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A Coleman power series encodes a norm-compatible sequence of local units in a cyclotomic tower by a single integral power series evaluated at ${\zeta }_{p^n}-1$. A logarithmic derivative and trace correction give a measure, relating the local images of cyclotomic units to p-adic L-functions.

The central binomial coefficient satisfies $4^m/(2m+1)\le \left(\genfrac{}{}{0ex}{}{2m}{m}\right)\le 4^m$. Prime factors between $m$ and $2m$ give a dyadic upper bound for the Chebyshev theta function. The prime-power exponents in the factorial quotient are sums of zeros and ones, so the central coefficient gives a linear lower bound for the Second Chebyshev function. Higher prime powers contribute only $O(\sqrt{x}\log x)$; hence the prime-counting function has upper and lower bounds of order $x/\log x$.

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

A character of a finite abelian group is a homomorphism into the complex unit circle. Equivalently it is a linear character; finite order forces each character value to be a root of unity. These are the elements of the character group of a finite abelian group and form the orthogonal Fourier basis of functions on the group.

If $H$ is a subgroup of a finite abelian group, $a\notin H$, and $k$ is the least positive integer with $ka\in H$, then each character of a finite abelian group on $H$ has exactly $k$ extensions to $H+⟨a⟩$. The displayed root choice makes the extension well-defined. Starting with the trivial subgroup and adjoining generators proves that the number of characters equals the group order without using a decomposition into cyclic groups.

A contracting homotopy on a chain complex is a degree-one map $h$ whose displayed identity makes the identity chain-homotopic to zero. It implies that all homology vanishes. The converse is not valid for arbitrary complexes of modules.

On a good cover, combine Čech degree $p$ and differential-form degree $q$ in the double complex of forms on intersections. The Čech differential and exterior derivative commute, so the total differential $\delta +(-1{)}^{p}d$ squares to zero. A partition of unity and the Poincare lemma identify its cohomology with both constant-sheaf and de Rham cohomology. The sign convention makes Čech-de Rham curvature descent explicit.

A phase-mixed population secularly forced by a distant eccentric planet has typical orbital eccentricity proportional to $e_{p}a/a_p$. Consequently a relative speed of order orbital eccentricity times Keplerian speed gives $v_{\mathrm{col}}^2\sim e_p^2(a/a_p)GM/a_p$. Comparing specific impact energy with a catastrophic disruption threshold gives a lower disruption radius $a/a_p\gtrsim Q_D^{\ast }{a}_p/(GM{e}_p^2)$, up to an order-one coefficient depending on velocity statistics and impact-energy convention. The planet mass sets the secular stirring time, but cancels from this energy scaling.

For a Cartan subalgebra $H$ of a finite-dimensional complex semisimple Lie algebra $L$, $H$ is abelian and $C_L(H)=H$, so it is maximal among abelian subalgebras. Starting from the nilpotent self-normalizing definition, the generalized-weight decomposition for a nilpotent Lie algebra and the Engel theorem show that the zero generalized weight space of its adjoint action is exactly $H$. Invariance of the Killing form makes $H$ orthogonal to every nonzero generalized weight space. The Lie theorem gives $B([H,H],H)=0$, and nondegeneracy then forces $[H,H]=0$. An element commuting with $H$ normalizes it, hence belongs to $H$.

For a relativistic particle, the Hamiltonian gauge parameter changes the worldline einbein by its time derivative. Fixing the einbein to a constant therefore gives a Faddeev-Popov determinant of ${\partial }_t$, represented by anticommuting FP ghosts. Gauge zero modes and the proper-time modulus are treated separately. The displayed phase convention gives the odd canonical bracket $\{b,c\}=-i$.

Using the graded Leibniz rule, $s^{2}A=D(sc)+g(Dc)\times c=0$ because $D(c\times c)=2(Dc)\times c$. The graded Jacobi identity gives $s^{2}c=g^2(c\times c)\times c/2=0$. BRST nilpotence on the antighost and Nakanishi-Lautrup field is immediate. No field equations are required.

Use gauge-fixing term $({\partial }^{\mu }b)\cdot A_{\mu }$, ghost term $-({\partial }^{\mu }\bar{c})\cdot D_{\mu }c$, and the left-acting BRST differential with $s\bar{c}=b$. Localizing the odd parameter on the left yields $\delta S=-\int ({\partial }_{\mu }ϵ)j_B^{\mu }$. Moving it through the antighost derivative determines the last sign. Changing the overall Noether convention reverses this current; rewriting gauge fixing by integration by parts changes its improvement term. Every summand has ghost number one.

The FP ghost square vanishes, so this BRST charge is nilpotent. On a ghost-number-zero wavefunction, BRST cohomology implements the Klein-Gordon equation. Closure of an unrestricted ghost-extended wavefunction does not independently constrain every component: a pure FP ghost component can lie in the kernel without satisfying the same scalar equation. The FP ghost sector is part of the physical prescription.

For standard one-dimensional Brownian motion and a fixed $h>0$, the Brownian reflection principle gives the survival probability $2\Phi (h/\sqrt{t})-1$. The standard normal distribution function has derivative $1/\sqrt{2\pi }$ at zero, giving the displayed asymptotic. Thus the persistence exponent is one half, with leading constant $h\sqrt{2/\pi }$.

In the completed natural Brownian filtration, a bounded continuously differentiable $\varphi$ with bounded derivative satisfiesThe integrand is unique up to $d\mathbb{P}dt$-almost everywhere equality. Gaussian integration by parts first gives the duality between the payoff and every square-integrable Itô integral. The Brownian martingale representation theorem then identifies the integrand as the projection of ${\varphi }^{\prime }(W_T)1_{\{t\le T\}}$ onto the predictable processes in the product $L^2$ space. This explains the conditional expectation in the formula, rather than a nonadapted terminal derivative.

For standard Brownian motion started at zero, let ${\tau }_x$ be its first exit from $(-x,x)$, with $x>0$. Bounded-time stopping of $B_t^2-t$ first proves integrability of the exit time and then gives its mean $x^2$. Stopping the quartic Hermite polynomial martingale gives a uniform bound on $\mathbb{E}(t\wedge {\tau }_x{)}^2$, proving second-moment integrability before passing to the limit. The resulting second moment is $5x^4/3$ and the variance is $2x^4/3$.

For standard Brownian motion started at zero and first exit ${\tau }_x$ from $(-x,x)$, the Laplace transform is $1/\cosh (x\sqrt{2\lambda })$ for $\lambda >0$. The stochastic process $e^{-\lambda t}\cosh (\sqrt{2\lambda }{B}_t)$ is the average of two copies of the Exponential martingale for Brownian motion. Its values stopped at $t\wedge {\tau }_x$ are bounded, so bounded-time application of the optional stopping theorem followed by dominated convergence evaluates the transform without assuming uniform integrability of an unstopped exponential martingale on the infinite horizon.

With $s=\sqrt{1+A^2}$, a dimensionless decaying first-order transverse sheet has $f=(s{e}^{-y}-e^{-sy})/(s-1)$, $f(0)=1$, $f^{\prime }(0)=0$, and pressure $-s(s+1)e^{-y}\cos (x-t)$. The coincident-root limit is $(1+y)e^{-y}$, the ordinary transverse mode of a Taylor swimming sheet.

The leading fixed-stroke speed of a transverse Brinkman swimming sheet is enhanced over the pure-fluid value by $\sqrt{1+(\alpha /k{)}^2}$. It follows from the mean displaced-boundary velocity and zero mean traction. The expansion is at fixed screening parameter and does not assert an enhancement at fixed available motor power.

For $s=\sqrt{1+(\alpha /k{)}^2}$, the leading work per projected area on one side of a transverse Brinkman swimming sheet is $\mu b^{2}k{\omega }^{2}s(s+1)/2$. Two fluid sides double it. Work includes viscous dissipation and drag against the matrix, giving a greater fixed-stroke cost than in the unscreened fluid.