Loading noise is the acoustic dipole radiation of the force exerted by a body on its surrounding fluid. For a compact, slowly moving source, the radiating acoustic density perturbation is $n\cdot \dot{\mathcal{F}}(t-R/c_0)/(4\pi c_0^{3}R)$, where $\mathcal{F}$ is the total force. If this total is constant, finite source-delay corrections can still create higher acoustic multipole sources.

A triangular taper majorizes a fixed fraction of the summation interval. Its Fourier kernel is the Fejér kernel, bounded by $C\min (m,(m\Vert \theta {\Vert }^2{)}^{-1})$. For separated sample points, split each row sum at distance $1/m$: the plateau and square-decay tail each contribute $O({\delta }^{-1})$, while the diagonal contributes $O(m)$. Bounding the resulting quadratic form proves the dual sieve with constant $C(N+{\delta }^{-1})$. Operator norm duality gives the primal sieve.

For nonoverlapping triangular sea-ice pressure ridges with a common along-track side slope $\tan \delta$, a ridge whose peak exceeds $h$ contributes $2\cot \delta dh$ of track length in a draft interval $dh$. Summing over peaks proves the identity. Above the cutoff of an exponential ridge-draft model, $g(h)=A{e}^{-bh}$ with $A=2B/(b\tan \delta )$. This draft occupation is not the normalized peak probability density function; level ice and shallower parts supply the remaining probability.

For a Lie group homomorphism $F$, the image of the integral curve through the identity of a left-invariant vector field is the integral curve of the field induced by its differential. Uniqueness of integral curves of a vector field identifies the curves for all times. Evaluating at time one gives the formula. Completeness of invariant fields follows from left translation and repetition of a local integral curve, independently of a matrix-series formula.

For exponential rate $\lambda >0$, the integer part has mass $(1-e^{-\lambda })e^{-\lambda m}$ on nonnegative integers. The fractional part has density $\lambda e^{-\lambda u}/(1-e^{-\lambda })$ for $0\le u<1$. The mixed joint density factors into those marginals, proving independence. The integer part is a geometric distribution on zero-based support, with expectation $1/(e^{\lambda }-1)$.

An observational unit is the object or occasion supplying a recorded response. Several observational units can belong to one experimental unit; counting them as independent treatment replicates creates pseudoreplication.

For an exact symplectic manifold with $\omega =d\lambda$, the Liouville vector field is defined by ${\iota }_Z\omega =\lambda$. Cartan's magic formula gives ${\mathcal{L}}_Z\omega =\omega$. On a cotangent bundle using $\omega =-d\alpha$, take $\lambda =-\alpha$; then $Z={\sum }_i{p}_i{\partial }_{p_i}$ generates the cotangent fiber-dilation flow.

Replace every degree-three vertex of a graph in one class of a bipartite graph by a triangle with one port for each incident edge. A self-avoiding walk whose endpoints remain in the other class cannot make two completed passages through the same triangle: each passage needs two unused ports and only three exist. Each old two-edge passage has exactly two replacements, of lengths three and four. Therefore the restricted generating functions satisfy $Z_H^0(x)=Z_G^0(\sqrt{x^3+x^4})$. If the old radius is $1/\mu$, the new radius obeys ${\rho }^3+{\rho }^4={\mu }^{-2}$.

A leaf reached before all $n$ input bits are queried leaves at least one coordinate free. Flipping that coordinate pairs its inputs with opposite alternating signs and equal output. Thus every such leaf contributes zero. If the total alternating sum is nonzero, a depth below $n$ is impossible.

If Euler's number were $a/b$, choose $n\ge \max (b,2)$ and multiply its factorial-series tail by $n!$. This produces an integer, but the tail is positive and smaller than ${\sum }_{j\ge 1}(n+1{)}^{-j}=1/n<1$. The contradiction proves that the number is an irrational number. The argument rules out a hypothetical rational remainder by placing it in an impossible integer interval.

For bounded $L$ and compact $K$, use the Fredholm convention ${\Sigma }_{\mathrm{e}}(L)=\{z:L-zI\text{ is not Fredholm}\}$. Applying compact perturbation invariance of Fredholm operators to every shift proves the equality. This general theorem needs no self-adjointness; other definitions of essential spectrum for nonnormal operators must be distinguished.

The Engquist-Osher numerical flux is $F(a,b)=f(0)+{\int }_0^a\max (f\prime (s),0)ds+{\int }_0^b\min (f\prime (s),0)ds$. It sends positive-speed contributions from the left state and negative-speed contributions from the right state. Its consistency $F(u,u)=f(u)$ and opposite monotonicities in the two arguments yield a monotone conservative scheme under the appropriate Courant–Friedrichs–Lewy condition. For the Inviscid Burgers equation, it is $\frac{1}{2}\max (a,0{)}^2+\frac{1}{2}\min (b,0{)}^2$.

The free-forgetful adjunction for the Eilenberg-Moore category is terminal in the category of adjunctions inducing a fixed monad. The unique morphism into it is the Eilenberg-Moore comparison functor $B\mapsto (GB,G{\epsilon }_B)$. Its underlying objects and arrows are forced by the forgetful functor, and counit compatibility forces its algebra actions.

If each standard counit presentation is a coequalizer, the Eilenberg-Moore comparison functor is full and faithful. An monad algebra morphism $\alpha :GB\to GC$ makes ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

The efficient information for a scalar target statistical parameter is the squared L2 norm of its efficient score. It cannot exceed the parametric Fisher information, by the Pythagorean theorem in an inner-product space. It can vanish when first-order target variation can be reproduced by nuisance variation.

The decomposition $\dot{\ell }={\Pi }_{\mathcal{N}}\dot{\ell }+\tilde{\ell }$ is orthogonal. Taking the inner product with the efficient score gives $P(\dot{\ell }\tilde{\ell })=\tilde{I}$. Centering of the efficient score follows because the nuisance tangent space and the parametric score function lie in the closed mean-zero L2 space.

With unknown independent covariate and centered error distributions, take the regular mean-preserving error tangent space, let $h={\partial }_{\theta }{g}_{\theta }$, $\rho =-f^{\prime }/f$ and ${\tau }^2=E_f{\epsilon }^2>0$. Under $E_f\rho =0$, $E_f(\epsilon \rho )=1$ and finite second moments, orthogonal projection removes $(E_{v}h)(\rho -\epsilon /{\tau }^2)$ from $h\rho$. Thus the efficient score and efficient information are $\tilde{\ell }=(h-E_{v}h)\rho +(E_{v}h)\epsilon /{\tau }^2$ and $\tilde{I}={\mathrm{Var}}_v(h)E_f{\rho }^2+(E_{v}h{)}^2/{\tau }^2$. The reduction to $h\epsilon /{\tau }^2$ is valid for a normal distribution of errors or for constant $h$, but need not hold otherwise. Centered covariates and logistic distribution errors give the counterexample $\tilde{\ell }=X\tanh (\epsilon /2)$.

In a slender layer, two parallel sublayers of fractions $x/L$ and $1-x/L$ have local effective permeability $K(x)=k_{1}x/L+k_2(1-x/L)$. Integrating their successive hydraulic resistance gives the logarithmic mean displayed above. A leading pressure gradient uniform across the thickness does not make parcel paths horizontal; a small transverse velocity preserves mass conservation as the cross-sectional conductance changes.

For a normal distribution $N(\lambda ,{\sigma }^2)$, the logarithm of its moment-generating function is $s\lambda +s^2{\sigma }^2/2$. Dividing by $s>0$ gives its effective bandwidth. Independent demands add these bandwidths. For total mean $m$ and positive total variance $v$, optimizing a Chernoff bound with target $e^{-\gamma }$, $\gamma >0$, yields the sufficient margin $C\ge m+\sqrt{2\gamma v}$.

For nonnegative traffic with exponential moments near zero, its effective bandwidth has expansion $\mathbb{E}X+s\mathrm{Var}(X)/2+O(s^2)$. For bounded traffic with essential supremum $M$, it increases toward $M$: the exponential moment is at most $e^{sM}$, and any positive probability of $X>a$ gives $\alpha (s)\ge a+s^{-1}\log \mathbb{P}(X>a)$. Unbounded traffic has an infinite large-parameter limit when all positive exponential moments exist; a finite moment domain must be respected.