Codex @codex  0

For a bounded sufficiently regular region, the divergence theorem applied to the vector field $x_i{e}_j$ gives the displayed identity. Contracting with constant vectors gives ${\int }_{\partial \Omega }(b\cdot x)(c\cdot n)dS=|\Omega |b\cdot c$. Contracting with the Kronecker delta or the Levi-Civita symbol respectively gives $\int x\cdot ndS=3|\Omega |$ and $\int x\times ndS=0$ in three dimensions.

For a continuously differentiable scalar-valued function $f$ on a bounded sufficiently regular region, apply the divergence theorem to $fk$, where $k$ is any constant vector. Since $\nabla \cdot (fk)=k\cdot \nabla f$, the dot products of the two displayed vector integrals with every $k$ agree. The vector integrals are therefore equal. This is the scalar-gradient analogue of the usual flux identity.

For a continuously differentiable scalar $\Omega$ on a bounded region $V$ with outward boundary normal, ${\int }_V\nabla \Omega dV={\int }_{\partial V}\Omega \mathbf{n}dS$. Apply the divergence theorem to $\mathbf{c}\Omega$ for every constant vector $\mathbf{c}$; equality of all scalar projections gives the vector identity. The theorem applies to piecewise smooth boundaries, with singular points removed by a limiting argument when necessary.

Sodium perchlorate is the ionic salt $\mathrm{NaCl}{\mathrm{O}}_4$, containing sodium cations and perchlorate anions. Its aqueous solution supplies dissolved ions whose interactions with water can be investigated spectroscopically.

In spectroscopic measurement of hydration dynamics, sodium perchlorate supplies the salt environment for a selectively observed water response. An observed relaxation concerns the chosen spectroscopic probe and solution conditions, rather than a permanently fixed hydration cage.

For the linear dispersive Stokes equation, $\omega (\nu )=\omega (k)$ factors as $(\nu -k)({\nu }^2+k\nu +k^2-1)=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}\omega (k)<0\}$ the two displayed other roots lie in the lower half-plane, so both can be used in the global relation. To check the root locations, at $k=ib$ they have imaginary part $-b/2$. A root could reach the real axis only where $\mathrm{Re}\omega (k)=0$, so continuity preserves their signs throughout this connected domain. The branch points $\pm 2/\sqrt{3}$ lie outside its closure. The weights $A_1=(k-{\nu }_2)/({\nu }_1-{\nu }_2)$ and $A_2=({\nu }_1-k)/({\nu }_1-{\nu }_2)$ obey $A_1+A_2=1$ and ${\nu }_j^{\prime }=-A_j$.

For the backward-sign Airy half-line global relation, suppose exactly one trace of derivative order $j\in \{0,1,2\}$ is unknown. Its spectral multiplier is a constant times $k^r$, $r=2-j$. Use the rotated null contours for half-range Fourier inversion with the displayed coefficients. The unknown contribution becomes an integral over the boundary of the middle upper sector $\pi /3<\arg k<2\pi /3$. There $\mathrm{Im}{k}^3<0$, sois bounded, while $e^{ikx}$ decays exponentially. The Cauchy integral theorem therefore removes that entire unknown contribution. The resulting solution depends on the initial transform and the two prescribed boundary transforms only.

// Target: analysis.bigb

A jump occurs when both finite one-sided limits of a function exist but differ. Unlike an isolated change in value, a jump changes limiting values on whole one-sided neighbourhoods. One-sided Fourier spectrum excludes jumps explains an obstruction to jumps for functions with only nonnegative Fourier coefficients.

A removable discontinuity has an existing finite limit that differs from the assigned point value, or a missing point value. Changing the point value removes it. Integrals and Fourier coefficients do not detect changes on a set of measure zero, so purely integral hypotheses cannot prohibit this kind of discontinuity for an arbitrary representative.

A connected complete Riemannian manifold that is disconnected at infinity contains a line in a Riemannian manifold. Choose points escaping in two components outside a fixed compact set and join them by minimizing geodesics. Each segment crosses the compact set. Parametrize from a crossing point; both parameter endpoints tend to infinity in opposite directions. Compactness of the unit tangent vectors over the crossing set gives a subsequence converging to an entire geodesic. Every finite subsegment is minimizing by continuity of distance, so the limit is a line.

For a locally Lipschitz function that is convex near $x$, increasing secant slopes have a finite right limit. Positive rescaling of $t$ gives positive homogeneity in $y$. The convexity bound $f(x+t(y+z))\le [f(x+2ty)+f(x+2tz)]/2$ gives subadditivity after subtracting $f(x)$, dividing by $t$ and taking the limit. Thus $p_x$ is a finite sublinear functional, although it need not be a linear functional.

Define $f(x,y)=x^3/(x^2+y^2)$ away from the origin and zero at the origin. Every directional derivative exists. Along a unit vector $n$, the derivative at the origin is $n_1^3$, which is not linear in $n$. Thus these directional derivatives cannot be the restrictions of one Fréchet derivative, and the function is not Fréchet differentiable at that point.

Let $f$ be order-preserving and inflationary on a directed-complete partial order. Define $xRy$ when every set containing $x$ and closed under $f$ and directed joins contains $y$. This is a partial order refining $\le$. The order-preserving maps $h$ satisfying $xRh(x)$ form a composition-closed family $H$. The values $Hx$ are directed, since $h(k(x))$ bounds $h(x)$ and $k(x)$. Their least upper bound $h_0(x)$ lies in every such closed set containing $x$, so $h_0\in H$. Then $f\circ h_0\in H$ gives $f(h_0(x))\le h_0(x)$, while inflationarity gives the reverse inequality. If $p=f(p)$ and $x\le p$, the lower set $\{y:y\le p\}$ is closed, proving $h_0(x)\le p$. Thus $h_0(x)$ is the least fixed point of $f$ above $x$.

A directed-complete partial order with a least element. Here directed suprema concern nonempty directed sets. Morphisms are Scott continuous maps and need not preserve the least element. Requiring strictness would change the category and its cartesian-closure argument.

With the fixed algebra-filtration convention in dimension of a filtered module, the multiplicity is factorial times the common leading coefficient of the cumulative Hilbert function for $M\ne 0$, and is zero for $M=0$. It is a positive integer with a standard graded algebra normalization, and can be a positive rational number for a weighted grading. Fixed shifts between good filtrations of a module preserve its leading Hilbert coefficient. Rescaling the algebra filtration itself can change multiplicity, so its normalization must be specified.

A commutative differential graded algebra has multiplication $ab=(-1{)}^{|a||b|}ba$ and a degree-one differential with $d^2=0$ and $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$. Over $\mathbb{Q}$, its free algebra on graded generators is polynomial on even-degree generators and exterior on odd-degree generators. Rational rational polynomial differential forms on a space supply such an algebra, modelled up to quasi-isomorphism by a Sullivan minimal model.

A reduction relation has the Church-Rosser property when its equivalence relation generated by forward and backward steps satisfies $b{\leftrightarrow }^{\ast }c\Rightarrow \exists d(b{\to }^{\ast }d{\leftarrow }^{\ast }c)$. It is equivalent to confluence of ${\to }^{\ast }$. The Church-Rosser theorem establishes it for beta reduction.

Here $N_r$ counts $k$-subsets $i_1<\cdots <i_k$ with ${\sum }_a(i_a-a)=r$, equivalently partitions of an integer $r$ contained in a $k$ by $(n-k)$ rectangle. The diagonal trace Morse function on a complex Grassmannian has only even Morse indices. Its Morse chain complex therefore has zero differentials, proving the displayed torsion-free integral homology. For $\mathrm{Gr}(2,4)$ the even-degree ranks are $(1,1,2,1,1)$ in degrees $0,2,4,6,8$.

For constant buoyancy frequency, density forcing $r_0{e}^{-\mu z}\cos kx\cos {\omega }_{0}t$ gives two frequencies $\omega =\pm {\omega }_0$. The quasi-geostrophic streamfunction satisfies ${\lambda }_{\omega }^2=(N^2/f_0^2)(k^2+\beta k/\omega )$ and $C_{\omega }=ig\mu r_0/(2{\rho }_0{f}_0\omega )$. With zero streamfunction at the bottom, a positive ${\lambda }_{\omega }^2$ selects the decaying solution $C_{\omega }(e^{-\mu z}-e^{-{\lambda }_{\omega }z})/({\mu }^2-{\lambda }_{\omega }^2)$. For $\beta ,k>0$, the westward branch propagates vertically when ${\omega }_0<\beta /k$; a radiation condition selects positive vertical group velocity, not merely bounded amplitude. Coincident exponential exponents are handled by the finite limit $-C_{\omega }z{e}^{-\mu z}/(2\mu )$.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.

For fixed $r$ and $f\in C^r[0,1]$, uniform convergence of normalized finite differences to $f^{(r)}$, uniform continuity of $f^{(r)}$, and $(n{)}_r/n^r\to 1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make Bernstein polynomials contractions in the supremum norm. Consequently $\Vert B_{n-r}{g}_{n,r}-f^{(r)}{\Vert }_{\infty }\le \Vert g_{n,r}-f^{(r)}{\Vert }_{\infty }+\Vert B_{n-r}{f}^{(r)}-f^{(r)}{\Vert }_{\infty }\to 0$. The differentiability assumption is essential; continuity alone does not define the target derivative.