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List each pair $(i,j)$ with multiplicity $a_{ij}$, sorted first by $i$ and then by $j$. Apply row insertion to the lower letters $j$, and record the upper letters $i$ in the new cells. The row insertion comparison is strictly greater than the carried entry. The result is a pair of Semistandard Young tableaux of the same shape. Reverse insertion removes the rightmost cell containing the largest recording entry, then replaces the rightmost strictly smaller entry in each preceding row. It reconstructs the ordered list, proving the bijection. The insertion and recording contents are respectively the column sums and row sums of $A$.

With $ϵ={\beta }^{\prime }/\beta \ll 1$, $\eta ={\mu }^{\prime }/\mu =O({ϵ}^2)$ and a fixed mode index, put $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The exact Love wave branch has $W=y\sqrt{1+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$ and $K=y\sqrt{{ϵ}^2+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$. Its minimum group velocity occurs near $y=(n+1/2)\pi$, with transition width $W-(n+1/2)\pi$ of order $({\eta }^2(n+1/2)\pi {)}^{1/3}$. Well above that width, the rigid-base approximation to Love waves captures both velocities. Density ratios and the mode index must be controlled in this estimate.

A càdlàg function is right-continuous and has a finite left limit at every positive time. The name abbreviates the French phrase /continue à droite, limites à gauche/.

On continuous piecewise smooth perpendicular fields with zero endpoints, the Riemannian index form is nonnegative exactly when no point strictly inside the geodesic segment is conjugate to its initial point. It is positive on every nonzero field exactly when no conjugate point occurs up to and including the final endpoint. At a first conjugate endpoint its nullspace consists of the endpoint-vanishing Jacobi fields. These statements distinguish nonnegative quadratic forms from strictly positive ones and do not assume global minimization.

On continuous piecewise smooth vector field along a curve with zero endpoints, the radical of a bilinear form of the Riemannian index form is exactly the space of endpoint-vanishing Jacobi fields. Integration by parts against tests supported inside each smooth piece gives the Jacobi equation; tests with arbitrary values at breakpoints force continuity of the covariant derivative, hence a global smooth Jacobi field. This characterization uses $I(V,W)=0$ for every test field $W$, not merely $I(V,V)=0$, which is insufficient for an indefinite form.

// Target: riemannian-geometry.bigb

In a parallel orthonormal frame along a geodesic, write the Jacobi equation as $A^{\prime \prime }+KA=0$, with $A(0)=0$, $A^{\prime }(0)=I$ and symmetric $K$. In the absence of conjugate points on $(0,b]$, $A$ is invertible there. The conserved quantity $A^T{A}^{\prime }-A^{\prime T}A=0$ makes $S=A^{\prime }{A}^{-1}$ symmetric, and $S^{\prime }+S^2+K=0$. For a field $u$ vanishing at both endpoints,Indeed the difference of the integrands is $(u^{T}Su{)}^{\prime }$; its boundary value at zero vanishes because $S=t^{-1}I+O(t)$ and $u=O(t)$. Equality forces $u=Ac$ and then $A(b)c=0$, so $u=0$. This proves strict positivity and the minimizing property of the Jacobi field with prescribed endpoint values.

// Target: riemannian-geometry.bigb

For likelihood $Y\mid \beta \sim N(X\beta ,{\sigma }^{2}I)$ and independent prior $\beta \sim N(0,{\tau }^{2}I)$, the posterior distribution is Gaussian with covariance $V$ as displayed and mean $(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$. Its mean and mode therefore coincide with the ridge regression coefficient estimate. A Gaussian prior alone does not imply this conclusion without the stated likelihood.

// Target: foundations-of-mathematics.bigb

A sphere approaching a locally compliant elastic coating has a parabolic lubrication gap increased by pressure-induced downward surface displacement. With a local compliance $\eta$, the dimensionless compliance is $\epsilon =\eta \mu a\hat{v}/{\hat{h}}^3$. The gap is $H=H_0+R^2/2+\epsilon P$ and its Reynolds lubrication equation is $12H_T=R^{-1}(R{H}^3{P}_R{)}_R$. The rigid term is $P_0=-3{\dot{H}}_0/(H_0+R^2/2{)}^2$. The first elastic correction contains both ${\dot{H}}_0^2$ and ${\ddot{H}}_0$: elastic gap evolution retains the time derivative of pressure even when fluid inertia is negligible.

For $ϵ=1-\alpha \ll 1$, the near-pole gap is $h=ϵ\Delta +r^2/(2R)$ with $R\simeq a^2/\Delta$, and horizontal rotation gives local sliding speed $V\simeq a\Omega$. The annulus $a\sqrt{ϵ}\ll r\ll a$ has $h\sim \Delta r^2/a^2$. Its shear dissipation is at least $\mu V^2\int dA/h\sim 2\pi \mu V^{2}R\log (1/ϵ)$, so a uniformly bounded torque of order $\mu \Omega a^4/\Delta$ is impossible. The exact local Reynolds lubrication equation solution $p=-(6\mu V/5)x/h^2$ gives the displayed logarithmic torque and a dominant horizontal holding-force magnitude $(12\pi /5)(\mu \Omega a^4/{\Delta }^2)\log (1/ϵ)$. The smallest patch and fixed-angle outer region each contribute only a bounded amount; the intermediate annulus supplies the logarithm. A fully flooded film is assumed.

Let $S=s/s_{\infty }$, let the interstellar medium have mass $g=s_{\infty }S(1-S)$, and retain fraction $f=S$ of returned material. For $0\le \beta <1$ put $q=(2-\beta )/(1-\beta )$. The integrating factor for the metal mass is $S^{1/(1-\beta )}/(1-S)$. A finite initial gas-phase metallicity removes the homogeneous solution and gives the displayed expression, equivalently $Z=y{\sum }_{n\ge 1}{S}^n/(q+n)$ for $0\le S<1$. The increasing initial gas mass requires pristine galactic gas inflow: gas accounting gives $d{M}_{\mathrm{in}}/ds=q(1-S)$. A fixed initial mass function makes the birth metallicity distribution function proportional to $ds/dZ$ when enrichment is monotone.

For constant stratification and $U=U_0+\Lambda z$, a decaying zero-interior-PV mode has vertical structure $e^{-Kz}$. Its topographic boundary condition gives $A^{\prime }+ik(U_0+\Lambda /K)A=ikϵN^2{U}_0/(Kf)$. Resonance requires $K{U}_0+\Lambda =0$ and nonzero forcing $kϵU_0$. With zero initial disturbance, $A$ grows linearly and is imaginary relative to cosine topography. Its meridional velocity is therefore opposite in phase to the relief for positive $f,U_0,ϵ$. A zero zonal wavenumber or zero boundary current removes the forcing and invalidates an unconditional claim of secular growth.

The displayed condition and the resolvent bound for a discrete SIR epidemic imply $\mathbb{E}Z=o(n)$. The Markov inequality then makes $Z/n\to 0$ in probability. A sufficient special case is a sublinear initial infected count $m=o(n)$ with a fixed positive gap $1-\beta \rho \ge \eta >0$. The inequality $\beta \rho <1$ at each finite population size alone does not control a shrinking gap or an initially macroscopic outbreak.

Resistor tolerance specifies the permissible manufacturing deviation of a resistor's electrical resistance from its nominal value under stated conditions. For nominal electrical resistance $R_{\mathrm{nom}}>0$ and fractional resistor tolerance $0\le \tau <1$, the specified range is $R_{\mathrm{nom}}(1-\tau )\le R\le R_{\mathrm{nom}}(1+\tau )$. This differs from measurement uncertainty: resistor tolerance describes a component specification, while measurement uncertainty describes knowledge of a measured value; the tolerance alone is not a probability distribution for that value.

Two nominally equal resistors can have different actual electrical resistances, changing their voltage divider ratio even without an external electrical load. In a loaded voltage divider, use the actual electrical resistances to calculate both the unloaded output and the Thévenin resistance governing voltage divider loading error. A mismatch from resistor tolerance and a change from voltmeter loading are distinct effects that can occur together.

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.

For a linear viscoelastic fluid, write the solvent stress explicitly as $2{\mu }_{0}E(t)$, or use the displayed kernel with a causal endpoint delta satisfying ${\int }_0^{\infty }{\delta }_{+}(s)f(s)ds=f(0)$. If a symmetric delta is integrated with half its weight at the endpoint, its coefficient must instead be $2{\mu }_0$. Mixing those conventions loses a factor of two. The Oldroyd-B model has the displayed exponentially decaying polymer part after linearization.

Take independent and identically distributed draws from the uniform three-outcome reference above. An observed sample mean at least $1/2$ is unusual because the source expected value is zero. For two draws, the only successful ordered pairs are $(0,1),(1,0),(1,1)$, giving probability $3/9=1/3$.

For long samples, the relevant empirical distributions have mean at least $1/2$. The method of types assigns a profile $q$ a leading probability cost $e^{-nD(q\Vert p)}$. The least costly admissible profile is the information projection on three outcomes: in this example the minimizing mean is on the boundary $1/2$, and the leading exponent is approximately $0.197378$ nats per draw. The exponential scale is not an exact finite-sample probability formula.^[ref]

The attraction of this connection is that a constrained optimization problem predicts the most probable way an unusual aggregate observation can occur. It links relative entropy, expected value constraints and rare empirical distributions without identifying a rare event with just one exact attainable type class.

For a subgroup $H$ of a group generated by a set $S$, choose representatives $t$ of right cosets $Ht$. If $\overline{tg}$ represents $Htg$, then $tg{\overline{tg}}^{-1}$ lies in $H$. These Schreier generators generate $H$: inserting successive coset representatives along a word telescopes the word into such factors. Lifted relators can be rewritten in the same factors.

For a real or complex matrix, the three displayed conditions are equivalent to being a regular matrix summability method. For necessity, first apply the Uniform boundedness principle to each row's finite partial sums on the convergent sequence space, obtaining absolute summability of that row. Apply the principle again to the full row functionals, obtaining a uniform bound. Coordinate vectors and the constant-one sequence give the remaining conditions. For sufficiency, write $x_j=L+(x_j-L)$ and split the error into a finite head, controlled by columnwise convergence, and a uniformly small tail, controlled by the absolute row bound.

For every regular summability matrix $A$, there is a bounded sequence whose transformed row sums do not converge. A gliding hump argument selects successive rows whose mass on already assigned coordinates is small and whose remaining tails are small. On each new finite block, choose signs agreeing or disagreeing with that row alternately. Since row sums tend to one, the transformed values have subsequences uniformly above and below zero. This does not contradict preservation of ordinary convergence, since the constructed sequence is divergent.

The summability domain is the inverse image of the convergent sequence space under the bounded matrix operator on the l-infinity sequence space. It is therefore a closed linear subspace. The bounded divergence for every regular summability matrix makes it proper, so it has empty interior and is a nowhere dense set. The Baire category theorem then ensures that countably many regular methods miss some common bounded sequence.