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Equation Index

Authors
Affiliations
University of British Columbia
University of British Columbia
s=1vs=\frac{1}{v}
ρ=1/σ\rho = 1/\sigma
μ=μ0(1+κ)\mu = \mu_0 (1 + \kappa)
F[m]=d\mathcal{F}[m] = d
dj=Fj[m]d_j=\mathcal{F}_j[m]
F[αf+βg]=αF[f]+βF[g]\mathcal{F}[\alpha f+\beta g]=\alpha\mathcal{F}[f]+\beta\mathcal{F}[g]
dj=abgj(x)m(x)dxd_j=\int^b_ag_j(x)m(x)dx
y=mwy = m \otimes w
y(tj)=m(τ)w(tjτ)dτy(t_j)=\int^{\infty}_{-\infty}m(\tau)w(t_j-\tau)d\tau
B(rj)=μ04πJ(r)×r^rjr2\vec{B}(\vec{r}_j)=\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}\prime)\times\hat{r}}{\left|\vec{r}_j-\vec{r}\prime\right|^2}
dj(1)=abgj(x)m1(x)dxd_j^{(1)}=\int^b_a g_j(x)m_1(x)dx
dj(2)=abgj(x)m2(x)dxd_j^{(2)}=\int^b_a g_j(x)m_2(x)dx
dj=abgj(x)[αm1(x)+βm2(x)]dj=αdj(1)+βdj(2)\begin{aligned}d_j=\int^b_a g_j(x)[\alpha & m_1(x) +\beta m_2(x)]\\ d_j=\alpha &d_j^{(1)}+\beta d_j^{(2)}\end{aligned}
dj=abgj(x)m2(x)dxd_j=\int^b_ag_j(x)m^2(x)dx
σV=Iδ(rr)\nabla\cdot\sigma\nabla V=-I\delta(\vec{r}-\vec{r}\prime)
Vrms2(tj)=1tj0tmaxvint2(u)H(tju)duV^2_{rms}(t_j)=\frac{1}{t_j}\int^{t_{max}}_0 v^2_{int}(u)H(t_j-u)du
Vrms2(t)=1t0tmaxvint2(u)H(tu)duV^2_{rms}(t)=\frac{1}{t}\int^{t_{max}}_0 v_{int}^2(u)H(t-u)du
vint=Vrms(t)(1+2tVrms(t)Vrms(t))12v_{int}=V_{rms}(t)\left(1+\frac{2tV'_{rms}(t)}{V_{rms}(t)}\right)^{\frac{1}{2}}
Vrms(t)=v0+asin(2πft)V_{rms}(t)=v_0+a\sin(2\pi ft)
Vrms(t)=2πafcos(2πft)V'_{rms}(t)=2\pi af\cos(2\pi ft)
dobs=dtrue+δdd^{obs}=d^{true}+\delta d
F1[dtrue]=mc\mathcal{F}^{-1}[d^{true}]=m_c
F1[dobs]=F1[dtrue+δd]=mc+δm\mathcal{F}^{-1}[d^{obs}]=\mathcal{F}^{-1}[d^{true}+\delta d]=m_c+\delta m
δvintftav0\delta v_{int}\propto \sqrt{\frac{fta}{v_0}}
P(mdobs)P(dobsm)P(m)P(m|d^{obs})\propto P(d^{obs}|m)P(m)
Minimize  ϕ=ϕd+βϕmsubject to  ml<m<mu,\begin{aligned}&\text{Minimize}~~\phi=\phi_d+\beta\phi_m\\ &\text{subject to}~~m_l<m<m_u,\end{aligned}
gj(x)=epjxcos(2πqjx)g_j(x)= e^{p_jx}\cos(2\pi q_jx)
dj=0x1gj(x)m1dx+x1x2gj(x)m2dx+=i=1M(xk1xkgj(x)dx)midj=gjm\begin{aligned} d_j&=\int_0^{x_1}g_j(x)m_1dx +\int_{x_1}^{x_2}g_j(x)m_2dx+\dots \\ &=\sum^M_{i=1}\left(\int_{x_{k-1}}^{x_k}g_j\left(x\right)dx\right)m_i\\ &\\ d_j &= \mathbf g_j \mathbf m\end{aligned}
d=Gm=[d1dN]\begin{aligned} \mathbf{d} = \mathbf{G}\mathbf{m} = \begin{bmatrix} d_1\\ \vdots\\ d_{N} \end{bmatrix}\end{aligned}
Gjk=xk1xkgj(x)dxG_{jk} = \int_{x_{k-1}}^{x_k} g_j(x) dx
dobs=d+n\mathbf{d}^{obs}=\mathbf{d}+\mathbf{n}
ϵj=%dj+νj\epsilon_j = \%|d_j| + \nu_j
ϕd=j=1N(djdjobsϵj)2\phi_d=\sum^N_{j=1}\left(\frac{d_j-d_j^{obs}}{\epsilon_j}\right)^2
ηj=djdjobsϵj\eta_j = \frac {d_j - d_j^{obs}} {\epsilon_j}
E[χ2]=NE[\chi^2] = N
Var[χ2]=2NVar[\chi^2] = 2N
ϕd=N\phi_d^*=N
ϕm=(mmref)2dx\phi_m=\int (m-m^{ref})^2 dx
ϕm=(d(mmref)dx)2dx\phi_m=\int \left(\frac{d(m-m^{ref})}{dx}\right)^2 dx
ϕm=αs(mmref)2dx+αx(d(mmref)dx)2dx\phi_m=\alpha_s\int (m-m^{ref})^2 dx+\alpha_x\int (\frac{d(m-m^{ref})}{dx})^2 dx
ϕm=mmref2=i=1M(mimiref)2\phi_m=\|\mathbf{m - m^{ref}}\|^2=\sum_{i=1}^M(m_i-m^{ref}_i)^2
ϕm=dmdx2=i=1M1(mi+1mi)2\phi_m=\|\frac{d\mathbf{m}}{dx}\|^2=\sum_{i=1}^{M-1}(m_{i+1}-m_i)^2
ϕ(m)=ϕd(m)+βϕm(m)\phi(m) = \phi_d(m)+\beta\phi_m(m)
ϕ=T+βF\phi=T+\beta F
gj(x)=epjxcos(2πqjx)Δx\mathbf{g}_j(\mathbf{x}) = e^{p_j\mathbf{x}} cos (2 \pi q_j \mathbf{x}) \Delta x
G=[g1gN]\begin{aligned} \mathbf{G} = \begin{bmatrix} \mathbf{g}_1\\ \vdots\\ \mathbf{g}_{N} \end{bmatrix} \end{aligned}