For $S\subseteq {\mathbb{F}}_p$ with $|S|\ge 2$, put $R=(S-S)/(S-S)$ using nonzero denominators. If $R$ is proper, some $r\in R+1$ lies outside $R$, since invariance under addition of one would give all of the prime field. The map $(x,y)\mapsto x+ry$ on $S^2$ is injective. Clearing the denominator in a representation of $r-1$ puts a dilate of its image inside $3SS-3SS$. If $R$ is the full field, averaging the collision count of $S+rS$ gives some $r$ with $|S+rS|\ge \frac{1}{2}\min (p,|S{|}^2)$. Clearing its denominator puts a dilate inside $2SS-2SS\subseteq 3SS-3SS$. Here $SS$ is a product set, and the prefactors denote repeated sumsets, not scalar multiplication.

Write $D(t)=|A+tA|/|A|$ and suppose some nonzero $b$ has $D(b)\le K$. The mixed-set Plünnecke-Ruzsa inequality gives $|3A-2A|\le K^5|A|$ and $|A+A|\le K^2|A|$. The Ruzsa covering lemma covers $tA$ by at most $D(t)$ translates of $A-A$. Covering two dilates proves the addition bound. First covering $sA$, then both copies of $rA$ in $r(A-A)$, proves the multiplication bound. The Ruzsa triangle inequality also gives $D(-r)\le K^{2}D(r)$. These bounds transfer controlled additive expansion through a bounded-complexity expression in scalars.

An operator has weak type $(p,q)$, with finite $q$, if the displayed bound holds for every input and every $\lambda >0$, with one constant independent of both. It controls the distribution of its output rather than requiring its strong $L^q$ norm to be finite.

In the functional-analytic sense used here, a superlinear functional is real valued, positively homogeneous and superadditive. Equivalently $-q$ is a sublinear functional. This definition concerns functional inequalities, rather than a comparison of asymptotic growth rates. Such a $q$ is concave, satisfies $q(0)=0$, and can serve as a lower support in the sandwich form of the Hahn-Banach theorem.

For $y^{\prime \prime }+\lambda x^{-2}y=0$ on $0<a<x<b$ with zero endpoint values, write $t=\log (x/a)$ and $y=\sqrt{x}w(t)$. The Euler-Cauchy equation becomes $w^{\prime \prime }+(\lambda -1/4)w=0$. Thus $y_n=\sqrt{x}\sin [n\pi \log (x/a)/\log (b/a)]$ and the exact spectrum is displayed above. Leading WKB quantization omits the constant $1/4$, while producing the same leading eigenfunction shape after quantization.

For a finite simple graph with at least two vertices, no triangles, and exactly three common neighbours for every nonadjacent pair, the graph is connected. For adjacent vertices $u,v$, the bipartite graph between $N(u)\setminus \{v\}$ and $N(v)\setminus \{u\}$ has degree two at every vertex, forcing $d(u)=d(v)$. Thus the graph is regular. Counting second neighbours gives $3(n-d-1)=d(d-1)$. The nontrivial adjacency eigenvalues are $(-3\pm \sqrt{4d-3})/2$, and the larger-root multiplicity is $[n-1+d^2/\sqrt{4d-3}]/2$. Integrality forces $t=\sqrt{4d-3}$ to be an odd integer dividing $9$, yielding the stated degrees. The complete bipartite graph $K_{3,3}$ realizes degree three. The one-vertex graph is a vacuous exception with degree zero.

An aligned source behind an axisymmetric gravitational lens produces a circular image at an Einstein radius. Its angular radius satisfies ${\theta }_E^2=4G{M}_{\mathrm{projected}}(<D_d{\theta }_E)D_{ds}/(c^2{D}_d{D}_s)$. Exact alignment gives a ring; displaced sources generally give arcs or distinct images rather than a complete ring.

A critical curve is a locus in the image plane where the Jacobian matrix of the thin gravitational lens equation becomes singular. Its formal point-source lensing magnification diverges, while finite source size regularizes the measured flux. Its mapped locus in the source plane is a gravitational-lensing caustic.

The two resolved quasar images of 0957+561 provide an observational question: are they separate sources, or different light paths from one source? In May 1979, spectrophotometry used the McDonald Observatory 2.7 m reflecting telescope, the UVITS spectrograph, and an intensified dissector scanner at the f/9 focus of the Cassegrain reflector. A 600-groove-per-millimetre diffraction grating produced 4100-7100 ångström scans sampled every 2 ångströms, with about 7 ångström resolution over most of that range. Two spectrograph slits, about 52 arcseconds apart, recorded object and sky background simultaneously for subtraction. Interleaving observations of the two images reduced instrumental and atmospheric gain changes.^[ref]

The readout was received light versus wavelength for each image. Closely matching absorption line redshifts supported a common source and absorber; differences in continua and line strengths motivated differential interstellar extinction. Spectral agreement supplied evidence for gravitational lensing, rather than proving the point-mass gravitational lens model. The authors discussed a physical double and discrepant optical and radio light ratios. Their note added in proof attributed the redder spectrum to foreground galaxy contamination and emphasized an extended mass distribution.^[ref]

A violation of the strong equivalence principle can make the gravitational-to-inertial mass ratio depend on a body's negative gravitational self-energy $E_G$. Bodies with different self-binding fractions would then fall differently in the same external field, even after composition-dependent weak-equivalence effects are removed. For the Earth and Moon this produces a polarization of the lunar orbit toward the Sun, measurable by lunar laser ranging. General relativity predicts ${\eta }_N=0$.

The length of a word over an alphabet is the number of symbol positions in that finite string, counting repetitions. Thus $\ell (abba)=4$ and $\ell (\epsilon )=0$ for the empty word. Concatenation satisfies $\ell (uv)=\ell (u)+\ell (v)$. This differs from the word length of a group element, which minimizes over representing words using a chosen generating set of a group.

Concatenation joins two strings end to end, retaining the order of their symbols. For example, joining $ab$ and $ba$ gives $abba$. It has the associative property: joining $(uv)w$ or $u(vw)$ gives the same word over an alphabet. The empty word is its identity element. All finite words over an alphabet with this binary operation form a free monoid.

For a strictly distance-decreasing map of a finite-dimensional Euclidean space, one bounded forward orbit suffices for existence of a fixed point. Its closure is compact by the Heine-Borel theorem and invariant by continuity, so strict distance decrease on a compact metric space applies. Strict distance decrease gives uniqueness throughout the whole space, not merely the orbit closure.

For $d>1$, the displayed stress-energy tensor is symmetric and conserved on the free massless scalar field equation $\square \varphi =0$. Its improvement is identically divergence-free. Since $\square {\varphi }^2=2(\partial \varphi {)}^2$ on shell, its trace vanishes. Contracting it with a Conformal Killing vector field therefore produces a conserved conformal current. In $d=2$ the improvement coefficient is zero: the canonical tensor already has zero trace.

These stress measures refer force or area to the reference configuration. Different index-order conventions distinguish a first Piola-Kirchhoff stress tensor from its transposed nominal stress tensor. The second Piola-Kirchhoff stress tensor has two material indices.

The Kirchhoff tensor is the Cauchy stress tensor multiplied by the local volume ratio. Its stress power is per reference volume. Pushing forward the second Piola-Kirchhoff stress tensor gives the displayed formula.

With the right Cauchy-Green deformation tensor $C=F^{T}F$ and the second Piola-Kirchhoff stress tensor $S$, this mixed material stress is generally nonsymmetric. In the material-first nominal stress tensor convention, $PF=SC=M^T$. Thus replacing $P{F}^{-T}$ by $PF$ changes the stress measure. The relation follows from $\mathcal{P}=FS$, without any assumption of material isotropy.

A stress measure $T$ is work-conjugate to a strain measure $E$ when their contraction gives the mechanical power per specified volume for every admissible deformation rate. The reference-volume power is $P_{Ii}{\dot{F}}_{iI}=\tau :D$, with Kirchhoff stress tensor $\tau$ and rate-of-strain tensor $D$. A strain constraint can leave the conjugate stress undetermined by a reaction component that does no admissible work; uniqueness should therefore not be assumed on a restricted set of rates.

The infinitesimal strain tensor is the symmetric part of a displacement gradient tensor. It is the linear approximation to a finite strain measure for small displacement gradients. An infinitesimal rigid rotation contributes no strain, so the elastic energy depends on the symmetric part rather than the entire gradient.

A constructible polygon is a polygon whose vertices can be obtained by a compass-and-straightedge construction from the specified initial data. The classical question for a regular polygon asks which numbers of sides permit an exact construction from a given side or circumscribed circle.