For real continuous functions $f,g$ on a compact interval with $g\ge 0$, some $\alpha$ in the interval satisfies $\int fg=f(\alpha )\int g$. If $\int g>0$, the weighted average lies between the minimum and maximum of $f$, and the intermediate value theorem supplies $\alpha$. If $\int g=0$, the bound $|\int fg|\le \sup |f|\int g$ makes both sides zero. Nonnegativity of the weight is essential to this argument.

Dropping the position-dependent source delay in the leading far-zone retarded field requires source size $d$ to be small compared with its variation time $\tau$, in units $c=1$. The retarded quadrupole field also requires $d/r\ll 1$. For a typical nonrelativistic bound source, $\tau \sim d/v$ implies this long-wavelength condition; an arbitrarily rapidly oscillating small displacement is not controlled merely by its instantaneous speed.

For $P(z)=a_N{z}^N+\cdots +a_0$ with $N\ge 1$ and $a_N\ne 0$, the unique retarded fundamental solution of $P(D)$ is $E=Hu$, whereThe distributional jump formula for a Heaviside product gives $P(D)E={\delta }_0$. Its explicit expression isThe residue theorem supplies the initial derivatives by the coefficients at infinity. Uniqueness follows because a difference of two retarded solutions is an analytic homogeneous solution vanishing on the negative half-line.

With inverse Fourier transform convention $e^{ix\xi }$, the equivalent contour representation integrates $e^{ix\xi }/P(i\xi )$ along $\mathrm{Im}\xi =-c$, with $c$ larger than the real parts of all roots of $P$. This line lies below all frequency poles. This is an inverse Laplace transform expressed in frequency coordinates: for $P(D)=D-a$, $a>0$, the solution $H(x)e^{ax}$ is a distribution but not a tempered distribution, so it has no ordinary tempered Fourier transform. The contour shift accommodates this exponentially growing case.

Measure the quantum typical subspace projector $\Pi$. Encode its successful sector isometrically and reserve one orthogonal compressed flag for failure; decode the flag to a fixed state. The composite channel is ${\mathcal{N}}_n(\tau )=\Pi \tau \Pi +\mathrm{Tr}[(I-\Pi )\tau ]|{\phi }_0⟩⟨{\phi }_0|$. It is trace preserving. The Kraus formula for entanglement fidelity gives the displayed lower bound, while convexity of the square gives the same bound for average pure-source fidelity. One flag adds only a vanishing asymptotic rate overhead.

At leading electromagnetic order and with negligible electron mass, the photon-channel relativistic cross-section is ${\sigma }_h=4{\pi }^2\alpha {\rho }_h=16{\pi }^3{\alpha }^2{\rho }_J/s$, in the stated hadronic photon spectral density and hadronic electromagnetic-current spectral density conventions. Initial electron and positron spins are averaged; final hadronic states are summed inclusively. Electroweak channels beyond photon exchange are separate contributions.

A pseudoclassical spinning particle supplements ordinary position and momentum by odd Grassmann variables. An even multiplier imposes $p^2=0$ and an odd multiplier imposes $p\cdot \zeta =0$. The odd symplectic term gives $\{{\zeta }^m,{\zeta }^n{\}}_D=-i{\eta }^{mn}$. Quantization represents these variables by a Clifford algebra, producing a spinor wavefunction.

For $x{y}^{\prime \prime }+a(x)y=0$ with analytic coefficient and a continuous solution having $a(0)y(0)\ne 0$, the equation gives $y^{\prime \prime }\sim -a(0)y(0)/x$. Integrating towards the endpoint gives the displayed logarithmic divergence of the derivative. A bounded solution value therefore does not imply a finite derivative at a regular singular point.

A regular formula is built using the constructors of regular logic, with a finite variable context. The image of a definable relation interprets an existential quantifier. Regular formulas remain regular after substitution, conjunction and existential quantification.

A regular theory has axioms that are sequents between regular formulas. It has a regular syntactic category and a classifying topos of sheaves for the regular coverage. Its regular formulas enjoy the disjunction property of regular theories when a finite disjunction is allowed as an outer coherent conclusion.

The regular coverage on a small regular category takes each regular epimorphism as a one-arrow cover. Pullback and composition stability make these a coverage basis. It is subcanonical because maps constant on the kernel pair of a regular epimorphism descend uniquely to its quotient.

At an abruptly truncated razor-thin disk with nonzero edge surface density of a disk, the midplane radial force diverges logarithmically. Locally, mass occupies a half-plane and the inward force contains a nonzero angular factor multiplying $G{\Sigma }_{\mathrm{edge}}{\int }_{ϵ}^{\ell }ds/s$. The opposite exterior half-plane is absent, so its cancellation does not occur. Finite thickness or a taper taking the edge density to zero regularizes the force. Therefore a finite-edge galaxy rotation curve requires more than the ideal scale-free Mestel disk normalization.

Starting from rest, compose $n$ positive collinear Lorentz boosts, each with the same finite rapidity $\alpha >0$. Each step has relative velocity $w=c\tanh \alpha$, measured in the preceding rest inertial frame. The relativistic velocity-addition formula givesThese are equal increments of rapidity, not equal increments of velocity measured in the original inertial frame. They can describe a chain of relatively moving inertial frames or successive idealized changes of a massive particle's motion; no particular acceleration schedule is assumed. ^[ref].

For a relative velocity $w=c/2$ at each step,The first four velocities are $u_0=0$, $u_1=c/2$, $u_2=4c/5$, and $u_3=13c/14$. At step eight, $u_8/c=3280/3281\approx 0.99969521$. The rapidities grow by the same amount at each step, while the successive velocity increments become smaller.

A short derivation applies the hyperbolic-function identity for $\tanh (a+b)$ to the recurrence, giving ${\theta }_{n+1}={\theta }_n+\alpha$. Substituting $\tanh x=(e^{2x}-1)/(e^{2x}+1)$ and $e^{2\alpha }=3$ gives the displayed closed expression. In particular,for every finite nonnegative integer $n$. The limit is $u_n\to c$ as $n\to \infty$, but no finite sequence reaches the speed of light. This illustrates the beauty of rapidity: unbounded additive parameters coexist with bounded physical velocities. Replacing every step by a negative rapidity gives the corresponding approach to $-c$.

Restricted randomization samples assignments from a specified subset preserving features such as block balance, fixed group sizes or allowable treatment sequences. Inference must respect the chosen allocation scheme rather than treating all label permutations as admissible.

Allocation concealment prevents those deciding whether to enroll an eligible participant from knowing the upcoming randomized assignment. It protects the randomized controlled trial against selection based on that assignment. It concerns access to the assignment before enrollment; keeping participants or assessors unaware after assignment is a separate design feature.

For a geometric distribution count on the positive integers, independent of independent and identically distributed random variables $X_i$, the moment-generating function of $S={\sum }_{i=1}^N{X}_i$ is the displayed expression. Condition on the count and sum a geometric series. For real $t$, the finite domain requires $M_X(t)<\infty$ and $(1-p)M_X(t)<1$. Positive summands always permit $t\le 0$, even when no positive exponential moment exists; this gives a Laplace transform of a nonnegative random variable. The aggregate has no mass at zero when all summands are strictly positive.

For independent uniform couplings on $[0,g_{\ast }]$ and equatorial initial bath spins,At fixed finite $N$, the mean tends to zero and the mean square tends to $2^{-N}$ as time grows. At any fixed nonzero time, the mean-square bracket is strictly below one; the Markov inequality proves coherence tends to zero in probability as $N$ grows. Individual finite realizations still exhibit finite spin-bath coherence recurrence.

For initially equatorial spins and uniform couplings on $[0,g_{\ast }]$, expand $\log \cos (2g_{k}t)=-2g_k^2{t}^2+O(g_k^4{t}^4)$. The strong law of large numbers then gives $\sum g_k^2\simeq N{g}_{\ast }^2/3$. On the scale $t=O((g_{\ast }\sqrt{N}{)}^{-1})$, the total fourth-order remainder tends to zero, giving the Gaussian coherence envelope. This approximation concerns early times and does not remove finite spin-bath coherence recurrence.

The whole space of infinite subsets of the natural numbers is a meagre set in the Ramsey cone topology. For $D_j=\{X:\min X=j\}$, the closure is $\{X:j\in X\}$, which has empty interior: every infinite ground set can be thinned to omit $j$. Thus all $D_j$ are nowhere dense, while their countable union is the whole space. In contrast, the whole space is not meagre in either the Ellentuck topology or the ordinary topology on infinite subsets.

For $L={\mathbb{Q}}_p({\zeta }_{p^n})$, the uniformiser ${\zeta }_{p^n}-1$ and the uniformizer criterion for lower ramification groups give the displayed valuation for $a\ne 1$. Thus $G_0$ is the full unit group modulo $p^n$, and $G_i$ consists of units congruent to one modulo $p^j$ when $p^{j-1}\le i\le p^j-1$, $1\le j<n$. All groups are trivial for $i\ge p^{n-1}$. The formula includes $p=2$, where the first congruence subgroup is already the full group.

For any aggregate with positive finite variance $v$, quota share reinsurance with retained fraction $\alpha$ has total party variance $v({\alpha }^2+(1-\alpha {)}^2)=v/2+2v(\alpha -1/2{)}^2$. Equal sharing uniquely minimizes this objective. The two party totals remain dependent; adding their individual variances is a different objective from the fixed variance of their sum.