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A circular elastic plate with bending stiffness $B$, constant excess pressure $p$ and clamped boundary conditions at radius $R$ obeys $B{\nabla }_r^{4}h=p$, where ${\nabla }_r^2=r^{-1}{\partial }_r(r{\partial }_r)$. Its regular solution is $h=p(R^2-r^2{)}^2/(64B)$. Consequently its enclosed volume is $V=\pi p{R}^6/(192B)$, and its edge curvature is $h_{rr}(R)=p{R}^2/(8B)=24V/(\pi R^4)$.

A fluid advancing under an elastic plate can be resisted mainly by lubrication theory flow near its front. For front gap $h_f$ and peeling length $l_p$, the elastic pressure gradient is of order $B{h}_f/l_p^5$, while the gap volume flux is of order $B{h}_f^4/(\mu l_p^5)$. Mass balance therefore gives peeling speed $U_p\sim B{h}_f^3/(\mu l_p^5)$.

For a thin plate with areal mass $m$ and bending stiffness $B$, small free transverse disturbances satisfy $m{\eta }_{tt}+B{\nabla }^4\eta =0$. A plane wave of tangential wavenumber $k$ therefore satisfiesThe frequency is quadratic in wavenumber, so these waves are dispersive. Matching this free-wave relation cancels the plate's inertial and bending response in an acoustic transmission problem.

For $k>2$ and a Dirichlet character modulo $N$ with $\chi (-1)=(-1{)}^k$, define $G_k(\chi ,z)={\sum }_{m,n\in \mathbb{Z},(n,N)=1}\chi (n)(mz+n{)}^{-k}$. This series converges absolutely and locally uniformly on the complex upper half-plane and has period $N$.

The Proca stress-energy tensor is the metric variation of the matter part of the Proca action. It consists of the electromagnetic stress-energy tensor plus the quadratic mass contribution $m^2(A_a{A}_b-g_{ab}{A}^2/2)$.

The unit-coupling magnetic solution of Einstein-Maxwell-dilaton theory has $d{s}^2=-Vd{t}^2+V^{-1}d{r}^2+R^{2}d{\Omega }_2^2$, magnetic potential $A=P\cos \theta d\varphi$, and dilaton $\varphi =-\frac{1}{2}\log (1-r_{-}/r)$, where $r_{-}=2P^2/r_{+}$ and $r_{+}>0$. Its physical asymptotic component has $r>r_{-}$. For $0\le r_{-}<r_{+}$ it has a regular Killing horizon at $r_{+}$, area $4\pi r_{+}(r_{+}-r_{-})$ and surface gravity $1/(2r_{+})$. Its asymptotic mass is $r_{+}/2$. For $r_{-}>r_{+}$ the zero-area singularity is naked. The limiting case $r_{-}=r_{+}$ is singular and must not be treated as a regular nondegenerate horizon.

For algebras for a monad $(A,\alpha ),(B,\beta )$ and underlying binary coproducts in a category, put $\kappa =[T{\nu }_1,T{\nu }_2]:TA+TB\to T(A+B)$. If the algebra pair $F(\alpha +\beta ),{\mu }_{A+B}F\kappa :F(TA+TB)\rightrightarrows F(A+B)$ has a coequalizer, that coequalizer is their algebra coproduct. An algebra map from $F(A+B)$ transposes to $[a,b]:A+B\to C$; equalizing the pair is exactly the two algebra-morphism equations for $a,b$. The pair is reflexive through $F({\eta }_A+{\eta }_B)$.

Expand an infinite first-order structure by Skolem functions, choose distinct elements $b_n$, and fix a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. For each finite subset $F$ of the required index order, form the ultrapower over ${\mathbb{N}}^F$ using the ordered Fubini product of ultrafilters. Pullback along coordinate projections gives coherent elementary maps by the Łoś theorem. In their directed limit of elementary embeddings, the classes of the coordinate functions $\mathbf{n}\mapsto b_{n_i}$ are distinct and form an order-indiscernible sequence; every first-order formula on increasing coordinates has the same nested ultrafilter test. Their Skolem hull gives an Ehrenfeucht-Mostowski model. An index-order automorphism sends a term in the generators to the same term in the transported generators; indiscernibility makes this well defined. Uniqueness is in the specified Skolem expansion, not necessarily in its reduct. If generators must respect a definable infinite order, first produce an increasing sequence by an ultrapower of finite increasing chains.

For an effective Cartier divisor $E$ and any Cartier divisor $D$, multiplication by a defining section of $E$ gives the displayed short exact sequence of sheaves. Its long exact sequence in sheaf cohomology compares sections and cohomology on $X$ with their restrictions to $E$.

An Algol variable is an eclipsing binary with well-separated eclipse minima and relatively little variation between eclipses. This light-curve classification includes detached systems as well as interacting Algol binaries; it does not by itself imply Roche-lobe overflow.

For a common semi-major axis $a$ and proper eccentricity $e_p$, randomly distributed proper longitudes of periapsis give an annulus of radii $a(1-e_p)$ and $a(1+e_p)$ about the point displaced from the star by minus $a$ times the forced eccentricity vector, to first order in orbital eccentricity.

An aligned eccentric ring consists of nested Kepler orbits with nearly common longitude of periapsis. Its width and surface density of a disk depend on both the distribution of semi-major axis values and the radial gradient of forced eccentricity.

A higher orbital line density at apoapsis does not always give a higher surface density of a disk: the local ring width also changes. In a narrow aligned eccentric ring, neglecting disc twist and keeping first-order orbital eccentricity, ${\Sigma }_{\mathrm{peri}}\propto 1+ade/da$ and ${\Sigma }_{\mathrm{apo}}\propto 1-ade/da$. Constant eccentricity therefore gives constant surface density to this order.

The discriminant of the Eady model with a sloping lower boundary isFor any $a<1$, choose the unique $M>0$ satisfying $M\tanh M=2-a$. Then $\Delta <0$, proving existence of an unstable band even when $a<0$. For large negative $a$, resonance occurs at large $M$ and the exponentially weak boundary-wave coupling makes the unstable band narrow.

The outer similarity solution approaches a plate edge as $h/h_0\sim [(4C/5)(1-x/L){]}^{1/4}$, where $C=B(4/5,1/2)$. Its slope becomes order one atThis region lies outside lubrication theory because its vertical and horizontal scales are comparable. The outer zero-thickness boundary condition is consistent at leading order since the matching height is $O(\ell )\ll h_0$, while the actual edge flow requires an inner solution. A finite outward flux survives even though the outer thickness vanishes, because the slope diverges.

Combining the lubrication gravity-current flux with vertical imbibition under a draining liquid layer givesThe injected volume is the integral of $h+\varphi l$, not just the surface current thickness. The model neglects horizontal subsurface flow and applies only where the substrate is wetted and the surface current supplies its drainage.

Layered convection consists of mixed convecting layers separated by more weakly transporting interfaces. In a giant planet, composition gradients can support such interfaces and reduce heat loss, contributing to delayed cooling of an inflated giant planet.

For a group $G$ and subgroup $\Gamma$ such that every double coset has finitely many orbits under left multiplication by $\Gamma$, this algebra consists of complex $\Gamma$-bi-invariant functions on $G$ supported on finitely many double cosets. Its convolution isThe double coset indicator functions form a basis. On invariant modular functions it acts on the right by $f\ast h={\sum }_{\Gamma x}h(x)f{|}_{k}x$. For $G=G{L}_2(\mathbb{Q}{)}^{+}$ and $\Gamma =S{L}_2(\mathbb{Z})$, rational conjugation of finite-index modular subgroups verifies the finiteness condition. The determinant-$n$ Hecke operator uses the indicator of all integral determinant-$n$ matrices, not only a single double coset when $n$ is composite.

An upcrossing of an interval $[a,b]$ consists of an observation at or below $a$ followed by a later observation at or above $b$. Successive completed upcrossings use disjoint buy-sell pairs.

For a smooth irreducible algebraic curve, send a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$ to the line bundle associated to a divisor defined locally by ${\mathcal{O}}_X(D{)}_P=t_P^{-n_P}{\mathcal{O}}_{X,P}$. A nonzero rational section of a line bundle gives its inverse construction, and changing that section changes its divisor by a principal divisor. Thus the divisor class group is the Picard group. The quotient of the sheaf of nonzero rational functions on an irreducible variety by the sheaf of units of the structure sheaf has stalk $\mathbb{Z}$ at each closed point, via the discrete valuation. Its global sections are finite sums of points. The corresponding long exact sequence in sheaf cohomology, and the vanishing for the rational-function sheaf, identify both groups with $H^1(X,{\mathcal{O}}_X^{\ast })$.