A normalized bipartite pure state of Schmidt rank $r$ has quantum fidelity at most $\sqrt{r/d}$ with a fixed maximally entangled state on $d\otimes d$. Its coefficient matrix has matrix rank $r$ and Hilbert-Schmidt norm one. Its overlap is the trace of that matrix divided by $\sqrt{d}$; trace duality and the Cauchy-Schwarz inequality bound this by $\sqrt{r/d}$. Uniform Schmidt coefficients on $r$ matching basis pairs attain equality.

The Schmidt rank of a bipartite pure state cannot increase under local operations, even in a nonzero selected branch. A coefficient matrix $C$ changes under a local product operator to $AC{B}^T$, whose rank is at most that of $C$. For deterministic exact conversions the same conclusion follows from Nielsen's pure-state conversion theorem: at the input rank $r$, the majorization inequality forces all output coefficients after $r$ to vanish.

For an orthonormal scaling function satisfying $\phi (x)=\sqrt{2}{\sum }_k{h}_k\phi (2x-k)$, its low-pass Fourier series symbol is $m(\xi )=2^{-1/2}{\sum }_k{h}_k{e}^{-ik\xi }$. The scaling refinement equation becomes $\hat{\phi }(2\xi )=m(\xi )\hat{\phi }(\xi )$. The symbol is initially an almost-everywhere defined periodic function; the assumption that the scaling function is Lebesgue integrable alone does not make it smooth. The orthonormal translates give the quadrature mirror filter identity.

For $dX=b(X)dt+dW$ with continuous drift, integrate the positive derivative displayed above. The resulting scale function of a one-dimensional diffusion solves $g^{\prime \prime }=-2b{g}^{\prime }$. The Itô formula removes the drift of $g(X)$, giving $dY=h(Y)dW$, where $h=g^{\prime }\circ g^{-1}$ on the interval $g(\mathbb{R})$. If the drift is bounded, $h^{\prime }=-2b\circ g^{-1}$ is bounded, so $h$ is Lipschitz on that interval.

For independent Brownian motions $W,B$, the Itô formula cancels the drift after the arctangent transform. Normalizing the two noise coefficients gives a continuous local martingale $Z$ with quadratic variation $t$, hence another Brownian motion by the Lévy characterization of Brownian motion.

With scalar charges $e$ and $2e$, both $\psi$ and ${\varphi }^2$ transform with phase $e^{2i\alpha }$. Therefore $M^2|\psi -b{\varphi }^2{|}^2$ is a gauge-invariant potential term. It contains a nonzero direct cubic interaction $-M^2(b{\psi }^{\ast }{\varphi }^2+h.c.)$. Adding positive quadratic and quartic modulus terms gives a real bounded-below potential. All expanded interactions have field degree at most four.

The Frobenius inner product of a symmetric matrix and an antisymmetric matrix is zero. Thus invariant contractions of an array with all symmetric second-rank tensors test only its symmetric part; an arbitrary basis-dependent antisymmetric part is invisible. A nonzero antisymmetric matrix in one basis and the zero matrix in another passes all such tests with scalar zero but is not a Cartesian second-rank tensor.

If an array is antisymmetric in every orthonormal basis and its contractions with every antisymmetric second-rank tensor are invariant scalars, it obeys the Cartesian second-rank tensor transformation law. The difference $D=R^{T}A\prime R-A$ is antisymmetric and orthogonal to every antisymmetric test matrix. Choosing the test matrix equal to $D$ gives $D:D=0$, hence $D=0$. The Frobenius inner product is nondegenerate on the antisymmetric subspace.

Let $G^{\prime }(z_0)=0$, $G^{\prime \prime }(z_0)\ne 0$, and let $F,G$ be analytic. Assume a contributing contour can be deformed onto two descent arms of the method of steepest descent with one clockwise semicircular avoidance of the pole at $z_0$, and that its other portions are negligible. For the integrand $F(z)e^{-i\lambda G(z)}/(z-z_0)$, a local coordinate with $G-G_0=-i{w}^2/2$ givesThe odd pole term has zero Cauchy principal value on the real arms and integral $-i\pi F_0$ on the clockwise indentation. The constant regular term integrates to $\sqrt{2\pi /\lambda }$ times its coefficient, proving the expansion and generic correction order. A degenerate saddle has a different local width and possibly a different indentation angle; analyticity and a single critical point alone do not imply this result. If $F_0=0$, retain the additive expansion rather than writing asymptotic equivalence to zero.

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.

Consider equal-density planetesimals with projectile differential number distribution $n(d)\propto d^{-\alpha }$, $\alpha >1$, and a fixed encounter speed. For projectiles small compared with a target of diameter $D$, the minimum projectile diameter needed to exceed specific impact energy $Q$ is proportional to $Q^{1/3}$. If this threshold is well below the upper projectile cutoff, integration over the projectile population gives a collision rate $R(>Q)\propto Q^{(1-\alpha )/3}$.

Write the catastrophic disruption threshold as $Q_D^{\ast }=Q_s+Q_g$, with strength term $Q_s=Q_a{D}^{-a}$ and gravity term $Q_g=Q_b{D}^b$. Let $R_{\mathrm{rub}}=R(>Q_s)$ count all strength-shattering impacts and $R_{\mathrm{cc}}=R(>Q_D^{\ast })$ count dispersive impacts. ThenWith this inclusive convention, the rate of rubblising collisions that leave a bound remnant is $R_{\mathrm{rub}}-R_{\mathrm{cc}}$, rather than $R_{\mathrm{rub}}$ alone.

For $a=1/2$, $b=3/2$ and $\alpha =3$, define $D_w$ as the actual minimum of $Q_D^{\ast }$ at the strength-gravity disruption transition. Differentiation gives $Q_b/Q_a=1/(3D_w^2)$, soin the gravity-dominated limit. Defining the crossover instead by $Q_s=Q_g$ changes the reference diameter and the apparent coefficient, but not the collision-rate scaling. Variable speeds, comparable projectile and target sizes, and finite-cutoff corrections require a different collision integral.

During row insertion into a near Young tableau, the successive carried entries strictly increase while the columns in which they are bumped weakly decrease. Every old cell keeps its entry or receives a smaller one. These facts follow from increasing rows and strictly increasing columns.

The zeroth velocity moment gives ${\partial }_t\rho +R^{-1}{\partial }_R(R\rho ⟨v_R⟩)+R^{-1}{\partial }_{\varphi }(\rho ⟨v_{\varphi }⟩)=0$. In the rotating-frame collisionless Boltzmann equation, velocity integration by parts supplies the geometric $\rho ⟨v_R⟩/R$ term. This equation expresses conservation of planar mass.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe curve is admissible only if ${\omega }^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive angular frequency squared is not a Hopf bifurcation. At zero angular frequency this curve meets the stationary curve for that fixed wavenumber in a double-zero limit.

For a finite $0$-$1$ matrix, list its occupied pairs by increasing top entry and decreasing bottom entry within a top-entry tie. Perform ordinary semistandard row insertion on the bottom entries and record top entries in the new cells. The result is a bijection with same-shape tableaux $T,U$ for which $T$ and $U^t$ are Semistandard Young tableaux. The types of $T$ and $U$ are respectively the column-sum and row-sum vectors of the matrix.

For a bounded operator on a complex Banach space, partition a disconnected spectrum into two nonempty compact pieces. The holomorphic function equal to one near the first piece and zero near the second defines a Riesz projection. The holomorphic spectral mapping theorem makes it neither zero nor identity. Its closed range is a proper nonzero invariant vector subspace. This need not be an orthogonal projection, and the argument does not require a normal operator.

For $2\le T\le x$, the truncated Perron formula and a contour shift give the displayed form of the Riemann–von Mangoldt explicit formula. The pole at one contributes $x$ and the Nontrivial zeros of the Riemann zeta function contribute the finite sum. One can choose a separated height in $[T,T+1]$ and then restore $T$ using the local zero count for the Riemann zeta function. The error absorbs the half-weight discrepancy at a prime power. Outside this range of $T$, one needs the fuller error with the near-integer terms retained.

In ridge regression, penalize slopes but ordinarily leave the regression intercept free. Centering gives $\hat{\beta }=(X_c^T{X}_c+\lambda I{)}^{-1}{X}_c^T{Y}_c$ and $\hat{\alpha }=\overline{Y}-{\overline{x}}^T\hat{\beta }$. An exactly centered predictor matrix has $\hat{\alpha }=\overline{Y}$ for every penalty. A table with a penalty-dependent intercept therefore cannot arise from exactly centered columns under this convention.

For a full-rank centred design, unscaled ridge regression and zero-mean errors with covariance ${\sigma }^{2}I$, put $S=X^{\mathsf{T}}X$ and $A=(S+\lambda I{)}^{-1}$. The ordinary-least-squares risk matrix minus the ridge risk matrix is $\lambda A(2{\sigma }^{2}I+\lambda {\sigma }^2{S}^{-1}-\lambda {\beta }^0{\beta }^{0\mathsf{T}})A$. It is positive definite whenever $\lambda \Vert {\beta }^0{\Vert }_2^2<2{\sigma }^2$, including every positive $\lambda$ when ${\beta }^0=0$. Thus one signal-dependent small penalty improves every nonzero-direction mean squared error.