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For two independent binomial counts, form the Fisher exact test rejection region from their conditional null hypergeometric distribution. Summing alternative joint binomial probabilities over that region gives its exact power. For equal group sizes, the null allocation is symmetric, so the two-sided probability-ordering test agrees with doubled lower-tail probabilities. Conservative test discreteness can make a normal-approximation sample size underpowered even when both samples are large but event counts are low.

A smooth scalar Landau free energy can end a two-phase phase coexistence curve by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic Landau free energy; the order-parameter jump then vanishes and the magnetic susceptibility diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.

A finitely generated cone is a closed convex cone. For a convergent sequence of represented vectors, use linearly independent active generators and pass to a subsequence with the same active set. Its coefficients converge through a fixed left inverse and retain nonnegativity. General linear images of closed cones need not be closed; finiteness of the generators supplies the missing argument.

Under the finite-valued assumptions, subgradients of an infimal convolution are obtained as its attained dual optimizers. Equivalently $z\in \partial (h^{\ast }+k^{\ast })(p)$. Splitting the latter subdifferential into two separate ones needs a subdifferential sum rule qualification; solving the aggregate dual objective avoids that extra assumption.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

For any group $G$, evaluation at a chosen generator of a group $x$ of $C_n$ gives a bijection between group homomorphisms $C_n\to G$ and elements satisfying $g^n=1$. A group homomorphism must send $x^j$ to $g^j$. Conversely, this formula is well-defined because exponents differing by a multiple of $n$ give equal powers, and it respects multiplication. The displayed correspondence is a bijection of sets; it is not claimed to be a group isomorphism for a nonabelian target. In a symmetric group, the order of a group element is the least common multiple of its disjoint cycle lengths, making this correspondence easy to enumerate.

Thickness noise is the sound generated by displacement of fluid by a moving impermeable body, represented by the surface acoustic monopole term. For a rigid body, ${\int }_S{v}_{n}dS=d\mathcal{V}/dt=0$, so its leading acoustic compact-source approximation has no net-volume monopole. Its local source need not vanish: a moving finite-volume body can radiate through higher spatial moments. A thin-body or negligible-volume approximation is needed when all such radiation is to be omitted.

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 illustrates how a rapidly convergent rational series can force a hypothetical rational remainder into 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.