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A Cox process can be directed by a random locally finite measure, which need not have a density. Conditional counts in disjoint sets are independent Poisson variables with that measure's masses. Uniform circular offspring displacements give a directing measure supported on random circles; its unconditional mean can still be a multiple of planar Lebesgue measure. A random intensity function is the special case in which the directing measure is absolutely continuous.

When each parent has a Poisson number of independently displaced offspring, the conditional offspring process has the displayed directing measure. For disjoint target sets, multinomial distribution splitting of a parent's Poisson count gives joint probability generating function $\exp (\mu {\sum }_j\kappa (A_j-z)(s_j-1))$. Multiply over independent parents to obtain independent Poisson counts conditional on the parents. Compactly supported displacement ensures local finiteness for a locally finite parent process.

// Destination: discrete-probability-distribution.bigb

For $d$ events tied within a risk set, put $u_i=\exp ({\beta }^T{z}_i)$, $S={\sum }_{i\in R}{u}_i$ and $U_D={\sum }_{i\in D}{u}_i$. The Efron approximation to Cox partial likelihood removes the average tied-event hazard multiplier from the denominator at each successive event. Unlike the Breslow approximation for tied event times, it accounts approximately for depletion of the risk set within the group. Factors independent of $\beta$ can be dropped for estimation. This is an approximation for coarsened continuous event times, not a full interval-observation likelihood.

In a discrete-time model with independent event indicators whose odds are $a{u}_i$, condition on exactly $d$ events in the risk set $R$. The common factor $a^d$ cancels and the conditional probability of the event set $D$ is the displayed ratio. The denominator is an elementary symmetric polynomial and can be evaluated by recursion rather than enumerating every subset. This conditional model is distinct from summing a continuous-time rank likelihood over all unobserved within-tie orders.

At an untied event time, the Cox partial likelihood weights the current risk set by $w_{jk}=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$. Differentiation gives the displayed score function and the negative Hessian as a sum of weighted covariances. The score is the sum of Schoenfeld functions, so Schoenfeld residuals sum to zero at a finite interior unpenalized fit. At arbitrary trial coefficients or boundary fits, a zero sum is not a general identity.

Assume $a^2+b^2>0$, so the strain magnitude $c=\sqrt{a^2+b^2}$ is strictly positive. For horizontal strain matrix $S=\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$ and zero vertical velocity, horizontal scalar gradients obey ${\dot{k}}_h=-S{k}_h$. Since $S^2=c^{2}I$ with $c=\sqrt{a^2+b^2}$, the growing projection is $(I-S/c)k_h(0)/2$. If nonzero it equals $k_0(\cos \varphi ,\sin \varphi )$ and grows as $e^{ct}$. Fixed vertical shears give ${\dot{k}}_3=-{\sigma }_{13}{k}_1-{\sigma }_{23}{k}_2$, so $k_3\sim -k_{0}D{e}^{ct}/c$ with $D={\sigma }_{13}\cos \varphi +{\sigma }_{23}\sin \varphi$. The ratio $\alpha =|k_3|/|k_h|$ tends to the displayed value. It measures scalar-isosurface orientation; zero growing projection or $D=0$ must not be described as generic exponential vertical-gradient growth.

Let the columns of an invertible matrix $B$ be a new basis expressed in the old basis. The same vector has old coordinate vector $x$ and new coordinate vector $x^{\prime }=B^{-1}x$. If a covector has old row $\ell$, its new row is ${\ell }^{\prime }=\ell B$, sinceThus the scalar dual pairing stays unchanged. If covector coefficients are stored as a column $c={\ell }^T$, their new column is $c^{\prime }=B^{T}c$; this is the inverse transpose of the transformation $x\mapsto B^{-1}x$ on vector coordinates. These conventions express the covariance and contravariance of vectors. ^[ref].

For the basis $b_1=(1,0)$, $b_2=(1,1)$,The linear functional $\phi (x,y)=2x-y$ has rows $\ell =\left(\begin{array}{cc}2& -1\end{array}\right)$ and ${\ell }^{\prime }=\left(\begin{array}{cc}2& 1\end{array}\right)$. Both calculations give five: $2\cdot 3-1\cdot 1=2\cdot 2+1\cdot 1$. Changing the basis changes the descriptions, not the measured value.

Every countable set $B$ of real numbers is contained in a countable set closed under taking all real roots of a polynomial with coefficients in that set. Let $\varphi (A)$ consist of these roots of a polynomial, excluding the zero polynomial. There are countably many finite coefficient lists over a countable set, and each nonzero polynomial has finitely many roots of a polynomial; hence $\varphi (A)$ is countable. Define $B_0=B$ and $B_{n+1}=B_n\cup \varphi (B_n)$. The countable union of countable sets theorem shows that $X={\bigcup }_n{B}_n$ is countable. Every finite coefficient list in $X$ belongs to one stage $B_N$, so its real roots of a polynomial belong to $B_{N+1}\subseteq X$. Retaining $B_n$ at the next stage matters: $\varphi (A)$ need not contain $A$.

For every fixed positive integer $d$, the polynomial ring with integer coefficients in $d$ indeterminates is countably infinite. Polynomials of bounded total degree have only finitely many possible monomials, so they are encoded by a finite Cartesian power of the integers. Taking a countable union of countable sets over degree bounds proves countability; constant polynomials supply an infinite subset. Nonzero one-variable polynomials have finitely many real roots, so the real algebraic numbers form a countable set as well.

Nondecreasing functions $\mathbb{N}\to \mathbb{N}$ form an uncountable set: every subset $A$ is encoded by the strictly increasing function $n+{\sum }_{k=1}^n{1}_A(k)$. In contrast, nonincreasing functions form a countable set because each is determined by a finite initial segment and its eventual constant value.

// Target: classical-mechanics.bigb

A finite subset $A$ of the natural numbers, numbered from one, has the binary code ${\sum }_{j\in A}{2}^{j-1}$. Distinct finite subsets give distinct codes, because the largest differing exponent exceeds the sum of all smaller powers of two. Thus the finite subsets form a countable set, although the entire power set is an uncountable set by the Cantor diagonal argument. An enumeration transfers this coding to any countable set.

A family of pairwise disjoint real intervals of positive length is countable. Each interval contains a rational number in its interior by the density of the rational numbers. Fix an enumeration of the rational numbers and select the first member lying inside each interval. Disjointness makes this selection an injection into a countable set. The argument also applies to disjoint nonempty open subsets of the real line.

For $\mathrm{Im}z>0$, writing cotangent as $i(e^{2\pi iz}+1)/(e^{2\pi iz}-1)$ and expanding a geometric series gives this Fourier series. Differentiating the cotangent partial-fraction expansion and this Fourier series $k-1$ times yields

The incremental cost-effectiveness ratio divides an intervention's additional mean cost by its additional mean effect relative to a comparator. Its units are cost per outcome unit. When both increments are positive, comparison with a threshold $\lambda$ is equivalent to checking whether the incremental net monetary benefit $\lambda \Delta E-\Delta C$ is positive. When $\Delta E<0$ the inequality reverses on division, and when $\Delta E=0$ the ratio is undefined. A ratio alone cannot identify the quadrant of the cost-effectiveness plane.

The cost-effectiveness plane plots an effect increment horizontally and a cost increment vertically. The southeast quadrant represents greater effectiveness at lower cost, while the northwest quadrant represents lower effectiveness at greater cost. A threshold $\lambda$ corresponds to the line $\Delta C=\lambda \Delta E$; points below the line have positive incremental net monetary benefit, in every quadrant. A joint confidence region or paired bootstrap cloud reveals uncertainty and preserves the dependence between cost and effect estimates.

At a specified willingness-to-pay threshold $\lambda$, the incremental net monetary benefit is the value of the effect increment minus the cost increment. It avoids dividing by a potentially near-zero or negative effect estimate. For estimates with variances $v_E,v_C$ and covariance $v_{CE}$, its estimated variance is ${\lambda }^2{v}_E+v_C-2\lambda v_{CE}$. A normal approximation therefore gives a confidence interval directly on the net-benefit scale. The economic sign criterion does not require a positive effect increment.

A light signal sent from a comoving observer to a comoving reflector at separation $\chi$, and immediately returned, consumes conformal time $2\chi$. The supremum of the initial physical reflector distances allowing a return is therefore half the cosmological event horizon radius. Equality generally requires infinite return time; finite-time reception requires a strictly smaller distance. For a cosmological domain wall network with $a\propto t^2$, the supremum is $t/2=1/H$.

For separately conserved pressureless matter, ${\rho }_m\propto a^{-3}$. The deceleration parameter identity $\dot{H}/H^2=-(1+q)$ and ${\Omega }_m\propto {\rho }_m/H^2$ give ${\dot{\Omega }}_m=H{\Omega }_m(2q-1)$. A dust-plus-vacuum universe has $q={\Omega }_m/2-{\Omega }_{\Lambda }$; conversion to conformal time yields the displayed flow. The familiar matter-only equation requires negligible cosmological constant. In a spatially flat dust-plus-vacuum universe, ${\Omega }_m+{\Omega }_{\Lambda }=1$ and ${\Omega }_m^{\prime }=-3\mathcal{H}{\Omega }_m{\Omega }_{\Lambda }$, so flatness does not require ${\Omega }_m=1$.

For a scale-invariant potential spectrum with ${\mathcal{A}}_{\Phi }=k^3{P}_{\Phi }/(2{\pi }^2)$ constant, the matter-era Sachs-Wolfe effect gives $C_{\ell }=(4\pi {\mathcal{A}}_{\Phi }/9)\int j_{\ell }(x{)}^{2}dx/x$. The logarithmic spherical Bessel square integral then gives $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{A}}_{\Phi }/9$. This plateau applies to large-angle temperature anisotropies, neglecting acoustic evolution and late evolving potentials; it is not a constant $C_{\ell }$ spectrum.

For an outward sightline, Fourier velocity $v_b^i=i{k}^i{\theta }_b$, and sudden last scattering, intrinsic photon temperature projects with a Spherical Bessel function, while the Doppler term projects with its derivative. Density and velocity acoustic sources are approximately a quarter cycle out of phase. Their amplitude is suppressed by photon diffusion, giving a power damping envelope $e^{-2k^2/k_D^2}$. Projection maps the sound scale to peak spacing approximately $\pi D_{\ast }/r_s$ and the damping scale to angular multipoles of order $k_D{D}_{\ast }$.