text:
\begin{document}

\begin{frontmatter}

\title{ LEO intermediary propagation as a feasible alternative to Brouwer's gravity solution\tnoteref{KePASSA} }
\tnotetext[KePASSA]{Presented in KePASSA 2014, Logro\~no, Spain, April 23--25, 2014}

\author
{ Martin Lara\fnref{GRUCACI}
}
\fntext[GRUCACI]{ GRUCACI, University of La Rioja, Spain }

\ead{mlara0@gmail.com}

\address
{Columnas de H\'ercules 1, ES-11000 San Fernando, Spain}

\date{}

\begin{abstract}
The performance of Brouwer's gravity solution is compared with Deprit's second-order radial intermediary. Taking the main problem of the artificial satellite as the test model, that is perturbations are limited to the effects of the second zonal harmonic, it is demonstrated that the intermediary solution provides an efficient alternative for the analytical propagation of low earth orbits in a range determined by non-impact orbits with eccentricities below one tenth.
\end{abstract}

\begin{keyword}

Brouwer's theory \sep SGP4 \sep intermediaries \sep elimination of the parallax

\end{keyword}

\end{frontmatter}

\section{Introduction}

Increasing needs of collision avoidance stir recent concern in reviewing SGP4 \citep{HootsRoehrich1980}, an orbit prediction model that is customarily used in the propagation of two-line element sets \citep[see also][and references therein]{HootsSchumacherGlover2004,ValladoCrawfordHujsakKelso2006}.
Among the reasons for this concern is the need of including uncertainties estimation in the predictions, but also the detected significant along-track errors which may be related to missing terms in the SGP4 implementation of the gravitational theory \citep{Kelso2007,Easthope2014}. These facts motivated me to search the literature for alternatives to the gravitational solution used by SGP4. I found that these alternatives do exist, can be refined to cope with more subtle short-period effects, and may provide increased efficiency in terms of precision and computation time when compared with the classical approach. In spite of comparisons are limited to the main problem of artificial satellite theory, the proposed alternative may be further explored to include more disturbing effects which are required for realistic propagation of space debris or orbits about asteroids.

\par

Brouwer's celebrated closed form solution of the earth's artificial satellite problem \citep{Brouwer1959} is in the roots of SGP4.
In spite of the undeniable merits of Brouwer's seminal approach, or its variants for properly dealing  with small eccentricities and inclinations \citep{Lyddane1963,CohenLyddane1981}, proceeding by averaging is not the unique possibility in obtaining analytical solutions by general perturbations, and the use of \emph{intermediary} orbits like useful approximate solutions of the problem of artificial satellite theory (AST) was proposed as early as in 1957 \citep{Sterne1958}.
\par

AST intermediaries are commonly obtained by reorganizing the terms of the disturbing function, a simple expedient that may be preceded by the simultaneous addition and subtraction to the geopotential of some smartly-chosen supplementary terms. After reorganization, a part of the Hamiltonian that admits a separable generating function is taken as the (zero order) integrable problem whereas the rest of the disturbing function is taken as the perturbation, which may be further neglected \citep[see][for instance]{Aksnes1965,Oberti2005}. Hence, \emph{common} intermediaries are formulated in the same variables as the original satellite problem ---traditionally in spherical variables.
\par

It is habitually accepted that useful intermediary solutions should retain all the first-order secular and long-period effects of the artificial satellite problem and as much as possible of the short-period effects \citep[cf.][]{GarfinkelAksnes1970}. Because of that, neither the Keplerian Hamiltonian nor the equatorial main problem\footnote{We call the \emph{equatorial main problem} to the perturbed Keplerian motion of a satellite in the main problem assumptions that is constrained to lie on the earth's equatorial plane.} are considered suitable intermediaries in spite of their integrability \citep{Jezewski1983}. Including some of the second order secular effects of the geopotential in the intermediary is highly desirable too, and hence solutions devised by \citet{Vinti1959,Vinti1961,Vinti1966} or equivalent ones based on the generalized problem of two fixed centers \citep[see][and references therein]{AksenovGrebenikovDemin1962,LukyanovEmeljanovShirmin2005} are considered remarkable achievements.\footnote{Vinti found a particular choice for the coefficients of the zonal potential

that makes it separable in oblate spheroidal coordinates. Vinti's analytical solution closely applies to earth's orbits where the zonal coefficient of degree 4 almost matches his choice. The equivalence between Vinti's model and the problem of two fixed centers was later recognized.}
\par

Eventually, the efforts of \citet{CidLahulla1969} in obtaining a competitive alternative to Brouwer's solution produced a major breakthrough in the search for efficient intermediaries. Proceeding in polar-nodal variables, the usual polar ones complemented with the argument of the ascending node and its conjugate momentum, Cid and Lahulla showed that, contrary to the two canonical transformations required in Brouwer's approach, a single contact transformation is enough for removing the argument of the latitude from the main problem Hamiltonian, which after that, and in view of the cyclic character of the argument of the node, results to be integrable.
\par

Up to the first order, Cid and Lahulla's Hamiltonian is formally equal to the radial ---and, therefore, integrable--- part of the main problem Hamiltonian, in this way highlighting its relationship with common intermediaries, on one hand, and disclosing the important role played by polar-nodal variables in the search for them, on the other. Besides, it incorporates all the first-order secular effects of the main problem \citep{DepritFerrer1987} as well as the short-period terms that affect the radial distance, thus fulfilling the traditional requisites for acceptable main problem intermediaries. Furthermore, Cid and Lahulla's solution also takes into account the remaining first-order periodic terms of the main problem, which are recovered by means of a contact transformation. All these facts inspired Deprit, who introduced the concept of \emph{natural} intermediaries, which are integrable after a contact transformation, and demonstrated that most of the common intermediaries can be ``naturalized'' by finding the transformation that turns the main problem into the intermediary \citep{Deprit1981}. Remarkably, Deprit also showed how these contact transformations may be extended to higher orders, and proposed his own radial intermediary which, as opposite to other existing intermediaries, does not rely on the evaluation of elliptic functions. Later efforts based on Deprit's approach, showed how the generation of intermediaries of AST can be systematized \citep{FerrandizFloria1991}.
\par

In spite of the efforts in improving the intermediaries' performance by including second order effects of the gravity potential \citep{Garfinkel1959,Aksnes1967,AlfriendDasenbrockPickardDeprit1977,DepritJGCD1981,CidFerrerSeinEchaluce1986,DepritFerrer1989}, increasing requirements on satellite orbit prediction accuracy soon led to a decline in interest in intermediaries, to favor instead analytical and semi-analytical theories based on averaging. Still, the use of intermediary orbits of AST is enjoying a revival these days, and new applications of intermediary-based solutions had been recently encouraged for different purposes, as for predicting long-term bounded relative motion \citep{LaraGurfil2012} or like a suitable choice for onboard orbit propagation as opposite to the usual numerical integration \citep{GurfilLara2014}.
\par

Here, the main problem of AST is taken as a test model to demonstrate that the use of intermediary orbits may provide an efficient alternative to Brouwer's gravity solution for the propagation of low earth orbits (LEO) in the short time intervals required by usual catalogue maintenance. Indeed, while natural intermediary closed-form solutions are unquestionably of higher complexity than Brouwer's secular terms and, in consequence, their evaluation becomes much more computationally costly, they only rely on one simplification transformation whose terms are definitely simpler than the corresponding Fourier series required by Brouwer's double averaging approach. Besides, AST intermediaries do not suffer from mathematical singularities at the critical inclination \citep[see reviews in][]{CoffeyDepritMiller1986,Jupp1988,Lara2014Rome} and hence do not need to rely upon the functional patches on which higher-order analytical theories by averaging unavoidably depend ---cf.~Section 7 of \citep{CoffeyNealSegermanTravisano1995} or appendix A.F.~in \citep{HootsSchumacherGlover2004}.\footnote{It must be emphasized that neither analytical theories based on a double averaging nor existing intermediaries can deal properly with second-order perigee effects, which are crucial in the study of frozen-perigee orbits. This special class of orbits is usually approached either from a semi-analytical perspective or a purely numerical one \citep{CuttingFrautnickBorn1978,Broucke1994,CoffeyDepritDeprit1994,Lara1999,Lara2008}.}
\par

On the other hand, efforts in solving higher-order intermediaries by separation of variables have been unsuccessful, in spite of their integrable character. This shortcoming imposes general perturbations algorithms to progress with an additional transformation in order to achieve the analytical solution of the intermediary to higher orders \citep{Aksnes1967,DepritJGCD1981,DepritRichardson1982}. However, this extra transformation may well be avoided in the case of LEO, an instance in which orbital eccentricity is small.
In particular, this is the case of Deprit's radial intermediary, or DRI in short \citep{Deprit1981}. For the lower-eccentricity orbits, which are a vast majority in a space catalogue of earth orbiting objects, DRI  can be solved up to higher-order effects by separation of the generating function \citep{DepritJGCD1981,Floria1993}.

\par

In the present research, a quasi-Keplerian intermediary has been constructed which admits standard closed-form solution and is of straightforward evaluation. The proposed intermediary is based on a simplification of DRI which is valid for low-eccentricity orbits and has been improved with the computation of those second order short-period corrections that are consistent with the simplification assumptions of the theory, in this way incorporating second-order secular as well as periodic terms of the main problem into the analytical solution. Also, a complete reordering of the periodic corrections has been carried out that notably fastens their evaluation.
\par

Taking the numerical integration of the main problem as the true solution, a variety of tests have been carried out in a representative set of LEO with inclinations encompassing from equatorial to polar. These tests show that the performance of the second-order intermediary is clearly better than that of Brouwer's analogous propagations in nonsingular variables for short time intervals. Besides, the intermediary-based LEO propagator remains very competitive for time intervals spanning up to several weeks.
\par

Propagation of space debris and orbits about asteroids undoubtedly requires more complete forces model than the one discussed here. In particular orbit propagation about highly inhomogeneous gravity fields definitely requires inclusion of tesserals effects, which can also be dealt with analytically \citep[see][and references therein]{Garfinkel1965,CeccaroniBiggs2013,LaraSanJuanLopezOchoa2013a} albeit the efficiency of the closed form approach may be questionable in some cases \citep[cf.][]{LaraSanJuanLopezOchoa2013}. The usefulness of replacing parts of the standard perturbation solution by analogous results based on intermediary orbits still remain to be explored.

\section{Deprit's radial intermediary}

Assume an inertial reference system defined by the earth's center of mass, the $z$ axis defined by the earth's rotation axis, and the $x$ and $y$ axes lying in the equatorial plane and defining a direct frame.
In the canonical set of polar-nodal variables $(r,\theta,\nu,R,\Theta,N)$ ---standing for the radial distance from the origin, the argument of the latitude, the longitude of the ascending node, the radial velocity, the modulus of the angular momentum vector, and the projection of the angular momentum over the $z$ axis, respectively--- the main problem Hamiltonian is written
\begin{equation} \label{hmp}
\mathcal{H}=\frac{1}{2}\left(R^2+\frac{\Theta^2}{r^2}\right)-\frac{\mu}{r}\left[
1-J_2\,\frac{\alpha^2}{r^2}\,P_2(s\sin\theta)\right],
\end{equation}
where $\mu$ is the earth's gravitational parameter, the scaling factor $\alpha$ is the earth's mean equatorial radius, and $J_2$ is the second zonal harmonic coefficient of the geopotential. Besides, $P_2$ is the Legendre polynomial of degree 2, $s\equiv\sin{i}$, and $c\equiv\cos{i}=N/\Theta$. In view of $\nu$ is ignorable in Eq.~(\ref{hmp}), its conjugate momentum $N$ is an integral of the motion, reflecting the symmetry of the main problem dynamics with respect to rotations about the $z$ axis.
\par

Deprit's radial intermediary is obtained after simplifying the main problem Hamiltonian by applying to it the elimination of the parallax, which is a canonical transformation
\begin{equation*}
(r,\theta,\nu,R,\Theta,N)\longrightarrow(r',\theta',\nu',R',\Theta',N')
\end{equation*}
that removes non essential short-periodic terms of the original Hamiltonian without reducing the number of degrees of freedom \citep{Deprit1981}.
\par

The elimination of the parallax from the main problem Hamiltonian is fully documented in the literature so details are not provided here. The interested reader may consult the original paper of \citet{Deprit1981} or my simpler re-derivation of this canonical simplification based on the use of Delaunay variables \citep{LaraSanJuanLopezOchoa2013b}.
\par

After eliminating the parallax, and up to the second order of $J_2$, the main problem Hamiltonian in new (prime) polar-nodal variables is
\begin{equation} \label{hpnp}
\mathcal{H}=H_0+H_1+\frac{1}{2}H_2+\mathcal{O}(J_2^3)
\end{equation}
where, dropping primes for brevity,
\begin{align} \allowdisplaybreaks \label{H00}
H_0 \;=&\; \frac{1}{2}\left(R^2+\frac{\Theta^2}{r^2}\right)-\frac{\mu}{r}, \displaybreak[0] \\ \label{H01}
H_1 \;=&\;-\varepsilon\,\frac{\Theta^2}{r^2}\left(1-3c^2\right) \displaybreak[0] \\ \label{H02}
H_2 \;=&\; \varepsilon^2\,\frac{\Theta^2}{r^2}\left\{ 1-21c^4-6\left(c^2-\frac{5}{8}s^4\right)(\kappa^2+\sigma^2) \right. \\ \nonumber
& \left. +21\left(1-\frac{15}{14}s^2\right)s^2\left[(\kappa^2-\sigma^2)\cos2\theta+2\kappa\,\sigma\sin2\theta\right] \right\}.
\end{align}
For abbreviation, it has been introduced the notation
\begin{equation}
\varepsilon\equiv\varepsilon(\Theta)=-\frac{1}{4}J_2\frac{\alpha^2}{p^2},
\end{equation}
where $p$ is the parameter of the conic, or \textit{semilatus rectum},
\begin{equation}
p=\Theta^2/\mu,
\end{equation}
and $\kappa$ and $\sigma$ are the projections of the eccentricity vector in the orbital frame
\begin{equation} \label{ksef}
\kappa=e\cos{f}, \qquad \sigma=e\sin{f},
\end{equation}
where $f$ is the true anomaly and $e$ is the eccentricity. Based on the conic solution of the Keplerian motion, these functions are easily expressed in polar-nodal variables as
\begin{equation} \label{ef}
\kappa\equiv\kappa(r,\Theta)=\frac{p}{r}-1, \qquad \sigma\equiv\sigma(R,\Theta)=\frac{p\,R}{\Theta}.
\end{equation}
Note that $f\equiv{f}(r,R,\Theta)$ and $e\equiv(r,R,\Theta)$.
\par

The appearance of $\theta$ in Eq.~(\ref{H02}) prevents integrability, and the transformed Hamiltonian in Eq.~(\ref{hpnp}) has the same degrees of freedom as the original main problem in addition to being yet more intricate. However, the new Hamiltonian is said to be ``simplified'' \citep{DepritFerrer1989}. Indeed, because the argument of perigee is $\omega=\theta-f$, it is simple to derive from Eq.~(\ref{ksef}) the relations
\begin{eqnarray} \label{ecg}
e\cos\omega
&=& \kappa\cos\theta+\sigma\sin\theta, \\ \label{esg}
e\sin\omega
&=& \kappa\sin\theta-\sigma\cos\theta,
\end{eqnarray}
from which
\begin{eqnarray}
e^2\cos2\omega &=& (\kappa^2-\sigma^2)\cos2\theta+2\kappa\sin2\theta,
\\ \label{e2}
e^2 &=& \kappa^2+\sigma^2.
\end{eqnarray}
Hence, Eq.~(\ref{H02}) is rewritten as
\begin{equation} \label{H02d}
H_2=\varepsilon^2\,\frac{\mu}{r}\,\frac{p}{r}\left[1-21c^4-6\left(c^2-\frac{5}{8}s^4\right)e^2
+21\left(1-\frac{15}{14}s^2\right)s^2e^2\cos2\omega\right],
\end{equation}
in this way showing that the maximum negative power of $r$ in Eq.~(\ref{hpnp}) is $-2$ contrary to the power $-3$ in Eq.~(\ref{hmp}). This fact makes a subsequent closed form elimination of the short-period terms from Eq.~(\ref{hpnp}) by means of the usual Delaunay normalization \citep{Deprit1982} much simpler than Brouwer's direct elimination, on the one hand, and eases the construction of higher order solutions, on the other.\footnote{The total number of terms of the short-period transformations resulting from the elimination of the parallax and a consequent normalization is claimed to be reduced to just one fourth of the number of terms required in a third-order solution in closed when computed by the classical, Brouwer style normalization \citep[cf.][]{CoffeyDeprit1982}.}
\par

In spite of the non-integrable character of the simplified Hamiltonian, if terms multiplied by $e^2\,J_2^2$, which will be of higher order when $e^2\sim{J}_2$ (say $e<0.1$ for the earth if we neglect terms of the order of $10^{-8}$), are neglected then Eq.~(\ref{H02}) is converted into
\[
H_2=\frac{\Theta^2}{r^2}\,\varepsilon^2\left(1-21c^4\right),
\]
and the argument of the latitude becomes cyclic as it is cancelled from Eq.~(\ref{H02d}). Therefore, Eq.~(\ref{hpnp}) is simplified to the radial intermediary
\begin{equation} \label{qks}
\mathcal{D}=\frac{1}{2}\left(R^2+\frac{\tilde\Theta^2}{r^2}\right)-\frac{\mu}{r},
\end{equation}
where
\begin{equation}
\tilde\Theta=\Theta\,\sqrt{1-\varepsilon\,(2-6c^2)+\varepsilon^2\,(1-21c^4)}.

\end{equation}
Equation (\ref{qks}) is a \emph{quasi-Keplerian} system with varying (depending on $J_2$) angular momentum $\tilde\Theta$, which is called here DRI.\footnote{In fact, in view of Eqs.~(\ref{e2}) and (\ref{ef}), the eccentricity does not depend on $\theta$ and integrability can be obtained by neglecting only from Eq.~(\ref{H02d}) the terms that are affected by $\omega$, in this way keeping in the intermediary more effects of the original problem. But the integration of this more complete intermediary requires the apparatus of perturbation theory \citep[cf.][]{DepritJGCD1981} which is avoided in the case of quasi-Keplerian systems.}
\par

The integration of Eq.~(\ref{qks}) can be done by the standard Hamilton-Jacobi reduction to Delaunay-similar variables. This procedure requires the introduction of auxiliary variables for performing the quadratures, the so-called \emph{anomalies}, and solving the Kepler equation. For completeness, the necessary formulae are detailed in \ref{s:qK}.
\par

However, the quasi-Keplerian system provides only the solution in the prime space, and the integration of DRI is not completed until the short-period terms removed by the elimination of the parallax are recovered. Sequences for recovering these terms from the direct transformation, as well as the inverse transformation required for computing initial conditions in the prime space, are provided in \ref{s:shp}. Note that, in spite of the transformation from and to prime variables is computed in closed form of the eccentricity, in consequence with the assumption made of neglecting terms of $\mathcal{O}(e^2)$ from the second order Hamiltonian after the elimination of the parallax, the transformation equations for the second order short-period effects have been analogously simplified. Note also that evaluation of the corrections only involves sine and cosine functions of arguments $2\theta$ and $4\theta$.

\section{Comparison with Brouwer's solution}

Brouwer's gravity solution is accomplished by finding the canonical transformations that reduce the satellite problem to its secular terms \citep{Brouwer1959}. These transformations are carried out in the set of Delaunay variables $(\ell,g,h,L,G,H)$,

which comprises three angles: the mean anomaly $\ell$, the argument of the perigee $g$, and the argument of the node $h$, as well as their  three canonical conjugate momenta: the Delaunay action $L=\sqrt{\mu\,a}$, conjugate to $\ell$, the total angular momentum $G=L\sqrt{1-e^2}$, conjugate to $g$, and the polar component of the angular momentum $H=G\cos{i}$, conjugate to $h$. Note that $g$ was previously noted $\omega$, and that $h$, $G$, and $H$ are noted $\nu$, $\Theta$, and $N$, respectively, in polar-nodal variables.
\par

The computation of ephemeris based on Brouwer's canonical transformations in Delaunay variables is known to introduce errors of the first order in the short-period corrections when the eccentricity approaches to zero. This happens because of the singularity of Delaunay variables in the case of zero eccentricity orbits.\footnote{Delaunay variables are also singular for zero inclination orbits} However, since these singularities are just virtual \citep{Henrard1974}, the trouble is avoided by expressing the perturbations of the elements in a different set of nonsingular variables.
\par

Thus, in order to make Brouwer's solution useful also for low-eccentricity and low-inclination orbits, instead of using Delaunay variables \cite{Lyddane1963} resorts to the set of Poincar\'e canonical elements
\begin{align} \allowdisplaybreaks
x_1 = & L \displaybreak[0] \\
x_2 = & \sqrt{2L}\,\sqrt{1-\eta}\cos(g+h) \displaybreak[0] \\
x_3 = & \sqrt{2L}\,\sqrt{\eta\,(1-c)}\cos{h} \displaybreak[0] \\
y_1 = & \ell+g+h \displaybreak[0] \\
y_2 = &-\sqrt{2L}\,\sqrt{1-\eta}\sin(g+h) \displaybreak[0] \\
y_3 = &-\sqrt{2L}\,\sqrt{\eta\,(1-c)}\sin{h}
\end{align}
where
\begin{equation}
\eta=\sqrt{1-e^2}=\frac{G}{L}
\end{equation}
is commonly called the ``eccentricity function''.
\par

Note that, in Lyddane's view, the perturbation theory is not constructed in Poincar\'e elements, but in Delaunay ones. After that, the perturbation theory is reformulated in the desired set of variables, which, furthermore, do not need to be canonical. Indeed, other different sets of nonsingular variables can be used, and, because of the axial symmetry of the main problem, the evaluation of Brouwer's analytical solution is found more expedite when using the non-canonical, nonsingular set:
\begin{equation} \label{FCShHL}
F=\ell+g, \quad C=e\cos{g}, \quad S=e\sin{g}, \quad h, \quad H, \quad L,
\end{equation}
proposed by \cite{DepritRom1970}.
\par

The locution ``Brouwer's solution'' is loosely applied to different versions of the perturbation solution to the problem of the artificial satellite found by \citet{Brouwer1959}. Here, this term is used to refer to an analytical solution obtained by double averaging, which  comprises the secular terms up to the second order, but is limited to first-order periodic terms ---in agreement with common implementations of this theory. In particular, because the concern is limited to the propagation of low-eccentricity orbits, Brouwer's solution is formulated in the nonsingular variables proposed by \citet{DepritRom1970}. Besides, Brouwer's solution is constrained to the main problem, and since it is compared with DRI both in accuracy and computing time, in order to obtain results that are as far as possible unbiased Brouwer's solution has been simplified to match the assumptions made in the analytical integration of DRI. Namely, it has been truncated by neglecting the order of $\mathcal{O}(J_2^2\,e^2)$ in the secular terms
and the $\mathcal{O}(J_2\,e^4)$ in the transformation equations.

\par

A set of numerical experiments that has been carried out to compare the relative efficiency of both approximate analytical solutions, Brouwer and DRI, is summarized in what follows. The reference orbits used as true solutions in the computation of the errors are obtained by the numerical integration of the main problem in Cartesian coordinates. In particular, the tests are based on a set of LEO with the following orbital elements \citep{GurfilLara2014}
\begin{equation} \label{iicc}
a=7000\,\mathrm{km}, \quad \omega=10^\circ, \quad \Omega=0, \quad f=15^\circ,
\end{equation}
and varying inclinations from equatorial to polar. The tests have been carried out for the case of almost-circular orbits, with $e=0.005$, and also, for  $e=0.075$, which is assumed to be the maximum eccentricity before re-entry for this kind of orbit.
\par

In general, it is found that DRI reduces the errors by about one order of magnitude when compared with Brouwer's solution. Because the purpose of this research is to asses the relative efficiency of both different solutions, it is illustrative enough to compare the errors in the radial distance and in the modulus of the velocity vector. These comparisons are displayed in Figs.~\ref{f:e005days7} and \ref{f:e075days7}, which show the one-week time history of the errors obtained when using the different analytical solutions for orbits with initial conditions in Eq.~(\ref{iicc}).
\par

\begin{figure}[htbp]
\centerline{
\includegraphics[scale=0.79]{e005i05dr.pdf} \quad \includegraphics[scale=0.75]{e005i05dv.pdf} \\[1.5ex]
}
\centerline{
\includegraphics[scale=0.79]{e005i55dr.pdf} \quad \includegraphics[scale=0.75]{e005i55dv.pdf} \\[1.5ex]
}
\centerline{
\includegraphics[scale=0.79]{e005i89dr.pdf} \quad \includegraphics[scale=0.75]{e005i89dv.pdf}
}
\caption{Propagation errors of Brouwer's solution (gray), first order of DRI (black) and second order of DRI (white, over the axis of abscissas). From top to bottom, $i=5$, $55$ and $89\deg$. Initial conditions in Eq.~(\ref{iicc}) for an almost-circular orbit with $e=0.005$}
\label{f:e005days7}
\end{figure}

\begin{figure}[htbp]
\centerline{
\includegraphics[scale=0.78]{e075i05dr.pdf} \quad \includegraphics[scale=0.77]{e075i05dv.pdf} \\[1.5ex]
}
\centerline{
\includegraphics[scale=0.78]{e075i55dr.pdf} \quad \includegraphics[scale=0.77]{e075i55dv.pdf} \\[1.5ex]
}
\centerline{
\includegraphics[scale=0.78]{e075i89dr.pdf} \quad \includegraphics[scale=0.77]{e075i89dv.pdf}
}
\caption{Propagation errors of Brouwer's solution (gray), first order of DRI (black) and second order of DRI (white, over the axis of abscissas), for, from top to bottom, $i=5$, $55$ and $89\deg$. Initial conditions in Eq.~(\ref{iicc}) for a slightly elliptic orbit with $e=0.075$}
\label{f:e075days7}
\end{figure}

Errors obtained when using the first order truncation of DRI, hereafter named DRI1, are also provided (black curves in Figs.~\ref{f:e005days7} and \ref{f:e075days7}). As observed in Fig.~\ref{f:e005days7}, in the case of the lower eccentricities DRI1 enjoys similar performance to Brouwer's solution for low- and high-inclination orbits. For medium inclination orbits, DRI1 clearly outperforms Brouwer's solution, and only performs slightly worse than DRI. The behavior of DRI1 remains the same for the higher-eccentricity orbits, as shown in Fig.~\ref{f:e075days7}, still performing better than Brouwer's solution in general, the case of high-inclination orbits being the unique where the performance of Brouwer's solution is comparable to that of DRI1.
\par

Similar computations have been carried out for time intervals spanning up to one month, which show analogous results. Thus, for 30 days propagation DRI always remain within less than 20 meter from the true (numerically computed) distance, and within less than 2\;cm/s of the true velocity for the lower-eccentricity orbits. Corresponding figures are 80\;m for distance and 4\;cm/s for velocity in the case of Brouwer's solution. These errors notably increase for higher eccentricities, but still errors obtained with DRI computations remain better than half a km in distance in all cases checked, while oscillations of the velocity errors remain below $\pm50$\;cm/s. In the case of Brouwer's solution, the errors after one month propagation may reach 5\;km in distance and 4\;m/s in velocity. Tests carried out include orbits with eccentricities in the range $0<e<0.1$.
\par

In addition to the higher precision obtained with DRI, because DRI equations are much simpler than Brouwer's equations, in all the cases tested the evaluation of DRI only requires about one fourth of the evaluation time required by Brouwer's solution, on average, in this way providing a definitive advantage of DRI over Brouwer's solution.
\par

Results in Figs.~\ref{f:e005days7} and \ref{f:e075days7} clearly show the importance of taking into account second-order effects in the short-period corrections. To further illustrate this issue, a version of Brouwer's solution which includes the more relevant terms of the second-order corrections of the short-period terms has also been computed \citep[see][for instance]{DepritRom1970}. Comparisons of this improved Brouwer's solution with DRI are presented in Fig.~\ref{f:brouwer2nd} limited to errors in distance. Now Brouwer's improved solution clearly outperforms DRI for the lower eccentricity orbits, but this solution notably deteriorates for high-inclination orbits when the ellipticity is higher. Besides, Brouwer's improved solution unavoidably requires yet more computational effort than the original one.

\begin{figure}[htbp]
\centerline{
\includegraphics[scale=1.075]{e005i05dr2nd.pdf} \quad \includegraphics[scale=1.07]{e075i05dr2nd.pdf} \\[1.5ex]
} \centerline{
\includegraphics[scale=1.075]{e005i55dr2nd.pdf} \quad \includegraphics[scale=1.07]{e075i55dr2nd.pdf} \\[1.5ex]
} \centerline{
\includegraphics[scale=1.075]{e005i89dr2nd.pdf} \quad \includegraphics[scale=1.07]{e075i89dr2nd.pdf}
}
\caption{Propagation errors of the second order of DRI (gray) and Brouwer's solution with second order corrections for the short-period effects (black). From top to bottom, $i=5$, $55$ and $89\deg$. The left column corresponds to $e=0.005$ and the right one to $e=0.075$.}
\label{f:brouwer2nd}
\end{figure}

\section{Conclusions}

When using Brouwer's gravity solution (in nonsingular elements) for short-term propagation, the more important source of errors comes from neglecting the contribution of second-order short-period effects in the computation of the theory, which in the case of LEO may result in the introduction of a concomitant uncertainty of tens of meters in the propagation of initial conditions. Besides, the computation of short-period corrections requires the evaluation of long Fourier series even when limited to first-order effects. Hence,  in order to speed computations, practical implementations of Brouwer's theory, as in SGP4, make additional simplifications by neglecting some of the first-order short-period corrections.
\par

On the contrary, first- and second-order short-period corrections are both obtained in a compact form of fast and straightforward evaluation when the gravity solution is approached with DRI, albeit the contribution of second-order secular and long-period terms is limited in DRI to the case of the lower eccentricity orbits. This fact makes that using DRI in the propagation of LEO leads to an increased precision when compared to the classical Brouwer's gravity solution, with the additional bonus of dramatically reducing the computation time. Hence, performances of existing orbit propagation software would benefit of replacing parts of the standard perturbation solution by analogous results based on intermediary orbits.
\par

While the model has been limited in this research to the main problem of artificial satellite theory, extension of the intermediary solution to other zonal models is encouraged by the quality of current results. The same assumptions made for integrating DRI as a quasi-Keplerian system apply also to models involving even zonal harmonics. In the case of odd zonal harmonics, separability of the intermediary obtained after the elimination of the parallax requires  making a stronger assumption by neglecting also long-period terms of the second order that are multiplied by the eccentricity. Nevertheless, the intermediary solution may still provide an acceptable accuracy for short-term propagation. The construction of a radial intermediary for a more complete zonal model is in progress, and corresponding results will be published elsewhere.

\appendix

\section{Solution of DRI} \label{a:DRI}

\subsection{Integration of the quasi-Keplerian system} \label{s:qK}
The quasi-Keplerian system is realized in the prime space; hence all variables of this appendix should be understood as prime variables.
\par
From initial conditions $(r_0,\theta_0,\nu_0,R_0,\Theta,N)$, where $\Theta$ and $N$ are constant in the prime space, the auxiliary constants
\begin{align} \allowdisplaybreaks
c \;=&\; \frac{N}{\Theta} \displaybreak[0] \\
\varepsilon \;=&\;-\frac{1}{4}J_2\,\frac{\alpha^2}{(\Theta^2/\mu)^2}, \displaybreak[0] \\
\tilde{\Theta} \;=&\; \Theta\,\sqrt{1-(2-6c^2)\,\varepsilon+(1-21c^4)\,\varepsilon^2}  \displaybreak[0] \\
\zeta \;=&\; \frac{\Theta}{\tilde\Theta}\left[1+\left(2-12c^2\right) \varepsilon-\left(3-105 c^4\right) \varepsilon ^2\right],

 \displaybreak[0] \\
\chi \;=&\; 6\varepsilon\left(1-7\varepsilon\,c^2\right)\frac{N}{\tilde\Theta},

 \displaybreak[0] \\
\tilde{p} \;=&\; \frac{\tilde{\Theta}^2}{\mu} \displaybreak[0] \\
a \;=&\; -\frac{\mu}{R_0^2+(\tilde{\Theta}^2/r_0^2) - 2\mu/r_0} \displaybreak[0] \\
e \;=&\; \sqrt{1-\frac{\tilde{p}}{a}},
\end{align}
as well as the initial values of the anomalies
\begin{align} \allowdisplaybreaks
f_0 \;\phantom{=}&\; \mathrm{from:} \qquad e\cos{f}_0=\frac{\tilde{p}}{r_0}-1, \quad e\sin{f}_0= R_0\,\sqrt{\frac{\tilde{p}}{\mu}} \displaybreak[0] \\
u_0 \;=&\; 2\arctan\left(\sqrt{\frac{1 - e}{1 + e}}\tan\frac{f_0}{2}\right), \displaybreak[0] \\
\ell_0 \;=&\; u_0-e\sin{u}_0,
\end{align}
are first computed.
Then, at a given time $t$, the anomalies
\begin{align} \allowdisplaybreaks
\ell \;=&\; \ell_0+ \sqrt{\frac{\mu}{a^3}}\,t, \displaybreak[0] \\
u \;\phantom{=}&\; \mathrm{from:} \qquad\ell=u-e\sin{u}, \displaybreak[0] \\
f \;=&\; 2\arctan\left(\sqrt{\frac{1+e}{1-e}}\tan\frac{u}{2}\right),
\end{align}
are computed first, and, then, the polar-nodal (prime) elements are obtained from the inverse sequence,
\begin{align} \allowdisplaybreaks
r(t) \;=&\; a\,(1-e\cos{u}), \displaybreak[0] \\
\theta(t) \;=&\; \theta_0+\zeta\,(f-f_0), \displaybreak[0] \\
\nu(t) \;=&\; \nu_0+\chi\,(f-f_0), \displaybreak[0] \\
R(t) \;=&\; \frac{\mu}{\tilde\Theta}\,e\sin{f},

\end{align}
which is completed with the constants $\Theta$ and $N$.

\subsection{Short-period transformation} \label{s:shp}

The transformation from prime polar-nodal variables to original ones, and vice-versa, requires the computation of the total corrections
\begin{equation} \label{pncorrections}
\Delta_T\xi=\delta\,\Delta_1\xi+\frac{1}{2}\delta^2\,\Delta_2\xi,
\qquad
\delta=-\frac{1}{2}J_2\,\frac{\alpha^2}{p^2},
\end{equation}
where $\xi$ denotes any of the polar-nodal variables $\xi\in(r,\theta,\nu,R,\Theta,N)$, and $\delta=-\frac{1}{2}J_2\,(\alpha/p)^2$. Each total correction $\Delta_T\xi$ is to be added to the corresponding polar-nodal variable $\xi$.
\par

The necessary corrections $\Delta_1\xi$ and $\Delta_2\xi$ are given below, where we recall that $\kappa\equiv\kappa(r,\Theta)$ and $\sigma\equiv\sigma(R,\Theta)$ are given in Eq.~(\ref{ef}). These corrections, as well as $\delta$, must be expressed in prime variables for the direct transformation $\Delta_{TD}\xi=\xi-\xi'$, and in original variables for the inverse transformation $\Delta_{TI}\xi=\xi'-\xi$.
\par

Remark that the first order corrections have been arranged in a form different from the original derivations of \citet{Deprit1981}, which is amenable for faster evaluation \citep{GurfilLara2014}. The second order corrections have been specifically derived and conveniently arranged for fast evaluation in the present work. For this sake, terms of the order of $e^2$ have been neglected from the expressions provided for the second order corrections, in agreement with the assumption made for the separability of DRI.

\subsubsection{First order corrections}

The first-order corrections are the same both for the direct and inverse transformations:
\begin{align} \allowdisplaybreaks
\Delta_1r \;=&\; p\left(1-\frac{3}{2}s^2-\frac{1}{2}s^2\cos2\theta\right),
\displaybreak[0] \\[1ex]
\Delta_1\theta \;=&\; \left[\frac{3}{2}-\frac{7}{4}s^2+ (2-3s^2)\,\kappa\right]\sin2\theta \\ \nonumber
& -\left[5-6s^2+(1-2s^2)\cos2\theta\right]\sigma,
\displaybreak[0] \\[1ex]
\Delta_1\nu \;=&\;  c\left[(3+\cos2\theta)\,\sigma-\left(\frac{3}{2}+2\kappa\right) \sin2\theta\right],
\displaybreak[0] \\[1ex]
\Delta_1R \;=&\; \frac{\Theta}{p}\,(1+\kappa)^2s^2\sin2\theta,
\displaybreak[0] \\[1ex]
\Delta_1\Theta \;=&\; -\Theta\,s^2\left[\left(\frac{3}{2}+2\kappa\right)\cos2\theta+\sigma\sin2\theta\right],
\displaybreak[0] \\[1ex]
\Delta_1N \;=&\; 0,
\end{align}
where the right member must be expressed in prime variables for the direct transformation, and in original variables in the case of the inverse one.

\subsubsection{Second order corrections of the direct transformation}
Terms of $\mathcal{O}(e^2)$ have been neglected from the corrections. The right member must be expressed in prime variables.
\begin{align} \allowdisplaybreaks
\Delta_2r
\;= &\; \left\{-8+15 s^2-\frac{23}{4}s^4+\left(-\frac{3}{2}+\frac{7}{2}s^2-\frac{41}{16}s^4\right)\kappa \right. \\ \nonumber
& -\left[13-14s^2-\left(\frac{65}{8}-\frac{153}{16}s^2\right)\kappa\right]s^2\cos2\theta
-\left(\frac{1}{4}-\frac{1}{16}\kappa\right)s^4\cos4\theta \\ \nonumber
& \left. +\left[\left(\frac{27}{8}-\frac{51}{16}s^2\right)s^2\sin2\theta+\frac{9}{32}s^4\sin4\theta\right]\sigma \right\}p \displaybreak[0]
\\[1ex]
\Delta_2\theta
\;= &\; \left[8-29s^2+\frac{85}{4}s^4+\left(32-\frac{803}{4}s^2+\frac{1419}{8}s^4\right)\kappa\right]\sin2\theta \\ \nonumber
& +\left[\frac{9}{4}-\frac{3}{8}s^2-\frac{17}{8}s^4+\left(6-3s^2-\frac{55}{16}s^4\right)\kappa\right]\sin4\theta \\ \nonumber
&+  \left[72-121s^2+\frac{327}{8}s^4+ \left(-56+\frac{989}{4}s^2-\frac{1609}{8}s^4\right)\cos2\theta \right. \\ \nonumber
& \left. +\left(-3+3s^2+\frac{1}{8}s^4\right)\cos4\theta \right]\sigma
\displaybreak[0] \\[1ex]
\Delta_2\nu \;= &\; \left\{
\left[\left(56-92s^2\right)\cos2\theta+\left(3-\frac{3}{2}s^2\right)(-9+\cos4\theta)\right]\sigma
   \right.\\ \nonumber
& \left. -\left[8-21s^2+\left(32-76s^2\right)\kappa\right]\sin2\theta
-\left(\frac{9}{4}+\frac{3}{4}s^2+6\kappa\right)\sin4\theta \right\}c
\displaybreak[0] \\[1ex]
\Delta_2R \;= &\;
\left\{ \left[16-16s^2+\left(\frac{237}{8}-\frac{437}{16}s^2\right)\kappa\right]s^2\sin2\theta \right. \\ \nonumber
& +\left(1+\frac{65}{32}\kappa\right)s^4\sin4\theta
+\left[-\frac{3}{2}-\frac{1}{2}s^2+\frac{71}{16}s^4 \right. \\ \nonumber
& \left. \left.
+\left(-\frac{95}{8}+\frac{231}{16}s^2\right) s^2 \cos2\theta
+\frac{17}{16}s^4\cos4\theta
\right]\sigma\right\}\frac{\Theta}{p}
\displaybreak[0] \\[1ex]
\Delta_2\Theta \;= &\; \Theta  \left\{\left(\frac{9}{2}-\frac{25}{4}s^2+6\left(2-3s^2\right)\kappa\right)s^2 \right. \\ \nonumber
& -\left[8-\frac{15}{2}s^2+32\left(1-s^2\right)\kappa\right]s^2\cos2\theta
-\frac{3}{4}s^4\cos4\theta \\ \nonumber
& \left. +\sigma  \left[\left(-56+64s^2\right)s^2\sin2\theta+\frac{3}{2}s^4\sin4\theta\right]
\right\}
\displaybreak[0] \\[1ex]
\Delta_2N =& 0.
\end{align}

\subsubsection{Second order corrections of the inverse transformation}

Terms of $\mathcal{O}(e^2)$ have been neglected from the corrections. The right member must remain in original variables
\begin{align} \allowdisplaybreaks
\Delta_2r
\;= &\; \left\{ 8-12s^2+s^4+\left(\frac{3}{2}+\frac{1}{2}s^2-\frac{71}{16}s^4\right)\kappa \right. \\ \nonumber
& +\left[28-32s^2+\left(\frac{95}{8}-\frac{231}{16}s^2\right)\kappa\right]s^2\cos2\theta
 -\left(1+\frac{17}{16}\kappa\right)s^4\cos4\theta \\ \nonumber
& \left. +\left[\left(-\frac{27}{8}+\frac{51}{16}s^2\right)s^2\sin2\theta-\frac{9}{32}s^4\sin4\theta\right]\sigma \right\}p \displaybreak[0] \\[1ex]
\Delta_2\theta
\;= &\; \frac{9}{4}-\frac{15}{8}s^2+2s^4+\left(6-3s^2-\frac{25}{16}s^4\right)\kappa +\left [-12+31s^2-\frac{73}{4}s^4 \right. \\ \nonumber
& \left. +\left(-40+\frac{819}{4}s^2-\frac{1371}{8}s^4\right)\kappa \right]\sin2\theta +\left[ -72+116s^2-\frac{243}{8}s^4 \right. \\  \nonumber
& \left.+ \left(26-\frac{1029}{4}s^2+\frac{1993}{8}s^4 \right)\cos2\theta + \left(-3+\frac{43}{8}s^4\right)\cos4\theta \right]\sigma \displaybreak[0] \\[1ex]
\Delta_2\nu \;= &\; \left\{ \left[12-21s^2+\left(40-76s^2\right)\kappa\right]\sin2\theta
 -\left(\frac{9}{4}-\frac{3}{4}s^2+6\kappa\right)\sin4\theta \right. \\ \nonumber
& \left. +\left[27-\frac{27}{2}s^2 + \left(-26+92s^2\right)\cos2\theta + \left(3+\frac{3}{2}s^2\right)\cos4\theta\right]\sigma \right\}c\displaybreak[0] \\[1ex]
\Delta_2R \;= &\;
\left\{\left[-20+22s^2 - \left(\frac{333}{8}-\frac{725}{16}s^2\right)\kappa\right]s^2\sin2\theta \right. \\ \nonumber
& +\left(1+\frac{95}{32}\kappa\right)s^4\sin4\theta +\left[\frac{3}{2}-\frac{7}{2}s^2+\frac{41}{16}s^4  \right. \\ \nonumber
& \left. \left.
+ \left(-\frac{65}{8}+\frac{153}{16}s^2\right)s^2\cos2\theta - \frac{1}{16}s^4\cos4\theta \right]\sigma \right\}\frac{\Theta}{p} \displaybreak[0] \\[1ex]
\Delta_2\Theta \;= &\; \Theta\left\{ \left[\frac{9}{2}-\frac{25}{4}s^2+\left(12-18s^2\right)\kappa\right]s^2 \right. \\ \nonumber
& +\left[12-\frac{27}{2}s^2+\left(40-44s^2\right)\kappa\right]s^2\cos2\theta + \frac{3}{4}s^4\cos4\theta \\ \nonumber
& \left. +\left[\left(26-28s^2\right)s^2\sin2\theta - \left(\frac{3}{2}+\frac{9}{4}\kappa\right)s^4\sin4\theta \right]\sigma \right\}
\displaybreak[0] \\[1ex]
\Delta_2N \;= &\; 0.
\end{align}


TITLE: dimension of space of modular functions using the Riemann-Roch theorem?
QUESTION [2 upvotes]: Let $H$ be the upper half-plane, and $M_k$ be the space of modular forms of weight $k$ on $H$ under the action of $SL(2,\mathbb{Z})$.
I have read (Koblitz, Introduction to Elliptic Curves and Modular forms) a computational approach to determining the dimension of $M_k$.
However, in multiple sources, it has claimed that one can compute these dimensions easily using the Riemann-Roch theorem. After reading the Riemann-Roch theorem (from Otto Forster's Lectures on Riemann Surfaces) I am no closer to understanding how this is done. This probably means I haven't really understood the theorem, but could someone please illustrate how it helps in finding $dim(M_k)$?

REPLY [4 votes]: First of all you speak about $M_k$ but only even $k$'s matter, as $M_k = 0$ for odd $k$. So I will look at $M_{2k}$.
Compactify $H$ with one point at infinity in $\overline{H}$, note $G = SL(2,\mathbb{Z})$, and consider the quotient left quotient $X = {}_{G\backslash} \overline{H}$. This is a compact Riemann surface, and any modular form $f$ of weight $2k$ invariant by the action of $G$ descends to an holomorphic function on $X$. Take such a non-zero $f$. Now note $z_1,\ldots,z_n$ the zeros of $f$ and $k_1,\ldots,k_n$ their respective ordres. (You should take care properly of how you defined the order of a zero here, as some points like $i$ for instance may have non trivial stabilizer's. If $z_i$ is such a point, $k_i$ should by replaced by its product by the order of the stabilizer.)
Now, let state something weaker than Riemann-Roch that will suffice to conclude that $M_{2k}$ is of finite dimension :
Theorem. Let $Z$ be a compact Riemann surface. Let $z_1,\ldots,z_n\in Y$ and (positive integers) $k_1,\ldots,k_n$. Let $M$ be the vector space of
meromorphic functions on $Y$ which are holomorphic except possibly at the the $z_i$'s where at a $z_i$ the situation is the following : they are either holomorphic or they have a pole of order $\leq k_i$. Then $M$ is of finite dimension $\leq k_1+\ldots+k_n +1$.
Proof. Take $z$ a coordinate around a $z_i$, then any element $f$ of $M$ has a Laurent expansion around $z_i$ of the form $$f(z) = a_{-k_i} z^{-k_i} + a_{-k_i+1} z^{-k_i+1} + \ldots$$ For each $i$ take all coefficents of negative indices. You have a total of $k = \leq k_1+\ldots+k_n$ such elements that gives you an element $\varphi(f)\in\mathbf{C}^k$. The map $\varphi : M\to \mathbf{C}^k$ is obviously linear. Now take $f_1,\ldots,f_l \in M$ with $l>k$. Then by a dimension argument you can find $\lambda_1,\ldots,\lambda_l\in\mathbf{C}$, not all zero, such that $$\lambda_1 \varphi(f_i) + \ldots + \lambda_l \varphi(f_l) = 0.$$ Then $g = \lambda_1 f_i + \ldots + \lambda_l f_l$ has no poles and is, by the maximum modulus principle, constant. This implies that the dimension of $M$ is $\leq k+1$. $\square$
Now going back to our modular form $f$. If $g\in M_{2k}$ then $\frac{g}{f}$ is a weakly modular form of weight $0$. Take $M$ to be the space associated thx to the theorem to our situation given by $f$. Then we see that $g\mapsto\frac{g}{f}$ is an isomorphism $M_{2k} \to M$, showing that $M_{2k}$ is finite dimensional, of dimension $\leq k_1+\ldots+k_n +1$.
To explicitely calculate the dimension, you will indeed need full Riemann-Roch, and you can have all details in this note :
http://www2.math.ou.edu/~kmartin/mfs/ch5.pdf
You will see, it is not hard, just technical. The result is the following : $$\textrm{dim}_{\mathbf{C}} M_k = \left\{ \begin{array}{cll} 0 & \textrm{if} & k\textrm{ is odd} \\
\left[ \frac{k}{24} \right] & \textrm{if} & k\textrm{ is even and }\textrm{if} & \frac{k}{2} = 2 \textrm{ mod } 12 \\
\left[ \frac{k}{24} \right] + 1 & \textrm{if} & k\textrm{ is even and }\textrm{if} & \frac{k}{2} \not= 2 \textrm{ mod } 12 \\\end{array}
\right.$$ where $[x]$ is the floor part of $x$.


text:
\subsection{Variation}
\label{sec:variation}

\begin{definition}
\label{def:var}
Let $f : X \to T$ be a flat morphism between normal projective varieties, with $-K_{X/T}$ $\bQ$-Cartier and $f$-ample. Let $q_0$ be an integer such that $q_0K_{X/T}$ is Cartier, and for all positive integers $q_0|q$, set $\sL_q:= \sO_X(-qK_{X/T})$.   As $\sL_q$ provides a relatively ample polarization, the $\Isom$ scheme $I:=\Isom_{T \times T} ( p_1^* f , p_2^* f)$ exists together with the two natural projections $q_i : I  \to T$. Let $I'$ be the image of $(q_1, q_2) : I \to T \times T$. Then, there is a non-empty open set $U \subseteq T$ where the fibers of $p_1|_{I'}: I' \to T$ have the same dimension, say $d$. This dimension is the dimension of a general isomorphism equivalence class of the fibers of $f$. As these isomorphism equivalence classes (at least general ones) would be exactly the fibers of any reasonable moduli map, one defines the \emph{variation} of $f$ as
\begin{equation}
\label{eq:var}
 \var (f):= \dim T - d.
\end{equation}
$f$ has \emph{maximal variation}, if $\var (f)=\dim T$.
\end{definition}

\subsection{Curve base}

\begin{notation}
\label{notation:CM_positivity}
In the situation of \autoref{notation:CM_semi_positivity}, assume that
\begin{enumerate}
 \item $\delta>1$, where $\delta= \delta\left(X_{\ot}, \Delta_{\ot}\right)$ for very general geometric points $\ot \in T$, and
 \item $\deg \lambda_{f, \Delta} =0$.
\end{enumerate}

\end{notation}

\begin{theorem}
\label{thm:no_divisor}
In the situation of \autoref{notation:CM_positivity}, for each ample $\bQ$-divisor $L$ on $T$, $|-K_{X/T} -\Delta -f^*L|_{\bQ}=\emptyset$.
\end{theorem}

\begin{proof}
 Assume that $\Gamma \in |-K_{X/T} -\Delta -f^*L|$. Using \cite[Thm 1.2]{Fujita_Openness_results_for_uniform_K-stability}, \autoref{def:K_stable} and \autoref{prop:delta_general_fiber}, choose a small rational number $\varepsilon>0$ such that for very general geometric points $\ot \in T$ we have $\delta\left(X_{\ot}, \Delta_{\ot} + \varepsilon \Gamma_{\ot}\right)>1$. Then,
 \begin{multline*}
 0 \geq
 \underbrace{ (-K_{X/T} - \Delta - \varepsilon \Gamma)^{n+1} }_{\textrm{by \autoref{thm:semi_positivity_curve}}}
 = (-K_{X/T} - \Delta + \varepsilon (K_{X/T} + \Delta + f^*L))^{n+1}

 =

 (-(1-\varepsilon) (K_{X/T} + \Delta) + \varepsilon f^* L )^{n+1}

 \\ =

 (1-\varepsilon)^n \left( (1-\varepsilon)(- K_{X/T} - \Delta)^{n+1} + (n+1)\varepsilon(-K_{X/T} - \Delta)^n f^* L \right)

\\  =

\underbrace{ (n+1)\varepsilon(1-\varepsilon)^n   (-K_{X_t} - \Delta_t)^n \deg L  }_{(- K_{X/T} - \Delta)^{n+1} =0} >0.
 \end{multline*}
This is a contradiction.
\end{proof}

\begin{notation}
\label{notation:CM_positivity_HN}
 In the situation of \autoref{notation:CM_positivity},
 \begin{enumerate}
  \item let $q_0>0$ be an integer such that $q_0 (-K_{X/T} - \Delta)$ is Cartier,
  \item for each integer $q_0 |q$, define $\sE_q:= f_* \sO_X(q(-K_{X/T} - \Delta))$, and set $0=\sF^0_q \subseteq \sF^1_q \subseteq \dots \subseteq \sF^{s_q-1}_q \subseteq \sF^{s_q}_q$ be the Harder-Narasimhan filtration of $\sE_q$. Set $\sG^i_q:= \sF^i_q/ \sF^{i-1}_q$,
  \item  let $g$ be the genus of $T$.
 \end{enumerate}

\end{notation}

\begin{lemma}
\label{lem:2g_slope_bound}
In the situation of \autoref{notation:CM_positivity_HN}, for every positive integer $q_0 | q$,  $\mu(\sF_q^1) \leq 2g$.
\end{lemma}

\begin{proof}
Assume the contrary, that is, $\mu(\sF_q^1)>2g$, and let $t \in T$ be an arbitrary closed point. According to \autoref{prop:semi_stable_globally_generated}, $\sF_q^1(-t)$ is globally generated. So, there is a $\Gamma'\in |q (-K_{X/T} - \Delta) - f^* L'|$, where $L'$ is the divisor determined by $t$ on $T$. Hence, for $\Gamma := \frac{\Gamma}{q}$ and $L:= \frac{L'}{q}$ we have $\Gamma \in |-K_{X/T} - \Delta - f^* L |_{\bQ}$. This contradicts \autoref{thm:no_divisor}.

\end{proof}

\begin{proposition}
\label{prop:no_positive_slopes}
In the situation of \autoref{notation:CM_positivity_HN}, for every positive integer $q_0 | q$, $\mu(\sF_q^1) \leq 0$.
\end{proposition}

\begin{proof}
Assume that $\mu\left( \sF_q^1\right) >0$, and let $\sH$ be the image of
\begin{equation*}
\xi: \left( \sF_q^1 \right)^{\otimes m }  \to \sE_{qm}
\end{equation*}
for some $m \gg 0$. \emph{We claim that  $\sH$ is not zero} because of the following: Let $\eta$ be the generic point of $T$. Then any $x \in \left( \sF_q^1\right)_\eta$ can be identified with some $\tx \in H^0\left( X_\eta,q \left( - K_{X_\eta} - \Delta_\eta \right) \right)$, in which case $\xi \left( x^{\otimes m}\right)$ gets identified with $\tx^m \in H^0\left( X_\eta, mq \left(- K_{X_\eta} - \Delta_\eta \right) \right)$. In particular, the following implications conclude our claim:
$
x \neq 0 \Rightarrow \tx \neq 0 \Rightarrow \tx^m \neq 0 \Rightarrow \xi\left( x^{\otimes m} \right) \neq 0$.

Let then $j$ be the smallest integer such that $\sF_{qm}^j$ contains $\sH$, and let $\sH'$ be the image of $\sH$ in $\sG_{qm}^j$. By the choice of $j$, $\sH' \neq 0$, and as $\sH'$ is a surjective image of  $ \left( \sF_q^1 \right)^{\otimes m }$:
\begin{equation*}
\mu\left(\sF_{mq}^1\right)

\geq
\underbrace{\mu \left(\sG_{mq}^j\right)}_{\parbox{57pt}{\tiny by the definition of the Harder-Narasimhan filtration}} \geq

\underbrace{\mu(\sH') }_{\parbox{45pt}{\tiny $\sG_{mq}^j$ is semi-stable}}
>

\underbrace{\mu \left( \left( \sF_q^1 \right)^{\otimes m } \right)}_{\parbox{65pt}{\tiny $ \left( \sF_q^1 \right)^{\otimes m }$ is semi-stable according to \autoref{prop:tensor_product_semi_stable}}}

=

\underbrace{m \mu \left(\sF_q^1\right)}_{\textrm{\autoref{prop:tensor_product_semi_stable}}}
>
\underbrace{ 2g}_{\parbox{53pt}{\tiny $m \gg 0$, and we assumed that $\mu\left( \sF_q^1 \right)>0$}} .
\end{equation*}
This contradicts \autoref{lem:2g_slope_bound}.
\end{proof}

\begin{theorem}
\label{thm:positivity_curve}
In the situation of \autoref{notation:CM_positivity}, if $q>0$ is an integer such that $-q(K_{X/T} + \Delta)$ is Cartier, then

$f_* \sO_X(-q (K_{X/T} + \Delta))$ is a semi-stable vector bundle of slope $0$.

\end{theorem}

\begin{proof}

First,
\autoref{thm:bounding_nef_threshold}  yields that $-K_{X/T} - \Delta$ is nef. Then, $f_* \sO_X(  q(-K_{X/T}- \Delta))$ is also nef, by \autoref{prop:semi_positivity_engine_downstairs} taking into account  the $\bQ$-linear equivalence
\begin{equation*}
q(-K_{X/T}- \Delta) \sim_{\bQ} K_{X/T} + \Delta + (q+1)(-K_{X/T}- \Delta).
\end{equation*}
Finally,  \autoref{prop:no_positive_slopes}, concludes our proof.

\end{proof}

\subsection{Ampleness lemma}

\autoref{thm:ampleness} is an extract of the argument of the ampleness lemma of \cite{Kollar_Projectivity_of_complete_moduli} (one assumption removed in \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). It will be one of the main technical ingredients for the proof of items \autoref{itm:semi_positive_no_boundary:big} and \autoref{itm:semi_positive_no_boundary:ample} of \autoref{thm:semi_positive_no_boundary} given in \autoref{sec:arbitrary_base_pos}.

\begin{theorem}
\label{thm:ampleness}
Let $V$ be a vector bundle of rank $v$ on a normal  projective variety $T$ over $k$, and let $\phi : W:=\Sym^d (V) \twoheadrightarrow Q$ be a surjective homomorphism onto another vector bundle, where the ranks are $w$ and $q$, respectively. Assume that there is an open set, where the map of sets $T(k) \to \Gr(w,k)/\GL(v, k)$ induced by $\phi$ is finite to one. Then, for each ample Cartier divisor $B$ on $T$ there is an integer $m>0$ and a non-zero homomorphism
\begin{equation*}
\Sym^{qm}\left( \bigoplus_{i=1}^w W  \right) \to \sO_T(-B) \otimes ( \det Q)^m.
\end{equation*}

\end{theorem}

\begin{proof}
The proof is contained in \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}, but not presented there ideally for our purposes, so we give a recipe of how to turn \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} into the above statement. First, specialize \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} to the case of projective base and the special choice of $W = \Sym^d (V)$ and  $G=\GL(v,k)$, where the latter two choices are  fine according to \cite[Rem 5.3]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}. At this point the assumptions of \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} become identical to ours, except that in \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} was assumed to be weakly positive. However, this assumption is not used until the last three lines of the proof. In particular, the existence of the non-zero homomorphism of the displayed equation \cite[(5.5.5)]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} exists even without the weakly positive assumption. This is exactly our statement, taking into account the isomorphism of the displayed equation   of the proof of \cite[Thm 5.5]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} after \cite[(5.5.5)]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}.
\end{proof}

\subsection{Arbitrary base}
\label{sec:arbitrary_base_pos}

\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}]
As in the proofs of \autoref{thm:semi_positivity_curve} and \autoref{thm:nef_threshold}, we may assume that $k$ is uncountable.
Let $\eta$ be the generic point of $T$.
\begin{enumerate}
\item Set  $n:= \dim X - \dim T$, $v:=K_{X_\eta}^n$, $\delta:= \delta\left(X_{\overline{\eta}}\right)$.
 \item Fix a rational number $\alpha$ such that  $\alpha > \max\left\{ 1, \frac{\delta}{(\delta -1)v(n+1)} \right\}$.
\end{enumerate}
Throughout the proof $\iota : C \to T$ denotes the normalization of a very general member of an arbitrary covering curve family of $T$. Very general here  means that it is not contained in  countably many divisors $S_i$, which we will specify during the proof explicitly. Set:
\begin{itemize}
 \item $\eta_C$ to be the generic point of $C$,
 \item $Z:=X_C$ (note that as the fibers of $f$ are reduced, and the general ones are normal, $Z$ is normal),
 \item $\sigma : Z \to X$ and $g : Z \to C$ be the induced morphisms,

\item $\lambda:=\lambda_g$.
\end{itemize}
Then the following holds:
\begin{itemize}
\item $\sigma^* K_{X/T} \cong K_{Z/C}$ by \autoref{rem:reduced_fibers_no_boundary}.\autoref{itm:relative_canonical_base_change:base_change}, and
  $\lambda=\lambda_f|_C$ by \autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}.
 \item a $\bQ$-Cartier divisor $L$ is pseudo-effective if and only if $L \cdot C \geq 0$ (for any such $C$),
 \item according to \autoref{prop:delta_general_fiber}, $\delta\left(X_{\overline{\eta_C}} \right) = \delta$ (assuming we add the countably many divisors to $S_i$, over which $\delta(X_t)<\delta$, which are given by \autoref{prop:delta_general_fiber}). In particular, as $\delta>1$ the very general fibers of $g$ are uniformly $K$-stable, and hence klt, see \cite[Theorem 1.3]{Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy}.

 \item in particular, by \autoref{thm:semi_positivity_curve}, $\deg \lambda \geq 0$,
 \item by \autoref{thm:bounding_nef_threshold}, $-K_{Z/C}  + \alpha g^* \lambda$ is nef and $g$-ample.
\end{itemize}
It is important that throughout the proof all constants (so all rational numbers) will be fixed independently of the particular choice of  $C$ (for which there are two choices, first one choses a covering family, and then a very general member of that). For this reason, whenever such a constant is fixed, we do it in a numbered list item (see points above and below).

Choose integers $r \geq 2$ and $d>0$ such that
\begin{enumerate}[resume]
 \item $rK_{X/T}$ and $r\alpha \lambda_f$ are Cartier,
 \item $h^i(X_t, -rK_{X_t})=0$ for all $i>0$ and all $t \in T$,
 \item $-rK_{X_t}$ is very ample for all $t \in T$,
 \item the multiplication maps  $W:=\Sym^d f_* \sO_X (-r K_{X/T}) \to f_* \sO_X (-dr K_{X/T})=:Q$ are surjective, and
 \item \label{itm:positivity:classifying_map} for all $t \in T$, $K_{t}:=\Ker \left( \Sym^d H^0\left(X_t, -r K_{X_t}\right) \to H^0\left(X_t,-d r K_{X_t}\right) \right)$ generates $\sI(d)$, where $\sI$ is the ideal of $X_t$ via the embedding $\varphi_{\left|-rK_{X_t} \right|} : X_t \to \bP^{v-1}$, where $v:= \rk f_* \sO_X (-r K_{X/T})$ and $\varphi_{\left|-rK_{X_t} \right|}$ is defined only up to the action of $\GL(v,k)$ on the target. Note that this is achievable because $\sI$ form a flat family as $t$ varies.

 In particular, if we set  $w:= \rk W$ and $q:= \rk Q$, then for every $t \in T(k)$, $K_{t} \subseteq W_t$ determines $X_t \hookrightarrow \bP^{v-1}$ up to the action of $\GL(v, k)$, which then means that the orbit of $K_{t}$ in $\Gr(w,q)/G(v,k)$ determines $X_t$ up to isomorphism. Therefore if we apply \autoref{thm:ampleness} for $W \to Q$, then the fibers of the classifying map $T(k) \to \Gr(w,q)/G(v,k)$ are contained in the isomorphism classes of the fibers of $f$ and hence, by the maximal variation assumption, there is an open set where these fibers are finite.
\end{enumerate}
As,
\begin{equation*}
r (- K_{Z/C}  +  2\alpha g^* \lambda)  = K_{Z/C}  +  \underbrace{ (r+1)( -K_{Z/C} +  \alpha g^* \lambda) +  \underbrace{(r-1) \alpha}_{>1} g^* \lambda}_{\textrm{nef and $g$-ample}},
\end{equation*}
by \autoref{prop:semi_positivity_engine_downstairs}, $g_* \sO_Z(r (- K_{Z/C}  +  2\alpha g^* \lambda)) $ is a nef vector bundle. Set
\begin{equation*}
M:= r( -K_{X/T} + 2 \alpha f^* \lambda_f).
\end{equation*}
Note that the conclusions of point \autoref{itm:positivity:classifying_map} about the finiteness of the classifying map hold also for $-rK_X$ replaced by  $M$, as $f_* \sO_X(M)$ and $f_* \sO_X(dM)$ differs from $f_* \sO_X(-rK_{X/T})$ and $f_* \sO_X(-rdK_{X/T})$ only by a twist with $r2 \alpha \lambda_f$ and $dr2 \alpha \lambda_f$, respectively. So,
\autoref{thm:ampleness} yields an ample divisor $B$ on $T$,  an integer $m>0$ and  a non-zero homomorphism as follows (see point \autoref{itm:positivity:classifying_map} above for the definition of $w$ and $q$):
\begin{equation*}
\xi : \Sym^{qm}\left( \bigoplus_{i=1}^w \Sym^d (f_* \sO_X(M)) \right) \to \sO_X(-B) \otimes \left(\det f_* \sO_X(dM) \right)^m.
\end{equation*}
As the target of $\xi$ is a line bundle, there exists a divisor, on the complement of which $\xi$ is surjective. Let us add this divisor to $S_i$. Then $\xi|_C$ is a non-zero homomorphism as follows:
\begin{equation*}

\xi_C : \Sym^{qm}\left( \bigoplus_{i=1}^w \Sym^d( g_* \sO_Z(M_C)) \right) \to \sO_C(-B_C) \otimes \left( \det g_* \sO_Z(dM_C) \right)^m.
\end{equation*}
Define
\begin{equation*}
\sA:=\det f_* \sO_X(dr ( -K_{X/T}  + 2 \alpha f^* \lambda_f))= \det f_* \sO_X(dM),
\end{equation*}
and let $A$ be a divisor corresponding to $\sA$.
As $g_* \sO_Z(M)$ is nef and hence so is every bundle that admits a generically surjective map from the left side of $\xi_C$, we obtain that
\begin{equation}
\label{eq:positivity:B_A}
\deg \sA|_C = \deg  \det g_* \sO_Z(dM_C)) \geq \frac{B \cdot C}{m} .
\end{equation}
Consider now, the natural  embedding:
\begin{equation*}
\alpha : \det f_* \sO_X(dM) \hookrightarrow \bigotimes_{i=1}^q f_* \sO_X(dM)  \cong f^{(q)}_* \sO_{X^{(q)}}\left(dM^{(q)}\right),
\end{equation*}
given by the embedding of representations $\det \to \bigotimes_{i=1}^q  $ of $\GL(q,k)$.
Hence, by adjunction of $f_*^{(q)}(\_)$ and $\left(f^{(q)}\right)^*(\_)$ one can write $\left(f^{(q)} \right)^* A + D = dM^{(q)}$, where $D$ is an effective divisior on $X^{(q)}$. Furthermore, as $\alpha$ is a morphism of vector bundles, and the formation of $f_* \sO_X(dM)$ is compatible with base-change, $D$ does not contain any fiber. By the continuity of log canonical threshold, there is a $0<\varepsilon < \frac{1}{rd}$ such that $\left(X_t^{(q)}, \varepsilon D_t\right)$ is klt for general closed points $t \in T$. In particular by the genericity of $C$ the same holds also for general $t \in C$.	 Then, if we define $N:= d r( -K_{X/T} + 3 \alpha f^* \lambda_f)$,
 according to \autoref{cor:semi_positivity_engine_upstairs}, the following divisor is nef ($Z^{(q)}$ is normal by \autoref{lem:normal_stable_pullback_fiber_product}.\autoref{itm:normal_stable_pullback_fiber_product:fiber_product}).
\begin{multline*}
 \underbrace{K_{Z^{(q)}/C}  + \varepsilon D_C }_{\parbox{80pt}{\tiny $\left(Z^{(q)}_t,  \varepsilon \left(D_C\right)_t \right)$ is klt for $t \in C$ general}}

+

 \underbrace{\left(dr + 1 - \varepsilon rd \right)\left(   -K_{Z^{(q)}/C}  + 2 \alpha q \left(g^{(q)}\right)^* \lambda \right) +  (dr-2) \alpha q \left(g^{(q)}\right)^* \lambda}_{\textrm{nef and $f$-ample  $(r \geq 2, d>0)$}}

 \\ \sim \left(  N^{(q)}_C - \varepsilon \left(g^{(q)} \right)^* A_C \right) = \left(N_C - g^* \frac{\varepsilon }{q} A_C \right)^{(q)}
\end{multline*}
Set $\varepsilon':= \frac{\varepsilon}{qr d}$. Then we have that $\frac{N_C}{dr} - \varepsilon' g^*A_C$ is nef according to \autoref{lem:product_nef}.
So,
\begin{multline*}
0 \leq  (-K_{Z/C}  + 3\alpha g^* \lambda - \varepsilon'  g^* A_C)^{n+1}

 =

 \left(\underbrace{-K_{Z/C}  +  \frac{g^*\lambda}{v(n+1)}}_{\parbox{90pt}{\tiny top self-intersection is $0$ by the definition of $\lambda$}}  + \left(3\alpha - \frac{1}{v(n+1)} \right) g^* \lambda -  \varepsilon' g^*A_C \right)^{n+1}

\\ =

(n+1) v \deg \left( \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda -  \varepsilon' A_C \right)

\leq

\underbrace{(n+1) v \deg \left( \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda -  \varepsilon' \frac{B_C}{m} \right)}_{\textrm{equation \autoref{eq:positivity:B_A}}}
\end{multline*}
Hence, $ \left(3\alpha - \frac{1}{v(n+1)} \right) \lambda_f -  \varepsilon' \frac{ B}{m}$ is pseudo-effective (as it dots to at least zero with each movable class). Therefore, $\lambda_f$ is the sum of an ample and a pseudo-effective $\bQ$-divisor, so  $\lambda_f$ is big.

\end{proof}

\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:ample} of \autoref{thm:semi_positive_no_boundary}]
By Nakai-Moishezon it is enough to prove that for all normal varieties $V$ mapping finitely to $X$, $\left( \lambda_f|_V \right)^{\dim V} >0$. However, using \autoref{prop:CM_base_change}, this we may obtain by replacing $f : X \to T$ with $f_V : X \times_T V \to V$, and applying point  \autoref{itm:semi_positive_no_boundary:big} to $f_V$.
\end{proof}

\begin{proof}[Proof of point \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}]
Let $m>0$ be the integer such that $L:=-mK_{X/T}$ is an $f$-very ample divisor, and let $N$ be the line bundle, which is the inverse of the leading term of the Knudsen-Mumford expansion for $L$ (\autoref{notation:Paul_Tian}). According to \cite[Thm 6.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds} it is enough to prove that $N$ is nef and that for all closed subvarieties $V$ of $T$ intersecting $U$, $\left(N|_V\right)^{\dim V}>0$. In fact, also nefness is a similar intersection question, that is, $\deg N|_C \geq 0$ for all curves $C$ of $T$.
Note that the normalization of $C$ is automatically a scheme, hence we may assume that $C$ is a smooth scheme.
Furthermore, $V$  has a finite cover by a scheme \cite[Tag 04V1]{stacks-project} and by Nagata's theorem and resolution of singularities we may also assume that $V$ is projective and smooth. Therefore, by replacing $V$ by this generically finite cover we may assume that $V$ is also a smooth scheme.
As $N$ is compatible with base-change (\autoref{lem:Knudsen_mumford_base_change}), by relaxing the isomorphism class assumption, we can replace $T$ by $C$ or $V$. Then, the base is smooth and projective, and  we have to prove that without any variation assumption $N$ is nef, and if furthermore the variation is maximal, then $N$ is even big. By \autoref{prop:2_defs_CM_same}.\autoref{itm:2_defs_CM_same:leading_term}, we can prove this for the CM line bundle, instead of $N$, which is then shown in points \autoref{itm:semi_positive_no_boundary:nef} and \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}.
\end{proof}

\begin{proof}[Proof of \autoref{thm:lambda_non_big}]
Choose $q$ big enough such that $-q(K_{X/T} + \Delta)$ is Cartier and without higher cohomology on the fibers.
Let $H_i \in |H|$ be general for $i=1,\dots, \dim T -1$, and set $C:= \bigcap_{i=1}^{\dim T -1} H_i$. By the above generic choices, $Z:=X_C$ is normal. Furthermore, $C$ lies in the smooth locus of $T$, hence for base-change properties along $C \to T$ we may assume that $T$ is smooth. In particular,  there is an induced boundary $\Delta_Z$ on $Z$ (\autoref{sec:base_change_relative_canonical_smooth_base}), for which $K_{X/T} + \Delta|_Z = K_{Z/C} + \Delta_Z$  (\autoref{prop:relative_canonical_base_change_normal}), and consequently
\begin{equation}
\label{eq:lambda_non_big:base_change}
f_* \sO_X(-q(K_{X/T} + \Delta))|_C \cong \left(f_C\right)_* \sO_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right).
\end{equation}
Furthermore,
\begin{equation*}

0 =

\underbrace{\lambda_{f,\Delta} \cdot H^{\dim T -1}}_{\textrm{assumption}}

= \deg \lambda_{f,\Delta}|_C

=
\underbrace{\deg \lambda_{f_C, \Delta_Z} .}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}}}
\end{equation*}
Therefore, according to \autoref{thm:positivity_curve},  $\left(f_C\right)_* \sO_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right)$ is a semi-stable vector bundle of slope $0$. However, then the isomorphism \autoref{eq:lambda_non_big:base_change} implies that $f_* \sO_X (- q(K_{X/T} + \Delta))$ is $H$-semi-stable of slope $0$: if it had a subsheaf $\sF$ of $H$-slope bigger than $0$, then for the saturation $\sF'$ of $\sF$, $\sF'|_C$ would be a subbundle of positive degree of $\left(f_C\right)_* \sO_{Z}\left(-q\left(K_{Z/C} + \Delta_Z \right) \right)$, which is a contradiction.
 \end{proof}

\begin{proof}[Proof of \autoref{cor:proper_base}]
The proof is very similar to that of  point \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary} above.
As in the above proof, $T$ has a generically finite cover by a smooth, projective scheme. By base-changing over this cover one may assume that the base is smooth and projective. By \autoref{prop:2_defs_CM_same}, we may replace $N$ by the CM-line bundle notion used in the present article (\autoref{def:CM}), and then points \autoref{itm:semi_positive_no_boundary:nef} and \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary} concludes the proof.

\end{proof}

config: arxiv


TITLE: Basic doubt related to property of Fourier transform
QUESTION [1 upvotes]: Let $f\in L^1$ such that $\hat{f}\in L^1.$

$\hat{f}\in L^1\implies f$ is continuous.

Define $g(x)=\begin{cases} 0 , \text{ if } x\in \mathbb{Q}\\ f(x), \text{ if }
x\in \mathbb{Q}^c  \end{cases} . $ Then $f=g$ a.e.

So  $g\in L^1$ and $\hat{f}=\hat{g}$.

So $\hat{g}\in L^1$ and hence $g$ is continuous.

So $g\equiv 0$ and hence $f\equiv 0$.

Please help me find the flaw(s) as
I know the above is not true since there are plenty of non-zero elements in Schwartz class.

REPLY [1 votes]: Perhaps item 1 should be written to imply that $f$ is equal, almost everywhere, to some continuous function $g$ (i.e., in the "equivalence class" (mod equality a.e.) of $f$, there is at least one continuous function.)


TITLE: Question about Column Space Matrix multiplication properties
QUESTION [1 upvotes]: If say two square matrices A,B have the same column space, will it also hold that some multiplication of these matrices with another matrix will have the same column space?
i.e  if Col(A) = Col(B)
does Col(AC) = Col(BC)    for some other matrix C of the same dimensions.
It seems to make sense to me, but I couldn't work out how to prove it.
Thanks,

REPLY [0 votes]: The counterexample @Bubububu has found is by taking a non-invertible $C$.
This wouldn't have occured with an invertible $C$.
Let us work with left multiplication by $C$ instead of right multiplication, because it is simpler (otherwise, transpose all). How can we prove that:
$$\tag{1}Col(A) = Col(B) \  \ \Longrightarrow  \ \ Col(CA) = Col(CB) $$
under the supplementary condition that $C$ is invertible ?
Some useful properties:

Lemma 1: Let $M_1 \cdots M_n$ be the columns of a certain matrix $M$;
$\sum_k a_k M_k = (M_1|\cdots|M_n)\begin{pmatrix}a_1\\\cdots\\a_n\end{pmatrix}=ML$ where $L$ is the column vector made of the coefficients of the linear combination.

as a corollary:

Lemma 2: $Col(M)$, being the set of all linear combinations of elements of $M$, is the (image) set of $M$, i.e., the set of all images $ML$ for any $M$.

as a corollary:

Lemma 3:  $Col(A)=Col(B) \Leftrightarrow \forall X, \exists L,L'  \ \text{such that} \   X=AL=BL'.$

With the help of this lemmas, (1) amounts to prove that
$$\forall Y, \exists L,L'  \ \text{such that} \  Y=CAL=CBL'$$
which is evident by multiplying by $C^{-1}$.


TITLE: A mathematical brain teaser in probability field
QUESTION [0 upvotes]: That is a brain teaser but it is also a mathematical problem in probability field.
One day, a man have been trapped into a building which has 18 floors. Now, he want to leave the building but there is a guard.
Assumptions:

if the man bribes the guard by 10 thousand dollors, he will down 1 stair (with probability 1).
If he invites the guard to dinner spending 20 thousand dollors, he will down 3 stairs with 80% probability while he will in vain with 20% probablity.
If he bribes the guard's wife by a 30 thousand cosmetic, he will down a half of stairs with 50% probability(e.g., 12 floor will down to 6 floor, 9 floor will down to 5 floor. Odd floor will down to $(n+1)/2$). Also, he will in vain with 50% probability.

The problem is: if the man has 150 thousand dollors, how does him spend money will has a highest probability to leave the building(Say, the floor number is less or equal to 0)?
This is similar to a programing problem but has probability. So I wonder if there some mathematical theory about it and how to solve the problem?

REPLY [3 votes]: Dynamic programming approach
Let $p_{i,j} = \{\text{probability of escaping given that the guard is in floor $i$ and I have $j$ dollars}\}$. You want to compute $p_{18,150}$. It is easy to see that $p_{0,j} = 1$ and $p(i, 0) = 0$ for any $i>0$, $j\geq 0$. There is a recurrence relation (you need to fill the dots with the 3rd case)
$p_{i, j} = \max{\{p_{i-1,j-10},\quad 0.8 p_{i-3, j-20} + 0.2 p_{i, j-20},\quad \cdots\}}$
This recurrence allows you to compute the term you need in $O(I*J)$ (I = maximum i, J = maximum j) if you store the terms you already computed in a table.


TITLE: Finding correlation in large data, non-numeric sets
QUESTION [4 upvotes]: Suppose I collect a lot of data from a group of persons, like

their height
their weight
color of eyes (chosen from eg the four categories blue/brown/black/other)
sex
day of the week the measurement was done

and I want to find which of the above data correlates with height.
For weight, I can image how that goes: just make the (x,y) set and measure stuff like Pearson's correlation. But I get stuck for things like 'colors of eyes' where there's no natural way to order them, or binary data like sex. Also for day of the week (where I'd like to find no correlation at all) but I wouldn't know how to order them.
How do I do this?

REPLY [3 votes]: There are lots of textbooks under headings like "Categorical Data Analysis", "Non-Parametric Statistics", et cetera, and all the standard statistical packages — GenStat, SAS, SPSS, and other names that will make me sound real ancient if I mention them — will have lots of practical advice about their usage on this kind of data.