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Translating an integer interval by $xy$ changes a sum of unit-modulus terms by at most $2xy$. Averaging gives the displayed formula. For $n\asymp N$, $Z=N^{2/5}$, expand $-t\log (1+xy/n)$ to degree $r=\lfloor 5.01\log t/\log N\rfloor$. Its remainder is at most $t{N}^{-(r+1)/5}<t^{-1/500}$. This reduces the logarithmic exponential sum to bilinear polynomial sums while keeping explicit boundary and approximation errors.

By orthogonality of integer Fourier modes, this integral counts pairs of $k$-tuples with the same first $r$ power sums. Diagonal pairs give $J_{k,r}(Z)\gg Z^k$. There are $O_{k,r}(Z^{r(r+1)/2})$ possible moment vectors; the Cauchy-Schwarz inequality gives $J_{k,r}(Z)\gg Z^{2k-r(r+1)/2}$. Upper bounds measure how much arithmetic coincidence remains beyond these necessary contributions, and enter the Vinogradov mean-value method for a bilinear exponential sum.

For $J(\epsilon )={\int }_0^{\infty }{e}^{-x-\epsilon /x}dx/x$, reciprocity $x\mapsto \epsilon /x$ and the small-argument expansion of the exponential integral give $J=-\log \epsilon -2\gamma +o(1)$. Differentiation and integration by parts give $\epsilon J^{\prime \prime }+J^{\prime }=J$. Its Frobenius method yieldsThe logarithm is a singular contribution that a naive termwise expansion at $x=0$ misses.

If every nonidentity element of a finite group has order equal to a prime $p$, its order is a power of $p$. For any prime divisor $q$ of the group order, Cauchy theorem for groups gives an element of order $q$, forcing $q=p$. This statement does not imply that the group is abelian; odd-characteristic upper unitriangular groups provide counterexamples.

A mean-zero function has expectation zero with respect to the specified probability measure. On a finite set with uniform measure this means ${\sum }_{x}f(x)=0$. A balanced subset indicator is an example. In Fourier analysis on a finite group, this is equivalent to vanishing of the component at the trivial representation.

For independent integrable variables, $\mathbb{E}{\prod }_i{X}_i={\prod }_i\mathbb{E}{X}_i$. Tonelli theorem first gives $\mathbb{E}{\prod }_i|X_i|={\prod }_i\mathbb{E}|X_i|<\infty$; then Fubini's theorem factors the signed joint integral.

An idealized self-gravitating interior with constant mass density. Negligible surface pressure and spherical hydrostatic equilibrium give $M=4\pi \rho R^3/3$ and the uniform-density planetary pressure profile. It is the $n=0$ polytrope interpretation, rather than a literal substitution into $P=K{\rho }^{1+1/n}$. It approximates weak compression and misses real compositional stratification.

Transit spectroscopy infers the properties of an exoplanet atmosphere from the wavelength dependence of an exoplanet transit. Its measured product is an exoplanet transmission spectrum, while an annulus model for transmission spectroscopy connects the spectrum to atmospheric optical depth. The observing technique is described by NASA.

A primary planetary atmosphere is a retained gas envelope acquired from the protoplanetary nebula, commonly rich in molecular hydrogen and helium. The solar-system giant planets retain such envelopes, while escape and subsequent processing can substantially change their composition.

A secondary planetary atmosphere is supplied or replenished after the original nebular-envelope stage, through outgassing, volatile delivery and processing. The present atmospheres of Earth, Venus and Mars are predominantly secondary. The distinction from a primary planetary atmosphere concerns origin, not merely current molecular weight.

A family $C\subseteq {\omega }^{\omega }$ such that every $f\in {\omega }^{\omega }$ is eventually dominated by some $g\in C$.

The least size of a family in ${\omega }^{\omega }$ having no common bound for eventual domination. Every countable family $(f_i)$ has the bound $g(n)=1+{\max }_{i\le n}{f}_i(n)$, so ${\aleph }_1\le \mathfrak{b}\le 2^{{\aleph }_0}$.

Every finite point set in the unit square admits a cyclic visiting order with ${\sum }_i\Vert x_{i+1}-x_i{\Vert }^2\le 4$. Split the square into two right triangles, apply the quadratic path bound in a right triangle, and remove auxiliary corners using the law of cosines. The Cauchy-Schwarz inequality then gives $\mathrm{TS}(X)\le 2\sqrt{|X|}$. The tour minimizing ordinary length need not minimize the sum of squared lengths.

For every configuration $x$ in the unit square there are nonnegative weights $a_i(x)$ such that ${\sum }_i{a}_i(x{)}^2\le 16$ andUse the sums of the two incident edge lengths in a tour from the squared edge bound for tours in the unit square. Runs of missing vertices can be added by the cheaper of two out-and-back detours. This connects Euclidean travelling salesman tours to Talagrand's convex distance inequality.

The open planar sector between two distinct Euclidean rays of positive slope consists of nonzero points whose directions lie strictly between their directions. Sectors between consecutive members of an ordered collection of Euclidean rays are disjoint.

If finite point sets $P,Q$ lie on two distinct Euclidean rays in a plane, the map $(p,q)\mapsto p+q$ is injective because the two direction vectors are linearly independent. Thus $|P+Q|=|P||Q|$. If all points are nonzero and the rays have positive slope, every sum lies in the open planar sector between those rays.

For a full-rank Euclidean lattice of covolume $A$ in ${\mathbb{R}}^d$, bounded fundamental cells compare lattice points in a radius-$R$ ball with the volumes of balls of radii $R\pm C$. Hence $N(R)=\mathrm{vol}(B_1)R^d/A+O(R^{d-1})$. A lattice norm multiset therefore determines the covolume, which is useful for the spectrum of a flat torus.

In a rank-two Euclidean lattice, a shortest nonzero vector $v$ and a shortest vector $w\notin \mathbb{Z}v$ form a basis. The shortest vector is primitive. If the vertical coordinate of $w$ has index $m\ge 2$ in the projection lattice, a vector with smaller positive vertical coordinate can be reduced horizontally to length squared at most $(|v{|}^2+|w{|}^2)/4<|w{|}^2$, a contradiction. This lemma need not hold in higher rank.

The covolume of a full Euclidean lattice is the Lebesgue measure of a fundamental parallelepiped in its ambient inner product space. In orthonormal coordinates it is $|\det B|$ for a lattice basis matrix $B$. Scaling the lattice by $t>0$ multiplies the covolume by $t^n$; the dual lattice has reciprocal covolume.

The probability of a market event under an equivalent martingale measure for a specified numéraire. It can differ from the physical probability. For a random numéraire, impose the martingale condition on asset-price ratios rather than treating the numéraire return as deterministic.