Geochemistry uses chemical and isotopic measurements to investigate Earth materials and the processes affecting them. Element abundances and isotope ratios help distinguish sources, reaction histories and mixing. A concentration change can arise from adding an element or from removing other material, so mass balance is essential.

Trace elements respond differently to melting and crystallization because minerals incorporate them unequally. Radioactive systems can constrain time, while isotope fractionation records some physical or chemical transformations. Several independent measurements are needed when alteration or mixing could reproduce the same pattern as the proposed original process.

The binomial transform of a sequence $a_k$ is $b_n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a_k$. When $a_k$ counts objects on a specified set of $k$ positions, $b_n$ counts the choices obtained by selecting those positions from $n$ and filling them with such an object. Its inverse is $a_n={\sum }_{k=0}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n}{k})b_k$, by binomial inversion.

A rational generating function is a formal power series equal to a quotient of two polynomials, with denominator nonzero at the origin. Multiplying by the denominator shows that its coefficients eventually satisfy a linear recurrence with constant coefficients.

The Euler transform of a sequence $(b_n{)}_{n\ge 1}$ has coefficients $a_n$ defined by ${\sum }_{n\ge 0}{a}_n{x}^n={\prod }_{n\ge 1}(1-x^n{)}^{-b_n}$. In unlabelled counting, $b_n$ may count connected components of size $n$, while $a_n$ counts multisets of components.

An algebraic generating function is a formal power series $F(x)$ satisfying a nonzero polynomial equation $P(x,F(x))=0$. Over a field of characteristic zero, it is D-finite.

For $a,b>0$, the integrand is even and integrable at zero and infinity. On the positive half-axis the change of variables $u=\sqrt{a}x-\sqrt{b}/x$ maps $(0,\infty )$ bijectively onto $\mathbb{R}$ and satisfies $a{x}^2+b/x^2=u^2+2\sqrt{ab}$. Solving the quadratic for the positive root givesAfter doubling this half-axis integral, the odd part of the transformed integrand integrates to zero and the remaining Gaussian integral yields the formula. The reciprocal substitution $x=1/t$ also givesThe positive parameters ensure that the substitutions and improper integrals are valid; the second identity can also be recovered by differentiating the first in $b$ under the integral sign.

Multiply the two exponential series in $e^{-(x-a{)}^2}=e^{-a^2}{e}^{2ax}{e}^{-x^2}$ to obtain the displayed Taylor expansion. Its first three nonzero terms depend on the real parameter $a$. For $a\ne 0,\pm 1/\sqrt{2}$, they end at degree two. At $a=0$, odd coefficients vanish and the first three terms are $1-x^2+x^4/2$. At $a=\pm 1/\sqrt{2}$, the quadratic coefficient vanishes, leaving $e^{-1/2}(1+2ax-4a{x}^3/3)$ as the first three nonzero terms. Parameter-dependent vanishing coefficients must be checked when a question counts nonzero terms rather than prescribing an expansion order.

For a Hermitian adjoint scalar field with kinetic term $\mathrm{tr}(D_{\mu }\Phi (D^{\mu }\Phi {)}^{†})$, use $D_{\mu }\Phi ={\partial }_{\mu }\Phi +ig[A_{\mu },\Phi ]$ and generators normalized by $\mathrm{tr}(t^a{t}^b)={\delta }^{ab}/2$. Expanding at a constant vacuum expectation value ${\Phi }_0$ and matching ${\mathcal{L}}_{\mathrm{mass}}=A_{\mu }^a(m^2{)}^{ab}{A}^{b\mu }/2$ gives the displayed expression. A real combination of generators $X=x_a{t}^a$ has quadratic form $2g^2\mathrm{tr}([X,{\Phi }_0{]}^{†}[X,{\Phi }_0])\ge 0$. Thus the massless directions are exactly the generators commuting with the vacuum.

For ${\Phi }_0=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, the two Hermitian generators associated with each matrix unit pair $E_{ij},E_{ji}$ have squared mass $g^2({\lambda }_i-{\lambda }_j{)}^2$. The surviving group is the centralizer of a Hermitian matrix in SU(n).

Gas can shock and radiate during a galaxy encounter, allowing it to change orbital energy in ways unavailable to an ideal collisionless stellar component. Gravitational torques can transport its angular momentum and fuel central star formation or accretion. The outcome depends on orbit, gas content and internal structure.

For $X\sim {\chi }_{\nu }^2$, the Wilson-Hilferty transformation uses the approximationThe cube root reduces the right skewness of the chi-squared distribution. Inverting gives the approximate $p$-quantile $\nu [1-2/(9\nu )+z_p\sqrt{2/(9\nu )}{]}^3$, where $z_p$ is a standard normal quantile. This is an approximation, especially imperfect for extreme tails or small degrees of freedom. For a Gamma distribution with shape $\alpha$ and rate $\beta$, apply it to $X=2\beta Y$, which has $\nu =2\alpha$.^[ref]

To leading order neglecting nuclear recoil, a measured gamma ray has energy equal to the difference between the upper and lower nuclear energy levels. For a freely recoiling atomic nucleus initially at rest, four-momentum conservation instead giveswhere $M_u$ and $M_l$ are the rest masses of the upper and lower nuclear states. Some transition energy becomes daughter kinetic energy, so high-precision level reconstruction must account for nuclear recoil.

Reconstructing a nuclear gamma ray radiation spectrum amounts to assigning measured photon energies to differences between ordered nuclear energy levels with such corrections where needed. Cascade energy sums constrain the graph, while population-flow balances constrain its orientation and feeding. Experimental central energies need not close exactly; a weighted least squares energy-network fit uses their uncertainties consistently. Multiple graphs can satisfy the same radiation spectrum when coincidence or lifetime information is absent.

A Silurian jawless vertebrate preserving continuous paired ventrolateral fins, interpreted as a precursor to more localized paired appendages.

For $\alpha >1$, write $[\mathrm{sech}x-\mathrm{sech}(\alpha x)]/x={\int }_1^{\alpha }\mathrm{sech}(xy)\tanh (xy)dy$. The integrand is nonnegative, so Tonelli's theorem permits reversal. Substitution $u=xy$ makes the inner $x$ integral $1/y$ because ${\int }_0^{\infty }\mathrm{sech}u\tanh udu=1$. Integrating $dy/y$ proves the formula. The original difference is $O(x^2)$ at zero, making division by $x$ integrable. For $0<\alpha <1$, substitution and reversal of the two terms give the same $\ln \alpha$ result; at $\alpha =1$ both sides vanish.

Over a commutative ring containing $\mathbb{Q}$, a formal power series $f=1+u$ with $u\in xR[[x]]$ has the formal logarithmEvery coefficient receives only finitely many contributions, since $u^k$ is divisible by $x^k$. No analytic convergence is involved. For example, $\log (1+x)=x-x^2/2+x^3/3-\cdots$.

The formal derivative satisfies $(\log f{)}^{\prime }=f^{\prime }/f$, and $\log (fg)=\log f+\log g$ for commuting series with constant coefficient one. This turns multiplicative identities for generating functions into additive ones. In positive characteristic, division by $k$ may fail, so this definition cannot be transferred unchanged.

Here an inverse of a formal power series means its multiplicative inverse: $f(x{)}^{-1}f(x)=1$. The formal power series reciprocal exists exactly when the constant coefficient of $f$ is a unit in the coefficient ring. Thus $1-x$ is invertible in $\mathbb{Z}[[x]]$, while $2+x$ is not, though it is invertible in $\mathbb{Q}[[x]]$.

A compositional inverse solves $f(g(x))=x$, which is a different problem. For instance, the multiplicative inverse of $1-x$ is $1+x+x^2+\cdots$, whereas the compositional inverse of $x+x^2$ starts $x-x^2+2x^3-\cdots$. Stating which operation is inverted prevents a common ambiguity.

A q-series is a series in a parameter $q$, treated either as a formal power series or as an analytic series, often for $|q|<1$. Its coefficients can encode a sequence, while identities involving infinite products record choices of parts in integer partitions.

For example,where $p(n)$ counts integer partitions of $n$. Expanding each factor as a geometric series chooses the number of parts of size $m$. Formally, each coefficient uses only finitely many factors. This product is a basic link between combinatorics, number theory, and modular forms.

A formal power series $f(x)$ over a field of characteristic zero is D-finite if it satisfies a nontrivial linear differential equation with polynomial coefficients:The polynomials are not all zero. Equivalently, the coefficients of $f$ satisfy a linear recurrence relation with polynomial functions of the index as coefficients.

For example, $e^x$ satisfies $f^{\prime }-f=0$, and every rational function regular at zero is D-finite. The class is closed under sums, products, and derivatives. This permits symbolic computation with many generating functions using finite equations rather than an infinite list of coefficients.

For a formal power series $A$ with zero constant term over a characteristic-zero field, its exponential is $\exp A={\sum }_{k\ge 0}{A}^k/k!$. Every coefficient is a finite sum because $A^k$ starts in degree at least $k$. In finite characteristic, truncated computations require the factorial denominators that occur to be invertible.

If the constant coefficient of $A$ is invertible, then for every formal power series $B$ there is a unique quotient $Q$ satisfying $AQ=B$. Comparing coefficients determines $Q$ recursively; multiplying $B$ by the formal power series reciprocal of $A$ computes the same quotient.