For a small planet on a nearly circular Kepler orbit, a stellar chord at normalized transit impact parameter $b$ has length $2R_{\star }\sqrt{1-b^2}$. Dividing by transverse orbital speed $2\pi a/P$ gives the duration. A central transit has the longest duration within this approximation. Eliminating $a$ with Kepler's third law yields $P/D^3\ge {\pi }^{2}G{\rho }_{\star }/3$ for known mean stellar mass density. Finite planet radius, appreciable orbital eccentricity, and $R_{\star }/a$ corrections change the bound.

The minimum projected planet-star separation during an exoplanet transit, divided by the stellar radius. It determines the stellar chord length and hence the transit duration of a small planet.

The eutectic composition is the liquid concentration at the eutectic temperature, where liquid and two solid phases coexist. It is a composition coordinate on a phase diagram, not the overall concentration of every sample in an eutectic system.

If $L$ is bounded and self-adjoint on a complex Hilbert space and $K$ is compact and self-adjoint, thenA singular Weyl sequence for $L$ stays singular for $L+K$, because compact operators send weak convergence to norm convergence and therefore $K{f}_n\to 0$. Applying the same argument with $-K$ proves the reverse inclusion. Finite-multiplicity isolated eigenvalues can move under such perturbations; the essential spectrum of a bounded self-adjoint operator is unchanged.

For offered load $a$ and capacity $C\ge 1$, the carried load equals the mean occupancy:Differentiating the truncated-Poisson equilibrium weights gives $m_C^{\prime }(a)={\mathrm{Var}}_a(N)/a>0$ for $a>0$. The carried load increases from zero to $C$. This monotonicity supplies the strictly increasing term in the convex potential for the Erlang fixed point.

For a top-hat planar jet in stationary fluid of the same mass density, odd and even velocity potentials have the same temporal growth rate. Put $s=\tanh (|k|L)$. Their dispersion relations use $s$ and $1/s$, but $\sqrt{s}/(1+s)=\sqrt{1/s}/(1+1/s)$. Their real phase velocities are respectively $Vs/(1+s)$ and $V/(1+s)$. The common growth rate is strictly below $|kV|/2$ for finite $|k|L>0$, approaching the isolated vortex sheet value when the sheets decouple.

In the unsquared quantum fidelity convention, operation fidelity compares a purification of a density operator with its image under a quantum channel acting on the source subsystem:Its square is entanglement fidelity. Thus this convention is distinct from the squared quantity $F_e$; the two should not be installed as synonyms.

If $\{K_a\}$ is any Kraus representation of a quantum channel $\Lambda$, thenFor a purification of a density operator, each branch overlap is $⟨{\Psi }_{\rho }|(I\otimes K_a)|{\Psi }_{\rho }⟩=\mathrm{Tr}(\rho K_a)$. Summing the squared branch overlaps gives entanglement fidelity.

For a quantum channel on a $d\ge 2$ dimensional system, purify the input to $|\Psi {⟩}_{QR}$ and set $\sigma =(\mathcal{N}\otimes \mathrm{id})(|\Psi ⟩⟨\Psi |)$. Its entanglement fidelity $F_e=⟨\Psi |\sigma |\Psi ⟩$ satisfies $S(\sigma )\le h(F_e)+(1-F_e){\log }_2(d^2-1)$. This is the entropy bound from overlap with a pure state in dimension $d^2$. For a trivial one-dimensional system the output entropy is zero.

For a finite-dimensional module over an algebraically closed field, write a Krull-Schmidt decomposition $X={⨁}_a{M}_a^{\oplus m_a}$ with distinct indecomposable types. Each endomorphism ring of $M_a$ is a local endomorphism ring with residue division algebra $k$, by the finite-dimensional division algebra over an algebraically closed field result. Modulo the Jacobson radical of $\mathrm{End}(X)$, the blocks of one type become $M_{m_a}(k)$ and all maps between different types vanish. A composite through a different indecomposable type cannot be invertible, since that would make one type a direct summand of the other. This yields the displayed product and the Levi decomposition of a quiver automorphism group.

An endomorphism ring is local when its nonunits form a proper two-sided ideal; commutativity is not required. The Fitting lemma gives this property for an indecomposable finite-length module. If two nonunits had an invertible sum, multiplication by its inverse would give $I=f+g$ with noninvertible $f,g$. Fitting makes $f$ nilpotent, so $I-f=g$ would be invertible, a contradiction. Products with a nonunit are nonunits by finite-length injectivity/surjectivity. The quotient by the nonunit ideal is a division ring.

An elliptic curve in characteristic zero admits no nonconstant rational map of projective varieties from the projective line. Over $\mathbb{C}$, write its equation as $y^2={\prod }_{j=1}^3(x-e_j)$ with distinct $e_j$. Substituting $x=u/v$ with coprime polynomials gives $(y{v}^2{)}^2=v{\prod }_j(u-e_{j}v)$. These four pairwise coprime factors must all be squares. The polynomial pencil with four square members forces $u,v$ constant, ruling out a nonconstant rational parametrization of an algebraic curve.

For a Higgs doublet with hypercharge $1/2$ and vacuum expectation value $(0,v/\sqrt{2}{)}^T$, its gauge-covariant kinetic term gives the neutral gauge-boson mass matrixin the $(W^3,B)$ basis. Its eigenvalues are zero and $(g^2+g^{\prime 2})v^2/4$, and the Weinberg angle rotation gives the photon and Z boson. The two charged electroweak gauge bosons have $m_W^2=g^2{v}^2/4$. The zero eigenvalue expresses the unbroken electric-charge gauge symmetry.

The condition $p_F=m_{e}c$ gives ${\rho }_{\ast }=[{\mu }_e{m}_u/(3{\pi }^2)](m_{e}c/\hslash {)}^3\simeq 9.74\times 10^5{\mu }_e\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. Degenerate electrons are nonrelativistic for $\rho \ll {\rho }_{\ast }$ and ultrarelativistic for $\rho \gg {\rho }_{\ast }$. This density threshold differs from thermal relativity in a nondegenerate gas.

In a thin Bottom Ekman layer, slowly evolving geostrophic flow drives upward Ekman pumping into a uniform-depth exterior layer. With $f>0$, $\alpha =\sqrt{f\nu /2}$ and $R_D=\sqrt{gH}/f$, its balanced height satisfies $({\nabla }_h^2-R_D^{-2}){\eta }_t=-(\alpha /H){\nabla }_h^2\eta$. Small-scale Fourier modes decay on $H/\alpha$; large-scale modes obey a diffusion equation with diffusivity $g\alpha /f^2$. The balanced approximation requires $|f|\tau \gg 1$.

Among families of a given size in the Boolean hypercube, a binary initial segment maximizes the number of contained two-dimensional cube faces. Under section compression in binary order, faces crossing the selected coordinate correspond to internal edges in the intersection of its two sections. The edge extremal theorem and mathematical induction control that contribution and the faces within the sections.

For a down-set of size $m$, maximize its edge boundary by minimizing the sum of the sizes of its members. Choose all sets in the smallest ranks, followed by any needed part of the next rank; this minimizes that sum among all families and is itself a down-set. With $M_j={\sum }_{i=0}^j\left(\genfrac{}{}{0ex}{}{n}{i}\right)$, $M_{-1}=0$, and $M_{r-1}\le m\le M_r$, the exact maximum is $h(m)=nm-2[{\sum }_{j<r}j(\genfrac{}{}{0ex}{}{n}{j})+r(m-M_{r-1})]$.

Linear eccentricity damping on one planet changes $\dot{z}=iAz$ to $\dot{z}=i[A+i\mathrm{diag}({\tau }^{-1},0)]z$. For a nondegenerate, weakly damped two-planet matrix, let $D=\sqrt{(A_{11}-A_{22}{)}^2+4A_{12}{A}_{21}}$. The modal damping rates are $g_{p,m}=[1\pm (A_{11}-A_{22})/D]/(2\tau )>0$. Both secular eigenmodes damp even though only one planet is directly affected; the coupling shares the dissipation.

Every doubly stochastic matrix is a convex combination of permutation matrices. Equivalently, the extreme points of the doubly stochastic convex polytope are exactly the permutation matrices. The extreme-point assertion follows from an alternating-cycle perturbation of a doubly stochastic matrix; the convex-hull assertion follows because a bounded finite-dimensional polytope is the convex hull of its vertices. Permutation matrices are extreme since their zero entries force the same zeros in every convex decomposition.

The total differential is $d_h+d_v$ for anticommuting differentials. For a double cochain complex bounded in both indices, both index filtrations are finite and have the same total cohomology as abutment.