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A translation and positive uniform scaling of a finite configuration $F$ in a real vector space. It preserves the ratios and directions of differences between its points. In an integer lattice, the Gallai theorem for an integer lattice guarantees a monochromatic homothetic copy of every finite pattern in any finite colouring, with positive integer scaling factor.

The points with integer coordinates in a real d-dimensional affine space with a chosen origin and coordinate basis. Translations by integer vectors and dilations by positive integers preserve this set. The Gallai theorem for an integer lattice concerns finite colourings of these points and homothetic copies of finite configurations of them.

Points $p_0,\dots ,p_k$ in an affine space are affinely independent if the only affine dependence displayed above has all coefficients zero, using any choice of origin to read the vector equation. Since the coefficients sum to zero, replacing the origin leaves that equation unchanged. Substituting ${\lambda }_0=-{\sum }_{i=1}^k{\lambda }_i$ shows that this condition is equivalent to linear independence of $p_1-p_0,\dots ,p_k-p_0$. It also says that an affine combination of the points has unique coefficients. In ${\mathbb{R}}^n$, at most $n+1$ points can be affinely independent; $n+1$ such points span a full-dimensional simplex.

For unaffected genetic carriers of a disease with complete penetrance and recessive inheritance, suppose parental marker-disease haplotype phases are equally likely and the marker is fully informative. Write $u=\theta (1-\theta )$ and $v=1/2-u$, where $\theta$ is the recombination fraction. Conditional on sharing a parental marker copy, the probability that both siblings inherit that parent's disease allele is $[(1-\theta {)}^2+{\theta }^2]/2=v$; conditional on inheriting different marker copies it is $\theta (1-\theta )=u$. Independent paternal and maternal transmissions therefore give $L_0=u^2,L_1=uv,L_2=v^2$. Conditional instead on both siblings being affected, the IBD probabilities are $s^2,2s(1-s),(1-s{)}^2$, where $s=2u\in [0,1/2]$.

// Target: biology.bigb

An elastic collision with a wall moving toward the particle at speed $V$ changes the particle speed by approximately $2V$ when recoil is negligible. The collision rate is approximately $v/\ell$ for $v\gg V$, and $\dot{\ell }=-2V$, yielding $\dot{v}/v=-\dot{\ell }/\ell$. For a freely moving finite wall of mass $M$, the neglected recoil changes the increment by order $(m/M)v$, so relative accuracy also requires $(m/M)(v/V)\ll 1$.

// Target: classical-mechanics.bigb

In a slowly changing spherical gravitational potential, angular momentum is exactly conserved and the radial action is an adiabatic invariant. An initially circular orbit remains circular in the adiabatic limit, so $J_r=0$ and $L^2=GrM(r)$ give the displayed shell relation. If dark-matter shells do not cross, their enclosed dark mass is also conserved. A density profile alone does not specify this orbital model: eccentric-orbit populations require their full action distribution.

Over a field $K$ containing ${\mathbb{F}}_q$, a q-linearized polynomial isThe Frobenius endomorphism gives $L(u+v)=L(u)+L(v)$ and $L(cu)=cL(u)$ for $c\in {\mathbb{F}}_q$, so its polynomial function is ${\mathbb{F}}_q$-linear. If $a_0\ne 0$, its formal derivative is the nonzero constant $a_0$. Thus it has exactly $q^d$ distinct roots of a polynomial in an algebraic closure, forming a vector space of dimension $d$ over ${\mathbb{F}}_q$.

The chosen coefficient field need not contain all those roots. For instance, $X^{q^2}-X$ has $q^2$ roots in ${\mathbb{F}}_{q^2}$ but only $q$ in ${\mathbb{F}}_q$. The Galois group of the splitting field over $K$ acts faithfully and linearly on the full root vector space, embedding in $GL(d,q)$. Faithfulness holds because the root vectors generate the splitting field.

If a subset $A$ of a cyclic group has normalized additive energy at most ${\alpha }^4+ϵ$, every nonconstant Fourier coefficient on a finite abelian group of its indicator has modulus at most ${ϵ}^{1/4}$. Expanding the count of $a+c=2b$ gives the frequency product $\widehat{1_A}(r)\widehat{1_C}(r)\widehat{1_B}(-2r)$. The Cauchy-Schwarz inequality and Parseval identity on a finite group bound its nonconstant part by the displayed quantity. Character doubling has multiplicity $\gcd (2,N)$, so no assumption that the modulus is odd is needed. The error is a normalized density error multiplied by $N^2$.

Two equal spins $j$ have the unique total-spin-zero stateIt is invariant under joint rotations.

A singlet state is the one-dimensional rotation representation; this construction supplies the angular-momentum-zero case for two equal spins.

For weighted binary data, a weak learner supplies a classifier whose weighted error is below one half. A uniform edge $\gamma >0$ for every weighting permits a quantitative AdaBoost training error bound. This training-weight condition is a separate assumption from merely having error below one half on the initial unweighted sample.

// Target: foundations-of-mathematics.bigb

At score $F$, the exponential classification risk weights observation $i$ proportionally to $e^{-y_{i}F(x_i)}$. A stage $\alpha G$ multiplies this weight by $e^{-\alpha y_{i}G(x_i)}$. For weighted error $ϵ$, the loss multiplier is $(1-ϵ)e^{-\alpha }+ϵe^{\alpha }$, minimized at the displayed value. Its minimum is $2\sqrt{ϵ(1-ϵ)}$. Thus errors gain weight relative to correct observations in the next stage.

// Target: foundations-of-mathematics.bigb

Acoustic directivity describes the angular dependence of radiation in the acoustic far field. For a time-harmonic source in $d$ spatial dimensions, separating the geometric spreading and oscillatory phase leaves a complex amplitude pattern $\mathcal{D}$. The angular intensity pattern is proportional to $|\mathcal{D}{|}^2$. Directivity can be normalized by a reference direction or total radiated power; a stated convention is needed when comparing absolute directivity factors.

For outgoing velocity potential $\varphi =F(r-ct)/r$, the radial velocity is $F^{\prime }/r-F/r^2$ and the pressure perturbation is ${\rho }_{0}c{F}^{\prime }/r$. The acoustic energy density separates into kinetic density ${\rho }_0(F^{\prime }/r-F/r^2{)}^2/2$ and compressional density ${\rho }_0(F^{\prime }/r{)}^2/2$. They are generally unequal near the source but agree at leading far-field order. The additional $r^{-2}$ velocity term is the spherical near field.

The conjugate base of acetic acid. It is a reaction product in the conductometric saponification assay.

For a Keplerian disk, let $S$ be the total loss through both faces per unit projected area and $\dot{M}>0$ the inward accretion rate. Annular mass conservation gives ${\dot{M}}^{\prime }=2\pi RS$. A circular orbit has specific orbital energy $-GM/(2R)$, so a marginally escaping wind requires additional energy $GM/(2R)$ per unit lost mass. Equating this sink to a fraction $f$ of the two-face viscous dissipation $(9/4)\nu \Sigma {\Omega }^2$ gives the displayed relation. If the wind carries the local specific angular momentum $h=\sqrt{GMR}$, conservation of angular momentum gives $(\dot{M}h-\mathcal{G}{)}^{\prime }=h{\dot{M}}^{\prime }$, with viscous torque in an accretion disk $\mathcal{G}=3\pi \nu \Sigma h$. Eliminating the torque yields $R^2{\dot{M}}^{\prime \prime }+\frac{3}{2}R{\dot{M}}^{\prime }=\frac{3}{2}f\dot{M}$.

For smooth radial variation on scale $r$, $c_s\sim \Omega H$, $\nu \sim \alpha \Omega H^2$, $u_r\sim \nu /r$ and $u_z\sim H{u}_r/r$. The exact point-mass vertical gravity differs from ${\Omega }_K^{2}z$ by relative order $(H/r{)}^2$. Radial pressure support changes the rotation and dominant shear heating by the same order. Meridional viscous forces, radial radiative-flux divergence, thermal advection and compressional work are at most order $(H/r{)}^2$ relative to their retained vertical terms for $\alpha \lesssim 1$. These estimates require secular disk evolution and exclude sharp radial boundary layers or independently rapid vertical motion.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

For $p=K{\rho }^2$, a point-mass gravitational potential and a zero-density free surface at $z=\pm ϵr$, the barotropic cylindrical rotation theorem gives ${\Omega }^2=GM/(r^3\sqrt{1+{ϵ}^2})$. Vertical balance gives $\rho =GM[1/\sqrt{r^2+z^2}-1/(r\sqrt{1+{ϵ}^2})]/(2K)$. Integrating over the finite height produces the displayed constant surface density. This is a local radial equilibrium family; infinite radial extent would have infinite mass.

For the accreting raindrop with quadratic drag, set $q=v^2/a$ and $B=7c/8+3k/4$. Its exact equation is $\dot{q}=2(v/a)(g-Bq)$. Dividing by $\dot{a}=cv/4$ gives $q(a)=g/B+(q(a_0)-g/B)(a_0/a{)}^{7+6k/c}$. Thus $q$ stays between its positive initial value and positive equilibrium. Since $d\sqrt{a}/dt=c\sqrt{q}/8$, the radius grows without finite-time divergence and tends to infinity. The power correction then vanishes, proving both limits. Positive drag makes the limiting acceleration strictly less than $g/7$.

If two multiplicative absolute values on a field satisfy a comparison with fixed positive constants $c,C$, apply it to $x^n$ and take $n$th roots. Letting $n\to \infty$ eliminates both constants. This stronger bounded comparison differs from equivalence merely as topologies: replacing an absolute value by its square generally preserves topology but not equality.

For finitely many pairwise inequivalent nontrivial absolute values on a field, an element of the field can approximate independently prescribed targets in the associated completion of a valued field at each place to arbitrary positive accuracies. Equivalently the diagonal map has dense image in the finite product. Requiring finitely many inequivalent places is essential; the conclusion controls finitely many inequalities, not infinitely many independently prescribed bounds at once, or incompatible targets at one place. Weak approximation for number fields is the class-field-theoretic instance.