Choose a topological generator $\gamma$ of the Galois group of a Zp-extension and set $T=\gamma -1$. The completed group algebra becomes the formal power series ring ${\mathbb{Z}}_p[[T]]$, a two-dimensional complete regular local integral domain with maximal ideal $(p,T)$. Its finite-layer quotient is $\Lambda /((1+T{)}^{p^n}-1)\cong {\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$.

The scalar stochastic differential equation $d{X}_t=\tanh X_{t}dt+d{W}_t$ has a pathwise unique global strong solution, since its coefficients are globally Lipschitz. The positive martingale $e^{t/2}/\cosh X_t$ gives a Girsanov theorem change of measure under which $X_t-X_0$ is Brownian motion. For $X_0=x$, its transition density isThis is a Doob h-transform with $h=\cosh$, and a mixture of $N(x+t,t)$ and $N(x-t,t)$ with weights $e^x/(2\cosh x)$ and $e^{-x}/(2\cosh x)$.

The neutral one-dimensional Wright–Fisher diffusion records a proportion in $[0,1]$, with absorbing endpoints, zero drift and generator $Lf(x)=x(1-x)f^{\prime \prime }(x)/2$. Its conditional variance rate is largest near one-half and vanishes at fixation. Mutation or selection variants add drift and must be specified separately.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.

A continuous injective real-valued function on a real interval is strictly monotone. A value outside the interval between two endpoint values would, by the intermediate value theorem, create a repeated value. Applying this observation to subintervals gives consistent strict ordering. If the left endpoint value is smaller than the right endpoint value, the function is strictly increasing; otherwise it is strictly decreasing.

A commuting square $vf=f^{\prime }u$ induces a unique map between their images by $I(u,v)p=p^{\prime }u$ and $i^{\prime }I(u,v)=vi$. The cokernel in a category property supplies existence, and cancellation of the epimorphic $p$ supplies uniqueness. These equations prove identity and composition laws, giving a functor from the arrow category to the abelian category.

Pull back the two arrows of an image factorization in an abelian category. Monomorphisms are preserved by pullback, and pullback stability of epimorphisms in an abelian category preserves its epic part. Pasting the squares gives the pulled-back composite. The resulting epi-mono factorization is therefore its image factorization by uniqueness.

For the renormalization-group transformation $f(u)=b^{-D}f(R_{b}u)+g(u)$, a background solving $a(u)=b^{-D}a(R_{b}u)+g(u)$ leaves a homogeneous remainder. In linear scaling coordinates $t^{\prime }=b^{{\lambda }_t}t$ and $h^{\prime }=b^{{\lambda }_h}h$, an analytic source term $g_{mn}{t}^m{h}^n$ is removed by $a_{mn}=g_{mn}/[1-b^{m{\lambda }_t+n{\lambda }_h-D}]$ unless the denominator vanishes. Such a resonance may produce a logarithm and invalidate a pure homogeneous power law. The scaling hypothesis for critical phenomena concerns the singular remainder. A regular field-dependent background cannot generally be represented by a function of temperature alone; an arbitrary analytic identity term proportional to $h^2$ makes this explicit.

On a coordinate trivialization of a principal bundle, the right-invariant vector fields $R_a(\gamma )=T_a\gamma$ satisfy $[R_a,R_b]=-c^d{}_{ab}{R}_d$. With $A=A_i^a{T}_{a}d{x}^i$, the displayed lifts project to the commuting coordinate fields and contract to zero with the principal connection. Taking their Lie brackets gives the local curvature of a principal connection coefficients $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i+[A_i,A_j]$. Thus these lifts commute exactly when the connection is flat. Arbitrary base fields need not commute, and flatness alone does not supply a global coordinate frame or remove holonomy of a connection.

For a vertically separable perturbation with normalized height profile $h(z)$, the midplane gravity reduction factor is $\int h(z)e^{-k|z|}dz$. Its small-$k$ expansion is $1-k⟨|z|⟩+o(k)$ when this first moment is finite. Match it to $e^{-kϵ}$ to get the displayed choice. For $h=e^{-|z|/H}/(2H)$ the exact factor is $(1+kH{)}^{-1}$, demonstrating why no single exponential matches every wavelength.

On a compact connected Riemannian manifold without boundary with nonnegative Ricci curvature, the Bochner formula for one-forms makes every harmonic one-form parallel. Evaluation at any point is then injective: a parallel form that is zero there stays zero along every path. Hence $\dim {\mathcal{H}}^1(M)\le \dim M$. Flat tori attain equality. Positive-definite Ricci curvature at even one point instead forces ${\mathcal{H}}^1=0$, because the Bochner integral makes its Ricci contraction vanish everywhere.

For a compact H-hull and a Borel subset of the intrinsic boundary of a simply connected domain, hydrodynamic normalization at infinity and the Poisson kernel for the upper half-plane give this limit as $y\to \infty$ with $x/y\to 0$. After multiplying the kernel by $\pi y$, it tends pointwise to one and is eventually uniformly bounded. Dominated convergence proves the finite-measure case and Fatou's lemma proves the infinite-measure case.

This capacity measures the asymptotic probability that planar Brownian motion hits a compact H-hull before the real axis. It scales linearly with length and is distinct from half-plane capacity. A vertical slit of height $h$ has capacity $2h$; a half-disc of radius $r$ has capacity $4r$. Monotonicity of Brownian hitting probabilities gives $\mathrm{cap}(K)\le 4\mathrm{rad}(K)$, where the radius is that of the least real-centred enclosing half-disc. For finite slit hulls the capacity is the length of the image of the two-sided absorbing boundary under the mapping-out function.

Finite total variation of a function on $\mathbb{R}$ gives the following error bound for averaging on dyadic cells of length $\delta =2^{-j}$:The cell error is at most its length times the oscillation on that cell, and the sum of the oscillations is at most the total variation. This proves that the difference is integrable even if $f$ is not globally in $L^1$. For an even function decreasing to zero on the positive half-line, $\mathrm{TV}(f)\le 2f(0)$ and the bound becomes $f(0)2^{1-j}$.

Global warming potential compares the time-integrated radiative forcing of equal-mass emission pulses of a greenhouse gas and carbon dioxide, over a stated time horizon $H$. It is not an instantaneous temperature ratio, and its numerical value depends on the horizon and included chemical and climatic responses.

For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.

If a time-dependent vector field satisfies ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, its flow maps satisfy ${\rho }_t^{\ast }{\alpha }_t=u_t{\alpha }_0$, where $u_t(x)=\exp ({\int }_0^t{h}_s({\rho }_s(x))ds)$. Thus the conformal factor is smooth and positive even when $h_t$ changes sign.

For any fixed odd length $\ell \ge 3$, positive triangle in a graph density forces positive density of $\ell$-cycles. A Szemerédi regularity lemma partition retains a triangle in a graph of regular dense pairs after the sparse and exceptional pairs are removed. Count embeddings of a proper three-colouring of $C_{\ell }$ into those three clusters. The constants depend on $\beta$ and $\ell$, not on graph order.

For $dX=\mu dt+\sigma dB$ on $[0,T]$, take $\theta =(\nu -\mu )/\sigma$ and use density $\mathcal{E}(\int \theta dB{)}_T$. If $\mu ,\nu$ are bounded and $\sigma$ is continuous and bounded below by a positive constant, the exponential has bounded bracket and defines an equivalent measure. Under it, $\tilde{B}=B-\int \theta dt$ is Brownian and $dX=\nu dt+\sigma d\tilde{B}$. Set $\theta =0$ after $T$ to define the density on the entire original sigma-algebra and the new Brownian motion on the whole time axis.

A family of definable arrows covers a formula when the theory proves that their images jointly exhaust that formula. Such covers encode disjunction and existential quantification. The resulting site presents the classifying topos of the geometric theory. Adding geometric axioms adds covering sieves, yielding the duality between geometric quotients and subtoposes.