n T n ( x − σ 1

1 4 , 1 2 , 3 4 , 5 6 {\displaystyle {\dfrac {1}{6}},{\dfrac {1}{4}},{\dfrac {1}{2}},{\dfrac {3}{4}},{\dfrac {5}{6}}} respectively. The random walk has

{\begin{cases}{\dfrac {4}{9}}{\dfrac {\sigma ^{2}}{r^{2}+\sigma ^{2}}}&{\mbox{for }}r^{2}\geq {\dfrac {5}{3}}\sigma ^{2},\\{\dfrac {4}{3}}{\dfrac {\sigma ^{2}}{r^{2}+\sigma

2 λ 1 + λ 2 x {\displaystyle {\dfrac {\lambda _{1}\cdot 1+\lambda _{2}\cdot (1+x)}{\lambda _{1}+\lambda _{2}}}={\dfrac {\lambda _{1}+\lambda _{2}+\lambda

transport rate diffusive transport rate {\displaystyle \mathrm {Pe} ={\dfrac {\mbox{advective transport rate}}{\mbox{diffusive transport rate}}}} For

>2\!} Median ≈ 1 ν ( 1 − 2 9 ν ) 3 {\displaystyle \approx {\dfrac {1}{\nu {\bigg (}1-{\dfrac {2}{9\nu }}{\bigg )}^{3}}}} Mode 1 ν + 2 {\displaystyle {\frac

\mathrm {B} (x+1,y)=\mathrm {B} (x,y)\cdot {\dfrac {x}{x+y}},\quad \mathrm {B} (x,y+1)=\mathrm {B} (x,y)\cdot {\dfrac {y}{x+y}}.} The positive integer values

y^{2}-2Rx+(K+1)x^{2}=0} alternately x = y 2 R + R 2 − ( K + 1 ) y 2 {\displaystyle x={\dfrac {y^{2}}{R+{\sqrt {R^{2}-(K+1)y^{2}}}}}} where R is the radius of curvature

{E} [X(t)\mid X(t-1)=i]&=ps{\dfrac {1-p}{ps+1}}+i\\\operatorname {Var} (X(t+1)\mid X(t)=i)&=p(1-p){\dfrac {(s+1)+(ps+1)^{2}}{(ps+1)^{2}}}\end{aligned}}}

  0 ≤ G L i ≤ 1 {\displaystyle GL_{i}={\dfrac {(X_{i}+M_{i})-\left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}}=1-{\dfrac {\left|X_{i}-M_{i}\right|}{X_{i}+M_{i}}}\qquad

9 = 56 % {\displaystyle {\begin{aligned}{\dfrac {PES}{PES-PED}}&={\dfrac {0.5}{0.5-(-.0.4)}}\\\\&={\dfrac {0.5}{0.9}}\\\\&=56\%\end{aligned}}} So 56%

{\displaystyle \mathbf {V} (\mathbf {r} )={\dfrac {1}{2\pi }}\sum _{i}\gamma _{i}\cdot \left[\mathbf {e} _{z}\times {\dfrac {\mathbf {r} -\mathbf {r} _{i}}{(\mathbf

}={\frac {{\dfrac {x_{1}}{y_{1}}}L_{1}+{\dfrac {x_{2}}{y_{2}}}L_{2}+\dots +{\dfrac {x_{n}}{y_{n}}}L_{n}}{{\dfrac {L_{1}}{y_{1}}}+{\dfrac {L_{2}}{y_{2}}}+\dots

= 3. {\displaystyle \nu _{2}\!{\binom {10}{3}}={\dfrac {S_{2}(3)+S_{2}(7)-S_{2}(10)}{2-1}}={\dfrac {2+3-2}{2-1}}=3.} Kummer's theorem can be generalized

}-{\overline {Y}}_{\cdot \cdot \cdot })^{2}} SS Tr s − 1 {\displaystyle {\dfrac {{\text{SS}}_{\text{Tr}}}{s-1}}} σ ε 2 + n σ π 2 + D Tr {\displaystyle \sigma

g:\left[a,{\dfrac {b+a}{2}}\right]\to \mathbb {R} } defined by g ( x ) = f ( x + b − a 2 ) − f ( x ) {\displaystyle g(x)=f\left(x+{\dfrac {b-a}{2}}\right)-f(x)}

number of edges are ( n 2 ) = n ( n − 1 ) 2 {\displaystyle {\binom {n}{2}}={\dfrac {n(n-1)}{2}}} and degree here is n − 1 {\displaystyle n-1} . So k = n −

{x}{1-e^{-x}}}=1+{\dfrac {x}{2}}+\sum _{i=1}^{\infty }{\frac {B_{2i}}{(2i)!}}x^{2i}=1+{\dfrac {x}{2}}+{\dfrac {x^{2}}{12}}-{\dfrac {x^{4}}{720}}+\cdots

{\displaystyle {\dfrac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}{\dfrac {2x^{\frac {1}{2}}}{\sqrt {\pi }}}={\frac {2}{\sqrt {\pi }}}{\dfrac {\Gamma (1+{\frac

&=\arccos \left({\dfrac {R^{2}-a^{2}}{Q}}\right),\\\tau &=\operatorname {arsinh} \left({\dfrac {2az}{Q}}\right),\\\phi &=\arctan \left({\dfrac {y}{x}}\right)

\right)\\&={\dfrac {(\alpha +\gamma -1)!}{(\alpha -1)!\,(\gamma -1)!}}{\dfrac {\Gamma (\gamma +n-k)\Gamma (\alpha +k)}{\Gamma (\alpha +\gamma +n)}}\\&={\dfrac {\left(\alpha

pressures: p = p A ⋆ n A + p B ⋆ n B + ⋯ n A + n B + ⋯ {\displaystyle p={\dfrac {p_{\text{A}}^{\star }n_{\text{A}}+p_{\text{B}}^{\star }n_{\text{B}}+\cdots

{\begin{aligned}y&\approx Y+YZ+{\dfrac {1}{42}}Y^{3}+{\dfrac {1}{2}}YZ^{2}\\[5pt]z&\approx Z-{\dfrac {1}{7}}Y^{2}-{\dfrac {1}{3}}Y^{2}Z\end{aligned}}} is

{\displaystyle k=\lim _{t\to \infty }{\dfrac {y(t)}{x(t)}}} Calculate r = t 25 t 75 {\displaystyle r={\dfrac {t_{25}}{t_{75}}}} P = − 18.56075 r + 0

{\ell }{t}}={\frac {1+{\dfrac {\ell }{s}}}{1+{\dfrac {d}{\ell }}}},\qquad {\frac {s}{t}}={\frac {1+{\dfrac {s}{\ell }}}{1+{\dfrac {d}{\ell }}}}.} These

{\begin{cases}{\dfrac {2}{\pi }}\left(\ln \left({\dfrac {z}{2}}\right)+\gamma \right)&{\text{if }}\alpha =0\\[1ex]-{\dfrac {\Gamma (\alpha )}{\pi }}\left({\dfrac

t 2 {\displaystyle Yield={\dfrac {{\dot {m}}*{\dfrac {bu}{56lb}}*{\dfrac {3600s}{hr}}}{w*v*{\dfrac {5280ft}{mi}}*{\dfrac {acre}{43560ft^{2}}}}}} Grain

x ) d d x ] + q ( x ) {\displaystyle \operatorname {L} ={\dfrac {d}{dx}}\left[p(x){\dfrac {d}{dx}}\right]+q(x)} and let D → {\displaystyle {\vec {\operatorname

\mathbf {\tilde {u}} ={\dfrac {\mathbf {u} }{u_{0}}};\quad \mathbf {\tilde {r}} ={\dfrac {\mathbf {r} }{L}};\quad {\tilde {t}}={\dfrac {t}{(L/u_{0})}};\quad

assumptions, and reads: P W V = E inc ⋅ h 2 r ρ {\displaystyle PWV={\sqrt {\dfrac {E_{\text{inc}}\cdot h}{2r\rho }}}} The Moens–Korteweg equation states that

}}={\begin{bmatrix}{\dfrac {\partial \sigma _{xx}}{\partial x}}+{\dfrac {\partial \sigma _{yx}}{\partial y}}+{\dfrac {\partial \sigma _{zx}}{\partial z}}\\{\dfrac {\partial

{\begin{vmatrix}{\dfrac {\partial x_{1}}{\partial x}}&{\dfrac {\partial x_{1}}{\partial y}}&{\dfrac {\partial x_{1}}{\partial z}}\\[12pt]{\dfrac {\partial x_{2}}{\partial

\Theta }(x,\theta )={\begin{cases}{\dfrac {4}{t\pi }}&:\ 0\leq x\leq {\dfrac {t}{2}},\ 0\leq \theta \leq {\dfrac {\pi }{2}}\\[4px]0&:{\text{elsewhere

-1){\dfrac {v_{x}^{2}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{y}}{v^{2}}}&(\gamma -1){\dfrac {v_{x}v_{z}}{v^{2}}}\\-\gamma v_{y}/c&(\gamma -1){\dfrac

,t)=-\left({\dfrac {\partial V}{\partial r}},{\dfrac {1}{r}}{\dfrac {\partial V}{\partial \theta }},{\dfrac {1}{r\sin \theta }}{\dfrac {\partial V}{\partial

given by f ( x , y ) = 2 x 2 y + y 3 x 2 + y 2 . {\displaystyle f(x,y)={\dfrac {2x^{2}y+y^{3}}{x^{2}+y^{2}}}.} If in addition we define f ( 0 , 0 ) = 0

1 {\displaystyle 1} 1.0000 1 Trivially optimal. 2 1 2 {\displaystyle {\dfrac {1}{2}}} 0.5000 0.25 Trivially optimal. 3 2 3 − 3 {\displaystyle 2{\sqrt

L = A d = n l n e n e = n l n e {\displaystyle L=Ad={\dfrac {nl{\sqrt {n_{e}}}}{n_{e}}}={\dfrac {nl}{\sqrt {n_{e}}}}} This is the average length a polymer

2 ) cos ⁡ ( a 2 − a x ) {\displaystyle -{\dfrac {1}{2}}\csc \left({\dfrac {a}{2}}\right)\cos \left({\dfrac {a}{2}}-ax\right)} cos ⁡ ( a ) + sin ⁡ ( a

{\dfrac {2\ell +1}{4\pi }}}\ P_{\ell }^{m}(\cos \theta )&{\text{if }}m=0\\[4pt]\left(-1\right)^{m}{\sqrt {2}}{\sqrt {{\dfrac {2\ell +1}{4\pi }}{\dfrac {(\ell

) Γ ( 1 − x 2 ) ) } , {\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma

{\displaystyle {\dfrac {1}{6+1}}} ), while for a 12-member electorate, a candidate requires 7.69% ( 1 12 + 1 {\displaystyle {\dfrac {1}{12+1}}} ). The

α ( θ α − 1 ) α {\displaystyle \kappa _{p}(\theta )={\dfrac {\alpha -1}{\alpha }}\left({\dfrac {\theta }{\alpha -1}}\right)^{\alpha }} , is the cumulant

{\begin{alignedat}{2}\left\lfloor {\dfrac {i-1}{2}}\right\rfloor =&\quad \left\lfloor {\dfrac {i-2}{2}}+{\dfrac {1}{2}}\right\rfloor \\=&\quad {\frac

) , {\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z),} E ν − 1 ( z ) − E ν + 1

R T p = n i n = x i , {\displaystyle {\frac {V_{i}}{V}}={\dfrac {\dfrac {n_{i}RT}{p}}{\dfrac {nRT}{p}}}={\frac {n_{i}}{n}}=x_{i},} where: p {\displaystyle

P=\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)} P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) {\displaystyle P=\left({\dfrac {mx_{2}+nx_{1}}{m+n}}

{\displaystyle {\dfrac {k}{n+1}}} of the voting sequences. That is, the power index of the strong member is k n + 1 {\displaystyle {\dfrac {k}{n+1}}} . The

J={\begin{bmatrix}{\dfrac {\partial \Delta P}{\partial \theta }}&{\dfrac {\partial \Delta P}{\partial |V|}}\\{\dfrac {\partial \Delta Q}{\partial \theta }}&{\dfrac {\partial

v e r a g e   i n v e n t o r y C O G S / D a y s {\displaystyle DII={\dfrac {average~inventory}{COGS/Days}}} where DII is days in inventory and COGS

}{\ln(\eta )-\ln(\xi )}}\\[6pt]&={\begin{cases}x&{\text{if }}x=y,\\[2pt]{\dfrac {y-x}{\ln y-\ln x}}&{\text{otherwise,}}\end{cases}}\end{aligned}}} for the

I(n,m):=\int _{0}^{1}{\dfrac {{\text{artanh}}^{n}x}{x^{m}}}dx=\int _{0}^{1}\log ^{n/2}{\Bigl (}{\dfrac {1+x}{1-x}}{\Bigr )}{\dfrac {dx}{x^{m}}}=\int _{0}^{\infty

P O = e n d i n g   A / P P u r c h a s e / d a y {\displaystyle DPO={\dfrac {ending~A/P}{Purchase/day}}} where ending A/P is the accounts payable balance

{\begin{aligned}{\dfrac {d^{2}x}{dt^{2}}}&=-\omega ^{2}x+2\Omega {\dfrac {dy}{dt}}\sin \varphi \\{\dfrac {d^{2}y}{dt^{2}}}&=-\omega ^{2}y-2\Omega {\dfrac {dx}{dt}}\sin

/ A ) , Γ = ( A ) ] {\displaystyle {\dfrac {{\dfrac {}{B\leftarrow B}}\qquad {\dfrac {}{A\leftarrow A}}}{\dfrac {B\leftarrow (B/A),\;\;A}{(B/A)\backslash

r)\leq {\begin{cases}{\dfrac {4}{9}}{\dfrac {\operatorname {Var} (X)}{r^{2}+\operatorname {Var} (X)}}&{\text{for }}r^{2}\geq {\dfrac {5}{3}}\operatorname

Schläfli symbol {∞}#{2}, {∞}#{}, or { 2 0 , 1 } {\displaystyle \left\{{\dfrac {2}{0,1}}\right\}} . In three dimensions the discrete regular apeirogons

{\displaystyle {\nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial

{\displaystyle P(X=n)={\begin{cases}e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r,\lambda \right)}},\quad n=0,1,2,\ldots

p o − p ) 2 2 σ 2 ) {\displaystyle g(p_{o}|p)={\dfrac {1}{\sqrt {2\pi \sigma }}}\exp {{\Big (}{\dfrac {-(p_{o}-p)^{2}}{2\sigma ^{2}}}{\Big )}}} since

{\displaystyle V(r)={\begin{cases}{\dfrac {Ze^{2}}{(4\pi \epsilon _{0})2R}}\left({\dfrac {r^{2}}{R^{2}}}-3\right),&r\leq R,\\-{\dfrac {Ze^{2}}{(4\pi \epsilon _{0})r}}

u → ∂ t + u → ⋅ ∇ u → ) = − ∇ p + μ ∇ 2 u → {\displaystyle \rho \left({\dfrac {\partial {\vec {u}}}{\partial t}}+{\vec {u}}\cdot \nabla {\vec {u}}\right)=-\nabla

t}}={\dfrac {(1-H(u-\theta _{v}))(v_{\infty }-v)}{\tau _{v}^{-}}}-{\dfrac {H(u-\theta _{v})v}{\tau _{v}^{+}}}&{\text{in }}\Omega \times (0,T)\\[5pt]{\dfrac

dfrac {m_{A}-m_{B}}{m_{A}+m_{B}}}v_{A1}+{\dfrac {2m_{B}}{m_{A}+m_{B}}}v_{B1}\\[.5em]v_{B2}&=&{\dfrac {2m_{A}}{m_{A}+m_{B}}}v_{A1}+{\dfrac

P = ( 1 − d a ) ( 1 − d A ) Q {\displaystyle P=(1-{\dfrac {d}{a}})(1-{\dfrac {d}{A}})Q\,} (2)

π r 3 ⋅ n ( r ) d r ∫ 0 ∞ π r 2 ⋅ n ( r ) d r . {\displaystyle r_{e}={\dfrac {\displaystyle \int _{0}^{\infty }\pi r^{3}\cdot n(r)\,dr}{\displaystyle

binding energy given by E B = − R h c n 2 , {\displaystyle E_{\text{B}}=-{\dfrac {Rhc}{n^{2}}},} where R {\displaystyle R} is the Rydberg constant, h {\displaystyle

this: ''my'' :: OTT ''friend'' :: T ''my friend'' :: T {\displaystyle {\dfrac {\text{''my'' :: OTT ''friend'' :: T}}{\text{''my friend'' :: T}}}} Rules

N)={\begin{cases}{\dfrac {\sin {\tfrac {1}{2}}N(x-x_{k})}{N\sin {\tfrac {1}{2}}(x-x_{k})}}{\text{ for }}x\neq x_{k}\\[10mu]\lim \limits _{x\to 0}{\dfrac {\sin {\tfrac

⋅ d a y ] {\displaystyle PE={\dfrac {BOD\ load\ from\ industry\ \left[{\dfrac {kg}{day}}\right]}{0.060\ \left[{\dfrac {kg}{inhab\cdot day}}\right]}}}

2 {\displaystyle K_{el}={\dfrac {\ln 2}{t_{1/2}}}} . Thus, C l t o t = ln ⁡ 2 ⋅ V d t 1 / 2 {\displaystyle Cl_{tot}={\dfrac {\ln 2\cdot V_{d}}{t_{1/2}}}}

{\begin{aligned}\gamma &={\dfrac {1}{\sqrt {1-\beta ^{2}}}}\\[1ex]&=\sum _{n=0}^{\infty }\beta ^{2n}\prod _{k=1}^{n}\left({\dfrac {2k-1}{2k}}\right)\\[1ex]&=1+{\tfrac

{6\left(D_{31}{\dfrac {\tau _{1}}{\tau _{3}}}+D_{21}{\dfrac {\tau }{\tau _{3}}}\right){r_{2}}^{3}+\mu D_{31}\left(\tau ^{2}-{\tau _{1}}^{2}\right){\dfrac {\tau

properties of the root system. S U F = J r T a c t {\displaystyle SUF={\dfrac {J_{r}}{T_{act}}}} J r {\displaystyle J_{r}} = the radial flow entering

{\dfrac {\vdash {\text{who}}:N\circ W\quad {\dfrac {{\text{x}}:W\vdash {\text{x}}:W\quad {\dfrac {\vdash {\text{did}}:(S\backslash W)/S\quad {\dfrac {\vdash

{\begin{aligned}{\dfrac {d}{dx}}\operatorname {arccot} x&={\dfrac {d}{dx}}\left({\dfrac {\pi }{2}}-\arctan x\right)\\&=-{\dfrac {1}{1+x^{2}}}\end{aligned}}} Let y = arcsec

_{n=2}^{\infty }{\dfrac {(-1)^{n}}{n+(-1)^{n}}}} and ∑ n = 1 ∞ ( − 1 ) n cos 2 ⁡ n n 2 . {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}{\dfrac {\cos ^{2}n}{n^{2}}}

N)={\begin{cases}{\dfrac {\sin {\tfrac {1}{2}}N(x-x_{k})}{N\sin {\tfrac {1}{2}}(x-x_{k})}}{\text{ for }}x\neq x_{k}\\[10mu]\lim \limits _{x\to 0}{\dfrac {\sin {\tfrac

{4\left(1+2i{\dfrac {z}{L_{NL}}}\right)}{1+4\left({\dfrac {t}{T_{0}}}\right)^{2}+4\left({\dfrac {z}{L_{NL}}}\right)^{2}}}\right]e^{\dfrac {iz}{L_{NL}}}}

{\dfrac {\vdash {\text{who}}:N\circ W\quad {\dfrac {{\text{x}}:W\vdash {\text{x}}:W\quad {\dfrac {\vdash {\text{did}}:(S\backslash W)/S\quad {\dfrac {\vdash

⁡ ϕ × sin ⁡ δ cos ⁡ ϕ × cos ⁡ δ {\displaystyle \cos \omega _{\circ }={\dfrac {\sin a-\sin \phi \times \sin \delta }{\cos \phi \times \cos \delta }}}

_{n=2}^{\infty }{\dfrac {(-1)^{n}}{n+(-1)^{n}}}} and ∑ n = 1 ∞ ( − 1 ) n cos 2 ⁡ n n 2 . {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}{\dfrac {\cos ^{2}n}{n^{2}}}

same kind for the L-function L ( s , x p ) . {\displaystyle L\left(s,{\dfrac {x}{p}}\right).\,} This is of considerable potential interest in number

( x 1 , … , x n ) ∂ ( x ¯ 1 , … , x ¯ n ) | , {\displaystyle J=\left|{\dfrac {\partial (x_{1},\ldots ,x_{n})}{\partial ({\bar {x}}^{1},\ldots ,{\bar

_{\mathbf {p} }\max _{Q}\sum _{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{ij}\log \left({\dfrac {Q_{ji}}{p_{i}}}\right)} where p {\displaystyle \mathbf {p} } and Q {\displaystyle

function of }}\left({\dfrac {\phi _{P}-\phi _{L}}{\phi _{R}-\phi _{L}}}\right),\\[10pt]&f_{r}^{-}{\text{ is a function of }}\left({\dfrac {\phi _{R}-\phi _{RR}}{\phi

c ) C ( z ) . {\displaystyle \mathbb {P} (\Gamma {\mathcal {C}}(z)=c)={\dfrac {z^{\omega (c)}}{C(z)}}.} If C = { c 1 , … , c r } {\displaystyle {\mathcal

y i = k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}=-{\dfrac {1}{2}}\ln \left({\dfrac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum

)={\begin{bmatrix}{\dfrac {\partial x}{\partial r}}&{\dfrac {\partial x}{\partial \theta }}\\[1em]{\dfrac {\partial y}{\partial r}}&{\dfrac {\partial y}{\partial

this ratio as a percentage, A n ( A m + A n ) × 100 % , {\displaystyle {\dfrac {A_{n}}{(A_{m}+A_{n})}}\times 100\%,} and thus may be more properly expressed

{\left({\dfrac {dr}{dt_{r}}}\right)_{\text{raindrop}}}{\left({\dfrac {dr}{dt_{r}}}\right)_{\text{light}}}}={\frac {\sqrt {\dfrac {2M}{r}}}{1+{\sqrt {\dfrac {2M}{r}}}}}<1

=\sigma _{s}{\begin{Vmatrix}{\dfrac {1}{1+\beta ^{2}}}&{\dfrac {-\beta }{1+\beta ^{2}}}\\{\dfrac {\beta }{1+\beta ^{2}}}&{\dfrac {1}{1+\beta ^{2}}}\end{Vmatrix}}}

⁡ ( 1 + μ ) , − 1 ≤ x ≤ 1 , {\displaystyle F(x)=\operatorname {sgn}(x){\dfrac {\ln(1+\mu |x|)}{\ln(1+\mu )}},\quad -1\leq x\leq 1,} where μ = 255 in the

{\displaystyle {\begin{cases}1,\alpha ,{\dfrac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1{\bmod {9}}\\1,\alpha ,{\dfrac {\alpha ^{2}}{k}}&{\text{othe

P(k)={\begin{cases}{\dfrac {2(1-p)}{3-p}}&{\mbox{ if }}k=1\\{\dfrac {(1-p)^{2}}{(2-p)(3-p)}}+{\dfrac {p}{2-p}}&{\mbox{ if }}k=2\\({\dfrac {2}{1-p}}+2)({\dfrac {2}{1-p}}+1)B(k

{\displaystyle q={\dfrac {1}{2\pi }}\int _{{\sqrt {2}}/4}^{+\infty }{\frac {dt}{1+t^{2}}}={\dfrac {\arctan 2{\sqrt {2}}}{2\pi }}={\dfrac {\arccos {\frac

{\displaystyle R(s)={\dfrac {1-W(s)}{1+W(s)}}} where, W ( s ) = 1 − Z ( s ) Z ( k ) 1 + Z ( s ) Z ( k ) ( k + s k − s ) {\displaystyle W(s)={1-{\dfrac {Z(s)}{Z(k)}}

2 g ( 1 − τ i ) {\displaystyle |G|={\dfrac {|J_{C}(\mathbb {F} _{q^{n}})|}{|J_{C}(\mathbb {F} _{q})|}}={\dfrac {\prod _{i=1}^{2g}(1-\tau _{i}^{n})}{\prod

{\begin{aligned}\lim _{n\rightarrow 0}{\dfrac {Z^{n}-1}{n}}&=\lim _{n\rightarrow 0}{\dfrac {e^{n\ln Z}-1}{n}}\\&=\lim _{n\rightarrow 0}{\dfrac {n\ln Z+{1 \over 2!}(n\ln

{\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\dfrac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf

\left({\dfrac {\mathbb {C} [x_{0},x_{1},x_{2},x_{3}]}{(x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4})}}\right)&\to &\operatorname {Proj} \left({\dfrac {\mathbb

{\bar {h}}={\dfrac {1}{n}}\sum _{i=1}^{n}h_{ii}={\dfrac {p}{n}}} (see property 2 above). That is, if h i i > 2 p n {\displaystyle h_{ii}>2{\dfrac {p}{n}}}

{\displaystyle s_{\mathrm {L} }^{\circ }={\sqrt {\alpha {\dot {\omega }}\left({\dfrac {T_{\mathrm {b} }-T_{\mathrm {i} }}{T_{\mathrm {i} }-T_{\mathrm {u} }}}\right)}}}

} 0.04 m 0.72 m {\displaystyle {\dfrac {0.04m}{0.72m}}} = {\displaystyle =} { Simplify } 1 18 {\displaystyle {\dfrac {1}{18}}} = {\displaystyle =} { Calculate

c^{2}\left\{1-{\dfrac {Z^{2}\alpha ^{2}}{2n^{2}}}\left[1+{\dfrac {Z^{2}\alpha ^{2}}{n}}\left({\dfrac {1}{|k|}}-{\dfrac {3}{4n}}\right)\right]\right\}\end{array}}} (The

d r   ρ ( r ) {\displaystyle S_{mn}\ {\stackrel {\mathrm {def} }{=}}\ {\dfrac {\int d\mathbf {r} \ \rho (\mathbf {r} )\ r_{m}r_{n}}{\int d\mathbf {r}

as hT. h T 2 = reaction rate diffusion rate {\displaystyle h_{T}^{2}={\dfrac {\mbox{reaction rate}}{\mbox{diffusion rate}}}} The derivation of the Thiele

^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac

sin 2 ⁡ α N . {\displaystyle R_{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}\,.} It is possible to combine the

+r_{n-1}+r_{n}=-{\dfrac {a_{n-1}}{a_{n}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{n})+\cdots +r_{n-1}r_{n}={\dfrac

the equation: S = 7 3 l o g 10 1 T L + 1 {\displaystyle S={\dfrac {7}{3}}log_{10}{\dfrac {1}{T_{L}}}+1} T L {\displaystyle T_{L}} is defined with respect

ρ 1 ≈ 2 f h 2 c {\displaystyle \rho _{1}\approx {\dfrac {2fh^{2}}{c}}} (3)

{\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\[1em]{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial

element. H i j = K S i j H i i + H j j 2 {\displaystyle H_{ij}=KS_{ij}{\dfrac {H_{ii}+H_{jj}}{2}}} K is the Wolfsberg–Helmholz constant, and is usually

l u m e a m o u n t × t e m p e r a t u r e {\displaystyle R={\frac {{\dfrac {\mathrm {force} }{\mathrm {area} }}\times \mathrm {volume} }{\mathrm {amount}

(x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x)\end{aligned}}}

{\displaystyle {\dfrac {\partial \mathbf {p} }{\partial t}}=-{\dfrac {\partial H}{\partial \mathbf {q} }}} ∂ q ∂ t = ∂ H ∂ p {\displaystyle {\dfrac {\partial

 is odd. {\displaystyle a(n)={\begin{cases}{\dfrac {2^{n+1}-2}{3}}&{\text{when }}n{\text{ is even,}}\\{\dfrac {2^{n+1}-1}{3}}&{\text{when }}n{\text{ is odd

f ≥ 1000. {\displaystyle m(f)={\begin{cases}{\dfrac {3f}{200}},&f<1000,\\15+27\log _{6.4}\left({\dfrac {f}{1000}}\right),&f\geq 1000.\end{cases}}} The

V_{eff}^{1-loop}={\dfrac {1}{64\pi ^{2}}}\operatorname {str} {\bigg [}M^{4}\ln {\Big (}{\dfrac {M^{2}}{\Lambda ^{2}}}{\Big )}{\bigg ]}={\dfrac {1}{64\pi ^{2}}}\operatorname

= V ⋅ S 1 + S K 1 + P K 2   {\displaystyle v=V\cdot {\dfrac {S}{1+{\dfrac {S}{K_{1}}}+{\dfrac {P}{K_{2}}}}}\ } where v {\displaystyle v} , S {\displaystyle

{\displaystyle D_{pq}=\sum _{ij}log\left[{\dfrac {P_{ij}(p)+\alpha F_{i}}{(1+\alpha )F_{i}}}\right]log\left[{\dfrac {\sum _{k\epsilon q}P_{ij}(k)+\alpha

dF(0,0;a,b)={\begin{cases}{\dfrac {F(\tau a,\tau b)-0}{\tau }}&(a,b)\neq (0,0),\\0&(a,b)=(0,0)\end{cases}}={\begin{cases}{\dfrac {a^{3}}{a^{2}+b^{2}}}&(a

\left({\tfrac {1}{2}}\right)^{2}\times \left({\tfrac {1}{2}}\right)^{2}\\&={\dfrac {3}{8}}.\end{aligned}}} What is probability that when three independent

{\begin{array}{cc}{\dfrac {\Gamma ,x:\sigma \vdash _{\text{CD}}M:\varphi }{\Gamma \vdash _{\text{CD}}\lambda x.\!M:\sigma \to \varphi }}(\to \!\!{\text{I}})&{\dfrac {\Gamma

i t y ( B ) {\displaystyle ILF(L)={\dfrac {Expected\ indemnity\ cost(L)}{Expected\ indemnity\ cost(B)}}={\dfrac {Expected\ frequency(L)*Expected\

⁡ ( 8 R a ) − 1 + Y ] {\displaystyle \mathbf {F} ={\dfrac {I^{2}}{c^{2}R}}\left[\ln \left({\dfrac {8R}{a}}\right)-1+Y\right]} where Y is the internal

b c = sin ⁡ 1 2 ( α − β ) cos ⁡ 1 2 γ {\displaystyle {\frac {a-b}{c}}={\dfrac {\sin {\frac {1}{2}}(\alpha -\beta )}{\cos {\frac {1}{2}}\gamma }}} as required

( E ⁡ ( Y 2 ∣ x 2 ) E ⁡ ( Y 1 ∣ x 1 ) ) = β {\displaystyle \log \left({\dfrac {\operatorname {E} (Y_{2}\mid x_{2})}{\operatorname {E} (Y_{1}\mid x_{1})}}\right)=\beta

R {\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\text{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\text{if }}r\geq R\end{cases}}} Newton's

expressed in the following form: d x d t = N v ( x ( p ) , p ) {\displaystyle {\dfrac {\bf {dx}}{dt}}={\bf {N}}{\bf {v}}({\bf {x}}(p),p)} The notation for the

C_{\mathrm {linear} }={\begin{cases}{\dfrac {C_{\mathrm {srgb} }}{12.92}},&C_{\mathrm {srgb} }\leq 0.04045\\[5mu]\left({\dfrac {C_{\mathrm {srgb} }+0.055}{1.055}}\right)^{\

1 a 1 + 1 a 2 + 1 a 3 + ⋯ , {\displaystyle r=a_{0}+{\dfrac {1}{a_{1}+{\dfrac {1}{a_{2}+{\dfrac {1}{a_{3}+\cdots }}}}}},} where ak are natural numbers

the doubling constant of A to be K = | A + A | | A | . {\displaystyle K={\dfrac {|A+A|}{|A|}}.} Let A and B be two subsets of an abelian group. We define

+ d d {\displaystyle \ {\dfrac {a+b}{b}}={\dfrac {c+d}{d}}} ,   a − b b = c − d d {\displaystyle \ {\dfrac {a-b}{b}}={\dfrac {c-d}{d}}} . If   a b = c

potential energy: U C = − e 2 4 π ε 0 r {\displaystyle U_{\text{C}}=-{\dfrac {e^{2}}{4\pi \varepsilon _{0}r}}} . The similarity of the effective potential

\Gamma _{\text{w}}=g\,{\frac {\left(1+{\dfrac {H_{\text{v}}\,r}{R_{\text{sd}}\,T}}\right)}{\left(c_{\text{pd}}+{\dfrac {H_{\text{v}}^{2}\,r}{R_{\text{sw}}\

{\displaystyle \psi _{0}(x)={\dfrac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }\left(-{\dfrac {\zeta '(s)}{\zeta (s)}}\right){\dfrac {x^{s}}{s}}\,ds=x-\sum

y)={\begin{cases}{\sqrt {\theta }}\,{\dfrac {k_{+}(|x|,|y|)}{|x|-|y|}}&{\text{if }}xy>0,\\[12pt]{\sqrt {\theta }}\,{\dfrac {k_{-}(|x|,|y|)}{x-y}}&{\text{if

{I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}} Expected shortfall

division rule is a b c d = a d b c . {\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.} Thus, dividing a b {\displaystyle {\tfrac

4,195,835 3,145,727 = 1.333820449136241002 {\displaystyle \textstyle {\dfrac {4{,}195{,}835}{3{,}145{,}727}}=1.333820449136241002} When converted to

V_{m})+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla u_{e})=A_{m}\left(C_{m}{\dfrac {\partial V_{m}}{\partial t}}+I_{\text{ion}}(V_{m},w)\right)+I_{\text{app}}&{\text{in

{\jmath }}}&{\boldsymbol {\hat {k}}}\\[5mu]{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial

{\displaystyle {\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\dfrac {f(x+h)-f(x)}{h}}-{\dfrac {f(x)-f(x-h)}{h}}}{h}}.} This limit can be viewed as a continuous

according to the handshaking lemma, it has 1 2   n ! ( n − 1 ) {\displaystyle {\dfrac {1}{2}}~n!\left(n-1\right)} edges. Pn can be constructed recursively from

{\displaystyle f={\frac {1}{\left({\dfrac {2W\left({\frac {\ln 10}{2a}}\,10^{\frac {b}{2a}}\right)}{\ln 10}}-{\dfrac {b}{a}}\right)^{2}}}} Additional, mathematically

\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{4}+4a^{4}}}={\dfrac {1}{8a^{4}}}+{\dfrac {\pi (\sinh(2\pi a)+\sin(2\pi a))}{8a^{3}(\cosh(2\pi a)-\cos(2\pi

{\displaystyle f_{L}={\frac {K_{1}+{\dfrac {4K_{2}}{\left(K_{1}+{\frac {K_{1}K_{2}}{K_{1}^{4}+3K_{2}}}\right)^{3}}}}{1+{\dfrac {3K_{2}}{\left(K_{1}+{\frac

2 3 + 29 − 3 93 2 3 3 {\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {93}}}{2}}}}{3}}} 1

− 1 x + c 2 x > 0 {\displaystyle F(x)={\begin{cases}-{\dfrac {1}{x}}+c_{1}&x<0\\[1ex]-{\dfrac {1}{x}}+c_{2}&x>0\end{cases}}} is the most general antiderivative

{\displaystyle {\begin{aligned}m_{1}&={\dfrac {1+5}{2}}=3&&\Rightarrow \;m_{1}^{2}=9\leq 19&&\Rightarrow \;I_{2}=[3,5]\\m_{2}&={\dfrac {3+5}{2}}=4&&\Rightarrow \;m_{2}^{2}=16\leq

{A\times T_{\text{old}}}{A+T_{\text{old}}}}={\frac {1}{{\dfrac {1}{T_{\text{old}}}}+{\dfrac {1}{A}}}}} where A is both the maximum tariff which is agreed

{\displaystyle {\text{VI}}={\begin{cases}100{\dfrac {L-U}{L-H}}&{\text{if VI}}\leq 100,\\100+{\dfrac {\exp \left({\frac {\log H-\log U}{\log Y}}-1\right)}{0

φ ( r ) = 1 1 + ( ε r ) 2 , {\displaystyle \varphi (r)={\dfrac {1}{1+(\varepsilon r)^{2}}},} (4)

}}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial

{S}}}{^{0}_{2}{\ce {S}}+{\ce {K1}}\left(1+{\dfrac {{^{1}_{2}}{\ce {S}}^{\beta }}{\ce {K2}}}+{\dfrac {{^{1}_{2}}{\ce {S}}^{\gamma }}{\ce {K2}}}\right)}}}

{\displaystyle \pi _{k}={\begin{cases}\pi _{0}{\dfrac {(c\rho )^{k}}{k!}},&{\mbox{if }}0<k<c\\[10pt]\pi _{0}{\dfrac {(c\rho )^{k}c^{c-k}}{c!}},&{\mbox{if }}c\leq

distinct de Bruijn sequences B(k, n) is ( k ! ) k n − 1 k n . {\displaystyle {\dfrac {\left(k!\right)^{k^{n-1}}}{k^{n}}}.} The sequences are named after the

R n ; u ( 0 , x ) = f ( x ) , x ∈ R n . {\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Au(t,x),&t>0,x\in \mathbf {R} ^{n};\\u(0

\mathrm {for} \ a\leq x\leq b,} where z = x − a b − a {\displaystyle z={\dfrac {x-a}{b-a}}} and I z ( α , β ) {\displaystyle I_{z}(\alpha ,\beta )} is

− ∞ < u < ∞ {\displaystyle {\begin{cases}x(u)=a\,{\dfrac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)=b\,{\dfrac {2u}{1+u^{2}}}\\[10mu]-\infty <u<\infty \end{cases}}}

0}{\frac {\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{1}+h_{2})-f(x+h_{1})}{h_{2}}}-\lim \limits _{h_{2}\to 0}{\dfrac {f(x+h_{2})-f(x)}{h_{2}}}}{h_{1}}}\end{aligned}}}

{1+{\dfrac {c^{2}\mu _{0}\rho }{B^{2}}}}}}} For the case of an Alfvén wave v = v A 1 + v A 2 c 2 {\displaystyle v={\frac {v_{A}}{\sqrt {1+{\dfrac {v_{A}^{2}}{c^{2}}}}}}}

represented by ( 1 2 π i z , 1 2 π i z ) . {\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).} This is really a restatement of Cauchy's

dfrac {f(x_{0})}{f'(x_{0})}}&=&0.5-{\dfrac {\cos 0.5-0.5^{3}}{-\sin 0.5-3\times 0.5^{2}}}&=&1.112\,141\,637\,097\dots \\x_{2}&=&x_{1}-{\dfrac

f={\dfrac {r}{2\tan {\dfrac {\theta }{2}}}}} f = r θ {\displaystyle f={\frac {r}{\theta }}} f = r 2 sin ⁡ θ 2 {\displaystyle f={\dfrac {r}{2\sin {\dfrac {\theta

∂ Q ∂ P ⇒ P E d ⋅ Q = ∂ P ∂ Q {\displaystyle E_{d}={\dfrac {\partial Q}{\partial P}}\cdot {\dfrac {P}{Q}}\Rightarrow E_{d}\cdot {\frac {Q}{P}}={\frac {\partial

\Lambda (d)={\begin{cases}{\dfrac {6GM^{2}}{5R_{0}\hbar }}\left({\dfrac {5}{3}}\lambda ^{2}-{\dfrac {5}{4}}\lambda ^{3}+{\dfrac {1}{6}}\lambda ^{5}\right)&{\text{when

from ANSI Z87.1, S = 7 3 l o g 10 1 T L + 1 {\displaystyle S={\dfrac {7}{3}}log_{10}{\dfrac {1}{T_{L}}}+1} , which bases luminous transmittance ( T L {\displaystyle

reduces to a Riccati equation. The substitution y = 1 u {\displaystyle y={\dfrac {1}{u}}} brings the Abel equation of the first kind to the "Abel equation

x ≠ 0 1 , x = 0 {\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}} Motivation of invention Telecommunication

{F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}}

rel = m 1 − v 2 c 2 . {\displaystyle m_{\text{rel}}={\frac {m}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}.} In the center of momentum frame, v = 0 {\displaystyle

{\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial

˙ = d x d t = − d V ( u , x ) d x {\displaystyle {\dot {x}}={\dfrac {dx}{dt}}=-{\dfrac {dV(u,x)}{dx}}} In the above equation, V {\displaystyle V} is referred

A visual representation of the area under the curve y = x2 over [0, 2]. Using antiderivatives this area is exactly 8 3 {\textstyle {\dfrac {8}{3}}} .

=\left({\dfrac {1}{{\dfrac {1}{\lambda _{1}}}+{\dfrac {1}{\lambda _{2}}}+...+{\dfrac {1}{\lambda _{n}}}}}\right)=\left({\dfrac {1}{\sum _{k=1}^{N}{\dfrac {1}{\lambda

h a + 1 h b + 1 h c . {\displaystyle r={\frac {1}{{\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}}.} The product of the incircle radius r

_{0}^{\infty }{\frac {1+{\dfrac {x^{2}}{(b+1)^{2}}}}{1+{\dfrac {x^{2}}{a^{2}}}}}\times {\frac {1+{\dfrac {x^{2}}{(b+2)^{2}}}}{1+{\dfrac {x^{2}}{(a+1)^{2}}}}}\times

π , π ) {\displaystyle \log R={\begin{cases}0&{\text{if }}\theta =0\\{\dfrac {\theta }{2\sin \theta }}\left(R-R^{\mathsf {T}}\right)&{\text{if }}\theta

d x 2 , ∂ y ∂ x {\displaystyle \mathbf {f'} (x),{\dfrac {dy}{dx}},{\dfrac {d^{2}y}{dx^{2}}},{\dfrac {\partial {y}}{\partial {x}}}} ص∂/س∂ ،د٢ص/ د‌س٢ ،د‌ص/

{\begin{matrix}{\textbf {Proj}}\left({\dfrac {\mathbb {C} [s,t]}{sf(p)+tg(p)}}\right)&\rightarrow &{\textbf {Proj}}\left({\dfrac {\mathbb {C} [s,t][x,y,z]}{(sf(x

{\begin{array}{r}\left.{\dfrac {\delta v_{1}}{v_{1}}}=\varepsilon _{x}^{1}{\dfrac {\delta x}{x}}+\varepsilon _{e_{1}}^{1}{\dfrac {\delta e_{1}}{e_{1}}}=0\right\}{\text{

cos ⁡ β 2 . {\displaystyle \tau _{\scriptscriptstyle J}(p)={-2}\arctan {\dfrac {p\sin {\frac {\alpha }{2}}+\sin {\frac {\beta }{2}}}{p\cos {\frac {\alpha

dfrac {1}{L_{0}}}\right)^{\frac {2(R_{\text{McF}}^{2})}{n}}\\[1.5em]R_{\text{McF}}^{2}=-{\dfrac {n}{2}}\cdot {\dfrac {\ln(1-R_{\text{CS}}^{2})}{\ln

{\displaystyle {\begin{aligned}U&={\dfrac {|\det {\left[Z-Z_{t}\right]}|}{\det {\left[Z+Z^{*}\right]}}}\\&={\dfrac {|Z_{12}-Z_{21}|^{2}}{4(\operatorname

1 ) ( S S W N − p ) {\displaystyle F={\frac {\left({\dfrac {SS_{A}}{p-1}}\right)}{\left({\dfrac {SS_{W}}{N-p}}\right)}}} where p is the number of groups

{\displaystyle {\bar {X}}\pm {\begin{cases}{\dfrac {|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|<1/2\\[8pt]{\dfrac {1-|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|\geq

{\displaystyle {\dfrac {dw}{dx}}} is small, as it is for an Euler–Bernoulli beam, we can make the approximation 1 ρ ≃ d 2 w d x 2 {\displaystyle {\dfrac {1}{\rho

divergence of J is zero: ∂ J α ∂ x α = ∂ ρ ∂ t + ∇ ⋅ j = 0 , {\displaystyle {\dfrac {\partial J^{\alpha }}{\partial x^{\alpha }}}={\frac {\partial \rho }{\partial

dividend or numerator ← divisor or denominator } ← quotient {\displaystyle {\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow

≤ 2 π , 0 if  | x − λ t | ≥ 2 π . {\displaystyle u(x,t)={\begin{cases}{\dfrac {4\lambda }{3}}\cos ^{2}((x-\lambda t)/4)&{\text{if }}|x-\lambda t|\leq

{\begin{aligned}{\dfrac {\sigma (360)}{360}}&={\dfrac {1170}{360}}={\dfrac {13}{4}},{\text{ }}\\{\dfrac {\sigma (2016)}{2016}}&={\dfrac {6552}{2016}}={\dfrac {13}{4}}=3

+ ( 36 m m ) 2 ( 32.9 m m ) 2 + ( 43.8 m m ) 2 ≈ 0.790 {\displaystyle {\dfrac {\sqrt {(24mm)^{2}+(36mm)^{2}}}{\sqrt {(32.9mm)^{2}+(43.8mm)^{2}}}}\approx

_{\mu }\nabla _{\nu }\psi +{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi =-g^{\mu \nu }\nabla _{\mu }(\partial _{\nu }\psi )+{\dfrac {m^{2}c^{2}}{\hbar ^{2}}}\psi

( m n + a ) p ) {\displaystyle \phi _{n}(a)=\sum _{m{\bmod {p}}}\left({\dfrac {m(m^{n}+a)}{p}}\right)} where p is prime and () is the Legendre symbol

− 1 ) {\displaystyle \left({\dfrac {n-3}{n-1}},2{\dfrac {n+1}{n-1}}\right)} n − 5 ± Δ n 2 − 2 n {\displaystyle {\dfrac {n-5\pm \Delta _{n}}{2-2n}}} (

{\dfrac {(2l+1)m\pi }{2n}}\cdot \Psi \!\left(\!{\frac {2l+1}{4n}}\right)-{\frac {\,1\,}{\,32\pi n^{2}\,}}\!\sum _{l=0}^{2n-1}(-1)^{l}\cos {\dfrac {(2l+1)m\pi

modeled as follows: B = d N d t = k n ( c − c ∗ ) n , {\displaystyle B={\dfrac {dN}{dt}}=k_{n}(c-c^{*})^{n},} where B is the number of nuclei formed per

}}_{2}&h_{3}{\hat {\mathbf {e} }}_{3}\\{\dfrac {\partial }{\partial q^{1}}}&{\dfrac {\partial }{\partial q^{2}}}&{\dfrac {\partial }{\partial

p , q , ¬ q c l o s e d ( i d ) ( ∨ ) {\displaystyle {\dfrac {\quad {\dfrac {\quad {\dfrac {\quad r\wedge \neg r,\;p\wedge ((\neg p\vee q)\wedge \neg

to a linear relation which describes the ideal situation: ψ 0 {\displaystyle \psi _{0}\!} = 1 V × {\displaystyle {\dfrac {1}{V}}\!\times } A constant

{E}}={\begin{bmatrix}{\dfrac {\partial v_{1}}{\partial s_{1}}}&\cdots &{\dfrac {\partial v_{1}}{\partial s_{m}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial v_{n}}{\partial

099 ) 1 0.45 V ≥ 0.081 {\displaystyle L={\begin{cases}{\dfrac {V}{4.5}}&V<0.081\\\left({\dfrac {V+0.099}{1.099}}\right)^{\frac {1}{0.45}}&V\geq 0.081\end{cases}}}

](2ax+b)^{2}&=b^{2}-4ac\\[3mu]2ax+b&=\pm {\sqrt {b^{2}-4ac}}\\[5mu]x&={\dfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.{\vphantom {\bigg )}}\end{aligned}}} This

{\overline {PC}}={\begin{cases}{\dfrac {{\overline {BC}}\sin y}{\sin \beta }}&{\text{if }}|\sin \beta |>|\sin \alpha |,\\[4pt]{\dfrac {{\overline {AC}}\sin x}{\sin

× 100 {\displaystyle {\dfrac {\mbox{b.hp.}}{\mbox{i.hp.}}}\times 100} b.hp. b.hp. + avg. f.hp. × 100 {\displaystyle {\dfrac {\mbox{b.hp.}}{{\mbox{b.hp

{P_{1}}{P_{2}}}={\frac {\operatorname {crd} {\dfrac {360^{\circ }}{n_{2}}}}{\operatorname {crd} {\dfrac {360^{\circ }}{n_{1}}}}}.} Using functional notation

8+{\sqrt {2}}} = 9.414... 11 5 + 3 2 + 1 3 6 {\displaystyle 5+3{\sqrt {2}}+{\dfrac {1}{3}}{\sqrt {6}}} = 10.059... 12 10.422... 13 10.798... 14 2 + 3 2 + 2

1 γ {\displaystyle {\dfrac {1}{\gamma }}} , then the basic reproduction number is R 0 = β γ ⟨ k ⟩ {\displaystyle R_{0}={\dfrac {\beta }{\gamma }}{\langle

= α V K T C ~ V . {\displaystyle \gamma ={\dfrac {\sum _{i}\gamma _{i}c_{V,i}}{\sum _{i}c_{V,i}}}={\dfrac {\alpha VK_{T}}{{\tilde {C}}_{V}}}.} Debye model

, f o r   Δ 0 , 50 ≥ 0 {\displaystyle a_{\mathit {wf}}={\begin{cases}{\dfrac {4\Delta _{0,50}^{2}}{1+4\Delta _{0,50}^{2}}},&\mathrm {for} \ \Delta _{0

Capacitance can be expressed by the equation: C = ε 0 K A d {\displaystyle C={\dfrac {\varepsilon _{0}KA}{d}}} Where C is the capacitance, ε0 is the permittivity

and (2). Flexural Rigidity D = E h e 3 12 ( 1 − ν 2 ) {\displaystyle D={\dfrac {Eh_{e}^{3}}{12(1-\nu ^{2})}}} E {\displaystyle E} = Young's Modulus h e

_{b_{1}}(N)}} \over {b_{2}{\log _{b_{2}}(N)}}}={\left({\dfrac {b_{1}\ln(N)}{\ln(b_{1})}}\right) \over \left({\dfrac {b_{2}\ln(N)}{\ln(b_{2})}}\right)}={{b_{1}\ln(b_{2})}

1 {\displaystyle {\dfrac {1}{12+1}}} ), while in a normal half-Senate election the quota is 14.28% ( 1 6 + 1 {\displaystyle {\dfrac {1}{6+1}}} ). While

{\begin{cases}g_{i}={\dfrac {q_{i}}{q_{r}}}=e_{x}\tan \left({\dfrac {\theta }{2}}\right)\\g_{j}={\dfrac {q_{j}}{q_{r}}}=e_{y}\tan \left({\dfrac {\theta

1 M 1 , b 1 = w 1 ( 1 − w 1 ) M 1 , {\displaystyle w_{1}={\frac {1}{1+{\dfrac {1}{b_{1}M_{1}}}}},\quad b_{1}={\frac {w_{1}}{(1-w_{1})M_{1}}},} where b1

du\\[5pt]L(z)&\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&p<1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&p\geq 1/2.\end{cases}}\\[5pt]{\text{or

Burger number Bu B u = ( R o F r ) 2 {\displaystyle \mathrm {Bu} =\left({\dfrac {\mathrm {Ro} }{\mathrm {Fr} }}\right)^{2}} meteorology, oceanography (density

{\displaystyle {\begin{aligned}x=a_{0}+{\dfrac {1}{a_{1}}}{{} \atop +}\\[28mu]\ \end{aligned}}\!{\begin{aligned}{\dfrac {1}{a_{2}}}{{} \atop +}\\[2mu]\ \end{aligned}}\

≤ x . {\displaystyle \quad {}y={\begin{cases}{\dfrac {1}{c}},&{\text{where }}0\leq x\leq b,\\{\dfrac {1}{x}},&{\text{where }}b\leq x.\end{cases}}} and

99653257887490811 ⋅ 10 13 ⋅ x − 18 ) {\displaystyle {\begin{array}{rcl}f(x)&\approx &{\dfrac {1}{x}}\cdot \left({\frac {\begin{array}{l}1+7.44437068161936700618\cdot

1 r . {\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.} The

_{c}{\sqrt {c}}\left({\dfrac {a}{c}}\right)&{\text{if}}\ c\equiv 1{\pmod {2}}\\(1+i)\varepsilon _{a}^{-1}{\sqrt {c}}\left({\dfrac {c}{a}}\right)&{\text{if}}\

\ln _{q}(x)={\begin{cases}\ln(x)&{\text{if }}x>0{\text{ and }}q=1\\[8pt]{\dfrac {x^{1-q}-1}{1-q}}&{\text{if }}x>0{\text{ and }}q\neq 1\\[8pt]{\text{Undefined

p_{u}={\begin{cases}{\dfrac {f_{u}}{\sum _{i=1}^{20}f_{i}\,+\,w\sum _{k=1}^{\lambda }\tau _{k}}},&(1\leq u\leq 20)\\[10pt]{\dfrac {w\tau _{u-20}}{\sum

{F_{w}}{F_{i}}}={\frac {1}{\sin \theta +\tan \phi \cos \theta }}={\dfrac {1}{\sin \theta +{\dfrac {\sin \phi }{\cos \phi }}\cos \theta }}={\frac {\cos \phi }{\sin

{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}\left({\dfrac {x_{i}}{\sigma _{i}^{2}}}\right)}{\sum _{i=1}^{n}{\dfrac {1}{\sigma _{i}^{2}}}}}={\frac {\sum

{\displaystyle \cos(\varphi )={\frac {r_{2}}{x}}={\frac {r_{2}}{\left({\dfrac {Pr_{2}}{r_{1}+r_{2}}}\right)}}={\frac {r_{1}+r_{2}}{P}}\,\!} ⇒ φ = arccos

of 1 γ {\displaystyle {\dfrac {1}{\gamma }}} , then the basic reproduction number is just R 0 = β γ {\displaystyle R_{0}={\dfrac {\beta }{\gamma }}} . Some

( k 2 ) , x ≥ 0 ; 0 , otherwise . {\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma \left({\frac {k}{2}}\right)}},&x\geq

d = 10 4 2 F d ρ v 2 A , {\displaystyle \Delta C_{\mathrm {d} }=10^{4}{\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\,,} where: F d {\displaystyle F_{\mathrm

{\displaystyle \beta _{n}={\begin{cases}{\dfrac {2^{3n+1}\,(n/2)!^{2}\,n!}{(n+1)\,(2n)!\,\pi }}&({\text{for even }}n)\\{\dfrac {2^{n+1}\,n!^{3}}{(n+1)\,((n-1)/2)

t c , ∇ → ) = ( ∂ t c , ∂ x , ∂ y , ∂ z ) = ∂ μ = , μ {\displaystyle {\dfrac {\partial }{\partial X^{\mu }}}=\left(\partial _{0},\partial _{1},\partial

{\displaystyle {\dfrac {}{\Gamma \cup \{A\}\vdash A}}{\mbox{ axiom}}}         Γ ∪ { A } ⊢ B Γ ⊢ A → B  intro {\displaystyle {\dfrac {\Gamma \cup \{A\}\vdash

2 d x {\displaystyle \int _{0}^{\infty }{\dfrac {a^{x}}{\Gamma (x)}}\,dx=ae^{a}+a\int _{0}^{\infty }{\dfrac {e^{-ax}}{\log ^{2}(x)+\pi ^{2}}}\,dx} Bessel–Clifford

{\bar {T}}_{\text{lost}}\approx {\begin{cases}{\dfrac {1}{m}},{\text{ if }}mN_{e}\ll 1\\[8pt]{\dfrac {\ln {(mN_{e})}+\gamma }{m}}{\text{ if }}mN_{e}\gg

\exp x} Derivative d d x ln ⁡ x = 1 x , x > 0 {\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0} Antiderivative ∫ ln ⁡ x d x = x ( ln ⁡ x − 1 ) +

+ ( i ω τ 0 ) 1 − α {\displaystyle \epsilon ^{*}-\epsilon _{\infty }={\dfrac {\epsilon _{0}-\epsilon _{\infty }}{1+(i\omega \tau _{0})^{1-\alpha }}}}

− n m o d i f i e d n ) n e s s e n t i a l {\displaystyle f=\left(1-{\dfrac {n_{\mathrm {modified} }}{n}}\right)^{n_{\mathrm {essential} }}} and so

2 ) . {\displaystyle D_{n}(x)={\frac {\sin \left((2n+1){\dfrac {x}{2}}\,\right)}{\sin {\dfrac {x}{2}}}}=U_{2n}\!\!\left(\cos {\frac {x}{2}}\right).} (The

C_{i_{m}}={\begin{cases}{\dfrac {m(n+1)(n+2)\cdots (n+m/2-1)}{2(n+m/2+2)(n+m/2+3)\cdots (n+m)}}C_{n+m/2},&m{\text{ even,}}\\[5pt]{\dfrac {m(n+1)(n+2)\cdots

{\displaystyle \alpha } , then X x m − 1 ∼ β ′ ( 1 , α ) {\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )} . If X has a Lomax distribution

2 a 2 b ] {\displaystyle (Df)(a,b)={\begin{bmatrix}{\dfrac {\partial f}{\partial x}}(a,b)&{\dfrac {\partial f}{\partial y}}(a,b)\end{bmatrix}}={\begin

dx=\ln {\left|\sec {x}+\tan {x}\right|}+C=\ln \left|\tan \left({\dfrac {x}{2}}+{\dfrac {\pi }{4}}\right)\right|+C} (See Integral of the secant function

i r − 1 W 2 + … + W k − 1 k  odd . {\displaystyle F_{i}={\begin{cases}{\dfrac {\lambda _{i}^{r}W_{0}+\lambda _{i}^{r-1}W_{2}+\ldots +W_{k}}{\lambda

parameter (cn) Ratio = c n − 1 − c n − 2 c n − c n − 1 {\displaystyle ={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}} 1 2 −0.75 — 2 4 −1.25 — 3 8 −1.3680989

}}t=0\\[5px]7,&{\mbox{if }}t=1\\[7px]{\dfrac {1+5\cdot 4^{n}}{3}},&{\mbox{if }}t{\mbox{ is even otherwise}}\\[7px]{\dfrac {10+11\cdot 4^{n}}{6}},&{\mbox{if

term can be described as: ϵ a = Δ a a Δ c {\displaystyle \epsilon _{a}={\dfrac {\Delta a}{a\Delta c}}} , where a is the lattice parameter of the material

) . {\displaystyle y=Y(u,v).} An equation for ∂ x ∂ u {\displaystyle {\dfrac {\partial x}{\partial u}}} can be found by applying Cramer's rule. Cramer's

x i ) = 0  for some  i , {\displaystyle u(\mathbf {x} )={\begin{cases}{\dfrac {\sum _{i=1}^{N}{w_{i}(\mathbf {x} )u_{i}}}{\sum _{i=1}^{N}{w_{i}(\mathbf

d B d s = − τ N {\displaystyle {\dfrac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}=\kappa \mathbf {N} ,\quad {\dfrac {\mathrm {d} \mathbf {N} }{\mathrm

}}+\mathbf {T} (x))}\,\operatorname {d} {\boldsymbol {\eta }}}\\&=h(x){\dfrac {f({\boldsymbol {\chi }},\nu )}{f({\boldsymbol {\chi }}+\mathbf {T} (x)

diameter of the lumen): σ θ = P r t   {\displaystyle \sigma _{\theta }={\dfrac {Pr}{t}}\ } where P is the blood pressure t is the wall thickness r is the

{\displaystyle n} th iteration, for some 0 ≤ a ≤ 1 3 . {\displaystyle 0\leq a\leq {\dfrac {1}{3}}.} Then, the resulting set has Lebesgue measure 1 − ∑ n = 0 ∞ 2 n

2 ) ( R − 0.15 m ) ] {\displaystyle V=C(gS)^{1/2}\left[R^{1/2}+\left({\dfrac {0.22}{m^{1/2}}}\right)\left(R-0.15m\right)\right]} where m {\displaystyle

q 2 sin 2 ⁡ ( θ c ) 1 − 1 n 2 {\displaystyle N={\dfrac {N_{c}q^{2}\sin ^{2}(\theta _{c})}{1-{\dfrac {1}{n^{2}}}}}} and the precision with which the mean

w=T_{0}{\frac {d}{ds}}{\frac {dy}{dx}}={\frac {T_{0}{\dfrac {d^{2}y}{dx^{2}}}}{\sqrt {1+\left({\dfrac {dy}{dx}}\right)^{2}}}}\,.} In terms of φ and the radius

+ 2 . {\displaystyle {\begin{aligned}J_{-1}^{(1)}&={\dfrac {1}{\sqrt {2}}}(J_{x}-iJ_{y})={\dfrac {J_{-}}{\sqrt {2}}},\\J_{0}^{(1)}&=J_{z},\\J_{+1}^{(1)}&=-{\frac

{\displaystyle {\text{Yield to maturity(YTM)}}={\sqrt[{\text{Time period}}]{\dfrac {\text{Face value}}{\text{Present value}}}}-1} Consider a 30-year zero-coupon

x ∈ R d ; u ( t , ⋅ ) → f , as  t → 0 ; {\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Fu(t,x),&t>0,x\in \mathbf {R} ^{d};\\u(t

+ ( 36 m m ) 2 ( 32.9 m m ) 2 + ( 43.8 m m ) 2 ≈ 0.790 {\displaystyle {\dfrac {\sqrt {(24mm)^{2}+(36mm)^{2}}}{\sqrt {(32.9mm)^{2}+(43.8mm)^{2}}}}\approx

and we have: θ E = 1 1 + ∑ i = 1 n K i P i . {\displaystyle \theta _{E}={\dfrac {1}{1+\sum _{i=1}^{n}K_{i}P_{i}}}.} Also, we can define θ j {\displaystyle

+ 1 , p − 1 ( t ) . {\displaystyle B_{i,p}(t):={\dfrac {t-t_{i}}{t_{i+p}-t_{i}}}B_{i,p-1}(t)+{\dfrac {t_{i+p+1}-t}{t_{i+p+1}-t_{i+1}}}B_{i+1,p-1}(t).}

3 2 k B T {\displaystyle \langle E_{k}\rangle ={\dfrac {3}{2}}m\langle {v_{z}}^{2}\rangle ={\dfrac {3}{2}}k_{\text{B}}T} Notice, the Maxwell–Boltzmann

0 / X ) + ( r + σ 2 / 2 ) ( n / 365 ) σ n / 365 {\displaystyle d_{1}={\dfrac {\ln(S_{0}/X)+(r+{\sigma ^{2}}/2)(n/365)}{\sigma {\sqrt {n/365}}}}} d 2

{f}}&{\text{(Excess charge)}}\\E_{n}&={\dfrac {Q_{n}^{2}}{2C_{s}}}&{\text{(Electrostatic energy)}}\\C_{s}&=4\pi \epsilon _{0}\epsilon _{w}{\dfrac {R^{2}}{L}}&{\text{(S

{{\text{force}}\cdot {\text{linear distance}}}{\text{time}}}\\[6pt]&={\frac {\left({\dfrac {\text{torque}}{r}}\right)\cdot (r\cdot {\text{angular speed}}\cdot

k λ ( 1 − 1 3 k + 0.2 ) , {\displaystyle {\frac {k}{\lambda }}\left(1-{\dfrac {1}{3k+0.2}}\right),} i.e. below the mean k λ . {\displaystyle {\frac {k}{\lambda

y d t = x . {\displaystyle \left\{{\begin{array}{l}{\dfrac {dx}{dt}}=\mu (1-y^{2})x-y,\\{\dfrac {dy}{dt}}=x.\end{array}}\right.} The Jacobian matrix associated

{\displaystyle \sum _{r=1}^{m-1}\zeta \left(s,{\frac {r}{m}}\right)\cos {\dfrac {2\pi rk}{m}}={\frac {m\Gamma (1-s)}{(2\pi m)^{1-s}}}\sin {\frac {\pi s}{2}}\cdot

in Einstein notation: ∭ V ∂ F i ∂ x i d V = {\displaystyle \iiint _{V}{\dfrac {\partial \mathbf {F} _{i}}{\partial x_{i}}}\mathrm {d} V=} S {\displaystyle

then d f f = d n ( n − 1 ) = 1 n , {\displaystyle {\dfrac {df}{f}}={\dfrac {dn}{(n-1)}}={\dfrac {1}{n}},} d n {\displaystyle dn} is called the dispersion

f(x\mid \mu ,\kappa )\approx {\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[{\dfrac {-(x-\mu )^{2}}{2\sigma ^{2}}}\right]} where σ2 = 1/ κ {\displaystyle \kappa

1 − ℓ 2 {\displaystyle g={\frac {8\pi ^{2}}{{\dfrac {T_{1}^{2}+T_{2}^{2}}{\ell _{1}+\ell _{2}}}+{\dfrac {T_{1}^{2}-T_{2}^{2}}{\ell _{1}-\ell _{2}}}}}}

{\displaystyle y_{t+h/2}^{1}={\dfrac {y_{t}+y_{t+h}^{1}}{2}}} and y t + h / 2 2 = y t + y t + h 2 2 {\displaystyle y_{t+h/2}^{2}={\dfrac {y_{t}+y_{t+h}^{2}}{2}}}

0 otherwise . {\displaystyle L(x)={\begin{cases}1&{\text{if}}\ x=0,\\{\dfrac {a\sin(\pi x)\sin(\pi x/a)}{\pi ^{2}x^{2}}}&{\text{if}}\ -a\leq x<a\ {\text{and}}\

, T ) u ( 0 ) = u 0  in  Ω × { 0 } {\displaystyle {\begin{cases}\rho {\dfrac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla )\mathbf

equation: P W V = E i n c ⋅ h 2 ⋅ r ⋅ ρ {\displaystyle \mathrm {PWV} ={\sqrt {\dfrac {E_{\mathrm {inc} }\cdot h}{2\cdot r\cdot \rho }}}} , characterises PWV

( l 2 − ( r sin ⁡ ( α ) + o ) 2 ) 3 2 − r cos ⁡ ( α ) {\displaystyle {\dfrac {r\sin \left({\alpha }\right)\left(r\sin \left({\alpha }\right)+o\right)-r^{2}\cos

Q t = C c O 2 − C a O 2 C c O 2 − C v O 2 {\displaystyle {\dfrac {Q_{s}}{Q_{t}}}={\dfrac {Cc_{O_{2}}-Ca_{O_{2}}}{Cc_{O_{2}}-Cv_{O_{2}}}}} Divide by Qt

( i ) ∩ V + ( j ) | | V + ( i ) ∪ V + ( j ) | {\displaystyle D_{ij}=1-{\dfrac {|V^{+}(i)\cap V^{+}(j)|}{|V^{+}(i)\cup V^{+}(j)|}}} Where this measure

\sum _{n=1}^{\infty }{\dfrac {n^{k}}{e^{2\pi n}-1}}={\begin{cases}{\dfrac {1}{24}}-{\dfrac {1}{8\pi }}&{\text{if}}\ k=1\\{\dfrac {\mathrm {B} _{k+1}}{2k+2}}&{\text{if}}\

i = 1 S p i j p i | j q − 1 q − 1 {\displaystyle {}^{q}\!D_{\alpha }={\dfrac {1}{\sqrt[{q-1}]{\sum _{j=1}^{N}{\sum _{i=1}^{S}p_{ij}p_{i|j}^{q-1}}}}}}

{\displaystyle v_{\text{g}}={\frac {v_{\text{p}}}{1-{\dfrac {\omega }{v_{\text{p}}}}{\dfrac {dv_{\text{p}}}{d\omega }}}}.} When dispersion is present

n X o p + Y i j Y k l Y m n Y o p ) {\displaystyle C_{2}=Tr[(XY)^{2}]-{\dfrac {1}{4}}Tr[XY]^{2}+{\frac {1}{96}}\epsilon _{ijklmnop}\left(X^{ij}X^{kl}

u y ∂ H ∂ y {\displaystyle {\dfrac {\partial H}{\partial t}}=u_{z}-u_{x}{\dfrac {\partial H}{\partial x}}-u_{y}{\dfrac {\partial H}{\partial y}}} This

∂ t + ∇ ) F = μ 0 c ( c ρ − J ) . {\displaystyle \left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)\mathbf {F} =\mu

{\displaystyle U^{*}={\dfrac {\sum \limits _{i=1}^{n}{\dfrac {g_{s,i}}{R}}*U_{pre,i}*\Delta E_{s,i}+I_{\text{app}}}{1+\sum \limits _{i=1}^{n}{\dfrac {g_{s,i}}{R}}U_{pre

β y α y β [ α , β = 1 , 2 ] {\displaystyle {\begin{array}{c}\cos ^{-1}{\dfrac {\sum a_{\alpha \beta }x_{\alpha }y_{\beta }}{{\sqrt {\sum a_{\alpha \beta

notation specifies the order in another way, like 2 3 / 4 {\displaystyle {\dfrac {2}{3/4}}} ). However, mathematicians agree on a particular order of evaluation

{\displaystyle \operatorname {sgn}(z)={\begin{cases}0&{\text{for }}z=0\\{\dfrac {z}{|z|}}=e^{i\varphi }&{\text{otherwise}}.\end{cases}}} This definition

_{ij}^{}{a_{ij}^{(1)}{\dfrac {\partial {y_{j}}}{\partial {x_{i}}}}}+{f_{1}}=0\\\vdots \\\sum \limits _{ij}^{}{a_{ij}^{(n)}{\dfrac {\partial {y_{j}}}{\partial

wire + N row C cell ) V dd V swing {\displaystyle P_{\text{bitlines}}={\dfrac {N_{\text{col}}}{2}}\cdot (L_{\text{col}}C_{\text{wire}}+N_{\text{row}}

R\left(x\right):=\lim _{a\to \infty }{\begin{cases}{\frac {1}{a}},\quad x=0\\{\dfrac {x}{1-e^{-ax}}},\quad x\neq 0\end{cases}}} It could approximated as close

{1}{2}}\pi {\bigr )}=q^{\lfloor n/2\rfloor }{\frac {1}{2^{n}n!}}\left({\dfrac {p}{q}}\right)^{n+{\frac {1}{2}}}\int _{0}^{1}(1-z^{2})^{n}\cos \left({\tfrac

with − μ L = e ℏ 2 m L = μ B L {\displaystyle -{\boldsymbol {\mu }}_{L}={\dfrac {e\hbar }{2m}}\mathbf {L} =\mu _{B}\mathbf {L} } . Additionally taking into

i , g ^ m ( x i ) h ^ m ( x i ) } i = 1 N {\displaystyle \left\{x_{i},{\dfrac {{\hat {g}}_{m}(x_{i})}{{\hat {h}}_{m}(x_{i})}}\right\}_{i=1}^{N}} by solving

sufficiently small, f | f | : ( S ε 2 n + 1 ∖ V f ) → S 1 {\displaystyle {\dfrac {f}{|f|}}\colon \left(S_{\varepsilon }^{2n+1}\setminus V_{f}\right)\to S^{1}}

{1}{2}}\hbar \omega _{\text{C}}\alpha ^{2}={\frac {1}{2m_{\text{e}}}}\left({\dfrac {\hbar }{a_{0}}}\right)^{2}={\frac {1}{2}}{\frac {e^{2}}{(4\pi \varepsilon

Compute the polynomial g ( x ) = f ( x ) ( x − a ) {\textstyle g(x)={\dfrac {f(x)}{(x-a)}}} , for example using polynomial long division or synthetic

this query time to O ( log ⁡ n log ⁡ log ⁡ n ) {\displaystyle O\left({\dfrac {\log n}{\log \log n}}\right)} for range counting, which matches a lower

+b\mathbf {j} +c\mathbf {k} )={\dfrac {1+\mathbf {i} +\mathbf {j} +\mathbf {k} }{2}}(a\mathbf {i} +b\mathbf {j} +c\mathbf {k} ){\dfrac {1-\mathbf {i} -\mathbf

{\displaystyle {\begin{aligned}P_{t+\Delta t}&=P_{t}+(rP_{t}-M_{N})\Delta t\\[12pt]{\dfrac {P_{t+\Delta t}-P_{t}}{\Delta t}}&=rP_{t}-M_{N}\end{aligned}}} If N is increased

}}\quad \Rightarrow \quad {\begin{cases}{\dfrac {\ell (\ell +1)}{r}}E=i\omega B^{r},\\{\dfrac {dE}{dr}}+{\dfrac {E}{r}}=i\omega B^{(1)}.\end{cases}}} Gauss'

&\propto \ &{\bigl (}\#\mathbb {Z} _{k}^{d,(-dn)}+\alpha _{k}{\bigr )}{\dfrac {\#\mathbb {W} _{v}^{k,(-dn)}+\beta _{v}}{\sum _{v'=1}^{V}(\#\mathbb {W}

n Γ ( α n + 1 ) . {\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+1)}}.} The two-parameter Mittag-Leffler function

→ P ( t ) , t ) x → P ( t 0 ) = x → P 0 {\displaystyle {\begin{cases}{\dfrac {d{\vec {x}}_{P}}{dt}}(t)={\vec {u}}_{P}({\vec {x}}_{P}(t),t)\\[1.2ex]{\vec

1 s + 1 + s s + 1 ⋅ m R m − s + 1 {\displaystyle {\dfrac {1}{s+1}}+{\dfrac {s}{s+1}}\cdot {\dfrac {mR}{m-s+1}}} During this step, our task focus on how

quadratic approximation P 2 ( x ) = 1 + x + x 2 2 {\displaystyle P_{2}(x)=1+x+{\dfrac {x^{2}}{2}}} (red) at a = 0 {\textstyle a=0} . Note the improvement in the

− T c ) 2 , T < T c 0 , T > T c {\displaystyle F-F_{0}={\begin{cases}-{\dfrac {a_{0}^{2}}{2b_{0}}}(T-T_{c})^{2},&T<T_{c}\\0,&T>T_{c}\end{cases}}} From

water is: F r = U g A B . {\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g{\dfrac {A}{B}}}}}.} For rectangular cross-sections with uniform depth d, the Froude

− 1 ( n − 1 ) x ( n ) } {\displaystyle {\dfrac {1}{\lambda }}\left\lbrace \mathbf {R} _{x}^{-1}(n-1)-{\dfrac {\mathbf {R} _{x}^{-1}(n-1)\mathbf {x} (n)\mathbf

0 , t ≠ a {\displaystyle \delta _{a}^{\mathbb {H} }(t)={\begin{cases}{\dfrac {1}{\mu (a)}},&t=a\\0,&t\neq a\end{cases}}} Integral equations and summation

x}}=-R_{\mathrm {g} }T\left({\frac {kp}{\mu }}+D_{\mathrm {K} }\right)^{-1}{\dfrac {p}{R_{\mathrm {g} }T}}q\,.} This equation can be rearranged into the following

i)=\sum _{g_{j}\in S,j\leq i}{\dfrac {|r_{j}|^{p}}{N_{R}}};\\[0.6ex]&P_{miss}(S,i)=\sum _{g_{j}\not \in S,j\leq i}{\dfrac {1}{N-N_{H}}};\\[0.6ex]&N_{R}=\sum

t + ∇ ) F = 1 ϵ 0 ( ρ − 1 c J ) . {\displaystyle \left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)\mathbf {F} ={\frac

that the equation ∇ ⋅ E = ρ ε 0 {\displaystyle \nabla \cdot \mathbf {E} ={\dfrac {\rho }{\varepsilon _{0}}}} is equivalent to the equation ∇ ⋅ D = ρ f r

5 1 / n i = n . {\displaystyle m(i)={\begin{cases}1-0.5^{1/n}&i=1\\\\{\dfrac {i-0.3175}{n+0.365}}&i=2,3,\ldots ,n-1\\\\0.5^{1/n}&i=n.\end{cases}}} The

division of two zeros to be equal to zero, i.e. 0 0 := 0 {\displaystyle {\dfrac {0}{0}}:=0} . B-splines that are generated in this way own both the partition

{\text{PI}}_{\text{child}}=0.1\times {\text{PI}}_{\text{adult}}=100\times {\dfrac {{\text{mass}}_{\text{g}}}{{\text{height}}_{\text{cm}}^{3}}}} Body adiposity

− a ( λ ) ( y ≤ x ) . {\displaystyle G_{\lambda }(x,y)={\begin{cases}{\dfrac {(\varphi _{\lambda }(x)+a(\lambda )\chi _{\lambda }(x))(\varphi _{\lambda

− 1 ) + ( x + y ) ( β − α γ ) + α 2 γ − β 2 {\displaystyle \psi (x,y)={\dfrac {xy(\beta \gamma -\alpha )+(x+y)\alpha \beta (1-\gamma )+\alpha \beta (\alpha

given by: β ( z ) = β ∗ + z 2 β ∗ {\displaystyle \beta (z)=\beta ^{*}+{\dfrac {z^{2}}{\beta ^{*}}}} where z is the distance along the nominal beam direction

, T ) u ( 0 ) = u 0  in  Ω × { 0 } {\displaystyle {\begin{cases}\rho {\dfrac {\partial {\boldsymbol {u}}}{\partial t}}+\rho ({\boldsymbol {u}}\cdot \nabla

k 1 ψ 1 ) ⋯ ( 1 − k n ψ n ) . {\displaystyle h_{g;k_{1},\dots ,k_{n}}={\dfrac {m!}{\#{\text{Aut}}(k_{1},\ldots ,k_{n})}}\prod _{i=1}^{n}{\frac {k_{i}^{k_{i}}}{k_{i}

( x i − x ¯ ) 2 n − 1 , {\displaystyle {\text{sample variance}}=s^{2}={\dfrac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}},} where s is the standard

_{0}^{z}e^{(\lambda _{2}-\lambda _{1})x_{1}}\,dx_{1}\\&={\begin{cases}{\dfrac {\lambda _{1}\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\left(e^{-\lambda

t ) ) {\displaystyle N\left(X_{t}\mid \mu _{i,t},\sigma _{i,t}\right)={\dfrac {1}{(2\pi )^{D/2}}}{1 \over |\sigma _{i,t}|^{1/2}}\exp \left(-{1 \over 2}(X_{t}-\mu

> 1   {\displaystyle E_{d}>1\!\ } , then d R d P < 0   {\displaystyle {\dfrac {dR}{dP}}<0\!\ } : price and total revenue move in opposite directions.

m i n , p j , m a x ] } {\displaystyle {\mathcal {P}}(s)=\left\lbrace {\dfrac {\prod _{i}(s+z_{i})}{\prod _{j}(s+p_{j})}},\forall z_{i}\in [z_{i,min}

Γ 0 = N T ⋅ n 0 = P ⋅ n 0 {\displaystyle \mathbf {t} _{0}=\mathbf {t} {\dfrac {d{\Gamma }}{d\Gamma _{0}}}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol

{\begin{array}{lll}\Delta G_{\text{bind}}=-RT\ln K_{\text{d}}\\[1.3ex]K_{\text{d}}={\dfrac {[{\text{Ligand}}][{\text{Receptor}}]}{[{\text{Complex}}]}}\\[1.3ex]\Delta

also generated by the set { n T ( n ) : n > 0 } . {\displaystyle \left\{{\dfrac {n}{T(n)}}:n>0\right\}.} The weak Collatz conjecture asserts the following:

{\displaystyle \left\{{\begin{matrix}\cos \theta &=&{\dfrac {\operatorname {tr} (R)-1}{2}}\\\sin \theta &=&-{\dfrac {\operatorname {tr} (K_{n}R)}{2}}\end{matrix}}\right

: ℓ j ( x ) = ℓ ( x ) w j x − x j {\displaystyle \ell _{j}(x)=\ell (x){\dfrac {w_{j}}{x-x_{j}}}} By factoring ℓ ( x ) {\textstyle \ell (x)} out from the

v t = 0   ; A v 1 + ∂ E ( φ ) ∂ φ 1 = 0 {\displaystyle {\begin{cases}&{\dfrac {d}{dt}}Av_{t}+(Dv_{t})^{T}Av_{t}+(DAv_{t})v_{t}+(\nabla \cdot v)Av_{t}=0\

{\displaystyle L(1,\chi _{D})={\begin{cases}{\dfrac {2\pi }{w_{D}{\sqrt {|D|}}}}\,h(D),&{\text{if }}D<0\\[.5em]{\dfrac {\log \varepsilon _{D}}{\sqrt {D}}}\,h(D)

&{\text{if }}\sum n_{\text{initial}}<n_{\text{ams}}\\n_{\text{ams}}\times {\dfrac {n_{\text{initial}}}{\sum n_{\text{initial}}}},&{\text{if }}\sum

{c} }}{k'\cdot c_{\mathrm {s} }}}} If h c k ′ c s = 1 {\displaystyle {\dfrac {h_{\mathrm {c} }}{k'c_{\mathrm {s} }}}=1} then the temperature of the drop

P l η Q {\displaystyle S={\cfrac {7d}{\rho \,(1-\epsilon \,)}}{\sqrt {\dfrac {\epsilon \,^{3}\pi \,\delta \,P}{l\eta \,Q}}}} where: S is specific surface

k 2 ) , x > 0 ; 0 , otherwise . {\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\\0

{1}{(z+k)^{2}}}\sim \underbrace {\int _{0}^{\infty }{\frac {1}{(z+k)^{2}}}\,dk} _{={\dfrac {1}{z}}}+{\frac {1}{2z^{2}}}+\sum _{t=1}^{\infty }{\frac {B_{2t}}{z^{2t+1}}}

}2^{i}&=n-k,\\\mathbb {B} :\ \sum _{j\in \mathbb {B} }2^{j}&=\left\lfloor {\dfrac {k-1}{2}}\right\rfloor .\\\end{aligned}}} One can mimic a bitwise AND operation

f}{\partial x}}(x,t)\,dt+f(x,b(x))b'(x)\right)-\left(\int _{t_{1}}^{a(x)}{\dfrac {\partial f}{\partial x}}(x,t)\,dt+f(x,a(x))a'(x)\right)\\[2pt]&=f(x,b(x))b'(x)-f(x

{\displaystyle \mathrm {PSF} (f,z)=I_{r}(0,z,f)\exp \left[-z\alpha (f)-{\dfrac {2\rho ^{2}}{0.36{\frac {cka}{{\text{NA}}f}}{\sqrt {{1+\left({\frac {2\ln

− 1 {\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}} where ωk,s is the

For example, in lim x → 1 x 2 − 1 x − 1 , {\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},} the separate limits of the numerator and denominator are

For example, X = − 1 1234567 = − e − e e 0.9711308 {\displaystyle X=-{\dfrac {1}{1234567}}=-e^{-e^{e^{0.9711308}}}} and its SLI representation is x =

t ) = k ( t ) 1 + k ( t ) k n ( t ) , {\displaystyle k_{d}(t)={\dfrac {k(t)}{1+{\dfrac {k(t)}{k_{n}(t)}}}},\quad } with k d ( t ) {\displaystyle k_{d}(t)}

)&L(\gamma )\\[5pt]\varphi &\gamma \\[8pt]{\dfrac {\partial \varphi }{\partial n}}&{\dfrac {\partial \gamma }{\partial n}}\\[8pt]\gamma &0\end{matrix}}\right)

x t n ) , n ∈ N , x ∈ R . {\displaystyle {\begin{aligned}f_{n}(x_{t})={\dfrac {1}{(1+x_{t}^{n})}}\,,\qquad n\in {\mathbb {N} },\;x\in {\mathbb {R} }.\end{aligned}}}

( v , w ) in  H w ( t = 0 ) = w 0 in  H {\displaystyle {\begin{cases}{\dfrac {\partial \mathbf {w} }{\partial t}}=\mathbf {F} (v,\mathbf {w} )&{\text{in

{\begin{aligned}y&=(2x+8)^{3}\\{\sqrt[{3}]{y}}&=2x+8\\{\sqrt[{3}]{y}}-8&=2x\\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}}} Thus the inverse function f −1

) , {\displaystyle [P(E)]={\sqrt {\frac {1}{N\pi J^{2}}}}\exp \left(-{\dfrac {E^{2}}{J^{2}N}}\right),} where [ ⋯ ] {\displaystyle [\cdots ]} denotes

w_{i-n+1}\cdots w_{i-1})\\[4pt]={}&{\begin{cases}d_{w_{i-n+1}\cdots w_{i}}{\dfrac {C(w_{i-n+1}\cdots w_{i-1}w_{i})}{C(w_{i-n+1}\cdots w_{i-1})}}&{\text{if

supply Elapsed days (inclusive of last prescription) {\displaystyle VMPR={\dfrac {\text{All days' supply}}{\text{Elapsed days (inclusive of last prescription)}}}}

_{2}+i\gamma _{2})-g^{2}}}\\\alpha _{2,ss}&={\frac {\omega _{2}}{\omega _{1}}}{\dfrac {-Fg}{(\Delta _{1}+i\gamma _{1})(\Delta _{2}+i\gamma _{2})-g^{2}}}\end{aligned}}}

{q_{2}}{2}}} q 2 = a − χ 2 b − q 1 2 {\displaystyle q_{2}={\frac {a-\chi }{2b}}-{\dfrac {q_{1}}{2}}} These functions describe each firm's optimal (profit-maximizing)

) = a n 2 + 1 a n . {\displaystyle a_{n+1}={\frac {1}{2}}\left(a_{n}+{\dfrac {2}{a_{n}}}\right)={\frac {a_{n}}{2}}+{\frac {1}{a_{n}}}.} Each iteration

kinetic energy K to its velocity: K = 1 2 m | v | 2 {\displaystyle K={\dfrac {1}{2}}m|\mathbf {v} |^{2}} . The primary difficulty with Mach's definition

{\frac {\left({\dfrac {\text{likelihood of null model}}{\text{likelihood of the saturated model}}}\right)}{\left({\dfrac {\text{likelihood of fitted

∂ S ∂ t ˙ + r ¨ ∂ S ∂ r ˙ + θ ¨ ∂ S ∂ θ ˙ {\displaystyle 0={\dot {t}}{\dfrac {\partial S}{\partial t}}+{\dot {r}}{\frac {\partial S}{\partial r}}+{\dot

divided by the number of intervals: R = E F S R 2 M − 1 , {\displaystyle R={\dfrac {E_{\mathrm {FSR} }}{2^{M}-1}},} where M is the ADC's resolution in bits

{F}}_{k}(\theta _{1},\theta _{2})&={\begin{bmatrix}\cos k\delta \sin k\mu \\{\dfrac {\sin k\delta }{\delta }}\cos k\mu \end{bmatrix}}.\end{aligned}}} The first

double integral ∬ | f ( x ) | x 2 + y 2 d x d y {\displaystyle \iint {\dfrac {\vert f({\textbf {x}})\vert }{\sqrt {x^{2}+y^{2}}}}\,dx\,dy} , extending

\operatorname {prox} \limits _{\lambda ,R,g}(w_{g})={\begin{cases}\left(1-{\dfrac {\lambda }{\left\|w_{g}\right\|_{2}}}\right)w_{g},&{\text{if

2 − 1 Γ ( k 2 ) , x ≥ 0 0 , x < 0 {\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma \left({\frac {k}{2}}\right)}},&x\geq

D r ‖ F 2 + λ ‖ r ‖ 1 {\displaystyle \min _{r\in \mathbb {R} ^{n}}\,\,{\dfrac {1}{2}}\,\,\|X-\mathbf {D} r\|_{F}^{2}+\lambda \,\,\|r\|_{1}} , where λ

2 − 1 2 t , t > 0 {\displaystyle {\begin{cases}x=\pm a{\dfrac {t^{2}+1}{2t}},\\[1ex]y=b{\dfrac {t^{2}-1}{2t}},\end{cases}}\qquad t>0} Through circular

t , p ) ( R × M ) {\displaystyle \mathbb {R} \times M\ni (t,p)\mapsto {\dfrac {\partial }{\partial t}}{\Biggl |}_{t}+X(p)\in T_{(t,p)}(\mathbb {R} \times

D r ‖ F 2 + λ ‖ r ‖ 1 {\displaystyle \min _{r\in \mathbb {R} ^{n}}\,\,{\dfrac {1}{2}}\,\,\|X-\mathbf {D} r\|_{F}^{2}+\lambda \,\,\|r\|_{1}} , where λ

{\displaystyle V_{i,j}={\begin{cases}0&{\text{if }}j<i-m_{\ell },\\[6pt]{\dfrac {(j-1)!}{(j-(i-m_{\ell }))!}}x_{i}^{j-(i-m_{\ell })}&{\text{if }}j\geq i-m_{\ell

\left[i\left(kz-\omega t\right)\right]\right\}\\\mathbf {B} (\mathbf {r} ,t)&={\dfrac {1}{c}}{\hat {\mathbf {z} }}\times \mathbf {E} (\mathbf {r} ,t)\end{aligned}}}

) − c ( s ) t − s : t ≠ s , | t | , | s | ≤ C } {\displaystyle \left\{{\dfrac {c(t)-c(s)}{t-s}}:t\neq s{,}|t|,|s|\leq C\right\}} is bounded in E , {\displaystyle

{cdf} _{\text{MJ}}(\gamma )={\frac {1}{\theta \operatorname {K} _{2}\left({\dfrac {1}{\theta }}\right)}}\int _{1}^{\gamma }{\gamma ^{\prime }}^{2}{\sqrt {1-{\frac

2 ⋅ 3 + ⋯ {\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot

\alpha =\tan ^{-1}\left({\dfrac {u}{v}}\right)} β = sin − 1 ⁡ ( v u 2 + v 2 + w 2 ) {\displaystyle \beta =\sin ^{-1}\left({\dfrac {v}{\sqrt {u^{2}+v^{2}+w^{2}}}}\right)}

u ∂ t + ( u ⋅ ∇ ) u = − ∇ p , ∇ ⋅ u = 0. {\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}+(u\cdot \nabla )u=-\nabla p,\\\nabla \cdot u=0

+ c + b + a e f g h . {\displaystyle {\frac {a\ b\ c\ d}{e\ f\ g\ h}}={\dfrac {d+{\cfrac {c+{\cfrac {b+{\cfrac {a}{e}}}{f}}}{g}}}{h}}.} If such a notation

momentum nonetheless which is given by: p = h λ = E p c , {\displaystyle p={\dfrac {h}{\lambda }}={\frac {E_{p}}{c}},} where p is momentum, h is Planck's constant

V_{\text{in}}={\frac {1}{Nt_{\Delta }}}R_{i}\left(-{\dfrac {N_{p}V_{\text{ref}}t_{\Delta }}{R_{p}}}+{\dfrac {N_{n}V_{\text{ref}}t_{\Delta }}{R_{n}}}-CV_{\text{out}}\right)}

coefficient cd c d = 2 F d ρ v 2 A , {\displaystyle c_{\mathrm {d} }={\dfrac {2F_{\mathrm {d} }}{\rho v^{2}A}}\,,} aeronautics, fluid dynamics (resistance

}k(x-y)Rn_{t}(y)dy} where R = d f d n | n = 0 {\displaystyle R=\left.{\dfrac {df}{dn}}\right|_{n=0}} . This can be written as the convolution n t + 1

= n 2 + 1 n ( 4 n 2 + 5 ) , {\displaystyle S(n)\approx {\dfrac {1}{4n+{\dfrac {1}{n+{\dfrac {1}{n}}}}}}={\frac {n^{2}+1}{n(4n^{2}+5)}},} attributed to

{\frac {k_{2}+k_{-1}}{k_{1}}}\\k_{cat}\ &{\stackrel {\mathrm {def} }{=}}\ {\dfrac {k_{3}k_{2}}{k_{2}+k_{3}}}\end{aligned}}} We see that for the limiting case

k − 1 k ) {\displaystyle DE_{SW}={\dfrac {1+\rho (ktn+bn-1)}{1+\rho \left({\frac {1}{2}}ktn+bn-1\right)}}*{\dfrac {3(1-\rho )}{2t\left(k-{\frac {1}{k}}\right)}}}

{\displaystyle B_{2k}={\begin{cases}c_{0}\,,&{\text{if }}k=0\,,\\[5px]{\dfrac {c_{k}}{k}}\,,&{\text{if }}k>0\,,\end{cases}}} where c k = ∑ j = 0 ∞ ( 2

for  i ≥ 3 ; − 1 2 x ( x + 1 ) for  i = 2. {\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac

0 3 π 2 , if  x < 0 {\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{if }}x>0\\-{\frac {\pi

{\begin{aligned}{\frac {dA}{dt}}&={\begin{bmatrix}{\dfrac {d\mathbf {e} _{1}}{dt}}&{\dfrac {d\mathbf {e} _{2}}{dt}}&{\dfrac {d\mathbf {e} _{3}}{dt}}\end{bmatrix}}

{Prismatic}}:\ {\dfrac {\textrm {length}}{{\textrm {cross-sectional}}\ {\textrm {area}}}}\\{\textrm {Cylinder}}:\ {\dfrac {\ln \left({\frac {\mathrm

⁡ 1 + x 1 − x {\textstyle \operatorname {artanh} x={\tfrac {1}{2}}\ln {\dfrac {1+x}{1-x}}} is the inverse hyperbolic tangent. Ideal points (points at

of n: m e k e Z e 2 m e r r = n ℏ , {\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar ,} so that the allowed orbit

error in the representation is | x − y x | ≤ ε mach . {\textstyle \left|{\dfrac {x-y}{x}}\right|\leq \varepsilon _{\text{mach}}.} The following proof is

t ∇ ⋅ B = 0 {\displaystyle {\begin{cases}\nabla \times \mathbf {E} &=-{\dfrac {\partial \mathbf {B} }{\partial t}}\\\nabla \cdot \mathbf {B} &=0\end{cases}}}

= α m n a c = α m n c / a {\displaystyle \omega _{mn}=\lambda _{mn}c={\dfrac {\alpha _{mn}}{a}}c=\alpha _{mn}c/a} . The values of α m n {\displaystyle

{\displaystyle \int {\frac {1}{ax+b}}\,dx={\begin{cases}{\dfrac {1}{a}}\ln(-(ax+b))+C^{-}&ax+b<0\\{\dfrac {1}{a}}\ln(ax+b)+C^{+}&ax+b>0\end{cases}}} is usually

1 + 1 n ) n = e , {\displaystyle \lim _{n\rightarrow \infty }\left(1+{\dfrac {1}{n}}\right)^{n}=e,} where n is the number of times the interest is to

> T / 2. {\displaystyle E(\tau )={\begin{cases}0&\tau \leq T/2\\[5pt]{\dfrac {T^{2}}{2\tau ^{3}}}&\tau >T/2.\end{cases}}} The variance is infinite. In

{\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)} all of which are continuous

t > 0 lim t → 0 + η ( t , x ) = δ ( x ) {\displaystyle {\begin{cases}{\dfrac {\partial }{\partial t}}\eta (t,x)=A\eta (t,x),\quad t>0\\[5pt]\displaystyle

2. {\displaystyle \phi _{kn}(r)=\max \left[0,\min \left(2r,\min \left({\dfrac {(1+2r)}{3}},2\right)\right)\right];\quad \lim _{r\to \infty }\phi _{kn}(r)=2

\sum _{r=1}^{m-1}\gamma _{1}{\biggl (}{\frac {r}{m}}{\biggr )}\cdot \cos {\dfrac {2\pi rk}{m}}=-\gamma _{1}+m(\gamma +\ln 2\pi m)\ln \left(2\sin {\frac {k\pi

Retrieved 13 April 2024. "DFRAC EXCLUSIVE: Fake handles from Pakistan misleading Twitteratis in the name of CPEC | DFRAC_ORG". dfrac.org. 14 May 2022. Retrieved

i ) {\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+\zeta _{i}+{\dfrac {K}{N}}\sum _{j=1}^{N}\sin(\theta _{j}-\theta _{i})} , where ζ i {\displaystyle

t} where γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\dfrac {v^{2}}{c^{2}}}}}}} is the Lorentz factor. Some authors generalize from

p l o y e d {\displaystyle {\mathrm {Official\ UK\ Unemployment\ Rate={\dfrac {All\ those\ aged\ 16\ and\ over\ classed\ as\ unemployed}{All\ those\ aged\

{\begin{array}{lcl}{\dfrac {dS_{1}}{dt}}&=&k_{1}X_{o}-k_{2}S_{1}\\[4pt]{\dfrac {dS_{2}}{dt}}&=&k_{2}S_{1}-k_{3}S_{2}\\[4pt]{\dfrac

Z_{0}=\left\{{\begin{matrix}Z_{\mathrm {f} }{\dfrac {\lambda _{\mathrm {g} }}{\lambda }}&{\text{(TE mode)}}\\\\Z_{\mathrm {f} }{\dfrac {\lambda }{\lambda _{\mathrm {g}

}=P_{0}\gamma ^{0}+P_{1}\gamma ^{1}+P_{2}\gamma ^{2}+P_{3}\gamma ^{3}={\dfrac {E}{c}}\gamma ^{0}-p_{x}\gamma ^{1}-p_{y}\gamma ^{2}-p_{z}\gamma ^{3}} appear

\Pr(U=u)={\begin{cases}p_{a}&{\text{if }}u=a;\\[8pt]p_{b}&{\text{if }}u=b\\[8pt]{\dfrac {1-p_{a}-p_{b}}{b-a-1}}&{\text{if }}a<u<b\\[8pt]0&{\text{otherwise}}\end{cases}}}

{z^{s}e^{-z}z^{k}}{s(s+1)\dots (s+k)}}=z^{s}e^{-z}\sum _{k=0}^{\infty }{\dfrac {z^{k}}{s^{\overline {k+1}}}}} where s k + 1 ¯ {\displaystyle s^{\overline

for  n ≠ 0 , 0 , for  n = 0 , {\displaystyle |c_{n}|^{2}={\begin{cases}{\dfrac {1}{n^{2}}},&{\text{for }}n\neq 0,\\0,&{\text{for }}n=0,\end{cases}}} and

definitions: E = − ∇ φ − ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla \varphi -{\dfrac {\partial \mathbf {A} }{\partial t}}} and B = ∇ × A {\displaystyle \mathbf

( $ ) {\displaystyle {\text{true price }}(\$)={\text{cost }}(\$)\cdot {\dfrac {{\text{consumption }}(\$)+{\text{depreciation }}(\$)}{{\text{credit

{0}}^{\operatorname {T} },&{\text{if }}{\vec {x}}={\vec {0}};\\{\dfrac {{\vec {x}}^{*}}{{\vec {x}}^{*}{\vec {x}}}},&{\text{otherwise}}.\end{cases}}}

j + 1 2 ) 2 − Z 2 α 2 ) 2 , {\displaystyle E={\frac {mc^{2}}{\sqrt {1+{\dfrac {Z^{2}\alpha ^{2}}{{\bigg (}{n-\left(j+{\frac {1}{2}}\right)+{\sqrt {\left(j+{\frac

otherwise . {\displaystyle f_{S}(s;\nu )\,\mathrm {d} s={\begin{cases}{\dfrac {\nu ^{\nu /2}\,s^{\nu -1}e^{-\nu \,s^{2}/2}\,}{2^{\left(\nu /2-1\right)}\Gamma

{\mathcal {E}}_{\text{D}}(\alpha ):={\dfrac {1}{2}}\langle \!\langle d\alpha ,d\alpha \rangle \!\rangle +{\dfrac {1}{2}}\langle \!\langle \delta \alpha

ζ ˙ j ∂ R ∂ ζ ˙ j ) , {\displaystyle {\frac {\partial R}{\partial t}}={\dfrac {d}{dt}}\left(R-\sum _{j=1}^{s}{\dot {\zeta }}_{j}{\frac {\partial R}{\partial