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For a BV space function, this is the set where no finite approximate mean limit of a function exists. Its difference from the jump set of a bounded-variation function has zero $(n-1)$-dimensional Hausdorff measure.

A BV space function is approximately differentiable at almost every point with respect to Lebesgue measure. Its approximate gradient is the density of the absolutely continuous part of its distributional derivative. This does not imply membership in a Sobolev space.

A BV space function has two one-sided approximate mean BV traces on a hypersurface at almost every point of an oriented countably rectifiable hypersurface, with respect to Hausdorff measure. They agree outside its approximate discontinuity set. At an approximate jump point, they are the jump values, up to the choice of orientation.

For a BV space function $a$, almost every restriction $a_y$ to a line parallel to $e_j$ has bounded variation on the corresponding domain slice. Directional derivative variation disintegrates as $|D_{j}a|={\mathcal{L}}^{n-1}\otimes |D{a}_y|$. Slice jump values agree with the appropriate oriented BV traces on a hypersurface for almost every crossing of a rectifiable jump surface. Integrating slice jump sums produces a surface integral with projection factor $|\nu \cdot e_j|$. Uniform essential bounds on the slices are not asserted for an unbounded function.

A Cauchy sequence $(u_n)$ in the BV space converges in $L^1$ to $u$. For fixed sufficiently large $n$, lower semicontinuity gives $|D(u_n-u)|(\Omega )\le {\mathrm{liminf}}_m|D(u_n-u_m)|(\Omega )$. The $L^1$ norm of the same difference converges, so the small full-norm Cauchy bound passes to $u_n-u$. The triangle inequality then puts $u$ in the BV space and proves convergence in its full norm. This argument turns completeness of $L^1$ and lower semicontinuity of variation into completeness of the combined norm.

A porous sheet of thickness $d$ and permeability of a porous medium $\Pi$ supplies a thin premelting film. The Darcy flow law equates water supply to ice growth for equal densities. With equal film and ice conductivities, negligible latent heat relative to the through-going conductive flux, and $a\ll h$, phase equilibrium gives $p_T=(\rho L\Delta T/T_m{)}^{3/4}(A/(6\pi h^3){)}^{1/4}$. The increasing gravitational load limits growth. A quasi-steady temperature field alone does not imply the negligible-latent-heat approximation.

A positive-strength parallel point stresslet below a flat shear-free impermeable plane is attracted by its identical reflected image. Its height satisfies $\dot{h}=-S/(32\pi \mu h^2)$ and formally reaches zero in finite time. Negative strength reverses the drift. Finite-body near-contact physics lies outside this singularity prediction.

For equal stresslets with height $h$ and axial separation $\ell \gg h$, the neighbour image doubles the leading axial repulsion. Its normal contribution is smaller than own-image attraction by order $(h/\ell {)}^3$, so leading height dynamics are unchanged. The approximation also neglects weak neighbour-induced orientation corrections.

A map is Fréchet differentiable at $x$ if it admits a bounded linear first-order approximation: $f(x+h)=f(x)+Ah+o(\Vert h\Vert )$ for a continuous linear map $A$.

A map on a subset of a normed vector space is Hadamard differentiable at $s$ if its directional difference quotients converge to a continuous linear map uniformly along all convergent sequences of directions: $t_j\to 0$, $a_j\to a$, and $s+t_j{a}_j$ in the domain imply $(\Phi (s+t_j{a}_j)-\Phi (s))/t_j\to {\dot{\Phi }}_s(a)$. Tangential differentiability restricts the limit directions to a specified subspace. This is the regularity used by the functional delta method.

A coframe dual to a local frame of the tangent bundle is the corresponding local frame of the cotangent bundle. Its elements are differential one-forms. Applying the coframe to covariant derivatives of the frame extracts the connection one-forms; applying it to the torsion tensor extracts the torsion forms.

For a complex symmetric three-dimensional $A=G+iB$ with real symmetric $B$ and positive-definite $G$, the reciprocal is a regular tempered distribution since $|x^{T}Ax|\ge x^{T}Gx$. Holomorphic continuation of the real quadratic transform proves the displayed expression. The determinant branch is the analytic determinant square root for accretive symmetric matrices. The other root has positive real part because $\mathrm{Re}({\xi }^T{A}^{-1}\xi )={\xi }^T{G}^{-1/2}(1+C^2{)}^{-1}{G}^{-1/2}\xi >0$ for nonzero real $\xi$. Both the original and transformed singularities are locally integrable, so the continued identity holds as distributions at the origin too.

The group velocity of a flexural-gravity wave diverges on the long-wave gravity branch and on the short-wave bending branch, and therefore has a positive minimum between them when the bending stiffness is positive. This minimum bounds the propagation speed of narrow wave packets within the ideal model. It does not by itself determine a wind-generation threshold: steady forcing requires matching phase velocity, while dissipation and coupling determine growth.

With objective $\Vert Y-Xb{\Vert }_2^2/(2n)+\lambda \Vert b{\Vert }_1$, the Gaussian score event for the Lasso $\Vert X^Tϵ/n{\Vert }_{\infty }\le \lambda /2$ gives $Q+\lambda \Vert {\delta }_{S^c}{\Vert }_1\le 3\lambda \Vert {\delta }_S{\Vert }_1$, where $Q=\Vert X\delta {\Vert }_2^2/n$. The Lasso cone condition activates the Compatibility condition for the Lasso. Then $Q+\lambda \Vert \delta {\Vert }_1\le 4\lambda \sqrt{sQ}/{\varphi }_0\le Q/2+8{\lambda }^{2}s/{\varphi }_0^2$. Thus even $Q+2\lambda \Vert \delta {\Vert }_1\le 16{\lambda }^{2}s/{\varphi }_0^2$. This controls prediction and coefficient error together, including a design with correlated columns.

A dilute suspension of identical rigid spheres raises the dynamic viscosity by a fraction $5\varphi /2$, where $\varphi$ is the particle volume fraction. At imposed trace-free rate-of-strain tensor $E$, a sphere of radius $a$ adds viscous dissipation $20\pi \mu a^3(E:E)/3$. Multiplication by the number density $3\varphi /(4\pi a^3)$ gives $5\mu \varphi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; hydrodynamic interactions enter at higher concentration.

A functor $\mathcal{D}\times \mathcal{C}\to \mathcal{E}$ is uniquely a functor $\mathcal{D}\to [\mathcal{C},\mathcal{E}]$: fix its first coordinate and use its first-coordinate arrows as natural transformations. Uncurrying evaluates those transformations and second-coordinate arrows. The adjunction unit inserts a fixed first coordinate, and the adjunction counit is evaluation. This makes the category of small categories cartesian closed.

For left M-sets, $B^A$ consists of equivariant maps of monoid sets $M\times A\to B$ for the diagonal action on the domain. Its action is $(m\cdot f)(w,a)=f(wm,a)$ and evaluation is $f(1,a)$. The curry of an equivariant $h:C\times A\to B$ is $\hat{h}(c)(w,a)=h(w\cdot c,a)$. This construction can have noninjective action maps even when both $A$ and $B$ are decidable.

The evaluation map is the counit representing application in an exponential object. For every $Z$, composition with evaluation gives the natural bijection $\mathrm{Hom}(Z,Y^X)\cong \mathrm{Hom}(Z\times X,Y)$. Its inverse is currying. Evaluation must be a morphism in the ambient category, such as an equivariant map in a category of group actions.

Currying transforms a morphism of two variables into a morphism taking values in an exponential object. The evaluation map of an exponential object recovers the original morphism. Its categorical universal property requires naturality and uniqueness, not just an underlying set bijection.

For independent and identically distributed random variables with law $N(-1,1)$, $e^{2S_n}$ is a nonnegative martingale, since the increment's moment-generating function equals one at parameter two. The Doob maximal inequality gives $\mathbb{P}(Z>t)\le e^{-2t}$ for the all-time maximum $Z\ge 0$. Integrating the tail gives the displayed bound. For $b\le 0$, $e^{bZ}\le 1$. The strong law of large numbers guarantees finiteness and attainment of the maximum because $S_n\to -\infty$.