Let $(w_m)$ be the orthonormal basis of Dirichlet eigenfunctions for a strictly positive Dirichlet realization of an elliptic operator, with eigenvalues ${\lambda }_m$. ThenFor integers $k$, these domains are the Sobolev domains of powers of an elliptic Dirichlet operator. At $k=0$ the condition is just Parseval identity and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/d{x}^2$ on $(0,\pi )$, the smooth Dirichlet function $u=x(\pi -x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.

For a constant-coefficient elliptic differential operator $P(D)$ of order $N\ge 1$, localizing the equation gives $P(D)(\chi u)=\chi P(D)u+[P(D),\chi ]u$. The commutator has order at most $N-1$, so a global Fourier transform estimate raises the known Local Sobolev space index by one until the forcing index plus $N$ is reached.

The number $m_n$ of monomials in $n$ variables with each exponent at most two and total polynomial degree at most $\lfloor 2n/3\rfloor$ satisfiesFor each exponent vector $a$ counted on the left, $1\le 2^{2n/3-{\sum }_i{a}_i}$. Summing and then allowing all $a\in \{0,1,2{\}}^n$ gives $m_n\le 2^{2n/3}(1+1/2+1/4{)}^n$. The base has cube $343/16<27$, so it is strictly less than three. This elementary weighted count gives the exponential saving in the polynomial method in combinatorics without an unstated probability estimate.

Steady conservative small-amplitude waves have divergence-free Eliassen–Palm flux by wave activity conservation. In a balanced mean state with homogeneous impermeable boundaries and no independently forced residual mean circulation, this gives zero residual circulation and zero mean acceleration. The Eulerian mean circulation can nevertheless be nonzero because the residual circulation includes an eddy-induced correction. Wave dissipation, time-dependent wave activity or boundary forcing can invalidate the conclusion.

A directed system of first-order structures and coherent elementary embeddings has a direct limit into which every member embeds elementarily. Represent an element by a pair consisting of a stage and an element at that stage; identify two representatives when their images agree at a common later stage. Interpret functions and relations at a common stage. Mathematical induction on first-order formulas proves elementarity: for an existential first-order formula true at the limit, its witness and parameters occur at a common stage, and elementarity pulls truth back to the stage containing the original parameters. This generalizes the elementary chain theorem beyond linearly ordered diagrams.

Because the electromagnetic field tensor is the exterior derivative $F=dA$ of a potential, $dF=d^{2}A=0$. For a torsion-free connection this iswhich is the covariant form of the two homogeneous Maxwell equations.

The source-free electromagnetic Lagrangian density is quadratic in the electromagnetic field tensor. Its Euler-Lagrange field equation is ${\partial }_{\mu }{F}^{\mu \nu }=0$. Antisymmetry of $F$ makes the density invariant under gauge transformations, and contraction of its indices makes it a Lorentz scalar.

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.