The complement of a comeagre set is a meagre set, a countable union of nowhere dense sets. A dense G-delta set is comeagre. The Baire category theorem ensures that two comeagre subsets of a nonempty complete metric space cannot be disjoint.

An uncountable subset of the real numbers with countable intersection with every meagre set. This is the category counterpart of a Sierpiński set.

An extendible geodesic is a geodesic segment that can be continued beyond at least one endpoint within the same manifold after retaining an affine parameter. An artificial truncation at a regular event is extendible even when its continuation eventually reaches a singularity.

Two rectangular matrices are related by invertible left and right multiplication exactly when their ranks agree. The rank-$r$ orbit in ${\mathrm{Mat}}_{b\times a}(k)$ has dimension $r(a+b-r)$: choose its image in a Grassmannian, then a surjective map to that image. These are the representation orbits of the one-arrow quiver.

An invertible complex Jordan block $\lambda I+N$, with $N^s=0$, has logarithm $\ell I+{\sum }_{k=1}^{s-1}(-1{)}^{k+1}(N/\lambda {)}^k/k$, where $e^{\ell }=\lambda$. Applying this construction blockwise and conjugating back proves surjectivity of the matrix exponential on the complex general linear group. There is no globally single-valued continuous choice of logarithm.

The Mathieu characteristic values are the parameter values $a$ for which the Mathieu equation has the even periodic mode traditionally denoted ${\mathrm{ce}}_n$, or the odd periodic mode ${\mathrm{se}}_n$. They are eigenvalues of $-d^2/d{t}^2+2q\cos 2t$ with the corresponding periodic or antiperiodic boundary conditions. Their small-$q$ expansions can be derived by projecting onto a Fourier series and perturbing the resulting tridiagonal matrix.

An ordinary massless spin-one particle has transverse vector polarizations under $SO(D-2)$. A covariant polarization satisfies $p\cdot ϵ=0$ and is identified under $ϵ\mapsto ϵ+\lambda p$. The quotient has $D-2$ positive polarizations.

A massive spin-one particle transforms as a vector under its massive particle little group $SO(D-1)$. Its $D-1$ polarizations are orthogonal to its timelike four-momentum. Under transverse $SO(D-2)$ rotations they split into a vector and a scalar.

For fixed cloud shape, gravitational energy scales as $-C_{g}G{M}^2/L$ and magnetic energy as $C_B{\Phi }^2/({\mu }_{0}L)$. The magnetic-criticality threshold for negligible pressure is $M/\Phi >\sqrt{C_B/C_g}/\sqrt{{\mu }_{0}G}$. Ideal flux freezing keeps this ratio fixed during homologous collapse; geometry determines the numerical coefficient.

For $T=-[\cos \varphi {]}_{-\infty }^{+\infty }/2$, complete the static energy into squares $({\varphi }_x\mp m\sin \varphi {)}^2$ and ${\sin }^2\varphi {\alpha }_x^2$. The boundary term is $\pm rmT$. A $T=1$ saturating kink is $\varphi =2\arctan e^{m(x-x_0)}$, with constant $\alpha$ and mass $rm$. This is a model-specific Bogomolny bound.

For $\alpha =\omega t+{\alpha }_0$ and $|\omega |<m$, the kink is $\varphi =2\arctan e^{\kappa (x-x_0)}$ with $\kappa =\sqrt{m^2-{\omega }^2}$. Its rest energy and mechanical internal charge are $M=r{m}^2/\kappa$ and $Q_0=r\omega /\kappa$. Eliminating the frequency gives the displayed mass relation. The orientation rotates while the energy profile remains stationary.

Locally this density is a total derivative, so it changes neither the bulk field equations nor the energy. Its contribution to the angular canonical momentum is a spatial boundary derivative. With ${ϵ}^{01}=+1$, it produces the theta-angle shift of a sigma-model kink charge.

For bounded measurable $a,b$, with $a$ symmetric positive semidefinite, an L-diffusion is a continuous adapted process $X$ such that $g(X_t)-g(X_0)-{\int }_0^{t}Lg(X_s)ds$ is a true martingale for every $g\in C_b^2$. The general local martingale problem may instead use compactly supported test functions and require only a local martingale. Specify the test class and the true or local convention when coefficients are unbounded.

No Markov property of finitely presented groups is decidable by an algorithm taking an arbitrary finite group presentation as input. The proof reduces the word problem for a group to property recognition using an effective construction which collapses when an input word is trivial and embeds the input group when it is nontrivial.

Let $y^{\ast }$ be a globally delay-minimizing feasible vector of link throughputs. For continuously differentiable strictly increasing delays, calibrated tollsmake the marginal social costs at $y^{\ast }$ equal the perceived route costs there. First-order route-exchange conditions make $y^{\ast }$ a tolled Wardrop equilibrium. The nonnegative nondecreasing penalty vanishes at the optimum; choosing ${\epsilon }_j>0$ makes the toll genuinely traffic-dependent. The perceived link delays remain strictly increasing, so their Beckmann potential gives unique equilibrium link loads. Every tolled Wardrop equilibrium therefore has the globally optimal $y^{\ast }$, even when the total-delay objective is not convex.

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.

For the compact H-hull $(x,x+ih]$, the square-root branch asymptotic to $z-x$ gives the hydrodynamic normalization at infinity of its mapping-out function of a compact H-hull. Its expansion has coefficient $h^2/2$, the half-plane capacity of a vertical slit.

The inverse of a hydrodynamically normalized mapping-out function has expansion $f(w)=w+O(1/w)$ at infinity. If bounded-domain points $z_n$ had unbounded images $w_n=g(z_n)$, then $z_n=f(w_n)=w_n+O(1/w_n)$ would be unbounded, a contradiction. Thus $g$ is bounded even near finite rough boundary points. Its own expansion at infinity then makes $g(z)-z$ uniformly bounded throughout the domain. Reflection is needed only near infinity, not along the entire hull boundary.

For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The maximum principle for harmonic functions applied to $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.