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For a spanning direction matrix $\Xi =({\xi }_1,\dots ,{\xi }_m)$ in ${\mathbb{R}}^d$, the box spline is the density of the pushforward of uniform measure on $[0,1{]}^m$ under $t\mapsto \Xi t$. Equivalently,Its support is the zonotope ${\sum }_j[0,1]{\xi }_j$. It is piecewise polynomial of total degree at most $m-d$. For integer directions, splitting each integration interval in half yields its binary refinement symbolup to the monomial specifying the choice of origin. If $r$ is the smallest number of directions whose removal leaves a nonspanning set, the usual box spline smoothness criterion gives $C^{r-2}$ regularity. Repeated directions increase smoothness while enlarging support.

A spline quasi-interpolant synthesizes a B-spline expansion from local bounded linear functionals, instead of solving a global interpolation system. Suppose each functional is bounded by $c_k\Vert f{\Vert }_{C[t_i,t_{i+k}]}$ and the operator reproduces every spline. On a knot cell $[t_j,t_{j+1}]$, only indices $j+1-k\le i\le j$ contribute. Their supports lie in $[t_{j+1-k},t_{j+k}]$, and subpartition of unity for B-splines proves local operator norm at most $c_k$. Reproduction of degree-$k-1$ polynomials and a local Taylor theorem remainder then give error at most $(1+c_k)(2k-1{)}^k{h}^k\Vert f^{(k)}{\Vert }_{\infty }/k!$, on the basic knot domain with the usual completed boundary basis.

In 1995, Conly L. Rieder, Richard W. Cole, Alexey Khodjakov and Greenfield Sluder tested whether an unattached kinetochore supplies the inhibitory signal delaying anaphase. Living PtK1 biological cells on glass coverslips were mounted in Rose chambers at $35$–$37{}^{\circ }\mathrm{C}$. An inverted microscope with a $60\times$ objective recorded differential interference contrast microscopy video. A pulsed $532\mathrm{nm}$ laser focused through that objective destroyed selected chromosome regions. The readout was chromosome movement and anaphase onset before and after surgery.^[ref]

Destroying the last unattached kinetochore permitted anaphase without that chromosome aligning. Cutting chromosome arms served as a damage control: progression still awaited attachment of the last naturally one-pole-attached chromosome. Correlative electron microscopy checked the absence of the targeted kinetochore. Together these results located an inhibitory activity in or near the unattached kinetochore, rather than establishing a requirement for alignment itself.^[ref]

This intervention localized control; it did not identify every signaling molecule or guarantee faithful inheritance after damage. The cultured PtK1 preparation differs from the hypothetical $2n=4$ cartoon. Neither the cartoon nor this experiment establishes identical control in every species.

For the exterior of a spherical volume conductor model, put $P=|\mathbf{r}-{\mathbf{r}}_0|$, $r=|\mathbf{r}|$ and $F=r{P}^2+P\mathbf{r}\cdot (\mathbf{r}-{\mathbf{r}}_0)$. The displayed Sarvas formula follows by taking the radial component of the Geselowitz formula and integrating the magnetic scalar potential from infinity. It is independent of a spherically symmetric conductivity profile. The sign shown uses $\mathbf{B}=+{\mu }_0\nabla U/(4\pi )$; conventions defining a magnetic potential with a minus sign reverse $U$. Sarvas's original paper derives the spherical-conductor field and its consequences for the inverse problem.

Set $\psi =\alpha \chi$, $p=p_0+P\chi$ and $\chi =F(r){\sin }^2\theta$. A regular family is $F=A[\cos (\alpha r)-\sin (\alpha r)/(\alpha r)]+c{r}^2$, where $P=-{\alpha }^{2}c/{\mu }_0$. A zero magnetic field at radius a requires $F(a)=F^{\prime }(a)=0$. For nonzero A this yields the displayed condition with $x=\alpha a$ and $c=-A[\cos x-\sin x/x]/a^2$. The boundary pressure is $p_0$, independent of angle. The trigonometric pair is tied by regularity and the radial differential equation; its coefficients are not independent.

On the unit sphere, write $x=A(V)$ and $s=4\pi$. The expression equals $|xA(D\setminus V)-(s-x)A(D\cap V)|$, at most $x(s-x)$. A constant-version spherical isoperimetric inequality therefore gives the displayed bound, uniformly in nonempty open $D$. A smooth positive area density $\theta$ gives a second metric; integrating this bound over $D_t=\{\theta >t\}$ by the layer cake representation bounds the discrepancy between the two normalized area fractions by $\kappa (\max \theta -\min \theta )L^2/(s\tilde{A}(S))$.

For fixed $t>0$ and large integer order, $y_n(t)\sim -(2n-1)!!/t^{n+1}$ while $j_n(t)$ is negligible. Stirling's formula gives $(2n-1)!!=(2n)!/(2^{n}n!)\sim \sqrt{2}(2n/e{)}^n$, proving the displayed asymptotic. In particular its magnitude is bounded above and below by constant multiples of $(2n/(et){)}^n$. This two-sided comparison is stronger than a one-sided big-O estimate and is what converts a finite spherical trace norm into a weighted condition on far-field coefficients.

Continuation of a far-field pattern to a spherical trace multiplies harmonic coefficients by the displayed factor, whose magnitude grows faster than exponentially with order. A data error of size $\delta$ in one normalized high-order harmonic therefore produces near-field error $\delta k|h_n^{(1)}(kR)|$, unbounded as its order increases. Some noisy square-summable far fields fail the range condition entirely. This proves unstable field continuation; bounds on actual obstacle geometry additionally require an admissible shape class and a stable boundary-extraction procedure.

For ideal isolated spherical collapse, energy conservation and the virial theorem give $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$ and ${\rho }_{\mathrm{vir}}=8{\rho }_{\mathrm{ta}}$ if the density-profile shape is unchanged. The cycloid gives $t_{\mathrm{coll}}=2t_{\mathrm{ta}}$ and ${\rho }_{\mathrm{ta}}/\bar{\rho }(t_{\mathrm{ta}})=9{\pi }^2/16$. Since the Einstein-de Sitter universe has $\bar{\rho }\propto t^{-2}$, the overdensity at the nominal collapse time is $32(9{\pi }^2/16)=18{\pi }^2$. Hence ${\rho }_{\mathrm{vir}}=18{\pi }^2{\rho }_0(1+z_{\mathrm{coll}}{)}^3$. The collapse time is that of the formal cold solution; actual virialization replaces its point collapse.

The similarity profiles $U=2\xi /(\gamma +1)$, $D=(\gamma +1)\xi /(\gamma -1)$ and $P=2{\xi }^3/(\gamma +1)$ solve the spherical blast equations at the displayed ambient exponent. For $\gamma =5/3$, $\beta =2$ and $R^3=3E{t}^2/(2\pi C)$. The dimensional fields are $u=r/(2t)$, $\rho =8\pi C^{2}r/(3E{t}^2)$ and $p=2\pi C^2{r}^3/(9E{t}^4)$ for $0<r<R$. Their kinetic and internal energies are equal. Substituting these profiles into all three equations and the shock conditions verifies the solution.

With symmetric traceless strain $\mathbf{E}$, these harmonic potentials solve the decaying disturbance around a stationary sphere in the convention $\mathbf{u}=\mathbf{\Phi }-\frac{1}{2}\nabla (\mathbf{x}\cdot \mathbf{\Phi }+\chi )$ and $p=-\mu \nabla \cdot \mathbf{\Phi }$. The vector potential supplies the leading stresslet; the scalar degree-two potential corrects no slip at finite radius. Matching the independent $\mathbf{Ex}$ and $(\mathbf{x}\cdot \mathbf{Ex})\mathbf{x}$ structures fixes both coefficients.

A map of topological spectra is a compatible family of based maps of the level spaces: $u_{j+1}{\sigma }_j^D={\sigma }_j^E(1\wedge u_j)$. It is distinct from its class in the stable homotopy category.

An Omega-spectrum has structure adjoints $E_j\to \Omega E_{j+1}$ that are weak homotopy equivalences. For such a model the level spaces represent the degrees of its represented cohomology theory.

A cell spectrum is obtained by attaching stable cells, with successive cofibers of spectra wedges of suspensions of spectra of the sphere spectrum. Cellular models provide the cofibrant objects needed to compute derived maps of topological spectra. Stable cells may occur in arbitrary integer degrees.

For a based space $X$, its suspension spectrum has level $j$ equal to $S^j\wedge X$. Its derived adjunction with the infinite loop space is $[{\Sigma }^{\infty }X,E{]}_{\mathrm{st}}=[X,{\Omega }^{\infty }E{]}_{\ast }$.

Suspension of spectra shifts the stable degree. Unlike reduced suspension of based spaces, it is invertible in the stable homotopy category, with inverse denoted ${\Sigma }^{-1}$.

The $k$th stable homotopy group of $E$ is ${\pi }_{k}E=[{\Sigma }^k\mathbb{S},E{]}_{\mathrm{st}}$, equivalently ${\mathrm{colim}}_j{\pi }_{k+j}{E}_j$ in a sequential model.

An indexed prespectrum over $U$ consists of based spaces $E(V)$ indexed by finite-dimensional $V\subset U$, with unital associative maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for inclusions $V\subset W$. The orthogonal differences make iterated suspension coordinates compose consistently.

A ring spectrum is a unital associative monoid object of the stable homotopy category under the smash product of spectra. It has $\mu :E\wedge E\to E$ and $\eta :\mathbb{S}\to E$, satisfying the associative and two unit identities. Structured ring spectra additionally specify coherent point-set or higher homotopies; a ring object alone does not assert those enhancements.

A connective spectrum has ${\pi }_{q}E=0$ for $q<0$. This makes the homological Atiyah-Hirzebruch spectral sequence first quadrant for a based space, and prevents negative coefficient degrees from affecting its first nonzero represented homology theory.