Pull back along loop evaluation $S^1\times \Omega B\to B$ and take the slant product with the circle's fundamental class. This degree-lowering operation kills positive-degree cup products. It commutes with stable cohomology operations such as Steenrod squares and identifies a transgressing base class with its fiber class in the path-loop fibration.

A dilute, vertically mixed particle-laden gravity current with no fluid entrainment, bed erosion or carrier-fluid leakage obeys volume conservation $q=uh=\text{constant}$. With downward settling velocity magnitude $v_s$, an absorbing bed gives $qd{g}^{\prime }/dx=-v_s{g}^{\prime }$. The variable-buoyancy shallow water equations conserve the kinematic momentum flux including hydrostatic pressure,These equations describe a steady interior supplied continuously from upstream, not the moving nose of a finite-volume release. The depth follows one of the depth branches of a steady depositing gravity current.

For slowly depositing particles, $B_x=-v_{s}B/(uh)$ and local adjustment gives $u\sim {\beta }_{\ast }{B}^{1/3}$ with the constant-speed entraining gravity current on a slope coefficient. At leading order $h\sim h_0+E(x-x_0)$ and $B^{1/3}=B_o^{1/3}-[v_s/(3{\beta }_{\ast }E)]\ln [1+E(x-x_0)/h_0]$. Its validity requires settling slow compared with the current and, for small $E$, compared with entrainment.

The integrated budget is ${\partial }_t(g^{\prime }h)+B_x=S_B-v_s{g}^{\prime }$. A steady spatial increase needs a source $S_B$ exceeding deposition, for example sediment resuspension or lateral particle input. In unsteady flow, local depletion can produce $B_x>0$ even without particle addition.

For fixed channel width $b(z)=\beta z$, cross-sectional area is $A=\beta h^2/2$. With reduced gravity $G\varphi$, uniform cross-sectional fields and downward settling velocity magnitude $w$, volume conservation, momentum and particle deposition flux giveThe hydrostatic pressure force contains ${\int }_0^h(h-z)\beta zdz=\beta h^3/6$. No streamwise widening term occurs in a prismatic channel. Deposition uses the projected boundary width $\beta h$ under the vertical-settling model.

For $A\subseteq M$, an A-type over first-order structure $M$ is a set of formulas in a fixed finite tuple of variables, with parameters in $A$, consistent with the complete theory of $M$ expanded by constants for $A$. Every finite subset is realized in $M$. A complete type decides all formulas in its variables over those parameters.

A mixed partial derivative takes partial derivatives in different variables, for example $f_{xy}={\partial }_y({\partial }_{x}f)$. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged: $f_{xy}=f_{yx}$. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

For the magnetized isothermal atmosphere, set $a=1/(2H)$, $s=v_s^2>0$, $b=v_a^2>0$ and $\Delta =i{k}_x{\xi }_x+i{k}_y{\xi }_y+(i{k}_z+a){\xi }_z$. After weighting the displacement by $e^{z/(2H)}$, the restoring Hermitian matrix $\mathsf{K}$ satisfiesThus $gH\le s$ makes the form nonnegative for every wavevector. If $gH>s$, choose $k_z=0$, $k_x\ne 0$, ${\xi }_z=1$, ${\xi }_x=-a/(i{k}_x)$ and ${\xi }_y=g/(si{k}_y)$. The squares vanish, and sufficiently small nonzero $k_y$ gives a negative form. The Rayleigh quotient proves instability when these wavelengths are admissible. Since $s=\gamma c_s^2$ and $\beta =2c_s^2/b$, the condition is precisely $\beta (\gamma -1)<1$.

For $q=(q_0,q_1,q_2,q_3)\in S^3$, the vectors $(-q_1,q_0,q_3,-q_2)$, $(-q_2,-q_3,q_0,q_1)$ and $(-q_3,q_2,-q_1,q_0)$ are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating $-i$ times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.

For $a>0$, polar-coordinate reduction and integration by parts give $m\sin (m||x{|}^2-a|)=2\delta (|x{|}^2-a)-\pi \cos (ma){\delta }_0+o_{{\mathcal{D}}^{\prime }}(1)$ in two dimensions. The circle term comes from the cusp in the folded phase, while the origin term is a radial endpoint contribution. A punctured-domain limit therefore need not extend across the quadratic critical point.

Here $Z$ has the standard normal distribution. Applied to the generating function $e^{sx-s^2/2}$, the Gaussian moment-generating function replaces $s$ by $e^{-t}s$, proving the Hermite eigenvalue identity. Positivity and preservation of Gaussian measure then give the $L^2$ contraction and identify the full Ornstein-Uhlenbeck semigroup by polynomial density.

On polynomials in $L^2$ of standard Gaussian measure, Gaussian integration by parts makes $D$ and $x-D$ adjoints, with commutator $[a^{-},a^{+}]=I$. The Probabilists' Hermite polynomials are obtained by applying $a^{+}$ repeatedly to $1$, and $a^{-}{\mathrm{He}}_n=n{\mathrm{He}}_{n-1}$. These give a Gaussian-space realization of creation and annihilation operators.

Differentiating the Mehler formula for the Ornstein-Uhlenbeck semigroup proves the identity for continuously differentiable functions with bounded derivative. Density extends it to the Gaussian Sobolev form domain. Together with a weighted Cauchy-Schwarz inequality, it yields the Ornstein-Uhlenbeck entropy dissipation identity proof of the Gaussian logarithmic Sobolev inequality.

Partial regression first projects a response and a selected predictor orthogonally off the remaining design columns. Regressing the residualized response on the residualized predictor yields its coefficient in the full ordinary least squares fit.

For a correctly specified full-rank fixed-design Gaussian linear regression with $p<n$, dividing the residual sum of squares by the residual degrees of freedom $n-p$ gives an unbiased estimator of the noise variance. The projection removes the whole deterministic mean, so the numerator's expectation is $(n-p){\sigma }^2$. A misspecified subset model need not have this property.

The Taylor expansion of the exact-solution residual for a linear multistep method has the same coefficient sequence asVanishing through degree $p$ is therefore equivalent to order at least $p$. Exact order $p$ additionally requires a nonzero coefficient at degree $p+1$. This symbol conveniently collects all polynomial consistency and order conditions.

The completion of a total order by its nonprincipal proper cuts, supplying least upper bounds. For a dense order it contains the original order densely, and preserves the interval countable chain condition and separability or nonseparability.

An order automorphism of an order-dense subset $D$ of the real line extends uniquely by $\tilde{\phi }(x)=\sup \{\phi (d):d\in D,d<x\}$. Order completeness supplies the extension, and the lower cuts in $D$ give uniqueness.

An order-dense subset with no nonidentity order automorphism. A transfinite diagonal construction marks an included point and an excluded image for every nonidentity real-line order automorphism, yielding such a subset of size $2^{{\aleph }_0}$.