\section{Introduction}

Soft-gamma repeaters (SGRs) are strongly-magnetized neutron stars that
produce frequent, short-duration bursts ($\hbox{${_{\displaystyle<}\atop^{\displaystyle\sim}}$} 1$ s) of $\hbox{${_{\displaystyle<}\atop^{\displaystyle\sim}}$} 10^{41}$
ergs in hard x-ray and soft gamma-rays. SGRs occasionally produce
giant flares that last $\sim 100$ s; the first giant flare to be
detected occurred in SGR 0526-66 on 5 March, 1979
\citep{barat_etal79,mazets_etal79,cline_etal80}, releasing
$\sim 2\times 10^{45}$ erg \citep{fenimore_etal96}. The August
27th 1998 giant flare from SGR 1900+14 liberated $\hbox{${_{\displaystyle>}\atop^{\displaystyle\sim}}$} 4\times
10^{43}$ erg, with a rise time of $<4$ ms
\citep{hurley_etal99,feroci_etal99}. The duration of the initial peak
was $\sim 1$ s \citep{hurley_etal99}. On December 27, 2004, SGR
1806-20 produced the largest flare yet recorded, with a total energy
yield of $\hbox{${_{\displaystyle>}\atop^{\displaystyle\sim}}$} 4\times 10^{46}$ ergs.\footnote{These energy estimates
assume isotropic emission.} In both short bursts and in giant flares,
the peak luminosity is reached in under 10 ms.  Measured spin down
parameters imply surface dipole fields of $6\times 10^{14}$ G for SGR
0526-66 \citep{tiengo_etal09}, $7\times 10^{14}$ G for SGR 1900+14
\citep{mereghetti_etal06}, and $2\times 10^{15}$ G for SGR 1806-20
\citep{nakagawa_etal08}, establishing these objects as magnetars.

The giant flares in SGR 1806-20 (hereafter SGR 1806) and SGR 1900+14
(hereafter SGR 1900) showed rotationally phase-dependent,
quasi-periodic oscillations (QPOs). QPOs in SGR 1806 were detected
at 18$\pm 2$ Hz, 26$\pm 3$ Hz, 30$\pm 4$ Hz, 93$\pm 2$ Hz, 150$\pm
17$ Hz, 626$\pm 2$ Hz, and 1837$\pm 5$ Hz
\citep{israel_etal05,ws06,sw06,hambaryan_etal11}.
QPOs in the giant flare of SGR 1900 were detected at 28$\pm 2$ Hz,
53 $\pm 5$ Hz, 84 Hz (width unmeasured), and 155$\pm 6$ Hz
\citep{sw05}.
Recently, oscillations at 57$\pm 5$ Hz were identified in the
short bursts of SGR 1806 \citep{huppenkothen_etal14a}, and
at 93$\pm 12$ Hz, 127$\pm 10$ Hz, and possibly 260 Hz in SGR J1550-5418
\citep{huppenkothen_etal14b}.\footnote{\citet{elib10} reported
evidence for oscillations in the
short, recurring bursts of SGR 1806, but this analysis was shown by
\citet{huppenkothen_etal13} to be flawed.}
To summarize, SGRs 1806 and 1900 have QPOs
that begin at about 20 Hz, with a spacing of some tens of
Hz below 160 Hz, and that are sharp with typical widths of 2-4
Hz.

The observed QPOs are generally attributed to oscillations of the star
excited by an explosion of magnetic origin that creates the flare. The
oscillating stellar surface should modulate the charge density in the
magnetosphere, creating variations in the optical depth for resonant
Compton scattering of the hard x-rays that accompany the flare
\citep{timokhin_etal08,dw12}.  In this connection, the problem of
finding the oscillatory modes for a strongly-magnetized neutron star
has received much attention, and has proven to be a formidable
problem. To make the problem tractable, most theoretical treatments of
the QPO problem have assumed smooth field geometries, usually dipolar
or variants
({\it e.g.}, \citealt{levin06,gsa06,levin07,sotani_etal08a,sotani_etal08b,cerda_etal09,cbk09,csf09,ck11,vl11,gabler_etal11,ck11,gabler_etal12,vl12,pl13,gabler_etal13,gabler_etal_sf13,gabler_etal14}).
Smooth field geometries support a problematic Alfv\'en continuum that
couples to the discrete natural spectrum of the crust. As
pointed out by \citet{levin06}, if energy is deposited in the crust at
one of the natural frequencies of the crust, and this frequency lies
within a portion of the core continuum, the energy is lost to the core
continuum in less than 0.1 s as the entire core continuum is
excited. The crust excitation is effectively damped through {\em
resonant absorption}, a familiar process in MHD; see
{\it e.g.}, \citet{gp04}. The problem has been addressed by assuming field
geometries with gaps in the Alfv\'en continuum. Under this assumption,
long-lived quasi-normal modes can exist inside the gaps or near the
edges of the Alfv\'en continuum. \cite{vl11} showed for a ``box''
neutron star that introduction of a magnetic tangle breaks the
Alfv\'en continuum. \cite{lv15} showed that for magnetic tangling in a
spherical neutron star the problematic Alfv\'en continuum disappears.\footnote{\cite{sotani15}
added general relativity in the treatment of the
magnetic field, and
confirmed some of the results of \cite{lv15}.}
They found that the star acquires discrete normal modes, and
quantified the mode spacing. It is clear from these investigations
that the unknown magnetic field geometry is the most important
ingredient in determining the oscillation spectrum of a magnetar.

As no model presented so far has provided good quantitative agreement
with observed QPOs, we take a new direction in this Letter. We begin
by arguing that stability considerations and observational evidence
show that magnetars do not possess the smooth fields considered in
most previous work, but rather have highly tangled fields. We propose
that magnetar QPOs represent torsional normal modes that are supported
by the magnetic tangle, and we present a simple model that supports
this hypothesis. Keeping the energy in the magnetic tangle as a free
parameter, we adjust this parameter to accommodate the 28 Hz QPO
observed in SGR 1900 while maintaining consistency with QPOs observed
at higher frequencies. We obtain a rough {\em measurement} of the
energy density in the tangled field to be $\sim 14$ times that in the
dipole field. Our model, though simple, is the first to give
reasonable quantitative agreement with the data. Our model also
provides useful scaling relations for the frequencies of the QPOs on
bulk neutron star parameters and provides insight into the problem
that might not emerge so clearly from more detailed numerical
simulations. In particular, the model shows that if strong field
tangling occurs, the normal-mode spectrum of a magnetar is determined
principally by field tangling, and less so by crust rigidity, the
dipole field, relativistic effects, and detailed stellar structure.
We conclude that the effects of a tangled field cannot be neglected in
the QPO problem, and we outline what we see to be interesting research
directions on this issue.

\section{Theoretical and observational evidence for field tangling}

A pure dipole field is unstable, and a strong toroidal field is needed
to stabilize the field
\citep{fr77,bs06}. Purely toroidal fields are also unstable
\citep{wright73,tayler73}.
There has been
considerable progress recently on the identification of magnetic
equilibria. \citet{bn06} found a ``twisted torus'' configuration,
which consists of torus of flux near the magnetic equator that
stabilizes the linked poloidal plus toroidal configuration. The
twisted torus is topologically distinct from any poloidal field, or
twisted poloidal field, in the sense that the twisted torus cannot be
continuously deformed into a dipole field - the field is {\em
tangled}. This topological complexity is required to
establish hydromagnetic stability.
Simulations by \citet{braithwaite08} show
that the evolution of the magnetic field from initially-turbulent
configurations can evolve to configurations other than the
twisted torus, generally non-axisymmetric equilibria
with highly tangled fields; see, {\it e.g.}, Fig. 12 of that
paper. \citet{braithwaite09} studied the relative strengths of the
poloidal and toroidal components in stable, axi-symmetric
configurations, and found that the energy in the toroidal
component typically exceeds that in the poloidal component.
By what factor the toroidal energy exceeds the poloidal energy
in an actual neutron star
depends on initial conditions and the equation of state;
\cite{braithwaite09} finds examples in which this ratio is 10-20, and he
argues that this ratio could plausibly be $\sim 10^3$ since a
proto-magnetar should be in a highly turbulent state that winds up the
natal field \citep{td93,brandenburg_etal05}. In this process, energy
injected at some scale propagates down to the dissipative scale as
well as up to large scales, giving a large-scale mean field with
complicated structure at many scales.

The chief conclusion of these theoretical studies is that a
topologically distinct tangle is needed to stabilize the dipolar
component. Most theoretical work on QPOs has assumed simple field geometries
that are demonstrably unstable.

Observational evidence that the internal fields of neutron stars
are highly tangled can be found in the `low-field SGRs'. In these
objects, the interior fields must be stronger than the inferred dipole
fields in order to power observed burst activity. SGR 0418+5729 has a
dipole field inferred from spin-down of $\sim 6\times 10^{12}$ G,
\citep{esposito_etal10,ret10,horst_etal10,turolla_etal11}. Two other
examples are Swift J1822-1606, with an inferred dipole field of $\sim
3\times 10^{13}$ G \citep{rea_etal12}, and 3XMM J1852+0033
\citep{rvi14}, with an inferred dipole field of less than $4\times
10^{13}$ G.  Energetics indicate that the interior field consists of
strong multipolar components, while stability considerations require
these components to be tangled \citep{braithwaite08,braithwaite09}.

\section{A simple model of QPOs}

\label{model}

Much of the work cited above on the QPO problem has included realistic
stellar structure, specific magnetic field geometries, and the effects
of general relativity.  The normal mode frequencies are determined
principally by the strength of the tangled field and bulk stellar
properties, with realistic stellar structure and general relativity
coming in as secondary effects. In \S 5 of the supplementary
materials \citep{lv16s}, we show that realistic
structure has a relatively small effect on the normal mode spectrum
for an isotropic tangle. Our approach, therefore, is to proceed with a
very simple model that elucidates the consequences of a tangled
field. We do not expect refinements of the simple model given here to
alter our chief conclusions.

We treat the magnetic field as consisting of a smooth dipolar
contribution $\Bbf_d$, plus a tangled component $\Bbf_t$ that
stabilizes the field. At location $\rbf$, the field is
\begin{equation}
\Bbf(\rbf)=\Bbf_d(\rbf)+\Bbf_t(\rbf).
\end{equation}
We assume that $\Bbf_t(\rbf)$ averages to nearly zero over a dimension
of order the stellar radius or smaller, that is,
$\ev{\Bbf_t(\rbf)}\simeq 0$ where $\ev{...}$ denotes a volume
average; see
the supplementary materials \citep{lv16s} for details.
The magnetic energy density in the tangle is
$\ev{B_t^2}/8\pi$.  We define a dimensionless measure of the strength
of the tangle as the ratio of the energy density in the tangle to that
in the dipolar component:
\begin{equation}
b_t^2\equiv \frac{\ev{B_t^2}}{B_d^2}.
\end{equation}
We regard
the magnetic tangle as approximately isotropic, and the dominant
source of magnetic stress, so that $b^2_t>>1$, as supported by the
simulations and arguments of \cite{braithwaite09}. In this limit, the
fluid core acquires an effective shear modulus given by:
\begin{equation}
\mu_B\equiv \frac{\ev{B_t^2}}{4\pi};
\end{equation}
We limit the analysis to low-frequency QPOs, as the approximation of
an isotropic tangle could break down at high frequencies.

Using realistic structure calculations (see \S \ref{crust}), the volume
averaged-shear modulus of the crust is $\bar{\mu}_c\simeq 3\times
10^{29}$ \hbox{\rm\hskip.35em  erg cm}$^{-3}$. The magnetic rigidity of the tangle will dominate
the material rigidity of the crust when
\begin{equation}
\frac{\bar{\mu}_c}{\mu_B}<<1 \longrightarrow
b_t^2 >> \left(\frac{\bar{\mu}_c}{3\times 10^{29}\mbox{ \hbox{\rm\hskip.35em  g cm}$^{-3}$}}\right)
\left(\frac{B_d}{10^{15}\mbox{ G}}\right)^{-2},
\label{crust_criterion}
\end{equation}
and so the crust is dynamically negligible in the same limit
(coincidentally) that the isotropic tangle becomes the dominant form
of magnetic stress for a typical magnetar dipole field of $10^{15}$ G;
we quantify the small effect of the crust in \S \ref{crust}.  We
ignore the crust in our simple model, and treat the star as a
self-gravitating, constant-density, magnetized fluid whose torsional
normal modes are determined by the isotropic stresses of the tangled
field. General relativity is included as a redshift factor that
reduces the oscillation frequencies observed at infinity by about
20\%. Electrical resistivity is negligible for the modes of interest,
and we work with ideal MHD.  Our normal-mode analysis is restricted
entirely to toroidal modes. The equations of motion are derived
using a mean-field formalism in \S 1 of the supplementary materials
\citep{lv16s}; the equation of motion for small displacements of the
fluid $\ubf$ is
\begin{equation}
c_t^2\nabla^2\ubf+\omega^2\ubf=0,
\label{waveeqn}
\end{equation}
where $c_t\equiv(\ev{B_t^2}/4\pi x_p\rho)^{1/2}$ is the Alfve\'n wave
speed through the tangle, $\rho$ is the mass density, $x_p\simeq 0.1$
is the proton mass fraction fraction, and $\omega$ is an
eigenfrequency. In evaluating $c_t$, we have assumed that the neutrons
are superfluid. If the protons are normal, the neutrons do not scatter
with the protons (ignoring scattering processes with the vortices of
the neutron superfluid).
If the protons are superconducting, the neutron
fluid is entrained by the proton fluid, but this effect is
negligible \citep{ch06}. In either case, the dynamical mass density is
essentially $x_p\rho$.

At this point we have reduced the normal mode problem to that of an
elastic sphere of constant density and rigidity.
Torsional modes have the form $\ubf = u_\phi(r,\theta) \hat{\phi}$
in spherical coordinates $(r,\theta,\phi)$ with the origin at the
center of the star.
The solutions to eq. (\ref{waveeqn}) are (see \S 3 of the
supplementary materials \citep{lv16s} for
further details):
\begin{equation}
u_\phi(r)=A j_l(kr)\,\frac{dP_l(\theta)}{d\theta},
\label{sol}
\end{equation}
where $k\equiv\omega/c_t$ and
$A$ is normalization. The eigenfunctions and associated
eigenfrequencies are determined by the boundary condition that the
traction vanish at the stellar surface:
\begin{equation}
\left[\frac{dj_l}{dr}-\frac{j_l}{r}\right]_{r=R}=0.
\label{bc_isotropic}
\end{equation}

For each value of $l$, eq. (\ref{bc_isotropic}) has solutions
$x_{ln}\equiv k_{ln} R$, where $n=0,1,2...$, the overtone number,
gives the number of nodes in $j_l(kr)$. The eigenfrequencies are
\begin{equation}
\omega_{ln}=z
\left(\frac{\ev{B_t^2}R}{3x_pM}\right)^{1/2} x_{ln},
\end{equation}
where a redshift factor $z\equiv(1-R_s(M)/R)^{1/2}$ has been
introduced; $R_s$ is the Schwarzchild radius.\footnote{For $l=1$,
eq. (\ref{waveeqn}) has a
solution $u_\phi =A\, r\, dP_1/d\theta= A\,r$ for $\omega=k=0$.  This solution,
which we label $n=0$, corresponds to rigid-body rotation, and so is of
no physical significance to the mode problem we are addressing. All
other modes have zero angular momentum.} In terms of fiducial
values
\begin{equation}
\nu_{ln}(\mbox{Hz})=\frac{\omega_{ln}}{2\pi}=
4.3\, \left(\frac{z}{0.77}\right)
\left(\frac{R}{10\mbox{ km}}\right)^{1/2}
\left(\frac{M}{1.4M_\odot}\right)^{-1/2}
\left(\frac{x_p}{0.1}\right)^{-1/2}
\left(\frac{\ev{B_t^2}^{1/2}}{10^{15}\mbox{ G}}\right)
\, x_{ln}\mbox{ Hz}.
\label{nu_isotropic}
\end{equation}

The energy in a torsional mode $(l,n)$ is
\begin{equation}
E_{ln}=\frac{1}{2} \int d^3r\, x_p \rho\, \omega^2 (u_\phi^{ln})^2.
\label{E}
\end{equation}
For the purpose of comparing the energies
of different normal modes of the same amplitude, we choose the
normalization $A$ in eq. (\ref{sol}) so that
\begin{equation}
\bar{u}^2=\int_{4\pi}d\Omega\, u_\phi(R)^2,
\label{norm}
\end{equation}
where $\bar{u}^2$ is the square of the mode amplitude averaged over
the stellar surface. We evaluate the energy in a mode $(l,n)$,
normalized by the $l=2$, $n=0$ mode energy, with $\bar{u}^2$ set equal
for both modes.

In the second row of Table \ref{nu_SGR1900_values}, we give example
frequencies for the fundamental of each $l$ for $M=1.4M_\odot$, $R=10$
km, and $x_p=0.1$. Numbers in parenthesis give the normalized energy
in the mode. By tuning the parameter
$b_t^2=\ev{B_t^2}/B_d^2$ to 14.4, the spectrum of the fundamentals for
each $l$ agrees with the QPOs seen in SGR 1900. A more accurate
analysis given in \S 4 of the supplementary materials \citep{lv16s}
changes $b_t^2$ slightly to 13.3.  Taking $B_d$ equal to
the inferred dipole field for SGR 1900 of $7\times 10^{14}$ G, this
value of $b_t^2$ corresponds to $\ev{B_t^2}^{1/2}=2.7\times 10^{15}$
G. We also show some of the eigenfrequencies for the first three
overtones. The overtones begin at higher frequencies; they also
require more energy to excite to the same root-mean square amplitude
(eq. \ref{norm}) than the fundamental modes and so are energetically
suppressed. To the extent that the surface amplitude of a mode
determines its observability through variations of magnetospheric
emission, the overtones might not be as important as the fundamental
modes. The amplitude of a given mode depends on the excitation
process, and we note that overtones could prove relevant in a more
detailed treatment that addresses the initial-value problem of mode
excitation; we discuss this issue further below.

The fundamental modes are
nearly evenly spaced in $l$, with frequencies given by
\begin{equation}
\nu_l(\mbox{Hz})\simeq 0.5\, l\, \nu_2,
\label{sequence}
\end{equation}
where $\nu_2$ is the frequency of the $l=2$ fundamental. The mode
spacing is about half of $\nu_2$. From eq. (\ref{nu_isotropic}), the
lowest-frequency mode and the mode spacing both scale
as
$z(M/R)^{-1/2}$.  The
observed QPO spacing follows eq. (\ref{sequence}), though only four of
the 12 modes in the range $2\le l\le 12$ are seen; we discuss this
point further below.

The highest fundamental frequency given in Table
\ref{nu_SGR1900_values} is 156 Hz for $l=12$. The wavelength of this
mode is $\simeq 0.5R$. Hence, for modes in the 28 to 156 Hz range, the
approximation of an isotropic tangle is required to hold over stellar
dimensions, as supported by the simulations of \cite{braithwaite09}.

This model of a tangle that dominates the dipole field does not
apply to SGR 1806. If we attempt to
explain the lowest-frequency QPO observed (18 Hz) as an $l=2$
fundamental mode for the inferred dipole field strength of $2\times
10^{15}$, eq. (\ref{nu_isotropic}) gives $b_t^2\simeq 0.5$, which is
inconsistent with the approximation of a strong, nearly isotropic
tangle. For this case, the magnetic stress of the smooth field must be
included. This problem is solved in \S 4 in the supplementary
materials \citep{lv16s}. A match to the 18 Hz QPO implies
$b_t^2=0.17$. The spectrum is very dense with a spacing of about two
Hz. If the dipole field has been overestimated by a factor several for
this object, not implausible, then the predicted spectrum is much less
dense, and more similar to SGR 1900. For example, taking $B_d$ equal
to 0.62 of the reported value, a value of $b_t^2=1.0$ matches the 18
Hz QPO, and the predicted spectrum is less dense, with a spacing of
about 7 Hz in the fundamentals.

\section{Effects of the crust are negligible}

\label{crust}

So far we have ignored the crust under the assumption that magnetic
stresses throughout the star dominate material stresses in the
crust. Here we show that crust rigidity increases the
eigenfrequencies calculated above for SGR 1900 by 3\% or less. A more
detailed treatment is given in \S 4 of the supplementary materials
\citep{lv16s}.

To estimate the effects of the crust, we use a two-zone crust plus
core model, assuming a nearly isotropic tangle ($b_t^2>>1$). The core
has constant density $\rho$ and constant effective shear modulus
$\mu_B$. The crust has inner radius $R_c$, outer radius $R$, thickness
$\Delta R$, average density $\bar{\rho}_c$, average material shear
modulus $\bar{\mu}_c$, and average effective shear modulus
$\bar{\mu}_{\rm crust}=\bar{\mu}_c+\mu_B$. \citet{chamel05,chamel12}
finds that the neutron fluid is largely entrained by ions in the inner
crust; in evaluating the wave speed in the crust, we use the total
mass density, so that the wave propagation speed in the crust is
$\bar{c}_{\rm crust}=\sqrt{\bar{\mu}_{\rm crust}/\bar{\rho}_c}$. In
the core the propagation speed is $c_t=\sqrt{\mu_B/x_p\rho}$, where
we take the proton mass fraction to be $x_p=0.1$ (see discussion after
equation \ref{waveeqn}).

In the core, the solution to the mode
problem is $u_{\rm core}=j_l(kr)$. The solution in the crust is
$u_{\rm crust}=aj_l(k^\prime r)+bn_l(k^\prime r)$, where $n_l$ are spherical
Neumann functions, and $a$ and $b$ are constants. The wavenumbers are
related through $\omega=c_tk=\bar{c}_{\rm crust}k^\prime$.
The boundary conditions are continuity in value and
traction at $r=R_c$, and vanishing traction at $r=R$:
\begin{equation}
u_{\rm core}(R_c)=u_{\rm crust}(R_c),
\end{equation}
\begin{equation}
\mu_B\left[\frac{du_{\rm core}}{dr}-\frac{u_{\rm core}}{r}
\right]_{r=R_c}=
\bar{\mu}_{\rm crust}\left[\frac{du_{\rm crust}}{dr}-\frac{u_{\rm crust}}{r}
\right]_{r=R_c},
\end{equation}
\begin{equation}
\left[\frac{du_{\rm crust}}{dr}-\frac{u_{\rm crust}}{r}
\right]_{r=R}=0.
\end{equation}
For $\bar{c}_{\rm crust}$ and $\bar{\mu}_{\rm crust}$, we use
volume-averaged quantities obtained in the following way. The shear
modulus in the crust as a function of density, ignoring magnetic
effects, is in cgs units
\citep{strohmayer_etal91}
\begin{equation}
\mu_c=0.1194\,\frac{n_i(Ze)^2}{a},
\end{equation}
where $n_i$ is the number density of ions of charge $Ze$, and $a$ is the
Wigner-Seitz cell radius given by $n_i4\pi a^3/3=1$.
For the composition of the inner crust, we use the results
of
\citet{dh01}, conveniently expressed analytically
by \citet{hp04}.
We solve for crust structure using the Newtonian equation for
hydrostatic equilibrium for a neutron star of 1.4 $M_\odot$. The
volume-averaged density and shear modulus in the crust are
$\bar{\rho}_c=0.06\rho$ and $\bar{\mu}_c=2.5\times 10^{29}$ \hbox{\rm\hskip.35em  erg cm}$^{-3}$,
respectively. The crust thickness is $\Delta
R=0.1R$. We take $\ev{B_t^2}^{1/2}=2.7\times 10^{15}$ G assumed
above for SGR 1900, corresponding to $\mu_B=5.8\times 10^{29}$
\hbox{\rm\hskip.35em  erg cm}$^{-3}$.

To evaluate the effects of the crust, we solve the two-zone model for
a particular eigenmode first taking $\bar{\mu}_{\rm
crust}=\bar{\mu}_c+\mu_B$, then taking $\bar{\mu}_{\rm crust}=\mu_B$,
and evaluating the difference between the two eigenfrequencies. We
find that the finite shear modulus of the crust increases the
frequencies of the fundamental normal modes by only about 3\% for
$l=2,3,4$, and 2\% for $l=12$. We have confirmed that the effects are
even smaller for overtones. The crust is nearly dynamically irrelevant
in this limit of a strong magnetic tangle, and neglect of the crust is
a good approximation.

\section{Discussion and Conclusions}

\label{conclusions}

Theoretical study of magnetar QPOs is seriously hampered by the fact
that we do not know the detailed magnetic structure within a
magnetar. Even if the magnetic structure were known or is
assumed, solution of the full problem is very difficult. Motivated by
stability considerations and observational evidence, we have argued
that the magnetic field should be highly tangled; {\em significant tangling
of the magnetic field is needed to stabilize the linked poloidal and
toroidal fields} \citep{bn06,braithwaite08}.  We have shown with a
simple model that a highly-tangled field, under the assumption that
the tangle is approximately isotropic, supports normal modes with
frequencies consistent with the QPOs observed in SGR 1900. In
comparison with data, we have obtained a rough measurement of the
ratio of the energy density in the tangle to that in the dipolar field
of $\sim 14$. Given the approximations we have made, this model should
not be taken as quantitatively very accurate, but what is significant
is that the assumption of a strong magnetic tangle leads naturally to a
normal mode spectrum with frequencies that lie in the range of
observed QPOs. Our model predicts about three times as many modes
below 160 Hz than are observed. That the normal mode spectrum is more
dense than the observed QPO spectrum is crucial, for if the
opposite were true, we would be forced to abandon the model as
unviable.

The main point that we would like to emphasize is that field tangling has
important effects that cannot be ignored in the QPO problem, and that
the magnetic tangle is likely to be the principal factor that
determines the normal-mode spectrum. Use of dipolar magnetic
geometries and their variants is not adequate, and we point out that
such magnetic geometries are unstable and therefore unphysical starting points
for study of normal modes of neutron stars. One simplifying feature of
magnetic tangling is that for a tangle of magnetar-scale strength,
the crust becomes dynamically unimportant.

We conclude by briefly describing several directions of future
research that we consider to be interesting and important. One issue
is the prediction that the normal-mode spectrum is denser than the
observed QPO spectrum.  In general, any excitation mechanism will give
preferential mode excitation. Determination of which modes are excited
requires solution of the initial-value problem.  Also, it is unknown
at present if the QPO emission is beamed or not. If the emission has a
beaming fraction less than unity, only that fraction of modes would be
potentially observable. We are currently studying the mode excitation
problem; we find that excitation in a localized region of the star can
lead to excitation of separate groups of modes.

Further development of the model into a quantitative tool will require
inclusion of realistic stellar structure and treatment of the case of
comparable energy densities in the magnetic tangle and the dipolar
component. For SGR 1806, the energy densities in the
tangle and the dipolar component appear to be comparable within our
interpretation; see \S 4 of the supplementary materials
\citep{lv16s}. In this case, the spectrum of normal modes could be very
dense, with a mode spacing of about 2-7 Hz.
If the spectrum is so dense that it begins to
assume the qualities of a quasi-continuum, resonant absorption might
be important as has been considered in previous work with smooth field
geometries. We will study this interesting problem in future work.
The magnetic field that occurs in nature may not be a
nearly isotropic tangle as we have assumed. It would be interesting to
explore how spatial variations in the magnetic tangle affect the
spectrum of normal modes.

\begin{acknowledgments}

We thank M. Gabler, D. Huppenkothen, Y. Levin, and A. Watts for very
helpful discussions, and the anonymous referee for useful criticism. This
work was supported by NSF Award AST-1211391 and NASA Award NNX12AF88G.

\end{acknowledgments}


\section{\label{SectionJammed}What is {}``jammed''?}

OSLN question whether the hard-sphere system is ``physical"
and therefore resort to studying particle systems
with soft-sphere interactions to mimic hard-particle packings.
The latter is inherently a geometrical problem.
In fact, there is a simple and rigorous geometrical approach
to jamming in hard-sphere systems that is not only well-defined, but, as we show
below, is closely related to OSLN's jamming point $J$.

Although the hard-sphere potential is an idealization, it is no less
physical than any soft-sphere potential, especially in regards to jamming.
Indeed, the singular nature of the hard-sphere potential is crucial because it
enables one to be precise about the concept of ``jamming.''
Recently, three hierarchically ordered jamming
categories have been introduced \cite{Torquato_jammed}: \emph{local},
\emph{collective} and \emph{strict} \emph{jamming}.
Each successive category progressively relaxes the boundary conditions
imposed on the particle displacements. These definitions are very
intuitive and completely \emph{geometric}, and are closely
linked to definitions of {}``rigid'' or {}``stable'' packings appearing
in the mathematics literature \cite{Connelly_disks,Connelly_packings}.

OSLN's definition of jamming simply states that
the configuration of particles is at a stable (strict) energy minimum.
Such a definition is dependent on the particular interparticle potential,
and thus it obscures the relevant packing geometry (exclusion-volume effects).
Furthermore, the distinctions between
different jamming categories is critical, especially if one is trying
to determine the density of the MRJ state \cite{Jamming_LP_results}.
Specifically, this density will generally be higher the more demanding is the
jamming category. Clearly, OSLN do not distinguish between different degrees
or levels of jamming.  We have recently demonstrated that
the distinction between collective and strict jamming is important even for
very large packings, especially in two dimensions \cite{Jamming_LP_results} .

For OSLN, a jammed configuration is one where there are no
zero-frequency modes of the Hessian matrix of the total potential energy
with respect to the positions of the particles (the dynamical
matrix), while keeping the periodic unit cell \emph{fixed}. Our definition of
strict jamming relaxes this requirement and includes the lattice vectors
as additional degrees of freedom \cite{Jamming_LP}. As explained
in detail in Ref. \cite{Connelly_Tensegrities}, the Hessian consists
of two parts, a negative definite \emph{stress matrix} and a positive
semidefinite \emph{stiffness matrix}. OSLN's
definition of jammed means simply that the Hessian
is positive definite at the energy minimum. A precise phrase for this
is \emph{a stable} or \emph{strict (local) energy minimum}, and we
see no point in redefining this elementary concept. In fact,
according to OSLN, any stable energy minimum represents a jammed configuration,
and it is not possible to relate this idea to \emph{packing} concepts
without numerous additional assumptions about the form of the pair potential
and the interparticle distances at the energy minimum.

Although OSLN point out themselves that our definitions
of jamming and MRJ are for \emph{hard-sphere packings}, they claim
to replace them with a {}``cleaner definition,'' which applies \emph{only}
to systems of \emph{soft spheres}. The two definitions
cannot directly be compared as they apply to different systems.  However,
OSLN themselves clearly imply that their {}``jammed'' soft-sphere
systems and {}``jammed'' hard-sphere packings are related, by referring
to other works on hard-sphere systems. For example, they claim a direct
relation between their special point $J$ and RCP of hard spheres
in Section IID of Ref. \cite{Jamming_ZeroT}.
The basic idea, as OSLN explain, is that one {}``can approach the
hard sphere by making the potential harder and harder...{[}to{]} produce
a limiting hard sphere value''. However, they question whether the
hard-sphere limit is well-defined and {}``would argue that hard spheres
are a singular limit and thus unphysical'' and that
{}``One should therefore concentrate
on softer potentials for which unambiguous definitions can be constructed.''

To demonstrate that the limit is well-defined, let us first define
a \emph{collectively jammed sphere packing} to be \emph{any nonoverlapping
configuration of hard spheres in which no subset of particles can
continuously be displaced so that its members move out of contact
with one another and with the remainder set}
(while maintaining nonoverlap) \cite{Torquato_jammed}.
The following theorem \cite{Connelly_Tensegrities} shows
that near the {}``jamming threshold'' $\phi_{c}$, as defined in
Section IIB in Ref. \cite{Jamming_ZeroT}, the jamming of particle
systems as defined by OSLN is directly related to this definition of
collective jamming in hard-sphere packings:

\noindent\textbf{Theorem:} {\sl Consider an interparticle potential that
is continuous and strictly monotonically decreasing
around $r_{ij}=D$, and vanishes for $r_{ij}>D+\delta$. If in a finite
configuration of particles interacting with such a potential, all
interacting (i.e., closer then $D+\delta$) particles are a distance
$D$ apart, and the configuration is a stable local energy minimum,
then the configuration corresponds to a collectively jammed packing
of hard spheres with diameter $D$.}

If one relaxes the condition that all interacting particles are exactly
at distance $D$ apart and instead asks only that the minimum interparticle
distance be $D$, then for a sufficiently small $\delta$ one can
prove \cite{Connelly_Energy} that the above sphere packing is almost
collectively jammed (i.e., it is trapped in a small neighborhood of
the initial configuration \cite{Jamming_LP}). This theorem implies
that the packings studied in Ref. \cite{Jamming_ZeroT} that are very
slightly above the {}``jamming threshold'' $\phi_{c}$ are indeed
closely related to collectively jammed ideal packings of spheres of
diameter $D=\sigma$ (polydispersity is trivial to incorporate).
All of these considerations call into question the value of a
definition of jamming that hinges on eigenvalues of dynamical matrices.

Finally, it is important to note that despite the fact that our definition of
collective jamming above calls for virtual displacing (groups of) particles, one can
in fact rigorously test for our hard-particle
jamming categories using linear programming
\cite{Jamming_LP}, without what OSLN call {}``shifting particles,''
even for very large disordered packings \cite{Jamming_LP_results}.
We have in fact communicated to OSLN the results \cite{footnote_2}
of our algorithm applied to several sample packings provided by them. In short, our
algorithm verified that  OSLN's systems near $\phi_{c}$ were indeed
nearly collectively jammed (within a very small tolerance) when viewed as
packings. However, they were not strictly jammed because OSLN keep
the lattice vectors fixed during energy minimization.

\section{\label{SectionRandom}What is {}``random''?}

We agree that the maximum of an appropriate ``entropic" metric
would be a potentially useful way to characterize the randomness of a packing
and therefore the MRJ state \cite{Anu_order_metrics}.
However, as pointed out in Ref. \cite{Anu_order_metrics},
a substantial hurdle to be overcome
is the necessity to generate all possible jammed states,
or at least a representative sample of such states,
in an unbiased fashion using a ``universal''
protocol in the large-system limit.
Even if such a protocol could be developed, however, the issue of what weights to assign
the resulting configurations remains.
Moreover, there are other fundamental problems with entropic
measures, as we discuss below, including its significance
for two-dimensional monodisperse hard disk packings.

According to OSLN, maximally random is defined by {}``where the entropy
of initial states is maximum'' and imply that this is a universal
measure of disorder. It is not clear exactly what the
authors mean by entropy and how (or whether) it can be measured for
{}``initial states''. It is not obvious that one
can  relate the {}``randomness'' of the \emph{final} configurations
(which is what OSLN are analyzing) to that of the \emph{initial} configurations.
It appears OSLN's rationale is that their algorithm goes to the {}``nearest'' energy
minimum from a given initial configuration.
Does this process preserve {}``entropy'' or randomness? Clearly
if one used, for example, global energy minimization, one would obtain
very different results. Furthermore, entropy is a concept inherently
related to \emph{distributions} of configurations. However, one classifies
\emph{particular} final configurations (packings) as random or disordered,
and by considering a given configuration, one can devise a procedure
for quantitatively measuring (using order metrics) how disordered
or ordered it is. This distinction between distributions of configurations
and particular configurations is an important one that OSLN do not
make.

The MRJ state is defined in \cite{Torquato_MRJ} as the jammed state
which minimizes a given order metric $\psi$. OSLN suggest their interpretation
of maximally random as superior because using order metrics {}``will
always be subject to uncertainty since one never knows if one has
calculated the proper order parameter.'' Therefore, OSLN believe that
they have identified the proper, unique, measure of order (related
to entropy). We wish to stress the difference between \emph{well-defined}
and \emph{unique}, as the two seem to be blurred in Ref. \cite{Jamming_ZeroT}.
The MRJ state is well-defined in that for a particular
choice of jamming category and order metric it can be identified unambiguously.
For a finite system, it will consist of a discrete set (possibly one)
of configurations,  becoming more densely populated as the system becomes larger.
At least for collective and strict
jamming in three dimensions, a variety of sensible order metrics seem
to produce an MRJ state near $\phi\approx0.64$ \cite{Anu_order_metrics}, the
traditionally accepted density of the RCP state.

However, the \emph{density} of the MRJ state should not be confused
with the MRJ \emph{state} itself.
It is possible to have a rather
ordered packing at this very same density; for example,
a jammed but diluted  vacancy FCC lattice packing  \cite{Anu_order_metrics}.
This is why the two-parameter description of packings in terms of the density
$\phi$ and order metric $\psi$, as in Ref. \cite{Torquato_MRJ},
is not only useful, but actually necessary.

OSLN's description of order implies a direct
relation between probability densities and randomness, i.e., that
the \emph{most probable} \cite{footnote_3}
configurations represent the most disordered state. In this sense,
one expects that the density of jammed configurations, when viewed
as a three-dimensional plot over the $\phi-\psi$ plane will be very
strongly peaked around the MRJ point for very large systems, just
as the probability distribution curves in Fig. 6 in Ref. \cite{Jamming_ZeroT}
are very peaked around $\phi\approx0.64$. As OSLN suggest themselves,
this might explain why several different packing procedures yield
similar hard-particle packings under appropriate conditions,
historically designated as RCP.

However, this is far from being a closed question \cite{footnote_4}.
Consider two-dimensional monodisperse circular disk packings as an example.
It is well-known that two-dimensional analogs of
three-dimensional computational and experimental protocols
that lead to putative RCP states result in disk
packings that are highly crystalline,
forming rather large triangular domains (grains) \cite{footnote_4a}.
 Because such highly ordered packings are the most probable for these
\emph{protocols}, OSLN's entropic measure would identify these as the most disordered,
a dubious proposition. An appropriate order metric, on the other hand,
is capable of identifying a particular configuration (not an ensemble
of configurations) of considerably lower density
(e.g., a jammed diluted triangular lattice)
that is consistent with our intuitive notions
of maximal disorder. However, typical packing protocols would
almost never generate such disordered disk configurations because
of their inherent bias toward undiluted crystallization.
This brings us to OSLN's claim that they have
devised an unbiased universal protocol, to which we now turn
our attention.

\section{\label{SectionAlgorithms}Universal, Hard and Soft Algorithms}

In this section, we focus on the algorithms used by OSLN and
point out why they are neither universal nor superior to other procedures.
We point out the close relations between OSLN's algorithm
for generating configurations near the onset of jamming and
the Zinchenko hard-sphere packing algorithm \cite{Zinchenko}.
Furthermore, we question OSLN's implication that using
one kind of interaction potential (with three different exponents)
and one algorithm amounts
to exploring the space of all jammed configurations in an unbiased
manner. This puts into doubt the claimed universality of the point $J$.

By fixing the interaction potential, initial density and energy
minimization (conjugate gradient) algorithm, OSLN obtain a well-defined collection
of final configurations with well-defined (not unique) properties.
In essence, OSLN make their algorithm devoid of tunable parameters
by simply choosing specific and fixed values for them.
Both the Zinchenko and OSLN algorithms are {}``dynamics independent'', in the sense
that there is no tunable parameter for the rate of compression, which
would be an analog of the cooling or quenching rate in molecular systems.
Both also imply that this makes their algorithm
universal or superior to other algorithms and that the (well-defined)
results they obtain are somehow special.
Most sensible algorithms will in fact produce
a well-defined density in the limit of large systems given a choice
of algorithmic parameters. For example, by changing the expansion
rate in the Lubachevsky-Stillinger algorithm, one can achieve final
densities for spheres anywhere in the range from $0.64$ (fast expansion) to $0.74$
(very slow expansion), as clearly illustrated in Fig. 2a in Ref. \cite{Torquato_MRJ}.
Therefore, if we followed the logic of OSLN, we could claim that any
number in that range represents a special point.
In our opinion, a good packing algorithm should be capable of generating
a variety of packings, in both density and the amount of order. How
can one ascertain that the packings one produces are {}``most random''
if there are no other jammed packings to compare to?

OSLN use two main procedures to generate final configurations.
The first procedure is to choose a density
and then use conjugate gradients to find a nearby energy minimum,
starting from a randomly-generated initial configuration ($T=\infty$),
as described in Section IIA in Ref. \cite{Jamming_ZeroT}.
Using this procedure, OSLN sampled inherent structures \cite{Inherent_Structures}
at fixed density to measure the fraction $f_j(\phi)$
of states that had nonzero bulk and shear moduli, and showed that
$f_j$ has a strong system-size dependence
with its derivative becoming a delta-function in the large system limit.
It is important to note that this procedure as such has little or nothing to do with
hard-sphere packings, especially for the kind of soft potentials ($\alpha\geq3/2)$
that OSLN study. Many stable energy  minima will be completely unrelated
to packings, and especially not to those designated as MRJ states.

OSLN used a second procedure to study the mechanical and structural
properties of systems near the onset of jamming $\phi_{c}$.
In this procedure, a configuration is compressed (or decompressed)
using very small steps in density until the bulk and shear
moduli vanished (or nonzero moduli develop), as described in Section IIB in Ref. \cite{Jamming_ZeroT}.
We now demonstrate that this procedure
is closely related to Zinchenko's  algorithm \cite{Zinchenko}
for generating hard-sphere packings. Start at low density with a set of nonoverlapping spheres
of diameter $\sigma$. Both algorithms then slowly grow the particles (OSLN
in small increments, Zinchenko continuously) while moving the particles
to avoid overlap \cite{footnote_7}.
In the Zinchenko algorithm, one strictly maintains the contact between
particles as soon as they touch, which requires solving a system of
ODE's containing the rigidity matrix of the packing \cite{Jamming_LP}
to find the necessary particle displacements. OSLN on the other hand,
use conjugate gradients (CG) to reminimize the potential energy, which
will simply push the particles just enough to keep them nonoverlapping,
i.e., almost in contact. This procedure continues in both algorithms
until no further densification is possible without inducing overlap.

Accordingly, it is not surprising
the packing configurations close to $\phi_{c}$
obtained in Ref. \cite{Jamming_ZeroT} closely resemble (in packing
fraction, amorphous character, coordination, etc.) packings generated
via a variety of bonafide  hard-sphere algorithms (and experiments
\cite{Experiment_RCP}). In particular, very similar packings are produced
with the Lubachevsky-Stillinger (LS) algorithm \cite{LS_algorithm,LS_algorithm_3D}
(with sufficiently high expansion rates) and the Zinchenko algorithm
\cite{Zinchenko}. OSLN criticize the LS algorithm
for changing the density in a dynamic fashion. The stated advantage
of the OSLN protocol is that one can {}``quench the system to the final
state within a fixed energy landscape'' since {}``the density is
always held constant''. We are very puzzled by this last claim in
light of their admission (in Section IIB of Ref. \cite{Jamming_ZeroT})
that they slowly change the density of the packing
to find $\phi_{c}$. In fact, OSLN do not seem to clearly distinguish
between the two rather different procedures they employ: the first
for finding inherent structures (at a fixed density) and  the second
for generating packings at the jamming threshold (which searches in
density). Fig. 6 of Ref. \cite{Jamming_ZeroT}, which
supposedly represents the distributions of jamming thresholds $\phi_{c}$,
\emph{defined} by the \emph{second} procedure, is obtained by differentiating the distribution
generated with the \emph{first} procedure, with no clear justification.

Most problematic of all is OSLN's claim that their results are universal.
Despite the statement that {}``Starting with randomly generated $T=\infty$
states guarantees that we sample \emph{all} {[}emphasis added{]} phase
space equally'', all that their first algorithm manages to explore
is the space of energy minima for the \emph{particular} chosen interaction
potential. By comparing three different exponents
$\alpha$, OSLN conclude that the exact form of the potential is not
important. However, a much more convincing picture would have been
made if they instead tried \emph{qualitatively} different kinds
of interaction potentials,
rather then simply changing the curvature of the potential at the
contact point. Otherwise, why focus on continuous interaction potentials
at all? Since it is geometry (i.e., the nonoverlap condition on
the spherical cores) that is crucial, the hard-sphere system
offers a far {}``cleaner'' system to study when trying to understand
the special point $J$.


\section{Introduction}

Already in \cite{Fontana} and more recently and independently in \cite{Ebulfez 2} we have rediscovered the Hochster's inverse topology (see \cite[Prop. 8]{Hochster}) on the prime spectrum by a new and purely algebraic method. We call it the flat topology (it is worthy to mention that during the writing \cite{Ebulfez 2} we were not aware of \cite{Fontana} and Hochster's work). Hence the flat topology and Hochster's inverse topology are exactly the same things. The flat topology behaves as the dual of the Zariski topology. Roughly speaking, for a given ring $R$, then the collection of subsets $\V(I)=\{\mathfrak{p}\in\Spec(R) : I\subseteq\mathfrak{p}\}$ where $I$ runs through the set of f.g. ideals of $R$ formes a basis for the opens of the flat topology, see \cite[Theorem 3.2]{Ebulfez 2}. We use f.g. in place of ``finitely generated". Note that, in general, there are tremendous differences between the two topologies even if the base ring $R$ be a noetherian ring. In fact, the two topologies on $\Spec(R)$ are the same if and only if every prime ideal of $R$ is maximal, see Corollary \ref{coro 9}. \\

Recall that a topological space is called a \emph{separated} (or Hausdorff or $T_{2}$) space if every two distinct points admit disjoint open neighbourhoods. In this article, we are concerned primarily with the problem of when the Zariski topology is separated, and the dual question of when the flat topology is separated. The flat topology, as Zariski, is not necessarily separated, see \cite[Corollary 3.6]{Ebulfez 2}. Because of its importance and geometric applications, characterizing the separability of the Zariski topology became the most urgent task to the author at that time.\\

Note that characterizing the separability of the Zariski topology as well as the flat topology is not as easy to understand as one may think at first. This is because we are used to the topology of locally Hausdorff spaces, but the flat and Zariski topologies in general are not locally Hausdorff. By applying some sophisticated properties of the absolutely flat (von-Neumann regular) rings and flat epimorphisms the separability of these topologies are completely understood, see Theorem \ref{theorem 2}. In the literature one can also find another versions or parts of this characterization. Corollary \ref{coro 9} can be considered as interesting application of Theorem \ref{theorem 2}. \\

In this article, by an epimorphism $\phi:R\rightarrow S$ we mean it is an epimorphism in the category of commutative rings. The class of injective ring maps is precisely coincide to the class of monomorphisms of rings; but surjective ring maps are just special cases of epimorphisms. As a specific example, the canonical ring map $\mathbb{Z}\rightarrow\mathbb{Q}$ is an epimorphism while it is not surjective. For more details on epimorphisms of rings please consider \cite[\S2]{ebulfez}.\\

The titles of the sections should be sufficiently explanatory. Throughout the article, all of the rings which are discussed are commutative.\\

\section{Preliminaries-absolutely flat rings}

Absolutely flat rings play a major role throughout this article. The authors's contributions in this section are including Corollaries \ref{epic-flat}, \ref{coro 6}, \ref{coro 700} and Proposition \ref{theorem 3}.\\

\begin{lemma}\label{tricky lemma} Let $R$ be a ring and let $I=\langle a_{1},...,a_{n}\rangle$ be a f.g. ideal of $R$. Suppose there is some $c_{i}\in R$ such that $a_{i}=a^{2}_{i}c_{i}$ for all $i$. Then $I$ can be generated by an idempotent element.\\
\end{lemma}

{\bf Proof.} Clearly $e_{i}=a_{i}c_{i}$ is an idempotent element and $I=\langle e_{1},...,e_{n}\rangle$. If $n=1$ then there is nothing to prove.
Suppose $n> 1$. Then, by the induction hypothesis, the ideal $\langle e_{1},...,e_{n-1}\rangle$ is generated by an idempotent element $e'\in R$. It follows that $I=\langle e\rangle$ where $e=e'+e_{n}-e'e_{n}$ is an idempotent element. $\Box$ \\

Recall that a ring $R$ is said to be absolutely flat if each $R-$module is flat.\\

\begin{proposition}\label{prop 3} Let $R$ be a ring. Then the following conditions are equivalent.\\
$\mathbf{(i)}$ The ring $R$ is absolutely flat. \\
$\mathbf{(ii)}$ Every ideal of $R$ is idempotent.\\
$\mathbf{(iii)}$ Each element $r\in R$ can be written as $r=r^{2}s$ for some $s\in R$.\\
$\mathbf{(iv)}$ Every f.g. ideal of $R$ is a direct summand of $R$.\\
\end{proposition}

{\bf Proof.} $\mathbf{(i)}\Rightarrow\mathbf{(ii)}$: Let $I$ be an ideal of $ R$.
The map $i\otimes1:I\otimes_{R}R/I\rightarrow R\otimes_{R}R/I$ induced by the canonical injection $i:I\rightarrow R$ is injective since $R/I$ is $R-$flat. But $\Ima(i\otimes1)=0$. Therefore $I=I^{2}$.\\
$\mathbf{(ii)}\Rightarrow\mathbf{(iii)}$: There is nothing to prove.\\
$\mathbf{(iii)}\Rightarrow\mathbf{(iv)}$: Let $I=\langle a_{1},...,a_{n}\rangle$ be a f.g. ideal of $R$. By Lemma \ref{tricky lemma}, there is an idempotent $e\in R$ such that $I=\langle e\rangle$. Let $J=\langle1-e\rangle$. Then clearly $I+J=R$ and $I\cap J=0$.\\
$\mathbf{(iv)}\Rightarrow\mathbf{(i)}$: Let $M$ be a $R-$module. By \cite[Theorem 7.7]{Matsumura1}, it suffices to show that for every f.g. ideal $I$ of $R$, the canonical map $I\otimes_{R}M\rightarrow M$ which maps each pure tensor $a\otimes m$ into $am$ is injective. By the hypothesis, there is an ideal $J$ of $R$ such that $R=I+J$ and $I\cap J=0$. It follows that the following sequence $\xymatrix{0\ar[r] & I \ar[r]^{i} & R\ar[r]^{p} &J \ar[r] & 0}$ is exact and split
where $i$ is the canonical injection and $p$ is the projection map. Therefore the following sequence is exact and split
$$\xymatrix{0\ar[r] & I\otimes_{R}M \ar[r]^{i\otimes1} & R\otimes_{R}M\ar[r]^{p\otimes1} &J\otimes_{R}M\ar[r] & 0}$$ because the exact and split sequences are left exact and split
by an additive functor. $\Box$ \\

\begin{corollary}\label{epic-flat} Absolutely flat rings are stable under taking  quotients and localizations. \\
\end{corollary}

{\bf Proof.} Let $R$ be an absolutely flat ring, let $I$ be an ideal of $R$ and let $S$ be a multiplicative subset of $R$. Each ideal $K$ of $R/I$ is of the form $J/I$ where $J$ is an ideal of $R$ which contains $I$. By Proposition \ref{prop 3}, $J=J^{2}$. Thus $K^{2}=J^{2}+I/I=K$. Therefore, by Proposition \ref{prop 3}, $R/I$ is absolutely flat. Suppose $r/s\in S^{-1}R$ then, by Proposition \ref{prop 3}, $r=r^{2}r'$ for some $r'\in R$. Thus $r/s=(r/s)^{2}(r's/1)$. Therefore by Proposition \ref{prop 3}, $S^{-1}R$ is absolutely flat. $\Box$ \\

\begin{proposition}\label{prop 987} A ring $R$ is absolutely flat if and only if $R_{\mathfrak{m}}$ is absolutely flat for all maximal ideals $\mathfrak{m}$ of $R$.\\
\end{proposition}

{\bf Proof.} The implication ``$\Rightarrow$" is an immediate consequence of Corollary \ref{epic-flat}. For the reverse implication, let $M$ be a $R-$module and let $\xymatrix{0 \ar[r] & N'\ar[r]^{f} & N}$ be an  exact sequence of $R-$modules. Denote $K$ the kernel of the morphism $f\otimes 1:N'\otimes_{R}M\rightarrow N\otimes_{R}M$. Let $\mathfrak{m}$ be a maximal ideal of $R$.  The sequence $\xymatrix{0 \ar[r] & N'_{\mathfrak{m}}\otimes_{_{R_{\mathfrak{m}}}}M_{\mathfrak{m}}\ar[r] & N_{\mathfrak{m}}}\otimes_{_{R_{\mathfrak{m}}}}M_{\mathfrak{m}}$ is exact since $R_{\mathfrak{m}}$ is absolutely flat. It follows that the map
$(f\otimes 1)_{\mathfrak{m}}:(N'\otimes_{R}M)_{\mathfrak{m}}\rightarrow (N\otimes_{R}M)_{\mathfrak{m}}$ is injective. This means that $K_{\mathfrak{m}}=0$. Therefore $K=0$. $\Box$ \\

\begin{corollary}\label{coro 6} Let $\{R_{i}\}$ be a family of rings. Then the direct product ring $A=\prod\limits_{i}R_{i}$ is absolutely flat if and only if each $R_{i}$ is so.\\
\end{corollary}

{\bf Proof.} The projection map $\pi_{i}:A\rightarrow R_{i}$ is surjective. Therefore, by Corollary \ref{epic-flat}, $R_{i}$ is absolutely flat. Conversely, pick $a=(r_{i})\in A$. For each $i$, by Proposition \ref{prop 3}, $r_{i}=r_{i}^{2}s_{i}$ for some $s_{i}\in R_{i}$. Thus $a=a^{2}b$ where $b=(s_{i})$. Therefore, by Proposition \ref{prop 3}, $A$ is absolutely flat. $\Box$ \\

\begin{corollary}\label{coro 700} Let $R$ be an absolutely flat ring. If one of the following conditions hold then $R$ is a field.\\
$\textbf{(i)}$ $R$ is local.\\
$\textbf{(ii)}$ $R$ is an integral domain.\\
$\textbf{(iii)}$ $R$ is a non-trivial noetherian ring with the trivial idempotents.\\
\end{corollary}

{\bf Proof.} Let $\textbf{(i)}$. Let $\mathfrak{m}$ be the maximal ideal of $R$.
For each $a\in\mathfrak{m}$, by Proposition \ref{prop 3}, there is some $b\in R$ such that $a(ab-1)=0$. But $ab-1$ is invertible in $R$. Therefore $a=0$.
If $\textbf{(ii)}$ holds. Then, by Proposition \ref{prop 3}, every non-zero element of $R$ is invertible. Suppose $\textbf{(iii)}$ holds. Let $I$ be a non-zero ideal of $R$. Therefore, by Proposition \ref{prop 3} and Lemma \ref{tricky lemma}, $I=R$. This, in particular, implies that every non-zero element of $R$ is invertible. $\Box$ \\

\begin{proposition}\label{theorem 3} A ring $R$ is absolutely flat if and only if each $R-$algebra is $R-$flat.\\
\end{proposition}

{\bf Proof.} The implication ``$\Rightarrow$" is clear. Conversely, let $M$ be a $R-$module. Consider the ring $S$ where the underlying set of this ring is the cartesian product $R\times M$ and its addition and multiplication are defined as $(r,m)+(r',m')=(r+r',m+m')$ and $(r,m).(r',m')=(rr', rm'+r'm)$, respectively. Clearly $S$ is a commutative ring whose identity element is $(1,0)$. Moreover the map $\phi: R\rightarrow S$ given by $r\rightsquigarrow(r,0)$ is a ring homomorphism. The $R-$module structure induced via $\phi$ on $S$ is the same as the usual $R-$module structure on the direct sum $R\bigoplus M$. By the hypothesis, $\phi$ is a flat morphism. It follows that $M$ is a flat $R-$module.  $\Box$ \\

\section{Preliminaries-pointwise rings}

In this section we develop the theory of pointwise rings which we need to it in the sequel. Here, the only author's contributions are Proposition \ref{prop 6}, part $\textbf{(iii)}$,  Lemma \ref{lemma 453} and the proofs of Lemma \ref{lemma 9} and Proposition \ref{pointwise}. The remaining results are well-known and can be found in the S\'{e}minaire Samuel \cite{Samuel} also see \cite{Wiegand} and \cite{Wiegand 2}. \\

If $R$ is an absolutely flat ring then each element $a\in R$, by Proposition \ref{prop 3}, can be written as $a=a^{2}b$ for some $b\in R$. This leads us to the following definition:\\

\begin{definition} Let $R$ be a ring and let $a\in R$. If there is an element $b\in R$ such that $a=a^{2}b$ and $b=b^{2}a$, then $b$ is said to be a pointwise inverse of $a$. \\
\end{definition}

\begin{lemma}\label{lemma 9} Let $a,b\in R$. Then $b$ is a pointwise inverse of $a$ if and only if $a\in Ra^{2}$. Moreover, if $b$ is a pointwise inverse of $a$ then there is an idempotent element $e\in R$ such that $(e+a)(e+b)=1$. Finally, the pointwise inverse, if it exists, is unique.   \\
\end{lemma}

{\bf Proof.} Suppose $a\in Ra^{2}$. We have $a=ra^{2}$ for some $r\in R$. Let $b=r^{2}a$. Then $b$ is a pointwise inverse of $a$. Clearly $e=1-ab$ is an idempotent element and $(e+a)(e+b)=1$.
Let $c\in R$ be another pointwise inverse of $a$. We have $b=ab^{2}=(ac)(ab^{2})=a^{2}c^{2}b=ac^{2}=c$. $\Box$ \\

The pointwise inverse of $a\in R$, if it exists, is usually denoted by $a^{(-1)}$.\\

\begin{lemma} Let $\phi: R\rightarrow S$ be a ring map. Suppos $a,b\in R$ have pointwise inverses in R. Then the pointwise inverses of $\phi(a)$ and $ab$ exist. Moreover $\phi(a)^{(-1)}=\phi(a^{(-1)})$ and $(ab)^{(-1)}=a^{(-1)}b^{(-1)}$. \\
\end{lemma}

{\bf Proof.} Easy. $\Box$  \\

The following result establishes the universal property of the poinwise rings.\\

\begin{proposition}\label{pointwise} Let $R$ be a ring and let $S$ be a subset of $R$. Then there exist a ring $S^{(-1)}R$ and a canonical ring map $\eta: R\rightarrow S^{(-1)}R$ such that for each $s\in S$, the pointwise inverse of $\eta(s)$ in $S^{(-1)}R$ exists and the pair $(S^{(-1)}R, \eta)$ satisfies in the following universal property: if there is a ring map $\phi: R\rightarrow R'$ such that for each $s\in S$ the pointwise inverse of $\phi(s)$ in $R'$ exists then there is a unique ring map $\psi:S^{(-1)}R\rightarrow R'$ such that $\phi=\psi\circ\eta$.\\
\end{proposition}

{\bf Proof.} Consider the polynomial ring $A=R[x_{s} : s\in S]$ and let $S^{(-1)}R=A/I$ where the ideal $I$ is generated by elements of the form $sx_{s}^{2}-x_{s}$ and $s^{2}x_{s}-s$ with $s\in S$. Let $\eta: R\rightarrow S^{(-1)}R$ be the canonical ring map. For each $s\in S$, the element $x_{s}+I$ is the pointwise inverse of $\eta(s)=s+I$. Let $\phi:R\rightarrow R'$ be a ring map such that for each $s\in S$, the pointwise inverse of $\phi(s)$ exists in $R'$. By the universal property of the polynomial rings, there is a (unique) homomorphism of $R-$algebras $\widetilde{\phi}:R[x_{s} : s\in S]\rightarrow R'$ such that $x_{s}\rightsquigarrow\phi(s)^{(-1)}$ for all $s\in S$. We have $\widetilde{\phi}(I)=0$. Denote by $\psi:S^{(-1)}R\rightarrow R'$ the ring map induced by $\widetilde{\phi}$. Clearly $\psi$ is the unique ring homomorphism such that $\phi=\psi\circ\eta$. Because suppose there is another such ring map $\psi':S^{(-1)}R\rightarrow R'$. Then we have $\psi(x_{s}+I)=\widetilde{\phi}(x_{s})=\phi(s)^{(-1)}=
\psi'\big(\eta(s)\big)^{(-1)}=\psi'\big(\eta(s)^{(-1)}\big)=\psi'(x_{s}+I)$ for all $s\in S$. Therefore $\psi=\psi'$. $\Box$ \\

We call $S^{(-1)}R$ the pointwise localization of $R$ with respect to $S$. \\

\begin{proposition}\label{prop 6} Let $R$ be a ring and let $S$ be a subset of $R$. Then the following are true. \\
$\mathbf{(i)}$  The canonical ring map $\eta: R\rightarrow S^{(-1)}R$ is an epimorphism.\\
$\mathbf{(ii)}$ The map $\eta^{\ast}:\Spec\big(S^{(-1)}R\big)\rightarrow\Spec(R)$ is bijective.\\
$\mathbf{(iii)}$ For each $s\in S$,  $(\eta^{\ast})^{-1}\big(V(s)\big)$ is a clopen subset of $\Spec\big(S^{(-1)}R\big)$ with respect to the flat (resp. Zariski) topology.\\
$\mathbf{(iv)}$ The ring $S^{(-1)}R$ is nontrivial if and only if $R$ is so.\\
$\mathbf{(v)}$ $\Ker(\eta)\subseteq\mathfrak{N}$ where $\mathfrak{N}$ is the nil-radical of $R$.\\
\end{proposition}

{\bf Proof.} $\mathbf{(i)}:$ This implies from the universal property of  Proposition \ref{pointwise}.\\
$\mathbf{(ii)}:$  The map $\eta^{\ast}$ is injective since $\eta$ is an epimorphism, see \cite[Theorem 3.3]{ebulfez}. Let $\mathfrak{p}$ be a prime ideal of $R$ and consider the canonical ring map $\pi: R\rightarrow\kappa(\mathfrak{p})$. The image of every element of $R$ under $\pi$ has a pointwise inverse in $\kappa(\mathfrak{p})$. Thus, by Proposition \ref{pointwise}, there is a (unique) ring map $\psi:S^{(-1)}R\rightarrow\kappa(\mathfrak{p})$ such that $\pi=\psi\circ\eta$.
Then $\mathfrak{p}=\eta^{\ast}(\mathfrak{q})$ where
$\mathfrak{q}=\psi^{-1}(0)$.\\
$\mathbf{(iii)}:$ We have $(\eta^{\ast})^{-1}\big(V(s)\big)=V\big(\eta(s)\big)$. Moreover $V\big(\eta(s)\big)=D\big(1-\eta(s)\eta(s)^{(-1)}\big)$. Therefore, by \cite[Corollary 3.12]{Ebulfez 2}, $(\eta^{\ast})^{-1}\big(V(s)\big)$ is both open and closed. \\
$\mathbf{(iv)}$ and $\mathbf{(v)}$: These are immediate consequences of $\mathbf{(ii)}$. $\Box$  \\

\begin{lemma}\label{lem 1} Let $\phi: R\rightarrow S$ be an epimorphism of rings where $S$ is a nontrivial ring with the trivial idempotents. Suppose $\phi(r)$ has a pointwise inverse in $S$ for all $r\in R$. Then $A=\Ima(\phi)$ is an integral domain and $S$ is its field of fractions.\\
\end{lemma}

{\bf Proof.} Suppose $\phi(r)\phi(r')=0$ for some elements $r,r'\in R$. If $\phi(r)\neq0$ then $\phi(r)\phi(r)^{(-1)}=1$ since $\phi(r)\phi(r)^{(-1)}$ is an idempotent element. Therefore $A$ is an integral domain. Let $K$ be the field of fractions of $A$. Since every non-zero element of $A$ is invertible in $S$ therefore by the universal property of the localization, there is a (unique) ring map $\psi:K\rightarrow S$ such that $i=\psi\circ j$ where $i:A\rightarrow S$
and $j:A\rightarrow K$ are the canonical injections. The map $\phi$ factors as $\phi=i\circ\phi'$ where $\phi':R\rightarrow A$ is the ring map induced by $\phi$. Since $\phi$ is an epimorphism thus $i$ and so $\psi$ are epimorphisms. By \cite[Corollary 2.3]{ebulfez}, $\psi$ is an isomorphism. $\Box$ \\

\begin{lemma}\label{lemma 453} Let $R$ be a ring. Then $\Spec\big(R^{(-1)}R\big)$ equipped with the Zariski (resp. flat) topology is separated.\\
\end{lemma}

{\bf Proof.} Let $\mathfrak{q}$ and $\mathfrak{q'}$ be distinct prime ideals of $R^{(-1)}R$. The ideals $\mathfrak{p}=\eta^{\ast}(\mathfrak{q})$ and $\mathfrak{p'}=\eta^{\ast}(\mathfrak{q'})$ are distinct since by Proposition \ref{prop 6}, $\eta^{\ast}$ is injective. Choose  $a\in\mathfrak{p}\setminus\mathfrak{p'}$. It follows that $\mathfrak{q}\in V\big(\eta(a)\big)$ and $\mathfrak{q'}\in D\big(\eta(a)\big)$. By Proposition \ref{prop 6}, $V\big(\eta(a)\big)$ is a clopen. $\Box$ \\

\begin{theorem}\label{theorem 1} Let $R$ be a ring and let $\eta:R\rightarrow R^{(-1)}R$ be the canonical ring map. Then the following are true. \\
$\mathbf{(i)}$ For each prime ideal $\mathfrak{q}$ of $R^{(-1)}R$, then $F=\big(R^{(-1)}R\big)_{\mathfrak{q}}$ is canonically isomorphic to $\kappa(\mathfrak{p})$ where $\mathfrak{p}=\eta^{\ast}(\mathfrak{q})$.\\
$\mathbf{(ii)}$ The ring $R^{(-1)}R$ is absolutely flat.\\
\end{theorem}

{\bf Proof.} $\mathbf{(i)}:$ For each prime ideal $\mathfrak{q}$ of $R^{(-1)}R$, the composed map $$\xymatrix{R \ar[r]^{\eta} & R^{(-1)}R\ar[r] & F}$$ satisfies all of the hypotheses of  Lemma \ref{lem 1}. Therefore $F$ is a field. Now consider the following commutative diagram $$\xymatrix{
R_{\mathfrak{p}} \ar[r]^{\eta_{\mathfrak{q}}=epic} \ar[d]^{} & F \ar[d]^{\simeq} \\ \kappa(\mathfrak{p})\ar[r]^{} & \kappa(\mathfrak{q}) } $$
where $\mathfrak{p}=\eta^{\ast}(\mathfrak{q})$. By \cite[Corollary 2.3]{ebulfez}, the map $\kappa(\mathfrak{p})\rightarrow\kappa(\mathfrak{q})$ is an isomorphism and we win.\\
$\mathbf{(ii)}:$ It is an immediate consequence of $\mathbf{(i)}$ and Proposition \ref{prop 987}. $\Box$  \\

By Proposition \ref{pointwise} and Theorem \ref{theorem 1}, the assignment $R\rightsquigarrow R^{(-1)}R$ is a covariant functor form the category of commutative rings into the category of absolutely flat rings. \\

\section{Main results}

\begin{lemma}\label{remark 1} Let $\phi: R\rightarrow S$ be a ring map, let $M$ and $N$ be $S-$modules and consider the canonical map $\eta:M\otimes_{R}N\rightarrow M\otimes_{S}N$ which maps each pure tensor $m\otimes_{R}n$ into $m\otimes_{S}n$. Then $\Ker(\eta)=\langle sm\otimes_{R}n-m\otimes_{R}sn : s\in S\setminus\Ima(\phi), m\in M, n\in N\rangle$.\\
\end{lemma}

{\bf Proof.} Let $K$ be the $R-$submodule of $M\otimes_{R}N$ generated by elements of the form $sm\otimes_{R}n-m\otimes_{R}sn$ with $s\in S\setminus\Ima(\phi)$, $m\in M$ and $n\in N$. Clearly $K\subseteq\Ker(\eta)$. Consider the map
$\overline{\eta}:P=M\otimes_{R}N/K\rightarrow M\otimes_{S}N$ induced by $\eta$. We have $\Ker(\overline{\eta})=\Ker(\eta)/K$. The scalar multiplication $S\times P\rightarrow P$ which is defined on pure tensors by $s.(m\otimes_{R}n+K)=sm\otimes_{R}n+K$ is actually well-defined and puts a $S-$module structure over $P$. By the universal property of the tensor products, the $S-$bilinesr map $M\times N\rightarrow P$ defined by $(m,n)\rightsquigarrow m\otimes_{R}n+K$ induces a (unique) $S-$homomorphism $M\otimes_{S}N\rightarrow P$ which maps each pure tensor $m\otimes_{S}n$ into $m\otimes_{R}n+K$. This implies that $\overline{\eta}$ is bijective. Therefore $\Ker(\eta)=K$.  $\Box$ \\

\begin{lemma}\label{lem 2} Let $\phi: R\rightarrow S$ be a flat ring map which has a factorization $\xymatrix{ R \ar[r]^{\psi} & A \ar[r]^{\phi'} & S}$ such that $\phi'$ is an injective ring map and $\psi$ is an epimorphism. Then $\phi'$ is a flat morphism.\\
\end{lemma}

{\bf Proof.}  For each $A-$module $M$, the canonical map $\eta_{M}:M\otimes_{R}S\rightarrow M\otimes_{A}S$ which maps each pure tensor $m\otimes_{R} s$ into $m\otimes_{A}s$ is injective because in $A\otimes_{R}A-$module $M\otimes_{R}S$ we have  $am\otimes_{R}s=(a\otimes_{R}1_{A}).(m\otimes_{R}s)=
(1_{A}\otimes_{R}a).(m\otimes_{R}s)=m\otimes_{R}a.s$ then apply Lemma \ref{remark 1}. In fact it is bijective. Now suppose $\xymatrix{0 \ar[r] & N \ar[r]^{f} & M}$ is an exact sequence of $A-$modules. The following diagram is commutative
$$\xymatrix{
 N\otimes_{R}S\ar[r]^{f\otimes_{R}1} \ar[d]^{\eta_{N}} & M\otimes_{R}S \ar[d]^{\eta_{M}} \\ N\otimes_{A}S \ar[r]^{f\otimes_{A}1} & M\otimes_{A}S} $$ and the map $f\otimes_{R}1$ is injective since $S$ is flat over $R$. Therefore $f\otimes_{A}1$ is injective as well. $\Box$  \\

\begin{lemma}\label{simplified lemma} Let $\phi: R\rightarrow S$ be a flat epimorphism of rings. Then for each prime $\mathfrak{p}$ of $R$ we have either $\mathfrak{p}S=S$ or that the canonical map $R_{\mathfrak{p}}\rightarrow T^{-1}S$ given by $r/s\rightsquigarrow\phi(r)/\phi(s)$ is bijective where $T=\phi(R\setminus\mathfrak{p})$. \\
\end{lemma}

{\bf Proof.} Suppose $\mathfrak{p}S\neq S$ for some prime $\mathfrak{p}$. The canonical map $R_{\mathfrak{p}}\rightarrow T^{-1}S$ is a flat epimorphism because flat morphisms and epics are stable under base change and composition (recall that the ring $ T^{-1}S$ is canonically isomorphic to $S_{\mathfrak{p}}$). It is also faithfully flat since $\mathfrak{p}S\neq S$. Therefore, by \cite[Corollary 2.2]{ebulfez}, it is bijective. $\Box$ \\

It is worthy to mention that the converse of Lemma \ref{simplified lemma} also holds.\\

For a given ring $R$, the quotient ring $R/\mathfrak{N}$ is denoted by $R_{\mathrm{red}}$ where $\mathfrak{N}$ is the nil-radical of $R$. For any ring map $\phi:R\rightarrow S$ the induced map $R_{\mathrm{red}}\rightarrow S_{\mathrm{red}}$ is denoted by $\phi_{\mathrm{red}}$.\\

\begin{theorem}\label{outstanding lemma} Let $\phi: R\rightarrow S$ be a flat epimorphism of rings. If $\phi_{\mathrm{red}}$ is surjective then so is $\phi$.\\
\end{theorem}

{\bf Proof.} The map $\phi$ factors as $\xymatrix{R \ar[r]^{\pi\:\:\:\:\:\:\:\:\:\:\:} & R/\Ker(\phi)\ar[r]^{\:\:\:\:\:\:\:\:\:\:\:\:\phi'} & S}$
where $\pi$ is the canonical ring map and $\phi'$ is induced by $\phi$. We have $\Ima(\phi)=\Ima(\phi')$, $\phi'$ is an epimorphism and  $\phi'_{\mathrm{red}}$ is surjective. Moreover, by Lemma \ref{lem 2}, $\phi'$ is flat. Therefore, without loss of generality, we may assume that $\phi$ is injective. It follows that $\phi_{\mathrm{red}}$ is an isomorphism and so $\phi^{\ast}: \Spec(S)\rightarrow\Spec(R)$ is bijective. Therefore  $\mathfrak{p}S\neq S$ for all primes $\mathfrak{p}$ of $R$ and so by Lemma \ref{simplified lemma}, the canonical map $R_{\mathfrak{p}}\rightarrow S_{\mathfrak{p}}$ is bijective. It follows that $S/\phi(R)\otimes_{R}R_{\mathfrak{p}}=0$ for all primes $\mathfrak{p}$. $\Box$ \\

\begin{theorem}\label{coro 3} Let $\phi: R\rightarrow S$ be an epimorphism of rings such that $R$ is absolutely flat. Then $\phi$ is surjective.\\
\end{theorem}

{\bf Proof.} The map $\phi$ factors as $\xymatrix{R \ar[r]^{\pi\:\:\:\:\:\:\:\:\:\:\:} & R/\Ker(\phi)\ar[r]^{\:\:\:\:\:\:\:\:\:\:\:\:\phi'} & S}$
where $\pi$ is the canonical ring map and $\phi'$ is the injective ring map induced by $\phi$. The quotient ring $R/\Ker(\phi)$ is absolutely flat. Moreover, $\Ima(\phi)=\Ima(\phi')$ and yet $\phi'$ is an epimorphism. Hence, without loss of generality, we may assume that $\phi$ is injective. In this case, $\phi$ is a faithfully flat morphism. Because, suppose $S\otimes_{R}M=0$ for some $R-$module $M$. From the following short exact sequence of $R-$modules $$\xymatrix{0 \ar[r] & R\ar[r]^{\phi} & S\ar[r]^{\pi} & S/R \ar[r] & 0}$$ we obtain the following long exact sequence of $R-$modules
$\xymatrix{... \ar[r] &}$\\$\xymatrix{\Tor^{R}_{1}(S/R, M) \ar[r] &R\otimes_{R}M\ar[r]^{\phi\otimes1_{M}} & S\otimes_{R}M \ar[r]^{\pi\otimes1_{M}} & S/R\otimes_{R}M \ar[r] & 0}$.\\
But $\Tor^{R}_{1}(S/R,M)=0$ since $S/R$ is $R-$flat, see \cite[Theorem 7.2]{Rotman}. Thus $M\simeq R\otimes_{R}M=0$. Therefore $\phi$ is a faithfully flat epimorphism and so by \cite[Corrollary 2.2]{ebulfez}, it is bijective. This means that, in our factorization $\phi=\phi'\circ\pi$, $\phi'$ is an isomorphism therefore the original $\phi$ is surjective. $\Box$ \\

\begin{lemma}\label{lemma 234} A ring $R$ is absolutely flat if and only if the canonical map $\eta:R\rightarrow R^{(-1)}R$ is bijective.\\
\end{lemma}

{\bf Proof.} Suppose $R$ is absolutely flat. Then, by Theorem \ref{coro 3}, $\eta$ is surjective. Pick $a\in\Ker(\eta)$. By Proposition \ref{prop 3}, there exists some $b\in R$ such that $a=ba^{2}$. It follows that $a=b^{n-1}a^{n}$ for all $n\geq1$. But $a$ is a nilpotent element, see Proposition \ref{prop 6}. Therefore $a=0$. The converse implies from Theorem \ref{theorem 1}. $\Box$ \\

\begin{theorem}\label{theorem 2} Let $R$ be a ring. Then the following conditions are equivalent.\\
$\mathbf{(i)}$ The ring $R_{\mathrm{red}}=R/\mathfrak{N}$ is absolutely flat where $\mathfrak{N}$ is the nil-radical of $R$.\\
$\mathbf{(ii)}$ Every flat epimorphism $\phi:R\rightarrow S$ is surjective.\\
$\mathbf{(iii)}$ For each prime ideal $\mathfrak{p}$ of $R$, the canonical map $R\rightarrow R_{\mathfrak{p}}$ is surjective.\\
$\mathbf{(iv)}$ Every prime ideal of $R$ is maximal.\\
$\mathbf{(v)}$ The patch and  Zariski topologies over $\Spec(R)$ are the same.\\
$\mathbf{(vi)}$ The set $\Spec(R)$ equipped with the Zariski topology is separated.\\
$\mathbf{(vii)}$ Every prime ideal of $R$ is minimal.\\
$\mathbf{(viii)}$ The set $\Spec(R)$ equipped with the flat topology is separated.\\
$\mathbf{(ix)}$ The patch and flat topologies over $\Spec(R)$ are the same.\\
\end{theorem}

{\bf Proof.} $\mathbf{(i)}\Rightarrow\mathbf{(ii)}:$ By Theorem \ref{outstanding lemma}, it suffices to show that $\phi_{\mathrm{red}}:R_{\mathrm{red}}\rightarrow S_{\mathrm{red}}$ is surjective. The following diagram is commutative $$\xymatrix{
R \ar[r]^{\phi} \ar[d]^{} & S \ar[d]^{} \\ R_{\mathrm{red}}\ar[r]^{\phi_{\mathrm{red}}} & S_{\mathrm{red}}.}$$
It follows that $\phi_{\mathrm{red}}$ is an epimorphism. Therefore, by Theorem \ref{coro 3}, $\phi_{\mathrm{red}}$ is surjective and we win.\\
$\mathbf{(ii)}\Rightarrow\mathbf{(iii)}:$ For each prime ideal $\mathfrak{p}$ of $R$, the canonical map $R\rightarrow R_{\mathfrak{p}}$ is a flat epimorphism.\\
$\mathbf{(iii)}\Rightarrow\mathbf{(iv)}:$ The canonical map $R/\mathfrak{p}\rightarrow\kappa(\mathfrak{p})$ is surjective.\\
$\mathbf{(iv)}\Rightarrow\mathbf{(i)}:$ Let $\mathfrak{m}'=\mathfrak{m}/\mathfrak{N}$ be a maximal ideal of $R_{\mathrm{red}}$ where $\mathfrak{m}$ is a maximal ideal of $R$. The ring $(R_{\mathrm{red}})_{\mathfrak{m}'}$ is canonically isomorphic to
$R_{\mathfrak{m}}/\mathfrak{N}R_{\mathfrak{m}}$.
Moreover, $\mathfrak{N}R_{\mathfrak{m}}=\mathfrak{N'}$ where $\mathfrak{N'}$ denotes the nil-radical of $R_{\mathfrak{m}}$. But $\mathfrak{N'}=\mathfrak{m}R_{\mathfrak{m}}$ since every prime ideal of $R$ is maximal. Thus, by Proposition \ref{prop 987}, $R_{\mathrm{red}}$ is absolutely flat.\\
$\mathbf{(v)}\Rightarrow\mathbf{(vi)}:$ It is obvious.\\
$\mathbf{(vi)}\Rightarrow\mathbf{(v)}:$ Consider the identity map $i: \big(\Spec(R),\mathscr{J}_{p}\big)\rightarrow
\big(\Spec(R),\mathscr{J}_{z}\big)$ where $\mathscr{J}_{p}$ and $\mathscr{J}_{z}$ denote the patch and Zariski topologies, respectively. By the hypothesis and \cite[Theorem 2.4]{Ebulfez 2}, $i$ is a homeomorphism, so $\mathscr{J}_{p}=\mathscr{J}_{z}$ .\\
$\mathbf{(vi)}\Rightarrow\mathbf{(iv)}:$ In every separated space the points are closed. \\
$\mathbf{(i)}\Rightarrow\mathbf{(vi)}$ and $\mathbf{(viii)}:$ The map $\pi^{\ast}:\Spec(R_{\mathrm{red}})\rightarrow\Spec(R)$ induced by the canonical map $\pi: R\rightarrow R_{\mathrm{red}}$ is a homeomorphism. By Lemma \ref{lemma 234}, $\Spec(R_{\mathrm{red}})$ is homeomorphic to $\Spec(A)$ where $A:=R_{\mathrm{red}}^{(-1)}R_{\mathrm{red}}$. The latter space, by Lemma \ref{lemma 453}, is separated.\\
The implications $\mathbf{(ix)}\Rightarrow\mathbf{(viii)}$ and $\mathbf{(vii)}\Leftrightarrow\mathbf{(iv)}$ are clear.\\
$\mathbf{(viii)}\Rightarrow\mathbf{(vii)}:$ See \cite[Corollary 3.6]{Ebulfez 2}.\\
$\mathbf{(viii)}\Rightarrow\mathbf{(ix)}:$ Using the similar argument as applied in the implication $(vi)\Rightarrow(v)$. $\Box$ \\

\begin{corollary}\label{coro 9} The flat and Zariski topologies on $\Spec(R)$ are the same if and only if every prime ideal of $R$ is maximal. \\
\end{corollary}

{\bf Proof.} It is an immediate consequence of Theorem \ref{theorem 2}. $\Box$ \\


\section{Introduction and Main Results}\label{Sec.Introduction}

Deep learning provides various models and algorithms to process data as efficiently
as biological nervous systems or neuronal responses in the human brain \cite{Lecun1998, Hinton2006, Krizhevsky2012, Goodfellow2016, LecunNature}.
It is based on deep neural network architectures and those structures bring essential tools for obtaining data features and function representations in practical applications.
A main concern about deep learning which has attracted much scientific attention and some criticism is its lack of
theories supporting its practical efficiency caused by its network structures, though there have been some theoretical attempts
from approximation theory viewpoints \cite{Mallat, Mallat2016, Poggio}.
In particular, for deep CNNs having convolutional structures without fully connected layers, it is unknown which kinds of functions can be approximated.
This paper provides a rigorous mathematical theory to answer this question and to illustrate the role of convolutions.

\bigskip

\noindent {\bf Notation and Concepts}

\medskip

\noindent {\bf Convolutional Filters and Matrices.}

\medskip

The deep CNNs considered in this paper have two essential ingredients: a rectified linear unit (ReLU) defined as a univariate nonlinear function $\sigma$ given by
$$\sigma (u) = (u)_+ = \max\{u, 0\}, \qquad u\in  {\mathbb R}$$
and a sequence of convolutional filter masks ${\bf w} =\{w^{(j)}\}_j$
inducing sparse convolutional structures. Here a filter mask $w=(w_k)_{k=-\infty}^{\infty}$ means a sequence of filter coefficients.
We use a fixed integer filter length $s \geq 2$ to control the sparsity, and assume that
$w^{(j)}_k \not=0$ only for $0\leq k \leq s$.
The convolution of such a filter mask $w$ with another sequence $v =(v_0, \ldots,v_D)$ is a sequence $w{*}v$ given by
$\left(w{*}v\right)_i = \sum_{k=0}^{D} w_{i-k} v_k$. This leads to
a $(D+s) \times D$ Toeplitz type convolutional matrix $T$
which has constant diagonals:
$$
T = \left[\begin{array}{llllllll}
w_0 & 0 & 0 & 0 & \cdot & 0  \\
w_1 & w_0 & 0 & 0 & \cdot & 0  \\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\
w_{s} & w_{s-1} & \cdots & \hspace*{-3mm} w_0 &  \hspace*{-3mm}  0 \cdots & 0  \\
0 & w_s & \cdots &  \hspace*{-3mm} w_1 &  \hspace*{-3mm} w_0 \cdots \ & 0 \\
\vdots & \ddots  & \ddots \ddots & \ddots  & \ddots \ddots & \vdots   \\
\cdots  & \cdots & 0 & \hspace*{-4mm} w_{s} & \cdots & w_0   \\
\cdots  & \cdots & \cdots &  \hspace*{-4mm} 0 &  \hspace*{-6mm} w_{s} \cdots & w_1   \\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots   \\
0 & \cdots  & \cdots \cdots &   \cdots  0  & \hspace*{6mm}  w_{s}  &  w_{s-1} \\
0 & \cdots & \cdots & \cdots &  \hspace*{4mm} \cdots 0 & w_{s}
\end{array}\right].
$$
Sparse matrices of this form induce deep CNNs which are essentially different from the classical neural networks involving full connection matrices.
Note that the number of rows of $T$ is $s$ greater than that of columns. This leads us to take a sequence of  linearly increasing widths $\{d_j = d + j s\}$ for the network,
which enables the deep CNN to represent functions of richer structures.

\medskip

\noindent {\bf Convolutional Neural Networks.}

\medskip

Starting with the action of $T$ on the input data vector $x \in {\mathbb R}^d$,
we can define a deep CNN of depth $J$ as a sequence of $J$ vectors
$h^{(j)}(x)$ of functions on ${\mathbb R}^d$ given iteratively by $h^{(0)}(x)=x$ and
$$
h^{(j)} (x) = \sigma \left(T^{(j)} h^{(j-1)}(x)  - b^{(j)}\right), \qquad j=1, 2, \ldots, J,
$$
where $T^{(j)} =\left(w^{(j)}_{i-k}\right)$ is a $d_j \times d_{j-1}$ convolutional matrix,
$\sigma$ acts on vectors componentwise,
and ${\bf b}$ is a sequence of bias vectors $b^{(j)}$.

Except the last iteration, we take $b^{(j)}$ of the form $[b_1 \ \ldots b_{s} \ b_{s+1} \ b_{s+1} \ \ldots \ b_{s+1} \ b_{d_j-s+1} \ \ldots b_{d_j}]^T$
with the $d_j - 2s$ repeated components in the middle. The sparsity of $T^{(j)}$ and the special form of $b^{(j)}$
tell us that the $j$-th iteration of the deep CNN involves $3 s +2$ free parameters. So in addition to the $2 d_J + s +1$ free parameters for $b^{(J)}, c \in {\mathbb R}^{d_J}, w^{(J)}$,
the total number of free parameters in the deep CNN is $(5 s +2) J + 2d -2 s -1$, much smaller than
that in a classical fully connected multi-layer neural network with full connection matrices $T^{(j)}$ involving $d_j d_{j-1}$ free parameters. It demonstrates the computational efficiency of deep CNNs.

\bigskip

\noindent {\bf Mathematical Theory of Deep CNNs}

\medskip

The hypothesis space of a learning algorithm is the set of all possible functions that can be represented or produced by the algorithm.
For the deep CNN of depth $J$ considered here, the hypothesis space is a set of functions defined by
$${\mathcal H}^{{\bf w}, {\bf b}}_J =\left\{\sum_{k=1}^{d_{J}} c_k
h^{(J)}_k (x): c \in {\mathbb R}^{d_J}\right\}.$$
This hypothesis space and its approximation ability depend completely on the sequence of convolutional filter masks ${\bf w} =\{w^{(j)}\}_{j=1}^J$ and the sequence of bias vectors ${\bf b}=\{b^{(j)}\}_{j=1}^J$.
Observe that each function in the hypothesis space ${\mathcal H}^{{\bf w}, {\bf b}}_J$
is a continuous piecewise linear function (linear spline) on any compact subset $\Omega$ of ${\mathbb R}^d$.
Our first main result verifies the universality of deep CNNs, asserting that any function $f\in C(\Omega)$,
the space of continuous functions on $\Omega$ with norm $\|f\|_{C(\Omega)} =\sup_{x\in \Omega} |f(x)|$,
can be approximated by ${\mathcal H}^{{\bf w}, {\bf b}}_J$ to an arbitrary accuracy when the depth $J$ is large enough.

\medskip

\noindent {\bf Theorem A.}
Let $2 \leq s\leq d$. For any compact subset $\Omega$ of ${\mathbb R}^d$ and any $f\in C(\Omega)$, there exist sequences ${\bf w}$ of filter masks, ${\bf b}$ of bias vectors and $f^{{\bf w}, {\bf b}}_J \in {\mathcal H}^{{\bf w}, {\bf b}}_J$ such that
$$
\lim_{J\to\infty} \|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} = 0.
$$

Our second main result presents rates of approximation by deep CNNs for functions in the Sobolev space $H^{r} (\Omega)$ with an integer index $r>2 + d/2$.
Such a function $f$ is the restriction to $\Omega$ of a function $F$ from the Sobolev space $H^{r} ({\mathbb R}^d)$ on ${\mathbb R}^d$ meaning that $F$ and all its partial derivatives up to order $r$ are square integrable on ${\mathbb R}^d$.

\medskip

\noindent{\bf Theorem B.}
Let $2 \leq s\leq d$ and $\Omega \subseteq [-1, 1]^d$. If $J \geq 2d/(s-1)$ and $f=F|_{\Omega}$ with $F \in H^{r} ({\mathbb R}^d)$ and an integer index $r>2 + d/2$,
then there exist ${\bf w}$, ${\bf b}$ and $f^{{\bf w}, {\bf b}}_J \in {\mathcal H}^{{\bf w}, {\bf b}}_J$ such that
$$\|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq c \left\|F\right\| \sqrt{\log J} \left(1/J\right)^{\frac{1}{2} + \frac{1}{d}},$$
where $c$ is an absolute constant and $\left\|F\right\|$ denotes the Sobolev norm of $F \in H^{r} ({\mathbb R}^d)$.

\medskip

According to Theorem B, if we take $s=\lceil 1+ d^\tau/2\rceil$ and $J= \lceil 4 d^{1-\tau}\rceil L$ with $0 \leq \tau \leq 1$ and $L \in {\mathbb N}$,
where $\lceil u\rceil$ denotes the smallest integer not smaller than $u$, then we have
$$ \|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq c \left\|F\right\| \sqrt{\frac{(1-\tau) \log d + \log L + \log 5}{4 d^{1-\tau} L}},$$
while the widths of the deep CNN are bounded by $12 L d$ and the total number of free parameters by
$$(5 s +2) J + 2d -2 s -1 \leq (73 L +2) d. $$
We can even take $L=1$ and $\tau=1/2$ to get a bound for the relative error
$$ \frac{\|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)}}{\left\|F\right\|} \leq \frac{c}{2}  d^{-\frac{1}{4}} \sqrt{\log (5\sqrt{d})} $$
achieved by a deep CNN of depth $\lceil 4 \sqrt{d}\rceil$ and at most $75 d$ free parameters, which decreases as the dimension $d$ increases.
This interesting observation is new for deep CNNs, and does not exist in the literature of fully connected neural networks.
It explains the strong approximation ability of deep CNNs.

A key contribution in our theory of deep CNNs is that
an arbitrary pre-assigned sequence $W=(W_k)_{-\infty}^{\infty}$ supported in $\{0, \ldots, {\mathcal M}\}$
can be factorized into convolutions of a mask sequence $\{w^{(j)}\}_{j=1}^{J}$.
It is proved by the same argument as in \cite{ZhouAA} for the case with the special restriction $W_0 \not=0$.
Convolutions are closely related to translation-invariance in speeches and images \cite{Mallat, Daubechies, Strang}, and also in some learning algorithms \cite{SmaleZhou, FanHuWuZhou}.

\medskip

\noindent{\bf Theorem C.}
Let $s \geq 2$ and $W=(W_k)_{-\infty}^{\infty}$ be a sequence supported in $\{0, \ldots, {\mathcal M}\}$ with ${\mathcal M} \geq 0$.
Then there exists a finite sequence of filter masks $\{w^{(j)}\}_{j=1}^{J}$ supported in $\{0, \ldots, s\}$ with $J < \frac{{\mathcal M}}{s-1}+1$
such that the convolutional factorization $W  = w^{(J)}{*}\ldots {*}w^{(2)}{*}w^{(1)}$ holds true.

\bigskip

\noindent{\bf Discussion}

\medskip

The classical shallow neural networks associated with an activation function $\sigma: {\mathbb R}  \to {\mathbb R}$ produce functions of the form
$$
f_N (x) =\sum_{k=1}^N c_k \sigma(\alpha_k \cdot x - b_k)
$$
with $\alpha_k \in {\mathbb R}^d, b_k, c_k \in {\mathbb R}$. A mathematical theory for approximation of functions by shallow neural networks was well developed three decades ago  \cite{Cybenko, Hornik, Barron, Mhaskar1993, Leshno1993, Pinkus1999}
and was extended to fully connected multi-layer neural networks shortly afterwards  \cite{Hornik, Mhaskar1993, Chui1996} .

The first type of results obtained in the late 1980s  are about universality,
asserting that any continuous function $f$ on any compact subset $\Omega$ of ${\mathbb R}^d$ can be approximated by some $f_N$ to an arbitrary accuracy when the number of hidden neurons $N$ is large enough.
Such results were given in  \cite{Cybenko, Hornik, Barron} when $\sigma$ is a sigmoidal function, meaning that $\sigma$ is a continuous strictly increasing function
satisfying $\lim_{u\to-\infty}\sigma(u)=0$ and $\lim_{u\to \infty}\sigma(u)=1$. A more general result with a locally bounded and piecewise continuous activation function $\sigma$
asserts \cite{Leshno1993, Pinkus1999} that universality holds if and only if $\sigma$ is not a polynomial.

The second type of results obtained in the early 1990s are about rates of approximation. When $\sigma$ is a $C^\infty$ sigmoidal function
and $f=F|_{[-1, 1]^d}$ for some $F\in L^2({\mathbb R}^d)$ with the Fourier transform $\hat F$  satisfying $|w| \hat F(w) \in L^1 ({\mathbb R}^d)$,
rates of type $\|f_N - f\|_{L^2_{\mu} ([-1, 1]^d)} =O(1/\sqrt{N})$  were given in \cite{Barron} where $\mu$ is an arbitrary probability measure $\mu$.
Analysis was conducted in \cite{Mhaskar1993} for shallow neural networks with more general continuous activation functions $\sigma$
satisfying a special condition with some $b\in{\mathbb R}$ that $\sigma^{(k)} (b)\not= 0$ for any nonnegative integer $k$ and a further assumption with some integer $\ell \not=1$ that
$\lim_{u\to-\infty} \sigma(u)/|u|^\ell=0$ and $\lim_{u\to \infty} \sigma(u)/u^\ell=1$.
The rates there are of type $\left\|f_N - f\right\|_{C([-1, 1]^d)} =O(N^{-r/d})$ for
$f \in C^r ([-1, 1]^d)$. Note that the ReLU activation function considered in this paper
does not satisfy the condition with $\sigma^{(k)} (b)\not= 0$ or the special assumption with $\ell\not= 1$.
To achieve the approximation accuracy $\left\|f_N - f\right\|_{C([-1, 1]^d)}  \leq \epsilon$,  when $r= \lceil \frac{d+1}{2} +2\rceil$ with $d/r \approx 2$,
the number of hidden neurons $N \geq \left(c_{f, d, \ell}/\epsilon\right)^{d/r}$ and the total number of free parameters is at least $\left(c_{f, d, \ell}/\epsilon\right)^{d/r} d$,
where the constant $ c_{f, d, \ell}$ depends on the dimension $d$ and might be very large.
To compare with our result, we take the filter length $s=\lceil 1+ d/2\rceil$ and depth $J= 4 L$ with $L \in {\mathbb N}$. We know from Theorem B that
the same approximation accuracy $ \|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq \epsilon$ with $0< \epsilon \leq c \left\|F\right\|$
can be achieved by the deep CNN of depth $J= 4 \lceil \frac{1}{\epsilon^2} \log  \frac{1}{\epsilon^2}\rceil$ having at most
$ \lceil \frac{75}{\epsilon^2} \log  \frac{1}{\epsilon^2}\rceil d$ free parameters, which does not depend on the dimension $d$.
Though a logarithmic term is involved, this dimension independence gives evidence for the power of deep CNNs.

A multi-layer neural network is a sequence of function vectors
$h^{(j)}(x)$ satisfying an iterative relation
$$
h^{(j)} (x) = \sigma \left(T^{(j)} h^{(j-1)}(x)  - b^{(j)}\right), \qquad j=1, 2, \ldots, J.
$$
Here $T^{(j)}$ is a full connection matrix without special structures.
So a deep CNN is a special multi-layer neural network with sparse convolutional matrices. This sparsity gives difficulty in developing a mathematical theory for deep CNNs,
since the techniques in the literature of fully connected shallow or multi-layer neural networks do not apply.
Our novelty to overcome the difficulty is
to factorize an arbitrary finitely supported sequence into convolutions of filter masks $\{w^{(j)}\}_{j=1}^J$ supported in $\{0, 1, \ldots, s\}$.
Our method can be applied to distributed learning algorithms \cite{LGZ, GLZ}.

Recently there have been quite a few papers \cite{Telgarsky2016, Eldan2016, Yarosky, Coifman, Grohs, Petersen}
on approximation and representation of functions by deep neural networks and benefit of depth, but all these results are for fully connected networks without pre-specified structures, not for deep CNNs.
In particular, it was shown in \cite{Grohs, Petersen} that the rate of approximaton of some function classes by multi-layer fully connected neural networks may be achieved by networks
with sparse connection matrices $T^{(j)}$, but the locations of the sparse connections are unknown. This sparsity of unknown pattern is totally different from that of deep CNNs, the latter enables
computing methods like stochastic gradient descent to learn values of the free parameters efficiently.

Deep CNNs are often combined with pooling, a small number of fully connected layers, and some other techniques for improving the practical performance of deep learning.
Our purpose to analyze purely convolutional networks is to demonstrate that convolution makes full use of shift-invariance properties of speeches and images for extracting data features efficiently.
Also, for processing an image, convolutions based on the 2-D lattice ${\mathbb Z}^2$ are implemented by taking inner products of $(s+1) \times (s+1)$ filter matrices with shifted patches of the image.
Though we do not consider such deep learning algorithms in this paper, some of our ideas can be used to establish mathematical theories for more general deep neural networks involving convolutions.

\bigskip

\noindent{\bf Methods}

\medskip

For approximation in $C(\Omega)$ we can only consider those Sobolev spaces which can be embedded into the space of continuous functions,
that is, those spaces with the regularity index $r > \frac{d}{2}$. To establish rates of approximation we require $r>\frac{d}{2} +2$ in Theorem B. In this case, the set $H^r (\Omega)$ is dense in $C(\Omega)$, so
Theorem A follows from Theorem B by scaling.

\bigskip

\noindent{\bf Proof of Theorem B.}
Let $J \geq \frac{2d}{s-1}$ and $m$ be the integer part of $\frac{(s-1)J}{d} -1\geq 1$. In our assumption, $f =F|_{\Omega}$ for some function $F \in H^{r} ({\mathbb R}^d)$ with the Fourier transform $\widehat{F}(\omega)$ giving the norm
$\left\|F\right\| =\left\|\left(1+ |\omega|^2\right)^{r/2} \widehat{F}(\omega)\right\|_{L^2}$.
By the Schwarz inequality and the condition $r>\frac{d}{2} +2$,
$v_{F, 2}:= \int_{{\mathbb R}^d} \|\omega\|_1^2 \left|\widehat{F}(\omega)\right| d\omega \leq c_{d, r} \left\|F\right\|$
where $c_{d, r}$ is the finite constant $\left\|\|\omega\|_1^2 \left(1+ |\omega|^2\right)^{-r/2}\right\|_{L^2}$.
Then we apply a recent result from \cite{Klusowski2018} on ridge approximation to $F|_{[-1, 1]^d}$ and know that there exists a linear combination of ramp ridge functions of the form
$$
F_m (x) = \beta_0 + \alpha_0 \cdot x + \frac{v}{m} \sum_{k=1}^m \beta_k \left(\alpha_k \cdot x - t_k\right)_+
$$
with $\beta_k \in [-1, 1], \|\alpha_k\|_1 =1, t_k \in [0, 1], \beta_0 = F(0), \alpha_0 = \nabla F(0)$ and $|v| \leq 2 v_{F, 2}$ such that
$$
\left\|F - F_m\right\|_{C([-1, 1]^d)} \leq c_0 v_{F, 2} \max\left\{\sqrt{\log m}, \sqrt{d}\right\} m^{-\frac{1}{2} - \frac{1}{d}}
$$
for some universal constant $c_0>0$.

Now we turn to the key step of constructing the filter mask sequence ${\bf w}$.
Define a sequence $W$ supported in $\{0, \ldots, (m+1) d-1\}$ by stacking the vectors $\alpha_0, \alpha_1, \ldots, \alpha_m$ (with components reversed) by
$$\left[W_{(m+1) d -1} \ \ldots \ W_{1} \ W_{0}\right] = \left[\alpha_m^T \ \ldots \ \alpha_1^T \ \alpha_0^T\right].$$
We apply Theorem C to the sequence $W$ with support in $\{0, 1, \ldots, (m+1) d\}$
and find a sequence of filter masks ${\bf w}=\{w^{(j)}\}_{j=1}^{\hat{J}}$ supported in $\{0, 1, \ldots, s\}$
with $\hat{J} < \frac{(m+1) d}{s-1}+1$ such that $W  = w^{(\hat{J})}{*}w^{(\hat{J}-1)}{*}\ldots {*}w^{(2)}{*}w^{(1)}.$
The choice of $m$ implies $\frac{(m+1) d}{s-1} \leq J$. So $\hat{J} \leq J$ and by taking $w^{(\hat{J}+1)} = \ldots = w^{(J)}$ to be the delta sequence, we have
$W  = w^{(J)}{*}w^{(J-1)}{*}\ldots {*}w^{(2)}{*}w^{(1)}.$
This tells us \cite{ZhouAA} that
$$T^{(J)} \ldots T^{(1)}= T^{W}$$
where
$T^{W}$ is the $d_J \times d$ matrix given by $[W_{\ell -k}]_{\ell=1, \ldots, d_J, k=1, \ldots, d}$. Observe
from the definition of the sequence $W$ that for $k =0, 1, \ldots, m$, the $(k+1)d$-th row of $T^{W}$ is exactly the transpose of $\alpha_k$.
Also, since $Js \geq (m+1) d$, we have $W_{Js}=0$ and the last row of $T^{W}$ is a zero row.

Then we construct ${\bf b}$. Denote $\|w\|_1=\sum_{k=-\infty}^{\infty} |w_k|$,
$B^{(0)} =\max_{x\in \Omega} \max_{k=1, \ldots, d} |x_k|$ and
$B^{(j)} = \|w^{(j)}\|_1 \ldots \|w^{(1)}\|_1 B^{(0)}$ for $j\geq 1$. Then we have
$$\left\|\left(T^{(j)} \ldots T^{(1)}  x\right)_k\right\|_{C(\Omega)} \leq B^{(j)}, \qquad \forall k=1, \ldots, d_j.$$
Take $b^{(1)} =- B^{(1)} {\bf 1}_{d_1} :=- B^{(1)} (1, \ldots, 1)^T$, and
$$b^{(j)} =
B^{(j-1)} T^{(j)} {\bf 1}_{d_{j-1}} - B^{(j)} {\bf 1}_{d_j}, \qquad j=1, \ldots, J-1. $$
Then for $j=1, \ldots, J-1$, we have
$$
h^{(j)} (x) = T^{(j)}\ldots T^{(1)} x + B^{(j)} {\bf 1}_{d_j}$$
and $b^{(j)}_{\ell} = B^{(j-1)} \sum_{k=0}^{s} w^{(j)}_{k} - B^{(j)}= b^{(j)}_{s+1}$ for $\ell=s+1, \ldots, d_j -s.$ Hence the bias vectors are of the required form.

Finally, we take the bias vector $b^{(J)}$ by setting $b^{(J)}_{\ell}$ to be
$$ \left\{\begin{array}{ll}
B^{(J-1)} (T^{(J)} {\bf 1}_{d_{J-1}})_{\ell} - B^{(J)}, & \hbox{if} \ \ell =d, d + Js, \\
B^{(J-1)} (T^{(J)} {\bf 1}_{d_{J-1}})_{\ell} + t_k, & \hbox{if} \ \ell =(k+1)d, \ 1 \leq k \leq m, \\
 B^{(J-1)}(T^{(J)} {\bf 1}_{d_{J-1}})_{\ell} + B^{(J)}, & \hbox{otherwise.} \end{array}\right. $$
Substituting this bias vector and the expression for $h^{(J-1)} (x)$ into the iterative relation of the deep CNN,
we see from the identity $T^{(J)} \ldots T^{(1)}= T^{W}$ and the definition of the sequence $W$ that the $\ell$-th component
$h^{(J)}_{\ell} (x)$ of $h^{(J)} (x)$ equals
$$ \left\{\begin{array}{ll}
\alpha_0 \cdot x + B^{(J)}, & \hbox{if} \ \ell =d,  \\
 B^{(J)}, & \hbox{if} \ \ell =d +Js,  \\
\left(\alpha_k \cdot x - t_k\right)_+, & \hbox{if} \ \ell =(k+1)d, \ 1 \leq k \leq m, \\
0, & \hbox{otherwise.} \end{array}\right. $$
Thus, we can take
$f^{{\bf w}, {\bf b}}_J = F_m|_{\Omega} \in \hbox{span}\{h^{(J)}_{k} (x)\}_{k=1}^{d_J} ={\mathcal H}^{{\bf w}, {\bf b}}_J$ and know that the error
$\|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq \left\|F - F_m\right\|_{C([-1, 1]^d)}$ can be bounded as
$$\|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq c_0 v_{F, 2} \max\left\{\sqrt{\log m}, \sqrt{d}\right\} m^{-\frac{1}{2} - \frac{1}{d}}.
$$
But $\frac{1}{2} (s-1)J \leq m d < (s-1)J$ and $2 r -d -4 \geq 1$. By a polar coordinate transformation,
$c_{d, r} d^{1 + \frac{1}{d}}
\leq \sqrt{\frac{d^6 \pi^{d/2}}{\Gamma (\frac{d}{2} +1)}} \left(1 + \frac{1}{\sqrt{2 r -d -4}}\right)$ which can be bounded by an absolute constant
 $c' :=\max_{\ell\in {\mathbb N}} 2\sqrt{\ell^6 \pi^{\ell/2}/\Gamma (\frac{\ell}{2} +1)}$. Therefore,
$$\|f-f^{{\bf w}, {\bf b}}_J \|_{C(\Omega)} \leq 2 c_0 c' \left\|F\right\| \sqrt{\log J} J^{-\frac{1}{2} - \frac{1}{d}}.
$$ This proves Theorem B by taking $c = 2 c_0 c'$.

\bigskip

Convolutional factorizations have been considered in our recent work \cite{ZhouAA} for sequences $W$ supported in $\{0, 1, \ldots, S\}$ with $S \geq d$ under the special restrictions $W_0>0$ and $W_{S}\not= 0$.
Theorem C gives a more general result by improving the bound for $J$ in \cite{ZhouAA} and removing the special restrictions on $W_0$ and $W_S$.

\bigskip

\noindent{\bf Proof of Theorem C.}
We apply a useful concept from the literature of wavelets \cite{Daubechies}, the symbol $\widetilde{w}$ of a sequence $w$ finitely supported in the set of nonnegative integers, defined as a polynomial on ${\mathbb C}$ by $\widetilde{w} (z) = \sum_{k=0}^\infty w_k z^k$.
The symbol of the convoluted sequence $a{*}b$ is given by $\widetilde{a{*}b} (z) = \widetilde{a} (z)\widetilde{b} (z)$.
Notice that the symbol $\widetilde{W}$ of  the sequence $W$ supported in $\{0, \ldots, {\mathcal M}\}$  is a polynomial of degree $M$ with real coefficients for some $0 \leq M \leq {\mathcal M}$.
So we know that complex roots $z_k =x_k + i y_k$ of $\widetilde{W}$ with $x_k \not=0$ appear in pairs
and by $(z-z_k)(z- \overline{z_k}) =z^2 - 2 x_k  z + \left(x_k^2 + y_k^2\right)$, the polynomial $\widetilde{W} (z)$ can be completely factorized as
$$
\widetilde{W} (z) =W_{M} \Pi_{k=1}^K \left\{z^2 - 2 x_k  z + \left(x_k^2 + y_k^2\right)\right\} \Pi_{k=2K+1}^{M} (z-x_k),
$$
where $2K$ is the number of complex roots with multiplicity, and $M - 2 K$ is the number of real roots with multiplicity.
By taking groups of up to $s/2$ quadratic factors (or $(s-1)/2$ quadratic factors with a linear factor) and $s$ linear factors in the above factorization,
we get $\widetilde{W} (z) = \widetilde{w^{(J)}}(z) \ldots \widetilde{w^{(2)}}(z) \widetilde{w^{(1)}}(z)$,  a factorization of $\widetilde{W}$ into polynomials of degree up to $s$, which yields a desired convolutional factorization $W  = w^{(J)}{*}w^{(J-1)}{*}\ldots {*}w^{(2)}{*}w^{(1)}$ and proves Theorem C.

\section*{Acknowledgments}
The author would like to thank Gilbert Strang and Steve Smale for their detailed suggestions and encouragement.
The work described in this paper is supported partially by the Research Grants Council of Hong Kong [Project No CityU 11306617] and by National Nature Science Foundation of China [Grant No 11461161006].

\bibliographystyle{abbrvnat}