The Mesozoic is the geological era from about 251.902 to 66.00 million years ago. It contains the Triassic, Jurassic and Cretaceous periods and precedes the Cenozoic. The chart gives uncertainty on the older numerical boundary.^[ref]

The Cenozoic is the geological era from 66.00 million years ago to the present. It comprises the Palaeogene, Neogene and Quaternary. Its subdivisions organize geological and climate records; their boundaries are defined stratigraphically, with numerical ages calibrated separately.^[ref]

The geological boundary at approximately 66 million years ago, associated with a major mass extinction. Deep lineage splits, ecological radiation and first fossil occurrence are distinct events in assessing mammalian history.

Unroll a circular cylinder of radius $R>0$ using the coordinate $u=R\varphi$. Its line element becomes $d{s}^2=d{u}^2+d{z}^2$, so a geodesic lifts to a straight line in the plane. An endpoint has infinitely many lifts differing by $2\pi Rk$. Minimize the ordinary Euclidean distance over those lifts to obtain the unrestricted shortest path. A fixed lift instead specifies a winding class and need not give the global shortest path.

For angular difference $0\le \delta \le 2\pi$, put $m=\min (\delta ,2\pi -\delta )$. Then $d=\sqrt{R^2{m}^2+(z_1-z_0{)}^2}$. At $\delta =\pi$, two different geodesics have the same shortest length. This is the source of the corner in the distance-versus-angle graph at the opposite generator of the circular cylinder.

A geochemist studies the distribution, chemical state and movement of elements and isotopes in Earth and planetary materials. Chemical reactions, phase equilibria and transport link measured composition with geological processes.

An isotope ratio can trace a source or constrain an age, but interpreting it requires a model of fractionation, mixing and possible later alteration. A measured number becomes a geological explanation only through those explicit physical and chemical assumptions.

For $\alpha >0$ and a nonnegative integer $m$, substitute $u=\alpha x^2$. Then $x^{2m+1}dx=u^{m}du/(2{\alpha }^{m+1})$, so the integral reduces to the gamma function value $\Gamma (m+1)=m!$. This proves the formula. The corresponding whole-line integral is zero by odd symmetry, and both integrals converge absolutely. A half-line integral of an odd function need not vanish.

A Gaussian prime is a nonzero nonunit Gaussian integer irreducible in $\mathbb{Z}[i]$. Since this ring is a Euclidean domain, irreducible and prime elements coincide. For $a,b\ne 0$, $a+bi$ is a Gaussian prime exactly when $a^2+b^2$ is an ordinary prime number. On a coordinate axis, it is prime exactly when the nonzero coordinate has absolute value an ordinary prime congruent to three modulo four.

Thus $1+i$, $2+i$, and $3$ are Gaussian primes, while $2=-i(1+i{)}^2$ and $5=(2+i)(2-i)$ are not. The multiplicative norm $N(a+bi)=a^2+b^2$ makes these factorizations and primality tests transparent.

The Gaussian Möbius function on nonzero Gaussian integers is zero when a square of a Gaussian prime divides the argument, and otherwise equals $(-1{)}^r$, where $r$ is the number of nonassociate prime factors. It takes value one on units and is unchanged by multiplying its argument by a unit.

To sum over divisors without counting associates four times, one may define it on nonzero principal ideals, or choose one generator from each associate class. Then ${\sum }_{\mathfrak{d}\mid \mathfrak{a}}\mu (\mathfrak{d})$ is one for the unit ideal and zero otherwise: for a product of $r$ distinct prime ideals the sum is $(1-1{)}^r$. This gives Möbius inversion in Gaussian arithmetic.

A nonzero Gaussian integer is squarefree if no square of a Gaussian prime divides it. Equivalently, every exponent in its Gaussian prime factorization is at most one, disregarding multiplication by units.

The Gaussian integer $2$ is not squarefree, because $2=-i(1+i{)}^2$. By contrast $5=(2+i)(2-i)$ is squarefree: its factors are nonassociate Gaussian primes. Thus squarefreeness in $\mathbb{Z}[i]$ need not agree with squarefreeness in $\mathbb{Z}$. This distinction is essential when using a Gaussian Möbius function to exclude repeated prime factors.

The product of a Gaussian integer with its complex conjugate. It is multiplicative: $N(zw)=N(z)N(w)$, which gives the Brahmagupta-Fibonacci identity.

2016 paper: Public original PDF, physical page 807. Student-answer availability for this paper remains unverified.

Historical Part III experimental condensed-matter physics papers. Availability of student answers has not been established; years with advertised answers are skipped when identified. The original public paper is linked from each solutions page.

For real wave impedances at normal incidence, the electric-field transmission coefficient $t$ gives transmitted power fraction $T=(Z_1/Z_2)|t{|}^2$. Reflection gives $R=|r{|}^2$. Enforcing both tangential electric and magnetic boundary conditions gives $r=(Z_2-Z_1)/(Z_2+Z_1)$ and $t=2Z_2/(Z_1+Z_2)$, which satisfy $R+T=1$.

For a ray meeting an interface at nonzero incidence angle, the plane spanned by the incoming wavevector and the surface normal. transverse electric polarization has electric field perpendicular to this plane, while transverse magnetic polarization has electric field in it. At exact normal incidence, the surface normal and ray alone do not select a unique plane.

The ratio of an envelope width to a fringe period counts spatial intervals. It need not equal the number of bright fringe centres strictly inside the envelope's zeros. For a symmetric overlap from phase index $-25$ to $25$, the width spans 50 periods, but the strictly interior integer centres are $-24$ through $24$, totaling 49. Endpoints where one aperture amplitude vanishes have no two-beam contrast. State the count convention rather than treating these as inconsistent answers.

2016 paper: Public original PDF, physical page 810. Student-answer availability for this paper remains unverified.

To solve $\log G=A$ with $A(0)=0$, start with $G=1$ and update $G\leftarrow G(1+A-\log G)$ at successively doubled truncation precision. The correction removes the current logarithmic error to first order. This is Newton iteration applied to the formal power series exponential, using a formal power series logarithm.

For low-density gas with Maxwellian electrons, the excitation-energy difference between the ${}^1{S}_0$ and ${}^1{D}_2$ levels of doubly ionized oxygen makes $(I_{4959}+I_{5007})/I_{4363}$ temperature sensitive. A constant-collision-strength approximation is $R\simeq 7.90e^{32900\mathrm{K}/T_e}$. Increasing temperature strengthens the higher-excitation auroral line and lowers $R$. Density, extinction and nonthermal electrons require separate treatment; the $5007/4959$ pair shares one upper level and is primarily a branching ratio rather than this thermometer.

The sign of the initial direct P wave motion after instrument, component and ray-direction conventions are calibrated. It constrains the sign of ${\hat{r}}^{\mathsf{T}}M\hat{r}$ at the source takeoff direction. Near-nodal arrivals and uncertain takeoff angles need special treatment.