An imagined sphere around a seismic point source on which source-to-station ray takeoff directions are plotted. A lower-hemisphere equal-area projection commonly displays first-motion polarities and nodal planes.

Interchanging the slip direction and fault normal leaves a double-couple moment tensor unchanged. First-motion polarities therefore cannot identify the physical fault plane without information such as mapped rupture, aftershock geometry or geodetic displacement.

The negative and positive principal directions of a double-couple moment tensor. They bisect the nodal-plane quadrants, but are not automatically the principal axes of regional stress. A stress inversion needs additional faulting assumptions and multiple mechanisms.

For integers $n\ge 0$, $m>0$, write $F(n,a,b,m)={\sum }_{i=0}^{n-1}\lfloor (ai+b)/m\rfloor$. Euclidean division gives $a=qm+r$ and $b=sm+t$, with $0\le r,t<m$, henceFor $0<a<m$ and $0\le b<m$, put $y=an+b$. Interchanging the two axes in the remaining lattice point problem givesIf $y<m$, the original sum is zero; if $a=0$ after reduction, it is also zero. Each swap lowers the modulus as in the Euclidean algorithm, enabling logarithmically many reduction stages. For example $F(5,3,1,7)=3$, and the swapped sum $F(2,7,2,3)=0+3$ gives the same answer.

For an arithmetic function $a$, changing the order of the finite summation gives ${\sum }_{d=1}^{n}a(d)\lfloor n/d\rfloor ={\sum }_{t=1}^n{\sum }_{d\le \lfloor n/t\rfloor }a(d)$. Both sides count pairs of positive integers $(d,t)$ with $dt\le n$, assigning weight $a(d)$ to each pair. This expresses a weighted floor sum in terms of a summatory function.

In the specified thin-sheet deep-water model, bending stiffness adds restoring force while sheet mass adds inertia. Their ratio to an uncovered gravity-capillary wave gives the displayed expression. Added inertia first lowers frequency at small wavenumber; at formally large wavenumber bending produces a frequency ratio proportional to $k^{1/2}$. That formal asymptote must not be extrapolated past the range where a thin-sheet continuum description is valid.

For a three-dimensional quadratic branch, the critical thermal density is $n_T=\zeta (3/2)[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Inserting this into the mean-field gap-closing condition gives $H_c(T)-H_g=(\nu /\eta )n_T\propto T^{3/2}$. This is a dilute low-temperature model; finite bandwidth, anisotropy and changes in interactions can alter its applicability outside that regime.

If the phenomenological single-density shift is continued into the condensed phase and the thermal population remains saturated at $n_T(T)$, the zero minimum requires $\nu [n_0+n_T]=\eta (H-H_g)$. At fixed temperature, $n_0=(\eta /\nu )[H-H_c(T)]$ above onset and zero below it. This prefactor belongs to that closure; a microscopic interaction model can distinguish normal and condensed self-energies. Continuing the normal parabola into negative energies without changing the state is not a consistent Bose distribution.

The complete wavefunction of identical fermions changes sign on exchange of two particles, including their spatial and internal coordinates. The Pauli exclusion principle follows because an antisymmetrized product with two identical complete spin orbitals is zero. Antisymmetry concerns the total state; a spin-antisymmetric singlet state can have a symmetric spatial factor.

The mass reached after consuming an idealized local annulus of solids. With full annular width $\Delta a=Ba[M_{\mathrm{iso}}/(3M_{\star }){]}^{1/3}$ and solid surface density ${\Sigma }_s$, the balance $M_{\mathrm{iso}}=2\pi a\Delta a{\Sigma }_s$ gives the displayed value. Radial replenishment, migration and embryo mergers can overcome local isolation; defining the width with a mutual rather than single-body Hill radius changes the numerical coefficient.

A layered neural network with affine maps and activation functions, usually with dense connections between successive layers. A dense layer with $n$ inputs and $m$ outputs has $m(n+1)$ parameters including biases.

For $Q=p-if(x)$, $H=Q^{†}Q/(2m)$ and $Q{\psi }_0=0$, the trial state ${\psi }_1=Q^{†}{\psi }_0$ obeys the displayed identity. It follows from $[Q,Q^{†}]=2\hslash f^{\prime }$ and $[p,f^{\prime }]=-i\hslash f^{\prime \prime }$. The coefficients are generally position-dependent multiplication functions, not constants. For linear positive $f$, the first term vanishes and the second becomes the harmonic-oscillator level spacing.

For equivalent scatterers on the four conventional fcc sites, the centering factor is $1+e^{i\pi (h+k)}+e^{i\pi (h+l)}+e^{i\pi (k+l)}$. It is four when the integer indices are all even or all odd, and zero otherwise. An additional atomic basis of a crystal can impose further extinctions. This rule concerns conventional cubic indexing.

For an exponential generating function $F(x)={\sum }_{n\ge 0}{a}_n{x}^n/n!$, the ordinary coefficient operator gives $[x^n]F(x)=a_n/n!$. To recover the enumerated quantity, multiply by $n!$:For $F(x)=e^{2x}$, the ordinary coefficient is $2^n/n!$, but the represented counting sequence is $a_n=2^n$. Products have the binomial-convolution rule $n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k{b}_{n-k}$. This follows by multiplying the two series and explains why exponential generating functions suit labeled combinatorial structures.

A fixed-efficiency closed box with permanent locking fraction $\alpha$ predicts exponential decay of gas and gross star formation. Backward extrapolation therefore grows exponentially with lookback time. It requires a stated present age and assumes the same gas reservoir and efficiency over the entire interval. Cosmological inflow, outflow, mergers and changing gas conditions generally invalidate such an extrapolation over many depletion times.

2016 paper: Public original PDF, physical page 804. Student-answer availability for this paper remains unverified.

The first-to-fourth-contact duration of an exoplanet transit by a spherical opaque planet of radius ratio $k=R_p/R_{\star }$ on a circular orbit. The transit impact parameter is $b=(a/R_{\star })\cos i$. The formula includes finite planet size and orbital curvature; it reduces to the transit duration of a small planet when $k\to 0$ and $R_{\star }/a\ll 1$. It describes contact geometry, not a uniquely specified light-curve threshold in the presence of limb darkening.

The probability that an isotropically oriented circular orbit produces an exoplanet transit, assuming $a>R_{\star }+R_p$. Uniformity of $|\cos i|$ gives the displayed result. Orbital eccentricity, a restriction to full rather than grazing crossings, or a non-isotropic orientation distribution changes the expression.

A condensation claim needs macroscopic low-energy occupation and spontaneous coherence together with a model of cooling, lifetime and pumping. Spectral narrowing, a nonlinear intensity threshold or long-range transport alone can have competing explanations.

Two identical uniform circular discs of angular diameter $\alpha$ and equal flux have $g_{\star }(Y)=2J_1(x)/x$, $x=\pi \alpha Y/\lambda$, with limiting value one at $Y=0$. Multiplication by $\cos (\pi \beta Y/\lambda )$ gives their signed binary coherence. Its first single-disc envelope zero is $Y=1.21967\lambda /\alpha$, while the binary modulation has a separate scale $\lambda /\beta$. Limb darkening and other nonuniform profiles change the envelope, not the elementary Fourier shift factor.