In a proof of the Szemerédi regularity lemma, suppose $k$ equal classes have size $L$ and witness splitting creates at most $2^k$ atoms per class. Divide every atom into blocks of size $\ell =\lfloor L/4^k\rfloor$, putting leftovers into the exceptional set. At most $k{2}^k\ell \le kL{2}^{-k}$ vertices are lost. When $L\ge 2\cdot 4^k$, at most $2k{4}^k$ blocks remain. For equitable regularity energy defined by omitting exceptional vertices, the loss from discarding a fraction $r$ is at most $2r$: only ordered pairs incident with discarded vertices are removed, and every squared edge density of a bipartite graph is at most one. This provides a quantitative way to restore equal class sizes after each energy increment.

For ${\eta }_k(z)=|z-k|$, the displayed flux is continuous at $z=k$ and has derivative $\mathrm{sgn}(z-k)f^{\prime }(z)$ almost everywhere. On a bounded state range with $|f^{\prime }|\le M$, its symmetric two-state version $Q(a,b)$ obeys $|Q(a,b)|\le M|a-b|$ and is Lipschitz in both arguments. The vanishing factor at $a=b$ removes any sign-function jump contribution.

For conservative binary mass transfer and $R_L\propto a(M_d/M{)}^{1/3}$,The conservation of angular momentum gives $d\log a/d\log M_d=-2(1-q)$. Since a mass loss has $d\log M_d<0$, the donor moves inside its lobe when ${\zeta }_{\mathrm{ad}}>{\zeta }_L$. An ideal fully convective donor with ${\zeta }_{\mathrm{ad}}=-1/3$ is therefore stable for $q<2/3$ and unstable for $q>2/3$ under these assumptions.

The stellar radius response exponent measures how a star's radius changes as it loses mass. The adiabatic exponent ${\zeta }_{\mathrm{ad}}$ describes loss too rapid for thermal adjustment; the equilibrium exponent ${\zeta }_{\mathrm{eq}}$ follows a sequence of thermally relaxed stars. Compare ${\zeta }_{\mathrm{ad}}$ with the Roche-lobe radius response exponent to test dynamical stability.

For a finite-dimensional irreducible representation $L(\lambda )$ of a complex semisimple Lie algebra, its dual Lie algebra representation has highest weight $-w_0\lambda$, where $w_0$ is the longest Weyl-group element. Indeed duality negates all weights, and $w_0\lambda$ is the lowest weight of the original module. In particular $w_0=-I$ makes every such irreducible module self-dual.

The conjugate Dolbeault operator is the type-$(1,0)$ component of the exterior derivative: $\partial :{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p+1,q}$. Complex conjugation interchanges $\partial$ and $\bar{\partial }$. On a complex manifold, $d=\partial +\bar{\partial }$, ${\partial }^2={\bar{\partial }}^2=0$, and $\partial \bar{\partial }+\bar{\partial }\partial =0$.

Let $U\subseteq D$ be open. Regard $H_{\mathrm{supp}}=H_0^1(U)$ as functions on $D$ by zero extension of H01, and let $H_{\mathrm{harm}}$ consist of the weakly harmonic Sobolev functions on $U$ belonging to $H_0^1(D)$. For $h\in H_{\mathrm{harm}}$, the Dirichlet inner product $(h,\varphi {)}_{\nabla }$ vanishes for each test function supported in $U$. Approximate any $u\in H_{\mathrm{supp}}$ by those test functions in the gradient norm and use the Cauchy-Schwarz inequality. This gives $\boxed{(h,u{)}_{\nabla }=0}$. The proof also works with the inhomogeneous zero-boundary Sobolev space convention, and with a homogeneous completion realized as weak functions.

For the dilaton equation from contracted Bianchi identity, put $F=e^{-2\phi }$. The chain rule gives ${\square }_{g}F=F(-2{\square }_g\phi +4|\nabla \phi {|}^2)=-2CF$. Thus the exponential obeys a linear Klein-Gordon equation on the given background, although the original dilaton equation is nonlinear and the string-frame metric remains coupled to the dilaton. With the opposite named-field convention $\phi =-\Phi$, this variable is $F=e^{2\Phi }$.

In flat target space, $\phi =q_a{X}^a$ has zero Hessian and therefore solves the leading metric-dilaton Weyl condition for every constant covector $q$. But $\square \phi -2|\nabla \phi {|}^2=-2q^2$, which need not vanish. This demonstrates the integration constant left by the contracted Bianchi identity. Full quantum Weyl invariance fixes the allowed $q^2$ through the central charge, as in a linear dilaton conformal field theory; the metric equation alone does not.

For a spherical droplet at fixed center height $h>R$ above a flat equilibrium reservoir, the diffusion-capacitance analogy and sphere-plane capacitance giveAt large separation, $c(R/h)=1+R/(2h)+O((R/h{)}^2)$, soThe nearby reservoir increases the total flux and shortens the lifetime. The spherical-shape assumption controls the geometry; the local current density is not uniform over the surface.

A quasi-static chemical potential outside a fixed-shape droplet solves the same Laplace equation as an electrostatic potential outside an equipotential conductor. If $\mu ={\mu }_s$ on the droplet and zero on the surrounding reservoirs, its outward diffusive current iswhere $C$ is the corresponding electrostatic capacitance in a medium of permittivity $ϵ$. The arbitrary $ϵ$ cancels against the capacitance's proportionality to $ϵ$. Thus electrostatic geometry directly determines evaporation rates.

Every real-linear derivation of an algebra $D:C^{\infty }(M)\to C^{\infty }(M)$ on a smooth manifold has the form $Dh=Z(h)$ for a unique smooth vector field $Z$. The product rule makes $D$ local: multiplying a function that vanishes near a point by a smooth cutoff function supported there gives zero derivative at that point. Locally write $h(x)=h(p)+{\sum }_i(x^i-x^i(p))h_i(x)$, where $h_i(p)={\partial }_{i}h(p)$. Applying $D$ at $p$ gives $Dh(p)={\sum }_{i}D(x^i)(p){\partial }_{i}h(p)$; the smooth coefficients $D(x^i)$ define the claimed vector field. No continuity hypothesis is needed. In dimension zero all such derivations vanish.

The endomorphism bundle connection of a rank-$r>0$ bundle is flat if and only if the original curvature satisfies $F_A=\omega \mathrm{id}$. By the curvature of an endomorphism bundle connection, flatness means each $F_A(X,Y)$ commutes with every fiber endomorphism. The centre of the full matrix algebra consists of scalar matrices, and $\omega =r^{-1}\mathrm{tr}{F}_A$ is the required smooth two-form. The original connection can still have nonzero scalar curvature.

The defining representation of ${\mathfrak{so}}_{2n+1}$ has weights ${\epsilon }_1,\dots ,{\epsilon }_n,0,-{\epsilon }_n,\dots ,-{\epsilon }_1$. Its crystal basis is a chain on these vertices, with lowering-edge colors $1,\dots ,n-1,n,n,n-1,\dots ,1$. The short-root string has length two through zero. With the crystal tensor-product rule, for $n\ge 2$ the tensor square has highest vertices $1\otimes 1$, $1\otimes 2$ and $1\otimes \bar{1}$, corresponding to the traceless symmetric square, the exterior square and the trivial Lie algebra representation. Their dimensions are $(2n+1)(n+1)-1$, $n(2n+1)$ and one.

The seven-dimensional irreducible representation of the G2 root system has highest weight ${\omega }_1=2{\alpha }_1+{\alpha }_2$. Its crystal basis is the chain of weights $2{\alpha }_1+{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1,0,-{\alpha }_1,-{\alpha }_1-{\alpha }_2,-2{\alpha }_1-{\alpha }_2$, with successive edge colors $1,2,1,1,2,1$. Each step subtracts its color's simple root. The nonzero weights are the six short roots, each of multiplicity one; the zero weight also has multiplicity one.

For a prime number $p$, integer $k\ge 1$, and $\gamma =C_{p^k}(\lambda )$, taking the $p^k$-core of a partition preserves the entries of the core tower of a partition at levels $0,\dots ,k-1$ and makes every level at least $k$ empty. Also the weight of a partition $w_{p^k}(\lambda )$ equals the total size at level $k$ of the quotient tower of a partition.

For the first claim, a removable rim hook of length $p^k$ is a bead move by $p^k$ on the abacus of a partition. It preserves the numbers of beads on the $p$ runners and hence the $p$-core. Under the abacus divisible-hook correspondence, it becomes removal of a $p^{k-1}$-hook in one quotient component; iteration preserves the cores at the next $k-1$ levels. After all $p^k$-hooks are removed, every component at quotient level $k$ is empty, so every higher core level is empty as well. The weight assertion follows from iterated quotient equals a power quotient up to permutation and the size formula for the quotient of a partition.

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the power seriesHere $(a{)}_n$ is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.

For circular planetary orbits with angular momenta $L_j=M_j\sqrt{G{M}_{\star }{a}_j}$, the inclination matrix has positive off-diagonal entries, zero row sums, and $L_j{B}_{jk}=L_k{B}_{kj}$. Positive diagonal symmetrization of a matrix with $W=\mathrm{diag}(L_j)$ makes $W^{1/2}B{W}^{-1/2}$ a symmetric matrix. Moreover,All eigenvalues are real and nonpositive. The zero eigenvalue is a common tilt of every orbital plane; nonzero modes describe relative nodal precession.

The Faddeeva function is an entire function built from the complementary error function. Its integral representation for $\mathrm{Im}z>0$ is $w(z)=(i\pi {)}^{-1}{\int }_{-\infty }^{\infty }{e}^{-t^2}/(t-z)dt$, as normalized in NIST's integral representation. Its reflection identity is $w(-z)=2e^{-z^2}-w(z)$, directly from the odd function property of the error function.

This category has commutative monoids and their homomorphisms. Pointwise addition makes each hom-set a commutative monoid, and composition is additive in both variables. Finite cartesian products are also coproducts, using the coordinate injections with zero in the other coordinates and summing images. Therefore it is a semi-additive category, but it is not an additive category. The reflexive pair from the submonoid $\{(m,n):m\le n\le 2m\}$ of ${\mathbb{N}}^2$ to $\mathbb{N}$, using the two projections and section $m\mapsto (m,m)$, has an arrow from $1$ to $2$ and none from $2$ to $1$ in its represented graph. Thus a reflexive pair here need not yield a groupoid.