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Two vertically unbounded regions with buoyancy frequencies $N_1,N_2$ and thermal wind shears ${\Lambda }_1,{\Lambda }_2$ support a localized wave if pressure and normal velocity are continuous at their common material interface. With decays ${\varphi }_1=A{e}^{{\mu }_{1}z}$ below and ${\varphi }_2=A{e}^{-{\mu }_{2}z}$ above, ${\mu }_j=N_{j}K/|f_0|$, matching givesWhen $N_1/N_2\to \infty$ with bounded shear, the lower layer becomes effectively rigid and the upper-layer Eady edge wave speed ${\Lambda }_2/{\mu }_2$ is recovered.

For a semi-infinite fluid above a rigid horizontal boundary, constant buoyancy frequency $N$, and thermal wind $U=\Lambda z$, a zero-interior-potential vorticity perturbation has vertical dependence $e^{-\mu z}$, where $\mu =NK/|f_0|$ and $K=(k^2+l^2{)}^{1/2}$. Material conservation of boundary buoyancy gives $c{\varphi }^{\prime }(0)+\Lambda \varphi (0)=0$. Thus its phase velocity relative to the boundary flow is $c=\Lambda /\mu$. The wave's time dependence resides in its boundary condition; the interior field follows by potential-vorticity inversion. MIT's Eady-edge-wave notes develop this boundary-wave interpretation.

The boundary of a planar domain $D$ is its boundary of a set. It is distinct from the boundary of a parametrizing interval: a curve can begin on $\partial D$ while all its positive-time points lie inside $D$.

At a step in buoyancy frequency without a jump in background mass density, vertical velocity and fluid pressure are continuous. For a fixed horizontal wavenumber and angular frequency, these are continuity of $W$ and $W^{\prime }$. They determine reflection and transmission between propagating internal gravity waves and evanescent waves.

A tangential boundary derivative differentiates the boundary restriction of a function along a tangent vector. Prescribing it fixes the boundary value only up to a constant on each connected boundary component. This differs from a Neumann boundary condition, which specifies a normal derivative.

In an unbroken supersymmetric vacuum, physical boson and fermion partners have equal masses because supercharges commute with four-momentum. For one canonical chiral superfield, fluctuations about $W^{\prime }(v)=0$ have mass $|W^{\prime \prime }(v)|$ for both the complex scalar field and Weyl spinor.

Relativistic spherical accretion is steady radial gas inflow in Schwarzschild spacetime, with conserved rest-mass flux and relativistic Bernoulli function. A regular solution passes a finite sonic point outside the event horizon. For an isentropic gas with $\gamma =5/3$ and cold asymptotic sound speed $c_{\infty }\ll c$, the critical radius is approximately $3GM/(4c{c}_{\infty })$ and the leading mass accretion rate is $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$. Relativistic corrections resolve the zero-radius critical point of the Newtonian $\gamma =5/3$ model without eliminating accretion.

In bond percolation, a percolation cluster is a connected component of a graph formed by the open edges, with every graph vertex retained. In site percolation, it is a connected component of a graph of the subgraph induced by the open graph vertices. Write $C(x)$ for the cluster containing $x$; for site percolation set $C(x)=\varnothing$ when $x$ is closed.

The independent edge law of bond percolation on ${\mathbb{Z}}^d$ is ergodic under lattice translations: every translation-invariant event has probability zero or one.

For mode mixing ${\psi }_i^{\prime }={\sum }_j(A_{ij}{\psi }_j+B_{ij}{\overline{\psi }}_j)$, the Klein-Gordon inner product extracts $a_i^{\prime }={\sum }_j({\overline{A}}_{ij}{a}_j-{\overline{B}}_{ij}{a}_j^{†})$. The canonical commutation relation is preserved exactly when $A{A}^{†}-B{B}^{†}=I$ and $A{B}^T=B{A}^T$. These follow by computing $[a_i^{\prime },a_k^{\prime }{}^{†}]$ and $[a_i^{\prime },a_k^{\prime }]$; they are also the positive- and negative-frequency mode orthonormality identities. In infinitely many modes, convergence and the existence of a common bosonic Fock space representation require additional analysis; the expected total particle number is finite only if $B$ is a Hilbert-Schmidt operator.

A function into a Banach space is strongly measurable if, outside a null set, it is a pointwise norm limit of measurable simple functions. A measurable function with values in a separable Banach space is strongly measurable: approximate values by a countable collection of balls of shrinking radii and then truncate the resulting countably valued approximants to simple functions. A strongly measurable function is Bochner integral integrable exactly when its norm is integrable. Finite second moments under a probability measure therefore imply existence of its Bochner integral expected value by the Cauchy-Schwarz inequality.

For a probability measure whose identity map is strongly measurable with integrable norm, its barycenter is the Bochner integral of that map. It is characterized by $\phi (b(\mu ))=\int \phi (x)d\mu (x)$ for every continuous linear functional. For a measure on a weakly compact set in a separable Banach space, its barycenter belongs to the norm-closed convex hull of that set.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$ over an algebraically closed field, with exceptional curve $E$, $K_{\tilde{X}}={\psi }^{\ast }{K}_X+E$ after choosing compatible representatives. In local coordinates $y=xu$, the differential $dx\wedge dy=xdx\wedge du$ has one additional zero along $E$. Thus the formula is valid in every characteristic.

Over a splitting field for finite group representations of characteristic $2$, put $a=(123)$. The block of a group algebra idempotents of $k{S}_3$ are $b_0=1+a+a^2$ and $b_1=a+a^2$. Their block algebras are $b_{0}k{S}_3\cong k{C}_2$ and $b_{1}k{S}_3\cong M_2(k)$, with a defect group of a block given by $C_2$ and $1$, respectively. The first is a local ring; for the second, the representation $a\mapsto \mathrm{diag}(\omega ,{\omega }^2)$, $(12)\mapsto \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ generates the full matrix algebra. The Cartan matrix of a group algebra is $\mathrm{diag}(2,1)$, and the ordinary trivial, sign, and two-dimensional characters give decomposition matrix $\left(\begin{array}{cc}1& 0\\ 1& 0\\ 0& 1\end{array}\right)$.

Over a field of characteristic three, the group algebra $A=k{S}_3$ has a single block of an Artinian algebra. Put $r=(123)$, $s=(12)$ and $a=r-1$. Then $a^3=0$, $sas=-a+a^2$, and $J(A)=aA$: the latter is a nilpotent ideal, with semisimple ring quotient $k{C}_2$. The center of an associative algebra is $k1\oplus k{a}^2\oplus k{a}^{2}s$, whose last two summands form a square-zero ideal. A central idempotent $d1+n$ satisfies $d^2=d$ and $(2d-1)n=0$, hence is zero or one. Thus no nontrivial central block decomposition exists.

For a finite-dimensional associative algebra, form a graph on its simple modules, joining $S,T$ when ${\mathrm{Ext}}^1(S,T)$ or ${\mathrm{Ext}}^1(T,S)$ is nonzero. Its connected components are exactly the simple modules belonging to each block of a finite-dimensional algebra. They are also the components generated by sharing a Jordan–Hölder factor occurrence in an indecomposable representation that is a projective module.

Indeed, Ext separation of finite-length modules would split the regular module of a block into canonical summands if that block had two graph components. Right multiplication preserves these summands, so its projection supplies a nontrivial central idempotent, contradicting the definition of a block. An indecomposable projective belongs to one block. Finally, a nonsplit extension $0\to U\to V\to W\to 0$ of simple modules is a quotient of the projective cover of $W$: a lift onto $W$ must contain $U$, since otherwise the extension splits. Thus its two endpoints share factors in an indecomposable projective.

The Thorne spin limit is the equilibrium spin produced by competition between coherent thin-disc matter accretion and capture of disc photons by a Kerr black hole. Preferential capture of counterrotating photon angular momentum prevents the standard radiatively efficient model from reaching $a_{\ast }=1$. The commonly quoted limit is approximately $0.998$, with some dependence on emission and accretion assumptions. Thorne's original calculation derives this result.

For $F(a)={\sum }_{j=0}^{a-1}{s}_2(j)$ and nonnegative integers $a,b$, the displayed inequality follows by induction from $F(2r)=2F(r)+r$ and $F(2r+1)=F(r)+F(r+1)+r$. Splitting a vertex set of a hypercube graph into its two coordinate sections then proves the edge-isoperimetric theorem for binary initial segments.

Let a tripartite graph have nonempty parts $X,Y,Z$, pair edge density of a bipartite graph values $\alpha ,\beta ,\gamma$ on $XY,YZ,XZ$, and $\Vert G(X,Y)-\alpha {\Vert }_{\square }\le c$. If every $z\in Z$ has exactly $\beta |Y|$ neighbors in $Y$, its normalized triangle count $\tau$ obeysThe constant-degree assumption makes the contribution of the constant $\alpha$ exactly $\alpha \beta \gamma$. For each fixed $z$, apply the bilinear correlation bound for the box norm to the indicator functions of its two vertex neighbourhoods. Their squared $L^2$ norms are $\beta$ and the relative $X$-degree of $z$. Average over $z$ and use the Cauchy-Schwarz inequality to bound the mean square root of that degree by $\sqrt{\gamma }$.

For $c=1-a-b$ with positive real parts $\mathrm{Re}a,\mathrm{Re}b,\mathrm{Re}c$, the integral converges at its two finite singularities and at infinity, and equalsTo prove it, use Schwinger parameterization with exponents $1-a$ and $1-b$. Integrating the resulting planar Gaussian integral gives $\pi /(\lambda +\rho )$ times $\exp [-\lambda \rho /(\lambda +\rho )]$. Set $q=\lambda +\rho$ and $x=\lambda /q$; the $q$ integral is a gamma function and the $x$ integral is $B(a,b)$, giving the stated ratio. This domain justifies the interchanges; elsewhere the answer is interpreted by analytic continuation. The string measure $d^{2}z=2dxdy$ multiplies the answer by two.