text:
\section{Introduction}

\subsection{Spectral breaks in jets and pulsar-wind nebulae}

Broad-band spectra from synchrotron radiation characterize a wide
variety of astrophysical sources, including active galactic nuclei
(AGN), shell supernova remnants (SNRs), and pulsar-wind nebulae
(PWNe).  When observed over a sufficiently broad frequency range, such
spectra almost invariably show steepening to higher energies.  Such
steepening can be attributed either to intrinsic spectral structure or
to the effects of radiative losses.  However, the power-law shape of
radio spectra ($S_\nu \propto \nu^{-\alpha}$), with $\alpha \sim 0.5 -
0.8$ for many sources, suggests an origin of the requisite
relativistic electrons in diffusive shock acceleration (DSA), which
produces a power-law (or near-power-law, for efficient nonlinear DSA)
spectrum of particles $N(E) \propto E^{-s}$ with $s = 2\alpha + 1$
depending on the shock compression ratio.  Shocks are clearly present
in these astrophysical objects: outer blast waves in SNRs (and perhaps
reverse shocks into ejecta as well, for younger objects);
relativistic-wind termination shocks in PWNe; and shocks in jets and
hotspots in active galaxies.  The absence of an obvious mechanism to
generate a broken power-law distribution, and the necessity of the
operation of radiative losses at some level, has led to the common
acceptance of synchrotron losses as the mechanism to bring about
spectral steepening.

The standard calculation of the effect of synchrotron losses (reviewed
in the next section) for a homogeneous source predicts a steepening of
the initial electron spectrum to a second power-law one power steeper
($s_2 = s + 1$) than the injection spectrum, implying a radiation
spectrum one-half power steeper ($\alpha_2 = \alpha + 0.5$).  This is
in fact rarely observed.  In shell supernovae, the maximum energies to
which electrons can be accelerated are limited by losses or by finite
shock age (or size), but produce an exponential cutoff in $N(E)$,
further broadened by inhomogeneities, as observed in a few cases in
which synchrotron X-ray emission can be identified (see Reynolds 2008
for a review), so a sharp spectral break to a steeper power-law is
neither expected nor observed.  In PWNe, radio spectra are almost all
flatter than $\alpha = 0.5$, a phenomenon not well explained at
present, but the steepenings $\Delta \equiv \alpha_{\rm high} -
\alpha_{\rm low}$ are almost always greater than 0.5 (ranging from 0.7
to 1, in seven of the eight cases tabulated in Chevalier 2005, using
updated values for B0540-693 from Williams et al.~2008).  Knots in
extragalactic jets, when observable in X-rays, show similar too-large
steepenings (e.g., M87: Perlman \& Wilson 2005; Cygnus A: Stawarz et
al.~2007).  One standard interpretation of knot and hot-spot spectra
invokes shock acceleration and subsequent spectral steepening in a
uniform post-shock region \citep{heavens87}, but it cannot explain
these larger values of $\Delta$.  Coleman \& Bicknell (1988) present
numerical hydrodynamic simulations and find larger values of $\Delta$,
which they apply to observations, but without analytic results
allowing the wider application of the results.  While there is
recognition of the possibility of values of $\Delta$ different from
0.5 (e.g., Kennel \& Coroniti 1984a, who find $\Delta = (4 +
\alpha)/9$ for the Crab Nebula, or Petre et al.~2007, in a qualitative
discussion of the broadband spectrum of the PWN B0540-693), there has
as yet been no simple characterization of conditions under which
values of $\Delta \ne 0.5$ can naturally arise.  That characterization
is the goal of this paper.

\subsection{Synchrotron losses}

The first widely known calculation of the behavior of a distribution
of electrons subject to synchrotron losses is that of Kardashev
(1962).  While most of these results are well known, it is important
to recall the particular conditions under which each is applicable, so
I shall beg the reader's indulgence for a brief review, which can also
serve to fix notation.  Kardashev solved the kinetic equation for a
distribution $N(E,t)$ of electrons subject to gains by first and
second-order Fermi acceleration, and losses due to radiation or
adiabatic expansion, with and without the assumption of new-particle
injection and stationarity.  He writes the energy-loss rate from a
single electron as
\begin{equation}
\dot E = -b B_\perp^2 E^2,
\label{edot}
\end{equation}
where $B_\perp \equiv
B\sin\theta$, $b \equiv (2/3)(e^4/m_e^4 c^7)= 2.37\times 10^{-3}$ cgs
\citep[e.g.,][]{pacholczyk70}, and $\theta$ is the electron pitch
angle between its velocity vector and the magnetic field.  Kardashev
pointed out that in a uniform magnetic field in the absence of
scattering, electrons preserve their value of $\theta,$ since they
radiate a beam pattern that is symmetric with respect to their
velocity vector.  An electron injected into $B$ at $t = 0$ with energy
$E_0$ has an energy $E$ after time $t$ given by the well-known result
\begin{equation}
E(t) = {E_0 \over 1 + E_0 b B_\perp^2 t}.
\label{eoft}
\end{equation}
An initially infinitely energetic electron is reduced after time $t$
to energy $E_{\rm max}(t, \theta) = 1/b B_\perp^2 t$.  A power-law
energy distribution of electrons $N(E_0) = K E_0^{-s}$, with a single
value of pitch angle, will be cut off at $E_{\rm max}(t, \theta)$.
The electron distribution $N(E)$ will evolve according to
$N(E)dE = N(E_0)dE_0$, so that
\begin{equation}
N(E) = K \left[E_0(E)\right]^{-s} {dE_0 \over dE},
\label{evoldist}
\end{equation}
with $E(E_0)$ given by Equation~\ref{eoft}.  If $s < 2$, the electrons
initially above $E_{\rm max}(t)$ are sufficiently numerous to pile up
in a ``bump'' just below $E_{\rm max}(t)$, while if $s > 2$, the
``bump'' disappears.  (Numerical solutions to Equation~\ref{evoldist}
are shown below for inhomogeneous models, illustrating the ``bump''
effect.)

An initially isotropic distribution of electrons suffers unequal
radiation losses, with electrons with large pitch angles being more
rapidly depleted.  For an initial isotropic power-law distribution,
after a time $t$ one finds no electrons with pitch angles greater than
$\theta_{\rm max}$ given by $\sin^2 \theta_{\rm max} = 1/(b E B^2 t)$.
Since for synchrotron radiation, an individual electron's radiation
pattern has an angular width of order $1/\gamma$ where $\gamma$ is the
individual Lorentz factor, and $\gamma \mathrel{\mathpalette\@versim>} 10^3$ for radio
emission and higher frequencies, we can approximate electrons as
radiating exactly in their directions of motion.  Then $\theta$ is
also the angle between the line of sight and the local magnetic field.
An initially isotropic distribution of electrons injected at $t = 0$
into a source with a uniform magnetic field making an angle $\chi$
with the line of sight, observed through its synchrotron emission,
would then disappear abruptly at time $t(\chi)$.  More realistically,
one might expect that the source has a tangled magnetic field, so that
all values of $\chi$ are achieved in some part of the source.  One
should then (for an unresolved source) perform an angular integration
over the electron distribution. The result is an electron distribution
that steepens by one power of $E$, i.e., $N(E) \propto E^{-s-1}$,
above the characteristic energy $E_b = 1/(aB^2 t)$.  (The synchrotron
emission from this distribution steepens above $\nu(E_b)$ by more than
the value of 0.5 in spectral index $\alpha$ ($S_\nu \propto
\nu^{-\alpha}$) because, in this time-dependent case without
continuous injection, a correlation exists between $E$ and $\theta$
such that more efficiently radiating electrons are depleted most
rapidly.)

This situation is still relatively unrealistic, as it ignores any
processes by which electrons could change their pitch angles.  If
electrons scatter in pitch angle on timescales much shorter than the
synchrotron-loss timescale, one should simply average the energy-loss
rate over angles:
\begin{equation}
\dot E = 1.57 \times 10^{-3} B^2 E^2 \equiv a B^2 E^2,
\label{edota}
\end{equation}
where $a \equiv b<\sin^2 \theta> = (2/3)b$.  Then the electron
distribution will remain isotropic, and will simply cut off at $E_{\rm
max}(t)$; a source synchrotron spectrum would then cut off
exponentially above $\nu(E_{\rm max} (t)$.

However, the result we all remember from graduate school is neither of
these.  A source that turns on at $t = 0$ with continuous, uniformly
spatially distributed injection of a power-law distribution of
electrons $q(E) \equiv J_0 E^{-s}$ electrons erg$^{-1}$ s$^{-1}$
cm$^{-3}$, develops a break at energy $E_b = 1/(aB^2t)$, where the
electron spectrum steepens by one power in $s$.  This corresponds to a
steepening of the synchrotron spectrum by one-half power in $\alpha$
at $\nu_b = c_1 B^{-3} t^{-2}$ with $c_1 = 1.12 \times 10^{24}$ cgs.
This relation is frequently used to deduce a magnetic-field strength
in a synchrotron source of known age.

The result that synchrotron losses (or inverse-Compton losses, which
have the same dependence on electron energy) result in the steepening
of the particle spectrum by one power and the steepening of the
emitted synchrotron spectrum by a half-power, is a widely held idea.
It is this application that will be generalized below.  It is
important to remember that the standard derivation assumes a
distributed injection of electrons in a homogeneous source.  For other
conditions, it is not correct, as we shall see.

\section{Basic Calculation}

Energy-loss breaks from a synchrotron source in which relativistic
electrons are advected systematically away from an injection region
(such as a wind-termination shock) can be described as being due to
shrinking of the effective source size with frequency.  At a high
enough observing frequency, electrons drop below the energy required
to emit at that frequency before they reach the edge of the object,
hence limiting its effective size at that frequency.  Thus all results
depend on the critical energy an electron may have after suffering
synchrotron losses and convecting at speed $v$.  Equation~\ref{edota}
gives the synchrotron loss rate from a single electron.  In a
constant-density flow (i.e., neglecting adiabatic losses), but
allowing the possibility of spatially varying $B$, we generalize
the homogeneous results slightly to obtain
\begin{equation}
E_c = \left[\int a \,B^2\,{dr \over v}\right]^{-1}
\label{ec0}
\end{equation}
for the energy an initially infinitely energetic electron would have
after moving at $v$ through a field $B$ for a distance $r$.  So any
injected electron distribution must cut off at this energy.  For a
one-dimensional flow of electrons streaming at constant speed $v$ in a
constant magnetic field $B$, the effective length $r(E)$ of the source
is found from
\begin{equation}
E(r) = \left( {v \over a B^2} \right) r^{-1}.
\end{equation}
An electron of this energy radiates chiefly at frequency
$\nu = c_{\rm m} E^2 B = c_{\rm m}(v^2/a^2 B^3) r^{-2}$ where
$c_{\rm m} = 1.82 \times 10^{18}$ cgs (e.g., Pacholczyk 1970).
Then at any frequency $\nu$
the source has an effective length
\begin{equation}
r(\nu) = c_{\rm m}^{1/2} \left( v \over a B^{3/2} \right) \nu^{-1/2}.
\end{equation}
For synchrotron emission with a spectral index $\alpha$, the
observed flux density would then vary as
\begin{equation}
S_\nu \propto \nu^{-\alpha} r(\nu) \propto \nu^{-\alpha-1/2}
\end{equation}
-- the famous steepening by one-half power in the spectral index.
(Note that at no position $\bf {r}$ in the source is there a break in
the electron energy distribution $N({\bf r}, E)$ to a new power-law,
although such a distribution does result after integrating over
the entire source volume to obtain $N(E)$.)

This kind of argument can be generalized to inhomogeneous sources, for
which the steepening may be larger or smaller than 0.5.  For instance,
if the synchrotron emissivity increases with distance from the center,
then as the effective source size shrinks, more emission will be lost
than if the source were homogeneous, and the steepening can be greater
than 0.5.

Here we consider a simple model in which electrons are injected at
some initial radius $r_0$ in a ``jet'' whose full width $w$ rises with
dimensionless distance $l \equiv r/r_0$ as $l^\epsilon$: $w =
w_0 l^\epsilon.$ A conical jet (or piece of spherical outflow) then has
$\epsilon = 1$; a confined jet has $\epsilon < 1.$ Then the
cross-sectional area increases as $A_\perp \propto l^{2\epsilon}$ (see
Figure~\ref{jetfig}).  We shall assume ad-hoc power-law dependences of
quantities on dimensionless length $l$, ignoring any transverse
variations -- so a one-dimensional problem.  Let the $l$-dependence
of basic quantities be given by
\begin{equation}
B = B_0 l^{m_b} \qquad v = v_0 l^{m_v} \qquad \rho = \rho_0 l^{m_\rho}.
\end{equation}
While this is completely general, physical constraints may couple the
$m$'s.  For instance, in the absence of some mechanism such as mass
loading or entrainment \citep{lyutikov03} to alter the effective
density $\rho$, conservation of mass gives
\begin{equation}
{\rm Mass \ conservation}\ \qquad  \rho v A_\perp = {\rm const}
    \Rightarrow m_\rho + m_v = -2\epsilon.
\label{masscons}
\end{equation}
In the absence of turbulent amplification or reconnection, magnetic flux
will be conserved, giving different relations for components of
$B$ parallel and perpendicular to the jet axis:
\begin{eqnarray}
{\rm Longitudinal \ (radial) \ field}\ \qquad  B_\parallel: BA_\perp = {\rm const}
    \Rightarrow & m_b = -2\epsilon\\
{\rm Transverse \ (toroidal) \ field} \ \qquad B_\perp: Bwv = {\rm const}
    \Rightarrow & m_b = -m_v - \epsilon = m_\rho + \epsilon
\label{fluxcons}
\end{eqnarray}
where the last form for transverse $B$ involves invoking mass conservation.
Of course, unless the magnetic field is exactly radial, any toroidal component
will eventually dominate, barring extremely peculiar and probably unphysical
behaviors (e.g., $m_\rho < -3$).

\begin{figure}
\epsscale{0.80}
\plotone{figure1.eps}
\caption{Schematic of flow geometry.  The flow occurs in a tube
whose cross-sectional width $w$ grows as a power $\epsilon$ of (normalized)
distance $l$ from the injection radius $r_0$ ($l \equiv r/r_0$).
\label{jetfig}}
\end{figure}

In this formalism, Equation~\ref{ec0} implies $E_c \propto l^{-(1 +
2m_b - m_v)}$ in the absence of other energy-loss mechanisms such as
adiabatic expansion losses (non-constant density). In the presence of
adiabatic losses, it can be shown
\citep[e.g.,][]{kennel84a,reynolds98} that the critical energy is
given by
\begin{equation}
E_c = \rho^{1/3}\left[ \int_1^l {a B^2 \over v} \rho^{1/3} r_0 dl\right]^{-1} =
l^{m_\rho/3}
\left[ \int_1^l a v_0^{-1} r_0 B_0^2 l^{2m_b -m_v + m_\rho/3} dl \right]^{-1}
\label{ec1}
\end{equation}
\begin{equation}
={v_0 \over ar_0 B_0^2} (1 + 2m_b - m_v + m_\rho/3)
l^{-(1 + 2m_b - m_v)}
\equiv A_E l^{m_E}
\end{equation}
where we have assumed the source is long enough that $l(E) \gg 1$.
Note that the effects of adiabatic losses have canceled out (except
for a small change in the integration constant), leaving $E_c$ with
the same $l$-dependence as in the constant-density case.  We have also
made the assumption that the integral in Equation~\ref{ec1} is
dominated by the upper limit at $l$, demanding that $m_c \equiv 2m_b
-m_v + m_\rho/3 > -1$.  Otherwise, $E_c \propto l^{m_\rho/3}$ and
radiative losses play no role in the spectral behavior, so that the
calculation is not self-consistent.  Even if this condition is met, we
still wish to exclude situations in which the gradients conspire to
arrange adiabatic {\sl gains} of electrons as they convect, i.e., $m_E
> 0$.  This situation appears both unphysical and unlikely.  Note that
$m_c = m_\rho/3 - m_E - 1$.  These two conditions rule out some volume
in the parameter space of $(\epsilon, m_i)$ and must be checked for
any particular choices of those parameters.  The constraints are
related; $m_c > -1$ requires $m_E < m_\rho/3$, so the we ultimately
require $m_E < {\rm min}(0, m_\rho/3)$.

Electrons with energy $E_c$ (at position $l$ where the magnetic
field strength is $B(l)$) radiate chiefly at a frequency
\begin{equation}
\nu(E_c) = c_{\rm m} \left( v_0^2 \over a^2 r_0^2 B_0^3\right)
  \left(1 + 2m_b - m_v + m_\rho/3\right)^2
l^{m_b-2(1 + 2m_b - m_v)}
\equiv A_\nu l^{m_\nu}
\label{nub}
\end{equation}
defining $A_\nu$ and $m_\nu$.  Note $m_{\nu} = m_b + 2m_E = -2 - 3m_b
+ 2m_v$.  Then
\begin{equation}
l(\nu) = A_\nu^{-1/m_\nu} \nu^{1/m_\nu} \equiv A_l \nu^{1/m_\nu}.
\label{lnu}
\end{equation}
We are focusing on conditions such that the source size shrinks with
increasing frequency, i.e., $m_\nu < 0$.  It can easily be shown that
the condition on $m_c$ above is equivalent to $m_\nu < (m_\rho + m_v)
-1$.  If mass conservation is assumed, $m_\rho + m_v = -2\epsilon$ and
$m_\nu < 0$ always.  Otherwise, this condition must be checked, but
for reasonable values of the parameters (such as those in all examples
described below) it is always fulfilled.

Now to discuss the synchrotron flux, we assume a power-law electron
distribution $N(E) = KE^{-s}$ between energies $E_l$ and $E_h \gg E_l$
(we take $s > 1$).  As the flow expands, conservation of electron
number gives
\begin{equation}
n_e = {K \over s-1} E_l^{1-s} \propto \rho \Rightarrow
  K \propto \rho E_l^{s-1} \propto \rho^{1 + (s-1)/3} = \rho^{(s+2)/3}
\equiv \rho^{(2\alpha + 3)/3}
\end{equation}
since adiabatic losses give individual-particle energies varying as $E
\propto \rho^{1/3}$, at least near $E_l$, an energy we assume to be too low
for radiative losses to be important.  We have taken $s$ enough larger
than 1 that $E_l^{1-s} \gg E_h^{1-s}$.
Then we can write the synchrotron emissivity
(following Pacholczyk 1970) as
\begin{equation}
j_\nu = c_j K B^{1+ \alpha} \nu^{-\alpha}
= c_j K_0 B_0^{1+\alpha} l^{(2\alpha + 3)m_\rho/3 + (1 + \alpha)m_b} \nu^{-\alpha}
\equiv A_j l^{m_j} \nu^{-\alpha}.
\end{equation}
Here $c_j(s) \equiv c_5(s) (2c_1)^\alpha$ in the notation of
Pacholczyk; for $s = 1.5$, $c_j = 1.34 \times 10^{-18}$ cgs, and we
have defined another important index, $m_j \equiv (2\alpha +
3)m_\rho/3 + (1 + \alpha)m_b$.  Assume for the time being that we view
the jet directly perpendicular to the axis.  Then the line-of-sight
path length through the jet at any position $l$ is just $w(l) = w_0
l^{\epsilon}$ (through the center; averaged over lines of sight
intersecting a circular cross-section, we obtain the mean
line-of-sight path of $(\pi/4)w$.  We recall that we are assuming all
jet quantities to be constant in cross-section, that is, along these
lines of sight.  So if the source is at distance $D$, the integrated
flux density $S_\nu$ is given by
\begin{equation}
S_\nu = \int I_\nu d\Omega
= {1 \over D^2}\int_1^{l_\nu} dA \int_0^w {\pi \over 4} j_\nu ds
= {\pi A_j \over 4 D^2}\int_1^{l_\nu} w \ r_0 dl (w \ l^{m_j})\nu^{-\alpha}
\label{flux1}
\end{equation}
\begin{equation}
={\pi A_j \over 4 D^2}\int_1^{l_\nu} r_0 w_0^2 \ l^{2\epsilon + m_j} \nu^{-\alpha}\ dl
={\pi A_j r_0 w_0^2 \over 4 D^2(1 + 2\epsilon + m_j)}
    l_\nu^{1 + 2\epsilon + m_j}\nu^{-\alpha}
\label{flux2}
\end{equation}
\begin{equation}
={\pi c_j K_0 B_0^{1+\alpha}r_0 w_0^2 \over 4 D^2
  \left[1 + 2\epsilon + (2\alpha + 3)m_\rho/3 + (1 + \alpha)m_b\right]}
A_l^{(1 + 2\epsilon + (2\alpha + 3)m_\rho/3 + (1 + \alpha)m_b)}
\nu^{- \alpha - \Delta}
\label{flux3}
\end{equation}
where the last equation defines $\Delta$, the amount of spectral
steepening:
\begin{equation}
\Delta = - {1 + 2\epsilon + m_j \over {m_b + 2m_E}} =
  {1 + 2\epsilon + m_j \over |m_\nu|} =
{1 + 2\epsilon + (2\alpha + 3)m_\rho/3 + (1 + \alpha)m_b
\over 2 + 3m_b -2m_v}.
\label{Delta}
\end{equation}
We have assumed that $m_\nu < 0$, and that the flux integral
depends on the outer, not the inner, limit of integration, that is,
that
\begin{equation}
1 + 2\epsilon + m_j > 0.
\end{equation}
This latter condition can be restrictive.

While we have assumed a jet seen from the side, since the emission is
optically thin the observed flux density should be independent of
viewing angle.  This can be shown explicitly for the case of jets seen
exactly end-on, for which the flux integral Equation~\ref{flux1}
becomes
\begin{equation}
S_\nu = {1 \over D^2} \int_0^{w_{\rm max}} {\pi \over 2} w dw
  \int_{l_i(w)}^{l(\nu)} c_j K_0 B_0^{1 + \alpha}r_0 \nu^{-\alpha} l^{m_j} dl.
\label{fluxend}
\end{equation}
Here the line-of-sight is parallel to the jet axis, beginning
at a value of $l \equiv l_i$ dependent on $w$ (if $w < w_0, l_i = 1$),
The upper limit $w_{\rm max}$ is just the value of $w$ at which
$l_i = l_\nu$, i.e., $w_{\rm max} = w(l(\nu)) \equiv [l(\nu)]^\epsilon$.
Now we need a slightly more restrictive condition for the emission
to be dominated by the outer limit of integration $l(\nu)$:
$1 + m_j > 0$.  If this is the case, the two integrals in
Equation~\ref{fluxend} decouple:
\begin{equation}
S_\nu = {\pi \over 2 D^2} c_j K_0 B_0^{1 + \alpha} r_0 \nu^{-\alpha}
  {w_{\rm max}^2 \over 2} {1 \over 1 + m_j} \left[ l(\nu) \right]^{1 + m_j}.
\end{equation}
But $w_{\rm max} = [l(\nu)]^\epsilon$, so
\begin{equation}
S_\nu \propto \left[ l(\nu)\right]^{1 + 2\epsilon + m_j} \nu^{-\alpha}
\propto \nu^{((1 + 2\epsilon + m_j)/m_\nu) - \alpha}
\end{equation}
just as in Equation~\ref{flux2}.

These power-law spectra can hold only over a frequency range related
to the size range of the source by $l(\nu)$.  For instance, for
conical, constant-velocity, mass-conserving flow with tangential
magnetic field, we have $\epsilon = 1$, $m_\rho = -2$, and $m_b = -1$,
giving $l(\nu) \propto \nu^{-1}.$ Thus a source showing the
corresponding value of $\Delta$ (in this case, $\Delta = 7\alpha/3$)
between frequencies $\nu_1$ and $\nu_2$ must shrink over a range of
sizes given by $r_1/r_2 = \nu_2/\nu_1$.  In general, sources whose
spectra are set by effective variations of size with frequency are
predicted to have sizes varying as
\begin{equation}
l(\nu) \propto \nu^{1/m_\nu} \equiv \nu^{1/(-2 - 3m_b + 2m_v)}.
\label{size}
\end{equation}
Equation~\ref{Delta} relates the size exponent $1/|m_\nu|$ to
$\Delta$: $1/|m_\nu| = \Delta/(1 + 2\epsilon + m_j)$.  (Thus a source
with observed $\Delta$ will have a smaller rate of shrinkage with
frequency for a larger value of $1 + 2\epsilon + m_j$.)  The source
subtends a solid angle on the sky of
\begin{equation}
\Delta \Omega = {1 \over D^2} \int_1^{l(\nu)} (w_0 l^\epsilon) (r_0 dl)
= {1 \over D^2} {r_0 w_0 \over 1 + \epsilon} \left[l(\nu)^{1 + \epsilon}\right].
\label{solidang}
\end{equation}
If the source is only marginally resolved, one may consider an ``average''
angular size $\langle\theta\rangle$ defined by
\begin{equation}
\langle\theta\rangle \equiv (\Delta \Omega)^{1/2}
= {\sqrt{r_0 w_0} \over D} {1 \over \sqrt{1 + \epsilon}} [l(\nu]^{(1 + \epsilon)/2}
\end{equation}
so that $\langle\theta\rangle \propto \nu^{(1 + \epsilon)/2m_\nu}$.
For spherical or conical flows, i.e., $\epsilon = 1$,
$\langle\theta\rangle \propto \nu^{1/m_\nu}$ as before, but for
confined flows ($\epsilon < 1$), the average angular size decreases
more slowly with frequency.  If a confined jet is seen end-on, the
size variation with frequency is reduced even further, as the apparent
diameter is now proportional to $w_{\rm max} \propto
[l(\nu)]^\epsilon$ instead of $l(\nu)$:
\begin{equation}
\theta \propto \nu^{\epsilon/m_\nu} = \nu^{\epsilon/(m_b + 2m_E)}
   =\nu^{\epsilon/(-2 - 3m_b + 2m_v)}.
\end{equation}
This may result in a significantly reduced size effect, since the
primary change in emitting volume is a shrinking along the line of sight.

\section{Special Cases}

We can examine a few special cases.  First, in the case of a plane
constant-velocity flow, we have $\epsilon = 0, m_\rho = 0$, and $m_b =
0$, and we recover $\Delta = 1/2$.  Next, consider tangential magnetic
field, and conical outflow ($\epsilon = 1$) with constant density and
assuming mass and flux conservation. This corresponds to the inner
parts of a Kennel \& Coroniti (KC) MHD spherical flow.  Then $m_\rho =0$,
$m_v = -2$, $m_b = m_\rho + \epsilon = 1$, and we have
\begin{equation}
\Delta = {4 + \alpha \over 9}
\end{equation}
which was derived in Kennel \& Coroniti (1984b, eq.~4.11b).  This
already indicates that values of $\Delta \ne 0.5$ can be obtained
from reasonable situations.  It also suggests that obtaining values
very different from $0.5$ may be difficult.  The size effect predicted
above is quite weak:  $m_\nu = -2 - 3m_b + 2m_v = -9$, so $l(\nu)
\propto \nu^{-1/9}$, as shown in Kennel \& Coroniti (1984b, eq.~4.10b).
This weak effect accounts for the very small deviation of $\Delta$
from the homogeneous value of 0.5.

Now KC models can be divided into two regions: an inner one as above,
and an outer one, a constant-velocity flow with $\rho \propto r^{-2}$
(that is, $m_v = 0$ and $m_\rho = -2$) and tangential field ($m_b =
m_\rho + 1 = -1$).
Mass conservation is assumed ($m_v = -2\epsilon -
m_\rho$).  At lower frequencies, the burnoff radius moves in through the
outer region, giving
\begin{equation}
\Delta_{\rm outer} = {7 \alpha \over 3}.
\end{equation}
However, this is not realized actually, as neither
condition $m_E < 0$ or $1 + 2\epsilon + m_j > 0$ is met.  First, $m_E
\equiv -(3 + 2m_b + m_\rho) = + 1$, indicating that $E_c$ rises
with $l$.  Further, the $m_j$ condition gives $3 -2(2\alpha +
3)/3 -1 - \alpha = -7\alpha/3$, so the flux is dominated by the inner
parts of the nebula.  At high enough frequencies or photon energies
that the burnoff radius is at the transition point and moves into the
$v \propto r^{-2}$ region, we obtain the above result
\begin{equation}
\Delta_{\rm inner} = {4 + \alpha \over 9}.
\end{equation}
Here, $1 + 2\epsilon + m_j = 4 + \alpha > 0$, so the consistency condition
is met -- the flux is dominated by regions near $l(\nu)$.
For KC's model of the Crab, $\alpha = 0.6$, so $\Delta_{\rm inner} = 0.51.$

It is straightforward to show directly that spherically symmetric
sources obey the same scaling laws with $\epsilon = 1$ -- that is,
the assumptions of thin jets perpendicular to the line of sight
still give the correct scaling for spheres.  Let the dimensionless
radius be $R \equiv r/r_0$, with injection at $R = 1$ and burnoff at
\begin{equation}
R(\nu) = A_R \nu^{1/m_\nu} \equiv A_R \nu^{1/(-2 -3m_b + 2m_v))}
\end{equation}
-- that is, the same expression as for $l(\nu)$.
(Sphericity shouldn't change the expression, only, perhaps, the values
of the $m$'s.)  Similarly, we should have the same dependence of
$j_\nu(R)$ in the spherical case as we had for $j_\nu(l)$ in the jet
case:
\begin{equation}
j_\nu(R) = c_j K_0 B_0^{1+\alpha} \nu^{-\alpha} R^{m_j}
\end{equation}
with
\begin{equation}
m_j \equiv {2\alpha + 3 \over 3}m_\rho + (1 + \alpha) m_b.
\end{equation}
Then the total flux from this spherically symmetric, optically thin
source is just
\begin{equation}
S_\nu = {4\pi \over D^2} \int_1^{R(\nu)} 4\pi j_\nu(R) R^2 dR
= {4\pi \over D^2}\left(4\pi c_j K_0 B_0^{1+\alpha}\nu^{-\alpha}\right)
{[R(\nu)]^{m_j + 3} \over {m_j +3}}
\end{equation}
which gives
\begin{equation}
\Delta \equiv - {m_j + 3 \over m_\nu} =
+ {(2\alpha + 3)m_\rho/3 + (1+\alpha)m_b + 3
  \over 2 + 3m_b - 2m_v}.
\end{equation}
This is the same result as can be obtained by setting $\epsilon = 1$ in
the previous expression for $\Delta$, Equation~\ref{Delta}.

\section{Breaks greater than 0.5}

Obtaining values of $\Delta$ considerably greater than 0.5, as seems
to be required by observations of most PWNe, requires relaxing some of
the assumptions.  The relation from mass conservation of $m_\rho =
-2\epsilon - m_v$ might not hold if some form of mass-loading occurs,
for instance by evaporation of material from thermal filaments, or
entrainment of material from a confining medium.  Flux-freezing for
the magnetic field will not hold in the presence of either turbulent
amplification of magnetic field or of reconnection.  As an example,
consider (for algebraic simplicity) the case $\alpha = 0$, but
conical, constant-density flow (so $m_v = -2$), still conserving mass.
Now in this case, $m_j = m_b$, and $m_E = -(3 + 2m_b)$, so
\begin{equation}
\Delta = - {{3 + m_b} \over {m_b -2(3 + 2m_b)}} = {3 + m_b \over 6 + 3m_b}
\ \Rightarrow \ m_b = -{6 \Delta - 3 \over 3\Delta - 1}.
\label{mb01}
\end{equation}
Now we can obtain $\Delta = 2/3$ with the value $m_b = -1$ in this
case: the magnetic field drops as the first power of distance (faster
than if frozen-in and tangential, slower than if longitudinal).  This
does not seem like an unreasonable possibility.  Note that the
conditions are met: $1 + 2\epsilon + m_j = 3 + m_b = 2 > 0$,
$m_E = -1 < 0$, and $m_c \equiv 2m_b - m_v + m_\rho/3 = 0 > -1$.
The source size would obey
\begin{equation}
l(\nu) \propto \nu^{1/(m_b + 2m_E)} = \nu^{1/(-1 - 2(1))} = \nu^{-1/3}.
\end{equation}
To obtain weaker size effects, one is driven to larger values of
$1 + 2\epsilon + m_j$, for the same observed $\Delta.$

For practical use, it is convenient to consider $m_b$ a dependent
variable, in terms of the other quantities.  First, solve the equation
for $\Delta$ for $m_b$, in terms of $\Delta$, $m_\rho$, $\epsilon$,
and $\alpha$, with no presumed relation between $m_b$ and $m_\rho$.
The result is
\begin{equation}
m_b = {\Delta(2m_v - 2) + (2\alpha + 3)m_\rho/3 + 1 + 2\epsilon
   \over 3\Delta - 1 - \alpha}.
\label{mb}
\end{equation}
Two of the conditions are satisfied if $m_E < {\rm min}(0, m_\rho/3)$, or
\begin{equation}
1 + 2m_b - m_v > {\rm max}(0, -m_\rho/3).
\end{equation}
Assuming mass conservation, this becomes
\begin{equation}
1 + 2\epsilon + 2m_b + m_\rho > {\rm max}(0, -m_\rho/3).
\label{condition}
\end{equation}
In addition, we still require $1 + 2\epsilon + m_j > 0.$

An application to a particular source (i.e., an object of known
$\alpha$ and $\Delta$, presumably) can then be made by inserting those
values.  Various possibilities for $\epsilon$ and $m_\rho$ can be
tried; for each value of $m_b$ obtained in this way, the condition on
$m_E$ must be checked by hand.  For example, consider B0540-693, with
a radio spectrum of $\alpha = 0.25$ (Manchester et al. 1993), and a
break with $\Delta \sim 1$ at around 20 $\mu$m \citep{williams08}.
A constant-density spherical (or conical) outflow cannot produce this
$\Delta$.  Inserting the values of $\alpha$ and $\Delta$ into
Equation~\ref{mb}, and assuming for simplicity $\epsilon = 1$,
we find
\begin{equation}
m_b = {1 + 2m_v + 1.2m_\rho \over 1.7}.
\end{equation}
Then at the price of abandoning mass conservation, we can choose
$m_\rho = 1$ and $m_v = -2$, giving $m_b = -1.06,$ or about $-1$.
The consistency conditions are all met:  $m_c = 1/3 > -1$,
$m_E \equiv -(1 + 2m_b - m_v) = -1 < 0$,, and $1 + 2\epsilon + m_j
= 2.9 > 0$.
The source effective radius decreases as
\begin{equation}
r \propto \nu^{1/(m_b + 2m_E)} = \nu^{-0.33}
\end{equation}
which may be a serious problem, since we need the slope of $-\alpha -
\Delta \cong -1.2$ to hold from about 20 $\mu$m to somewhere in the blue
or near UV -- say 0.2 $\mu$m, requiring that the source shrink between
these two wavelengths by a factor of 4.6 -- perhaps unlikely.  (Though
what is shrinking is really the region containing the dominant flux;
there could be a faint halo contributing a small amount of flux in
which a brighter, shrinking core is embedded).  A numerical
calculation of this model is shown in Figure~\ref{b0540}, along with
observations.  The physics which could cause these values of $m_\rho$
and $m_b$ is, of course, completely unknown.

\begin{figure}
\epsscale{1.0}
\plottwo{figure2a.eps}{figure2b.eps}
\caption{Left: Electron distribution $N(E, l)$ at five positions in
the flow model for the PWN B0540-693: $l \equiv r/r_0 = 10, 30, 50,
70,$ and $90$.  Right: Model spectral-energy distribution and
observations for B0540-693, reproduced from Williams et al.~2008.
Radio: Manchester et al.~1993.  IR: Williams et al.~2008. Optical:
Serafimovich et al.~2004.  X-ray: Kaaret et al.~2001.
\label{b0540}}
\end{figure}

Figures~\ref{mb1} and~\ref{mb0p5} plot $m_b$ vs.~$\Delta$ for
Equation~\ref{mb01} and its generalizations to the pairs $(\alpha,
\epsilon) = (0.3, 1),$ $(0, 0.5)$, and $(0.3, 0.5)$, respectively.
Mass conservation is still invoked.  The consistency condition on
$m_c$ places upper limits on $Delta$ (lower limits on $m_b$) shown as
the squares on curves on each plot.  (The condition $1 + 2\epsilon +
m_j > 0$ is met for all curves shown.)  It is difficult to obtain
values of $\Delta > 0.7$; flows with rapidly dropping density (such as
$m_\rho = -2\epsilon$, for constant-velocity mass-conserving flows)
seem unable to do so.  Substantial deceleration seems to be required,
as well as rapid decreases in the magnetic-field strength.  While
there is some parameter space available for accomplishing this,
especially for sources with very flat radio spectra, the most
physically reasonable way to bring about the required deceleration
seems to be mass-loading, which also considerably expands the
available parameter space of source gradients.

\begin{figure}
\epsscale{1.0}
\plottwo{figure3a.eps}{figure3b.eps}
\caption{Left:  magnetic-field index vs.~$\Delta$ for $\alpha = 0$,
$\epsilon = 1,$ for several values of $m_\rho$.  Mass conservation
is assumed. Right:  Same, for $\alpha = 0.3$.  Only values of
$\Delta$ less than the square symbols on each curve satisfy the consistency
requirement.
\label{mb1}}
\end{figure}

\begin{figure}
\epsscale{1.0}
\plottwo{figure4a.eps}{figure4b.eps}
\caption{As in Figure~\ref{mb1}:  Left, for $\alpha = 0$ and $\epsilon = 0.5$;
right, for $\alpha = 0.3$ and $\epsilon = 0.5$.
\label{mb0p5}}
\end{figure}

\section{Inferring physical parameters}

Energy-loss spectral breaks are commonly used to infer source
magnetic-field strengths in pulsar-wind nebulae.  One requires a
source age $t$; for PWNe in SNR shells, one can use modeling of the
shell emission to estimate an age, while in some cases, a source size
divided by a mean flow speed (estimated one way or another) can give
an age estimate.  Then one simply assumes a homogeneous source for
which $E_c = (a B^2 t)^{-1} = 637/B^2 t$ and from Equation~\ref{ec0},
\begin{equation}
B_h = \left(c_{\rm m} \over a^2\right)^{1/3} \nu_b^{-1/3} t^{-2/3}
= 0.90 \ \nu_{\rm GHz}^{-1/3} \ t_{\rm yr}^{-2/3} \ {\rm G}.
\label{bhom}
\end{equation}
(where we have averaged over pitch angles).  It is of
interest to compare this to the magnetic field that would be inferred
for a flow model assuming (incorrectly) that the source is
homogeneous.  Let the source have a size $L$ and break frequency
$\nu_b$ (so that $L = r_0 l(\nu_b)$).  For a flow model, we can deduce
the initial magnetic field $B_0$ from Equation~\ref{nub} above:
\begin{equation}
\left(L \over r_0\right)^{m_\nu} = {a^2 \over c_{\rm m}}
 \left(r_0^2 \over v_0^2\right) {1\over f^2} B_0^3 \nu_b
\end{equation}
where $f \equiv 1 + 2m_b -m_v + m_\rho/3$.  This implies
\begin{equation}
B_0 = \left( L \over r_0 \right)^{m_\nu/3} \left(c_{\rm m} \over a^2 \right)^{1/3}
\nu_b^{-1/3} \left(v_0^2 \over r_0^2\right)^{1/3} f^{2/3}.
\end{equation}
Then
\begin{equation}
{B_h \over B_0} = \left(r_0 \over v_0 t\right)^{2/3} f^{-2/3}
  \left( L \over r_0\right)^{-m_\nu/3}.
\end{equation}

Of course, with substantial magnetic-field gradients, $B_h/B_0$ can
range widely either below or above 1.  Some source properties, such as
the initial ratio of energy input in magnetic field to that in particles
(KC's $\sigma$ parameter), require knowledge of $B_0$.  For those
properties, estimation of source magnetic field from the homogeneous
assumption can lead to significant error.  However, it is possible to
show that $B_h$ does give a good approximation to the mean magnetic
field averaged over the lifetime of a particle moving with the flow, as
of course it must.  The total magnetic-field energy in a flow
model is
\begin{equation}
U_B = {1 \over 8\pi}\int_1^{L/r_0} r_0 dl \ \pi \left({w(l)\over 2}\right)^2
    B_0^2 l^{2m_b}
\end{equation}
\begin{equation}
= {1 \over 32} \ \left[{w_0^2 B_0^2 r_0 \over 1 + 2\epsilon + 2m_b} \right] \
   \left(L \over r_0\right)^{1 + 2\epsilon + 2m_b}
\end{equation}
where we have asumed $1 + 2\epsilon + 2m_b > 0$ and $L \gg r_0$.  The
total volume in the flow is
\begin{equation}
V = \int_1^{L/r_0} \pi \left( {w(l)\over 2}\right)^2 r_0 dl
   = {\pi r_0 w_0^2 \over 4(1 + 2\epsilon)}
   \left(L \over r_0\right)^{1 + 2\epsilon}.
\end{equation}
Then the mean magnetic energy density $\langle u_B \rangle$ is
\begin{equation}
\langle u_B \rangle \equiv {U_B \over V} = {B_0^2 \over 8 \pi}
   {(1 + 2\epsilon) \over 1 + 2\epsilon + 2m_b} \left(L \over r_0 \right)^{2m_b}
\end{equation}
and the homogeneous energy density $u_B({\rm hom}) \equiv B_h^2/8\pi$
satisfies
\begin{equation}
{u_B({\rm hom}) \over \langle u_B \rangle} =
  f^{-4/3} \left(r_0 \over v_0 t\right)^{4/3}
  \left(1 + 2\epsilon + 2m_b \over 1 + 2\epsilon\right)
  \left(L \over r_0\right)^{4(1 - m_v)/3}
\label{bratio}
\end{equation}
where the exponent of $L/r_0$ has been rewritten using $m_\nu = 2m_v - 2 - 3m_b$.

If the value of $t$ used in the homogeneous relation
Equation~\ref{bhom} is the actual transit time of an electron from
$r_0$ to $L$, one obtains a similar result.  That $t$ is given by
\begin{equation}
t_{trans} \equiv \int {r_0 dl \over v} = {r_0 \over v_0} \int_1^{L/r_0} l^{-m_v} dl
= {r_0 \over v_0} {1 \over 1 - m_v} \left(L\over r_0\right)^{1 - m_v}.
\end{equation}
Then
\begin{equation}
\left(r_0 \over v_0 t_{\rm trans}\right)^{4/3} \left(L \over r_0\right)^{4(1 - m_v)/3}
  = (1 - m_v)^{4/3}
\end{equation}
independent of physical parameters.  (We are assuming $m_v \le 0$, i.e.,
we exclude {\sl accelerating} flows.) This means that all factors in
Equation~\ref{bratio} are of order unity, so that there is not a large
discrepancy between the true mean magnetic-field energy density and
that inferred under the assumption that the source is homogeneous.
However, if the source lifetime is used for $t$ (which may differ
substantially from $t_{\rm trans}$), or an estimate of $t_{\rm trans}$
is made from the measured expansion velocity of the outer boundary of
the PWN, serious errors may be made in inferring $B$.

\section{Numerical calculations and applications to observed sources}

These results can easily be confirmed by numerical integration of the
appropriate equations.  In particular, Equation~\ref{evoldist} can be
used to find the detailed particle distribution, accounting for the
pileup of particles at energies just below $E_{\rm max}(t)$ as they
migrate down in energy (for $s < 2$).  This pileup can produce a
detectable ``bump'' in the spectrum just below $\nu_b$.  The numerical
calculations can also show how sharp a break can be achieved in
practice.

I illustrate these effects with several models.  The example
parameters for B0540-693 mentioned above $(\alpha = 0.25, \epsilon =
1, m_\rho = 1, m_b = -1, m_v = -2)$ produce distribution functions
$N(E, l)$ at various points in the flow shown in Figure~\ref{b0540},
at positions $l \equiv r/r_0 = 10, 30, 50, 70,$ and $90$.  (The
energies are in units of a fiducial energy $E_f \equiv aB_0^2r_0/v_0$,
the energy an initially infinitely energetic electron would have after
radiating for a time $r_0/v_0$ in a magnetic field $B_0$.)  The sharp
cutoff energy $E_c$, decreasing down the flow, is apparent, as is the
spike just below it of electrons formerly above $E_c.$  The rising
density produces adiabatic {\sl gains} in the density of electrons of
too low energy to be subject to radiative losses. Spatial integration
over these electron distributions produces the model spectral-energy
distribution also shown in Figure~\ref{b0540}, reproduced from
\cite{williams08}, which fits the data surprisingly well, apart from
the anomalous X-ray flux.  (A technical problem, ``pileup'' in the
{\it Chandra} detectors due to the bright X-ray pulsar in B0540-693,
makes the absolute determination of the X-ray flux of the nebula
difficult; see Petre et al.~2007.) For the relatively steep
low-frequency spectrum of B0540-693 ($\alpha = 0.25$), the ``bump''
from integrating over the spikes of Figure~\ref{b0540} is barely
noticeable, but it is much more obvious for a flatter input spectrum.
The model assumes a source radius $L \sim 1.3$ pc \citep{williams08},
but $r_0$ is a free parameter.  If $r_0$ is the pulsar wind
termination shock, we might expect $v_0 = c/3$ \citep{kennel84a}.  The
break frequency (Equation~\ref{nub}) constrains the remaining
combination $r_0^2 B_0^3.$ The model of Figure~\ref{b0540} takes $r_0
= L/100$ and $B_0 = 2.4 \times 10^{-3}$ G.  The radio flux
(Equation~\ref{flux3}) then sets the combination $w_0^2 K_0$; the
model takes $w_0 = r_0/10$ and $K_0 = 2.1 \times 10^{-8}$ cm$^{-3}$
erg$^{0.5}$.  The break frequency calculated from Equation~\ref{nub}
is about $6 \times 10^{12}$ Hz, about a factor of 5 lower than
the intersection of the extrapolations from low and high frequencies.
This is due to the approximation that electrons radiate entirely
at their peak frequency $\nu_m$, an approximation not made in
the analytic calculations; in general, break frequency predictions
will be low by a factor of several, depending somewhat on the
value of $s$.

\begin{figure}
\epsscale{1.0}
\plottwo{figure5a.eps}{figure5b.eps}
\caption{Left:  Integrated spectrum for Kes 75 model.  Right:  Same for
MSH 15--5{\it 2}.  Flux scales are arbitrary.
\label{fluxes}}
\end{figure}

Chevalier (2005) summarizes spectral indices for several PWNe,
including Kes 75 ($\alpha = 0 \Rightarrow s = 0$, $\Delta = 1$) and
MSH 15--5{\it 2} ($\alpha = 0.2 \Rightarrow s = 1.4$, $\Delta =
0.85$).  Such large values of $\Delta$ typically require relaxing
either mass or flux conservation, or both.  For Kes 75, the values
$\epsilon = 1$, $m_\rho = 1$, $m_v = -2$, and $m_b = -1$ (the same as
for the B0540-693 model, except $\alpha = 0$) predict $\Delta = 1.0.$
The consistency condition $1 + 2m_b - m_v > {\rm max}(0, -m_\rho/3)$
is met.  Figure~\ref{fluxes} (left) illustrates the integrated
spectrum.  The ``bump'' is quite prominent; the flux at the peak
around $10^{13.5}$ Hz is 4.4 times that at 1 GHz.  The slope above the
break is $0.96$ between 0.4 and 4 keV, close to the analytic value of
1.0.  Equation~\ref{size} gives the frequency-dependence of the source
size as $l_{\rm max} \propto \nu^{-1/3}$, sufficiently slow that it
might be hard to detect.  For MSH 15--5{\it 2}, Figure~\ref{fluxes}
(right) shows a calculation for $s = 1.4$, $\epsilon = 1$, $m_\rho =
1$, $m_v = -2.22$, and $m_b = -1,$ predicting $\Delta = 0.85.$ The
consistency condition is again met.  The ``bump'' is still
perceptible.  The predicted value of $\Delta$ is reproduced exactly.
The size effect is even slighter: $l_{\rm max} \propto \nu^{-0.29}$.
A factor of 11 frequency range would be required to see the source
shrink by a factor of 2.  For an actual well-resolved source, the
``size'' would need to be measured with some relatively coarse
quantity such as the 50\% enclosed power radius or FWHM, as cited for
the Crab by \cite{kennel84b}.

\section{Summary of results}

Here I collect the principal results and consistency requirements.
The basic result is the expression for $\alpha_2 - \alpha_1 \equiv \Delta$,
the amount of spectral steepening, Equation~\ref{Delta}:
\begin{equation}
\Delta = - {1 + 2\epsilon + m_j \over {m_b + 2m_E}} =
{1 + 2\epsilon + (2\alpha + 3)m_\rho/3 + (1 + \alpha)m_b
\over 2 + 3m_b -2m_v}.
\end{equation}
This expression holds if several consistency requirements are met:
$m_E < {\rm min}(0, m_\rho/3)$, where $m_E \equiv -1-2m_b+m_v$, so
that the burnoff energy at position $l$ depends on $l$, and $E_c$
drops with $l$; and $1 + 2\epsilon + (2\alpha + 3)m_\rho/3 + (1 +
\alpha)m_b > 0$, so that the integrated flux density $S_\nu$ depends
on the outer limit of integration.  Finally, the effective source size
should shrink with frequency: $m_\nu < 0,$ a condition always met in
the presence of mass conservation, and almost always met for
reasonable parameters otherwise.  If the conditions are met, the
source size (some measure of the region from which the bulk of the
emission originates) decreases with frequency as
\begin{equation}
l(\nu) \propto \nu^{1/m_\nu} \equiv \nu^{-1/(2 + 3m_b - 2m_v)}.
\end{equation}
if seen more or less from the side; if the flow is nearly along the
line of sight, the expression becomes
\begin{equation}
\theta \propto \nu^{\epsilon/m_\nu} = \nu^{-\epsilon/(2 + 3m_b - 2m_v)}.
\end{equation}
(which is the same for spherical or conical flows where $\epsilon = 1$).

\section{Conclusions}

My basic conclusion is just that synchrotron-loss spectral breaks
differing from 0.5 can be produced naturally in inhomogeneous sources.
I have treated the inhomogeneities resulting from flows, which seem
most natural, using simple power-law parameterizations, but more
complex functional dependencies can be treated the same way.  Other
types of inhomogeneities may be possible as well.  These results are
most straightforwardly applied to PWNe or knots in extragalactic jets,
but may have applications wherever bulk flows of relativistic material
are involved.  In particular, energy-loss breaks seen in gamma-ray
burst afterglows (e.g., Sari et al.~1998; Galama et al.~1998; and much
later work) may provide opportunities for the application of these
results.  For nearby sources, the simplest test of the models is the
detection of the size effect; every model predicts both a particular
$\Delta$ and some rate of decrease of size with frequency (really of
volume, since the size decrease may take place along the line of
sight).  For the same $\Delta$, some variation in the strength of the
size effect is possible.

Fortunately, assuming that an inhomogeneous source is actually
homogeneous does not drastically alter the inferred mean magnetic
field (averaged over the history of a fluid element), but since there
are by assumption large gradients of most quantities, local values of
the magnetic-field strength, such as those at the injection radius,
may depart substantially from the mean values.  This may affect
inferences of the KC magnetization parameter $\sigma$.  Furthermore,
the inference requires knowledge of the actual flow time across the
source -- knowledge that may be hard to come by in the presence of
large velocity gradients.

\acknowledgments

I gratefully acknowledge the hospitality of the Arcetri Observatory of
the University of Florence, where this work was begun.  This work was
also supported by NASA through Spitzer Guest Observer grants RSA
1264893 and RSA 1276758.

redpajama_set_name: RedPajamaArXiv


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The Writer's Workshop Series
the forgotten soldier poem
He takes us on a journey through two brutal Russian winters, being bombarded by artillery, taking part in battGuy Sajer was a sixteen year old boy in 1942 who was brought up in France by a French father and a German mother. He was among those Alsatians who joined the Wehrmacht following the incorporation of Alsace (one of France's eastern regions) into the Third Reich following France's defeat in June 1940. Hard to feel sorry for any of the Krauts in WWII, but this is as close as you will get. Things quickly go from bad to worse as you follow him retreating back to Fatherland with the broken remnants of the Master Race. For those who live a nightmare reality, sleep is a black hole, lost in time, like death.""Then there was the war, and I married it because there was nothing else when I reached the age of falling in love." It is used in most commonwealth nations to remember lost soldiers, it is fitting with the poem "A Soldier's Cemetery".The poem "A Soldiers Cemetery" depicts the forgotten soldiers of the first world war, it uses. Peace has brought me many pleasures, but nothing as powerful as the passion for surviving in wartime, that faith in love, and that sense of absolutes. Later he joins the Gross Deutschland division as an infantryman in order to qualify for some leave and participates in many of the big battles of the eastern front. Hes getting older. Honestly, I could probably have given this two stars but I think that my harsh criticism is at least in part due to the fact that I'm outgrowing the point in my life where military biographies will do it for me and I think that had this been more my cup of tea recently I would have rated it at least 4 stars. I read a review of this book around 1971 and my Dad and I eagerly awaited its arrival at the library. An intense first hand experience of WWII in eastern front given by GUY SAJER. I read a review of this book around 1971 and my Dad and I eagerly awaited its arrival at the library.
Endlessly repetitive yet often psychologically dissociative.
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The Cannibals of Kitsap by Jonny Eberle
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redpajama_set_name: RedPajamaCommonCrawl


If you are a traditional kind of guy you will love our blue stripe pyjamas. Crafted in pure cotton, they feature a classic button front with a breast pocket and long sleeves and the matching bottoms have an elasticated waist. Team with one of our dressing gowns and slippers for a complete nightwear look.


text:
Bryan's Story
A life or death ski accident dramatically changed Bryan's perspective on life. That moment created a ripple in his life that transformed the way we do business as MiiR. It's taken him near and far from Boise Idaho to Buchanan Liberia where's he's been inspired by the joy and resiliency of those he has met.
Bryan Papé
On April 15, 2006, lying against that tree not knowing if I would live or die, I thought, what would people say about me at my funeral? And that question cut to the core of who I was.
Having inherited his grandfather's entrepreneurial spirit, it wasn't until a life-or-death ski accident at the age of 20 that Bryan Papé fully realized his vision to begin building a legacy of generosity. That clarity, coupled with design opportunity in active lifestyle products, led to MiiR's launch in 2010.
A Product to Project™ company committed to trackable giving, MiiR gives 3% of revenue to partner organizations with an aim to provide people with access to a better future. Every product we design is built in-house with an emphasis on innovation, timelessness, simplicity and sustainability.
MiiR partners with some of the most respected nonprofits in the world to create sustainable giving projects in the clean water, health and food sectors that address both domestic and international issues. To date, we have given over $600,000 and empowered the lives of 100,000+ people worldwide, and invite customers to track the impact of their purchase through our Give Code™ registration process.
By being authentic, generous and transparent, we've been able to improve our business results, proving that there need not be a disconnect between business and philanthropy.
Outside of MiiR, Bryan has many passions including teaching his daughter to explore, bird hunting with his vizsla and building his family's cabin in the middle of nowhere. He enjoys skiing and mountaineering, and has successfully summited Mount Rainier on four occasions, two of them with his wife Rebecca. Born and raised in Boise, Idaho, he considers the Pacific Northwest home after having lived in Seattle for close to fifteen years.
In 2016, Bryan was named a 40 Under 40 business leader by the Puget Sound Business Journal, the same publication that named MiiR the third fastest-growing private company in Washington in the same year. He received the first Young Alum of the Year award from his alma mater Seattle Pacific University in 2009, and in 2015 was awarded the first Young Alum of the Year in the SPU School of Business and Economics. Bryan and Rebecca were honored as Supporters of the Year by One Day's Wages in 2013.
Bryan's broken femur
Bryan's wife Rebecca
Repaired femur with stainless steel rod and screws
On April 15th, 2006 I was filming for Stevens Pass after 12" of freshies fell overnight. Despite being a fairly advanced skier, I took a bad turn trying to post up for a shot of my good friend Chris Rudolph (may he rest in peace). My right ski hit a hidden stump and flicked my leg open, where upon I smacked my inner thigh against an elderly pine. It won. I cracked my femur in half and lay in shock against another tree. The thing is, when you break your femur, the bone shards can hit your femoral artery and you can bleed to death internally within 10-15 minutes. I knew this because a friend did something similar a few years before.
I sat against a tree not knowing if I would live or die. As I contemplated death, I distinctly remember thinking two things. The first being I needed to marry my then girlfriend (and now wife of nine years) if I survived the accident. The second was if I died today what would people say about me at my funeral. This thought shocked me. People would certainly come as I was a funny guy/class clown (put a VW bug on our high school's roof) but NOBODY would say I cared about the people around me or in my community. Certainly nobody would say I left a legacy or inspired anyone. This struck me because I had grown up admiring my grandparents and the legacy they were leaving through business and philanthropy. I committed at that tree to start leading a more generous life serving others over myself. I began to noodle on what it would look like to merge business and philanthropy.
Bryan at Little Hotties Warmers
In the summer of 2006 I started working at my mentor's best friend's new startup Little Hotties Warmers. I became Rick Wood's (the founder) first employee and earned sweat equity. At 21 I was running supply chain and marketing for the brand. Most things in small business I learned from Rick. He is a world class strategist and salesman. Under his wing I learned first hand how to negotiate with vendors, suppliers and close the biggest accounts in the world from REI to Costco.
We had the unique opportunity to sell the business for a premium at the height of the recession. I knew this was my opportunity to start my own business.
MiiR Takes Form
Design drawings from MiiRs first bottle line.
MiiR's first bottle line.
In 2009 I was frustrated with the lack of a simple and functional water bottle. Sounds funny but I had two brands I also used. One was a giant plastic bottle with a wide mouth and lid attached with a strap. When I drank from the bottle I would normally spill water all over my face because the opening was too big. If I didn't spill water, the lid was always hitting my face. If that wasn't enough to drive you crazy, the width of the entire bottle was so large it never fit in cup holders which meant it rolled around in my back seat while driving. The other bottle I used was an aluminum bottle, apparently it used to be a camping stove fuel canister (yikes!). The lid took about 100 twists to unscrew and if the liner cracked (which it did) it exposed my water to the aluminum body, which happens to be toxic.
I thought, why not make a super simple bottle. One twist opening, fit in cup holders, perfect sized opening to chug but not drown your face. I'd also make it from a stainless steel, a safer/non-toxic material. After all, I had a stainless steel rod in my body.
Today, we design all our products from the ground up and most are patented or patent pending. We've made a commitment to never sell off-the-shelf product that is poorly designed and copied quickly.
More About The Founder


text:
Kelsey Warner
SoftBank eyes another 100 companies in second Vision Fund
Number is an increase on the 86 companies targeted for the first fund
Japan's SoftBank is looking at increasing the number of companies it will invest in once as it looks to roll out a second fund. Reuters
Amid a $9.5 billion (Dh34.9bn) bailout of portfolio company WeWork, Japan's SoftBank plans to invest in 100 companies through its second Vision Fund, up from the 86 in its first round, a senior executive said.
"We have 86 companies in our portfolio in fund one, with another 100 in fund two," Rajeev Misra, chief executive of SoftBank Investment Advisers said at the Future Investment Initiative in Riyadh on Wednesday. "So those 186 companies interacting with each other and doing business with each other will be immense."
Mr Misra's comments come less than a week after the Wall Street Journal reported SoftBank would "scale back" its investment strategy and focus on corporate governance at its portfolio companies.
SoftBank's first Vision Fund invested $85bn since 2017, with about $15bn more earmarked for investment. The follow-up Vision Fund 2 announced in July is aiming to raise $108bn. SoftBank said it plans to commit $38bn of its own capital in the new fund. Apple, Goldman Sachs, Microsoft, Foxconn and the sovereign wealth fund of Kazakhstan are expected to invest, and seven Japanese financial institutions have signed preliminary agreements to participate.
SoftBank's WeWork takeover plan could cause Adam Neumann's exit
Troubled WeWork said to cut 500 tech jobs
Both Abu Dhabi's strategic investment company Mubadala and Saudi Arabia's sovereign Public Investment Fund invested in the first Vision Fund.
Last week, WeWork accepted $9.5bn from SoftBank Group, its largest investor, in exchange for 80 per cent stake in the company.
WeWork was valued at $47bn at its peak in January and scrapped a high-profile initial public offering attempt last month. It is now valued at less than $8bn in the bailout.
WeWork founder Adam Neumann will leave the company's board as part of the package, to be replaced by SoftBank executive and newly-appointed executive chairman Marcelo Claure. Mr Neumann is set to walk away from the deal with as much as $1.2bn in WeWork stock, a $500 million credit line from SoftBank and a consulting fee of about $185m, Bloomberg reported.
The bailout grants WeWork's parent We Company a bit more runway as it faced running out of money as early as next month, according to Bloomberg.
WeWork "has a pretty high probability of being a zero for the equity, as well as for the debt," said billionaire investor Bill Ackman, according to the Financial Times. "As someone who has put good money after bad, I think this looks like putting good money after bad, and SoftBank should have walked away," the newspaper cited him as saying.
When asked how start-ups balance rapid expansion, and if the culture is for companies to grow too fast, Mr Misra of SoftBank said: "At the end of the day, the business has to make sense, unit economics have to make sense."
He added, "The business has to make sense in a certain town, in a certain country" and the technology and business model have to be extensively tested before a company expands geographically. The tech sector too often spread across geographies before things have been tested and been proven to work.
His comments are in stark contrast to the Vision Fund's previous mandate that prioritised large investments to achieve rapid geographic growth over generating a profit in the short-term.
More in Future
Google CEO calls for aligned AI regulation as US and EU plans diverge
New AI toolkit from the World Economic Forum is promising because it's free
The AI talent war must be won at home and abroad
How the UAE could save $3bn by using blockchain
Mohammed Bin Rashid Innovation Fund selects 11 firms for latest accelerator round
Consumers access finance apps over 1 trillion times in 2019, says report
Abu Dhabi Investment Office allocates Dh60m to region's tech start-ups
More than $11bn lost in cryptocurrency hacking since 2011, report says
Apple 'to start making lower-cost iPhone next month'
Dubai's Careem decides to pull out of Oman
Huawei takes to Twitter to support CFO ahead of first hearing
Hub71 launches region's first Microsoft Reactor programme to build technical skills
World's first company using tech to create real milk secures KBW Ventures backing
China urges Canada to release Huawei CFO Meng


Paula & Nate might be the sweetest love story ever told.
The comic book tells the story of Paula, a bee living in Toronto's Queen West neighbourhood, and her love for a flower named Nate.
The book is part of a larger project, led by Sid Lee Toronto and local urban beekeeping company Alvéole, to raise awareness of urban beekeeping practices at a time when bee populations worldwide are facing an up-wind battle against climate change. It's being distributed to local shops throughout the Queen West neighbourhood.
In addition to the comic, the agency harvested honey from its rooftop hive and created custom jars and swag featuring the Paula & Nate brand.


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He feels no fear. He knows no pain. And he is so very, very hungry . . .
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One Lap Dance Can Change Your Life!
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Bikini Waxes for Interstellar Peace!
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What's a Murder or Two Between Friends?
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"And the Children . . .the Children are Screaming!"
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The World is About to End . . .


Wow great stuff, so that is mine as it was then? I know there has been some non genuine ones around! Any idea what year it was roughly and where this was taken?
Great boats , lucky you. It hasn't got large plates on the transom has it ? Just a thought when you said single inboard.


text:
The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes)
Joseph D. Wisk, Keith Cooper
School of Environmental and Biological Sciences, Biochemistry & Microbiology
Embryos of the Japanese medaka were individually exposed to varying concentrations of [3H]2,3,7,8‐tetrachlorodibenzo‐p‐dioxin (TCDD) in a static, nonrenewal system. The EC50 with 95% confidence interval (C.I.) to prevent hatching was 14 (11–17) nanograms (ng) of TCDD equivalents per liter (L) of water (parts per trillion). The LC50 with 95% C.I. for survival to 3 d posthatch was 9 (6–12) ng TCDD equivalents/L. The EC50 with 95% C.I. for embryos with minor lesions and severe, life‐threatening lesions were 3.5 (1.3–5.7) ng TCDD equivalents/L and 14 (12.4–15.6) ng TCDD equivalents/L, respectively. In a separate experiment that was terminated prior to the embryos hatching or dying, the EC50 for lesions was calculated to be 2.2 (1.4–3.0) ng TCDD equivalents/L. Based on the amount of TCDD equivalents recovered from dechorionated embryos, the ED50 with 95% C.I. for lesions was calculated to be 0.24 picograms of TCDD equivalents per milligram of dechorionated embryo weight (parts per billion). When the embryos were exposed to [3H]TCDD within 1 to 2 h after fertilization, no concentration dependent increase in visible lesions was observed until after the formation of the liver rudiment (day 4 of development). By exposing Japanese medaka embryos to lethal concentrations of TCDD beginning on different days of embryonic development, it was demonstrated that the sensitive period for toxicity was during liver formation on day 4 or 5 of development. The sensitive period for development was not caused by differences in TCDD absorption across the chorion. When embryos were exposed to [3H]TCDD prior to, during or after liver formation, there was no statistical difference in the dose of TCDD equivalents that crossed the chorion and entered the yolk and embryo.
Environmental Toxicology and Chemistry
https://doi.org/10.1002/etc.5620090907
Oryzias
lesion
Polychlorinated Dibenzodioxins
embryonic development
Medaka
Wisk, J. D., & Cooper, K. (1990). The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes). Environmental Toxicology and Chemistry, 9(9), 1159-1169. https://doi.org/10.1002/etc.5620090907
Wisk, Joseph D. ; Cooper, Keith. / The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes). In: Environmental Toxicology and Chemistry. 1990 ; Vol. 9, No. 9. pp. 1159-1169.
@article{1118eddffa4e40e6a9776c8d08581643,
title = "The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes)",
abstract = "Embryos of the Japanese medaka were individually exposed to varying concentrations of [3H]2,3,7,8‐tetrachlorodibenzo‐p‐dioxin (TCDD) in a static, nonrenewal system. The EC50 with 95{\%} confidence interval (C.I.) to prevent hatching was 14 (11–17) nanograms (ng) of TCDD equivalents per liter (L) of water (parts per trillion). The LC50 with 95{\%} C.I. for survival to 3 d posthatch was 9 (6–12) ng TCDD equivalents/L. The EC50 with 95{\%} C.I. for embryos with minor lesions and severe, life‐threatening lesions were 3.5 (1.3–5.7) ng TCDD equivalents/L and 14 (12.4–15.6) ng TCDD equivalents/L, respectively. In a separate experiment that was terminated prior to the embryos hatching or dying, the EC50 for lesions was calculated to be 2.2 (1.4–3.0) ng TCDD equivalents/L. Based on the amount of TCDD equivalents recovered from dechorionated embryos, the ED50 with 95{\%} C.I. for lesions was calculated to be 0.24 picograms of TCDD equivalents per milligram of dechorionated embryo weight (parts per billion). When the embryos were exposed to [3H]TCDD within 1 to 2 h after fertilization, no concentration dependent increase in visible lesions was observed until after the formation of the liver rudiment (day 4 of development). By exposing Japanese medaka embryos to lethal concentrations of TCDD beginning on different days of embryonic development, it was demonstrated that the sensitive period for toxicity was during liver formation on day 4 or 5 of development. The sensitive period for development was not caused by differences in TCDD absorption across the chorion. When embryos were exposed to [3H]TCDD prior to, during or after liver formation, there was no statistical difference in the dose of TCDD equivalents that crossed the chorion and entered the yolk and embryo.",
keywords = "Embryo, Medaka, Stage, TCDD, Toxicity",
author = "Wisk, {Joseph D.} and Keith Cooper",
doi = "10.1002/etc.5620090907",
journal = "Environmental Toxicology and Chemistry",
Wisk, JD & Cooper, K 1990, 'The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes)', Environmental Toxicology and Chemistry, vol. 9, no. 9, pp. 1159-1169. https://doi.org/10.1002/etc.5620090907
The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes). / Wisk, Joseph D.; Cooper, Keith.
In: Environmental Toxicology and Chemistry, Vol. 9, No. 9, 01.01.1990, p. 1159-1169.
T1 - The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes)
AU - Wisk, Joseph D.
AU - Cooper, Keith
N2 - Embryos of the Japanese medaka were individually exposed to varying concentrations of [3H]2,3,7,8‐tetrachlorodibenzo‐p‐dioxin (TCDD) in a static, nonrenewal system. The EC50 with 95% confidence interval (C.I.) to prevent hatching was 14 (11–17) nanograms (ng) of TCDD equivalents per liter (L) of water (parts per trillion). The LC50 with 95% C.I. for survival to 3 d posthatch was 9 (6–12) ng TCDD equivalents/L. The EC50 with 95% C.I. for embryos with minor lesions and severe, life‐threatening lesions were 3.5 (1.3–5.7) ng TCDD equivalents/L and 14 (12.4–15.6) ng TCDD equivalents/L, respectively. In a separate experiment that was terminated prior to the embryos hatching or dying, the EC50 for lesions was calculated to be 2.2 (1.4–3.0) ng TCDD equivalents/L. Based on the amount of TCDD equivalents recovered from dechorionated embryos, the ED50 with 95% C.I. for lesions was calculated to be 0.24 picograms of TCDD equivalents per milligram of dechorionated embryo weight (parts per billion). When the embryos were exposed to [3H]TCDD within 1 to 2 h after fertilization, no concentration dependent increase in visible lesions was observed until after the formation of the liver rudiment (day 4 of development). By exposing Japanese medaka embryos to lethal concentrations of TCDD beginning on different days of embryonic development, it was demonstrated that the sensitive period for toxicity was during liver formation on day 4 or 5 of development. The sensitive period for development was not caused by differences in TCDD absorption across the chorion. When embryos were exposed to [3H]TCDD prior to, during or after liver formation, there was no statistical difference in the dose of TCDD equivalents that crossed the chorion and entered the yolk and embryo.
AB - Embryos of the Japanese medaka were individually exposed to varying concentrations of [3H]2,3,7,8‐tetrachlorodibenzo‐p‐dioxin (TCDD) in a static, nonrenewal system. The EC50 with 95% confidence interval (C.I.) to prevent hatching was 14 (11–17) nanograms (ng) of TCDD equivalents per liter (L) of water (parts per trillion). The LC50 with 95% C.I. for survival to 3 d posthatch was 9 (6–12) ng TCDD equivalents/L. The EC50 with 95% C.I. for embryos with minor lesions and severe, life‐threatening lesions were 3.5 (1.3–5.7) ng TCDD equivalents/L and 14 (12.4–15.6) ng TCDD equivalents/L, respectively. In a separate experiment that was terminated prior to the embryos hatching or dying, the EC50 for lesions was calculated to be 2.2 (1.4–3.0) ng TCDD equivalents/L. Based on the amount of TCDD equivalents recovered from dechorionated embryos, the ED50 with 95% C.I. for lesions was calculated to be 0.24 picograms of TCDD equivalents per milligram of dechorionated embryo weight (parts per billion). When the embryos were exposed to [3H]TCDD within 1 to 2 h after fertilization, no concentration dependent increase in visible lesions was observed until after the formation of the liver rudiment (day 4 of development). By exposing Japanese medaka embryos to lethal concentrations of TCDD beginning on different days of embryonic development, it was demonstrated that the sensitive period for toxicity was during liver formation on day 4 or 5 of development. The sensitive period for development was not caused by differences in TCDD absorption across the chorion. When embryos were exposed to [3H]TCDD prior to, during or after liver formation, there was no statistical difference in the dose of TCDD equivalents that crossed the chorion and entered the yolk and embryo.
KW - Embryo
KW - Medaka
KW - Stage
KW - TCDD
KW - Toxicity
U2 - 10.1002/etc.5620090907
DO - 10.1002/etc.5620090907
JO - Environmental Toxicology and Chemistry
JF - Environmental Toxicology and Chemistry
Wisk JD, Cooper K. The stage specific toxicity of 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin in embryos of the japanese medaka (Oryzias latipes). Environmental Toxicology and Chemistry. 1990 Jan 1;9(9):1159-1169. https://doi.org/10.1002/etc.5620090907
10.1002/etc.5620090907
Rutgers, The State University of New Jersey contact form


text:
site: media | arena: nfl | pageType: stories | section: | slug: bears-announce-coach-matt-nagy-gm-ryan-pace-will-return-after-impressive-finish-to-2020 | sport: football | route: article_single.us | 6-keys: media/spln/nfl/reg/free/stories
Bears announce coach Matt Nagy, GM Ryan Pace will return after 'impressive' finish to 2020
Chairman George McCaskey lauded the duo's resiliency during the team's 8-8 season
By Cody Benjamin
Despite a second straight non-winning season, in which they started 5-1 before finishing 8-8 and failing to advance past their first playoff game for the second time in three years, the Chicago Bears are running it back in 2021. Chairman George McCaskey and president Ted Phillips announced Wednesday that coach Matt Nagy and general manager Ryan Pace will both return for another season, citing an "impressive" finish to the 2020 campaign -- the third under Nagy's guidance and sixth under Pace.
"Have mistakes been made? Yes," McCaskey said, per reporters. "But I think Ryan and Matt are learning and growing in their roles ... I understand your frustration. We're frustrated, too ... (but) sticking with these guys is the best route for the Bears at this time ... We all want things to get better, and a lot of times, you have to go through tough times to get better. The way they've handled adversity the last two seasons leads us to the conclusion that they're the best people to lead us through 2021."
Nagy opened his career as the Bears' head coach with high marks, leading Chicago to a 12-4 finish and NFC North title in 2018 before falling to the Eagles in the Wild Card Round. Since then, however, he's overseen two straight 8-8 seasons, not to mention the regression of former Pro Bowl quarterback Mitchell Trubisky, who was briefly benched early in 2020. The Bears finished strong amid a rebound by Trubisky, winning three of their last four to enter the playoffs, but still lost their opening postseason game 21-9.
Pace, meanwhile, has seemingly been on the hot seat for even longer. Hired as GM in 2015 after 14 years in the Saints' personnel department, he's overseen just one winning season in six years at his post. The Bears went 13-34 in his first three seasons when John Fox served as head coach, and infamously missed on some of the 2018 draft's top QBs, selecting Trubisky over lauded talents like Deshaun Watson and Patrick Mahomes.
Watch Now: Report: Chuck Pagano To Retire, Matt Nagy To Stay (2:57)
Bears retain Nagy, Pace for 2021 season


Who's in the lead? What's the latest bird to arrive? Where are you in the checklist rankings? Check out the latest results by visiting the links to Vermont eBird reports below and choosing the county of interest.
A bar chart displaying each county's results up to the minute!
How do you compare with the top birders in your county?


The font on Canvas, especially on quizzes, is often far too small. It would be a great feature to add to change the default font size. I have decent vision and oftentimes the quizzes are difficult to see even for me. I cannot imagine how stressful tests must be for people with poor vision.
An alternate option for you, for the time being, is to use the zoom feature on your browser. Have you tried that before? You can press the "Ctrl" (I think the Mac equivalent is "Cmd"?) key and either "+" or "-" on your keyboard, to zoom in/out. This should work on most web browsers. This would change the size for just you. Your students would also need to use the keyboard shortcuts if they wanted to zoom in/out, too.
I hope this information is helpful to you, Joe.


TURN CUPCAKES INTO EGG-SHAPES FOR EASTER! Why settle for decorating round cupcakes when you can kick it up a notch and give your Easter cupcakes an egg-shape, too! Here's a neat trick for turning ordinary round-top cupcakes into EGG-SHAPED EASTER CUPCAKES. In our last blog post we added a vanilla wafer to a cupcake to create an elongated shape for our SHREDDED RABBITS. Here we use the same trick to craft egg shapes for decorating to look like EASTER EGGS!
1. Spread some frosting on top of a cupcake, to one side. Place a vanilla wafer, flat side down, on the frosting, allowing it to overhang the edge by 3/4 inch. Pop the cupcake in the freezer for 15 minutes to set.
2. Add a mound of frosting to the chilled cupcake and spread into a smooth egg shape, using the frosting to fill any gaps and create a smooth transition between the wafer and the top of the cupcake.
3. Place several different colors of sugar each in a small bowl—make sure it is large enough to dip a cupcake. Hold the cupcake by the liner and dip one end in a colored sugar (try to dip up to a straight line). Turn the cupcake around and dip the opposite end in a second color, leaving a strip of frosting in the middle uncoated. Using a small spoon add a third color to the uncoated strip. Use a small brush to clean up any stray sugar. Pipe lines of frosting where the colors meet to create a clean line and hide imperfections.
4. Use dots of frosting to add additional candy decorations to the colored areas. Pipe dots and patterns as desired.
Our newest book Make It Easy, Cupcake! has over 100 SIMPLE 4-STEP DESIGNS that are just as easy and just as much fun as our EGG-SHAPED EASTER CUPCAKES!
HELLO CUPCAKE CLASSIC: HIGH-HEEL CUPCAKES!!!
ANOTHER QUICK LAST MINUTE HOLIDAY IDEA: OUR "ODE TO JOY" CAKE!!!
APRIL FOOL'S DAY—OUR FAVORITE HOLIDAY!


There are only 7 Dojo listed in Douglas County, Nevada. Jiu-Jitsu (Jujutsu/Jujitsu), Kempo/Kenpo, Tae Kwon Do, Kosho ryu Kenpo Jiu Jitsu & Kodenkan Danzan ryu Jujitsu are the top Martial Arts Styles offered in Douglas County.
Chart based on style information available from 7 Dojo in Douglas County.
Distance is showed from the centroid (center) of Douglas County, Nevada. Click on 'Map' next to a school to see it on the Map.
These are the top cities in Douglas County Nevada along with the number of Martial Arts Schools listed in them.


Simonson was an old-school journalist, mentoring young people into a business in which not enough talented young people are persisting.
"It is safe to say that over the past 25 years no one has spent more time telling the stories of northern Wisconsin than Mike Simonson and it isn't likely that anyone ever will again," Superior regional manager John Munson said.
One of his students was Aaron J. Brown, who writes about him today at his Minnesota Brown blog.
I worked for Mike at KUWS when I was in college. Mike was the somehow the toughest and kindest mentor I've ever had. Mike took 30-word sentences and made them 12-word bulldozers. He'd call you into the studio, ostensibly property of the radio station, but his de facto empire of reel-to-reel tape and scripts, each one always printed in Comic Sans because it was such a terrible font that it made you read slow and deliberate like Edward R. Murrow.
If you went into the studio with unanswered questions, excuses, sloppy copy or a hint of mush he'd mark up your script and scrawl the names of three more people to call to get the story right. You'd stay until it was done, because tomorrow's biology test didn't matter. The news mattered. Being right mattered. Mike taught a legion of students this way. They work across the country now.
See, Mike didn't yell at you. It's just that his talking sounded like yelling, and it took a few months to figure that out. By then, you were a functional cub reporter and you didn't turn in crap to Mike Simonson. You knew better than that.
For the longest time I never knew if Mike liked me. Then it was time for us to go to a conference in the Twin Cities. He drove the van full of budding reporters. I had shotgun. Some awkward moments, but then we got on the road and "Midnight Train to Georgia" came on the radio. I reached down to turn it up. He looked at me, and from that point forward he treated me as an equal. That. Precise. Moment.
Newer › When is a good age to die?
Dang, I was just listening to WPR on Saturday during a drive up to Hayward.
It sure makes one stop and think about the fragility of life when someone dies so suddenly and at such a young age.


I must confess the stuffing that comes with the roast chicken or turkey at Christmas has never been one of my favourites. I tend to pass on it when its offered around the table.
This new recipe is a flavoursome twist on an old recipe. Rather than cooking the stuffing in the roast, you cook it in a separate roasting dish. One of the advantages of this is that you get a lot more stuffing, so the folks who do like their Christmas stuffing will have plenty on their plate. The grated parmesan was enough for me to decide I will bake a tray of this stuffing for Christmas 2017.
Finely chop half the herbs. Heat butter and oil in a large frypan over medium heat. Add chopped herbs, onion, garlic, and pancetta, cook for 2-3 minutes until softened. Transfer to a large mixing bowl and toss with remaining ingredients. Set aside to soak for 10 minutes.
Preheat oven to 180C. Line a 20 x 30cm baking dish with baking paper. Add the mixture.
Bake for 1 hour until golden. It could go in the oven about the same time that you put the pork crackling in if you roast your pork crackling separately.
Sasha Cottman has kindly offered 3 e-book copies of A Scottish Duke for Christmas up as prizes to win in the AusRom Today Christmas Extravaganza.


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Tech Finance Strategy Retail Executive Lifestyle BI Prime Intelligence
Media 2020-01-21T20:30:00Z
LIVE: Netflix beat expectations in Q4 thanks to big-hit shows like 'The Witcher' — but it also showed signs of weakness
Netflix reported earnings for its fourth quarter on Tuesday after the markets closed.
Wall Street was mostly optimistic that Netflix would reassert its streaming dominance during the period, when new rivals like Disney Plus and Apple TV Plus launched and threatened Netflix's standing.
Netflix crushed its international subscriber-growth targets during the fourth quarter, but missed in the US where rivals like Disney Plus threatened its dominance.
Chinese scientists recommend quarantine of Wuhan to contain virus
Fearing Bolton testimony, Trump impeachment team draws emergency plans
Trump railed against exec for installing marble he chose himself
Science 2020-01-21T19:05:00Z
The deadly Wuhan virus has spread to the US — authorities have confirmed a case in Washington state
The coronavirus, which can be spread among humans, likely originated at a seafood market in Wuhan, China.
Politics 2020-01-21T20:34:00Z
Mitch McConnell pulled a 'Machiavellian' move to swing Trump's impeachment trial in his favor
Constitutional scholars told Insider McConnell's maneuver is "cynically Machiavellian" and a "hyper-partisan" manipulation of Senate norms.
BI Prime Markets 2020-01-21T19:47:00Z
A quantitative futurist explains how Amazon, Google, and Apple will 'completely dismantle' the healthcare industry — and says the wheels are already in motion
Amy Webb, a professor of strategic foresight at NYU, expects tech giants like Amazon and Apple to have a big influence on how we go to the doctor.
Tech 2020-01-21T19:15:12Z
Inside Slack's wild San Francisco headquarters, where each floor is designed to look like nature — from rocky deserts to soothing waterfalls
Kim Kardashian West and Nordstrom just announced a major deal
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Executive Lifestyle 2020-01-21T18:01:00Z
I rode Africa's first superfast bullet train that could cover the distance from New York to Washington, DC, in 90 minutes — and I understand why it's controversial
While the $2 billion train is impressive, it's hard not to think of public debate in Morocco that has left critics questioning its economic viability.
Finance 2020-01-21T19:38:19Z
Billionaires and heads of state have descended on a ski town in Switzerland. Here's what's happening in Davos, explained in 60 seconds.
If you've heard of "Davos" but never understood what it is or why it's important, here's what you need to know about the World Economic Forum.
BI Prime Tech 2020-01-21T19:45:20Z
eBay lays off at least 200 employees as restructuring continues under interim CEO
The layoffs are part of eBay's continued restructuring efforts as it struggles to gain share in the highly competitive e-commerce space.
Strategy 2020-01-21T20:40:00Z
The very first person counted in the 2020 US Census lives in a tiny Alaskan fishing town. Now, only a few hundred million left to go.
The decennial count of every person living in the US just started in the remote Alaskan village of Toksook Bay.
BI Prime Media 2020-01-21T18:11:51Z
A pair of 30-year-old video producers blew up on TikTok and are charging brands thousands of dollars for sponsored posts
'Signs of excess are building': 4 Wall Street giants explain why the stock market's rally may have gone too far — and share their advice for what happens next
BI Prime Careers 2020-01-21T16:28:00Z
The short question you should email a hiring manager who's rejected you for a job if you want to work at the same company in another role someday
Business Insider Intelligence
Personal Finance 2020-01-21T21:14:00Z
8 credit cards that get you elite hotel status just by having them in your wallet
You don't have to be a road warrior to earn hotel elite status. You can get it just for holding one of these credit cards.
All of the important primary, convention, and debate dates you need to know for the 2020 presidential election
Iowans will officially kick off the long-awaited 2020 presidential primaries in less than two weeks on February 3.
News 2020-01-21T21:10:16Z
From Clinton investigator to Trump impeachment defender: The life of controversial attorney Ken Starr
Kenneth Starr became a household name in the 1990s when he investigated then-President Bill Clinton in the Whitewater and Monica Lewinsky scandals.
THE US TELEHEALTH MARKET: The market, drivers, threats, and opportunities for incumbents and newcomers
In a new 2018 report, Business Insider Intelligence defines the opaque US telehealth market, forecasts the market growth potential and value, outlines the key drivers behind usage and adoption, and evaluates the opportunity telehealth solutions will afford all stakeholders.
This is the first time the Super Bowl is streaming in 4K resolution, but it won't be true 4K
Fox Sports is going to produce Super Bowl LIV in standard 1080p resolution, and it'll "upscale" the game to 4K resolution.
Markets 2020-01-21T20:51:08Z
Boeing slides to its lowest level in 13 months after pushing the 737 Max's ungrounding to mid-2020
Boeing's new return-to-service estimate "is informed by our experience to date with the certification process," according to a statement.
Military & Defense Contributors 2020-01-21T20:46:17Z
Officials say US Space Command passed its first combat test by detecting Iranian missiles
US Space Command's early-warning systems helped detect Iranian missile launches at US bases in Iraq this month, the new command said this week.
Google launched 3 new apps to help you put your phone down
Two of the apps promote awareness about how much time is spent on the phone, while the other prompts users to actually put their phone in an envelope.
Transportation 2020-01-21T20:33:00Z
The cargo ship that mysteriously capsized off the coast of Georgia in September with thousands of cars on board is still there
A cargo ship carrying around 4,000 Hyundai and Kia cars capsized off the coast of Georgia in September and is now only starting to be removed.
Warren Buffett tops list of America's biggest givers after donating nearly $15 billion in 5 years
The Berkshire Hathaway boss beat out Bill and Melinda Gates, George Soros, Michael Bloomberg, and the Walton family.
BI Prime Strategy 2020-01-21T17:58:36Z
Straight couples are more likely to divorce after a wife's promotion, but there are simple steps you can take to avoid that fate
Chances for divorce increase significantly when a woman is promoted. Relationship experts share how straight couples can avoid this trend.
Meryl Streep just sold her NYC penthouse at a 36% discount. Take a look inside the $15.8 million Tribeca condo, which spans an entire floor and comes with a wraparound terrace.
Meryl Streep was asking $18.25 million for the Manhattan penthouse she's owned for 14 years.
An adorable grocery robot is celebrating his first birthday with a party in hundreds of stores — and it shows a dystopian truth about the future of retail
Marty, an adorable clean-up assisting robot, was rolled out to 172 Giant Food Stores and 325 Stop and Shop locations beginning last January.
The 10 unhealthiest American cities to live in, where residents are stressed, worried about money, and unsatisfied with their health
The unhealthiest city in the US is Oklahoma City, Oklahoma, followed by Detroit, Mesa, Philadelphia, and San Antonio.
Trump and Conor McGregor congratulate each other on Twitter
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ClassPass is offering a one-month free trial for the New Year — here's how it works
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These are the 3 things business leaders must do to grow diversity in the workplace, according to a tech exec working to achieve gender parity by 2030
At the World Economic Forum, Dell's head of diversity Brian Reaves said it was a "business imperative" for tech companies to grow workforce diversity.
BI Prime Finance 2020-01-21T20:24:27Z
JPMorgan is hiring 20-30 people to its team that designs tech used by bankers and clients — the growing user-experience focus shows fintech's huge impact on Wall Street
Debra Herschmann, who leads JPMorgan CIB's user experience efforts, spoke to Business Insider about the importance of putting design first.
Sports 2020-01-21T20:24:13Z
The 2020 NFL Draft in Las Vegas is shaping up to be a spectacle unlike anything sports has seen, with draft picks being boated across the Bellagio fountains
Joe Burrow has a boat ride across the Bellagio fountains in his near future, as he's expected to be taken with the first overall pick by the Bengals.
56 practical and unique gifts for Dad — all under $50
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One astonishing statistic shows just how behind American millennials are when it comes to homeownership
Millennials own 4% of real estate value, compared to 32% of baby boomers at their age thanks to rising housing costs and lack of starter homes.
NASA received over 28,000 essays from kids who want to name the next Mars rover. The robot will search for signs of alien life and stash samples.
ThirdLove's nursing bra is the most comfortable one I own — their online sizing quiz and fit experts make it easy for new moms to shop from home
You can order a comfortable, supportive nursing bra from ThirdLove that actually fits, without ever leaving the comfort of your own home.
US threatens to slap tariffs on Italy and Britain in digital tax fight
The warning came a day after the US reached a truce over a similar tax that was passed in France last year.
I went to a trendy axe-throwing bar and saw why the activity is blowing up with urban millennials
Trendy axe-throwing bars are becoming all the rage in major cities around the nation, including New York, Philadelphia, and Washington, DC.
Hotel 50 Bowery brings approachable luxury to New York City's vibrant Chinatown
Hotel 50 Bowery from Hyatt's Joie de Vivre line is a sleek boutique hotel in Chinatown, and is a vibrant alternative to nearby SoHo or TriBeCa.
Here's what happens if Trump gets convicted by the Senate
If Trump is convicted, he would be removed effective immediately. The Senate would also vote on whether to bar him from holding office again.
This Mercedes-Benz Sprinter was turned into a tiny home on wheels using wood from a 300-year-old Redwood tree — see inside
Camper van converter Advanture Company created the Redwood King, a tiny home on wheels built on a 2015 Mercedes-Benz Sprinter.
A popular men's underwear company launched a women's line after numerous Facebook requests — and the styles are worth every penny
Tommy John continues to deliver with a line of women's underwear that doesn't bunch, roll, pill, or show lines. Here's our review.
Disney Plus now has gift subscriptions for Valentine's Day — here's how to buy one for the Disney fan in your life
Disney now offers a gift subscription for one year of Disney Plus. The gift code is conveniently emailed directly to your recipient.
We visited a Pier 1 store that's preparing to shutter and found its closing sale confusing and disorganized
We visited a closing Pier 1 store in Queens, New York, and saw uncompetitive discounts along with confusing layouts and signs.
China is behind on production of its most advanced fighter jet
China was thought to have built about 50 J-20s by the end of 2019, but problems with the J-20 stealth jets' engines have delayed production plans.
Last year, Southwest ran a blockbuster credit card offer that scored travelers 2-for-1 travel, but the airline's current deal could be even better
The sign-up bonus for up to 75,000 points gets you more than halfway toward the Companion Pass. If you qualify, you'll get it for more than a year.
Small Business 2020-01-21T19:55:27Z
Uniplaces founder has learned how not to communicate with employees as his company grows from 1 to 100
Uniplaces has learned that effective communication is key as it has grown to 100 employees. Uniplaces helps students find accommodations in Europe.
Dell itself recently partnered with 20 companies to form a new Women in Technology Consortium, aiming to reach gender parity in tech.
Uber surges 6% after offloading its Indian food-delivery business
"India remains an exceptionally important market to Uber and we will continue to invest in growing our local Rides business," said Uber's CEO.
AB InBev exec reveals how the company plans to dominate the seltzer market and what the brewing giant really thinks about White Claw
"It's more about the consumer than it is about the competition," AB InBev general counsel John Blood told Business Insider.
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This week's word for Tuesday Tales is necklace. This is still the reunion story I've been working on. Be sure to check out the other tales here.
After grabbing some chicken sandwiches to go at Chick-fil-A and eating them on the way across the bridge into Pensacola, Duncan drove to the venue of the dinner and dance. It was held at the Naval Air Station Mustin Beach Officers Club which was where the senior prom had been. Renee braced herself at the doors before entering. If things had panned out twenty years ago, she would have been at this very door with a corsage on her wrist and this man beside her in a tuxedo.
She let out a little sound.
"What is it?" Duncan looked down at her.
"I'm underdressed. Why didn't I take a minute to put on something better than this before we left the house?" She nervously ran the charm on her necklace along the chain.
"You shouldn't be so flip. Being a priest and all that." Renee couldn't help but laugh.
"Who cares? She has no idea what happened to your fancy dress and it's none of her business anyway." Duncan opened the door and allowed Renee to enter ahead of him.