A map of topological spectra inducing isomorphisms on every stable homotopy group is a stable equivalence of spectra.

The subgroup generated by elementary transvections in finite matrix sizes, with identity stabilization. A third distinct index expresses each transvection as a commutator; stable block identities show that this subgroup is normal and that its quotient in the stable general linear group is abelian. Thus it is precisely the commutator subgroup.

For a steady-state equivariant dynamical system, an axial subgroup is a stabilizer subgroup whose real fixed-point subspace of a group action has dimension one. Requiring an actual stabilizer subgroup matters: a smaller subgroup can have the same fixed line without being the full symmetry of any point on that line. The fixed line reduces the steady-state bifurcation equation to one real amplitude.

In an equivariant Hopf bifurcation, a complex axial isotropy subgroup is instead a stabilizer subgroup for the spatial symmetry group combined with the common phase circle action, with one-dimensional complex fixed-point subspace of a group action. Distinct conjugacy classes of such subgroups can describe distinct types of bifurcating branches.

For a group action, the pointwise stabilizer of a set $S$ is the subgroup fixing each of its points individually. It is the intersection of their stabilizer subgroups. This is stronger than merely mapping the set to itself.

A stabilizer group excludes $-I$, so each element squares to $I$ and is Hermitian; an imaginary-phase Pauli element would square to $-I$. A nonidentity stabilizer contains at least one nonidentity local Pauli matrix and has zero trace, since the trace of a tensor product is the product of local traces. Its eigenvalues are therefore $\pm 1$ with equal multiplicities. This trace fact also gives $\dim {\mathcal{X}}_S=2^n/|S|$ from the stabilizer-projector formula.

For positive streamwise wavenumber $k$, replace a three-dimensional Orr-Sommerfeld equation by a two-dimensional one with the displayed parameters. Both the transverse operator and the coefficient $1/(ikR)$ remain unchanged, preserving the eigenfunction and phase-speed eigenvalue. This proves the modal comparison in Squire's theorem.

For every nonnegative integer $n$,For example, $1^3+2^3+3^3+4^3=1+8+27+64=100=T_4^2$. A short proof uses the triangular number recurrence and the triangular number square-complement identity:Summing this difference of squares identity gives a telescoping sum; all intermediate squared triangular numbers cancel. Nicomachus describes a related grouping of the odd integers: take one, then the next two, then the next three, producing $1,3+5,7+9+11,\dots$, which are $1^3,2^3,3^3,\dots$. This historical grouping is recorded in Introduction to Arithmetic by Nicomachus (Hahn teacher edition) (Book II, Chapter 20, §5, printed p. 75).

A split diagram has two nonempty parts separated by a simple closed curve disjoint from the diagram. For a two-component link, this is equivalent to disconnectedness of its projection graph. An alternating link diagram representing a split link cannot have a connected projection graph, as its Jones value at minus one detects a nonzero spanning-tree count.

A quadratic spline with simple knots is a continuously differentiable function that is a polynomial of degree at most two between successive knots. Its first derivative is continuous and piecewise linear, while its second derivative is piecewise constant and can jump at knots. Constant tails impose vanishing first derivatives at the outermost knots. Such a spline arises from first-derivative roughness penalization when observations are interval averages, as in the quadratic smoothing spline for interval averages.

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.