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Outside both poles, the two geometric series use negative powers:HenceThe coefficient of $z^{-1}$ vanishes, as it must since the rational function starts with $z^{-2}$ at infinity.

In the middle annulus, expand the pole at one in negative powers and the pole at two in positive powers:Their geometric series convergence conditions are $|z|>1$ and $|z|<2$, respectively. The required Laurent series is

The coefficient ratio isThus the ratio test again gives radius of convergence $\boxed{R=1}$. On $|z|=1$, however, the terms have absolute value $n+2^{-n}$, which does not tend to zero. The necessary term condition for a convergent series fails at every boundary point. Therefore this power series diverges everywhere on $|z|=1$.

Use dots for derivatives with respect to proper time $s$ and the given unit curvature radius. Translation symmetry in the four brane coordinates gives constantsFor a future-directed timelike curve $E>0$. The unit-tangent constraint isIt follows that $K^2=E^2-{\mathbf{p}}^2>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Set $w=e^{-y}$. Thenwith solution $w=K\cos (s-s_0)$ on an interval where the cosine is positive. A phase shift includes either sign of the initial radial velocity. Integrating the four conserved-momentum equations gives the most general trajectory within this patch:The arbitrary constants encode the initial location and every unit timelike initial velocity. For the past-directed orientation reverse the tangent. The radial turning point is $y_{\min }=-\log K$; the two limits $y\to \infty$ are patch horizons, not the endpoints of the complete globally extended geodesic. This is the timelike geodesics in a warped Poincaré patch solution. As a check, $\ddot{y}=K^2{e}^{2y}$ and ${\ddot{x}}^{\mu }-2\dot{y}{\dot{x}}^{\mu }=0$, which are the explicit geodesic equations.

To measure the acceleration of the $y=0$ surface, take an observer at rest in its flat coordinates: $u^A=(1,0,0,0,0)$ there. The relevant Christoffel coefficient isThereforeIt points toward decreasing $y$. More generally any observer moving inertially within the brane satisfies $e^{-2y}{\eta }_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$, and ${\Gamma }_{\mu \nu }^y=e^{-2y}{\eta }_{\mu \nu }$ gives the same normal acceleration. The brane's extrinsic curvature for normal $+{\partial }_y$ is $K_{\mu \nu }=-h_{\mu \nu }$, consistent with this result. Restoring curvature radius $L$ gives warped-brane proper acceleration $|a|=1/L$. Intrinsically the induced brane metric is Minkowski and these observers have zero four-dimensional proper acceleration; the nonzero acceleration is relative to the bulk.

A low-frequency example is Galactic synchrotron emission: relativistic electrons spiral in the Galactic magnetic field and emit a broad nonthermal spectrum. In antenna-temperature units its approximate spectrum is $T_A\propto {\nu }^{-{\beta }_s}$ with ${\beta }_s$ around3, varying with the electron population and sky direction. It is a major, often dominant contamination at frequencies below roughly20–40GHz; radio point sources and bremsstrahlung can contribute in the same low-frequency region.

A high-frequency example is Galactic thermal dust emission. Grains heated by starlight have approximate modified-blackbody spectrum $I_{\nu }\propto {\nu }^{{\beta }_d}{B}_{\nu }(T_d)$, with grain temperatures of order tens of kelvin. Their emissivity-weighted emission rises rapidly through the microwave/submillimeter range and commonly dominates anisotropy contamination above roughly100–200GHz, especially at higher frequencies. Dusty extragalactic galaxies provide a related high-frequency foreground.

Synchrotron is the low-frequency contaminant and thermal dust the high-frequency contaminant. There is no universal crossover frequency at which either exceeds CMB anisotropy everywhere: sky masking, angular scale, polarization and source amplitude matter. Cleaner cosmological windows lie between these regimes, and multi-frequency observations distinguish the foreground spectra from a thermodynamic CMB temperature perturbation. The separation into low-frequency emission and high-frequency dust is also used in the Planck astrophysical-component products.

At two derivatives, specify the gauge group and matter representations, a real Kähler potential $K(\Phi ,\bar{\Phi })$, a holomorphic superpotential $W(\Phi )$ and a symmetric holomorphic gauge kinetic function $f_{ab}(\Phi )$. Gauge couplings and topological angles are contained in $f$; global theories can also have allowed Fayet–Iliopoulos terms for Abelian factors. Local theories require the gravitational coupling and consistent gauging: a constant Fayet–Iliopoulos term is not an independent addition to arbitrary supergravity data. A constant in $W$ affects supergravity although it is irrelevant in the global theory. $K$ is not holomorphic or protected; $W$ has no perturbative Wilsonian renormalization; holomorphic $f$ is perturbatively one-loop exact. Nonperturbative corrections can modify $W$ and $f$, and wave-function renormalization makes physical couplings run despite superpotential non-renormalization. Higher-derivative actions require further data.

Dependence on Grassmann variables alone does not specify a superfield transformation law. A superfield is a function on superspace, with components transforming under the induced supertranslation and Lorentz transformation action. Any suitable superfunction can be an unconstrained superfield once equipped with that action; a fixed element of a Grassmann algebra with independently assigned components need not be covariant. A chiral superfield, for example,also obeys ${\bar{D}}_{\dot{\alpha }}\Phi =0$. Keeping only a nonconstant $A(x)$ and forbidding the other components is not a closed supermultiplet: applying a supercharge generates Grassmann-dependent derivative terms. A real unconstrained superfield can instead describe a vector superfield.

Use the standard $L_{\infty }$-oriented partition normalization of B-splines, in contrast to the unit-integral normalization. Thus $N_i\ge 0$ and ${\sum }_i{N}_i(t)\le 1$ on $[0,1]$, with equality for a complete basis on its basic knot interval. This terminology does not mean that each basis function's maximum is exactly one; for example an order-three cardinal spline has maximum $3/4$. Assume the basis has positive support on the integration interval and is continuous, as is needed for a projection taking values in $C[0,1]$.

LetThe ordinary Gram matrix $H$ is positive definite by linear independence. The appropriate row-normalized, or mixed-normalization spline Gram matrix, isHere $M_i$ is a unit-integral nonnegative spline. Unlike $H$, $G$ need not be symmetric. If the full support lies inside the interval, $d_i=(t_{i+k}-t_i)/k$; otherwise use the actual interval integral rather than that full-support formula.

The normal equations for the $L_2$ orthogonal projection $P_{\mathcal{S}}f={\sum }_i{a}_i{N}_i$ say $Ha=c$, with $c_i=\int f{N}_i$. Divide each row by $d_i$ to obtainNonnegativity and unit integral give $|b_i|\le \Vert f{\Vert }_{\infty }$. The subpartition of unity for B-splines then givesTaking the supremum over $t$ and then over nonzero $f$ proves the uniform norm bound for B-spline orthogonal projection:The row normalization is essential; replacing $G$ with the ordinary Gram matrix would not be the same asserted estimate. The operator is defined on $C[0,1]$ and maps into its finite-dimensional spline subspace, onto $\mathcal{S}$, not onto all of $C[0,1]$ as one phrase in the PDF says. Its range follows both from the formula and from reproducing every $s\in \mathcal{S}$.

If “$L_{\infty }$-normalized” were instead read literally as individually peak-one splines, their sum can exceed one. The basis-dependent factor $K=\Vert {\sum }_i{N}_i{\Vert }_{\infty }$ must then be retained: $\Vert P_{\mathcal{S}}{\Vert }_{\infty }\le K\Vert (D^{-1}H{)}^{-1}{\Vert }_{{\ell }^{\infty }}$. Equivalently the displayed bound without an outside factor uses $G=K^{-1}{D}^{-1}H$. The standard partition-normalized convention has $K\le 1$ and yields the sharper bound proved above.

The change of time is $d\tau /dt=u$, which is positive in the half-plane under consideration. Dividing the two approximate equations by $u$ givesSet $w=v-\mu$. The inhomogeneous system becomes $(u,w{)}^{\prime }=(u-w,\lambda u)$, with eigenvalues $(1\pm \sqrt{1-4\lambda })/2$. For $\lambda >1/4$, write $\omega =\sqrt{4\lambda -1}/2$. Integrating with the prescribed initial data givesandThese expressions also verify both initial values and their initial derivatives. For fixed $\mu >0$ and small $h,v_0$ on the outgoing branch, the sine coefficient in $u$ is positive and dominates the initial cosine term. The first return to small $u$ is near the first positive zero, $\omega \tau \simeq \pi$. HenceThe factor is a half-turn of the growing spiral, not its full rotation period. After this excursion the approximate trajectory approaches the small-$u$, large-$v$ region. In the full system the term $-\lambda v$, omitted during the large excursion, allows the motion to descend along that region and return.

As $\lambda \downarrow 1/4$, the rescaled excursion time diverges and the factor $e^{\tau /2}$ produces arbitrarily large excursions. At the critical value the matrix is a nontrivial Jordan block, and the limiting formula isFor the small outgoing initial data this is positive and unbounded: the unstable manifold no longer turns back. Below $1/4$ the two eigenvalues are real and positive, so there is again no oscillatory return. This quadratic escape desingularization explains loss of the periodic orbit at infinity, together with the unstable-manifold escape, as $\lambda$ decreases through $1/4$. It is not a finite homoclinic bifurcation to the origin. The divergent $\tau$ travel time must not be identified with the physical period without integrating $dt=d\tau /u$; the time change becomes singular near $u=0$.

Here $\mu >{\mu }_H=\lambda (\lambda -1)>0$. The origin is a saddle equilibrium, $P_{+}$ is repelling, and $P_{-}$ lies in $u<0$. Thus there is no saddle equilibrium in the interior of $u>0$ to which this periodic orbit could become homoclinic. The origin cannot supply the missing loop: the line $u=0$ is invariant and is precisely its local stable manifold. An orbit in $u>0$ cannot lie on that stable manifold, so the positive periodic orbit cannot close into a homoclinic orbit to the origin either.

The outgoing trajectories do not escape to infinity in this regime. To see the large-amplitude turning mechanism, put $w=v/u$ once $v\ge 0$. Direct differentiation givesSince $\lambda >1$, the quadratic $w^2-w+\lambda$ is uniformly positive. At large $u$ while $0\le w\le 1$, $\dot{w}$ is bounded below by a positive multiple of $u$, whereas $\dot{u}/u\le \mu +u$. Thus $w$ rises by a fixed amount per bounded increase in $\log u$: the trajectory reaches $w>1$ and then $w>1+\mu /u$, where $\dot{u}<0$, before $u$ can diverge. The same estimate bounds repeated outgoing excursions; large positive $v$ also decreases when $v>u^2$. Initially negative $v$ increases until it becomes nonnegative. These nullcline and ratio estimates give bounded forward excursions.

The repelling upper equilibrium is therefore surrounded by recurrent trajectories, and the Poincaré-Bendixson alternative supplies periodic motion rather than a saddle equilibrium connection. The supercritical attracting periodic orbit persists in the displayed regime, with trajectories spiralling away from the upper equilibrium and approaching it. The last panel of the portrait figure shows this behavior; the boundary stable manifold of the origin is kept separate.