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.

The K-correction converts an observed photometric passband to a specified emitted photometric passband, accounting for cosmological redshift and the source spectrum. For a narrow same-band measurement, $K_R=-2.5{\log }_{10}[(1+z)L_{\nu }((1+z){\nu }_R)/L_{\nu }({\nu }_R)]$. Thus its sign depends on the spectrum, not on redshift alone. For $L_{\nu }\propto {\nu }^{\alpha }$, it is $-2.5(1+\alpha ){\log }_{10}(1+z)$. Finite-band corrections use the actual detector response and magnitude zero point.

For ${\eta }_t+\mathbf{c}\cdot \nabla \eta =\mu \eta +\nabla \cdot (\Gamma \nabla \eta )$ with symmetric positive definite $\Gamma$, the dispersion relation is $\omega =\mathbf{c}\cdot \mathbf{k}+i(\mu -{\mathbf{k}}^T\Gamma \mathbf{k})$. Its zero-group velocity saddle is ${\mathbf{k}}_0=-i{\Gamma }^{-1}\mathbf{c}/2$. The anisotropic heat kernel has fixed-position exponential rate $\mu -{\mathbf{c}}^T{\Gamma }^{-1}\mathbf{c}/4$, so this accessible saddle gives the actual threshold. At fixed speed $q$, the threshold ranges from $q^2/(4d_{\max })$ to $q^2/(4d_{\min })$, where $d_{\min },d_{\max }$ are the eigenvalues of $\Gamma$. Absence of absolute hydrodynamic instability in every direction requires the upper bound $\mu \le q^2/(4d_{\max })$.

The pure Abelian Chern--Simons theory has zero metric stress. For a source written as $-c\int \star J\wedge A$, holding the covariant components of the one-forms $J,A$ fixed makes the source equal to $-c\int \sqrt{-g}{g}^{\mu \nu }{J}_{\mu }{A}_{\nu }{d}^{3}x$. Variation using $\delta \sqrt{-g}=-\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }/2$ gives the displayed stress-energy tensor. If the independent source is instead the fixed two-form $\mathcal{J}=\star J$, the source action is metric-independent and its stress vanishes. The variational data must therefore be stated.

If $A(x)$ counts a set of positive integers and $\eta >0$, Abel summation gives ${\sum }_{n\in A,n\le X}1/n=A(X)/X+{\int }_1^{X}A(t)t^{-2}dt$. A bound $A(t)\ll t/(\log t{)}^{1+\eta }$ for $t\ge 3$ makes the integral convergent, since the substitution $u=\log t$ gives ${\int }_{\log 3}^{\infty }{u}^{-1-\eta }du<\infty$.

For a primitive positive solution of $x^n+y^n=z^n$ with $n\ge 4$, the three powers are pairwise coprime and their radical is at most $xyz<z^3$. The abc conjecture with $ϵ=1/6$ gives the displayed inequality, hence $z\le C_{1/6}^2$ uniformly in $n$. The inequality $z^n\le 2\max (x,y{)}^n$ then bounds $n$. Thus there are finitely many primitive triples and exponents. Homogeneity would produce infinitely many unscaled solutions from any one nonzero solution, so a primitive arithmetic height bound alone must not be described as finite many arbitrary multiples.

Put $M=x^p$ for positive bases $x,y\ge 2$. The coprime triple $(y^q,1,x^p)$ has radical at most $xy<M^{2/3}$. The abc conjecture with $ϵ=1/6$ gives $M\le C_{1/6}{M}^{7/9}$, proving the bound. Both bases and both exponents are then bounded, since $2^p\le M$ and $2^q\le M-1$. Unlike a homogeneous Fermat equation, no common-scaling ambiguity occurs.

A 2-opt schedule move reverses a consecutive block $v_p,\dots ,v_q$, reconnecting its boundary arcs. With directed costs, its change isOmitting the internal sum is valid only for symmetric costs. A simulated annealing step accepts an increase with probability $\exp (-\Delta /T)$; a strictly positive temperature permits escape from local minima.

For a connected graph whose 2-core contains a vertex of degree of a vertex at least three, suppress each maximal graph path with degree-two internal vertices to a single edge, allowing its two endpoints to coincide. Here suppression is the reverse of graph subdivision. The resulting multigraph, retaining self-loops and parallel edges, is its graph kernel. Each suppressed internal vertex removes one vertex and one edge, preserving graph excess; counting a self-loop twice toward degree of a vertex, every remaining vertex has degree of a vertex at least three. A purely cyclic 2-core is a separate unicyclic case: it has no branching vertex to anchor this construction.

For the barbell in bridges and connectivity in the 2-core, the graph kernel has two vertices $c,d$, one self-loop at each, and a single joining edge $cd$. Suppressing $x,y$ contracts the joining graph path to that bridge in a graph; suppressing the two other vertices on each triangle turns the triangle into a self-loop. The graph excess stays $9-8=3-2=1$. Thus forming a graph kernel compresses degree-two chains rather than deleting everything outside graph cycles.

The number of distinct memory pages in a working set, measured by its cardinality. It depends on both the current memory reference index $t$ and the declared window $\Delta$, not just on the resident set capacity.