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\section{Introduction}
Spheroidal systems (such as elliptical galaxies, lenticular galaxies and early-type spiral galaxies with bulges) are commonly believed to host in the central region super-massive black holes with masses in the range $M_{\mathrm{BH}}= 10^6 - 10^9 M_\odot$ \cite{rich}. During the early stages of galaxy evolution these SMBH accrete matter at high rates and are observed as bright QSOs. The radiative output at low energy (e.g. optical) decays from redshift z$>$3 to z=0 by almost 2 orders of magnitude. Therefore, the majority of SMBH in the local universe are not embedded in dense radiation fields. This enables VHE $\gamma$-rays\ to escape from the nuclear region without suffering from strong absorption via $\gamma$-photon pair absorption.
Several models \cite{gal-cen,lev,neron,slane} are proposed for the production of VHE $\gamma$-rays\ emission from these passive AGN. In all cases a large mass of the central object is the most important characteristic for generating a high VHE flux. H.E.S.S.\ has already observed nine nearby galaxies whose black hole mass is measured \cite{mag,pel}. Only the case of NGC 1399 is considered here. Constraints on the physical parameters of the system (e.g. the magnetic field \textbf{B}) are derived using several of the aforementioned models.
\section{Acceleration Mechanism}
If the central black hole is accreting matter from a disk that also carries magnetic flux, it will develop a magnetosphere similar to those surrounding neutron stars. If the charge density is not too high in the magnetosphere of the spinning black hole, it is possible to have a non-zero component of the electric field \textbf{E} parallel to the magnetic field \textbf{B}. In this configuration field gaps are created, where acceleration of particles can take place \cite{slane}.
Various methods can be used to estimate the magnetic field B. For example, B is estimated:
\begin{itemize}
\item assuming equipartition
\begin{equation}\label{eqn:magn}
\frac{B^2}{8\pi} = \frac{1}{2} \rho(r_0)v^2_r(r_0),
\end{equation}
where $\rho$ is the mass density and $v_r$ is the radial infall velocity of the accreting matter (both being a function of $r_0$, the distance to the inner edge of the disk);
\item from the angular momentum as in \cite{bick}
$$ B = 3.1\times10^3 \frac{\dot{m}^{1/2}}{M_{10}^{1/2}}\left(\frac{r}{r_\mathrm{g}}\right)^{-5/4} \textrm{Gauss,}$$
where $\dot{m}$ is the mass accretion rate in units of the Eddington mass accretion rate, $r_\mathrm{g}$ is the gravitational radius of the black hole and $M_{10}=(M_{\mathrm{BH}}/10^{10}M_{\odot})$.
\end{itemize}
In the model of \cite{slane} protons accelerated in the outer part of the black hole magnetosphere will collide with other protons present in the accretion disk producing pions some of which decay into VHE $\gamma$-rays\ .
The available power is
\begin{equation}\label{eq:power}
W_{\mathrm{max}} \sim 10^{27} \left(M_{\mathrm{BH}}\right)^2 \left(B_4\right)^2 \textrm{ ergs s}^{-1},
\end{equation}
where $B_4=(B/10^4 \textrm{ Gauss})$.
Here it is assumed that the magnetic energy density is in equipartition with the accretion energy density, which depends on various properties of the accretion disk (see Eq.~\ref{eqn:magn}).
In other models \cite{gal-cen,lev,neron} VHE $\gamma$-rays\ originate from electromagnetic processes such as synchrotron or curvature emission. Following the analogous arguments given in \cite{gal-cen} for the Galactic Center, synchrotron emission is not feasible due to a cut-off for protons and electrons at $\epsilon_{\gamma,\mathrm{max}} \simeq 0.3$ TeV and $\epsilon_{\gamma,\mathrm{max}} \simeq 0.16$ GeV respectively. These cut-offs are independent of the magnetic field strength.
The energy of curvature photons (when curvature losses are the dominant ones) does not depend on the mass of the particle, so it is the same for electron or proton originated photons. The emission spectrum from curvature radiation can extend up to VHE energies, with a cut-off at:
\begin{equation}\label{eq:cutoff}
E_{\mathrm{max}}\simeq14\left(M_{10}\right)^{1/2}\left(B_4\right)^{3/4} \textrm{TeV.}
\end{equation}
\section{VHE Observation of NGC 1399}
The giant elliptical galaxy NGC 1399 is located in the central region of the Fornax cluster at a distance of $20.3$ Mpc. An SMBH of $ M_{\mathrm{BH}} = 1.06\times10^9 M_\odot$ resides in the central region.
The nucleus of this galaxy is well known for its low emissivity at all wavelengths \cite{oconn}. Considering also the visibility of candidate sources for H.E.S.S., NGC 1399 therefore emerged as the best candidate for this study.
NGC 1399 was observed with the H.E.S.S.\ array of imaging atmospheric-Cherenkov telescopes for a total of 22.4 h (53 runs of $\sim$28 min each). After applying the standard H.E.S.S.\ data-quality selection criteria a total of 13.9 hours live time remain. The mean zenith angle is $Z_{\mathrm{mean}} = 22^\circ$.
The data were reduced using the standard analysis tools and selection cuts \cite{benb} and the Reflected-Region method \cite{berge} for the estimation of the background. This leads to a post-analysis threshold of 200 GeV at $Z_{\mathrm{mean}}$. No significant excess (-29 events, -1$\sigma$) is detected from NGC 1399 (see Fig. \ref{fig:thetasq} and Fig.~\ref{fig:skymap}). Results are consistent with independent analysis in the collaboration.
Assuming a photon index of $\Gamma$=2.6, the upper limit (99$\%$ confidence level; \cite{fc}) on the integral flux above 200 GeV is:
$$I \left(>200\textrm{GeV}\right) < 2.3 \times 10^{-12} \textrm{ cm}^{-2}\textrm{s}^{-1},$$
or ~1\% of the Crab Nebula flux.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{icrc0470_fig01.eps}
\caption{\small Distribution of squared angular distance from NGC 1399 for gamma-ray-like events in the ON region (dots) and in the OFF region (filled area, normalized). The dotted line represents the cut for point-like sources. Preliminary. \normalsize}
\label{fig:thetasq}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{icrc0470_fig02.eps}
\caption{\small The smoothed (smoothing radius r=0.09˚) VHE excess in the region centered on NGC 1399. The yellow star indicates the position of the optical centre of NGC 1399. Preliminary.\normalsize}
\label{fig:skymap}
\end{figure}
\section{Constraints from NGC 1399 Observations}
As can be seen from the spectral energy distribution (SED) of NGC 1399 in Fig.~\ref{fig:SED}, the VHE fraction of its total energy budget is potentially not-negligible. The H.E.S.S.\ limit on the isotropic VHE $\gamma$-ray\ luminosity is:
$$L_\gamma < 9.6 \times 10^{40} \textrm{ erg s}^{-1}.$$
Here it is assumed that the $\gamma$-ray\ emission originates solely from the nucleus, even though the entire galaxy is point-like considering the angular resolution of H.E.S.S.\
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\textwidth]{icrc0470_fig03.eps}
\caption{\small The SED of NGC 1399. All the data are for the core region. The archival points are VLA radio data (red triangles; \cite{sadl}), HST optical data (green squares; \cite{oconn}), and Chandra X-ray upper limits (solid line; \cite{lowen}). The blue dot is the H.E.S.S. upper limit derived from the 2005 observations. Preliminary. \normalsize}
\label{fig:SED}
\end{figure}
In the case of NGC 1399 photon-photon pair absorption would not hide any possible VHE emission. The cross section $\sigma_{\gamma\gamma}$ of this process depends on the product of the energies of the colliding photons. In the case of VHE photons, the most effective interaction is with background photons of energy:
$$\epsilon_{\mathrm{IR}} \approx \left(E/\mathrm{1TeV}\right)^{-1} \textrm{ eV.} $$
The optical depth resulting from this absorption, in a source of luminosity $L$ and radius $R$, reads:
\begin{eqnarray}
\tau\left(E,R_{\mathrm{IR}}\right) & = & \frac{L_{\mathrm{IR}}\sigma_{\gamma\gamma}}{4\pi R_{\mathrm{IR}} \epsilon_{\mathrm{IR}}}
\nonumber\\
& \simeq & 1 \left[\frac{L_{\mathrm{IR}}\left(\epsilon \right)}{10^{-7} L_{\mathrm{Edd}}}\right] \left[\frac{R_\mathrm{S}}{R_{\mathrm{IR}}} \right]\left[\frac{E}{\textrm{1 TeV}}\right],
\nonumber
\end{eqnarray}
where $R_\mathrm{S}$ is the Schwarzschild radius of the black hole and $L_{\mathrm{Edd}}$ is the Eddington luminosity.
In the system here presented, the visibility of a 200 GeV photon requires $L_{\mathrm{IR}} <7.9 \times 10^{40} \textrm{ ergs s}^{-1}$, a condition that seems to be satisfied.
In the p-p interaction scenario, assuming that all the available power (Eq. \ref{eq:power}) will be radiated in the VHE domain, the following limit for the magnetic field is obtained from the H.E.S.S. result:
$$ B < 92.6 \textrm{ Gauss.}$$
In order to maintain gaps in the magnetosphere, as is essential for particle acceleration, pair production should be avoided. Translating this condition into an upper limit for the magnetic field yields:
$$B < 3.6 \times 10^4 \left(M_{10}\right)^{-2/7} = 6.8\times10^4 \textrm{ Gauss.}$$
Therefore the H.E.S.S. NGC 1399 data allow plausible values of the magnetic field.
Considering the production of a 1 TeV photon via curvature emission (Eq. \ref{eq:cutoff}) requires in the case of NGC 1399:
$$B=1.3\times10^3 \textrm{ Gauss.} $$
The non-detection of NGC 1399 does not constrain the magnetic field. In all the aforementioned scenarios, hadronic and/or leptonic, no clear constraints on the magnetic field are derived.
\section{Conclusions}
VHE emission from passive SMBH is plausible either via leptonic or hadronic processes. In order to detect this emission the giant elliptical galaxy NGC 1399 was observed by H.E.S.S. in 2005. NGC 1399 is not detected in these observations. The corresponding upper limit does not allow a firm estimation of the circumnuclear magnetic field.
\subsubsection*{Acknowledgments}
The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment. \\This work has been supported by the International Max Planck Research School (IMPRS) for Astronomy \& Cosmic Physics at the University of Heidelberg.
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{"url":"https:\/\/liusson.com\/homework-solution-the-next-7-questions-pertain-to-a-general-normalized-floating-point-system-gb-p\/","text":"# Homework Solution: The next 7 questions pertain to a general, normalized floating-point system G(B, p,\u2026\n\nThe next 7 questions pertain to a general, normalized floating-point system G(B, p, m,M). Express answers in terms of B, p, m, and M, as needed. You may assume that m is negative and M is positive, and that M greaterthanorequalto p. Show your work where needed (you can assume as given anything from the slides). What is the smallest positive number in G? What is the largest positive number in G? What is the spacing between the numbers of G in the range [B^e, B^e+1)? How many numbers of G are there in the interval [B^e, B^e+l)? How many positive numbers are there in G? What is the smallest positive integer not representable in G?\n\nThe direct 7 questions pertain to a open, normalized floating-point scheme G(B, p, m,M). Express exculpations in provisions of B, p, m, and M, as needed. You may claim that m is disclaiming and M is enacted, and that M greaterthanorequalto p. Show your achievement where needed (you can claim as ardent everything from the slides). What is the last enacted sum in G? What is the largest enacted sum in G? What is the spacing among the sums of G in the rank [B^e, B^e+1)? How manifold sums of G are there in the space-among [B^e, B^e+l)? How manifold enacted sums are there in G? What is the last enacted integer referable representable in G?\n\n## Expert Exculpation\n\nThe normalized floating-point sums scheme G is a 4-tuple (BpmM) where\n\n\u2022 B\u00a0is the deep of the scheme,\n\u2022 p\u00a0is the accuracy of the scheme to\u00a0p\u00a0numbers,\n\u2022 m\u00a0is the last propounder representable in the scheme,\n\u2022 and\u00a0M\u00a0is the largest propounder used in the scheme.\n\n6.\u00a0Last enacted FP sum in G must possess a 1 as the promotive digit and 0 restraint the cherishing digits of the significand, and the last feasible compute restraint the propounder, which is correspondent to\u00a0$B^{m}$.\n\n7.\u00a0Largest floating-point sum\u00a0\u00a0must possess\u00a0B\u00a0\u2212 1 as the compute restraint each digit of the significand and the largest feasible compute restraint the propounder,which is correspondent to\u00a0$(1-B^{-p})(B^{M+1})$\u00a0.\n\n8.\u00a0The sums among\u00a0$\\[B^{e},B^{e+1})$\u00a0are correspondently disconnected by\u00a0$B^{-p}$\u00a0units.\n\n9.\u00a0Total sums among space-between\u00a0$\\[B^{e},B^{e+1})$\u00a0is ardent by\u00a0$\\frac{B^{e+1}-B^{e}}{B^{-p}} = B^{e+p}*(B-1)$\n\n10.Total +ve sums among in G are\u00a0$2*(B-1)(B^{p-1})(M-m+1)+1$","date":"2020-10-01 23:02:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 7, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8709075450897217, \"perplexity\": 2331.939126233179}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600402132335.99\/warc\/CC-MAIN-20201001210429-20201002000429-00670.warc.gz\"}"}
| null | null |
Montejo de Tiermes è un comune spagnolo di 233 abitanti situato nella comunità autonoma di Castiglia e León.
Nel territorio comunale si trovano i resti dell'antica città Celtiberica e romana di Tiermes (Termes).
Altri progetti
Comuni della provincia di Soria
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801. Tell Me When It's Over
daron | January 12, 2017
I finally talked to Ziggy for a good stretch on the phone around lunchtime in St. Louis. It had been like seven in the morning when we'd pulled in, Flip made sure I was tucked in and passed out before he went to bed himself, but he'd apparently slept through all the drama on the bus the night before so he got up in the morning and was not there when I woke up.
I ordered room service and caught Ziggy at home.
"Are you watching the news?" he said. "Bush just made a statement about the death of the Communist Party in Russia."
"What?" The state of world politics was pretty far from my mind at that point. Bush was the first Bush, by the way, the one who'd been veep under Reagan. We were in our twelfth regressive year in a row under conservative Republican rule at that point, the party that had been in power since I was in grade school.
"The U.S.S.R. just banned the Communist Party. Or disbanded it. Or something."
"I don't know, but it's good," he said. "How are you?"
"You are not going to believe what's been going on here."
I turned on the TV under the theory that I could watch the crawl on CNN while talking to him but the first thing I saw was the video for a Bryan Adams song I was utterly sick of at that point so I turned it off again. "My mother lives in some kind of conservative Christian enclave a couple in Kansas of hours from where we played last night. Court made plans to go visit her."
"Yeah, she had said she was thinking about trying to do that."
"Well, she did. And she brought my mother to see a show."
"Oh. And were there…fireworks?"
"No. Though my head did explode."
"Oh, dear one. I'm so sorry. Do you want to tell me about it?"
There was something so formal about how he said that; it rubbed me the wrong way. It was like I knew he was going through the motions to be supportive but somehow it didn't feel supportive. But was that because he wasn't "really" supportive or because I was too messed up right then to be able to appreciate it? Was it possible to be sincere about being willing to listen without being sincere about wanting to?
I decided to wait. "In a little while, maybe. Tell me how you're doing."
"Doing fine, but bored and missing you. You had all your shots?"
"What? Oh you mean before traveling to South America? Yeah, a while ago. And the band were supposed to all get them when they signed their contracts."
"Just checking."
"I'm sure Barrett and Carynne are on top of that kind of thing. Are you really thinking about stuff like that?"
"I guess I'm just looking out for whatever could go wrong." He sighed.
"You're clearly bored." I sat back on the bed. Everything in the hotel room was some shade of brown. "Memorized all the choreography already?"
"Yes. I've done everything I can do without you here."
"You sound resentful I'm not there."
"I am resentful you're not here. I'm okay, and I'm okay with you being on the road, but that doesn't stop me from feeling this way."
Well, that was honest. Ziggy always contained multitudes. "I'll be there soon. Tuesday, I think. Today's…" I had to get off the bed to look at the day sheet to be sure. "Thursday. I'm in St. Louis."
"When I was a kid I thought Kansas was a made-up place because of the Wizard of Oz," he said, in what might have sounded like a non-sequitur, but was actually obviously meant to bring me back to what I was saying before about my mother.
I took my cue this time. "It's real, but it's real strange. Like, Claire had to get her husband's permission to go anywhere?"
"I think that's how marriages are in a lot of the Third World," Ziggy said, "as well as traditional and rural parts of Europe."
"It's creepy."
"Agreed. But it's how a lot of people live."
"Are you saying it's okay?"
"No. The same people who think that's a wholesome lifestyle would like to see you and I dead." His voice was rather matter of fact, almost emotionless for such a strong sentiment.
Or maybe it was that through the phone there was only so much I could feel. "Anyway. She came to the show and then Remo and I went to see her backstage and… and…" I found myself at a loss to describe the encounter. "It was weird. She was…so…fake."
"Fake how?"
"I mean, like putting on such a show of… manners, I guess."
"So the opposite of Digger who would have cursed you out, spat on the floor, and left a shit stain if he could?"
That made me chuckle a little. "I guess. They do say opposites attract. The thing is she really only said like two sentences maybe to me before I had a concussion flare up."
"Maybe that's for the best. If all she said was something nice, I'd hate to see what happened if she said what she really felt."
"Yeah…" And yet. "I still somehow feel sort of let down, though."
"Well, you've built her up as Cruella DeVille in your mind. Of course it's disappointing if she's not wearing a skinned dalmatian."
"I suppose." I cradled the phone with my shoulder and began to do the gentle hand exercises I was supposed to. The first one was using the fingers of my left hand to carefully stretch and straighten the fingers on the right. "Then Remo drove her home and that made Melissa go off because she thought he went off to sleep with her."
"He didn't?"
"No! Besides, would a woman who asks her husband's permission to go see a concert commit adultery?"
"If she thought she could get away with it, maybe? But presumably Remo's too much of a goody two shoes for that."
"Pretty sure he is," I growled.
"Why are you being so defensive? I'm not criticizing."
"Sure you aren't."
"You're really in a state, aren't you? It's not like you to bite my head off for no reason."
Why did it upset me that he didn't sound the slightest bit upset? I knew rationally that he had no reason to be–and that I didn't want him to be upset at me–but somehow I was irked that his sympathy only extended to dispassionate listening and not participating in my pain. Which is unfair and counterproductive but that's how I felt.
Maybe it was the distance. "I have the day off tomorrow," I said.
"What are you planning to do?"
"Nothing. What are you doing this weekend?" I was trying for casual small talk.
His voice sounded like a shrug. "Everyone's off for the holiday weekend. No rehearsals, no nothing."
Right. Labor Day Weekend. "Come to St. Louis."
"Get on a plane. You can be here by showtime tonight I bet."
"You're serious?"
"I know I'm going to see you in a few days but–"
"Can you get someone to pick me up at the airport?"
"Page me your flight info when you have it and I'll see if someone from the crew can do it. If not I'm sure there's a limo service. We're at a hotel downtown. The venue's not far. St. Louis is a city, in case you were thinking we're in some cow pasture somewhere." I was stomping hard on the little voice in the back of my head that said it was going to take a thousand dollars minimum to get him there, what with needing to buy last minute tickets. I could live off that for a month.
"You're really really sure about this," he said.
"Yes. It'll be good for us to see each other."
I was so sure about that.
Categories: Daron's Guitar Chronicles Tags: another hotel room, another phone call, st. louis, ziggy | Comments
I really do think the time apart has made Ziggy resentful, and I don't blame him, not really. I got the same vibe as you. I think it may even still be weird when he first gets there, because it's still not permanent until next week, but hopefully not. I hope you can both just feel each other's presence and nothing else. But that's just me being your cheerleader, as usual.
I'm also hoping you are not foreshadowing something bad with that last line. It's been a while since I've been all angsty with you two so here's hoping those last three lines are just what they are and nothing else. Stay positive, G.
Mark Treble says:
ctan is the master of vorimitation. We're due for a crisis. If it's the one I've been suspecting, involving Carynne, it may be existential.
Would just "I was sure of that" be better? Heh.
That last line sounds ominous.
Also, typo:"My mother lives in some kind of conservative Christian enclave a couple in Kansas of hours from where we played last night.
Wow yeah I really made that into word salad. It's the right words, just tossed around wrong…
Just remember that indirect communication can create inaccurate filters…don't make assumptions. Even if Ziggy isn't sounding as supportive as you'd like and just going through the motions, he's present and responding. Daron, you may be a bit hypersensitive after having to deal with both your mom and Remo's issues. Focus on the fact that Ziggy is willing to come to you… just PLEASE enjoy it… no ANGST!
I totally feel this. You and s really brought me to a better place!
And Ziggy being Ziggy means filters anyway. And I'm sure I'm not exactly at my clearest right now. But yeah at least he's coming.
All these downer comments, and I'm over here like, "Daron, you just made Ziggy's day!" Look how quickly he said yes. He said he needs to feel like he's important to you, and you just said the magic words. And you'll feel so much better seeing him. I'm feeling pretty good about this…don't fuck it up. Lol
s, I did so sound like a downer! I must have just had a bad moment with students. Anyway, I totally see your side. I do understand Ziggy being resentful that Daron is not with him, but you're right, he did jump on that idea. I also like what chris wrote about Daron being hypersensitive after the Remo and Claire situations. He's probably stretched thin emotionally already (our Daron tends to do that sometimes). Thanks for the better perspective!
Being away from Ziggy gets to Daron more than he says. I fully except him to melt into a puddle when he sees him.
That said, they'll probably fight and make me cry. But right now, I'm feeling good. Lol
Honestly, I don't want to rain on anyone's parade, but I may just do that. You remember "I get lonely?" And, "I can do that!"
Let's assume Daron has completely misread Ziggy's tone, just focus on the words. He is resentful, and the words around Claire are those of someone going through the motions.
Daron, please call Carynne and ask her about Ziggy. General questions are all. You know her far better than you know Ziggy (shit, thr whole band knows more about quantum physics than ANYONE knows about Ziggy). Listen to phrasing and tone. And do this, please, before Ziggy arrives.
LOL me use the phone any more than absolutely necessary? Not likely.
Shit. I can't keep up. 800 chapters?!?! Wow. Congrats, Cecilia and Daron! 💜
Thank you! I can't believe it's been this long, either!
Feral says:
Thanks for getting the RSS back up! You're wonderful.
You're welcome! I'm still not sure what combination of software updates and hardware updates did it, but here's hoping the situation holds!
← 800. I Think I Love You Too Much
802. Run to You →
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\section{Introduction}
This article is devoted to the study of the `quantization commutes
with reduction' principle of Guillemin-Sternberg
\cite{Guillemin-Sternberg82}. The object of this paper is twofold.
The first goal is to give a K-theoretic approach to this problem
which provides a new
proof of results obtained by Meinrenken \cite{Meinrenken98},
Meinrenken-Sjamaar \cite{Meinrenken-Sjamaar} and Tian-Zhang
\cite{Tian-Zhang98}. The second goal is to define an extension
to the {\em non-symplectic} case.
In the Kostant-Souriau framework
one considers a prequantum line bundle $L$ over a compact
symplectic manifold $(M,\omega)$ : $L$ carries a
Hermitian connection $\nabla^{L}$
with curvature form equal to $-\imath\omega$. Suppose now that a
compact Lie group $G$, with Lie algebra $\ensuremath{\mathfrak{g}}$,
acts on $L\to M$, living the data
$(\omega,\nabla^{L})$ invariant. Then the $G$-action on $(M,\omega)$
is Hamiltonian with moment map $f_{_{G}}:M\to\ensuremath{\mathfrak{g}}^*$ given by
the Kostant formula :
$\ensuremath{\mathcal{L}}^L(X)-\nabla^L_{X_M}=\imath \langle f_{_G},X\rangle, \ X\in\ensuremath{\mathfrak{g}}$.
Here $\ensuremath{\mathcal{L}}^L(X)$ is the infinitesimal action of $X$ on the section
of $L\to M$ and $X_M$ is the vector field on $M$ generated by
$X\in \ensuremath{\mathfrak{g}}$.
Choose now an invariant almost complex structure $J$ on $M$
that is compatible with $\omega$, in the sense that $\omega(-,J-)$
defines a Riemannian metric. It defines a quantization map
$$
RR^{^{G,J}}(M,-): K_G(M)\to R(G)\ ,
$$
from the equivariant $K$-theory of complex vector bundles over $M$ to
the character ring of $G$. The
`quantization commutes with reduction' Theorem tells us how the
multiplicities of $RR^{^{G,J}}(M,L)$ behave (see Theorem {\bf C}).
\medskip
Here our main goal is to compute the multiplicity of the trivial
representation in $RR^{^{G,J}}(M,L)$, when the data $(L,J)$ are
not associated to a symplectic form.
\medskip
We consider a compact manifold $M$ on which a compact Lie group $G$
acts, and which carries a $G$-invariant almost complex structure $J$.
Let $L\to M$ be a $G$-equivariant Hermitian line bundle over $M$,
equipped with a Hermitian connection $\nabla^L$ on $L$.
This defines a map $f_{_L}:M\to \ensuremath{\mathfrak{g}}^*$ by the equation
\begin{equation}\label{eq.moment.L}
\ensuremath{\mathcal{L}}^L(X)-\nabla^L_{X_M}=\imath \langle f_{_L},X\rangle,\quad
X\in\ensuremath{\mathfrak{g}}\ .
\end{equation}
(see \cite{B-G-V}[section 7.1]). The map $f_{_L}$ is an {\em abstract
moment map} in the sense of Karshon \cite{Karshon.98}, since $f_{_L}$
is equivariant, and for any $X\in\ensuremath{\mathfrak{g}}$, the function $\langle
f_{_L},X\rangle$ is locally constant on the submanifold $M^X:=\{m\in
M,\ X_{M}(m)=0\}$.
If $0$ is a regular value of $f_{_L}$, $\ensuremath{\mathcal{Z}}:= f_{_L}^{-1}(0)$ is a
smooth submanifold of $M$ which carries a locally free action of $G$.
We consider the orbifold reduced space $\ensuremath{\mathcal{M}}_{red}=\ensuremath{\mathcal{Z}}/G$ and we
denote $\pi :\ensuremath{\mathcal{Z}}\to\ensuremath{\mathcal{M}}_{red}$ the projection. In
Lemma \ref{lem.spinc.induit} we show that the almost complex structure
$J$ induces an orientation $o_{red}$ on $\ensuremath{\mathcal{M}}_{red}$ and a Spin$^{\rm
c}$ structure on $(\ensuremath{\mathcal{M}}_{red},o_{red})$. Let $\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},-):
K_{orb}(\ensuremath{\mathcal{M}}_{red})\to \ensuremath{\mathbb{Z}}$ be the quantization map defined by the
Spin$^{\rm c}$ structure and let $L_{red}\to\ensuremath{\mathcal{M}}_{red}$ be the
orbifold line bundle induced by $L$.
We obtain the following `quantization commutes with reduction'
theorem.
\medskip
{\bf Theorem A}\hspace{5mm}{\it Let $L\to M$ be a $G$-equivariant
Hermitian line bundle over $M$, equipped with a Hermitian connection
$\nabla^L$ on $L$. Let $f_{_L}:M\to \ensuremath{\mathfrak{g}}^*$ be the corresponding
abstract moment map. If $0$ is a regular value of
$f_{_{L}}$, we have
\begin{equation}\label{eq.theorem.A}
\left[ RR^{^{G,J}}(M,L^{\stackrel{k}{\otimes}})\right]^G=
\ensuremath{\mathcal{Q}}\Big(\ensuremath{\mathcal{M}}_{red},L_{red}^{\stackrel{k}{\otimes}}\Big), \quad
k\in\ensuremath{\mathbb{N}}-\{0\},
\end{equation}
if any of the following hold:
(i) $G=T$ is a torus; or
(ii) $k\in \ensuremath{\mathbb{N}}$ is large enough , so that the ball
$\{ \xi\in\ensuremath{\mathfrak{g}}^{*},\, \parallel\xi\parallel\leq\frac{1}{k}
\parallel\theta\parallel \}$
is contained in the set of regular values of $f_{_L}$.
Here $\theta=\sum_{\alpha>0}\alpha$ is
the sum of the positive roots of $G$, and $\parallel\cdot\parallel$
is a $G$-invariant Euclidean norm on $\ensuremath{\mathfrak{g}}^*$.
Here, for $V\in R(G)$, we denote $[V]^G\in \ensuremath{\mathbb{Z}}$ the multiplicity of the
trivial representation.
}
\medskip
A similar result was proved by Jeffrey-Kirwan \cite{Jeffrey-Kirwan97}
in the Hamiltonian setting when one relaxes the condition of
positivity of $J$ with respect to the symplectic form. See also
\cite{CdS-K-T} for a similar result in the Spin$^{\rm c}$ setting,
when $G=S^1$.
\medskip
As an example, let us apply Theorem {\bf A} to the counterexemple
due to Vergne which shows that quantization does not always commute
with reduction. Let $G=SU(2)$ and let $M$ be the $SU(2)$-coadjoint
orbit passing through the
unique positive root $\theta$. Thus $M$ is the projective line bundle
$\ensuremath{\mathbb{C}}\mathbb{P}^{1}$ with $\omega$ equal to twice the standard
K\"ahler form. The prequantum line bundle is $L=\mathcal{O}(2)$ and
$RR^{^G}(M,L^{-1})=[RR^{^G}(M,L^{-1})]^G=-1$. Since
$\ensuremath{\mathcal{M}}_{red}=\emptyset$ we have $[RR^{^G}(M,L^{-1})]^G\neq$ \break
$\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},(L^{-1})_{red})$ : the condition $ii)$ of
Theorem {\bf A} does not hold since $\theta$ is not a regular value of
the moment map $M\ensuremath{\hookrightarrow}\ensuremath{\mathfrak{g}}^*$. But if we take
$(L^{-1})^{\stackrel{k}{\otimes}}$ with $k>1$ the condition $ii)$ is
satisfied, and thus $[RR^{^G}(M,(L^{-1})^{\stackrel{k}{\otimes}})]^G
=0$ for $k>1$. In fact a direct computation shows that
$-\,RR^{^G}(M,(L^{-1})^{\stackrel{k}{\otimes}})$ is the character of the
irreducible $SU(2)$-representation with highest weight $(k-1)\theta$ for
all $k\geq 1$.
\medskip
The result of Theorem {\bf A} can be rewritten when $J$ defines an almost
complex structure $J_{red}$ on $\ensuremath{\mathcal{M}}_{red}$. It happens when
the following decomposition holds
\begin{equation}\label{eq.J.red}
\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}\oplus J(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}})\quad {\rm with}\quad
\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}:=\{X_{\ensuremath{\mathcal{Z}}},\, X\in\ensuremath{\mathfrak{g}}\}\ .
\end{equation}
First we note that (\ref{eq.J.red}) always holds in the
Hamiltonian case when $J$ is compatible with the symplectic form.
Condition (\ref{eq.J.red}) already appears in the works of
Jeffrey-Kirwan \cite{Jeffrey-Kirwan97}, and Cannas da Silva-Karshon-Tolman
\cite{CdS-K-T}.
In all this paper we fix a $G$-invariant
scalar product on $\ensuremath{\mathfrak{g}}^*$ which induces an identification
$\ensuremath{\mathfrak{g}}\simeq\ensuremath{\mathfrak{g}}^*$. Thus $f_{_G}$ can be considered as a map from
$M$ to $\ensuremath{\mathfrak{g}}$, and we define the
endomorphism $\ensuremath{\mathcal{D}}$ of the bundle $\ensuremath{\mathfrak{g}}\times\ensuremath{\mathcal{Z}}$ by :
$\ensuremath{\mathcal{D}}(X)=-d\,f_{_G}(J(X_{\ensuremath{\mathcal{Z}}}))$, for $X\in\ensuremath{\mathfrak{g}}$. Condition
(\ref{eq.J.red}) is then equivalent to :
$\det\ensuremath{\mathcal{D}}(z)\neq 0$ for all $z\in\ensuremath{\mathcal{Z}}$. The endomorphism $\ensuremath{\mathcal{D}}$
defines a complex stucture $J_{\ensuremath{\mathcal{D}}}$ on $\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$,
so the vector bundle $\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$ inherits two irreducible
complex spinor bundles $\ensuremath{\mathcal{Z}}\times\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$
and $\ensuremath{\mathcal{Z}}\times\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$ related by
$$
\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}} =
\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}\otimes \pi^*L_{\ensuremath{\mathcal{D}}}
$$
where $L_{\ensuremath{\mathcal{D}}}\to\ensuremath{\mathcal{M}}_{red}$ is a line bundle (see (\ref{eq.L.D})).
In this case we prove in Proposition \ref{prop.RR.T.0.bis}
that (\ref{eq.theorem.A}) has the following form
\begin{equation}\label{eq.theorem.A.bis}
\left[ RR^{^{G,J}}(M,L^{\stackrel{k}{\otimes}})\right]^G
=\pm RR^{^{J_{red}}}
\Big(\ensuremath{\mathcal{M}}_{red},L_{red}^{\stackrel{k}{\otimes}}\otimes
L_{\ensuremath{\mathcal{D}}}\Big)\ ,
\end{equation}
where $\pm$ is the sign of $\det\ensuremath{\mathcal{D}}$, and where
$RR^{^{J_{red}}}(\ensuremath{\mathcal{M}}_{red},-)$ is the Riemann-Roch character
defined by $J_{red}$.
\medskip
In this paper, we start from an abstract moment
map $f_{_G}:M\to\ensuremath{\mathfrak{g}}^*$ , and we extend the result of Theorem {\bf A}
to the $f_{_G}$-moment bundles, and the
$f_{_G}$-positive bundles. These notions were
introduced in the Hamiltonian setting by Meinrenken-Sjamaar
\cite{Meinrenken-Sjamaar} and Tian-Zhang \cite{Tian-Zhang98}.
Let us recall the definitions.
Let $H$ be a maximal torus of $G$ with Lie algebra $\ensuremath{\mathfrak{h}}$.
\begin{defi}\label{moment.bundle}
A $G$-equivariant line bundle over $M$ is called a $f_{_G}$-moment
bundle if for all components $F$ of the fixed-point set $M^H$
the weight of the $H$-action on $L\vert_F$ is equal to $f_{_G}(F)$.
\end{defi}
It is easy to see that the definition is independent of the choice of
the maximal torus. Note that $f_{_G}(F)\in \ensuremath{\mathfrak{h}}^*=(\ensuremath{\mathfrak{g}}^*)^H$,
since $f_{_G}$ is equivariant. Any Hermitian line bundle $L$ is
tautologically a moment bundle relative to the abstract moment
map $f_{_L}$.
For any $\beta\in\ensuremath{\mathfrak{g}}$, we denote by $\ensuremath{\mathbb{T}}_{\beta}$
the torus of $G$ generated by $\exp_{G}(t.\beta),\, t\in \ensuremath{\mathbb{R}}$, and
$M^{\beta}$ the submanifold of points fixed by $\ensuremath{\mathbb{T}}_{\beta}$.
\begin{defi}\label{eq.mu.positif}
A complex $G$-vector bundle $E$ is called $f_{_G}$-{\em positive}
if the following hold: for
any $m\in M^{\beta}\cap f_{_G}^{-1}(\beta)$, we have
$$
\langle \xi,\beta\rangle \geq 0
$$
for any weights $\xi$ of the $\ensuremath{\mathbb{T}}_{\beta}$-action on $E_m$.
A complex $G$-vector bundle $E$ is called $f_{_{G}}$-{\em strictly
positive} when furthermore the last inequality
is strict for any $\beta\neq 0$.
For any $f_{_{G}}$-{\em strictly positive} complex vector bundle $E$,
and any $\beta\in\ensuremath{\mathfrak{g}}$ such that
$M^{\beta}\cap f_{_G}^{-1}(\beta)\neq \emptyset$, we define
$\eta_{_{E,\beta}}=\inf_{\xi}\langle \xi,\beta\rangle$,
where $\xi$ runs over the set of weights for the
$\ensuremath{\mathbb{T}}_{\beta}$-action on the fibers of each complex vector
bundle $E\vert_{\ensuremath{\mathcal{Z}}}$, $\ensuremath{\mathcal{Z}}$ being a connected component
of $M^{\beta}$ that intersects $f_{_G}^{-1}(\beta)$.
\end{defi}
It is not difficult to see that a $f_{_G}$-moment bundle $L$ is
$f_{_{G}}$-strictly positive with $\eta_{_{L,\beta}}=\parallel
\beta\parallel^2$, for any $\beta\in\ensuremath{\mathfrak{g}}$ such that
$M^{\beta}\cap f_{_G}^{-1}(\beta)\neq\emptyset$ (see Lemma \ref{L-a-positif}).
The bundle $M\times\ensuremath{\mathbb{C}}\to M$ is the trivial
example of $f_{_{G}}$-positive complex vector bundle over $M$.
\medskip
Let $\ensuremath{\mathfrak{h}}_{+}$ be the choice of some positive Weyl chamber in $\ensuremath{\mathfrak{h}}$.
We prove in Lemma \ref{lem.C.f.G} that the set $\ensuremath{\mathcal{B}}_{_{G}}:=
\{\beta\in\ensuremath{\mathfrak{h}}_{+},\ M^{\beta}\cap f_{_{G}}^{-1}(\beta)\neq\emptyset\}$
is finite.
\medskip
{\bf Theorem B}\hspace{5mm}{\it
Let $f_{_{G}}:M\to\ensuremath{\mathfrak{g}}^{*}$ be an abstract moment map with $0$ as
regular value. Let $E$ be a $f_{_{G}}$-{\em strictly positive} $G$-complex
vector bundle over $M$ (see Def. \ref{eq.mu.positif}). We have
\begin{equation}\label{eq.theorem.B}
\left[ RR^{^{G,J}}(M,E^{\stackrel{k}{\otimes}})\right]^G=
\ensuremath{\mathcal{Q}}\Big(\ensuremath{\mathcal{M}}_{red},E_{red}^{\stackrel{k}{\otimes}}\Big),
\quad k\in\ensuremath{\mathbb{N}}-\{0\},
\end{equation}
if any of the following hold:
(i) $G=T$ is a torus; or
(ii) $k$ is large enough, so that
$k.\eta_{_{E,\beta}}>\sum_{\alpha>0}\langle \alpha,\beta\rangle$, for
any $\beta\in \ensuremath{\mathcal{B}}_{_{G}}-\{0\}$; here the sum $\sum_{\alpha>0}$
is taken over the positive roots of $G$.
Moreover if (\ref{eq.J.red}) holds, (\ref{eq.theorem.B}) becomes
$$
\left[ RR^{^{G,J}}(M,E^{\stackrel{k}{\otimes}})\right]^G=\pm
RR^{^{J_{red}}}\Big(\ensuremath{\mathcal{M}}_{red},E_{red}^{\stackrel{k}{\otimes}}
\otimes L_{\ensuremath{\mathcal{D}}}\Big)\ .
$$
}
\bigskip
Let us explain why Theorem {\bf B} applied to a $G$-hermitian line
bundle $L$ with the abstract moment map $f_{_{G}}=f_{_{L}}$ implies
Theorem {\bf A}. It is sufficient to prove that condition $ii)$ of
Theorem {\bf A} implies condition $ii)$ of Theorem {\bf B}. The
curvature of $(L,\nabla^L)$ is $(\nabla^L)^2=-\imath\, \omega^L$,
where $\omega^L$ is a differential $2$-form on $M$. From the
equivariant Bianchi formula (see Proposition 7.4 in \cite{B-G-V}) we
get $\langle d f_{_{L}},X\rangle=-\omega^{L}(X_{M},-)$ for any
$X\in\ensuremath{\mathfrak{g}}$. So, for any $\beta\in \ensuremath{\mathcal{B}}_{_{G}}-\{0\}$, and $m\in
M^{\beta}\cap f_{_{L}}^{-1}(\beta)$, the last equality gives $\langle
d f_{_{L}}\vert_{m},\beta\rangle= 0$, hence $\beta$ is a critical
value of $f_{_{L}}$. Suppose now that $k\in\ensuremath{\mathbb{N}}$ is large enough so
that the ball $\{ \xi\in\ensuremath{\mathfrak{g}}^{*},\, \parallel\xi\parallel\leq
\frac{1}{k} \parallel\theta\parallel \}$ is included in the set of regular
values of $f_{_{L}}$. This gives first $\parallel\beta\parallel>
\frac{1}{k} \parallel\theta\parallel$ and then
$\eta_{_{L,\beta}}=\parallel\beta\parallel^2>
\frac{1}{k}\langle \theta,\beta\rangle$,
for any $\beta\in \ensuremath{\mathcal{B}}_{_{G}}-\{0\}$. $\Box$
\medskip
In the last section of this paper, we restrict ourselves to the
Hamiltonian case. In this situation, the
abstract moment map $f_{_G}$ and the almost complex structure
$J$ are related by means of a $G$-invariant symplectic $2$-form $\omega$ :
\begin{itemize}
\item $f_{_{G}}$ is the moment map associated to a Hamiltonian action
of $G$ over $(M,\omega)$ : $ d\langle f_{_G},X\rangle=-\omega(X_M,-)$ ,
for $X\in\ensuremath{\mathfrak{g}}$, and
\item the data $(\omega, J)$ are {\em compatible} :
$(v,w)\to \omega(v,Jw)$ is a Riemannian metric on $M$.
\end{itemize}
When $0$ is a regular value of $f_{_G}$, the compatible
data $(\omega, J)$ induce compatible data
$(\omega_{red}, J_{red})$ on $\ensuremath{\mathcal{M}}_{red}$. We have then a map
$RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},-)$. If $0$ is not a regular
value of $f_{_G}$, we consider elements $a$ in the principal
face $\tau$ of the Weyl chamber (see subsection \ref{non-regulier}). For
generic elements $a\in\tau\cap f_{_{G}}(M)$, the set
$\ensuremath{\mathcal{M}}_{a}:=f_{_G}^{-1}(G\cdot a)/G$ is a symplectic orbifold and one
can consider the quantization map $RR^{J_{a}}(\ensuremath{\mathcal{M}}_{a},-)$
relative to the choice of compatible almost complex structure $J_{a}$.
In this situation, we recover the results of
\cite{Meinrenken98,Meinrenken-Sjamaar,Tian-Zhang98}.
\medskip
{\bf Theorem C}\ (Meinrenken, Meinrenken-Sjamaar, Tian-Zhang).\
{\it Let $f_{_{G}}$ be the moment map
associated to a Hamiltonian action of $G$ over $(M,\omega)$, and let
$J$ be a $\omega$-compatible almost complex structure. Let
$E\to M$ be a $G$-vector bundle.
\noindent If $0\notin f_{_{G}}(M)$
and $E$ is $f_{_{G}}$-strictly positive, we have $[ RR^{^{G,J}}(M,E)]^G=
0$.
\noindent If $0\in f_{_{G}}(M)$ then :
\begin{enumerate}
\item[i)] If $0$ is a regular value, we have
$[ RR^{^{G,J}}(M,E)]^G=RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},E_{red})$, if
$E$ is $f_{_G}$-positive.
\item[ii)] If $0$ is not a regular value of $f_{_G}$ and $E=L$ is
a $f_{_G}$-moment bundle, we have
$[ RR^{^{G,J}}(M,L)]^G=RR^{J_{a}}(\ensuremath{\mathcal{M}}_{a},L_{a})$,
for every generic value $a$ of $\tau\cap f_{_{G}}(M)$ sufficiently
close to $0$. Here $L_a$ is the orbifold line bundle
$L\vert_{f_{_G}^{-1}(G\cdot a)}/G$.
\end{enumerate}
}
\bigskip
We now turn to an introduction of our method. We associate to the
abstract moment map $f_{_{G}}:M\to\ensuremath{\mathfrak{g}}$ the vector field
$$
\ensuremath{\mathcal{H}}^{^G}_{m}=[f_{_{G}}(m)]_{M}.m,\quad m\in M\ ,
$$
and we denote by $C^{f_{_{G}}}$ the set where $\ensuremath{\mathcal{H}}^{^G}$ vanishes.
There are two important cases. First, when the map $f_{_{G}}$ is
constant, equal to an element $\gamma$ in the center of $\ensuremath{\mathfrak{g}}$, the
set $C^{f_{_{G}}}$ corresponds to the submanifold $M^{\gamma}$.
Second, when $f_{_{G}}$ is the moment map associated with a
Hamiltonian action of $G$ over $M$. In this situation, Witten
\cite{Witten} introduces the vector field $\ensuremath{\mathcal{H}}^{^G}$ to propose, in
the context of equivariant cohomology, a localization on the
set of critical points of the function $\parallel f_{_{G}}\parallel^2$
: here $\ensuremath{\mathcal{H}}^{^G}$ is the Hamiltonian vector field of
$\frac{-1}{2}\parallel f_{_{G}}\parallel^2$, hence
$\ensuremath{\mathcal{H}}^{^G}_m=0\Longleftrightarrow d(\parallel
f_{_{G}}\parallel^2)_m=0$. This idea has been developed by the author
in \cite{pep1,pep2}.
\medskip
Using a deformation argument in the context of transversally elliptic
operator introduced by Atiyah \cite{Atiyah.74} and Vergne
\cite{Vergne96}, we prove in
section \ref{sec.general.procedure} that the
map\footnote{We fix one for once a
G-invariant almost complex structure $J$ and denote by $RR^{^{G}}$ the
quantization map.} $RR^{^{G}}$ can be localized near
$C^{f_{_{G}}}$. More precisely, we have the finite
decomposition $C^{f_{_{G}}}=\cup_{\beta\in\ensuremath{\mathcal{B}}_{_{G}}}C_{\beta}^{^G}$
with $C_{\beta}^{^G}=G(M^{\beta}\cap f_{_{G}}^{-1}(\beta))$, and
\begin{equation}\label{RR.G.decompose}
RR^{^G}(M,E)= \sum_{\beta\in\ensuremath{\mathcal{B}}_{_{G}}} RR^{^G}_{\beta}(M,E)\ .
\end{equation}
Each term $RR^{^G}_{\beta}(M,E)$ is a generalized character of $G$
that only depends on the behaviour of the data $M,E,J,f_{_{G}}$ near
the subset $C_{\beta}^{^G}$. In fact, $RR^{^G}_{\beta}(M,E)$ is the
index of a transversally elliptic operator defined in an open
neighbourhood of $C_{\beta}^{^G}$.
Our proof of Theorems {\bf B} and {\bf C} is in two steps. First we
compute the term $RR^{^G}_{0}(M,E)$ which is the Riemann-Roch
character localized near $f_{_{G}}^{-1}(0)$. After, we prove that
$[RR^{^G}_{\beta}(M,E)]^{G}=0$ for every $\beta\neq 0$.
For this purpose, the analysis of the localized Riemann-Roch
characters $RR^{^G}_{\beta}(M,-):K_{G}(M)\to
R^{-\infty}(G)$ is divided in three cases\footnote{$G_{\beta}$ is the
stabilizer of $\beta$ in $G$.} :
\medskip
\noindent{\bf Case 1 :} $\beta=0$
\noindent{\bf Case 2 :} $\beta\neq 0$ and $G_{\beta}=G$,
\noindent{\bf Case 3 :} $G_{\beta}\neq G$.
\medskip
We work out {\bf Case 1} in subsection \ref{subsec.RR.G.O}. We compute the
generalized character $RR^{^G}_{0}(M,E)$ when $0$ is a regular value
of $f_{_{G}}$. We prove in particular that the multiplicity of the
trivial representation in $RR^{^G}_{0}(M,E)$ is
$\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},E_{red})$. This last quantity is equal to $\pm
RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},E_{red}\otimes L_{\ensuremath{\mathcal{D}}})$ when
(\ref{eq.J.red}) holds.
\medskip
{\bf Case 2} is studied in section \ref{sec.loc.M.beta} for the
particular situation where $f_{_{G}}$ is constant, equal to a
G-invariant element $\beta\in\ensuremath{\mathfrak{g}}$. Then
$C^{f_{_{G}}}=C_{\beta}^{^G}=M^{\beta}$, and (\ref{RR.G.decompose})
becomes $RR^{^G}(M,E)= RR^{^G}_{\beta}(M,E)$. We prove then a
localization formula (see (\ref{eq.localisation.1})) in the spirit of
the Atiyah-Segal-Singer formula in equivariant K-theory
\cite{Atiyah-Segal68,Segal68}. Let us sketch out the result.
The normal bundle $\ensuremath{\mathcal{N}}$ of $M^{\beta}$ in $M$ inherits a canonical
complex structure $J_{\ensuremath{\mathcal{N}}}$ on the fibers. We denote by
$\overline{\ensuremath{\mathcal{N}}}\to M^{\beta}$ the complex vector bundle with the
opposite complex structure. The torus $\ensuremath{\mathbb{T}}_{\beta}$ is included in
the center of $G$, so the bundle $\overline{\ensuremath{\mathcal{N}}}$ and the virtual
bundle $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}:=\wedge_{\ensuremath{\mathbb{C}}}^{even}
\overline{\ensuremath{\mathcal{N}}}\stackrel{0}{\to}\wedge_{\ensuremath{\mathbb{C}}}^{odd}\overline{\ensuremath{\mathcal{N}}}$
carry a $G\times \ensuremath{\mathbb{T}}_{\beta}$-action: they can be considered as
elements of $K_{G\times \ensuremath{\mathbb{T}}_{\beta}}(M^{\beta})\, =
\,K_{G}(M^{\beta})\otimes R(\ensuremath{\mathbb{T}}_{\beta})$. Let
$K_{G}(M^{\beta})\,\widehat{\otimes} \,R(\ensuremath{\mathbb{T}}_{\beta})$ be the vector
space formed by the infinite formal sums $\sum_{a} E_{a}\, h^a$ taken
over the set of weights of $\ensuremath{\mathbb{T}}_{\beta}$, where $E_{a}\in
K_{G}(M^{\beta})$ for every $a$. The Riemann-Roch character $RR^{^G}$
can be extended to a map $RR^{^{G\times {\rm T}_{\beta}}}$ which
satisfies the commutative diagram
\[
\xymatrix@C=2cm{
K_{G}(M^{\beta}) \ar[r]^{RR^{^G}} \ar[d] & R(G)\ar[d]^{k}\\
K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})
\ar[r]^{RR^{^{G\times {\rm T}_{\beta}}}} &
R(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})\ .
}
\]
The arrow $k:R(G)\to R(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$
is the canonical map defined by $k(\phi)(g,h):=\phi(gh)$. We shall
notice that $[k(\phi)]^{G\times \ensuremath{\mathbb{T}}_{\beta}}=[\phi]^{G}$.
In Section 5, we define an inverse, denoted by $\left[
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}$, of
$\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}$ in
$K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ which is
polarized by $\beta$. It means that $\left[ \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}= \sum_{a} N_{a}\, h^a$ with
$N_{a}\neq 0$ only if $\langle a, \beta\rangle\geq 0$. We can state
now our localization formula as the following equality in
$R(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ :
\begin{equation}\label{eq.localisation.1}
RR^{^G}(M,E)=RR^{^{G\times {\rm T}_{\beta}}}
\left(M^{\beta},E_{\vert M^{\beta}}\otimes
\left[ \wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\right)\ ,
\end{equation}
for every $E\in K_{G}(M)$.
In subsection \ref{subsec.RR.G.beta} we work out {\bf Case 2} for the
general situation. The map \break $RR^{^G}_{\beta}(M^{\beta},-)$ is
the Riemann-Roch character on the $G$-manifold $M^{\beta}$, localized
near $M^{\beta}\cap f_{_{G}}^{-1}(\beta)$, and we extend it to a map
$RR^{^{G\times {\rm T}_{\beta}}}_{\beta}(M^{\beta},-):$
$K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta}) \to
R^{-\infty}(G)\,\widehat{\otimes}\,R(\ensuremath{\mathbb{T}}_{\beta})$.
We prove then the following localization formula
\begin{equation}\label{eq.localisation.2}
RR^{^G}_{\beta}(M,E)=RR^{^{G\times {\rm T}_{\beta}}}_{\beta}
\left(M^{\beta},E_{\vert M^{\beta}}\otimes
\left[ \wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\right)\ ,
\end{equation}
as an equality in
$R^{-\infty}(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$. With
(\ref{eq.localisation.2}) in hand, we see easily that \break
$[RR^{^G}_{\beta}(M,E)]^{G}=0$ if the vector bundle $E$ is
$f_{_{G}}$-strictly positive.
\medskip
Subsection \ref{subsec.induction.G.H} is devoted to {\bf Case 3}. The
abstract moment map $f_{_{G}}:M\to\ensuremath{\mathfrak{g}}$ for the $G$-action on $M$
induces abstract moment maps $f_{_{G'}}:M\to\ensuremath{\mathfrak{g}}'$ for every closed
subgroup $G'$ of $G$. For every $\beta\in\ensuremath{\mathcal{B}}_{_{G}}$, we consider
the Riemann-Roch characters $RR^{^G}_{\beta}(M,-)$,
$RR^{^{G_{\beta}}}_{\beta}(M,-)$, and $RR^{^H}_{\beta}(M,-)$ localized
respectively on $G(M^{\beta}\cap f_{_{G}}^{-1}(\beta))$,
$M^{\beta}\cap f_{_{G}}^{-1}(\beta)$, and $M^{\beta}\cap
f_{_{H}}^{-1}(\beta)$. The major result of subsection
\ref{subsec.induction.G.H} is the induction formulas proved in Theorem
\ref{th.induction.G.H} and Corollary \ref{coro.induction.G.G.beta},
between these three characters. I will explain briefly this result.
Let $W$ be the Weyl group associated to $(G,H)$.
The choice of a Weyl chamber $\ensuremath{\mathfrak{h}}^+$ in $\ensuremath{\mathfrak{h}}$ determines
a complex structure on the real vector space $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$.
Our induction formulas make a crucial use of the holomorphic induction
map ${\rm Hol}^{^G}_{_H}: R(H)\to R(G)$ (see (\ref{eq.holomorphe.G.H})
in Appendix B). Recall that ${\rm Hol}^{^G}_{_H}(h^{\lambda})$ is, for any
weight $\lambda$, either equal to zero or to the
character of an irreducible representation
of $G$ (times $\pm 1$). In Theorem \ref{th.induction.G.H} we prove the
following relation between $RR^{^{G}}_{\beta}(M,-)$ and
$RR^{^{H}}_{\beta}(M,-)$
\begin{eqnarray}\label{eq.relation.induction.1}
RR^{^{G}}_{\beta}(M,E)
&=& \frac{1}{\vert W_{\beta}\vert}
{\rm Hol}^{^G}_{_H}\left(\sum_{w\in W}w.RR^{^H}_{\beta}(M,E)\right)\\
&=&\frac{1}{\vert W_{\beta}\vert}
{\rm Hol}^{^G}_{_H}\left(RR^{^H}_{\beta}(M,E)\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}\right)
\nonumber \ ,
\end{eqnarray}
where $W_{\beta}$ is the stabilizer of $\beta$ in $W$.
In Corollary \ref{coro.induction.G.G.beta} we get the other relation:
\begin{equation}\label{eq.relation.induction.2}
RR^{^{G}}_{\beta}(M,E)={\rm Hol}^{^G}_{_{G_{\beta}}} \left(RR^{^{G_{\beta}}}_{\beta}(M,E)\,
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}}\right).
\end{equation}
Let us compare (\ref{eq.relation.induction.1}), with the
Weyl integration formula\footnote{See Remark \ref{wedge.C-wedge.R}.}:
for any $\phi\in R(G)$ we have
$\phi={\rm Hol}^{^G}_{_H}\left(\phi_{\vert H}\right)={\rm Hol}^{^G}_{_H}\left(\phi^{+}_{\vert H}
\, \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}\right)$,
where $\phi_{\vert H}$ is the restriction of $\phi$ to $H$, and
$\phi^{+}_{\vert H}=\sum_{\lambda}m(\lambda)\, h^{\lambda}$ is the
unique element in $R(H)\otimes\ensuremath{\mathbb{Q}}$ such that
$\sum_{w\in W} w.\phi^{+}_{\vert H}= \phi_{\vert H}$
and $m(\lambda)\neq 0$ only if $\lambda\in \ensuremath{\mathfrak{h}}^+$. In
(\ref{eq.relation.induction.1}),
the $W$-invariant element
$\frac{1}{\vert W_{\beta}\vert}\sum_{w\in W}w.RR^{^H}_{\beta}(M,E)$
plays the role of the restriction to $H$ of
the character $\phi=RR^{^{G}}_{\beta}(M,E)$, and
$\frac{1}{\vert W_{\beta}\vert}RR^{^H}_{\beta}(M,E)$ plays the role
of $\phi^{+}_{\vert H}$.
Since $\beta$ is a $G_{\beta}$-invariant element,
(\ref{eq.relation.induction.2}) reduces
the analysis of {\em Case 3} to the one of {\em Case 2}.
From the result proved in {\em Case 2}, we have
$[RR^{^{G_{\beta}}}_{\beta}(M^{\beta},E)]^{G_{\beta}}=0$
if the vector bundle $E$ is $f_{_{G}}$-strictly positive.
But this does not implies in general that
$[RR^{^G}_{\beta}(M,E)]^{G}=0$. We have
to take the tensor product of $E$ (so that
$E^{\stackrel{k}{\otimes}}$ becomes more and more
$f_{_{G_{\beta}}}$-strictly positive) to see that
$[RR^{^G}_{\beta}(M,E^{\stackrel{k}{\otimes}})]^{G}=0$, when
$\eta_{E^{\stackrel{k}{\otimes}},\beta}=
k.\eta_{_{E,\beta}}>\sum_{\alpha>0}\langle \alpha,\beta\rangle$.
\medskip
In the Hamiltonian setting considered in Section \ref{sec.Hamiltonien},
our strategy
is the same, but at each step we obtain considerable refinements that
are the principal ingredients of the proof of Theorem {\bf C}.
{\bf Case 1 :} When $0$ is a regular value of $f_{_{G}}$, we show
that the Spin$^{\rm c}$ structure on $\ensuremath{\mathcal{M}}_{red}$ is defined by
$J_{red}$, hence $\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},-)=RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},-)$.
When $0$ is not a regular value of $f_{_{G}}$,
we use the `shifting trick' to compute the
$G$-invariant part of $RR^{^G}_{0}(M,E)$
(see subsection \ref{non-regulier}).
{\bf Case 2 :} For any $G$-invariant element $\beta\in\ensuremath{\mathcal{B}}_{_G}$
with $\beta\neq 0$, we prove that the inverse
$\left[ \wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}$
is of the form $\sum_{a} N_{a}\, h^a$ with $N_{a}\neq 0$ only if
$\langle a, \beta\rangle > 0$ (in general we have only
$\langle a, \beta\rangle \geq 0$).
{\bf Case 3 :} For $\beta\in\ensuremath{\mathcal{B}}_{_{G}}$ with $G_{\beta}\neq G$, we consider
the open face $\sigma$ of the Weyl chamber which contains $\beta$,
and the corresponding symplectic slice $\ensuremath{\mathcal{Y}}_{\sigma}$
which is a $G_{\beta}$-symplectic submanifold
of $M$. The localized Riemann-Roch characters
$RR^{^{G}}_{\beta}(M,E)$ and $RR^{^{G_{\beta}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},-)$
are related by the following induction formula
$$
RR^{^{G}}_{\beta}(M,E)=
{\rm Hol}^{^G}_{_{G_{\beta}}}\left(RR^{^{G_{\beta}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},
E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})\right)\ .
$$
\bigskip
{\bf Acknowledgments.} I am grateful to Mich\`ele Vergne for her
interest in this work, especially for the useful discussions and
insightful suggestions on a preliminary version of this paper.
\bigskip
\begin{center}
{\bf Notation}
\end{center}
Throughout the paper $G$ will denote a compact, connected Lie group,
and $\ensuremath{\mathfrak{g}}$ its Lie algebra. We let $H$ be a maximal torus in $G$,
and $\ensuremath{\mathfrak{h}}$ be its Lie algebra. The integral lattice
$\Lambda\subset\ensuremath{\mathfrak{h}}$ is defined as the kernel of $\exp:\ensuremath{\mathfrak{h}}\to H$,
and the real weight lattice $\Lambda^* \subset\ensuremath{\mathfrak{h}}^*$ is defined by :
$\Lambda^*:=\hom(\Lambda,2\pi\ensuremath{\mathbb{Z}})$. Every $\lambda\in\Lambda^*$
defines a 1-dimensionnal $H$-representation, denoted $\ensuremath{\mathbb{C}}_{\lambda}$,
where $h=\exp X$ acts by
$h^{\lambda}:=e^{\imath\langle\lambda,X\rangle}$. We let $W$ be the
Weyl group of $(G,H)$, and we fix the positive Weyl chambers
$\ensuremath{\mathfrak{h}}_+\subset\ensuremath{\mathfrak{h}}$ and $\ensuremath{\mathfrak{h}}^*_+\subset\ensuremath{\mathfrak{h}}^*$. For any dominant
weight $\lambda\in\Lambda^*_+:=\Lambda^*\cap\ensuremath{\mathfrak{h}}^*_+$, we denote by
$V_{\lambda}$ the $G$-irreducible representation with highest weight
$\lambda$, and $\chi_{_{\lambda}}^{_G}$ its character. We denote by
$R(G)$ (resp. $R(H)$) the ring of characters of finite-dimensional
$G$-representations (resp. $H$-representations). We denote by
$R^{-\infty}(G)$ (resp. $R^{-\infty}(H)$) the set of generalized
characters of $G$ (resp. $H$). An element $\chi\in R^{-\infty}(G)$
is of the form $\chi=\sum_{\lambda\in\Lambda^{*}_{+}}m_{\lambda}\,
\chi_{_{\lambda}}^{_G}\,$, where $\lambda\mapsto m_{\lambda},
\Lambda^{*}_{+}\to\ensuremath{\mathbb{Z}}$ has at most polynomial growth. In the same way,
an element $\chi\in R^{-\infty}(H)$ is of the form
$\chi=\sum_{\lambda\in\Lambda^{*}}m_{\lambda}\, h^{\lambda}$, where
$\lambda\mapsto m_{\lambda}, \Lambda^{*}\to\ensuremath{\mathbb{Z}}$ has at most
polynomial growth.
Some additional notation will be introduced later :
\begin{itemize}
\item[]$G_{\gamma}$ : stabilizer subgroup of $\gamma\in \ensuremath{\mathfrak{g}}$
\item[]$\ensuremath{\mathbb{T}}_{\beta}$ : torus generated by $\beta\in \ensuremath{\mathfrak{g}}$
\item[]$M^{\gamma}$ : submanifold of points fixed by $\gamma\in\ensuremath{\mathfrak{g}}$
\item[]$\ensuremath{\hbox{\bf T}} M$ : tangent bundle of $M$
\item[]$\ensuremath{\hbox{\bf T}}_{G} M$ : set of tangent vectors orthogonal to the
$G$-orbits in $M$
\item[]$\ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$ : set of generalized functions on $G$, invariant
by conjugation
\item[]${{\rm Ind}^{^G}_{_{G_{\gamma}}}}:\ensuremath{\mathcal{C}^{-\infty}}(G_{\gamma})^{G_{\gamma}}
\to \ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$ : induction map
\item[]${{\rm Hol}^{^G}_{_{G_{\gamma}}}}: R(G_{\gamma})
\to R(G)$ : holomorphic induction map
\item[]$RR^{^G}_{\beta}(M,-)$ : Riemann-Roch character localized on
$G.(M^{\beta}\cap f_{_{G}}^{-1}(\beta))$
\item[]$\ensuremath{\hbox{\rm Char}}(\sigma)$ : characteristic set of the symbol $\sigma$
\item[]$\ensuremath{\hbox{\rm Thom}}_{G}(M,J)$ : Thom symbol
\item[]$\ensuremath{\hbox{\rm Thom}}^{\gamma}_{G}(M)$ : Thom symbol
localized near $M^{\gamma}$
\item[]$\ensuremath{\hbox{\rm Thom}}_{G,\beta}^f(M)$ : Thom symbol
localized near $G.(M^{\beta}\cap f_{_{G}}^{-1}(\beta))$.
\end{itemize}
\bigskip
\section{Quantization of compact manifolds}\label{sec.quantization}
\medskip
Let $M$ be a compact manifold provided with an action of a compact
connected Lie group $G$. A $G$-invariant almost complex structure $J$
on $M$ defines a map $RR^{^{G,J}}(M,-) : K_{G}(M)\to R(G)$ from the
equivariant $K$-theory of complex vector bundles over $M$ to the
character ring of $G$.
Let us recall the definition of this map. The almost complex
structure on $M$ gives the decomposition
$\wedge \ensuremath{\hbox{\bf T}}^{*} M \otimes \ensuremath{\mathbb{C}} =\oplus_{i,j}\wedge^{i,j}\ensuremath{\hbox{\bf T}}^* M$
of the bundle of differential forms. Using Hermitian structure in the tangent
bundle $\ensuremath{\hbox{\bf T}} M$ of $M$, and in the fibers of $E$, we define a twisted
Dirac operator
$$
\ensuremath{\mathcal{D}}^{+}_{E}:\ensuremath{\mathcal{A}}^{0,even}(M,E)\to\ensuremath{\mathcal{A}}^{0,odd}(M,E)
$$
where $\ensuremath{\mathcal{A}}^{i,j}(M,E):=\Gamma(M,\wedge^{i,j}\ensuremath{\hbox{\bf T}}^{*}M\otimes_{\ensuremath{\mathbb{C}}}E)$
is the space of $E$-valued forms of type $(i,j)$. The Riemann-Roch
character $RR^{^{G,J}}(M,E)$ is defined as the index of the elliptic
operator $\ensuremath{\mathcal{D}}^{+}_{E}$:
$$
RR^{^{G,J}}(M,E)= [Ker\ensuremath{\mathcal{D}}^{+}_{E}] - [Coker\ensuremath{\mathcal{D}}^{+}_{E}].
$$
In fact, the virtual character $RR^{^{G,J}}(M,E)$ is independent of
the choice of the Hermitian metrics on the vector bundles $\ensuremath{\hbox{\bf T}} M$
and $E$.
If $M$ is a compact complex analytic manifold, and $E$ is an
holomorphic complex vector bundle, we have
$RR^{^{G,J}}(M,E)=\sum_{q=0}^{q=dimM}(-1)^q [\ensuremath{\mathcal{H}}^q(M,\ensuremath{\mathcal{O}}(E))]$,
where $\ensuremath{\mathcal{H}}^q(M,\ensuremath{\mathcal{O}}(E))$ is the $q$-th cohomology group of the
sheaf $\ensuremath{\mathcal{O}}(E)$ of the holomorphic sections of $E$ over $M$.
In this paper, we shall use an equivalent definition of the map
$RR^{^{G,J}}$.
We associate to an invariant almost complex structure $J$
the symbol $\ensuremath{\hbox{\rm Thom}}_{G}(M,J)\in K_{G}(\ensuremath{\hbox{\bf T}} M)$ defined as follows. Consider
a Riemannian structure $q$ on $M$ such that the endomorphism $J$ is
orthogonal relatively to $q$, and let $h$ be the following Hermitian
structure on $\ensuremath{\hbox{\bf T}} M$ : $ h(v,w)=q(v,w) -\imath q(Jv,w)$ for $v,w\in
\ensuremath{\hbox{\bf T}} M$.
Let $p:\ensuremath{\hbox{\bf T}} M\to M$ be the canonical projection. The symbol
$\ensuremath{\hbox{\rm Thom}}_{G}(M,J):p^{*}(\wedge_{\ensuremath{\mathbb{C}}}^{even} \ensuremath{\hbox{\bf T}} M)\to p^{*}
(\wedge_{\ensuremath{\mathbb{C}}}^{odd} \ensuremath{\hbox{\bf T}} M)$ is equal, at
$(x,v)\in \ensuremath{\hbox{\bf T}} M$, to the Clifford map
\begin{equation}\label{eq.thom.complex}
Cl_{x}(v)\ :\ p^{*}(\wedge_{\ensuremath{\mathbb{C}}}^{even} \ensuremath{\hbox{\bf T}} M)\vert_{(x,v)}
\longrightarrow p^{*}(\wedge_{\ensuremath{\mathbb{C}}}^{odd} \ensuremath{\hbox{\bf T}} M)\vert_{(x,v)},
\end{equation}
where $Cl_{x}(v).w= v\wedge w - c_{h}(v).w$ for $w\in
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \ensuremath{\hbox{\bf T}}_{x}M$. Here $c_{h}(v):\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\ensuremath{\hbox{\bf T}}_{x}M\to\wedge^{\bullet -1} \ensuremath{\hbox{\bf T}}_{x}M$ denotes the
contraction map relatively to $h$ : for $w\in \ensuremath{\hbox{\bf T}}_{x}M$ we have
$c_{h}(v).w=h(w,v)$. Here $(\ensuremath{\hbox{\bf T}} M, J)$ is considered as a
complex vector bundle over $M$.
The symbol $\ensuremath{\hbox{\rm Thom}}_{G}(M,J)$ determines the Bott-Thom
isomorphism $\ensuremath{\hbox{\rm Thom}}_{J}:$\break
$K_{G}(M)\longrightarrow K_{G}(\ensuremath{\hbox{\bf T}} M)$
by $\ensuremath{\hbox{\rm Thom}}_{J}(E):=\ensuremath{\hbox{\rm Thom}}_{G}(M,J)\otimes p^*(E), \ E\in K_{G}(M)$. To
make the notation clearer, $\ensuremath{\hbox{\rm Thom}}_{J}(E)$ is the symbol
$\sigma^{_{E}} : p^{*}(\wedge_{\ensuremath{\mathbb{C}}}^{even} \ensuremath{\hbox{\bf T}} M\otimes E)\to
p^{*}(\wedge_{\ensuremath{\mathbb{C}}}^{odd} \ensuremath{\hbox{\bf T}} M\otimes E)$ with
\begin{equation}
\sigma^{_{E}}(x,v):= Cl_{x}(v)\otimes Id_{E_{x}}\ ,\quad (x,v)\in \ensuremath{\hbox{\bf T}} M,
\label{eq:thom-iso-b}
\end{equation}
where $E_{x}$ is the fiber of $E$ at $x\in M$.
Consider the index map $\ensuremath{\hbox{\rm Index}}_{M}^{G} : K_{G}(\ensuremath{\hbox{\bf T}}^{*}M)\to R(G)$
where $\ensuremath{\hbox{\bf T}}^{*}M$ is the cotangent bundle of $M$. Using a $G$-invariant
auxiliary metric on $\ensuremath{\hbox{\bf T}} M$, we can identify the vector bundle
$\ensuremath{\hbox{\bf T}}^{*}M$ and $\ensuremath{\hbox{\bf T}} M$, and produce an `index' map $\ensuremath{\hbox{\rm Index}}_{M}^{G} :
K_{G}(\ensuremath{\hbox{\bf T}} M)\to R(G)$. We verify easily that this map is independent
of the choice of the metric on $\ensuremath{\hbox{\bf T}} M$.
\begin{lem}\label{lem.quantization}
We have the following commutative diagram
\begin{equation}
\xymatrix@C=2cm{
K_{G}(M)\ar[r]^{\ensuremath{\hbox{\rm Thom}}_{J}} \ar[dr]_{RR^{^{G,J}}} &
K_{G}(\ensuremath{\hbox{\bf T}} M) \ar[d]^{\ensuremath{\hbox{\rm Index}}_{M}^{G}}\\
& R(G)\ .}
\end{equation}
\end{lem}
{\em Proof} : If we use the natural identification
$(\wedge^{0,1}\ensuremath{\hbox{\bf T}}^{*}M,\imath)\cong
(\ensuremath{\hbox{\bf T}} M,J)$ of complex vector bundles over $M$, we see that the
principal symbol of the operator $\ensuremath{\mathcal{D}}^{+}_{E}$ is equal to
$\sigma^{_E}$ (see \cite{Duistermaat96}).
\medskip
We will conclude with the following Lemma. Let $J^{0}, J^{1}$ be two
$G$-invariant almost complex structures on $M$, and let $RR^{^{G,J^{0}}},
RR^{^{G,J^{1}}}$ be the respective quantization maps.
\medskip
\begin{lem} \label{lem.inv.homotopy}
The maps $RR^{^{G,J^{0}}}$ and $RR^{^{G,J^{1}}}$ are identical in
the following cases:
\noindent i) There exists a $G$-invariant section $A\in
\Gamma(M,\ensuremath{\hbox{\rm End}}(\ensuremath{\hbox{\bf T}} M))$, {\em homotopic to the identity} in
$\Gamma(M,\ensuremath{\hbox{\rm End}}(\ensuremath{\hbox{\bf T}} M))^{G}$ such that $A_{x}$ is invertible,
and $A_{x}.J^{0}_{x}=J^{1}_{x}.A_{x}$ for every $x\in M$.
\noindent ii) There exists an homotopy $J^t,\ t\in[0,1]$ of
$G$-invariant almost complex structures between $J^{0}$ and $J^{1}$.
\end{lem}
\medskip
{\em Proof of i)} : Take a Riemannian structure $q^{1}$ on $M$ such
that $J^{1}\in O(q^{1})$ and define another Riemannian structure
$q^{0}$ by $q^{0}(v,w)=q^{1}(Av,Aw)$ so that $J^{0}\in O(q^{0})$.
The section $A$ defines a bundle unitary map $\underline{A}:
(\ensuremath{\hbox{\bf T}} M,J^{0},h^{0})\to(\ensuremath{\hbox{\bf T}} M,J^{1},h^{1}),\ (x,v)\to (x,A_{x}.v)$,
where $h^l(.,.):=q^l(.,.)-\imath q^l(J^l.,.),\
l=0,1$. This gives an isomorphism $A^{\wedge}_{x}:
\wedge_{J^{0}}\ensuremath{\hbox{\bf T}}_{x}M\to \wedge_{J^{1}}\ensuremath{\hbox{\bf T}}_{x}M$ such that the
following diagram is commutative
\[
\xymatrix@C=2cm{
\wedge_{J^{0}}\ensuremath{\hbox{\bf T}}_{x}M\ar[r]^{Cl_{x}(v)} \ar[d]_{A^{\wedge}_{x}} &
\wedge_{J^{0}}\ensuremath{\hbox{\bf T}}_{x}M\ar[d]^{A^{\wedge}_{x}}\\
\wedge_{J^{1}}\ensuremath{\hbox{\bf T}}_{x}M \ar[r]^{Cl_{x}(A_{x}.v)} &
\wedge_{J^{1}}\ensuremath{\hbox{\bf T}}_{x}M\ .
}
\]
Then $A^{\wedge}$ induces an isomorphism between the symbols
$\ensuremath{\hbox{\rm Thom}}_{G}(M,J^{0})$ and \break $\underline{A}^{*}(\ensuremath{\hbox{\rm Thom}}_{G}(M,J^{1}))
\ :\ (x,v)\to \ensuremath{\hbox{\rm Thom}}_{G}(M,J^{1})(x,A_{x}.v)$. Here
$\underline{A}^{*}:K_{G}(\ensuremath{\hbox{\bf T}} M)\to K_{G}(\ensuremath{\hbox{\bf T}} M)$ is the map induced by
the isomorphism $\underline{A}$. Thus the complexes \break
$\ensuremath{\hbox{\rm Thom}}_{G}(M,J^{0})$ and $\underline{A}^{*}(\ensuremath{\hbox{\rm Thom}}_{G}(M,J^{1}))$
define the same class in $K_{G}(\ensuremath{\hbox{\bf T}} M)$. Since $A$ is
homotopic to the identity, we have $\underline{A}^{*}={\rm Identity}$.
We have proved that $\ensuremath{\hbox{\rm Thom}}_{G}(M,J^{0})= \ensuremath{\hbox{\rm Thom}}_{G}(M,J^{1})$ in
$K_{G}(\ensuremath{\hbox{\bf T}} M)$, and by Lemma \ref{lem.quantization} this shows that
$RR^{^{G,J^{0}}}= RR^{^{G,J^{1}}}$.
{\em Proof of ii)} : We construct $A$ as in $i)$. Take first
$A^{1,0}:= Id -J^{1}J^{0}$ and remark that $A^{1,0}.J^{0}=J^{1}.A^{1,0}$.
Here we consider the homotopy $A^{1,0}_{u}:= Id -uJ^{1}J^{0},\ u\in [0,1]$.
If $-J^{1}J^{0}$ is close to $Id$, for example
$\vert Id + J^{1}J^{0}\vert\leq 1/2$, the bundle map $A^{1,0}_{u}$ will be
invertible for every $u\in [0,1]$. Then we can conclude with Point
$i)$. In general we use the homotopy $J^t,\ t\in[0,1]$. First,
we decompose the interval $[0,1]$ in
$0=t_{0}<t_1<\cdots<t_{k-1}<t_{k}=1$ and we consider the maps
$A^{t_{l+1},t_{l}}:= Id -J^{t_{l+1}}J^{t_{l}}$, with the corresponding
homotopy $A^{t_{l+1},t_{l}}_{u},\ u\in [0,1]$, for $l=0,\ldots,k-1$.
Because $-J^{t_{l+1}}J^{t_{l}}\to Id$ when $t\to t'$, the bundle maps
$A^{t_{l+1},t_{l}}_{u}$ are invertible for all $u\in [0,1]$ if
$t_{l+1}-t_{l}$ is small enough. Then we take the $G$-equivariant
bundle map $A:= \Pi_{l=0}^{k-1}A^{t_{l+1},t_{l}}$ with the homotopy
$A_{u}:= \Pi_{l=0}^{k-1}A^{t_{l+1},t_{l}}_{u},\ u\in [0,1]$. We have
$A.J^{0}=J^{1}.A$ and $A_{u}$ is invertible for every $u\in [0,1]$,
hence we conclude with the point $i)$.
$\Box$
\medskip
\section{Transversally elliptic symbols}\label{sec.T.G.M}
\medskip
We give here a brief review of the material we need in the next
sections. The references are
\cite{Atiyah.74,B-V.inventiones.96.1,B-V.inventiones.96.2,Vergne96}.
Let $M$ be a {\em compact} manifold provided with a $G$-action. Like
in the previous section, we identify the tangent bundle $\ensuremath{\hbox{\bf T}} M$ and the
cotangent bundle $\ensuremath{\hbox{\bf T}}^{*}M$ via a $G$-invariant metric $(.,.)_{_{M}}$
on $\ensuremath{\hbox{\bf T}} M$. For any $X\in \ensuremath{\mathfrak{g}}$, we denote by $X_{M}$ the following
vector field : for $m\in M$, $X_{M}(m):= \frac{d}{dt}\exp(-tX).m
|_{t=0}$.
If $E^{0},E^{1}$ are $G$-equivariant vector bundles over $M$, a
morphism \break
$\sigma \in \Gamma(\ensuremath{\hbox{\bf T}} M,\hom(p^{*}E^{0},p^{*}E^{1}))$ of $G$-equivariant
complex vector bundles will be called a symbol.
The subset of all $(x,v)\in \ensuremath{\hbox{\bf T}} M$ where $\sigma(x,v): E^{0}_{x}\to
E^{1}_{x}$ is not invertible will be called the characteristic set
of $\sigma$, and denoted $\ensuremath{\hbox{\rm Char}}(\sigma)$.
We denote by $\ensuremath{\hbox{\bf T}}_{G}M$ the following subset of $\ensuremath{\hbox{\bf T}} M$ :
$$
\ensuremath{\hbox{\bf T}}_{G}M\ = \left\{(x,v)\in \ensuremath{\hbox{\bf T}} M,\ (v,X_{M}(m))_{_{M}}=0 \quad {\rm for\ all}\
X\in\ensuremath{\mathfrak{g}} \right\} .
$$
A symbol $\sigma$ will be called {\em elliptic} if $\sigma$ is
invertible outside a compact subset of $\ensuremath{\hbox{\bf T}} M$ ($\ensuremath{\hbox{\rm Char}}(\sigma)$ is
compact), and it will be called {\em transversally elliptic} if the
restriction of $\sigma$ to $\ensuremath{\hbox{\bf T}}_{G}M$ is invertible outside a compact
subset of $\ensuremath{\hbox{\bf T}}_{G}M$ ($\ensuremath{\hbox{\rm Char}}(\sigma)\cap \ensuremath{\hbox{\bf T}}_{G}M$ is compact). An
elliptic symbol $\sigma$ defines an element of $K_{G}(\ensuremath{\hbox{\bf T}} M)$, and the
index of $\sigma$ is a virtual finite dimensional representation of
$G$
\cite{Atiyah-Segal68,Atiyah-Singer-1,Atiyah-Singer-2,Atiyah-Singer-3}.
A transversally elliptic symbol $\sigma$ defines an element of
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$, and the index of $\sigma$ is defined (see
\cite{Atiyah.74} for the analytic index and
\cite{B-V.inventiones.96.1,B-V.inventiones.96.2} for the cohomological
one) and is a trace class virtual representation of $G$. Remark that
any elliptic symbol of $\ensuremath{\hbox{\bf T}} M$ is transversally elliptic, hence we have
a restriction map $K_{G}(\ensuremath{\hbox{\bf T}} M)\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ which makes the
following diagram
\begin{equation}\label{indice.generalise}
\xymatrix{
K_{G}(\ensuremath{\hbox{\bf T}} M)\ar[r]\ar[d]_{\ensuremath{\hbox{\rm Index}}_{M}^G} &
K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)\ar[d]^{\ensuremath{\hbox{\rm Index}}_{M}^G}\\
R(G)\ar[r] & R^{-\infty}(G)\ .
}
\end{equation}
commutative.
\subsection{Index map on non-compact manifolds}
\label{subsec.indice.ouvert}
Let $U$ be a non-compact $G$-manifold. Lemma 3.6 and Theorem 3.7
of \cite{Atiyah.74} tell us that for any open G-embedding
$j:U\ensuremath{\hookrightarrow} M$ into a compact manifold we have a pushforward map
$j_{*}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}U)\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ such that the
composition
$$
K_{G}(\ensuremath{\hbox{\bf T}}_{G}U)\stackrel{j_{*}}{\longrightarrow} K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)
\stackrel{\ensuremath{\hbox{\rm Index}}_{M}^G}{\longrightarrow} R^{-\infty}(G)
$$
is independent of the choice of $j:U\ensuremath{\hookrightarrow} M$.
\begin{lem}
Let $U$ be a G-invariant open subset of a
$G$-manifold $\ensuremath{\mathcal{X}}$. If $U$ is relatively compact,
there exists an open G-embedding
$j:U\ensuremath{\hookrightarrow} M$ into a compact $G$-manifold.
\end{lem}
{\em Proof :} Here we follow the proof given by Boutet de Monvel in
\cite{Boutet.70}. Let $\chi\in\ensuremath{\mathcal{C}^{\infty}}(\ensuremath{\mathcal{X}})^{G}$ be a function with
compact support,
such that $0\leq \chi\leq 1$ and $\chi=1$ on $U$.
Let $q:\ensuremath{\mathcal{X}}\times \ensuremath{\mathbb{R}}\to \ensuremath{\mathbb{R}}$
be the function defined by $q(m,t)=\chi(m)-t^2$. The
interval $(-\infty, 1]$ is the image of $q$, and the fibers
$q^{-1}(\ensuremath{\varepsilon})$ are compact for every $\ensuremath{\varepsilon} > 0$. According to Sard's
Theorem there exists a regular value $0<\ensuremath{\varepsilon}_{0}<1$ of $q$.
The set $q^{-1}(\ensuremath{\varepsilon}_{0})$ is then a compact $G$-invariant submanifold
of $\ensuremath{\mathcal{X}}\times \ensuremath{\mathbb{R}}$, and $j:U\to q^{-1}(\ensuremath{\varepsilon}_{0})$,
$m\mapsto (m,\sqrt{1-\ensuremath{\varepsilon}_{0}})$ is an open embedding. $\Box$
\begin{coro}\label{hyp.indice}
The index map $\ensuremath{\hbox{\rm Index}}_{U}^{G}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}U)\to R^{-\infty}(G)$
is defined when $U$ is a $G$-invariant relatively compact
open subset of a $G$-manifold.
\end{coro}
\subsection{Excision lemma}\label{subsec.excision}
Let $j:U\hookrightarrow M$ be the inclusion map of a $G$-invariant
open subset on a compact manifold, and let
$j_{*}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}U)\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ be the pushforward map. We have
two index maps $\ensuremath{\hbox{\rm Index}}_{M}^{G}$, and $\ensuremath{\hbox{\rm Index}}^{G}_{U}$ such that
$\ensuremath{\hbox{\rm Index}}_{M}^{G}\circ j_{*}= \ensuremath{\hbox{\rm Index}}^{G}_{U}$. Suppose that
$\sigma$ is a transversally elliptic symbol on $\ensuremath{\hbox{\bf T}} M$ with
characteristic set contained in $\ensuremath{\hbox{\bf T}} M |_{U}$. Then, the restriction
$\sigma |_{U}$ of $\sigma$ to $\ensuremath{\hbox{\bf T}} U$ is a transversally elliptic
symbol on $\ensuremath{\hbox{\bf T}} U$, and
\begin{equation}
j_{*}(\sigma|_{U})=\sigma \quad {\rm in}\quad K_{G}(\ensuremath{\hbox{\bf T}}_{G}M).
\label{eq:excision}
\end{equation}
In particular, it gives
$\ensuremath{\hbox{\rm Index}}_{M}^{G}(\sigma)=\ensuremath{\hbox{\rm Index}}^{G}_{U}(\sigma|_{U})$.
\medskip
\subsection{Locally free action}\label{subsec.free.action}
Let $G$ and $H$ be compact Lie groups and let $M$ be a {\em compact}
$G\times H$ manifold
In a first place, we suppose that $G$ acts freely on $M$, and we
denote by $\pi : M\to M/G$ the principal fibration. Note that the map $\pi$ is
$H$-equivariant. In this situation we have $\ensuremath{\hbox{\bf T}}_{G\times H}M\widetilde{=}
\pi^{*}(\ensuremath{\hbox{\bf T}}_{H}(M/G))$, and thus an isomorphism
\begin{equation}
\pi^{*}\ :\ K_{H}(\ensuremath{\hbox{\bf T}}_{H}(M/G))\longrightarrow
K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}M)\ .
\label{eq:free.action}
\end{equation}
We rephrase now Theorem 3.1 of Atiyah in \cite{Atiyah.74}.
Let $\{W_{a}, a\in \hat{G} \}$ be a completed set of
inequivalent irreducible representations of $G$.
For each irreducible $G$-representation $V_{\mu}$,
we associate the complex vector bundle
$\underline{V}_{\mu}:=M\times_{H}V_{\mu}$ on $M/G$ and denote by
$\underline{V}_{\mu}^{*}$
its dual. The group $H$ acts trivially on $V_{\mu}$, this makes
$\underline{V}_{\mu}^{*}$ a $H$-vector bundle.
\begin{theo}[Atiyah]\label{thm.atiyah.1}
If $\sigma\in K_{H}(\ensuremath{\hbox{\bf T}}_{H}(M/G))$, then we have the following
equality in $R^{-\infty}(G\times H)$
\begin{equation}\label{eq:atiyah.1}
\ensuremath{\hbox{\rm Index}}^{G\times H}_{M}(\pi^{*}\sigma)\ =\ \sum_{\mu\in \Lambda^{*}_{+}}
\ensuremath{\hbox{\rm Index}}^{H}_{M/G}(\sigma\otimes \underline{V}_{\mu}^{*}) . V_{\mu}\quad.
\end{equation}
In particular the $G$-invariant part of $\ensuremath{\hbox{\rm Index}}^{G\times
H}_{M}(\pi^{*}\sigma)$ is $\ensuremath{\hbox{\rm Index}}^{H}_{M/G}(\sigma)$.
\end{theo}
A classical example is when $M=G$, $G=G_{r}$ acts by right
multiplications on $G$, and $G=G_{l}$ acts by left multiplications
on $G$. Then the zero map $\sigma_{0}:G\times \ensuremath{\mathbb{C}}\to G\times\{0\}$
defines a $G_{r}\times G_{l}$-transversally elliptic symbol
associated to the zero differential operator $\ensuremath{\mathcal{C}^{\infty}}(G)\to 0$.
This symbol is equal to the pullback of
$\ensuremath{\mathbb{C}}\in K_{G_{r}}(\ensuremath{\hbox{\bf T}}_{G_{r}}\{{\rm point}\})\widetilde{=}R(G_{r})$.
In this case $\ensuremath{\hbox{\rm Index}}^{G_{r}\times G_{l}}_{G}(\sigma_{0})$ is equal
to $L^{2}(G)$, the $L^{2}$-index
of the zero operator on $\ensuremath{\mathcal{C}^{\infty}}(G)$. The $G_{r}$-vector bundle
$\underline{V}^{*}_{\mu}\to\{{\rm point}\}$ is just the vector space
$V_{\mu}^{*}$ with the canonical action of $G_{r}$. Finally,
(\ref{eq:atiyah.1}) is the Peter-Weyl decomposition of $L^{2}(G)$ in
$R^{-\infty}(G_{r}\times G_{l})$:
$\ L^2(G)=\sum_{\mu\in \Lambda^{*}_{+}}V_{\mu}^{*}\otimes V_{\mu}$.
\medskip
We suppose now that $G$ acts locally freely on $M$. The quotient
$\ensuremath{\mathcal{X}}:=M/G$ is an orbifold, a space with finite-quotient
singularities. One considers on $\ensuremath{\mathcal{X}}$ the $H$-equivariant {\em proper}
orbifold vector bundles and the corresponding $R(H)$-module $K_{orb,H}(\ensuremath{\mathcal{X}})$
\cite{Kawasaki81}. In the same way we consider the $H$-equivariant
proper elliptic
symbols on the orbifold $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$ and the corresponding $R(H)$-module
$K_{orb,H}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$. The principal fibration $\pi: M\to \ensuremath{\mathcal{X}}$ induces
isomorphisms $K_{orb,H}(\ensuremath{\mathcal{X}})\simeq K_{G\times H}(M)$ and
$K_{orb,H}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})\simeq K_{G\times H}(\ensuremath{\hbox{\bf T}}_{H}M)$ that we
both denote by $\pi^{*}$. The index map
\begin{equation}\label{index-orbifold}
\ensuremath{\hbox{\rm Index}}^{H}_{\ensuremath{\mathcal{X}}}: K_{orb,H}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})\to R(H)
\end{equation}
is defined by the following equation: for any $\sigma\in K_{orb,H}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$,
$\ensuremath{\hbox{\rm Index}}^{H}_{\ensuremath{\mathcal{X}}}(\sigma):=[\ensuremath{\hbox{\rm Index}}^{G\times
H}_{M}(\pi^{*}\sigma)]^{G}$.
We are particularly interested in the case where the bundle
$\ensuremath{\hbox{\bf T}}_{G}M\to M$ carries a $G\times H$-equivariant almost complex structure $J$.
Taking the quotient by $G$, it defines a $H$-equivariant almost complex structure
$J_{\ensuremath{\mathcal{X}}}$ on the orbifold tangent bundle $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}}\to\ensuremath{\mathcal{X}}$. Like in the
smooth case, we have the Thom symbol
$\ensuremath{\hbox{\rm Thom}}_{H}(\ensuremath{\mathcal{X}},J_{\ensuremath{\mathcal{X}}})\in K_{orb,H}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$ and a Riemmann-Roch
character $RR^{^H}:K_{orb,H}(\ensuremath{\mathcal{X}})\to R(H)$ related as in Lemma
\ref{lem.quantization}.
\medskip
\subsection{Induction}\label{subsec.induction.def}
Let $i:H\ensuremath{\hookrightarrow} G$ be a
closed subgroup with Lie algebra $\ensuremath{\mathfrak{h}}$, and $\ensuremath{\mathcal{Y}}$ be a
$H$-manifold (as in Corollary
\ref{hyp.indice}). We have two principal bundles $\pi_{1}:
G\times\ensuremath{\mathcal{Y}}\to \ensuremath{\mathcal{Y}}$ for the $G$-action, and $\pi_{2}:
G\times\ensuremath{\mathcal{Y}}\to \ensuremath{\mathcal{X}}:=G\times_{H}\ensuremath{\mathcal{Y}}$ for the diagonal $H$-action.
The map $i_{*}: K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$ is well defined
by the following commutative diagram
\begin{equation}\label{eq:G.H.induction}
\xymatrix@C=25mm{
K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}})\ar[r]^{\pi_{1}^{*}} \ar[dr]_{i_{*}} &
K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}(G\times\ensuremath{\mathcal{Y}})) \ar[d]^{(\pi_{2}^{*})^{-1}}\\
& K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\ ,}
\end{equation}
since $\pi_{1}^{*}$ and $\pi_{2}^{*}$ are isomorphisms.
Let us show how to compute $i_{*}(\sigma)$, for an $H$-transversally elliptic
symbol $\sigma \in \Gamma( \ensuremath{\hbox{\bf T}} Y, \hom(E^{0},E^{1}))$,
where $E^{0}, E^{1}$ are $H$-equivariant vector bundles
over $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}$. First we notice\footnote{\label{eq.espace.tangent}
These identities come from the following $G\times H$-equivariant
isomorphism of vector bundles over $G\times\ensuremath{\mathcal{Y}}$:
$\ensuremath{\hbox{\bf T}}_{H}(G\times\ensuremath{\mathcal{Y}})\to G\times(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}} \oplus \ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}}),
(g,m;\frac{d}{dt}_{\vert t=0}(g.e^{tX})+ v_{m})\mapsto
(g,m; pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}(X)+v_{m})$. Here $pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}:
\ensuremath{\mathfrak{g}}\to \ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ is the orthogonal projection.}
that $\ensuremath{\hbox{\bf T}}(G\times_{H}\ensuremath{\mathcal{Y}})\widetilde{=}
G\times_{H}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}} \oplus \ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}})$, and
$\ensuremath{\hbox{\bf T}}_{G}(G\times_{H}\ensuremath{\mathcal{Y}})\widetilde{=}G\times_{H}( \ensuremath{\hbox{\bf T}}_{H} \ensuremath{\mathcal{Y}})$.
So we extend trivially
$\sigma$ to $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\oplus\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}$, and we define
$i_{*}(\sigma)\in \Gamma(G\times_{H}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}} \oplus \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}),
\hom(G\times_{H}E^{0},G\times_{H}E^{1}))$ by
$i_{*}(\sigma)([g;\xi,x,v]):=\sigma(x,v)$ for
$g\in G$, $\xi\in\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ and $(x,v)\in \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}$.
To express the $G$-index of $i_{*}(\sigma)$ in terms of the $H$-index
of $\sigma$, we need the induction map
\begin{equation}
{\rm Ind}^{^G}_{_H} : \ensuremath{\mathcal{C}^{-\infty}}(H)^{H}\longrightarrow \ensuremath{\mathcal{C}^{-\infty}}(G)^{G}\ ,
\label{eq:fonction.induction}
\end{equation}
where $\ensuremath{\mathcal{C}^{-\infty}}(H)$ is the set of generalized functions on $H$, and
the $H$ and $G$ invariants are taken with the conjugation action.
The map ${\rm Ind}^{^G}_{_H}$ is defined as follows : for $\phi\in\ensuremath{\mathcal{C}^{-\infty}}(H)^{H}$,
we have $\int_{G}{\rm Ind}^{^G}_{_H}(\phi)(g)f(g)dg= {\rm cst}
\int_{H}\phi(h)f|_{H}(h)dh$,
for every $f\in\ensuremath{\mathcal{C}^{\infty}}(G)^{G}$, where
${\rm cst}=\ensuremath{\hbox{\rm vol}}(G,dg)/\ensuremath{\hbox{\rm vol}}(H,dh)$.
We can now recall Theorem 4.1 of Atiyah in \cite{Atiyah.74}.
\begin{theo}\label{thm.atiyah.2}
Let $i:H\to G$ be the inclusion of a closed subgroup, let $\ensuremath{\mathcal{Y}}$ be a
$H$-manifold satisfying the hypothesis of Corollary \ref{hyp.indice}, and set
$\ensuremath{\mathcal{X}}=G\times_{H}\ensuremath{\mathcal{Y}}$. Then we have the commutative diagram
\[
\xymatrix{
K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}})\ar[r]^{i_{*}}\ar[d]_{\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Y}}}^H} &
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\ar[d]^{\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}^G}\\
\ensuremath{\mathcal{C}^{-\infty}}(H)^{H}\ar[r]_{{\rm Ind}^{^G}_{_H}} & \ensuremath{\mathcal{C}^{-\infty}}(G)^{G}\ .
}
\]
\end{theo}
\subsection{Reduction}\label{subsec.reduction}
Let us recall a multiplicative property of the index for the product
of manifold. Let a compact Lie group $G$ acts on two
manifolds $\ensuremath{\mathcal{X}}$ and $\ensuremath{\mathcal{Y}}$, and assume that another compact
Lie group $H$
acts on $\ensuremath{\mathcal{Y}}$ commuting with the action of $G$.
The external product of complexes on $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}}$ and $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}}$ induces
a multiplication (see \cite{Atiyah.74} and \cite{Vergne96}, section 2):
\begin{eqnarray}
K_{G}(\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}})\times K_{G\times H}(\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}})&\longrightarrow &
K_{G\times H}(\ensuremath{\hbox{\bf T}} (\ensuremath{\mathcal{X}}\times \ensuremath{\mathcal{Y}})) \\
(\sigma_{1},\sigma_{2})&\longmapsto & \sigma_{1}\odot
\sigma_{2} \ .\nonumber
\label{eq:produit.usuel}
\end{eqnarray}
Let us recall the definition of this external product. Let
$E^{\pm},F^{\pm}$ be $G\times H$-equivariant Hermitian vector bundles over
$\ensuremath{\mathcal{X}}$ and $\ensuremath{\mathcal{Y}}$ respectively, and let $\sigma_{1}:E^{+}\to E^{-}$,
$\sigma_{2}:F^{+}\to F^{-}$ be $G\times H$-equivariant symbols. We consider
the $G\times H$-equivariant symbol
$$
\sigma_{1}\odot \sigma_{2}:E^{+}\otimes F^{+}\oplus E^{-}\otimes F^{-}
\longrightarrow E^{-}\otimes F^{+} \oplus E^{+}\otimes F^{-}
$$
defined by
\begin{equation}\label{eq:produit.externe}
\sigma_{1}\odot \sigma_{2}=
\left(
\begin{array}{cc}
\sigma_{1}\otimes I & -I\otimes \sigma_{2}^{*}\\
I \otimes \sigma_{2} & \sigma_{1}^{*}\otimes I
\end{array}
\right)\ .
\end{equation}
We see that the set $\ensuremath{\hbox{\rm Char}}(\sigma_{1}\odot \sigma_{2})\subset
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}}\times\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}$ is equal to
$\ensuremath{\hbox{\rm Char}}(\sigma_{1})\times \ensuremath{\hbox{\rm Char}}(\sigma_{2})$.
This exterior product defines the $R(G)$-module structure on
$K_G(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$, by taking $\ensuremath{\mathcal{Y}}=point$ and $H=\{e\}$. If we take
$\ensuremath{\mathcal{X}}=\ensuremath{\mathcal{Y}}$ and $H=\{e\}$, the product on $K_G(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$ is defined
by
\begin{equation}\label{eq.produit.anneau}
\sigma_{1}\tilde{\odot}\, \sigma_2:=s_{_{\ensuremath{\mathcal{X}}}}^*(\sigma_{1}\odot
\sigma_{2}) \ ,
\end{equation}
where $s_{_{\ensuremath{\mathcal{X}}}}:\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\to\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\times\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$ is the diagonal
map.
In the transversally elliptic case we need to be careful in the
definition of the exterior product, since
$\ensuremath{\hbox{\bf T}}_{G\times H}(\ensuremath{\mathcal{X}}\times \ensuremath{\mathcal{Y}})\neq\ensuremath{\hbox{\bf T}}_{G} \ensuremath{\mathcal{X}}\times\ensuremath{\hbox{\bf T}}_{H} \ensuremath{\mathcal{Y}}$.
\begin{defi}
Let $\sigma$ be a $H$-transversally elliptic symbol on $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}}$.
This symbol is
called $H$-{\em transversally-good} if the characteristic set
of $\sigma$ intersects $\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}}$ in a compact subset of $\ensuremath{\mathcal{Y}}$.
\end{defi}
Recall Lemma 3.4 and Theorem 3.5 of Atiyah in \cite{Atiyah.74}.
Let $\sigma_{1}$ be a $G$-transversally
elliptic symbol on $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$, and $\sigma_{2}$ be a $H$-transversally
elliptic symbol on $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{Y}}$ that is $G$-equivariant. Suppose
furthermore that $\sigma_{2}$ is $H$-{\em transversally-good},
then the product $\sigma_{1}\odot \sigma_{2}$ is
$G\times H$-transversally
elliptic. Since every class of $K_{G\times H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}})$ can be
represented by an $H$-{\em transversally-good} elliptic symbol, we
have a multiplication
\begin{eqnarray}\label{eq:produit.transversal}
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\times K_{G\times H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{Y}})&\longrightarrow &
K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}(\ensuremath{\mathcal{X}}\times \ensuremath{\mathcal{Y}}) )\\
(\sigma_{1},\sigma_{2})&\longmapsto & \sigma_{1}\odot
\sigma_{2} \ .\nonumber
\end{eqnarray}
Suppose now that the manifolds $\ensuremath{\mathcal{X}}$ and $\ensuremath{\mathcal{Y}}$ satisfy
the condition of Corollary \ref{hyp.indice}. So, the index maps
$\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}^{G}$, $\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Y}}}^{G\times H}$, and
$\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}\times \ensuremath{\mathcal{Y}}}^{G\times H}$ are well defined.
According to Theorem 3.5 of \cite{Atiyah.74}, we know that
\begin{equation} \label{eq:formule.G.H.produit}
\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}\times \ensuremath{\mathcal{Y}}}^{G\times H} (\sigma_{1}\odot \sigma_{2})=
\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}^{G}(\sigma_{1})\cdot\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Y}}}^{G\times H}
(\sigma_{2})
\quad{\rm in}\quad \ R^{-\infty}(G\times H)\ ,
\end{equation}
for any $\sigma_{1}\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$ and
$\sigma_{2}\in K_{G\times H}(\ensuremath{\hbox{\bf T}}_{H}(\ensuremath{\mathcal{X}}\times H))$.
\bigskip
{\em In the rest of this subsection we suppose that the subgroup
$H\subset G$ is the stabilizer of an element $\gamma\in\ensuremath{\mathfrak{g}}$.
The manifold $G/H$ carries a $G$-invariant complex structure
$J_{\gamma}$ defined by the element $\gamma$: at
$e\in G/H$, the map $J_{\gamma}(e)$ equals
$ad(\gamma).(\sqrt{- ad(\gamma)^{2}})^{-1}$
on $\ensuremath{\hbox{\bf T}}_{e}(G/H)=\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$.
}
\bigskip
We recall now the definition of the map
$r^{\gamma}_{_{G,H}}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\to K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$
introduced by Atiyah in \cite{Atiyah.74}. We consider the manifold
$\ensuremath{\mathcal{X}}\times G$ with two actions of $G\times H$:
for $(g,h)\in G\times H$ and $(x,a)\in \ensuremath{\mathcal{X}}\times G$,
we have $(g,h).(x,a):=(g.x,gah^{-1})$ on
$\ensuremath{\mathcal{X}}\stackrel{1}{\times} G$, and we have $(g,h).(x,a):=(h.x,gah^{-1})$ on
$\ensuremath{\mathcal{X}}\stackrel{2}{\times} G$.
The map $\Theta :\ensuremath{\mathcal{X}}\stackrel{2}{\times} G\to
\ensuremath{\mathcal{X}}\stackrel{1}{\times} G,\ (x,a)\mapsto (a.x,a)$
is $G\times H$-equivariant, and induces $\Theta^{*}:
K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}(\ensuremath{\mathcal{X}}\stackrel{1}{\times}G))\to
K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}(\ensuremath{\mathcal{X}}\stackrel{2}{\times} G))$.
The $G$-action is free on $\ensuremath{\mathcal{X}}\stackrel{2}{\times} G$, so the quotient map
$\pi :\ensuremath{\mathcal{X}}\stackrel{2}{\times} G\to\ensuremath{\mathcal{X}}$ induces
an isomorphism $\pi^{*}:K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})
\to K_{G\times H}(\ensuremath{\hbox{\bf T}}_{G\times H}(\ensuremath{\mathcal{X}}\stackrel{2}{\times}G)) $.
We denote by $\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}\in K_{G\times H}(\ensuremath{\hbox{\bf T}}_{H}G)$
the pullback of the Thom class $\ensuremath{\hbox{\rm Thom}}_{G}(G/H,J_{\gamma})\in
K_{G}(\ensuremath{\hbox{\bf T}}(G/H))$, via the quotient map $G\to G/H$.
Consider the manifold $\ensuremath{\mathcal{Y}}=G$ with the action of $G\times H$
defined by $(g,h).a=gah^{-1}$ for $a\in G$, and $(g,h)\in G\times H$.
Since the symbol $\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$ is $H$-transversally
good on $\ensuremath{\hbox{\bf T}} G$, the product by $\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$
induces, by (\ref{eq:produit.transversal}), the map
\begin{eqnarray*}
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})&\longrightarrow &
K_{G\times H}(T_{G\times H}(\ensuremath{\mathcal{X}}\stackrel{1}{\times}G) )\\
\sigma&\longmapsto & \sigma\odot
\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}} \ .
\end{eqnarray*}
\begin{defi}[Atiyah]\label{def.G.H.restriction}
Let $H$ the stabilizer of
$\gamma\in \ensuremath{\mathfrak{g}}$ in $G$. The map $r^{\gamma}_{_{G,H}}:
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\to K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$ is
defined for every $\sigma\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$ by
$$
r^{\gamma}_{_{G,H}}(\sigma):=(\pi^{*})^{-1}
\circ\Theta^{*}(\sigma\odot \sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}})\ .
$$
\end{defi}
Theorem 4.2 in \cite{Atiyah.74} tells us that the following diagram is
commutative
\begin{equation}\label{indice.r.G.H}
\xymatrix@C=2cm@R=10mm{
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\ar[r]^{r^{_\gamma}_{_{G,H}}}
\ar[d]_{\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}^G} &
K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})\ar[d]^{\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}^H}\\
\ensuremath{\mathcal{C}^{-\infty}}(G)^{G} & \ensuremath{\mathcal{C}^{-\infty}}(H)^{H}\ar[l]_{{\rm Ind}^{^G}_{_H}}.
}
\end{equation}
\medskip
We show now a more explicit description of the map
$r^{\gamma}_{_{G,H}}$. Consider the moment map
$$
\mu_{_G}:\ensuremath{\hbox{\bf T}}^{*}\ensuremath{\mathcal{X}}\to \ensuremath{\mathfrak{g}}^{*}
$$
for the (canonical) Hamiltonian action of $G$ on the
symplectic manifold $\ensuremath{\hbox{\bf T}}^{*}\ensuremath{\mathcal{X}}$. If we identify $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$ with
$\ensuremath{\hbox{\bf T}}^{*}\ensuremath{\mathcal{X}}$ via a $G$-invariant metric, and $\ensuremath{\mathfrak{g}}$ with
$\ensuremath{\mathfrak{g}}^{*}$ via a $G$-invariant scalar product, the `moment map' is a
map $\mu_{_G}:\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\to \ensuremath{\mathfrak{g}}$ defined as follows. If
$E^{1},\cdots,E^{l}$ is an orthonormal basis of $\ensuremath{\mathfrak{g}}$, we have
$\mu_{_G}(x,v)=\sum_{i}(E^{i}_{M}(x),v)_{_{M}}E^i$ for
$(x,v)\in\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$. The moment map admits the decomposition
$\mu_{_G}=\mu_{_H}+\mu_{_{G/H}}$, relative to the $H$-invariant
orthogonal
decomposition of the Lie algebra $\ensuremath{\mathfrak{g}}=\ensuremath{\mathfrak{h}}\oplus\ensuremath{\mathfrak{h}}^{\perp}$.
It is important to note that $\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}}=\mu_{_G}^{-1}(0)$,
$\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}}=\mu_{_H}^{-1}(0)$, and $\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}}=
\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}}\cap\mu_{_{G/H}}^{-1}(0)$.
The real vector space $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ is endowed with the complex
structure defined by $\gamma$. Consider over $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$ the
$H$-equivariant symbol
\begin{eqnarray*}
\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}} : \ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\times
\wedge_{\ensuremath{\mathbb{C}}}^{even}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}&
\longrightarrow&
\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\times\wedge_{\ensuremath{\mathbb{C}}}^{odd}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\\
(x,v;w)&\longrightarrow& (x,v;w')\ ,
\end{eqnarray*}
with $w'=Cl(\mu_{_{G/H}}(x,v)).w$. Here $\ensuremath{\mathfrak{h}}^{\perp}\simeq
\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$, and $Cl(X):\wedge_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}
\to\wedge_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$, $X\in\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$, denotes the Clifford
action. This symbol has $\mu_{_{G/H}}^{-1}(0)$ for characteristic set.
For any symbol $\sigma$ over $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}}$, with characteristic set
${\rm Char}(\sigma)$, the product $\sigma\,\tilde{\odot}\,
\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}}$,
defined at (\ref{eq.produit.anneau}),
is a symbol over $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$ with characteristic set
${\rm Char}(\sigma\,\tilde{\odot}\,\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}})=
{\rm Char}(\sigma)\cap\mu_{_{G/H}}^{-1}(0)$. Then, if $\sigma$
is a $G$-transversally elliptic symbol over $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$, the product
$\sigma\,\tilde{\odot}\,\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}}$ is a
$H$-transversally elliptic symbol.
\medskip
\begin{prop}\label{prop.restriction.bis}
The map $r^{\gamma}_{_{G,H}}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})\to
K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$ has the
following equivalent definition: for every $\sigma\in
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$
$$
r^{\gamma}_{_{G,H}}(\sigma)=\sigma\,\tilde{\odot}\,\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}}
\quad {\rm in} \quad K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}}).
$$
\end{prop}
\medskip
{\em Proof} : We have to show that for every $\sigma\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$,
$\sigma\tilde{\odot}\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}}=$ \break
$(\pi^{*})^{-1}\circ\Theta^{*}(\sigma\odot\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}})$
in $K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$. Let $p_{_G}:\ensuremath{\hbox{\bf T}} G\to G$
and $p_{\ensuremath{\mathcal{X}}}:\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}\to \ensuremath{\mathcal{X}}$ be the canonical projections.
The symbol $\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}:p^{*}_{_G}(G\times
\wedge_{\ensuremath{\mathbb{C}}}^{even}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})\to p^{*}_{_G}(G\times\wedge_{\ensuremath{\mathbb{C}}}^{odd}
\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$ is defined by
$\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}(a, Z)=Cl(Z_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}})$ for $(a,Z)\in\ensuremath{\hbox{\bf T}}
G\simeq G\times\ensuremath{\mathfrak{g}}$, where $Z_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$ is the
$\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$-component of $Z\in \ensuremath{\mathfrak{g}}$.
Consider $\sigma:p_{\ensuremath{\mathcal{X}}}^{*}E_{0}\to p_{\ensuremath{\mathcal{X}}}^{*}E_{1}$, a
$G$-transversally elliptic symbol on $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}}$, where $E_{0}, E_{1}$ are
$G$-complex vector bundles over $\ensuremath{\mathcal{X}}$. The product $\sigma\odot
\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$ acts on the bundles
$p_{\ensuremath{\mathcal{X}}}^{*}E_{\bullet}\otimes p^{*}_{G}(G\times\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$ at $(x,v;a,Z)\in \ensuremath{\hbox{\bf T}}(\ensuremath{\mathcal{X}}\times G)$ by
$$
\sigma(x,v)\odot Cl(Z_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}) .
$$
The pullback $\sigma_{o}:=\Theta^{*}(\sigma\odot \sigma_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}})$ acts
on the bundle $G\times (p_{\ensuremath{\mathcal{X}}}^{*}E_{\bullet}\otimes \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$ (here we identify $\ensuremath{\hbox{\bf T}}(\ensuremath{\mathcal{X}}\times G)$ with
$G\times(\ensuremath{\mathfrak{g}}\oplus\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{X}})$).
At $(x,v;a,Z) \in \ensuremath{\hbox{\bf T}}(\ensuremath{\mathcal{X}}\times G)$ we have
$$
\sigma_{o}(x,v;a,Z)=\sigma\odot
\sigma^{_\gamma}_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}(a.x,v';a,Z'),\quad {\rm with}
$$
$(v',Z')=\left([\ensuremath{\hbox{\bf T}}_{(x,a)}\Theta]^{*}\right)^{-1}(v,Z)$. Here
$\ensuremath{\hbox{\bf T}}_{(x,a)}\Theta:\ensuremath{\hbox{\bf T}}_{(x,a)}(\ensuremath{\mathcal{X}}\times G)\to\ensuremath{\hbox{\bf T}}_{(a.x,a)}(\ensuremath{\mathcal{X}}\times G)$ is the
tangent map of $\Theta$ at $(x,a)$, and $[\ensuremath{\hbox{\bf T}}_{(x,a)}\Theta]^{*}:\ensuremath{\hbox{\bf T}}_{(a.x,a)}
(\ensuremath{\mathcal{X}}\times G)\to\ensuremath{\hbox{\bf T}}_{(x,a)}(\ensuremath{\mathcal{X}}\times G)$ its transpose. A small
computation shows that $Z'=Z +\mu_{_{G}}(v)$ and $v'=a.v$. Finally, we get
$$
\sigma_{o}(x,v;a,Z)=\sigma(a.x,a.v)\odot Cl(Z_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}
+\mu_{_{G/H}}(v)) .
$$
Hence, the symbol $(\pi^{*})^{-1}(\sigma_{o})$ acts on the bundle
$p_{\ensuremath{\mathcal{X}}}^{*}E_{\bullet}\otimes \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ by
$$
(\pi^{*})^{-1}(\sigma_{o})(x,v)=\sigma(x,v)\odot Cl(\mu_{_{G/H}}(v)).
$$
$\Box$
\medskip
For any $G$-invariant function
$\phi\in\ensuremath{\mathcal{C}^{\infty}}(G)^{G}$, the Weyl integration formula can be
written\footnote{See Remark \ref{wedge.C-wedge.R}.}
\begin{equation}\label{eq.Weyl.integration}
\phi={\rm Ind}^{^G}_{_H}\left(\phi_{\vert H}\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\right)\
{\rm in}\ \ensuremath{\mathcal{C}^{-\infty}}(G)^{G}\ .
\end{equation}
where $\phi_{\vert H}\in\ensuremath{\mathcal{C}^{\infty}}(H)^{H}$ is the restriction to $H=G_{\gamma}$.
Equality (\ref{eq.Weyl.integration}) remains true for any
$\phi\in\ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$ that admits a restriction to $H$.
\medskip
\begin{lem}\label{restriction.function.G.H}
Let $\sigma$ be a $G$-transversally elliptic symbol. Suppose
furthermore that $\sigma$ is $H$-transversally elliptic. This
symbol defines two classes $\sigma\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{X}})$ and
$\sigma_{\vert H}\in K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$ with
the relation\footnote{Here we note
$\sigma_{\vert H}\otimes\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ for the
difference $\sigma_{\vert H}\otimes\wedge^{even}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\,-\,
\sigma_{\vert H}\otimes\wedge^{odd}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$.}
$r^{\gamma}_{_{G,H}}(\sigma)=
\sigma_{\vert H}\otimes\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$.
Hence for the generalized character
$\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{X}}}(\sigma)\in R^{-\infty}(G)$ we have a
`Weyl integration' formula
\begin{equation}\label{eq.restriction.function.G.H}
\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{X}}}(\sigma)=
{\rm Ind}^{^G}_{_H}\left(\ensuremath{\hbox{\rm Index}}^{H}_{\ensuremath{\mathcal{X}}}(\sigma_{\vert H})
\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\right)\ .
\end{equation}
\end{lem}
{\em Proof :} If $\sigma$ is $H$-transversally elliptic, the
symbol $(x,v)\to \sigma(x,v)\odot Cl(\mu_{_{G/H}}(v))$ is homotopic to
$(x,v)\to \sigma(x,v)\odot Cl(0)$ in $K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$. Hence
$\sigma_{\vert H}\odot\sigma^{\ensuremath{\mathcal{X}}}_{_{G,H}}=
\sigma_{\vert H}\otimes\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ in
$K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{X}})$. (\ref{eq.restriction.function.G.H})
follows from the diagram (\ref{indice.r.G.H}).
$\Box$
\begin{coro}\label{coro.restriction.G.H}
Let $\sigma$ be a $G$-transversally elliptic symbol which
furthermore is $H$-transversally elliptic, and let
$\phi\in\ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$ which admits a restriction to $H$. We have
$$
\phi =\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{X}}}(\sigma)\Longleftrightarrow
\phi_{\vert H} =\ensuremath{\hbox{\rm Index}}^{H}_{\ensuremath{\mathcal{X}}}(\sigma_{\vert H})\ .
$$
\end{coro}
In fact, if we come back to the definition of the analytic index given by
Atiyah \cite{Atiyah.74}, one can show the following stronger result.
If $\sigma$ be a $G$-transversally elliptic symbol which is also
$H$-transversally elliptic, then
$\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{X}}}(\sigma)\in\ensuremath{\mathcal{C}^{-\infty}}(G)^G$ admits a {\em restriction} to
$H$ equal to $\ensuremath{\hbox{\rm Index}}^{H}_{\ensuremath{\mathcal{X}}}(\sigma_{\vert H})\in\ensuremath{\mathcal{C}^{-\infty}}(H)^H$.
\medskip
\section{Localization - The general procedure}\label{sec.general.procedure}
\medskip
We recall briefly the notations. Let $(M,J,G)$ be a compact $G$-manifold
provided with a $G$-invariant almost complex structure. We denote by
$RR^{^{G,J}} : K_{G}(M)\to R(G)$ (or simply $RR^{^{G}}$), the corresponding
quantization map. We choose a $G$-invariant Riemannian
metric $(.,.)_{_{M}}$ on $M$. We define in this section a general procedure
to localize the quantization map through the use of a
$G$-equivariant vector field $\lambda$. This idea of localization
goes back, when $G$ is a circle group, to Atiyah \cite{Atiyah.74}
(see Lecture 6) and Vergne \cite{Vergne96} (see part II).
We denote by $\Phi_{\lambda}:M\to\ensuremath{\mathfrak{g}}^{*}$ the map defined by
$\langle\Phi_{\lambda}(m),X\rangle:= (\lambda_{m}, X_{M}\vert_{m})_{_{M}}$
for $X\in\ensuremath{\mathfrak{g}}$. We denote by $\sigma^{_E}(m,v),\ (m,v)\in \ensuremath{\hbox{\bf T}} M$
the elliptic symbol associated to
$\ensuremath{\hbox{\rm Thom}}_{G}(M)\otimes p^*(E)$ for $E\in K_{G}(M)$ (see section
\ref{sec.quantization}).
Let $\sigma^{_E}_{1}$ be the following $G$-invariant elliptic symbol
\begin{equation}
\sigma^{_E}_{1}(m,v):=\sigma^{_E}(m,v-\lambda_{m}),\quad (m,v)\in \ensuremath{\hbox{\bf T}} M.
\label{eq:sigma.1}
\end{equation}
The symbol $\sigma^{_E}_{1}$ is obviously homotopic to $\sigma^{_E}$,
so they define the same class in $K_{G}(\ensuremath{\hbox{\bf T}} M)$. The characteristic
set $\ensuremath{\hbox{\rm Char}}(\sigma^{_E})$ is $M\subset \ensuremath{\hbox{\bf T}} M$, but we see easily
that $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{1})$ is equal to the graph of the vector field
$\lambda$, and
$$
\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{1})\cap\ensuremath{\hbox{\bf T}}_{G}M=\left\{(m,\lambda_{m})\in\ensuremath{\hbox{\bf T}} M, \ \ m\in
\{\Phi_{\lambda}=0\} \right\}.
$$
We will now decompose the elliptic symbol $\sigma^{_E}_{1}$ in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$
near
$$
C_{\lambda}:=\{\Phi_{\lambda}=0\}\ .
$$
If a $G$-invariant subset $C$ is a union of
{\em connected components} of $C_{\lambda}$
there exists a $G$-invariant open neighbourhood $\ensuremath{\mathcal{U}}^{c} \subset M$
of $C$ such that $\ensuremath{\mathcal{U}}^c \cap C_{\lambda}=C$ and
$\partial\ensuremath{\mathcal{U}}^c \cap C_{\lambda}=\emptyset$. We associate to the
subset $C$ the symbol $\sigma^{_E}_{^{C}}:=\sigma^{_E}_{1}\vert_{\ensuremath{\mathcal{U}}^c}\in
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^c)$
which is the restriction of $\sigma^{_E}_{1}$ to $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}^c$. It is
well defined since $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{1}\vert_{\ensuremath{\mathcal{U}}^c})\cap\ensuremath{\hbox{\bf T}}_{G}
\ensuremath{\mathcal{U}}^{c} =\{(m,\lambda_{m})\in\ensuremath{\hbox{\bf T}} M,\ m\in C\}$ is compact.
\medskip
\begin{prop}\label{prop.localisation}
Let $C^a,a\in A$, be a finite collection of disjoint $G$-invariant subsets of
$C_{\lambda}$, each of them being a union of connected components
of $C_{\lambda}$, and let $\sigma^{_E}_{^{C^a}}\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^a)$
be the localized symbols. If $C_{\lambda}=\cup_{a}C^a$, we have
$$
\sigma^{_E}=\sum_{a\in A} i^{a}_{*}(\sigma^{_E}_{^{C^a}})\quad
{\rm in}\quad K_{G}(\ensuremath{\hbox{\bf T}}_{G}M),
$$
where $i^{a}:\ensuremath{\mathcal{U}}^a \hookrightarrow M$ is the inclusion and
$i^a_{*}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^a)\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ is the
corresponding direct image.
\end{prop}
\medskip
{\em Proof :} This is a consequence of the property of excision
(see subsection \ref{subsec.excision}). We
consider disjoint neighbourhoods $\ensuremath{\mathcal{U}}^a$ of $C^a$, and take
$i:\ensuremath{\mathcal{U}}=\cup_{a}\ensuremath{\mathcal{U}}^a \hookrightarrow M$. Let $\chi_{a}\in\ensuremath{\mathcal{C}^{\infty}}(M)^{G}$ be a
test function (i.e. $0\leq \chi_{a}\leq 1$) with compact support on $\ensuremath{\mathcal{U}}^a$
such that $\chi_{a}(m)\neq 0$ if $m\in C^a$. Then the function
$\chi:=\sum_{a}\chi_{a}$ is a $G$-invariant test function with support in
$\ensuremath{\mathcal{U}}$ such that $\chi$ never vanishes on $C_{\lambda}$.
Using the $G$-equivariant symbol
$\sigma^{_E}_{^{\chi}}(m,v)
:=\sigma^{_E}(m,\chi(m)v-\lambda_{m}),\ (m,v)\in \ensuremath{\hbox{\bf T}} M$,
we prove the following :
\noindent i) the symbol $\sigma^{_E}_{^{\chi}}$ is $G$-transversally
elliptic and $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{^{\chi}})\subset\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{U}}}$,
\noindent ii) the symbols $\sigma^{_E}_{^{\chi}}$ and $\sigma^{_E}_{1}$
are equal in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$, and
\noindent iii) the restrictions $\sigma^{_E}_{^{\chi}}\vert_{\ensuremath{\mathcal{U}}}$ and
$\sigma^{_E}_{1}\vert_{\ensuremath{\mathcal{U}}}$ are equal in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}})$.
With Point i) we can apply the excision property to
$\sigma^{_E}_{^{\chi}}$, hence $\sigma^{_E}_{^{\chi}}=
i_{*}(\sigma^{_E}_{^{\chi}}\vert_{\ensuremath{\mathcal{U}}})$. By ii) and iii), the last
equality gives $\sigma^{_E}_{1}=i_{*}(\sigma^{_E}_{1}\vert_{\ensuremath{\mathcal{U}}})=
\sum_{a} i^{a}_{*}(\sigma^{_E}_{^{C^a}})$.
{\em Proof of i)}. The point $(m,v)$ belongs to $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{^{\chi}})$
if and only if $\chi(m)v=\lambda_{m}(*)$. If $m$ is not included in $\ensuremath{\mathcal{U}}$,
we have $\chi(m)=0$ and the equality $(*)$ becomes
$\lambda_{m}=0$. But $\{\lambda =0\}\subset C_{\lambda}\subset
\ensuremath{\mathcal{U}}$, thus $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{^{\chi}})\subset \ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{U}}}$.
The point $(m,v)$ belongs to $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{^{\chi}})\cap \ensuremath{\hbox{\bf T}}_{G}M$
if and only if $\chi(m)v=\lambda_{m}$ and $v$ is orthogonal to the
$G$-orbit in $m$. This imposes $m\in C_{\lambda}$, and finally we
see that $\ensuremath{\hbox{\rm Char}}(\sigma^{_E}_{^{\chi}})\cap \ensuremath{\hbox{\bf T}}_{G} M\simeq C_{\lambda}$
is compact because the function $\chi$ never vanishes on $C_{\lambda}$.
{\em Proof of ii)}. We consider the symbols $\sigma^{_E}_{t},\, t\in[0,1]$
defined by
$$
\sigma^{_E}_{t}(m,v)=\sigma^{_E}(m,(t+(1-t)\chi(m))v-\lambda_{m}).
$$
We see as above that
$\sigma^{_E}_{t}$ is an homotopy of
$G$-transversally elliptic symbols on $\ensuremath{\hbox{\bf T}} M$.
{\em Proof of iii)}. Here we use the homotopy
$\sigma^{_E}_{t}\vert_{\ensuremath{\mathcal{U}}},\ t\in[0,1]$.
$\Box$
Because $RR^{^G}(M,E)=\ensuremath{\hbox{\rm Index}}_{M}^{G}(\sigma^{_E})\in R(G)$, we obtain from
Proposition \ref{prop.localisation} the following decomposition
%
\begin{equation}\label{eq:decomposition.RR}
RR^{^G}(M,E)= \sum_{a\in A}\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{U}}^a}^{G}(
\sigma^{_E}_{^{C^a}})\quad {\rm in}\quad R^{-\infty}(G).
\end{equation}
The rest of this article is devoted to the description, in some
particular cases, of the Riemann-Roch character localized near $C^a$:
\begin{eqnarray}\label{eq:RR.localise}
RR_{C^a}^{^G}(M,-) \ :\ K_{G}(M)&\longrightarrow& R^{-\infty}(G)\\
E\ \ &\longmapsto &
\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{U}}^a}^{G}(\sigma^{_E}_{^{C^a}}).\nonumber
\end{eqnarray}
\medskip
\section{Localization on \protect $M^{\beta}$}\label{sec.loc.M.beta}
\medskip
Let $(M,J,G)$ be a compact $G$-manifold provided with a $G$-invariant
almost complex structure. Let $\beta$ be an element in the {\em center}
of the Lie algebra of $G$, and consider the $G$-invariant vector field
$\lambda:=\beta_{M}$ generated by the infinitesimal action of $\beta$.
In this case we have obviously
$$
\{\Phi_{\beta_{M}}=0\}=\{\beta_{M}=0\}=M^{\beta}\ .
$$
In this section, we compute the localization of the quantization map
on the submanifold $M^{\beta}$ following the technique explained in
section \ref{sec.general.procedure}. We first need to understand
the case of a vector space.
The principal results of this section, i.e. Proposition
\ref{prop.indice.thom.beta.V} and Theorem \ref{th.localisation.pt.fixe}
were obtained by Vergne \cite{Vergne96}[Part II], in the Spin case for
an action of the circle group.
\subsection{Action on a vector space}
Let $(V,q,J)$ be a real vector space equipped with a complex
structure $J$ and an euclidean metric $q$ such that $J\in O(q)$.
Suppose that a compact Lie group $G$ acts on $(V,q,J)$ in a unitary
way, and that there exists $\beta$ in the center of $\ensuremath{\mathfrak{g}}$
such that
$$
V^{\beta}=\{0\}.
$$
We denote by $\ensuremath{\mathbb{T}}_{\beta}$ the torus generated by $\exp(t.\beta),
t\in\ensuremath{\mathbb{R}}$, and $\ensuremath{\mathfrak{t}}_{\beta}$ its Lie algebra.
The complex $\ensuremath{\hbox{\rm Thom}}_{G}(V,J)$ does not define an element in
$K_{G}(\ensuremath{\hbox{\bf T}} V)$ because its characteristic set is $V$.
\begin{defi}
Let $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}V)$ be the
$G$-transversally\footnote{One can verify that $\ensuremath{\hbox{\rm Char}}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V))\cap
\ensuremath{\hbox{\bf T}}_{G}V=\{(0,0)\}$.} elliptic complex defined by
$$
\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)(x,v):=\ensuremath{\hbox{\rm Thom}}_{G}(V)(x,v- \beta_{V}(x)) \quad
{\rm for}\quad (x,v)\in \ensuremath{\hbox{\bf T}} V.
$$
\label{def.thom.beta.V}
\end{defi}
Before computing the index of $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)$ explicitely, we
compare it with the pushforward $j_{!}(\ensuremath{\mathbb{C}})\in K_{G}(\ensuremath{\hbox{\bf T}} V)$ where
$j: \{0\}\ensuremath{\hookrightarrow} V$ is the inclusion and $\ensuremath{\mathbb{C}}\to\{0\}$ is the trivial
line bundle. Recall that $\ensuremath{\hbox{\rm Index}}_{V}^G(j_{!}(\ensuremath{\mathbb{C}}))=1$.
We denote by $\overline{V}$ the real vector space $V$ endowed with
the complex structure $-J$, and
$\wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{V}:=\wedge_{\ensuremath{\mathbb{C}}}^{even}
\overline{V}-\wedge_{\ensuremath{\mathbb{C}}}^{odd} \overline{V} $ the
corresponding element in $R(G)$.
\begin{lem}\label{lem.indice.wedge.V.inverse}
We have $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{V}\, .\, \ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)=
j_{!}(\ensuremath{\mathbb{C}})$ in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}V)$, hence
$$
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{V}.\,\ensuremath{\hbox{\rm Index}}_{V}^G(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V))=1\quad
{\rm in} \quad R^{-\infty}(G).
$$
\end{lem}
{\em Proof} : The class $j_{!}(\ensuremath{\mathbb{C}})$ is represented by the symbol
$\sigma_{o}:\ensuremath{\hbox{\bf T}} V\times \wedge^{even}_{\ensuremath{\mathbb{C}}}(V\otimes\ensuremath{\mathbb{C}})\to
\ensuremath{\hbox{\bf T}} V\times \wedge^{odd}_{\ensuremath{\mathbb{C}}}(V\otimes\ensuremath{\mathbb{C}}),\ (x,v,w)\mapsto
(x,v,Cl(x+\imath v).w)$. If we use the following
isomorphism of complex $G$-vector spaces
\begin{eqnarray*}
V\otimes\ensuremath{\mathbb{C}} &\longrightarrow &V\oplus \overline{V}\\
x +\imath v &\longmapsto &(v-J(x), v+J(x))\ ,
\end{eqnarray*}
we can write $\sigma_{o}=\sigma_{-}\odot\sigma_{+}$, where
the symbols\footnote{$V_{+}=V$ and $V_{-}=\overline{V}$.}
$\sigma_{\pm}$ act on $\ensuremath{\hbox{\bf T}} V\times \wedge^{\bullet}_{\ensuremath{\mathbb{C}}}V_{\pm}$
through the Clifford maps $\sigma_{\pm}(x,v)=Cl(v \mp J(x))$.
Finally we see that the following $G$-transversally elliptic
symbols on $\ensuremath{\hbox{\bf T}} V$ are homotopic
\begin{eqnarray*}
Cl(v + J(x))&\odot& Cl(v - J(x))\\
Cl(v + J(x))&\odot& Cl(v - \beta_{V}(x))\\
Cl(0)&\odot& Cl(v - \beta_{V}(x)) \ .
\end{eqnarray*}
The Lemma is proved since $(x,v)\to Cl(0)\odot Cl(v - \beta_{V}(x))$
represents the class $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet} \overline{V}\, .\,
\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)$ in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}V)$. $\Box$
\medskip
We compute now the index of
$\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)$. For $\alpha\in \ensuremath{\mathfrak{t}}_{\beta}^{*}$, we define
the $G$-invariant subspaces\footnote{We denote by $z\cdot v:=x.v+y.J(v),\
z=x+\imath y\in \ensuremath{\mathbb{C}}$, the action of $\ensuremath{\mathbb{C}}$ on the complex vector space
$(V,J)$, and $zw=v\otimes zz',\ w=v\otimes z'\in V\otimes\ensuremath{\mathbb{C}}$
the canonical action of $\ensuremath{\mathbb{C}}$ on $V\otimes\ensuremath{\mathbb{C}}$.}
$V(\alpha):=\{v\in V,\ \rho(\exp X)(v)=
e^{\imath\langle\alpha,X\rangle}\cdot v,\ \forall X\in
\ensuremath{\mathfrak{t}}_{\beta}\}$, and
$(V\otimes\ensuremath{\mathbb{C}})(\alpha):=\{v\in V\otimes\ensuremath{\mathbb{C}},\ \rho(\exp X)(v)=
e^{\imath\langle\alpha,X\rangle}v,\ \forall X\in
\ensuremath{\mathfrak{t}}_{\beta}\}$.
An element $\alpha \in \ensuremath{\mathfrak{t}}_{\beta}^{*}$, is called a weight for
the action of $\ensuremath{\mathbb{T}}_{\beta}$ on $(V,J)$ (resp. on $V\otimes\ensuremath{\mathbb{C}}$)
if $V(\alpha)\neq 0$ (resp. $(V\otimes\ensuremath{\mathbb{C}})(\alpha)\neq 0$). We denote by
$\Delta(\ensuremath{\mathbb{T}}_{\beta},V)$ (resp. $\Delta(\ensuremath{\mathbb{T}}_{\beta},V\otimes\ensuremath{\mathbb{C}})$)
the set of weights for the action of $\ensuremath{\mathbb{T}}_{\beta}$ on $V$
(resp. $V\otimes\ensuremath{\mathbb{C}}$). We shall note that
$\Delta(\ensuremath{\mathbb{T}}_{\beta},V\otimes\ensuremath{\mathbb{C}})=\Delta(\ensuremath{\mathbb{T}}_{\beta},V)
\cup -\Delta(\ensuremath{\mathbb{T}}_{\beta},V)$.
\begin{defi}
We denote by $V^{+,\beta}$ the following $G$-stable subspace of $V$
$$
V^{+,\beta}:=\sum_{\alpha\in \Delta_{+}(\ensuremath{\mathbb{T}}_{\beta},V)}V(\alpha)\ ,
$$
where $\Delta_{+}(\ensuremath{\mathbb{T}}_{\beta},V)=\{\alpha\in\Delta(\ensuremath{\mathbb{T}}_{\beta},V),
\ \langle\alpha,\beta\rangle > 0\}$. In the same way, we denote by
$(V\otimes\ensuremath{\mathbb{C}})^{+,\beta}$ the following $G$-stable subspace of
$V\otimes\ensuremath{\mathbb{C}}$:
$(V\otimes\ensuremath{\mathbb{C}})^{+,\beta}
:=\sum_{\alpha\in \Delta_{+}(\ensuremath{\mathbb{T}}_{\beta},V\otimes\ensuremath{\mathbb{C}})}
(V\otimes\ensuremath{\mathbb{C}})(\alpha)$, where $\Delta_{+}(\ensuremath{\mathbb{T}}_{\beta},V\otimes\ensuremath{\mathbb{C}})
=\{\alpha\in\Delta(\ensuremath{\mathbb{T}}_{\beta},V\otimes\ensuremath{\mathbb{C}}),\
\langle\alpha,\beta\rangle > 0\}$.
\end{defi}
For any representation $W$ of $G$, we denote by $\det W$ the
representation $\wedge_{\ensuremath{\mathbb{C}}}^{max}W$. In the same way, if $W\to M$ is
a $G$ complex vector bundle we denote by $\det W$ the corresponding line
bundle.
\begin{prop}\label{prop.indice.thom.beta.V}
We have the following equality in $R^{-\infty}(G)$ :
$$
\ensuremath{\hbox{\rm Index}}^{G}_{V}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V))=(-1)^{\dim_{\ensuremath{\mathbb{C}}}V^{+,\beta}}\
\det V^{+,\beta}\otimes\sum_{k\in \ensuremath{\mathbb{N}}}
S^k((V\otimes\ensuremath{\mathbb{C}})^{+,\beta})\ ,
$$
where $S^k((V\otimes\ensuremath{\mathbb{C}})^{+,\beta})$ is the $k$-th symmetric
product over $\ensuremath{\mathbb{C}}$ of $(V\otimes\ensuremath{\mathbb{C}})^{+,\beta}$.
\end{prop}
Proposition \ref{prop.indice.thom.beta.V} and Lemma
\ref{lem.indice.wedge.V.inverse} give the two important properties
of the generalized function
$\chi:=\ensuremath{\hbox{\rm Index}}_{G}^{V}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V))$.
First $\chi$ is an inverse, in $R^{-\infty}(G)$, of the function
$g\in G\to \det_{V}^{\ensuremath{\mathbb{C}}}(1-g^{-1})$ which is the trace of
the (virtual) representation
$\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{V}$. Second, the decomposition
of $\chi$ into irreducible characters of $G$ is of the form
$\chi=\sum_{\lambda} m_{\lambda}\chi_{_{\lambda}}^{_{G}}$ with
$m_{\lambda}\neq 0
\Longrightarrow \langle \lambda,\beta\rangle \geq 0$.
\begin{defi}\label{wedge.V.inverse}
For any $R(G)$-module $A$, we denote by
$A\,\widehat{\otimes}\,R(\ensuremath{\mathbb{T}}_{\beta})$, the $R(G)\otimes
R(\ensuremath{\mathbb{T}}_{\beta})$-module formed by the infinite formal sums
$\sum_{\alpha} E_{\alpha}\, h^{\alpha}$ taken over the set
of weights of $\ensuremath{\mathbb{T}}_{\beta}$, where $E_{\alpha}\in A$
for every $\alpha$.
We denote by $\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{V}\,
\right]^{-1}_{\beta}$
the infinite sum $(-1)^{r}\,\det V^{+,\beta}\otimes\sum_{k\in \ensuremath{\mathbb{N}}}
S^k((V\otimes\ensuremath{\mathbb{C}})^{+,\beta})$, with $r=\dim_{\ensuremath{\mathbb{C}}}V^{+,\beta}$.
It can be considered either as an element of $R^{-\infty}(G)$,
$R(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$, or
$R^{-\infty}(\ensuremath{\mathbb{T}}_{\beta})$.
Let $\ensuremath{\mathcal{V}}\to \ensuremath{\mathcal{X}}$ be a $G$-complex vector bundle such that
$\ensuremath{\mathcal{V}}^{\beta}=\ensuremath{\mathcal{X}}$. The torus $\ensuremath{\mathbb{T}}_{\beta}$ acts on the
fibers of $\ensuremath{\mathcal{V}}\to \ensuremath{\mathcal{X}}$, so we can polarize the
$\ensuremath{\mathbb{T}}_{\beta}$-weights and define the vector bundles
$\ensuremath{\mathcal{V}}^{+,\beta}$ and $(\ensuremath{\mathcal{V}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}$.
In this case, the infinite sum
$\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{V}}}\,
\right]^{-1}_{\beta}:=(-1)^{\dim_{\ensuremath{\mathbb{C}}}\ensuremath{\mathcal{V}}^{+,\beta}}\,
\det \ensuremath{\mathcal{V}}^{+,\beta}\otimes\sum_{k\in \ensuremath{\mathbb{N}}}
S^k((\ensuremath{\mathcal{V}}\otimes\ensuremath{\mathbb{C}})^{+,\beta})$
is an inverse of $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{V}}}$
in \break
$K_G(\ensuremath{\mathcal{X}})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$.
\end{defi}
The rest of this subsection is devoted to the proof of Proposition
\ref{prop.indice.thom.beta.V}. The case $V^{+,\beta}=V$ or
$V^{+,\beta}=\{0\}$ is considered by Atiyah
\cite{Atiyah.74} (see Lecture 6) and Vergne
\cite{Vergne96} (see Lemma 6, Part II).
Let $H$ be a maximal torus of $G$ containing $\ensuremath{\mathbb{T}}_{\beta}$.
The symbol $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(V)$ is also $H$-transversally
elliptic and let $\ensuremath{\hbox{\rm Thom}}^{\beta}_{H}(V)$ be the corresponding
class in $K_{H}(\ensuremath{\hbox{\bf T}}_{H}V)$.
Following Corollary \ref{coro.restriction.G.H}, we can reduce the
proof of Proposition \ref{prop.indice.thom.beta.V} to the case where
the group $G$ is equal to the torus $H$.
\medskip
\underline{Proof of Th. \ref{prop.indice.thom.beta.V} for a
torus action.}
\medskip
We first recall the index theorem proved by Atiyah in Lecture 6 of
\cite{Atiyah.74}. Let $\ensuremath{\mathbb{T}}_{m}$ the circle
group act on $\ensuremath{\mathbb{C}}$ with the representation $t^m,\ m>0$. We have two classes
$\ensuremath{\hbox{\rm Thom}}^{\pm}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}})\in K_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))$
that correspond
respectively to $\beta=\pm \imath\in Lie(S^{1})$. Atiyah denotes these
elements $\overline{\partial}^{\pm}$.
\begin{lem}[Atiyah]\label{lem.atiyah.1}
We have, for $m>0$, the following equalities in $R^{-\infty}(\ensuremath{\mathbb{T}}_{m})$:
\begin{eqnarray*}
\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{m}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\rm Thom}}^{+}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))
&= \left[\frac{1}{1-t^{-m}}\right]^{+}=&-t^m.\sum_{k\in\ensuremath{\mathbb{N}}}(t^m)^k\\
\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{m}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\rm Thom}}^{-}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))
&=\left[\frac{1}{1-t^{-m}}\right]^{-}=&\sum_{k\in\ensuremath{\mathbb{N}}}(t^{-m})^k\ .
\end{eqnarray*}
\end{lem}
Here we follow the notation of Atiyah: $[\frac{1}{1-t^{-m}}]^{+}$ and
$[\frac{1}{1-t^{-m}}]^{-}$ are the
Laurent expansions of the meromorphic function
$t\in\ensuremath{\mathbb{C}}\to\frac{1}{1-t^{-m}}$ around $t=0$ and $t=\infty$ respectively.
From this Lemma we can compute the index of
$\ensuremath{\hbox{\rm Thom}}^{\pm}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}})$ when $m<0$. Suppose $m<0$
and consider the morphism $\kappa : \ensuremath{\mathbb{T}}_{m}\to\ensuremath{\mathbb{T}}_{\vert m\vert},
t\to t^{-1}$. Using the induced morphism $\kappa^{*}: K_{\ensuremath{\mathbb{T}}_{\vert m\vert}}
(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\vert m\vert}}(\ensuremath{\mathbb{C}}))\to
K_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))$, we see that
$\kappa^{*}(\ensuremath{\hbox{\rm Thom}}^{\pm}_{\ensuremath{\mathbb{T}}_{\vert m\vert}}(\ensuremath{\mathbb{C}}))=
\ensuremath{\hbox{\rm Thom}}^{\mp}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}})$. This gives
$\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{m}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\rm Thom}}^{+}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))$ $=
\kappa^*(\sum_{k\in\ensuremath{\mathbb{N}}}(t^{-\vert m\vert})^k)=
\sum_{k\in\ensuremath{\mathbb{N}}}(t^{-m})^k$ and
$\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{m}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\rm Thom}}^{-}_{\ensuremath{\mathbb{T}}_{m}}(\ensuremath{\mathbb{C}}))=
\kappa^*(-t^{\vert m\vert}.\sum_{k\in\ensuremath{\mathbb{N}}}(t^{\vert m\vert})^k)
=-t^{m}\sum_{k\in\ensuremath{\mathbb{N}}}(t^{m})^k$.
We can summarize these different cases as follows.
\begin{lem}\label{lem.atiyah.}
Let $\ensuremath{\mathbb{T}}_{\alpha}$ the circle group act on $\ensuremath{\mathbb{C}}$ with the
representation $t\to t^{\alpha}$ for $\alpha\in\ensuremath{\mathbb{Z}}\setminus\{0\}$.
Let $\beta \in Lie(\ensuremath{\mathbb{T}}_{\alpha})\simeq \ensuremath{\mathbb{R}}$ a non-zero element.
We have the following equalities in $R^{-\infty}(\ensuremath{\mathbb{T}}_{\alpha})$:
$$
\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{\alpha}}_{\ensuremath{\mathbb{C}}}
\left(\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\alpha}}(\ensuremath{\mathbb{C}})\right)(t)
= \left[\frac{1}{1-u^{-1}}\right]^{\ensuremath{\varepsilon}}_{u=t^{\alpha}}\ ,
$$
where $\ensuremath{\varepsilon}$ is the sign of $\langle\alpha,\beta\rangle$.
\end{lem}
We decompose now the vector space $V$ into an orthogonal sum
$V=\oplus_{i\in I}\ensuremath{\mathbb{C}}_{\alpha_{i}}$, where $\ensuremath{\mathbb{C}}_{\alpha_{i}}$ is
a $H$-stable subspace of dimension 1 over $\ensuremath{\mathbb{C}}$ equipped with the
representation $t\in H\to t^{\alpha_{i}}\in \ensuremath{\mathbb{C}}$. Here the set $I$ parametrizes
the weights for the action of $H$ on $V$, counted with their
multiplicities.
Consider the circle group $\ensuremath{\mathbb{T}}_{i}$ with the trivial action on
$\oplus_{k\neq i}\ensuremath{\mathbb{C}}_{\alpha_{k}}$ and with the canonical action on
$\ensuremath{\mathbb{C}}_{\alpha_{i}}$.
We consider $V$ equipped with the action of
$H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}$. The symbol $\ensuremath{\hbox{\rm Thom}}_{H}^{\beta}(V)$ is
$H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}$-equivariant and is either
$H$-transversally elliptic, $H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}$-transversally
elliptic (we denote by $\sigma_{B}$ the corresponding class), or
$\Pi_{k}\ensuremath{\mathbb{T}}_{k}$-transversally elliptic (we denote by $\sigma_{A}$
the corresponding class). We have the following canonical morphisms :
\begin{eqnarray}\label{morphismes.1.2}
K_{H}(\ensuremath{\hbox{\bf T}}_{H}V)\longleftarrow
&K_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{H}V)&\longrightarrow
K_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)\\
\ensuremath{\hbox{\rm Thom}}^{\beta}_{H}(V)
\longleftarrow &\sigma_{B_{1}}&
\longrightarrow \sigma_{B}\ ,\nonumber
\end{eqnarray}
\begin{eqnarray*}
K_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)
\leftarrow& K_{H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)&
\rightarrow K_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)\\
\sigma_{B}\leftarrow &\sigma_{B_{2}}&\rightarrow \sigma_{A} \ .
\end{eqnarray*}
We consider the following characters:
\noindent - $\phi(t)\in R^{-\infty}(H)$ the $H$-index of
$\ensuremath{\hbox{\rm Thom}}_{H}^{\beta}(V)$,
\noindent - $\phi_{B}(t,t_{1},\cdots,t_{l})\in
R^{-\infty}(H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k})$ the $H\times\Pi_{k}\ensuremath{\mathbb{T}}_{k}$-index
of $\sigma_{B}$ (the same for $\sigma_{B_{1}}$ and $\sigma_{B_{2}}$).
\noindent - $\phi_{A}(t_{1},\cdots,t_{l})\in R^{-\infty}(\Pi_{k}\ensuremath{\mathbb{T}}_{k})$
the $\Pi_{k}\ensuremath{\mathbb{T}}_{k}$-index of $\sigma_{A}$.
They satisfy the relations
\noindent i) $\phi(t)=\phi_{B}(t,1,\cdots,1)$ and
$\phi_{B}(1,t_{1},\cdots,t_{l})=\phi_{A}(t_{1},\cdots,t_{l}).$
\noindent ii) $\phi_{B}(tu,t_{1}u^{-\alpha_{1}},\cdots,t_{l}u^{-\alpha_{1}})=
\phi_{B}(t,t_{1},\cdots,t_{l})$, for all $u\in H$.
Point i) is a consequence of the morphisms (\ref{morphismes.1.2}).
Point ii) follows from the fact
that the elements $(u,u^{-\alpha_{1}},\cdots,u^{-\alpha_{l}}),\ u\in
H$ act trivially on $V$.
The symbol $\sigma_{A}$ can be expressed through the map
\begin{eqnarray*}
K_{\ensuremath{\mathbb{T}}_{1}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{1}}\ensuremath{\mathbb{C}}_{\alpha_{1}})
\times K_{\ensuremath{\mathbb{T}}_{2}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{2}}\ensuremath{\mathbb{C}}_{\alpha_{2}})\times\cdots
\times K_{\ensuremath{\mathbb{T}}_{l}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{l}}\ensuremath{\mathbb{C}}_{\alpha_{l}})&\longrightarrow&
K_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)\\
(\sigma_{1},\sigma_{2},\cdots,\sigma_{l})\longmapsto
\sigma_{1}\odot\sigma_{2}\odot\cdots\odot\sigma_{l}\ .
\end{eqnarray*}
Here we have $\sigma_A=\odot_{k=1}^l\ensuremath{\hbox{\rm Thom}}_{\ensuremath{\mathbb{T}}_{k}}^{\ensuremath{\varepsilon}_{k}}
(\ensuremath{\mathbb{C}}_{\alpha_{k}})$ in $K_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}(\ensuremath{\hbox{\bf T}}_{\Pi_{k}\ensuremath{\mathbb{T}}_{k}}V)$,
where $\ensuremath{\varepsilon}_{k}$ is the sign of $\langle\alpha_{k},\beta\rangle$.
Finally, we get
\begin{eqnarray*}
\phi(u)&=&\phi_{B}(u,1,\cdots,1) =
\phi_{B}(1,u^{\alpha_{1}},\cdots,u^{\alpha_{1}})\\
&=& \phi_{A}(u^{\alpha_{1}},\cdots,u^{\alpha_{1}})=
\Pi_{k}\left[\frac{1}{1-t^{-1}}\right]^{\ensuremath{\varepsilon}_{k}}_
{t=u^{\alpha_{k}}}.
\end{eqnarray*}
To finish the proof, it suffices to note that the following
identification of $H$-vector spaces holds : $V^{+,\beta}\simeq
\oplus_{\ensuremath{\varepsilon}_{k}> 0}\ensuremath{\mathbb{C}}_{\alpha_{k}}$ and
$(V\otimes\ensuremath{\mathbb{C}})^{+,\beta}\simeq \oplus_{k}\ensuremath{\mathbb{C}}_{\ensuremath{\varepsilon}_{k}\alpha_{k}}$. $\Box$
\medskip
\subsection{Localization of the quantization map on \protect $M^{\beta}$}
\label{sec.loc.application.moment}
Let $\beta\neq 0$ be a $G$-invariant element of $\ensuremath{\mathfrak{g}}$.
The localization formula that we prove for the
Riemann-Roch character $RR^{^{G}}(M,-)$ will hold
in\footnote{An element of $\widehat{R}(G)$ is simply a
formal sum $\sum_{\lambda}m_{\lambda}\chi_{\lambda}^{_{G}}$ with
$m_{\lambda} \in \ensuremath{\mathbb{Z}}$ for all $\lambda$.}
$\widehat{R}(G):=\hom_{\ensuremath{\mathbb{Z}}}(R(G),\ensuremath{\mathbb{Z}})$.
Let $\ensuremath{\mathcal{N}}$ be the normal bundle of $M^{\beta}$ in $M$.
For $m\in M^{\beta}$, we have the decomposition $\ensuremath{\hbox{\bf T}}_{m}M=
\ensuremath{\hbox{\bf T}}_{m}M^{\beta}\oplus\ensuremath{\mathcal{N}}\vert_{m}$. The linear action of $\beta$
on $T_{m}M$ precises this decomposition. The map $\ensuremath{\mathcal{L}}^{M}(\beta):
\ensuremath{\hbox{\bf T}}_{m}M\to\ensuremath{\hbox{\bf T}}_{m}M$ commutes with the map $J$ and
satisfies $\ensuremath{\hbox{\bf T}}_{m}M^{\beta}=\ker(\ensuremath{\mathcal{L}}^{M}(\beta))$. Here we take
$\ensuremath{\mathcal{N}}\vert_{m}:={\rm Image}(\ensuremath{\mathcal{L}}^{M}(\beta))$. Then the almost
complex structure $J$ induces a $G$-invariant almost complex structure
$J_{\beta}$ on $M^{\beta}$, and a complex structure $J_{\ensuremath{\mathcal{N}}}$ on
the fibers of $ \ensuremath{\mathcal{N}}\to M^{\beta}$. We have then a quantization map
$RR^{^G}(M^{\beta},-):K_{G}(M^{\beta})\to R(G)$. The torus $\ensuremath{\mathbb{T}}_{\beta}$
acts linearly on the fibers of the complex vector bundle $\ensuremath{\mathcal{N}}$.
Thus we associate the polarized complex $G$-vector bundles
$\ensuremath{\mathcal{N}}^{+,\beta}$ and $(\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}$ (see Definition
\ref{wedge.V.inverse}).
\begin{theo}\label{th.localisation.pt.fixe}
For every $E\in K_{G}(M)$, we have the following equality in \break
$\widehat{R}(G)$ :
$$
RR^{^{G}}(M,E)=(-1)^{r_{\ensuremath{\mathcal{N}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR^{^{G}}(M^{\beta},E\vert_{M^{\beta}}
\otimes\det\ensuremath{\mathcal{N}}^{+,\beta}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}) \ ,
$$
where $r_{\ensuremath{\mathcal{N}}}$ is the locally constant function on $M^{\beta}$
equal to the complex rank of $\ensuremath{\mathcal{N}}^{+,\beta}$.
\end{theo}
\medskip
Before proving this result let us rewrite this localization formula
in a more synthetic way. The $G\times\ensuremath{\mathbb{T}}_{\beta}$-Riemann-Roch character
$RR^{^{G \times {\rm T}_{\beta}}}(M^{\beta},-)$ is extended canonically
to a map from $K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ to
$R(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ (see Definition
\ref{wedge.V.inverse}).
Note that the surjective morphism $G\times \ensuremath{\mathbb{T}}_{\beta}\to G,
(g,t)\mapsto g.t$ induces maps $R(G)\to R(G)\otimes R(\ensuremath{\mathbb{T}}_{\beta})$,
$K_{G}(M)\to K_{G\times\ensuremath{\mathbb{T}}_{\beta}}(M)$, both denoted $k$, with
the tautological relation $k(RR^{^{G}}(M,E))=
RR^{^{G \times {\rm T}_{\beta}}}(M,k(E))$. To simplify, we
will omit the morphism $k$ in our notations.
Let $\overline{\ensuremath{\mathcal{N}}}$ be the normal bundle $\ensuremath{\mathcal{N}}$ with the opposite
complex structure. With the convention of Definition \ref{wedge.V.inverse}
the element $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\in
K_{G\times \ensuremath{\mathbb{T}}_{\beta}}(M^{\beta})\simeq
K_{G}(M^{\beta})\otimes R(\ensuremath{\mathbb{T}}_{\beta})$ admits a polarized inverse
$\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\, \in\,
K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$.
Finally the result of Theorem \ref{th.localisation.pt.fixe} can be
written as the following equality in $R(G)\,\widehat{\otimes}\,
R(\ensuremath{\mathbb{T}}_{\beta})$ :
\begin{equation}\label{eq.loc.M.beta.simplifie}
RR^{^{G}}(M,E)=RR^{^{G \times {\rm T}_{\beta}}}
\left(M^{\beta},E\vert_{M^{\beta}}\otimes
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\right)\ .
\end{equation}
\medskip
Note that Theorem \ref{th.localisation.pt.fixe} gives a proof
of some rigidity properties
\cite{Atiyah-Hirzebruch70,Meinrenken-Sjamaar}. Let $H$ be a maximal
torus of $G$. Following Meinrenken
and Sjamaar, a $G$-equivariant complex vector bundle
$E\to M$ is called {\em rigid} if the action of $H$ on
$E\vert_{M^H}$ is trivial. Take $\beta\in\ensuremath{\mathfrak{h}}$ such that $M^{\beta}=
M^{H}$, and apply Theorem \ref{th.localisation.pt.fixe}, with
$\beta$ and $-\beta$, to $RR^{^H}(M,E)$, with $E$ rigid.
If we take $+\beta$, Theorem \ref{th.localisation.pt.fixe} shows that
$h\in H\to RR^{^H}(M,E)(h)$ is of the form $h\in H\to
\sum_{a\in\hat{H}}n_{a}h^a$ with
$n_{a}\neq 0\Longrightarrow \langle a,\beta\rangle\geq 0$.
(see Lemma \ref{lem.multiplicites.tore}). If we take $-\beta$,
we find $RR^{^H}(M,E)(h)= \sum_{a\in\hat{H}}n_{a}h^a$, with
$n_{a}\neq 0 \Longrightarrow -\langle a,\beta\rangle\geq 0$. Comparing
the two results, and using the genericity of $\beta$, we see that
$RR^{^H}(M,E)$ is a {\em constant} function on $H$, hence $RR^{^G}(M,E)$ is
then a constant function on $G$. We can now
rewrite the equation of Theorem \ref{th.localisation.pt.fixe},
where we keep on the right hand side
the {\em constant} terms:
\begin{equation}\label{prop.E.rigid}
RR^{^G}(M,E)=\sum_{F\subset M^{H,+}}RR(F,E\vert_{F})\ .
\end{equation}
Here the summation is taken over all connected components $F$ of $M^{H}$
such that $\ensuremath{\mathcal{N}}_{F}^{+,\beta}=0$ (i.e. we have
$\langle \xi,\beta\rangle < 0$ for all weights $\xi$ of the $H$-action
on the normal bundle $\ensuremath{\mathcal{N}}_{F}$ of $F$).
\medskip
{\em Proof of Theorem \ref{th.localisation.pt.fixe} :}
\medskip
Let $\ensuremath{\mathcal{U}}$ be a $G$-invariant tubular neighborhood\footnote{To simplify the
notation, we keep the notation $M^{\beta}$ even if we work in fact
on a connected component of the submanifold $M^{\beta}$.} of
$M^{\beta}$ in $M$. We know from section \ref{sec.general.procedure} that
$RR^{^G}(M,E)=\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{U}}}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(M,J)\otimes
E\vert_{\ensuremath{\mathcal{U}}})$ where
$$
\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(M,J)(m,w):=\ensuremath{\hbox{\rm Thom}}_{G}(\ensuremath{\mathcal{V}},J)
(m,w-\beta_{\ensuremath{\mathcal{N}}}(m)), \quad {\rm }\quad (m,w)\in\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}.
$$
Let $\phi:\ensuremath{\mathcal{V}}\to \ensuremath{\mathcal{U}}$ be $G$-invariant diffeomorphism with a $G$-invariant
neighbourhood $\ensuremath{\mathcal{V}}$ of $M^{\beta}$ in the normal bundle $\ensuremath{\mathcal{N}}$.
We denote by $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{V}},J)$ the symbol
$\phi^{*}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(M,J))$.
Here we still denote by $J$ the almost complex structure transported on
$\ensuremath{\mathcal{V}}$ via the diffeomorphism $\ensuremath{\mathcal{U}}\simeq\ensuremath{\mathcal{V}}$.
Let $p:\ensuremath{\mathcal{N}}\to M^{\beta}$ be the canonical projection. The choice of
a $G$-invariant connection on $\ensuremath{\mathcal{N}}$ induces an isomorphism of $G$-vector bundles
over $\ensuremath{\mathcal{N}}$:
\begin{eqnarray}\label{eq:trivialisation.T.N}
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}&\tilde{\longrightarrow}&
p^{*}\left(\ensuremath{\hbox{\bf T}} M^{\beta}\oplus \ensuremath{\mathcal{N}}\right)\\
w&\longmapsto& \ensuremath{\hbox{\bf T}} p(w)\oplus (w)^{V}\nonumber
\end{eqnarray}
Here $w\to (w)^{V},\ \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}\to p^{*}\ensuremath{\mathcal{N}}$ is the
projection which associates to a tangent vector its {\em vertical}
part (see \cite{B-G-V}[section 7] or \cite{pep1}[section 4.1]).
The map $\widetilde{J}:=p^{*}(J_{\beta}\oplus J_{\ensuremath{\mathcal{N}}})$
defines an almost complex structure on the manifold $\ensuremath{\mathcal{N}}$ which
is constant over the fibers of $p$. With this
new almost complex structure $\widetilde{J}$ we construct
the $G$-transversally elliptic symbol over $\ensuremath{\mathcal{N}}$
$$
\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{N}})(n,w)=\ensuremath{\hbox{\rm Thom}}_{G}(\ensuremath{\mathcal{N}},\widetilde{J})
(n,w-\beta_{\ensuremath{\mathcal{N}}}(n)), \quad {\rm }\quad (n,w)\in\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}.
$$
We denote by $i: \ensuremath{\mathcal{V}}\to\ensuremath{\mathcal{N}}$ the inclusion map, and
$i_{*}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{N}})$ the induced map.
\begin{lem}\label{lem.J.modifie}
We have
$$
i_{*}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{V}},J))=
\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{N}})
\quad {\rm in} \quad K_{G}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}).
$$
\end{lem}
{\em Proof} : We proceed as in Lemma
\ref{lem.inv.homotopy}. The complex
structure $J_{n},\ n\in\ensuremath{\mathcal{V}}$ and $\widetilde{J}_{n},\
n\in\ensuremath{\mathcal{N}}$ are equal on $M^{\beta}$, and are related by the
homotopy $J^t_{(x,v)}:=J_{(x,t.v)},\ u\in[0,1]$ for $n=(x,v)\in \ensuremath{\mathcal{V}}$.
Then, as in
Lemma \ref{lem.inv.homotopy}, we can construct an invertible bundle
map $A\in\Gamma(\ensuremath{\mathcal{V}},\ensuremath{\hbox{\rm End}}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}}))^{G}$, which is homotopic
to the identity and such that $A.J= \widetilde{J}.A$ on $\ensuremath{\mathcal{V}}$.
We conclude as in Lemma \ref{lem.inv.homotopy} that the symbols
$\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{V}},J)$ and
$\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{N}})\vert\ensuremath{\mathcal{V}}$ are equal in $K_{G}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}})$.
Then the Lemma follows from the excision property. $\Box$
\medskip
Since $E\simeq p^{*}(E\vert_{M^{\beta}})$, for any $G$-complex vector
bundle $E$ over $\ensuremath{\mathcal{N}}$, the former Lemma tells us that
$RR^{^G}(M,E)=\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{N}}}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{N}})\otimes
p^{*}(E\vert_{M^{\beta}}))$.
We consider now the Hermitian vector bundle $\ensuremath{\mathcal{N}}\to M^{\beta}$ with the
action of $G\times \ensuremath{\mathbb{T}}_{\beta}$. First we use the decomposition
$\ensuremath{\mathcal{N}}=\oplus_{\alpha}\ensuremath{\mathcal{N}}^{\alpha}$ relatively to the unitary
action of $\ensuremath{\mathbb{T}}_{\beta}$ on the fibers of $\ensuremath{\mathcal{N}}$. Let
$N^{\alpha}$ be an Hermitian vector space of dimension equal to the
rank of $\ensuremath{\mathcal{N}}^{\alpha}$, equipped with
the representation $t\to t^{\alpha}$ of $\ensuremath{\mathbb{T}}_{\beta}$.
Let $U$ be the group of $\ensuremath{\mathbb{T}}_{\beta}$-{\em equivariant unitary
maps} of the vector space $N:= \oplus_{\alpha}N^{\alpha}$, and let $R$
be the $\ensuremath{\mathbb{T}}_{\beta}$-equivariant unitary frame of
$(\ensuremath{\mathcal{N}},J_{\ensuremath{\mathcal{N}}})$ framed on $N$.
Note that $R$ is provided with a
$U\times G$-action and a trivial action of
$\ensuremath{\mathbb{T}}_{\beta}$ : for $x\in M^{\beta}$, any element of $R\vert_{x}$
is a $\ensuremath{\mathbb{T}}_{\beta}$-equivariant unitary map from $N$
to $\ensuremath{\mathcal{N}}\vert_{x}$.
The manifold $\ensuremath{\mathcal{N}}$ is isomorphic to
$R\times_{U}N$, where $G$ acts on $R$ and
$\ensuremath{\mathbb{T}}_{\beta}$ acts on $N$.
We denote by $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})$ the
$G\times\ensuremath{\mathbb{T}}_{\beta}$ canonical extension of
$\ensuremath{\hbox{\rm Thom}}^{\beta}_{G}(\ensuremath{\mathcal{N}})$. It can be considered
as a $G$, $G\times\ensuremath{\mathbb{T}}_{\beta}$, or $\ensuremath{\mathbb{T}}_{\beta}$-transversally
elliptic symbol. Here we consider $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})$
as an element of $K_{G\times\ensuremath{\mathbb{T}}_{\beta}}
(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}}(R\times_{U}N))$.
Recall that we have two isomorphisms
\begin{equation}
\pi_{N}^{*}:
K_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}}(R\times_{U}N))
\tilde{\longrightarrow}
K_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}
(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(R\times N)),
\label{eq:pi.N.iso}
\end{equation}
\begin{equation}
\pi^{*}:
K_{G}(\ensuremath{\hbox{\bf T}} M^{\beta})
\widetilde{\longrightarrow}
K_{G\times U}(\ensuremath{\hbox{\bf T}}_{U}R),
\label{eq:pi.iso}
\end{equation}
where $\pi_{N}:R\times N\to R\times_{U}N\simeq \ensuremath{\mathcal{N}}$ and
$\pi:R\to R/U\simeq M^{\beta}$ are the quotient maps relative
to the free $U$-action.
Following (\ref{eq:produit.transversal}), we have a product
\begin{equation}
K_{G\times U}(\ensuremath{\hbox{\bf T}}_{U}R)
\times
K_{\ensuremath{\mathbb{T}}_{\beta}\times U}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}}N)
\longrightarrow
K_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}
(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(R\times N)) \ .
\label{eq:G.T.beta.U}
\end{equation}
The following Thom classes
\noindent - $\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})\in
K_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}}(R\times_{U}N))$,
\noindent - $\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N)\in
K_{\ensuremath{\mathbb{T}}_{\beta}\times U}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}} N)$, and
\noindent - $\ensuremath{\hbox{\rm Thom}}_{G}(M^{\beta})\in K_{G}(\ensuremath{\hbox{\bf T}} M^{\beta})$
are related by the following equality in $K_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}
(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(R\times N))$ :
\begin{equation}
\pi^{*}_{N}\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})=
(\pi^{*}\ensuremath{\hbox{\rm Thom}}_{G}(M^{\beta}))\odot
\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N).
\label{eq:Thom.egalite}
\end{equation}
We will justify (\ref{eq:Thom.egalite}) later. Every $E\in
K_{G}(M)$, when restrict to $M^{\beta}$, admit the decomposition
$E\vert_{M^{\beta}}=
\sum_{a\in\widehat{\ensuremath{\mathbb{T}}_{\beta}}} E^a\otimes \ensuremath{\mathbb{C}}_{a}$ in
$K_{G\times\ensuremath{\mathbb{T}}_{\beta}}(M^{\beta})\simeq
K_{G}(M^{\beta})\otimes R(\ensuremath{\mathbb{T}}_{\beta})$. Multiplication of
(\ref{eq:Thom.egalite}) by $E$ gives
$$
\pi^{*}_{N}(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})
\otimes E\vert_{M^{\beta}}) =
\sum_{a\in\widehat{\ensuremath{\mathbb{T}}_{\beta}}}
\pi^{*}(\ensuremath{\hbox{\rm Thom}}_{G}(M^{\beta})\otimes E^a)\odot
(\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N)\otimes \ensuremath{\mathbb{C}}_{a}).
$$
Following (\ref{eq:formule.G.H.produit}) and Theorem (\ref{thm.atiyah.1}),
the last equality gives, after taking the index and the $U$-invariant :
\begin{eqnarray}\label{eq:indice.egalite}
\lefteqn{RR^{^{G\times{\rm T}_{\beta}}}(M,E)=}\nonumber\\
& & \sum_{a}\left[\sum_{i\in\widehat{U}} RR^{^G}(M^{\beta},E^a\otimes
\underline{W}_{i}^{*})\cdot W_{i}\cdot\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{\beta}\times U}
\left(\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N)\right)\cdot
\ensuremath{\mathbb{C}}_{a}\right]^U\ .
\end{eqnarray}
Here we used that $RR^{^{G\times{\rm T}_{\beta}}}(M,E)$
is equal to the $U$-invariant part of \break
$\ensuremath{\hbox{\rm Index}}^{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}(\pi^{*}_{N}
(\ensuremath{\hbox{\rm Thom}}^{\beta}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(\ensuremath{\mathcal{N}})\otimes E\vert_{M^{\beta}}))$,
and the index of
$\pi^{*}(\ensuremath{\hbox{\rm Thom}}_{G}(M^{\beta})\otimes E^a)$ is
equal to $\sum_{i\in\widehat{U}}RR^{^G}(M^{\beta},E^a\otimes
\underline{W}_{i}^{*}).W_{i}$.
Now we observe that for any $L\in R(U)$, the $U$-invariant part of
\break
$\sum_{i\in\widehat{U}}RR^{^G}(M^{\beta},E\vert_{M^{\beta}}\otimes
\underline{W}_{i}^{*}).W_{i}\otimes L$
is equal to $RR^{^G}(M^{\beta},E\vert_{M^{\beta}}\otimes \underline{L})$
with $\underline{L}=R\times_{U}L$. With the computation of
$\ensuremath{\hbox{\rm Index}}^{\ensuremath{\mathbb{T}}_{\beta}\times U}(
\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N))$ given in
Proposition \ref{prop.indice.thom.beta.V} we obtain finally
$$
RR^{^{G\times{\rm T}_{\beta}}}(M,E)=
(-1)^{r_{\ensuremath{\mathcal{N}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR^{^{G\times{\rm T}_{\beta}}}
\Big(M^{\beta},E\vert_{M^{\beta}}\otimes\det \ensuremath{\mathcal{N}}^{+,\beta}
\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta})\Big) \
$$
which implies the equality of Theorem \ref{th.localisation.pt.fixe}.
\medskip
We give now an explanation for (\ref{eq:Thom.egalite}),
which is a direct consequence of the fact that
the almost complex structure $\widetilde{J}$ admits the decomposition
$\widetilde{J}=p^{*}(J_{\beta}\oplus J_{\ensuremath{\mathcal{N}}})$.
Hence $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{N}}$ equipped with the map
$Cl_{n}(v-\beta_{\ensuremath{\mathcal{N}}}(n)),\ v\in \ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{N}}$ is isomorphic
to $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\hbox{\bf T}}_{x}M^{\beta}\otimes
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathcal{N}}\vert_{x}$
equipped with $Cl_{x}(v_{1})\odot Cl_{x}(v_{2}-\beta_{\ensuremath{\mathcal{N}}}(n))$
where $x=p_{a}(n)$, and the vector $v\in\ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{N}}$
is decomposed, following the isomorphism (\ref{eq:trivialisation.T.N}),
in $v=v_{1}+v_{2}$ with $v_{1}\in \ensuremath{\hbox{\bf T}}_{x}M^{\beta}$ and $v_{2}\in \ensuremath{\mathcal{N}}\vert_{x}$.
Note that the vector $w=\beta_{\ensuremath{\mathcal{N}}}(n)\in \ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{N}}$ is
vertical, i.e. $w=(w)^{V}$. $\Box$
\medskip
\section{Localization via an abstract moment map}\label{sec.Localisation.f}
\medskip
Let $(M,J,G)$ be a compact $G$-manifold provided with a $G$-invariant
almost complex structure. We denote by $RR^{^{G}} : K_{G}(M)\to R(G)$ the
quantization map. Here we suppose that the $G$-manifold is equipped
with an {\em abstract moment map} \cite{G-G-K3,Karshon.98}.
\begin{defi}\label{moment-map}
A smooth map $f_{_{G}}:M\to\ensuremath{\mathfrak{g}}^{*}$ is called an
{\rm abstract moment map} if
i) the map $f_{_{G}}$ is equivariant for the action of the group $G$, and
ii)\footnote{Condition ii) is equivalent to the following :
for every $X\in\ensuremath{\mathfrak{g}}$, the fonction $\langle f_{_{G}},X \rangle$
is locally constant on $M^X$.} for every Lie subgroup $K\subset G$ with
Lie algebra $\ensuremath{\mathfrak{k}}$, the induced map $f_{_{K}}:M\to \ensuremath{\mathfrak{k}}^{*}$ is
locally constant on the submanifold
$M^{K}$ of fixed points for the $K$-action (the map
$f_{_{K}}$ is the composition of $f_{_{G}}$ with the projection
$\ensuremath{\mathfrak{g}}^{*}\to\ensuremath{\mathfrak{k}}^{*}$).
\end{defi}
The terminology ``moment map'' is usually used when we work in the
case of a Hamiltonian action. More precisely, when the manifold is
equipped with a symplectic $2$-form $\omega$ which is $G$-invariant,
a {\em moment map} $\Phi:M\to\ensuremath{\mathfrak{g}}^{*}$ relative to
$\omega$ is a $G$-equivariant map satisfying
$d\langle\Phi,X\rangle=-\omega(X_{M},-),\ X\in\ensuremath{\mathfrak{g}}$.
\medskip
For the rest of this paper we make the choice of a $G$-invariant
scalar product over $\ensuremath{\mathfrak{g}}^{*}$. This defines an identification
$\ensuremath{\mathfrak{g}}^{*}\simeq\ensuremath{\mathfrak{g}}$, and we work with a given abstract moment map
$f_{_{G}}: M\to \ensuremath{\mathfrak{g}}$.
\begin{defi}\label{def.H.vector}
Let $\ensuremath{\mathcal{H}}^{^{G}}$ be the $G$-invariant vector field over $M$ defined by
$$
\ensuremath{\mathcal{H}}^{^{G}} _{m}:=(f_{_{G}}(m)_{M})_{m},\quad \forall \ m\in M.
$$
\end{defi}
The aim of this section is to compute the localization, as in section
\ref{sec.general.procedure}, with the $G$-invariant vector field
$\ensuremath{\mathcal{H}}^{^{G}}$. We know that the Riemann-Roch character is localized near
the set $\{\Phi_{\ensuremath{\mathcal{H}}^{^{G}}}=0\}$, but we see that
$\{\Phi_{\ensuremath{\mathcal{H}}^{^{G}}}=0\}=\{\ensuremath{\mathcal{H}}^{^{G}}=0\}$.
We will denote by $C^{f_{_{G}}}$ this set.
Let $H$ be a maximal torus of $G$, with Lie algebra $\ensuremath{\mathfrak{h}}$, and let
$\ensuremath{\mathfrak{h}}_{+}$ be a Weyl chamber in $\ensuremath{\mathfrak{h}}$.
\begin{lem}\label{lem.C.f.G}
There exists a finite subset $\ensuremath{\mathcal{B}}_{_{G}}\subset\ensuremath{\mathfrak{h}}_{+}$, such that
$$
C^{f_{_{G}}}=\bigcup_{\beta\in\ensuremath{\mathcal{B}}_{_{G}}}C^{^{G}}_{\beta},\quad
{\rm with}\quad
C^{^{G}}_{\beta}=G.(M^{\beta}\cap f_{_{G}}^{-1}(\beta)).
$$
\end{lem}
{\em Proof} : We first observe that $\ensuremath{\mathcal{H}}^{^{G}} _{m}=0$ if and only
if $f_{_{G}}(m)=\beta'$ and $\beta'_{M}\vert_{m}=0$, that is
$m\in M^{\beta'}\cap f_{_{G}}^{-1}(\beta')$, for some
$\beta'\in\ensuremath{\mathfrak{g}}$. For every $\beta'\in\ensuremath{\mathfrak{g}}$, there exists
$\beta\in\ensuremath{\mathfrak{h}}_{+}$, with $\beta'=g.\beta$ for some $g\in G$. Hence
$M^{\beta'}\cap f_{_{G}}^{-1}(\beta')=g.(M^{\beta}\cap
f_{_{G}}^{-1}(\beta))$. We have shown that
$C^{f_{_{G}}}=\bigcup_{\beta\in\ensuremath{\mathfrak{h}}_{+}}C^{^{G}}_{\beta}$, and we
need to prove that the set $\ensuremath{\mathcal{B}}_{_{G}}:=\{\beta\in\ensuremath{\mathfrak{h}}_{+},\
M^{\beta}\cap f_{_{G}}^{-1}(\beta)\neq\emptyset\}$ is finite. Consider the set
$\{H_{1},\cdots,H_{l}\}$ of stabilizers for the action of the torus $H$
on the compact manifold $M$. For each $\beta\in\ensuremath{\mathfrak{h}}$ we denote by
$\ensuremath{\mathbb{T}}_{\beta}$ the subtorus of $H$
generated by $\exp(t.\beta),\ t\in\ensuremath{\mathbb{R}}$, and we observe that
\begin{eqnarray*}
M^{\beta}\cap f_{_{G}}^{-1}(\beta)\neq\emptyset
&\Longleftrightarrow&
\exists H_{i}\ {\rm such\ that}\
\ensuremath{\mathbb{T}}_{\beta}\subset H_{i}\ \ {\rm and}\ \ M^{H_{i}}\cap
f_{_{G}}^{-1}(\beta)\neq\emptyset \\
&\Longleftrightarrow&
\exists H_{i}\ {\rm such\ that}\ \beta\in f_{_{G}}(M^{H_{i}})\cap
Lie(H_{i}).\\
\end{eqnarray*}
But $f_{_{G}}(M^{H_{i}})\cap Lie(H_{i})\subset
f_{_{H_{i}}}(M^{H_{i}})$ is a finite set after Definition \ref{moment-map}.
The proof is now completed. $\Box$
\begin{defi}\label{def.thom.beta.f}
Let $\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,\beta}})$ defined
by
$$
\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)(x,v):=\ensuremath{\hbox{\rm Thom}}_{G}(M)(x,v-\ensuremath{\mathcal{H}}^{^{G}}_{x}), \quad
{\rm for}\quad (x,v)\in \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}^{^{G,\beta}}\ .
$$
Here $i^{^{G,\beta}}:\ensuremath{\mathcal{U}}^{^{G,\beta}}\ensuremath{\hookrightarrow} M$ is any $G$-invariant
neighbourhood of $C^{^{G}}_{\beta}$ such that
$\overline{\ensuremath{\mathcal{U}}^{^{G,\beta}}}\cap C^{f_{_{G}}}= C^{^{G}}_{\beta}$.
\end{defi}
\begin{defi}\label{def.RR.beta}
For every $\beta\in \ensuremath{\mathcal{B}}_{_{G}}$, we denote by $RR_{\beta}^{^{G}}(M,-):
K_{G}(M)\to R^{-\infty}(G)$ the localized Riemann-Roch character
near $C^{^{G}}_{\beta}$, defined as in (\ref{eq:RR.localise}), by
$$
RR_{\beta}^{^{G}}(M,E)=\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}
\left(\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)\otimes E_{\vert\ensuremath{\mathcal{U}}^{^{G,\beta}}}\right)\ ,
$$
for $E\in K_{G}(M)$. Note that the map $RR_{\beta}^{^{G}}(M,-)$ is
well defined on a {\em non-compact} manifold $M$ when the abstract
moment map is proper, since we can take
$\ensuremath{\mathcal{U}}^{^{G,\beta}}$ relatively compact and the index map
$\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$ is then defined (see Corollary
\ref{hyp.indice}).
\end{defi}
According to Proposition \ref{prop.localisation}, we have the partition
$RR^{^{G}}(M,-)=$ \break
$\sum_{\beta\in\ensuremath{\mathcal{B}}_{_{G}}}RR_{\beta}^{^{G}}(M,-)$, and
the rest of this article is devoted to the analysis of the maps
$RR_{\beta}^{^{G}}(M,-),\ \beta\in \ensuremath{\mathcal{B}}_{_{G}}$.
In subsections \ref{subsec.RR.G.beta} and \ref{subsec.induction.G.H}
we prove that $[RR_{\beta}^{^{G}}(M,E)]^G=0$, when $E$ is
$f_{_{G}}$-{\em strictly positive} with
$\eta_{_{E,\beta}}>\langle \theta,\beta\rangle$ (see Def.
\ref{eq.mu.positif} for the notion of
$f_{_{G}}$-{\em positivity}). The next two subsections are
devoted to the computation of
$RR^{^{G}}_{0}(M,-)$ when $0$ is a regular value of
the abstract moment map $f_{_{G}}$.
\subsection{Induced ${\rm Spin}^{\rm c}$ structures} \label{subsec.spinc}
In this subsection we first review the notion of ${\rm Spin}^{\rm c}$-structures
(see \cite{Lawson-Michel,Duistermaat96,Schroder}). After we show that the
almost complex structure $J$ on $M$ induces a ${\rm Spin}^{\rm c}$-structure on
$\ensuremath{\mathcal{M}}_{red}$.
The group ${\rm Spin}_n$ is the connected double cover of the group
${\rm SO}_n$. Let $\eta :{\rm Spin}_n\to{\rm SO}_n$ be the covering map, and let
$\ensuremath{\varepsilon}$ be the element who generates the kernel. The group ${\rm Spin}^{\rm c}_n$
is the quotient ${\rm Spin}_n\times_{\ensuremath{\mathbb{Z}}_2}{\rm U}_1$, where $\ensuremath{\mathbb{Z}}_2$ acts by
$(\ensuremath{\varepsilon},-1)$. There are two canonical group homomorphisms
$$
\eta:{\rm Spin}^{\rm c}_n\to{\rm SO}_n\quad ,\quad {\rm Det} :{\rm Spin}^{\rm c}_n\to {\rm U}_1\
$$
such that $\eta^{\rm c}=(\eta,{\rm Det}):{\rm Spin}^{\rm c}_n\to{\rm SO}_n\times {\rm U}_1$ is a
double covering map.
Let $p:E\to M$ be an oriented Euclidean vector bundle of rank $n$, and
let ${\rm P}_{{\rm SO}}(E)$ be its bundle of oriented orthonormal frames. A
${\rm Spin}^{\rm c}$-structure on $E$ is a ${\rm Spin}^{\rm c}_n$-principal bundle
${\rm P}_{{\rm Spin}^{\rm c}}(E)\to M$, together with a ${\rm Spin}^{\rm c}$-equivariant map
${\rm P}_{{\rm Spin}^{\rm c}}(E)\to{\rm P}_{{\rm SO}}(E)$. The line bundle
$\ensuremath{\mathbb{L}}:={\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Det}}\ensuremath{\mathbb{C}}$ is called the determinant line bundle
associated to ${\rm P}_{{\rm Spin}^{\rm c}}(E)$. Whe have then a double covering
map\footnote{If $P$, $Q$ are principal bundle over $M$ respectively
for the groups $G$ and $H$, we denote simply by $P\times Q$ their
fibering product over $M$ which is a $G\times H$ principal bundle
over $M$.}
\begin{equation}\label{eq.spin.covering}
\eta^{\rm c}_E \, : \, {\rm P}_{{\rm Spin}^{\rm c}}(E)\longrightarrow{\rm P}_{{\rm SO}}(E)\times{\rm P}_{{\rm U}}(\ensuremath{\mathbb{L}}) \ ,
\end{equation}
where ${\rm P}_{{\rm U}}(\ensuremath{\mathbb{L}}):={\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Det}}{\rm U}_1$ is the associated
${\rm U}_1$-principal bundle over $M$.
A ${\rm Spin}^{\rm c}$-structure on an oriented Riemannian manifold is a
${\rm Spin}^{\rm c}$-structure on its tangent bundle. If a group $K$ acts on the
bundle $E$, preserving the orientation and the Euclidean structure,
we defines a $K$-equivariant ${\rm Spin}^{\rm c}$-structure by requiring ${\rm P}_{{\rm Spin}^{\rm c}}(E)$
to be a $K$-equivariant principal bundle, and (\ref{eq.spin.covering})
to be $(K\times{\rm Spin}^{\rm c}_n)$-equivariant.
\medskip
We assume now that $E$ is of even rank $n=2m$.
Let $\Delta_{2m}$ be the irreducible complex Spin representation of
${\rm Spin}^{\rm c}_{2m}$. Recall that $\Delta_{2m}=\Delta_{2m}^+\oplus\Delta_{2m}^-$
inherits a canonical Clifford action ${\bf c} :\ensuremath{\mathbb{R}}^{2m}\to\ensuremath{\hbox{\rm End}}_{\ensuremath{\mathbb{C}}}(\Delta_{2m})$
which is ${\rm Spin}^{\rm c}_{2m}$-equivariant, and which interchanges the
graduation : ${\bf c}(v):\Delta_{2m}^{\pm}\to\Delta_{2m}^{\mp}$, for
every $v\in\ensuremath{\mathbb{R}}^{2m}$. Let
\begin{equation}\label{eq.spinor.bundle}
\ensuremath{\mathcal{S}}(E):={\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Spin}^{\rm c}_{2m}}\Delta_{2m}
\end{equation}
be the irreducible complex spinor bundle over $E\to M$. The
orientation on the fibers of $E$ defines a graduation
$\ensuremath{\mathcal{S}}(E):=\ensuremath{\mathcal{S}}(E)^+\oplus\ensuremath{\mathcal{S}}(E)^-$. Let $\overline{E}$ be the bundle
$E$ with opposite orientation. A ${\rm Spin}^{\rm c}$ structure on $E$ induces a
${\rm Spin}^{\rm c}$ on $\overline{E}$, with the same determinant line bundle, and such that
$\ensuremath{\mathcal{S}}(\overline{E})^{\pm}=\ensuremath{\mathcal{S}}(E)^{\mp}$.
More generaly, we associated
to an Euclidean vector bundle $p: E\to M$ its Clifford bundle
${\rm Cl}(E)\to M$. A complex vector bundle $\ensuremath{\mathcal{S}}\to M$ is
called a complex spinor bundle over $E\to M$ if it is a
left-${\rm Cl}(E)$-module; moreover $\ensuremath{\mathcal{S}}$ is called irreducible if
${\rm Cl}(E)\otimes\ensuremath{\mathbb{C}}\simeq\ensuremath{\hbox{\rm End}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\mathcal{S}})$. In fact the notion
of ${\rm Spin}^{\rm c}$-structure (in terms of principal bundle) on a Euclidean
bundle $E\to M$ is equivalent to the existence of an irreducible
complex spinor bundle over $E\to M$ \cite{Schroder}.
Since $E={\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Spin}^{\rm c}_{2m}}\ensuremath{\mathbb{R}}^{2m}$, the bundle $p^*\ensuremath{\mathcal{S}}(E)$
is isomorphic to \break
${\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Spin}^{\rm c}_{2m}}(\ensuremath{\mathbb{R}}^{2m}\oplus\Delta_{2m})$.
\begin{defi}
Let $\ensuremath{\hbox{\rm S-Thom}}(E): p^*\ensuremath{\mathcal{S}}(E)^+\to p^*\ensuremath{\mathcal{S}}(E)^-$ be the symbol
defined by
\begin{eqnarray*}
{\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Spin}^{\rm c}_{2m}}(\ensuremath{\mathbb{R}}^{2m}\oplus\Delta_{2m}^+)
&\longrightarrow&
{\rm P}_{{\rm Spin}^{\rm c}}(E)\times_{{\rm Spin}^{\rm c}_{2m}}(\ensuremath{\mathbb{R}}^{2m}\oplus\Delta_{2m}^-)\\
{} [p;v,w]&\longmapsto &[p,v,{\bf c}(v)w]\ .
\end{eqnarray*}
When $E$ is the tangent bundle of a manifold $M$, the symbol
$\ensuremath{\hbox{\rm S-Thom}}(E)$ is denoted by $\ensuremath{\hbox{\rm S-Thom}}(M)$. If a group $K$ acts equivariantly on
the ${\rm Spin}^{\rm c}$-stucture, we denote by $\ensuremath{\hbox{\rm S-Thom}}_{K}(E)$ the equivariant
symbol.
\end{defi}
The characteristic set of $\ensuremath{\hbox{\rm S-Thom}}(E)$ is $M\simeq\{{\rm zero\ section\
of}\ E\}$, hence it defines a class in $K(E)$ if $M$ is compact. When
$E=\ensuremath{\hbox{\bf T}} M$, the symbol $\ensuremath{\hbox{\rm S-Thom}}(M)$ corresponds to the {\em principal symbol}
of the ${\rm Spin}^{\rm c}$ Dirac operator associated to the ${\rm Spin}^{\rm c}$-structure
\cite{Duistermaat96}. When $M$ is compact, we define a quantization map
$\ensuremath{\mathcal{Q}}(M,-):K(M)\to\ensuremath{\mathbb{Z}}$ by the relation
$\ensuremath{\mathcal{Q}}(M,E):=\ensuremath{\hbox{\rm Index}}_{M}(\ensuremath{\hbox{\rm S-Thom}}(M)\otimes E)$ : $\ensuremath{\mathcal{Q}}(M,E)$ is the
index of the ${\rm Spin}^{\rm c}$ Dirac operator on $M$ twisted by $E$.
These notions extend to the orbifold case. Let $M$ be a manifold with
a locally free action of a compact Lie group $G$. The quotient
$\ensuremath{\mathcal{X}}:=M/G$ is an orbifold, a space with finite quotient
singularities. A ${\rm Spin}^{\rm c}$ structure on $\ensuremath{\mathcal{X}}$ is by definition a
$G$-equivariant ${\rm Spin}^{\rm c}$ structure on the bundle $\ensuremath{\hbox{\bf T}}_{G}M\to M$; where
$\ensuremath{\hbox{\bf T}}_{G}M$ is identified with the pullback of $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}}$ via the
quotient map $\pi:M\to\ensuremath{\mathcal{X}}$. We define in the same way
$\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{X}})\in K_{orb}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})$, such that $\pi^{*}\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{X}})=
\ensuremath{\hbox{\rm S-Thom}}_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$. The pullback by $\pi$ induces an isomophism $\pi^{*}:
K_{orb}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{X}})\simeq K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$. The quantization map
$\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{X}},-)$ is defined by : $\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{X}},\ensuremath{\mathcal{E}})=
\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{X}}}(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{X}})\otimes\ensuremath{\mathcal{E}})$.
\begin{lem}\label{lem.q.change}
Let $E\to M$ be an oriented $G$-bundle . Let $g_0,g_1$ be two
$G$-invariant metric on the fibers of $E$, and suppose that
$(E,g_0)$ admits an equivariant ${\rm Spin}^{\rm c}$-stucture denoted by
${\rm P}_{{\rm Spin}^{\rm c}}(E,g_0)$. The trivial homotopy
$g_t=(1-t).g_0 +t.g_1$ between the metrics, induces an equivariant
homotopy between the principal bundles ${\rm P}_{{\rm SO}}(E,g_0)$, ${\rm P}_{{\rm SO}}(E,g_1)$ which
can be lift to an equivariant homotopy between ${\rm P}_{{\rm Spin}^{\rm c}}(E,g_0)$ and a
${\rm Spin}^{\rm c}$-bundle over $(E,g_1)$. When the base $M$ is compact, the
corresponding symbols $\ensuremath{\hbox{\rm S-Thom}}_G(E,g_0)$ and $\ensuremath{\hbox{\rm S-Thom}}_G(E,g_1)$ define
the same class in $K_G(E)$.
\end{lem}
{\em Proof :} Let $\ensuremath{\mathcal{S}}$ be the irreducible complex spinor bundle
associated to ${\rm P}_{{\rm Spin}^{\rm c}}(E,g_0)$. We denote by
${\bf c}_0:{\rm Cl}(E,g_0)\to\ensuremath{\hbox{\rm End}}_{\ensuremath{\mathbb{C}}}(\ensuremath{\mathcal{S}})$ the
corresponding Clifford action. Let $A_t$ be the unique
$g_0$-symmetric endomorphism of $E$ such that
$g_t(v,w)=g_0(A_t(v),A_t(w))$. The composition ${\bf c}_0\circ A_t$ is
then a Clifford action of $(E,g_t)$ on $\ensuremath{\mathcal{S}}$. It defines a
${\rm Spin}^{\rm c}$-structure on the bundle $(E,g_t)$ which is homotopic to
${\rm P}_{{\rm Spin}^{\rm c}}(E,g_0)$. $\Box$
\medskip
Consider now the case of a {\em complex} vector bundle $E\to M$, of
complex rank $m$. The orientation on the fibers of $E$ is given by the
complex structure $J$.
Let ${\rm P}_{{\rm U}}(E)$ be the bundle of unitary frames on $E$. We have a
morphism ${\rm j}:{\rm U}_m\to{\rm Spin}^{\rm c}_{2m}$ which makes the
diagram\footnote{Here ${\rm i}:{\rm U}_m\ensuremath{\hookrightarrow}{\rm SO}_{2m}$ is the
canonical inclusion map.}
\begin{equation}
\xymatrix@C=2cm{
{\rm U}_m\ar[r]^{\rm j} \ar[dr]_{{\rm i}\times\det} &
{\rm Spin}^{\rm c}_{2m}\ar[d]^{\eta^{\rm c}}\\
& {\rm SO}_{2m}\times{\rm U}_1\ .}
\end{equation}
commutative \cite{Lawson-Michel}. Then
\begin{equation}\label{eq.J.spin}
{\rm P}_{{\rm Spin}^{\rm c}}(E):={\rm Spin}^{\rm c}_{2m}\times_{\rm j}{\rm P}_{{\rm U}}(E)
\end{equation}
defines a ${\rm Spin}^{\rm c}$-structure over $E$, with bundle of irreducible
spinors $\ensuremath{\mathcal{S}}(E)=\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}E$ and determinant line bundle
equal to $\det_{\ensuremath{\mathbb{C}}}E$.
\begin{rem} Let $M$ be a manifold equipped with an almost complex
structure $J$. The symbol $\ensuremath{\hbox{\rm S-Thom}}(M)$ defined by the ${\rm Spin}^{\rm c}$-structure
(\ref{eq.J.spin}), and the Thom symbol $\ensuremath{\hbox{\rm Thom}}(M,J)$ defined in section
\ref{sec.quantization} coincide.
\end{rem}
\bigskip
Consider our case of interest, where $M$ is a compact $G$-manifold
equipped with an equivariant almost complex structure $J$ and with an
abstract moment map $f_{_G}:M\to\ensuremath{\mathfrak{g}}^*$. Here we assume that $0$ is a
regular value of $f_{_G}$ : $\ensuremath{\mathcal{Z}}:=f_{_G}^{-1}(0)$ is a smooth
submanifold of $M$ with a locally free action of $G$. Let
$\ensuremath{\mathcal{M}}_{red}:= \ensuremath{\mathcal{Z}}/G$ be the corresponding `reduced' space, and
let $\pi:\ensuremath{\mathcal{Z}}\to \ensuremath{\mathcal{M}}_{red}$ be the projection map.
On $\ensuremath{\mathcal{Z}}$ we have an exact sequence
$0 \longrightarrow \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}} \longrightarrow
\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}} \stackrel{df_G}{\longrightarrow}
\ensuremath{\mathfrak{g}}^*\times\ensuremath{\mathcal{Z}} \longrightarrow 0$,
and $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}=\ensuremath{\hbox{\bf T}}_G\ensuremath{\mathcal{Z}}\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}$ where
$\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\simeq\ensuremath{\mathfrak{g}}\times\ensuremath{\mathcal{Z}}$
denotes the trivial bundle corresponding to the subspace of $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}$
formed by the vector field generated by the infinitesimal action of
$\ensuremath{\mathfrak{g}}$. So $\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}$ admits the decomposition
\begin{equation}\label{eq.TM.decompose}
\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\ensuremath{\hbox{\bf T}}_G\ensuremath{\mathcal{Z}}\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\oplus
\ensuremath{\mathfrak{g}}^*\times\ensuremath{\mathcal{Z}} \ .
\end{equation}
The bundle $\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})$ is identified with $\ensuremath{\hbox{\bf T}}_G\ensuremath{\mathcal{Z}}$.
Thus the decomposition (\ref{eq.TM.decompose}) can be rewritten
\begin{equation}\label{eq.TM.decompose.bis}
\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}} \ .
\end{equation}
with the convention $\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}=(\ensuremath{\mathfrak{g}}\otimes i\ensuremath{\mathbb{R}})\times\ensuremath{\mathcal{Z}}$ and
$\ensuremath{\mathfrak{g}}^*\times\ensuremath{\mathcal{Z}}=(\ensuremath{\mathfrak{g}}\otimes \ensuremath{\mathbb{R}})\times\ensuremath{\mathcal{Z}}$.
\begin{lem} \label{lem.spinc.induit}
The data $(J,f_{_G})$ induce :
\begin{itemize}
\item an orientation $o_{red}$ on $\ensuremath{\mathcal{M}}_{red}$,
\item a ${\rm Spin}^{\rm c}$-structure ${\rm Q}_{red}$ on $(\ensuremath{\mathcal{M}}_{red},o_{red})$.
\end{itemize}
Moreover, the irreducible complex spinor bundle
$\wedge^{\bullet}_{J}\ensuremath{\hbox{\bf T}} M$, when
restricted to $\ensuremath{\mathcal{Z}}$, defines a complex spinor bundle over
$\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ which is
homotopic to $\pi^{*}\ensuremath{\mathcal{S}}(\ensuremath{\mathcal{M}}_{red})\otimes
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$.
\end{lem}
{\em Proof} : Since $\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ is canonically oriented by
the complex multiplication by $i$, the orientation $o(J)$ on $M$
determines an orientation $o(\ensuremath{\mathcal{M}}_{red})$ on $\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{M}}_{red}$ such
that $o(J)= o(\ensuremath{\mathcal{M}}_{red})\,o(\imath)$.
Let $g_0$ be the Riemannian metric on $\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}$ equal to the
restriction to $\ensuremath{\mathcal{Z}}$ of the Riemannian metric on $M$ (which is taken
compatible with $J$). If ${\rm P}$ is the ${\rm Spin}^{\rm c}$-structure on $M$
determined by $J$ (see \ref{eq.J.spin}), the restriction ${\rm
P}\vert_{\ensuremath{\mathcal{Z}}}$ is then a ${\rm Spin}^{\rm c}$-structure on $(\ensuremath{\hbox{\bf T}}
M\vert_{\ensuremath{\mathcal{Z}}},g_0)$.Let $g_1$ be a $G$-invariant metric on the bundle
$\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}$ which makes (\ref{eq.TM.decompose.bis}) an
orthogonal sum, and which is constant on the the trivial bundle
$\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$. We know from Lemma \ref{lem.q.change} that
the ${\rm Spin}^{\rm c}$-structure ${\rm P}\vert_{\ensuremath{\mathcal{Z}}}$ on $(\ensuremath{\hbox{\bf T}}
M\vert_{\ensuremath{\mathcal{Z}}},g_0)$ is homotopic to ${\rm Spin}^{\rm c}$-structure ${\rm P}_1$ on
$(\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}},g_1)$ (both are $G$-equivariant).
The ${\rm SO}_{2k}\times{\rm U}_{l}$-principal bundle
${\rm P}_{{\rm SO}}(\pi^{*}(\ensuremath{\hbox{\bf T}} \ensuremath{\mathcal{M}}_{red}))\times{\rm P}_{{\rm U}}(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})$
is a reduction\footnote{Here $2n=\dim M$, $2k=\dim \ensuremath{\mathcal{M}}_{red}$ and
$l=\dim(\ensuremath{\mathfrak{g}})$, so $n=k+l$.} of the ${\rm SO}_{2n}$ principal bundle
${\rm P}_{{\rm SO}}(\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})$, thus
we have the commutative diagram
\begin{equation}\label{diagram.Q.seconde}
\xymatrix@C=2cm{
{\rm Q}\ar[r]\ar[d] &
{\rm P}_{{\rm SO}}(\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red}))\times{\rm P}_{{\rm U}}(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})
\times{\rm P}_{{\rm U}}(\ensuremath{\mathbb{L}}\vert_{\ensuremath{\mathcal{Z}}})\ar[d]\\
{\rm P}_1\ar[r] &
{\rm P}_{{\rm SO}}(\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})
\times{\rm P}_{{\rm U}}(\ensuremath{\mathbb{L}}\vert_{\ensuremath{\mathcal{Z}}})\ ,
}
\end{equation}
where $\ensuremath{\mathbb{L}}=\det_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\bf T}} M,J)$. Here ${\rm Q}$ is a
$(\eta^{\rm c})^{-1}({\rm SO}_{2k}\times{\rm U}_{l})
\simeq{\rm Spin}^{\rm c}_{2k}\times{\rm U}_{l}$-principal bundle.
Finally we see that
${\rm Q}_{red}={\rm Q}/({\rm U}_{l}\times G)$ is a ${\rm Spin}^{\rm c}$ structure
on $\ensuremath{\mathcal{M}}_{red}$ with determinant line bundle
$\ensuremath{\mathbb{L}}_{red}=\det_{\ensuremath{\mathbb{C}}}(\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}})/G$.
The irreducible complex spinor bundle $\wedge^{\bullet}_{J}\ensuremath{\hbox{\bf T}} M$,
when restricted to $\ensuremath{\mathcal{Z}}$, is homotopic to $\ensuremath{\mathcal{S}}'={\rm
P}_1\times_{{\rm Spin}^{\rm c}_{2n}}\Delta_{2m}$. Using (\ref{diagram.Q.seconde})
we get
\begin{eqnarray*}
\ensuremath{\mathcal{S}}' &=& {\rm Q}\times_{({\rm Spin}^{\rm c}_{2k}\times{\rm U}_l)}
\Big(\Delta_{2k}\otimes\wedge^{\bullet}\ensuremath{\mathbb{C}}^{l}\Big)\\
&=& \Big(({\rm Q}/{\rm U}_l)\times_{{\rm Spin}^{\rm c}_{2k}}\Delta_{2k}\Big)\otimes
\Big(({\rm Q}/{\rm Spin}^{\rm c}_{2k})\times_{{\rm U}_l}\wedge^{\bullet}\ensuremath{\mathbb{C}}^{l}\Big)\\
&=& \pi^{*}\ensuremath{\mathcal{S}}(\ensuremath{\mathcal{M}}_{red})\otimes
(\wedge^{\bullet}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}})\times\ensuremath{\mathcal{Z}} \ .
\end{eqnarray*}
Here we have used the identifications
${\rm Q}/{\rm Spin}^{\rm c}_{2k}={\rm P}_{{\rm U}}(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})$ and
${\rm P}_{{\rm U}}(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})\times_{{\rm U}_l}\wedge^{\bullet}\ensuremath{\mathbb{C}}^{l}=
(\wedge^{\bullet}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}})\times\ensuremath{\mathcal{Z}}$. $\Box$
\bigskip
We shall consider the particular case where $J$ defines an almost
complex structure on $\ensuremath{\mathcal{M}}_{red}$. It happens when the
following decomposition holds
\begin{equation}\label{eq.J.induit}
\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}\oplus J(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}})\ .
\end{equation}
With (\ref{eq.J.induit}), $\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}$ decomposes in
$\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus
\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\oplus J(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}})$ : let us denote by
$pr:\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}\to \pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})$ the corresponding
projection. Since
$\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\oplus J(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}})$ is invariant by $J$, the
endomorphism $J_{red}:=pr\circ J$ is a $G$-invariant almost complex
structure on $\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})$.
Using the identification $\ensuremath{\mathfrak{g}}\simeq\ensuremath{\mathfrak{g}}^*$, one considers the
endomorphism $\ensuremath{\mathcal{D}}$ of the trivial bundle $\ensuremath{\mathfrak{g}}\times\ensuremath{\mathcal{Z}}$ defined
by
\begin{equation}\label{eq.Dcal}
\ensuremath{\mathcal{D}}(X)=-df_{_G}(J(X_{\ensuremath{\mathcal{Z}}}))\ ,\quad {\rm for}\quad X\,\in\,\ensuremath{\mathfrak{g}}.
\end{equation}
Condition (\ref{eq.J.induit}) is then equivalent to :
$\det\ensuremath{\mathcal{D}}(z)\neq 0$ for all $z\in\ensuremath{\mathcal{Z}}$. We shall use the normalized
map $\ensuremath{\mathcal{D}}(\ensuremath{\mathcal{D}}^t\ensuremath{\mathcal{D}})^{-1/2}$ which is an orthogonal map for the
fixed Euclidean structure on $\ensuremath{\mathfrak{g}}$ (to simplify we keep the same
notation $\ensuremath{\mathcal{D}}$ for it). Let $J_{\ensuremath{\mathcal{D}}}$ be the complex structure
on the trivial bundle $\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ defined by the
following matrix
$$
J_{\ensuremath{\mathcal{D}}}:=\left(
\begin{array}{cc}
0 & -\ensuremath{\mathcal{D}} \\ \ensuremath{\mathcal{D}}^{-1} & 0
\end{array}
\right)\ .
$$
\begin{lem} \label{lem.J.induit}
Suppose that the decomposition
(\ref{eq.J.induit}) holds. On\footnote{Here we use the decompostion
(\ref{eq.TM.decompose.bis}) of $\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}$.} $\ensuremath{\hbox{\bf T}}
M\vert_{\ensuremath{\mathcal{Z}}}=$ \break
$\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ the almost complex
structure $J$ is homotopic to $J_{red}\times J_{\ensuremath{\mathcal{D}}}$. Hence the
irreducible complex spinor bundle $\wedge^{\bullet}_{J}\ensuremath{\hbox{\bf T}} M$,
when restricted to $\ensuremath{\mathcal{Z}}$, defines a complex spinor bundle over
$\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ which is
homotopic to \break $\pi^{*}(\wedge^{\bullet}_{J_{red}}\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})
\otimes \wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$.
\end{lem}
{\em Proof :} Trough the decomposition
$\ensuremath{\hbox{\bf T}} M\vert_{\ensuremath{\mathcal{Z}}}=\pi^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\oplus \ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\oplus
J(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}})$, the map $J$ is described by the matrix
$$
\left(
\begin{array}{cc}
J_{red} & 0 \\ A & \imath
\end{array}
\right)\ ,
$$
hence $J$ is homotopic to
$$
J'=\left(
\begin{array}{cc}
J_{red} & 0 \\ 0 & \imath
\end{array}
\right)\ .
$$
In the decomposition (\ref{eq.TM.decompose.bis}), $J'$ has the following
matrix
$$
\left(
\begin{array}{cc}
J_{red} & B \\ 0 & C
\end{array}
\right)\ ,
$$
with $C\in\ensuremath{\hbox{\rm End}}(\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})$ of the form
$$
\left(
\begin{array}{cc}
-\ensuremath{\mathcal{D}} b\ensuremath{\mathcal{D}}^{-1}& -\ensuremath{\mathcal{D}} \\ b^2\ensuremath{\mathcal{D}}^{-1} + \ensuremath{\mathcal{D}}^{-1} & b
\end{array}
\right)\ .
$$
Hence $J'$ is tied to $J_{red}\times J_{\ensuremath{\mathcal{D}}}$ through
the homotopies $t\to t\, B$ and $t\to t\, b$, $0\leq t\leq 1$.
$\Box$
\subsection{The map \protect $RR^{^{G}}_{0}$}\label{subsec.RR.G.O}
The map $RR^{^{G}}_{0}(M,-): K_{G}(M)\to R^{-\infty}(G)$ is
the Riemann-Roch character localized near $C^{^G}_{0}=
f_{_{G}}^{-1}(0)$ (see Definition \ref{def.RR.beta}). In particular,
$RR^{^{G}}_{0}(M,-)$ is the zero map if $0$ does not belong to $f_{_{G}}(M)$.
In this subsection, we assume that $0\in f_{_{G}}(M)$ is a regular value of
$f_{_{G}}$. We have proved in the past subsection that $J$ induces an
orientation $o(\ensuremath{\mathcal{M}}_{red})$ on the reduced space $\ensuremath{\mathcal{M}}_{red}$
together with a ${\rm Spin}^{\rm c}$-structure on $(\ensuremath{\mathcal{M}}_{red},o(\ensuremath{\mathcal{M}}_{red}))$.
Let $\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red})$ be the elliptic symbol defined by this
${\rm Spin}^{\rm c}$-structure and let $\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},-)$ be the corresponding
quantization map.
\begin{prop}\label{prop.RR.T.0}
For every $G$-equivariant vector bundle $E\to M$, we have
\begin{equation}\label{eq.RR.G.0}
RR_{0}^{^{G}}(M,E)=\sum_{\mu\in\Lambda^{*}_{+}}
\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},E_{red}\otimes\underline{V_{\mu}}^{*}) . V_{\mu}
\quad {\rm in}\quad R^{-\infty}(G) \ ,
\end{equation}
Here $E_{red}=E/G$ is the orbifold vector bundle on
$\ensuremath{\mathcal{M}}_{red}$ induced by $E$, and $\underline{V_{\mu}}=
\ensuremath{\mathcal{Z}}\times_{G}V_{\mu}$. In particular, the
$G$-invariant part of $RR_{0}^{^{G}}(M,E)$ is equal to \break
$\ensuremath{\mathcal{Q}}(\ensuremath{\mathcal{M}}_{red},E_{red})\in \ensuremath{\mathbb{Z}}$.
\end{prop}
Equality (\ref{eq.RR.G.0}) is
obtained by Vergne \cite{Vergne96}[Part II] in the case of a Hamiltonian
action of the circle group on a compact symplectic manifold.
\medskip
Suppose now that the decomposition (\ref{eq.J.induit}) holds. The
trivial bundle $\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ has two irreducible complex
spinor bundles $\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$ and
$\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$. Thus
\begin{equation}\label{eq.L.D}
\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}} =
\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}\otimes \pi^*L_{\ensuremath{\mathcal{D}}}
\end{equation}
where $\pi^*L_{\ensuremath{\mathcal{D}}}\to\ensuremath{\mathcal{Z}}$ is the line bundle equal to
$\ensuremath{\hbox{\rm Hom}}_{Cl_{\ensuremath{\mathbb{C}}}}(\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}},
\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})$ : at $z\in\ensuremath{\mathcal{Z}}$,
$\pi^*L_{\ensuremath{\mathcal{D}}}\vert_{z}$ is the complex vector space of linear
maps $\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\to\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}(z)}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$
commuting with the Clifford actions (see \cite{Schroder}). Note that
$\wedge^{\pm}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}} =
\wedge^{\pm}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}\otimes \pi^*L_{\ensuremath{\mathcal{D}}}$ if the
orientation of $J_{\ensuremath{\mathcal{D}}}$ coincide with those defined by $\imath$ (i.e.
$\det\ensuremath{\mathcal{D}}>0$). If $\det\ensuremath{\mathcal{D}}<0$, we have
$\wedge^{\pm}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}} =
\wedge^{\mp}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}\otimes \pi^*L_{\ensuremath{\mathcal{D}}}$.
\begin{prop}\label{prop.RR.T.0.bis}
Suppose that the decomposition (\ref{eq.J.induit}) holds, and let
\break
$RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},-)$ be the quantization map given by
$J_{red}$. For every $G$-equivariant vector bundle $E\to M$, we have
\begin{equation}\label{eq.RR.G.0.bis}
\left[RR_{0}^{^{G}}(M,E)\right]^G =\pm\,
RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},E_{red}\otimes L_{\ensuremath{\mathcal{D}}})\ ,
\end{equation}
where $\pm$ is the sign of $\det\ensuremath{\mathcal{D}}$.
\end{prop}
{\em Proof of Proposition \ref{prop.RR.T.0}} : Following Definition
\ref{def.RR.beta}, the map $RR_{0}^{^{G}}(M,-)$ is defined by
$\ensuremath{\hbox{\rm Thom}}_{G,[0]}^f(M)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,0}})$, where
$\ensuremath{\mathcal{U}}^{^{G,0}}$ is a (small) neighbourhood of $\ensuremath{\mathcal{Z}}$ in $M$. Since
$0$ is a regular value of $f_{_{G}}$, $\ensuremath{\mathcal{U}}^{^{G,0}}$ is
diffeomorphic to $\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}$, and the moment map is equal
to the projection $f:\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}\to \ensuremath{\mathfrak{g}}^{*}$ in a
neighbourhood of $\ensuremath{\mathcal{Z}}$ in $\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}$. We denote by
$\sigma_{\ensuremath{\mathcal{Z}}}\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}))$ the symbol
corresponding to $\ensuremath{\hbox{\rm Thom}}_{G,[0]}^f(M)$ through the diffeomorphism
$\ensuremath{\mathcal{U}}^{^{G,0}}\cong \ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}$. Let
$\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}}^{G}: K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times
\ensuremath{\mathfrak{g}}^{*}))\to R^{-\infty}(G)$ be the index map on $\ensuremath{\mathcal{Z}}\times
\ensuremath{\mathfrak{g}}^{*}$. The map $RR_{0}^{^{G}}(M,-)$ is defined by
$RR_{0}^{^{G}}(M,E)=\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}}^{G}
(\sigma_{\ensuremath{\mathcal{Z}}}\otimes f^{*}(E_{\vert\ensuremath{\mathcal{Z}}}))$.
Following Atiyah \cite{Atiyah.74}[Theorem 4.3], the inclusion map
$j:\ensuremath{\mathcal{Z}}\ensuremath{\hookrightarrow} \ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}$ induces an $R(G)$-module morphism
$j_{!}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}))$, with
the commutative diagram
\begin{equation}\label{j.point.bis}
\xymatrix@C=2cm{
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})\ar[r]^{j_{!}} \ar[dr]_{\ensuremath{\hbox{\rm Index}}^{G}_{\ensuremath{\mathcal{Z}}}} &
K_{G}(\ensuremath{\hbox{\bf T}}_{G}( \ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}))
\ar[d]^{\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}}^{G}}\\
& R^{-\infty}(G)}.
\end{equation}
More generally, the map $i_{!}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Y}})$ is
defined by Atiyah for any embedding $i:\ensuremath{\mathcal{Z}}\ensuremath{\hookrightarrow}\ensuremath{\mathcal{Y}}$ of $G$-manifolds
with $\ensuremath{\mathcal{Z}}$ compact.
Consider now the case where $i$ is the zero-section of a
$G$-vector bundle $\ensuremath{\mathcal{E}} \to \ensuremath{\mathcal{Z}}$.
In general the map $i_{!}$ is {\em not} an isomorphism.
If furthermore the $G$-action is {\em locally free} over $\ensuremath{\mathcal{Z}}$, then
$\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}}$, $\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{E}}$ are respectively
subbundles of $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}\to \ensuremath{\mathcal{Z}}$, $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{E}}\to \ensuremath{\mathcal{E}}$, and
the projection $\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{E}}\to\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}}$ is a vector bundle
isomorphic to $s^{*}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{E}})$ (where $s:\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}}\ensuremath{\hookrightarrow} \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}$ is
the inclusion). Hence the vector bundle $\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{E}}\to\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}}$
inherits a complex structure over the fibers (coming from the complex
vector bundle $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{E}}\to\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Z}}$). In this situation, the map
$i_{!}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{E}})$ is the
Thom isomorphism.
In the case of the (trivial) vector bundle $\ensuremath{\mathcal{Z}}\times
\ensuremath{\mathfrak{g}}^{*}\to\ensuremath{\mathcal{Z}}$, the map $j_{!}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})\to$
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^{*}))$ is then an {\em isomorphism}.
Take $\tilde{\sigma}_{\ensuremath{\mathcal{Z}}}=(j_{!})^{-1}(\sigma_{\ensuremath{\mathcal{Z}}})$, and from
the commutative diagram (\ref{j.point.bis}) we have
$RR_{0}^{^{G}}(M,E)= \ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Z}}}^{G}
\left(\tilde{\sigma}_{\ensuremath{\mathcal{Z}}}\otimes E\vert_{\ensuremath{\mathcal{Z}}}\right)$. From
Theorem \ref{thm.atiyah.1} we get $$
\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Z}}}^{G}(\tilde{\sigma_{\ensuremath{\mathcal{Z}}}}\otimes E\vert_{\ensuremath{\mathcal{Z}}})=
\sum_{\mu\in\Lambda^{*}_{+}}\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{M}}_{red}}(\sigma^{red}\otimes
E_{red} \otimes\underline{V_{\mu}}^{*}) . V_{\mu}\ , $$ where
$\sigma^{red}\in K_{orb}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})$ corresponds to
$\tilde{\sigma}_{\ensuremath{\mathcal{Z}}}=(j_{!})^{-1}(\sigma_{\ensuremath{\mathcal{Z}}})$ through the
isomorphism $\pi^{*}:K_{orb}(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{Z}})$.
Proposition \ref{prop.RR.T.0.bis} follows immediately from the
\begin{lem}\label{lem.RR.O.canonique}
We have
$$
j_{!}\circ(\pi)^{*}\Big(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red})\Big)=
\sigma_{\ensuremath{\mathcal{Z}}}
$$
in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^{*}))$.
\end{lem}
{\em Proof :} Let $\ensuremath{\mathcal{S}}(M)$ the irreducible spinor bundle defined by
the almost complex structure $J$. Let $\widetilde{J}$ be the almost
complex structure on $\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^*$, equal to $J$ on $\ensuremath{\mathcal{Z}}$,
and which is constant on the fibers of the projection $\ensuremath{\mathcal{Z}}\times
\ensuremath{\mathfrak{g}}\to \ensuremath{\mathcal{Z}}$. Since the almost complex structures $J$ and
$\widetilde{J}$ are {\em homotopic} near $\ensuremath{\mathcal{Z}}$, the complex
$\sigma_{\ensuremath{\mathcal{Z}}}$ can be defined on $\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}$ with
$\widetilde{J}$ : we take $\ensuremath{\mathcal{S}}(M)\vert_{\ensuremath{\mathcal{Z}}}\times \ensuremath{\mathfrak{g}}^*$ for
bundle of spinors over $\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^*$. Following
(\ref{eq.TM.decompose.bis}) and (\ref{eq.TM.decompose}), for
$(z,\xi)\in\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^*$ a vector $v\in
\ensuremath{\hbox{\bf T}}_{(z,\xi)}(\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^*)$ decomposes into $v=v_1 +X+\imath Y$,
where $v_1\in\pi^*(\ensuremath{\hbox{\bf T}} M_{\xi})$, and $X+\imath Y\in \ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}$. The
map $\sigma_{\ensuremath{\mathcal{Z}}}(z,\xi;v)$ acts on $\ensuremath{\mathcal{S}}(M)_z$ by the Clifford
action pushed by the vector field\footnote{The tangent vector
$\ensuremath{\mathcal{H}}^{^G}(z,\xi)\in\ensuremath{\mathfrak{g}}_{\ensuremath{\mathcal{Z}}}\vert_z$ is equal to
$\imath\xi\in\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$.} $\ensuremath{\mathcal{H}}^{^G}(z,\xi)=\imath\,\xi$ :
$$
\sigma_{\ensuremath{\mathcal{Z}}}(z,\xi;v)={\rm Cl}_z(v_1 +X+\imath \,(Y-\xi))\ .
$$
Using now Lemma \ref{lem.spinc.induit}, we see that $\sigma_{\ensuremath{\mathcal{Z}}}$ is homotopic to
the symbol $\sigma'_{\ensuremath{\mathcal{Z}}}$ which acts on the product
$(\pi^{*}\ensuremath{\mathcal{S}}(\ensuremath{\mathcal{M}}_{red})\otimes
\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})\times\ensuremath{\mathfrak{g}}^*$ by
$$
\sigma'_{\ensuremath{\mathcal{Z}}}(z,\xi;v)={\rm Cl}_z(v_1)\odot {\rm Cl}(X+\imath (Y-\xi))\ .
$$
Now we see that the map
${\rm Cl}_{z}(v_{1})\odot {\rm Cl}(X+\imath(Y-\xi))$ is homotopic, as a
$G$-transversally elliptic symbol, to
${\rm Cl}_{z}(v_{1})\odot {\rm Cl}(\xi+\imath X)$. The $K$-theory class of
this former symbol is equal to $(\pi)^{*}(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red}))\odot
k_!(\ensuremath{\mathbb{C}})$ (where $k:\{0\}\ensuremath{\hookrightarrow}\ensuremath{\mathfrak{g}}^*$) which is the symbol
map of $j_{!}\circ(\pi)^{*}\left(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red})\right)$
(see the construction of the map $j_{!}$ in \cite{Atiyah.74}[Lecture 4]).
We have shown that $j_{!}\circ(\pi)^{*}\left(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red})\right)
=\sigma_{\ensuremath{\mathcal{Z}}}$ in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times \ensuremath{\mathfrak{g}}^*))$.
$\Box$
\bigskip
{\em Proof of Proposition \ref{prop.RR.T.0.bis}} : Here the proof is
similar to the former proof but we use Lemma \ref{lem.J.induit}
instead of Lemma \ref{lem.spinc.induit}. One as to show that
$$
j_{!}\circ(\pi)^{*}\Big(\ensuremath{\hbox{\rm S-Thom}}(\ensuremath{\mathcal{M}}_{red})\otimes L_{\ensuremath{\mathcal{D}}}\Big)=\pm
\sigma_{\ensuremath{\mathcal{Z}}}
$$
in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}(\ensuremath{\mathcal{Z}}\times\ensuremath{\mathfrak{g}}^{*}))$, where $\pm$ is the sign of
$\det\ensuremath{\mathcal{D}}$. By Lemma \ref{lem.J.induit}, we see as before that
$\sigma_{\ensuremath{\mathcal{Z}}}$ is homotopic to the product
\begin{equation}\label{eq.clif.J.red}
{\rm Cl}_{z}(v_{1})\odot {\rm Cl}_{J_{\ensuremath{\mathcal{D}}}}(\xi+\imath X)
\end{equation}
acting on $(\wedge^{\bullet}_{J_{red}}\pi^*(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})
\otimes\wedge^{\bullet}_{J_{\ensuremath{\mathcal{D}}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}})\times\ensuremath{\mathfrak{g}}^*$.
Now we use the isomorphism of irreducible complex spinor bundles
(\ref{eq.L.D}) where we have two different orientations
$o(J_{\ensuremath{\mathcal{D}}})$ and $o(\imath)$ on $\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}$:
$o(J_{\ensuremath{\mathcal{D}}})=\pm o(\imath)$ where $\pm$ is the sign of $\det\ensuremath{\mathcal{D}}$.
Hence the transversally elliptic symbol (\ref{eq.clif.J.red}) is
equal to
$$
\pm\
{\rm Cl}_{z}(v_{1})\odot {\rm Cl}(\xi+\imath X)\odot {\rm Id}_{L_{\ensuremath{\mathcal{D}}}}
$$
acting on $(\wedge^{\bullet}_{J_{red}}\pi^*(\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{M}}_{red})
\otimes\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}_{\ensuremath{\mathbb{C}}}\times\ensuremath{\mathcal{Z}}\otimes
L_{\ensuremath{\mathcal{D}}})\times\ensuremath{\mathfrak{g}}^*$. $\Box$
\subsection{The map \protect $RR^{^{G}}_{\beta}$ when $G_{\beta}=G$}
\label{subsec.RR.G.beta}
When $\beta\in \ensuremath{\mathcal{B}}_{_{G}}-\{0\}$ is in the center of $\ensuremath{\mathfrak{g}}$,
the map $RR^{^G}_{\beta}(M,-)$ is the Riemann-Roch character localized near
$M^{\beta}\cap f_{_{G}}^{-1}(\beta)$. In this subsection we prove
that $[RR^{^G}_{\beta}(M,E)]^G=0$ if $E$ is a $f_{_{G}}$-strictly positive
complex vector bundle.
The almost complex structure $J$ and the abstract
moment map $f_{_{G}} : M\to \ensuremath{\mathfrak{g}}$ restrict on $M^{\beta}$ to an almost complex
structure $J_{\beta}$ and a abstract moment map $f_{_{G}}\vert_{M^{\beta}}$.
The set $M^{\beta}\cap f_{_{G}}^{-1}(\beta)=
(f_{_{G}}\vert_{M^{\beta}})^{-1}(\beta)$ is a component
of the critical set of $C^{f_{_{G}}\vert_{M^{\beta}}}$, and
we denote by $RR_{\beta}^{^{G}}(M^{\beta},-):K_{G}(M^{\beta})\to
R^{-\infty}(G)$ the Riemann-Roch character on $M^{\beta}$
localized near the component $(f_{_{G}}\vert_{M^{\beta}})^{-1}(\beta)$ (see
Definition \ref{def.RR.beta}).
Here we proceed as in section \ref{sec.loc.M.beta}.
Let $p:\ensuremath{\mathcal{N}}\to M^{\beta}$ be the normal bundle of $M^{\beta}$ in $M$.
The torus $\ensuremath{\mathbb{T}}_{\beta}\ensuremath{\hookrightarrow} G$ acts linearly on the fibers of the
complex vector bundle $\ensuremath{\mathcal{N}}$, thus we associate, as in Theorem
\ref{th.localisation.pt.fixe}, the polarized complex
$G$-vector bundles $\ensuremath{\mathcal{N}}^{+,\beta}$ and $(\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}$.
\begin{prop}\label{prop.loc.RR.G.beta}
For every $E\in K_{G}(M)$, we have the following equality in
\break $\widehat{R}(G)$ :
$$
RR_{\beta}^{^{G}}(M,E)=(-1)^{r_{\ensuremath{\mathcal{N}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR_{\beta}^{^{G}}(M^{\beta},E\vert_{M^{\beta}}\otimes\det
\ensuremath{\mathcal{N}}^{+,\beta}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}) \ ,
$$
where $r_{\ensuremath{\mathcal{N}}}$ is the locally constant function on $M^{\beta}$
equal to the complex rank of $\ensuremath{\mathcal{N}}^{+,\beta}$.
\end{prop}
Consider the $G\times\ensuremath{\mathbb{T}}_{\beta}$-Riemann-Roch character
$RR_{\beta}^{^{G\times{\rm T}_{\beta}}}(M^{\beta},-)$ localized near
$M^{\beta}\cap f_{_{G}}^{-1}(\beta)$. It
can be extended trivially to a map, still denoted by
$RR_{\beta}^{^{G\times{\rm T}_{\beta}}}(M^{\beta},-)$, from
$K_{G}(M^{\beta})\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ to
$R^{-\infty}(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$.
Following Definition \ref{wedge.V.inverse}
the element $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\in
K_{G\times \ensuremath{\mathbb{T}}_{\beta}}(M^{\beta})\simeq
K_{G}(M^{\beta})\otimes R(\ensuremath{\mathbb{T}}_{\beta})$ admits a polarized inverse
$\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\, \in\,
K_{G}(M^{\beta})\,\widehat{\otimes}\,R(\ensuremath{\mathbb{T}}_{\beta})$.
Finally the result of Proposition \ref{prop.loc.RR.G.beta} can be
written as the following equality in
$R^{-\infty}(G)\,\widehat{\otimes}\, R(\ensuremath{\mathbb{T}}_{\beta})$ :
\begin{equation}\label{eq.loc.RR.G.beta.simplifie}
RR^{^G}_{\beta}(M,E)=RR^{^{G\times{\rm T}_{\beta}}}_{\beta}
\left(M^{\beta},E\vert_{M^{\beta}}\otimes
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}\right)\ .
\end{equation}
\medskip
Consider the decomposition of $RR_{\beta}^{^G}(M,E)
=\sum_{\lambda}m_{\beta,\lambda}(E)\,\chi_{_{\lambda}}^{_G}$ in
irreducible characters $\chi_{_{\lambda}}^{_G},\
\lambda\in\Lambda_{+}^{*}$. Let $E$ be a $f_{_{G}}$-strictly positive
complex vector bundle over $M$, and let $\eta_{_{E,\beta}}> 0$ be the
constant defined in Definition \ref{eq.mu.positif}. If $\ensuremath{\mathcal{Z}}$ is a
connected component of $M^{\beta}$ which intersects
$f_{_{G}}^{-1}(\beta)$, every weight $a$ of the $\ensuremath{\mathbb{T}}_{\beta}$-action
on the fibers of the complex vector bundle
$E^{\stackrel{k}{\otimes}}\vert_{\ensuremath{\mathcal{Z}}}\otimes\det
\ensuremath{\mathcal{N}}^{+,\beta}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta})$ satisfy
$\langle a,\beta\rangle\geq k.\eta_{_{E,\beta}}$. Lemma
\ref{lem.multiplicites.tore} and Corollary
\ref{coro.multiplicites.tore}, applied to this situation, show that
\begin{equation}\label{eq.m.E.k}
m_{\beta,\lambda}(E^{\stackrel{k}{\otimes}})\neq 0 \ \Longrightarrow \
\langle \lambda,\beta\rangle \geq\, k.\eta_{_{E,\beta}}\ .
\end{equation}
In particular $[RR_{\beta}^{^G}(M,E)]^G=m_{\beta,0}(E)=0$, so we
have proved the
\begin{coro}\label{coro.G.beta.egal.G}
Let $E$ be a $f_{_{G}}$-{\em strictly positive} complex vector bundle
over $M$ (see Def. \ref{eq.mu.positif}). For any
$\beta\in\ensuremath{\mathcal{B}}_{_{G}}-\{0\}$, with $G_{\beta}=G$,
the $G$-invariant part of $RR_{\beta}^{^{G}}(M,E)$ is equal to $0$.
\end{coro}
\medskip
{\em Proof of Proposition \ref{prop.loc.RR.G.beta}} :
\medskip
Here we proceed
as in the proof of Theorem \ref{th.localisation.pt.fixe}. The almost
complex structure $J$ induces an almost complex structure $J_{\beta}$
on $M^{\beta}$ and a complex structure $J_{\ensuremath{\mathcal{N}}}$ on the fibers of
$\ensuremath{\mathcal{N}}$. The
$G\times\ensuremath{\mathbb{T}}_{\beta}$-vector bundle
$p:\ensuremath{\mathcal{N}}\to M^{\beta}$ is isomorphic
to $R\times_{U}N\to M^{\beta}=R/U$, where $R$ is the
$\ensuremath{\mathbb{T}}_{\beta}$-equivariant unitary frame of $(\ensuremath{\mathcal{N}},J_{\ensuremath{\mathcal{N}}})$
framed on $N$.
Let $\ensuremath{\mathcal{U}}^{^{G,\beta}}$ be a neighbourhood of $C^{^G}_{\beta}$
in $M$, and consider the $G$-transversally elliptic symbol
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,\beta}})$ introduced in
Definition \ref{def.thom.beta.f}. Here we choose $\ensuremath{\mathcal{U}}^{^{G,\beta}}$
diffeomorphic to an open subset of $\ensuremath{\mathcal{N}}$ of the form
$\ensuremath{\mathcal{V}}:=\{ n=(x,v)\in\ensuremath{\mathcal{N}},\ x\in \ensuremath{\mathcal{U}}\ {\rm and}\ \vert
v\vert<\ensuremath{\varepsilon}\}$, where $\ensuremath{\mathcal{U}}$ is a neighbourhood of
$(f_{_{G}}\vert_{M^{\beta}})^{-1}(\beta)$ in $M^{\beta}$.
The moment map $f_{_{G}}$, the vector field $\ensuremath{\mathcal{H}}^{^G}$, and
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M)$ are transported by this
diffeomorphism to $\ensuremath{\mathcal{V}}$ (we keep the same symbol for these
elements).
We define now the homogeneous vector field $\widetilde{\ensuremath{\mathcal{H}}}^{^G}$ on
$\ensuremath{\mathcal{N}}$ by
\begin{equation}\label{def.H.tilde}
\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}:=\Big(f_{_{G}}(p(n))\Big)_{\ensuremath{\mathcal{N}}}(n),
\ n\ \in\ \ensuremath{\mathcal{N}}\ .
\end{equation}
Using the isomorphism $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}\tilde{\to} p^{*}
(\ensuremath{\hbox{\bf T}} M^{\beta}\oplus \ensuremath{\mathcal{N}})$ (see (\ref{eq:trivialisation.T.N}))
the manifold $\ensuremath{\mathcal{N}}$ is endowed with the almost complex structure
$\widetilde{J}:=p^{*}(J_{\beta}\oplus J_{\ensuremath{\mathcal{N}}})$. With the data
$(\widetilde{J},\ \widetilde{\ensuremath{\mathcal{H}}}^{^G})$, we construct
the following $G$-transversally elliptic symbol over $\ensuremath{\mathcal{N}}$ :
\begin{equation}\label{Thom.G.f.modifie}
\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})(n,w):=
\ensuremath{\hbox{\rm Thom}}_{G}(\ensuremath{\mathcal{N}},\widetilde{J})(n,w-\widetilde{\ensuremath{\mathcal{H}}}^{^{G}}_{n}),\quad
{\rm for}\quad (n,w)\ \in\ \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{N}}\ .
\end{equation}
Let us now verify that
$$
\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M)=\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})
\quad {\rm in}\ K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}})\ .
$$
The invariance of the Thom class after the modification of the almost
complex structure is carried out in Lemma \ref{lem.J.modifie} : the class of
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M)$ is equal in $K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}})$ to the class of
the symbol
$$
\sigma_{1}(n,w):=\ensuremath{\hbox{\rm Thom}}_{G}(\ensuremath{\mathcal{N}},\widetilde{J})(n,w-\ensuremath{\mathcal{H}}^{^G}_{n}),\quad
(n,w)\ \in\ \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}}\ .
$$
Using now the family of vectors field
$\ensuremath{\mathcal{H}}_{t}^{^G}(n):=\Big(f_{_{G}}(x,t.v)\Big)_{\ensuremath{\mathcal{V}}}(n)$, $\, t\in[0,1]$,
$\, n=(x,v)\in \ensuremath{\mathcal{V}}$, we construct the homotopy
$$
\sigma_{t}(n,w):=\ensuremath{\hbox{\rm Thom}}_{H}(\ensuremath{\mathcal{N}},\widetilde{J})(n,w-\ensuremath{\mathcal{H}}_{t}^{^G}(n)),\quad
(n,w)\ \in\ \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}}
$$ of $G$-transversally elliptic symbol between
$\sigma_{1}$ and $\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})$ (one easily verifies that
$\ensuremath{\hbox{\rm Char}}(\sigma_{t})\cap\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}} = C^{^G}_{\beta}$ for every
$t\in[0,1]$). Finally, we have shown that
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})=\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M)$ in
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}})$, thus
$$
RR_{\beta}^{^G}(E)=\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{N}}}^{G}\left(\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})\otimes
p^{*}(E_{\vert M^{\beta}})\right)
$$
for every $E\in K_{G}(M)$.
\medskip
Now we proceed as follows. For every $(n,w)\in\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}}$, the Clifford
action \break
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})(n,w)=Cl_{n}(w-\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n})$
on $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{V}}$ is equal to
the exterior product
\begin{equation}\label{eq.produit.clifford}
Cl_{x}(w_{1}-[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{1})
\odot Cl_{x}(w_{2}-[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{2})
\end{equation}
acting on $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\hbox{\bf T}}_{x}M^{\beta}\otimes
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathcal{N}}\vert_{x}$, where
$x=p(n)$. Here $w\to w_{1},\ \ensuremath{\hbox{\bf T}}_{n}\ensuremath{\mathcal{V}}\to\ensuremath{\hbox{\bf T}}_{x}M^{\beta}$
is the tangent map $\ensuremath{\hbox{\bf T}} p\vert_{n}$, and $w\to w_{2}=[w]^{V},
\ T_{n}\ensuremath{\mathcal{V}}\to\ensuremath{\mathcal{N}}\vert_{x}$ is the `vertical' map. We see that
$[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{1}=\ensuremath{\mathcal{H}}^{^G}_{x}$ is the vector field on
$M^{\beta}$ generated by the moment map $f_{_{G}}\vert_{M^{\beta}}$
(see Definition \ref{def.H.vector}).
Suppose that the exterior product (\ref{eq.produit.clifford}) can be
modified in
\begin{equation}\label{eq.produit.clifford.bis}
Cl_{x}(w_{1}-\ensuremath{\mathcal{H}}^{^G}_{x})
\odot Cl_{x}(w_{2}-\beta_{\ensuremath{\mathcal{N}}}\vert_{n}),
\end{equation}
without changing the K-theoretic class. This will prove a modified version of
(\ref{eq:Thom.egalite}) in
$K_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}
(\ensuremath{\hbox{\bf T}}_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}(R\times N))$ :
\begin{equation}
\pi^{*}_{N}\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})=
\pi^{*}\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M^{\beta})\odot
\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N)\ ,
\label{eq:Thom.egalite.bis}
\end{equation}
where $\pi_{N}:R\times N\to R \times_{U}N=\ensuremath{\mathcal{N}}$,
$\pi:R\to R/U= M^{\beta}$ are the quotient maps relative
to the free $U$-action, and $\odot$ is the product
\begin{equation}\label{eq:G.T.beta.U.bis}
K_{G\times U}(\ensuremath{\hbox{\bf T}}_{G\times U}R)
\times
K_{\ensuremath{\mathbb{T}}_{\beta}\times U}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}}N)
\longrightarrow
K_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}
(\ensuremath{\hbox{\bf T}}_{G\times\ensuremath{\mathbb{T}}_{\beta}\times U}(R\times N)) .
\end{equation}
The symbols $\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})$,
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(M^{\beta})$ and $\ensuremath{\hbox{\rm Thom}}^{\beta}_{\ensuremath{\mathbb{T}}_{\beta}\times U}(N)$
belong respectively to
$K_{G\times\ensuremath{\mathbb{T}}_{\beta}}
(\ensuremath{\hbox{\bf T}}_{G\times\ensuremath{\mathbb{T}}_{\beta}}(R\times_{U} N))$,
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}(R/U))$, and
$K_{\ensuremath{\mathbb{T}}_{\beta}\times U}(\ensuremath{\hbox{\bf T}}_{\ensuremath{\mathbb{T}}_{\beta}\times U}N)$. The Proposition
\ref{prop.loc.RR.G.beta} follows after taking the index,
and the $U$-invariants, in (\ref{eq:Thom.egalite.bis}).
\medskip
Finally we explain why the change of $[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{2}$
in $\beta_{\ensuremath{\mathcal{N}}}\vert_{n}$ can be done in
(\ref{eq.produit.clifford}) without changing the class of
$\ensuremath{\hbox{\rm Thom}}^f_{G,[\beta]}(\ensuremath{\mathcal{N}})$.
\medskip
Let $\mu^{\ensuremath{\mathcal{N}}}:\ensuremath{\mathfrak{g}}\to \Gamma(M^{\beta},\ensuremath{\hbox{\rm End}}(\ensuremath{\mathcal{N}}))$ be the `moment'
relative to the choice of a connection on $\ensuremath{\mathcal{N}}\to M^{\beta}$
(see Definition 7.5 in \cite{B-G-V}). Then, for every $X\in\ensuremath{\mathfrak{g}}$ we have
$$
[X_{\ensuremath{\mathcal{N}}}(x,v)]^{V}=-\mu^{\ensuremath{\mathcal{N}}}(X)\vert_{x}.v,\quad
(x,v)\ \in\ \ensuremath{\mathcal{N}}
$$
(see Proposition 7.6 in \cite{B-G-V}). When $X=\beta$, the vector
field $\beta_{\ensuremath{\mathcal{N}}}$ is vertical, hence we have $\mu^{\ensuremath{\mathcal{N}}}(\beta)\vert_{x}.v
=\ensuremath{\mathcal{L}}^{\ensuremath{\mathcal{N}}}(\beta)\vert_{x}.v=-\beta_{\ensuremath{\mathcal{N}}}(x,v)$,
where $\ensuremath{\mathcal{L}}^{\ensuremath{\mathcal{N}}}(\beta)$ is the infinitesimal action of $\beta$
on the fiber of $\ensuremath{\mathcal{N}}\to M^{\beta}$. We have also
$[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{2}=-\mu^{\ensuremath{\mathcal{N}}}(f_{_{G}}(x))\vert_{x}.v$,
for every $n=(x,v)\in\ensuremath{\mathcal{N}}$.
Note that the quadratic form
$v\in\ensuremath{\mathcal{N}}_{x}\to\vert\ensuremath{\mathcal{L}}^{\ensuremath{\mathcal{N}}}(\beta)\vert_{x}.v\vert^{2}$
is positive definite for $x\in M^{\beta}$. Hence, for every
$X\in\ensuremath{\mathfrak{g}}$ close enough to $\beta$, the quadratic form
$v\in\ensuremath{\mathcal{N}}_{x}\to(\mu^{\ensuremath{\mathcal{N}}}(\beta)\vert_{x}.v,
\mu^{\ensuremath{\mathcal{N}}}(X)\vert_{x}.v)$
is positive definite for $x\in M^{\beta}$.
Consider now the homotopy
$$
\sigma^{t}(n,w):= Cl_{x}(w_{1}-\ensuremath{\mathcal{H}}^{^G}_{x})
\odot Cl_{x}(w_{2}-t.[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{2}
-(1-t).\beta_{\ensuremath{\mathcal{N}}}\vert_{n}),\quad (n,v)\in\ensuremath{\mathcal{V}}\quad t\in[0,1].
$$
We see that $(n,w)\in\ensuremath{\hbox{\rm Char}}(\sigma^{t})\cap\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}}$ if and only if
\noindent i) $w_{1}=\ensuremath{\mathcal{H}}^{^G}_{x}$, $w_{2}=t[\widetilde{\ensuremath{\mathcal{H}}}^{^G}_{n}]_{2}
+(1-t)\beta_{\ensuremath{\mathcal{N}}}(n)$, and
\noindent ii) $(w_{1},X_{M^{\beta}}(x))
+(w_{2},[X_{\ensuremath{\mathcal{N}}}(x,v)]^{V})=0$ for all $X\in\ensuremath{\mathfrak{g}}$.
Take now $X=f_{_{G}}(x)$ in ii). Using i), we get
\begin{equation}\label{eq.somme.positive}
\left| \ensuremath{\mathcal{H}}^{^G}_{x}\right|^{2}+
t.\left| \mu^{\ensuremath{\mathcal{N}}}(f_{_{G}}(x))\vert_{x}.v\right|^{2}+
(1-t).\Sigma(x,v)= 0\ ,
\end{equation}
with $\Sigma(x,v):=
(\mu^{\ensuremath{\mathcal{N}}}(\beta)\vert_{x}.v,\mu^{\ensuremath{\mathcal{N}}}(f_{_{G}}(x))\vert_{x}.v)$.
If $x\in M^{\beta}$ is sufficiently close to
$(f_{_{G}}\vert_{M^{\beta}})^{-1}(\beta)$ , the term $\Sigma(x,v)$ is
positive for all $v\in\ensuremath{\mathcal{N}}_{x}$.
In this case, (\ref{eq.somme.positive}) gives
$\ensuremath{\mathcal{H}}^{^G}_{x}=0$ and $\Sigma(x,v)=0$, which insures
that $x\in C^{^G}_{\beta}$ and $v=0$.
We have proved that $\ensuremath{\hbox{\rm Char}}(\sigma^{t})\cap\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{V}}=C^{^G}_{\beta}$
for every $t\in [0,1]$ if $\ensuremath{\mathcal{V}}$ is `small' enough.
Hence $\sigma^{t}$ is an homotopy of $G$-transversally
elliptic symbols over $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{V}}$ between the exterior products
(\ref{eq.produit.clifford}) and (\ref{eq.produit.clifford.bis}).
$\Box$
\subsection{Induction formula}\label{subsec.induction.G.H}
This section is concerned by an induction formula which compare
the map $RR_{\beta}^{^{G}}(M,-)$ with the similar
localized Riemann-Roch characters defined for the maximal torus, and
the stabilizer $G_{\beta}$.
The idea of this induction comes from a previous paper of the author
\cite{pep2} where a similar induction formula in the context of
equivariant cohomology was proved.
\medskip
Consider the restriction $f_{_{H}}:M\to \ensuremath{\mathfrak{h}}$ of the moment map
$f_{_{G}}$ to the maximal torus $H$. In this situation we use the
vector field $\ensuremath{\mathcal{H}}^{^{H}}\vert_{m}=
f_{_{H}}(m)_{M}\vert_{m}, m\in M$ to decompose the map
$RR^{^{H}}(M,-):K_{H}(M)\to R(H)$ near the set
$C^{f_{_{H}}}=\{\ensuremath{\mathcal{H}}^{^{H}}=0\}$. From Lemma \ref{lem.C.f.G}
there exists a finite subset $\ensuremath{\mathcal{B}}_{_{H}}\subset \ensuremath{\mathfrak{h}}$, such that
$C^{f_{_{H}}}=\bigcup_{\beta\in\ensuremath{\mathcal{B}}_{_{H}}}C^{^{H}}_{\beta}$, with
$C^{^{H}}_{\beta}= M^{\beta}\cap f_{_{H}}^{-1}(\beta)$.
As in Definition \ref{def.RR.beta}, we define for
every $\beta\in \ensuremath{\mathcal{B}}_{_{H}}$, the map $RR_{\beta}^{^{H}}(M,-):
K_{H}(M)\to R^{-\infty}(H)$ which is
the Riemann-Roch character localized near $C^{^{H}}_{\beta}$.
Let $W$ be the Weyl group of $(G,H)$. Note that $\ensuremath{\mathcal{B}}_{_{H}}$ is a
$W$-stable subset of $\ensuremath{\mathfrak{h}}$, and that $\ensuremath{\mathcal{B}}_{_{G}}\subset\ensuremath{\mathcal{B}}_{_{H}}
\cap\ensuremath{\mathfrak{h}}_{+}$.
\begin{theo}\label{th.induction.G.H}
We have, for every $\beta\in\ensuremath{\mathcal{B}}_{_{G}}$, the following induction
formula between $RR_{\beta}^{^{G}}(M,-)$ and $RR_{\beta}^{^{H}}(M,-)$.
For every $E\in K_{G}(M)$, we have\footnote{See Equations
(\ref{eq.holomorphe.G.H}) and (\ref{eq:Hol-G-beta}) in Appendix B
for the definition of the holomorphic induction maps
${\rm Hol}^{^G}_{_H}$ and ${\rm Hol}^{^G}_{_{G_{\beta}}}$.}
\begin{eqnarray*}
RR_{\beta}^{^{G}}(M,E)&=&\frac{1}{\vert W_{\beta}\vert}{\rm Hol}^{^G}_{_H}\left(
RR_{\beta}^{^{H}}(M,E)\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}\right)\\
&=&\frac{1}{\vert W_{\beta}\vert}\sum_{w\in W}{\rm Hol}^{^G}_{_H}\left(
w.RR_{\beta}^{^{H}}(M,E)\right)\\
&=& \sum_{\beta'\in W.\beta}{\rm Hol}^{^G}_{_H}\left(RR_{\beta'}^{^{H}}(M,E)\right)
\end{eqnarray*}
where $W_{\beta}$ is the stabilizer of $\beta$ in $W$.
\end{theo}
We can use the previous induction formula between $G$ and $H$ index
maps to produce an induction formula between $G$ and $G_{\beta}$ index
maps. Consider the restriction $f_{_{G_{\beta}}}:M\to\ensuremath{\mathfrak{g}}_{\beta}$
of the moment map to the stabiliser $G_{\beta}$ of $\beta$ in $G$.
Let $RR_{\beta}^{^{G_{\beta}}}(M,-)$ be
the Riemann-Roch character localized near $C^{^{G_{\beta}}}_{\beta}=
M^{\beta}\cap f_{_{G}}^{-1}(\beta)$\footnote{Note that
$M^{\beta}\cap f_{_{G_{\beta}}}^{-1}(\beta)=
M^{\beta}\cap f_{_{G}}^{-1}(\beta)$ because $f_{_{G_{\beta}}}=
f_{_{G}}$ on $M^{\beta}$.}.
\begin{coro}\label{coro.induction.G.G.beta}
For every $\beta\in\ensuremath{\mathcal{B}}_{_{G}}$ and every $E\in K_{G}(M)$, we have
$$
RR_{\beta}^{^{G}}(M,E)={\rm Hol}^{^G}_{_{G_{\beta}}}\left(
RR_{\beta}^{^{G_{\beta}}}(M,E)\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}}\right)
\quad{\rm in}\quad R^{-\infty}(G)\ .
$$
\end{coro}
{\em Proof of the Corollary }: It comes immediately by applying the
induction formula of Theorem \ref{th.induction.G.H} to the couples
$(G,H)$ and $(G_{\beta},H)$.
\begin{coro}\label{coro.multiplicites.beta}
Let $E$ be a $f_{_{G}}$-{\em strictly positive} complex vector bundle over
$M$ (see Def. \ref{eq.mu.positif}). We have
$[RR_{\beta}^{^{G}}(M,E^{\stackrel{k}{\otimes}})]^{G}=0$,
if $k.\eta_{_{E,\beta}}>\langle \theta,\beta\rangle$. Here
$\theta=\sum_{\alpha>0}\alpha$ is the sum of the
positive roots of $G$, and $\eta_{_{E,\beta}}$ is the strictly
positive constant defined in Definition \ref{eq.mu.positif}.
\end{coro}
\medskip
{\em Proof of Corollary \ref{coro.multiplicites.beta}} :
Let us first write the decomposition\footnote{We choose a set
$\Lambda_{+,\beta}^{*}$ of dominant weight for $G_{\beta}$ that
contains the set $\Lambda_{+}^{*}$ of
dominant weight for $G$.}
$RR_{\beta}^{^{G_{\beta}}}(M,E^{\stackrel{k}{\otimes}})=
\sum_{\lambda\in\Lambda_{\beta}^{+}}
m_{\lambda,\beta}(E^{\stackrel{k}{\otimes}})
\chi_{_{\lambda}}^{^{G_{\beta}}}$,
in irreducible character of $G_{\beta}$.
We know from (\ref{eq.m.E.k}) that
$m_{\lambda,\beta}(E^{\stackrel{k}{\otimes}})\neq 0 \
\Longrightarrow \
\langle \lambda,\beta\rangle \geq k.\eta_{_{E,\beta}}$.
Each irreducible character
$\chi_{_{\lambda}}^{^{G_{\beta}}}$ is equal to ${\rm Hol}^{^{G_{\beta}}}_{_H}(h^{\lambda})$,
so from Corollary \ref{coro.induction.G.G.beta} we have
$RR_{\beta}^{^{G}}(M,E^{\stackrel{k}{\otimes}})=
{\rm Hol}^{^G}_{_H}\Big((\sum_{\lambda}m_{\lambda,\beta}(E^{\stackrel{k}{\otimes}})\,
h^{\lambda})\Pi_{\alpha\in\Delta(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta})}(1-h^{-\alpha})\Big)$
where $\Delta(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta})$ is the set of $H$-weight on
$\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}$\footnote{The complex structure on
$\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}$ is defined by $\beta$, so that
$\langle\alpha,\beta\rangle>0$ for all $\alpha\in
\Delta(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta})$.}. Finally , we see that
$RR_{\beta}^{^{G}}(M,E^{\stackrel{k}{\otimes}})$
is a sum of terms of the form
$m_{\lambda,\beta}(E^{\stackrel{k}{\otimes}})\,
{\rm Hol}^{^G}_{_H}(h^{\lambda-\alpha_{I}})$ where $\alpha_{I}=\sum_{\alpha\in
I}\alpha$ and $I$ is a subset of $\Delta(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta})$.
We know from Appendix B that
${\rm Hol}^{^G}_{_H}(h^{\lambda'})$ is either $0$ or the character of an
irreducible representation; in particular ${\rm Hol}^{^G}_{_H}(h^{\lambda'})$ is
equal to $\pm 1$ only if
$\langle\lambda',X\rangle\leq 0$ for every $X\in\ensuremath{\mathfrak{h}}_{+}$
(see Remark \ref{hol.egale.1}). So
$[RR_{\beta}^{^{G}}(M,E^{\stackrel{k}{\otimes}})]^G\neq 0$ only if there
exists a weight $\lambda$ such that
$m_{\lambda,\beta}(E^{\stackrel{k}{\otimes}})\neq 0$ and
${\rm Hol}^{^G}_{_H}(h^{\lambda-\alpha_{I}})=\pm 1$. The first condition imposes
$\langle \lambda,\beta\rangle \geq k.\eta_{_{E,\beta}}$ and the second gives
$\langle \lambda,\beta\rangle \leq \langle \alpha_{I},\beta\rangle$,
and combining the two we end with $k.\eta_{_{E,\beta}}\leq \langle
\alpha_{I},\beta\rangle\leq
\sum_{\alpha\in \Delta(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta})}\langle\alpha,\beta\rangle
=\langle \theta,\beta\rangle$. We have proved that
$[RR_{\beta}^{^{G}}(M,E^{\stackrel{k}{\otimes}})]^G = 0$
if $k.\eta_{_{E,\beta}} >\langle \theta,\beta\rangle$. $\Box$
\medskip
{\em Proof of Theorem \ref{th.induction.G.H}} :
\medskip
The first two equalities of the Theorem can be deduced from the third
one, that is $RR_{\beta}^{^{G}}(M,E)= \sum_{\beta'\in
W.\beta}{\rm Hol}^{^G}_{_H}\left(RR_{\beta'}^{^{H}}(M,E)\right)$. First, it is easy
to see that \break $RR_{w.\beta}^{^{H}}(M,E)=w.RR_{\beta}^{^{H}}(M,E)$
for every $w\in W$ and $\beta\in \ensuremath{\mathcal{B}}_{_{H}}$. After, the relation
${\rm Hol}^{^G}_{_H}(\phi\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}})=\sum_{w\in
W} {\rm Hol}^{^G}_{_H}(w.\phi)$, which is true for every $\phi\in R^{-\infty}(H)$
(see Remark \ref{wedge.C-wedge.R}), gives the first equality of the
Theorem.
The map $RR_{\beta}^{^{G}}(M,-)$ is defined
through the symbol
$\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,\beta}})$ where
$i^{^{G,\beta}}:\ensuremath{\mathcal{U}}^{^{G,\beta}}\to M$ is any $G$-invariant neighbourhood
of $C^{^{G}}_{\beta}$ such that $\overline{\ensuremath{\mathcal{U}}^{^{G,\beta}}}\cap
C^{f_{_{G}}}=C^{^{G}}_{\beta}$ (see Definition \ref{def.thom.beta.f}).
We define in the same way the localized Thom complex
$\ensuremath{\hbox{\rm Thom}}_{H,[\beta]}^f(M)\in K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}}^{^{H,\beta}})$.
For notational convenience, we will note in the same way the direct image of
$\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)$ (resp. $\ensuremath{\hbox{\rm Thom}}_{H,[\beta]}^f(M)$) in
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ (resp. $K_{H}(\ensuremath{\hbox{\bf T}}_{H}M)$) via
$i^{^{G,\beta}}_{*}: K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,\beta}})\to
K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)$ (resp. $i^{^{H,\beta}}_{*}: K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}}^{^{H,\beta}})\to
K_{H}(\ensuremath{\hbox{\bf T}}_{H}M)$).
Then we have $RR_{\beta}^{^{G}}(M,E)=\ensuremath{\hbox{\rm Index}}^{G}_{M}(\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)
\otimes E)$ for $E\in K_{G}(M)$. The Weyl group acts on $K_{H}(\ensuremath{\hbox{\bf T}}_{H}M)$
and we remark that $w.\ensuremath{\hbox{\rm Thom}}_{H,[\beta]}^f(M)=\ensuremath{\hbox{\rm Thom}}_{H,[w.\beta]}^f(M)$
for every $\beta\in \ensuremath{\mathcal{B}}_{_{H}}$, and $w\in W$. After taking the
index we see that $RR_{w.\beta}^{^{H}}(M,E)=w.RR_{\beta}^{^{H}}(M,E)$
for every $G$-vector bundle $E$.
Consider the map $r^{\gamma}_{_{G,H}}:
K_{G}(\ensuremath{\hbox{\bf T}}_{G}M)\to K_{H}(\ensuremath{\hbox{\bf T}}_{H}M)$ defined with $\gamma\in\ensuremath{\mathfrak{h}}$ in the
interior of the Weyl chamber, so that $G_{\gamma}=H$
(see subsection \ref{subsec.reduction}). The third equality of the Theorem
is an immediate consequence of the next Lemma.
\begin{lem}\label{lem.rest.thom.beta.f}
We have
$$
r^{\gamma}_{_{G,H}}\left(\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M)\right)=
\sum_{\beta'\in W.\beta}\ensuremath{\hbox{\rm Thom}}_{H,[\beta']}^f(M)\otimes
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}
\quad {\rm in}\quad K_{H}(\ensuremath{\hbox{\bf T}}_{H}M)\ .
$$
\end{lem}
\medskip
{\em Proof of Lemma \ref{lem.rest.thom.beta.f} :}
Consider a
$G$-invariant open neighbourhood $\ensuremath{\mathcal{U}}^{^{G,\beta}}$ of $C^{^{G}}_{\beta}$
such that $\overline{\ensuremath{\mathcal{U}}^{^{G,\beta}}}\cap C^{f_{_{G}}}=
C^{^{G}}_{\beta}$. We know from Proposition \ref{prop.restriction.bis}
that the class $r^{\gamma}_{_{G,H}}(\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^f(M))$ is
represented by the restriction to $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}^{^{G,\beta}}$ of the symbol
$$
\sigma_{I}(m,v)=Cl_{m}(v-\ensuremath{\mathcal{H}}^{^{G}}_{m})\odot Cl(\mu_{_{G/H}}(v)),\quad
\quad (m,v)\in \ensuremath{\hbox{\bf T}} M\ .
$$
Here $\mu_{_{G/H}}:\ensuremath{\hbox{\bf T}} M\to \ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ is the $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ part of
the Hamiltonian moment map $\mu_{_{G}}:\ensuremath{\hbox{\bf T}} M\to \ensuremath{\mathfrak{g}}$. Let $f_{_{G/H}}:
M\to\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ (resp. $f_{_{H}}:M\to\ensuremath{\mathfrak{h}}$) be
the $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$-part (resp. the $\ensuremath{\mathfrak{h}}$-part) of the moment
map $f_{_G}$. We will use in our proof the relation
\begin{equation}\label{relation.p.s}
(\mu_{_{G/H}}(\ensuremath{\mathcal{H}}^{^{G}}), f_{_{G/H}})_{_{\ensuremath{\mathfrak{g}}}}=
||\ensuremath{\mathcal{H}}^{^{G}}||^{2}_{_{M}}-(\ensuremath{\mathcal{H}}^{^{G}},\ensuremath{\mathcal{H}}^{^{H}})_{_{M}}\ .
\end{equation}
Consider the family of $H$-equivariant symbols
$\sigma_{\theta},\ \theta\in [0,1]$
defined on $\ensuremath{\hbox{\bf T}} M$ by
$$
\sigma_{\theta}(m,v)=Cl_{m}(v-\ensuremath{\mathcal{H}}^{^{G}}_{m})\odot
Cl\left(\theta\mu_{_{G/H}}(v)+(1-\theta)f_{_{G/H}}(m)\right),\quad
\quad (m,v)\in \ensuremath{\hbox{\bf T}} M\ .
$$
We see that $(m,v)\in\ensuremath{\hbox{\rm Char}}(\sigma_{\theta}) \Longleftrightarrow
v=\ensuremath{\mathcal{H}}^{^{G}}_{m}\ {\rm and}\
\theta\mu_{_{G/H}}(\ensuremath{\mathcal{H}}^{^{G}}_{m})+(1-\theta)f_{_{G/H}}(m)=0$.
Combining (\ref{relation.p.s}) with the fact that the vector field
$\ensuremath{\mathcal{H}}^{^{H}}$ belongs to the $H$-orbits, we see that
$\ensuremath{\hbox{\rm Char}}(\sigma_{\theta})\cap\ensuremath{\hbox{\bf T}}_{H}M\subset \{\ensuremath{\mathcal{H}}^{^G}=0\}$, for every
$\theta\in [0,1]$. By this way we have proved that
$\sigma_{I}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$ is homotopic to the
$H$-transversally elliptic symbol
$\sigma_{II}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$ where
$$
\sigma_{II}(m,v)=Cl_{m}(v-\ensuremath{\mathcal{H}}^{^{G}}_{m})\odot
Cl(f_{_{G/H}}(m)),\quad \quad (m,v)\in \ensuremath{\hbox{\bf T}} M\ .
$$
We transform now $\sigma_{II}$ via the following homotopy of
$H$-transversally elliptic symbols
$$
\sigma^u(m,v):=Cl_{m}(v-\ensuremath{\mathcal{H}}^{^{H}}_{m}- u.\ensuremath{\mathcal{H}}^{^{G/H}}_{m} )
\odot Cl(f_{_{G/H}}(m)),\quad (m,v)\in \ensuremath{\hbox{\bf T}} M\ ,
$$
for $u\in [0,1]$. Here $\ensuremath{\hbox{\rm Char}}(\sigma^u)\cap\ensuremath{\hbox{\bf T}}_{H}M=
\{\ensuremath{\mathcal{H}}^{^G}=0\}\cap\{ f_{_{G/H}}=0\}$ for all $u\in [0,1]$, hence
$\sigma_{II}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$ is homotopic to the
$H$-transversally elliptic symbol
$\sigma_{III}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$
where
$$
\sigma_{III}(m,v)=Cl_{m}(v-\ensuremath{\mathcal{H}}^{^H}_{m})\odot
Cl(f_{_{G/H}}(m)),\quad
\quad (m,v)\in \ensuremath{\hbox{\bf T}} M\ .
$$
At this stage we have proved that
$\sigma_{I}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}=
\sigma_{III}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}$ in
$K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}}^{^{G,\beta}})$. Note that
\begin{eqnarray*}
\ensuremath{\hbox{\rm Char}}(\sigma_{III}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}})\cap\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}}^{^{G,\beta}}
&=& G.(M^{\beta}\cap f_{_{G}}^{-1}(\beta))\bigcap \{ f_{_{G/H}}=0\}\\
&=& W.(M^{\beta}\cap f_{_{G}}^{-1}(\beta))\ ,
\end{eqnarray*}
because $G.\beta\cap \ensuremath{\mathfrak{h}}= W.\beta$. Let
$i: \ensuremath{\mathcal{U}}^{^{G,\beta}}\ensuremath{\hookrightarrow}\ensuremath{\mathcal{U}}$ be a $H$-invariant neighbourhood
of $W.(M^{\beta}\cap f_{_{H}}^{-1}(\beta))$
such that $\overline{\ensuremath{\mathcal{U}}}\cap \{\ensuremath{\mathcal{H}}^{^H}=0\}=
W.(M^{\beta}\cap f_{_{H}}^{-1}(\beta))$. The
symbol $\sigma_{III}\vert_{\ensuremath{\mathcal{U}}}$ is $H$-transversally elliptic and
\begin{equation}\label{eq.U.G.beta.U}
i_{*}(\sigma_{III}\vert_{\ensuremath{\mathcal{U}}})=
\sigma_{III}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}=\sigma_{I}\vert_{\ensuremath{\mathcal{U}}^{^{G,\beta}}}
\quad {\rm in}
\quad K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}}^{^{G,\beta}})\ .
\end{equation}
As in the proof of Proposition
\ref{prop.localisation}, (\ref{eq.U.G.beta.U}) is an
immediate consequence of the excision property.
The symbol
$(m,v) \to Cl_{m}(v-\ensuremath{\mathcal{H}}^{^{H}}_{m})$ is $H$-transversally
elliptic on $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}$, and equal (by definition) to
$\sum_{\beta'\in W.\beta}\ensuremath{\hbox{\rm Thom}}_{H,[\beta']}^f(M)$.
Hence $\sigma_{III}\vert_{\ensuremath{\mathcal{U}}}$ is
homotopic, in $K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}})$, to
$(m,v) \to Cl_{x}(v-\ensuremath{\mathcal{H}}^{^{H}}_{m})
\odot 0_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$, where
$0_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$ is the zero map from $\wedge_{\ensuremath{\mathbb{C}}}^{even}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$
to $\wedge_{\ensuremath{\mathbb{C}}}^{odd}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$. Finally we have shown that
$\sigma_{III}\vert_{\ensuremath{\mathcal{U}}}=\sum_{\beta'\in W.\beta}
\ensuremath{\hbox{\rm Thom}}_{H,[\beta']}^f(M)\otimes \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$ in
$K_{H}(\ensuremath{\hbox{\bf T}}_{H}\ensuremath{\mathcal{U}})$, and then (\ref{eq.U.G.beta.U})
finishes the proof.
$\Box$
\bigskip
\section{The Hamiltonian case}\label{sec.Hamiltonien}
\medskip
In this section, we assume that $(M,\omega)$ is a compact
symplectic manifold with a Hamiltonian action of a compact
connected Lie group $G$. The corresponding moment map
$\mu_{_{G}} :M\to \ensuremath{\mathfrak{g}}^{*}$
is defined by
\begin{equation}\label{eq:def.application.moment}
d\langle\mu_{_G},X\rangle=-\,\omega(X_{M},-),\quad \forall\ X\in\ensuremath{\mathfrak{g}}.
\end{equation}
The symplectic $2$-form $\omega$ insures the existence of a
$G$-invariant almost complex structure $J$ {\em compatible} with
$\omega$, i.e, such that :
$$
(v,w)\to\omega_{x}(v,J_{x}w),\quad v,w \in \ensuremath{\hbox{\bf T}}_{x}M\
$$
is symmetric and positive definite for all $x\in M$.
We fix once and for all a $G$-invariant {\em compatible} almost complex
structure $J$, and we denote by $(-,-)_{_{M}}:=\omega(-,J-)$ the
corresponding Riemannian metric. Let
$RR^{^{G}}(M,-)$ be the quantization map defined with
the {\em compatible} almost complex structures $J$. Since two
compatible almost complex structure are homotopic
\cite{Salamon-McDuff}, the map $RR^{^{G}}(M,-)$ does not depend of
this choice (see Lemma \ref{lem.inv.homotopy}).
Here the vector field $\ensuremath{\mathcal{H}}^{^{G}}$ is the Hamiltonian vector field
of the function\footnote{Equality \ref{eq:def.application.moment}
gives $\frac{-1}{2}d||\mu_{_{G}}||^{2}=\omega(\ensuremath{\mathcal{H}}^{^{G}},-)$}
$\frac{-1}{2}||\mu_{_{G}}||^{2}:M\to\ensuremath{\mathbb{R}}$,
and $\{\ensuremath{\mathcal{H}}^{^{G}}=0\}$ is the set of critical points of
$||\mu_{_{G}}||^2$. We know from
the beginning of section \ref{sec.Localisation.f} that we
have the decomposition $RR^{^{G}}(M,-)=$ \break
$\sum_{\beta\in\ensuremath{\mathcal{B}}_{G}}RR^{^{G}}_{\beta}(M,-)$,
where $RR^{^{G}}_{\beta}(M,-): K_{G}(M)\to R^{-\infty}(G)$ is
the Riemann-Roch character localized near the critical set
$C_{\beta}^{^{G}}=G(M^{\beta}\cap\mu_{_G}^{-1}(\beta))$.
In this section we prove the following
Theorem for the $\mu_{_{G}}$-{\em positive} vector bundles
(see Def. \ref{eq.mu.positif}).
\begin{theo}\label{QR=RQ-regulier}
Let $E\to M$ be a $G$-equivariant vector bundle over $M$.
For all $\beta\in\ensuremath{\mathcal{B}}_{_G}-\{0\}$, the $G$-invariant part of
$RR^{^{G}}_{\beta}(M,E)$ is equal to $0$ if $E$ is
$\mu_{_{G}}${\em -positive} and $\mu_{_{G}}^{-1}(0)\neq\emptyset$, or
if $E$ is $\mu_{_{G}}$-{\em strictly positive}. If $0$ is a regular
value of $\mu_{_{G}}$, the $G$-invariant part of $RR^{^{G}}_{0}(M,E)$
is equal to $RR(\ensuremath{\mathcal{M}}_{red},E_{red})$.
\end{theo}
In subsection \ref{non-regulier}, we consider the general case where
$0$ is not necessarily a regular value of $\mu_{_{G}}$, and
$E=L$ a moment bundle for $\mu_{_{G}}$ (see Def. \ref{moment.bundle}).
With our $K$-theoritic approach we recover the following
\begin{theo} [Meinrenken-Sjamaar]\label{QR=RQ-singulier}
Let $L\to M$ be a $\mu_{_{G}}$-moment bundle, and let $\tau$ be the
principal face of $M$. The
$G$-invariant part of $RR^{^{G}}(M,L)$ is equal to
$RR(\ensuremath{\mathcal{M}}_{a},L_{a})$ for every generic value of
$\tau\cap\mu_{_{G}}(M)$ sufficiently close to $0$ (see
subsection \ref{non-regulier} for the notations).
\end{theo}
\medskip
\subsection{The map $RR^{^{G}}_{0}$.}\label{subsec.RR.O.hamilton}
We assume that $0$ is a regular value of $\mu_{_{G}}$. The orbifold
space $\ensuremath{\mathcal{M}}_{red}:=\mu_{_{G}}^{-1}(0)/G$ inherits a symplectic
structure $\omega_{red}$. Let $\ensuremath{\mathcal{D}}(X)=-d\,\mu_{_{G}}(J(X_{M}))$ be
the endomorphism of the trivial bundle $\mu_{_{G}}^{-1}(0)\times
\ensuremath{\mathfrak{g}}$ defined in (\ref{eq.Dcal}). The compatibility of $J$ with
$\omega$ gives
$$
(\ensuremath{\mathcal{D}}(X),X)= \omega(X_{M},J(X_{M}))_{_{M}}
=\parallel X_{M}\parallel^{2}\ ,
$$
thus decomposition (\ref{eq.J.induit}) holds. A small check shows that
the induced almost complex structure $J_{red}$ on $\ensuremath{\mathcal{M}}_{red}$ is
compatible with $\omega_{red}$. Moreover
$t\mapsto t\ensuremath{\mathcal{D}} +(1-t)Id$ is an homotopy of invertible maps between
$\ensuremath{\mathcal{D}}$ and the identity, hence the line bundle $L_{\ensuremath{\mathcal{D}}}\to\ensuremath{\mathcal{M}}_{red}$
defined in (\ref{eq.L.D}) is trivial. The map $RR^{^G}_{0}$ is
determined by the Proposition \ref{prop.RR.T.0.bis} ; in particular
$$
\left[RR_{0}^{^{G}}(M,E)\right]^G =
RR^{J_{red}}(\ensuremath{\mathcal{M}}_{red},E_{red})\ ,
$$
for any $E\in K_{G}(M)$.
\subsection{The map $RR^{^{G}}_{\beta}$ when $G_{\beta}=G$.}
When $\beta\in\ensuremath{\mathcal{B}}_{_{G}}-\{0\}$ is in the center of $\ensuremath{\mathfrak{g}}$, we
proved in Corollary \ref{coro.G.beta.egal.G}, that the $G$-invariant
part of $RR_{\beta}^{^{G}}(M,E)$ is equal to $0$ when $E$ is
$\mu_{_{G}}$-{\em strictly} positive. In the Hamiltonian case we
extend this result for the $\mu_{_{G}}$-positive bundles.
\begin{lem}\label{lem.N.beta.plus}
Let $(\ensuremath{\mathcal{X}},\omega)$ be a connected symplectic manifold with a
$G$-action, and a proper moment map
$\mu :\ensuremath{\mathcal{X}}\to \ensuremath{\mathfrak{g}}$. Let $J$ be a $G$-invariant almost complex
structure on $\ensuremath{\mathcal{X}}$ compatible with $\omega$. Let $\beta$ be
a $G$-invariant element in a Weyl chamber $\ensuremath{\mathfrak{h}}_{+}$ of the
Lie group $G$, such that
$\ensuremath{\mathcal{X}}^{\beta}\cap \mu^{-1}(\beta)\neq\emptyset$.
Let $\ensuremath{\mathcal{N}}^{+,\beta}$ be the polarized normal bundle of
$\ensuremath{\mathcal{X}}^{\beta}$ in $\ensuremath{\mathcal{X}}$ (see Def. \ref{wedge.V.inverse} and
Theorem \ref{th.localisation.pt.fixe}).
If $\ensuremath{\mathcal{N}}^{+,\beta}=0$, we have
$$
\mu(\ensuremath{\mathcal{X}})\cap\ensuremath{\mathfrak{h}}_{+}\subset\{ X\in\ensuremath{\mathfrak{h}}_{+},\ (X,\beta)\geq
\parallel \beta\parallel^2\, \}\ ,
$$
implying in particular that $\parallel \beta\parallel^2$ is the minimal
value of $\parallel \mu\parallel^2$ on $\ensuremath{\mathcal{X}}$.
\end{lem}
{\em Proof of the Lemma :} Let $\ensuremath{\mathcal{Z}}$ be a
connected component of $\ensuremath{\mathcal{X}}^{\beta}$ which intersects
$\mu^{-1}(\beta)$, and consider the
set of weights $\{\alpha_{i},\ i\in I\}$ for the action of
$\ensuremath{\mathbb{T}}_{\beta}$
on the fibers of the vector bundle $\ensuremath{\mathcal{N}}\to \ensuremath{\mathcal{Z}}$. We have then the
following description of the function $(\mu,\beta)$ in the
neighbourhood of $\ensuremath{\mathcal{Z}}$. For $v\in\ensuremath{\mathcal{N}}_{x}$, with the
decomposition $v=\oplus_{i}v_{i}$, we have for $\vert v\vert$ small
enough
$(\mu,\beta)_{(x,v)}=\vert\beta\vert^{2}-\frac{1}{2}
\sum_{i\in I}\langle\alpha_{i},\beta\rangle \vert v_{i}\vert^{2}$.
If $\langle\alpha_{i},\beta\rangle<0$ for every $i\in I$, we have
\begin{equation}\label{eq.mu.beta}
(\mu,\beta)\geq\parallel \beta\parallel^2\quad {\rm in\ a\
neighbourhood}\ \ensuremath{\mathcal{V}}\ {\rm of}\ \ensuremath{\mathcal{Z}}.
\end{equation}
As $\mu^{-1}(\beta)$ is connected and intersect $\ensuremath{\mathcal{Z}}$, the last
inequality imposes $\mu^{-1}(\beta)\subset \ensuremath{\mathcal{Z}}$. Take
$X\in \mu(\ensuremath{\mathcal{X}})\cap\ensuremath{\mathfrak{h}}_{+}$, and consider
$\ensuremath{\mathcal{K}}:=\mu^{-1}([X,\beta])$. From the convexity Theorem
\cite{Atiyah.82,Guillemin-Sternberg82.bis,Kirwan.84.bis,L-M-T-W}, the
set $\ensuremath{\mathcal{K}}$ is connected. Then $\ensuremath{\mathcal{V}}\cap\ensuremath{\mathcal{K}}$ contains, but is not
equal to $\mu^{-1}(\beta)$ : there exists $m\in \ensuremath{\mathcal{V}}\cap\ensuremath{\mathcal{K}}$ with
$\mu(m)\in [X,\beta)$. So $\mu(m)=\beta + t(X-\beta)$ with $t>0$, and
$(\mu(m),\beta)\geq\parallel \beta\parallel^2$. This two conditions
imply that $(X,\beta)\geq\parallel \beta\parallel^2$. $\Box$
\begin{lem}\label{lem.RR.beta.G.invariant}
Let $\beta\in\ensuremath{\mathcal{B}}_{_{G}}-\{0\}$ be a G-invariant element such
that $\parallel \beta\parallel^2$ is not the minimal value of
$\parallel \mu_{_{G}}\parallel^2$ on $M$. Then for every
$\mu_{_{G}}$-positive vector bundle $E$ over $M$ we have
the decomposition
$RR_{\beta}^{^{G}}(M,E)=\sum_{\lambda}m_{\beta,\lambda}(E)\,
\chi_{_{\lambda}}^{_G}$ in irreducible characters with
$$
m_{\beta,\lambda}(E)\neq 0\Longrightarrow
\langle\lambda,\beta\rangle >0\ .
$$
In particular, if $\mu_{_{G}}^{-1}(0)$ is not empty, the
$G$-invariant part of $RR_{\beta}^{^{G}}(M,E)$ is equal to $0$ for
every $G$-invariant $\beta\in\ensuremath{\mathcal{B}}_{_{G}}-\{0\}$.
The result remains when $M$ is non-compact, and the moment map
$\mu_{_{G}}$ is proper.
\end{lem}
{\em Proof :} Recall the localization formula
on $M^{\beta}$ obtained in Proposition
\ref{prop.loc.RR.G.beta}. For every complex $G$-vector bundle $E$
over $M$, we have the
following equality in $\widehat{R}(G)$
\begin{equation}\label{eq.non.compacte}
RR_{\beta}^{^{G}}(M,E)=(-1)^{r_{\ensuremath{\mathcal{N}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR_{\beta}^{^{G}}(M^{\beta},E\vert_{M^{\beta}}\otimes\det
\ensuremath{\mathcal{N}}^{+,\beta}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}) \ .
\end{equation}
Suppose that $M$ is non-compact and that the moment map
$\mu_{_{G}}$ is proper as a map from a $G$-invariant open neighborhood
of $\mu_{_{G}}^{-1}(\beta)$ in $M$ to a $G$-invariant open neighborhood
of $\beta$ in $\ensuremath{\mathfrak{g}}$. Each terms of (\ref{eq.non.compacte})
are well defined and the equality remains valid in this case (It is
not difficult to extend the proof given in subsection \ref{subsec.RR.G.beta}
to this situation).
If $\parallel \beta\parallel^2$ is not the minimal value of
$\parallel \mu_{_{G}}\parallel^2$, we know from Lemma \ref{lem.N.beta.plus},
that the vector bundle $\ensuremath{\mathcal{N}}^{+,\beta}$ is not trivial over
each connected component $\ensuremath{\mathcal{Z}}$ of $M^{\beta}$ that intersects
$\mu^{-1}(\beta)$. Then every $\ensuremath{\mathbb{T}}_{\beta}$-weight $a$
on the fibers of the complex vector bundle
$E\vert_{\ensuremath{\mathcal{Z}}}\otimes\det\ensuremath{\mathcal{N}}^{+,\beta}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+,\beta}$
satisfies $\langle a,\beta\rangle>0$.
Lemma \ref{lem.multiplicites.tore} and Corollary
\ref{coro.multiplicites.tore}, applied to this situation, show that
$RR_{\beta}^{^{G}}(M,E)=\sum_{\lambda}m_{\beta,\lambda}(E)\,
\chi_{_{\lambda}}^{_G}$ with $m_{\beta,\lambda}(E)\neq 0$
only if $\langle\lambda,\beta\rangle >0$. $\Box$
\medskip
\subsection{The map $RR^{^{G}}_{\beta}$ when $G_{\beta}\neq G$.}
Let $\sigma$ be the unique open face of $\ensuremath{\mathfrak{h}}_{+}$ which contains
$\beta$. The stabilizer subgroup $G_{\xi}$ does not depend on the
choice of $\xi\in\sigma$, and is denoted by $G_{\sigma}$. Let
$\ensuremath{\mathfrak{g}}_{\sigma}$ be the Lie algebra of $G_{\sigma}$, and let
$U_{\sigma}$ the $G_{\sigma}$-invariant open subset of
$\ensuremath{\mathfrak{g}}_{\sigma}$ defined by $U_{\sigma}=G_{\sigma}\cdot
\{y\in \ensuremath{\mathfrak{h}}_{+}\vert G_{y}\subset G_{\sigma}\}$.
The symplectic cross-section Theorem
\cite{Guillemin-Sternberg84,L-M-T-W}
asserts that the pre-image $\ensuremath{\mathcal{Y}}_{\sigma}=\mu_{_{G}}^{-1}(U_{\sigma})$
is a symplectic submanifold of $M$ provided with a Hamiltonian action
of $G_{\sigma}$. We denote by $\omega_{\sigma}$ the symplectic $2$-form on
$\ensuremath{\mathcal{Y}}_{\sigma}$, and $\mu_{\sigma}:\ensuremath{\mathcal{Y}}_{\sigma}\to
\ensuremath{\mathfrak{g}}_{\sigma}$ the moment map. Let $J_{\sigma}$ be a
$G_{\sigma}$-invariant almost complex structure on $\ensuremath{\mathcal{Y}}_{\sigma}$,
which is compatible with $\omega_{\sigma}$.
The vector field $\ensuremath{\mathcal{H}}^{\sigma}$
on $\ensuremath{\mathcal{Y}}_{\sigma}$ generated by $\mu_{\sigma}$ vanishes on
$C^{\sigma}_{\beta}:=
\mu^{-1}_{\sigma}(\beta)\cap (\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}
=\mu^{-1}_{_G}(\beta)\cap M^{\beta}$
(see Definition \ref{def.H.vector}). We denote by\footnote{
For a non-compact $G$-manifold $\ensuremath{\mathcal{X}}$, we denote by
$\tilde{K}_{G}(\ensuremath{\mathcal{X}})$ the equivariant $K$-theory of $\ensuremath{\mathcal{X}}$ with
non-compact support.}
$$
RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},-):
\tilde{K}_{G_{\sigma}}(\ensuremath{\mathcal{Y}}_{\sigma})\to R^{-\infty}(G_{\sigma})
$$
the Riemann-Roch character on $\ensuremath{\mathcal{Y}}_{\sigma}$ localized near
the {\em compact} subset $C^{\sigma}_{\beta}$ by the vector filed
$\ensuremath{\mathcal{H}}^{\sigma}$. It
is well defined even since $\mu_{\sigma}$ is a proper map
(see Definition \ref{def.RR.beta}).
\medskip
\begin{theo}\label{th.RR.G.beta.hamilton}
For every $E\in K_{G}(M)$, we have
$$
RR^{^{G}}_{\beta}(M,E)={\rm Hol}^{^G}_{_{G_{\sigma}}}\left(
RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})
\right)\quad {\rm in}\quad R^{-\infty}(G)\ ,
$$
\end{theo}
\medskip
\begin{coro} Let $\beta\in\ensuremath{\mathcal{B}}_{_{G}}$ with $G_{\beta}\neq G$.
If $\mu_{_{G}}^{-1}(0)\neq \emptyset$, we have
$[ RR^{^{G}}_{\beta}(M,E)]^{G}=0$, for every
$\mu_{_{G}}$-positive vector bundle $E\to M$. In general,
$[ RR^{^{G}}_{\beta}(M,E)]^{G}=0$, for every
$\mu_{_{G}}$-{\em strictly} positive vector bundle $E$.
\end{coro}
\medskip
{\em Proof of the Corollary :}
The moment map $\mu_{\sigma}$ is proper
as a map from a $G_{\sigma}$-invariant open neighborhood
of $\mu_{\sigma}^{-1}(\beta)$ in $\ensuremath{\mathcal{Y}}_{\sigma}$ to a
$G_{\sigma}$-invariant open neighborhood of $\beta$ in $\ensuremath{\mathfrak{g}}_{\sigma}$.
If $0\in \mu_{_{G}}(M)$ we see that $t\beta\in
\mu_{\sigma}(\ensuremath{\mathcal{Y}}_{\sigma})$ for any $0<t<1$, hence
$\parallel \beta\parallel^2$ is not the minimal value of
$\parallel \mu_{\sigma}\parallel^2$.
Proposition \ref{lem.RR.beta.G.invariant} can be
used for the map $RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},-)$.
For any $\mu_{_{G}}$-positive vector bundle $E$, we have
$RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})=
\sum_{\lambda}m_{\beta,\lambda}(E)\,\chi_{_{\lambda}}^{_{G_{\sigma}}}$
with $m_{\beta,\lambda}(E)\neq 0$ only if
$\langle\lambda,\beta\rangle >0$ (the same holds when
$0\notin \mu_{_{G}}(M)$ and $E$ is $\mu_{_{G}}$-{\em strictly} positive).
With the induction formula of Theorem \ref{th.RR.G.beta.hamilton} we
get\footnote{${\rm Hol}^{^G}_{_{G_{\sigma}}}(\chi_{_{\lambda}}^{_{G_{\sigma}}})={\rm Hol}^{^G}_{_H}(h^{\lambda})$
since $\chi_{_{\lambda}}^{_{G_{\sigma}}}={\rm Hol}^{^{G_{\sigma}}}_{_H}(h^{\lambda})$.}
$RR^{^{G}}_{\beta}(M,E) =\sum_{\lambda}m_{\beta,\lambda}(E)\,
{\rm Hol}^{^G}_{_H}(h^{\lambda})$. But ${\rm Hol}^{^G}_{_H}(h^{\lambda})=\pm 1$ only if
$\langle \lambda,X\rangle\leq 0$ for every $X$ in the Weyl chamber
(see Remark \ref{hol.egale.1}).
This shows
$$
{\rm Hol}^{^G}_{_H}(h^{\lambda})=\pm 1\Longrightarrow \langle
\lambda,\beta\rangle\leq 0
\Longrightarrow m_{\beta,\lambda}(E)=0\ .
$$
We have then proved that $[RR^{^{G}}_{\beta}(M,E)]^{G}=0$. $\Box$
\bigskip
\underline{\bf Proofs of Theorem \ref{th.RR.G.beta.hamilton} :}
We propose here two different proofs for this induction formula. Both
of them use the same technical remark.
The set $G\cdot \ensuremath{\mathcal{Y}}_{\sigma}\cong
G\times_{G_{\sigma}}\ensuremath{\mathcal{Y}}_{\sigma}$ is a $G$-invariant open
neighborhood of the critical set $C_{\beta}^{^G}$ in $M$.
The symplectic form $\omega$, when restricted to
$G\times_{G_{\sigma}}\ensuremath{\mathcal{Y}}_{\sigma}$,
can be written in terms of the moment map
$\mu_{\sigma}$ and the symplectic form $\omega_{\sigma}$:
\begin{equation}\label{eq.omega.slice.bis}
\omega_{[g,y]}(X+v,Y+w)=-(\mu_{\sigma}(y),[X,Y])+
\omega_{\sigma}\vert_{y}(v,w)\ ,
\end{equation}
where $X,Y\in \ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}$, and $v,w\in\ensuremath{\hbox{\bf T}}_{y}
\ensuremath{\mathcal{Y}}_{\sigma}$\footnote{We use here the identification
$\ensuremath{\hbox{\bf T}}(G\times_{G_{\sigma}}\ensuremath{\mathcal{Y}}_{\sigma})\cong G\times_{G_{\sigma}}
(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\oplus \ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}_{\sigma})$
(see (\ref{eq.espace.tangent})).}.
With the complex structure
$J_{G/G_{\sigma}}$ on $G/G_{\sigma}$ determined by $\beta$,
we form the almost complex structure $\widetilde{J}:=
J_{G/G_{\sigma}}\times J_{\sigma}$ on
$G\times_{G_{\sigma}}\ensuremath{\mathcal{Y}}_{\sigma}$. Equation
(\ref{eq.omega.slice.bis}) shows that $\widetilde{J}$ is compatible
with $\omega$ in a neighborhood of $C_{\beta}^{^G}$,
hence $\widetilde{J}$ is homotopic
to $J$ in a neighborhood of $C_{\beta}^{^G}$ in
$G\times_{G_{\sigma}}\ensuremath{\mathcal{Y}}_{\sigma}$.
\begin{rem}\label{rem.J.tilde}
The almost complex structures $J$ and $\widetilde{J}$ are
homotopic in a neighborhood of $C_{\beta}^{^G}$, so as in Lemma
\ref{lem.inv.homotopy} we see that the computation of the localized
Riemann-Roch character $RR^{^{G}}_{\beta}(M,E)$ can be done with
$\widetilde{J}$ instead of $J$.
\end{rem}
\medskip
{\bf First proof of Theorem \ref{th.RR.G.beta.hamilton} :}
We will show here that
Theorem \ref{th.RR.G.beta.hamilton} is a
consequence of the induction formula proved in Theorem
\ref{th.induction.G.H}
and of the localization formula obtained in Proposition \ref{prop.loc.RR.G.beta}.
The induction of Corollary \ref{coro.induction.G.G.beta} shows that
$RR^{^{G}}_{\beta}(M,E)={\rm Hol}^{^G}_{_{G_{\sigma}}}(RR^{^{G_{\sigma}}}_{\beta}(M,E)
\wedge^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}})$.
So we have to prove the following equality
\begin{equation}\label{equation1.th.7.4}
RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})=
RR^{^{G_{\sigma}}}_{\beta}(M,E)
\wedge^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\ .
\end{equation}
First we use the localization formula on both
sides of the equality. For the map
$RR^{^{G_{\sigma}}}_{\beta}(M,-)$
this gives
\begin{equation}\label{equation2.th.7.4}
RR^{^{G_{\sigma}}}_{\beta}(M,E)=
RR^{^{G_{\sigma}\times{\rm T}_{\beta}}}_{\beta}
\left(M^{\beta},E\vert_{M^{\beta}}\otimes
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,
\right]^{-1}_{\beta}\right)\ ,
\end{equation}
and for $RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},-)$ we have
\begin{equation}\label{equation3.th.7.4}
RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})=
RR^{^{G_{\sigma}\times{\rm T}_{\beta}}}_{\beta}
\left((\ensuremath{\mathcal{Y}}_{\sigma})^{\beta},E\vert_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}}
\otimes\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\,
\right]^{-1}_{\beta}
\right)\ .
\end{equation}
Here $\ensuremath{\mathcal{N}}$ and $\ensuremath{\mathcal{N}}'$ are respectively the normal bundle of
$M^{\beta}$ in $M$, and the normal bundle of
$(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$ in $\ensuremath{\mathcal{Y}}_{\sigma}$. The complex
structures on the fibers of $\ensuremath{\mathcal{N}}$ and $\ensuremath{\mathcal{N}}'$ are induced
respectively by the almost complex
structure $\widetilde{J}$, and by the almost complex structure
$J_{\sigma}$ (see Remark \ref{rem.J.tilde}).
Now we remark that $(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$ is an
open neighborhood of $M^{\beta}\cap\mu_{_{G}}^{-1}(\beta)$
in $M^{\beta}$, thus we have
$RR^{^{G_{\sigma}}}_{\beta}(M^{\beta},F)=
RR^{^{G_{\sigma}}}_{\beta}((\ensuremath{\mathcal{Y}}_{\sigma})^{\beta},
F\vert_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}})$ for any equivariant vector bundle
$F$. So
(\ref{equation2.th.7.4}) and (\ref{equation3.th.7.4})
shows us that (\ref{equation1.th.7.4}) is equivalent
to the following
\begin{eqnarray}\label{equation4.th.7.4}
\lefteqn{RR^{^{G_{\sigma}\times{\rm T}_{\beta}}}_{\beta}
\left((\ensuremath{\mathcal{Y}}_{\sigma})^{\beta},E\vert_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}}
\otimes\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,
\right]^{-1}_{\beta}\otimes \left[\wedge^{\bullet}
\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\,\right]\right)
=}\\
& & RR^{^{G_{\sigma}\times{\rm T}_{\beta}}}_{\beta}
\left((\ensuremath{\mathcal{Y}}_{\sigma})^{\beta},E\vert_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}}
\otimes\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\,
\right]^{-1}_{\beta}\right)\ ,\nonumber
\end{eqnarray}
where $\left[\wedge^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\,\right]$
is the trivial bundle
$\wedge^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\times
(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}\to (\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$.
To finish the proof, we notice that the normal bundle $\ensuremath{\mathcal{N}}\to
M^{\beta}$, when restricted to $(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$, can be
decomposed as
$\ensuremath{\mathcal{N}}\vert_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}}=\ensuremath{\mathcal{N}}'
\oplus[\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}]$.
Here $[\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}]\to (\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$ is
the trivial complex vector bundle defined by
$[\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}]_{m}=\{ X_{(\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}}\vert_{m},\
X\in \ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\}$ for any $m\in (\ensuremath{\mathcal{Y}}_{\sigma})^{\beta}$.
This decomposition gives first
the equality $\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}=
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\otimes
[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\,]$
and after\footnote{The product of
$\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\,\right]^{-1}_{\beta}$
and $\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}
\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\,\right]^{-1}_{\beta}$
is well defined in $\tilde{K}_{G_{\sigma}}((\ensuremath{\mathcal{Y}}_{\sigma})^{\beta})
\widehat{\otimes}\,R(\ensuremath{\mathbb{T}}_{\beta})$ since these elements are
polarized by $\beta$: each of them is a sum over the set of
weights of $\ensuremath{\mathbb{T}}_{\beta}$ of the form
$\sum_{\alpha}E_{\alpha}h^{\alpha}$ such that
$E_{\alpha}\neq 0$ only if $\langle\alpha,\beta\rangle\geq 0$,
and for any $\delta'>\delta\geq 0$ the sum
$\sum_{\delta\leq\langle\alpha,\beta\rangle\leq\delta'}
E_{\alpha}h^{\alpha}$ is
finite (see definition \ref{wedge.V.inverse}).}
$\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}=
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\,\right]^{-1}_{\beta}
\otimes\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}
\,\right]^{-1}_{\beta}\ ,
$
which implies
$
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}}\,\right]^{-1}_{\beta}
\otimes
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}}\,\right]=
\left[\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathcal{N}}'}\,\right]^{-1}_{\beta}\ .
$
(\ref{equation4.th.7.4}) is then proved. $\Box$
\medskip
{\bf Second proof of Theorem \ref{th.RR.G.beta.hamilton} :}
A $G$-invariant neighborhood $\ensuremath{\mathcal{U}}^{^{G,\beta}}$ of the critical set
$C_{\beta}^{^G}$ in $M$ can be taken of the form $\ensuremath{\mathcal{U}}^{^{G,\beta}}=
G\times_{G_{\sigma}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}$ where
$\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}$ a relatively compact $G_{\sigma}$-invariant
neighborhood of $\mu^{-1}_{_G}(\beta)\cap M^{\beta}$ in
$\ensuremath{\mathcal{Y}}_{\sigma}$ such that
$\overline{\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}}\cap\{\ensuremath{\mathcal{H}}^{\sigma}=0\}=
\mu^{-1}_{_G}(\beta)\cap M^{\beta}$.
The maps $RR^{^{G}}_{\beta}(M, -)$ and
$RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},-)$ are respectively
defined by the localized Thom symbols
$\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^{\mu}(M)\in K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}^{^{G,\beta}})$ and
$\ensuremath{\hbox{\rm Thom}}_{G_{\sigma},[\beta]}^{\mu}(\ensuremath{\mathcal{Y}}_{\sigma})
\in K_{G_{\sigma}}(\ensuremath{\hbox{\bf T}}_{G_{\sigma}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}})$
(see Definition \ref{def.thom.beta.f}). The inclusion
$i:G_{\sigma}\ensuremath{\hookrightarrow} G$ induces an isomorphism
$i_{*}:K_{G_{\sigma}}(\ensuremath{\hbox{\bf T}}_{G_{\sigma}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}})\to
K_{G}(\ensuremath{\hbox{\bf T}}_{G}(G\times_{G_{\sigma}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}))$
(see subsection \ref{subsec.induction.def}).
\begin{lem}\label{lem.induction.slice}
We have the following equality
$$
i_{*}\left(\ensuremath{\hbox{\rm Thom}}_{G_{\sigma},[\beta]}^{\mu}(\ensuremath{\mathcal{Y}}_{\sigma})
\, \wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\right)
=\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^{\mu}(M)\ .
$$
\end{lem}
\medskip
This Lemma, combined with Theorem \ref{thm.atiyah.2}, shows that
$RR^{^{G}}_{\beta}(M, E)=$\break
${\rm Ind}^{^G}_{_{G_{\sigma}}}\left(RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})
\,\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\right)=
{\rm Hol}^{^G}_{_{G_{\sigma}}}\left(RR^{^{G_{\sigma}}}_{\beta}(\ensuremath{\mathcal{Y}}_{\sigma},
E\vert_{\ensuremath{\mathcal{Y}}_{\sigma}})\right)$
for any $G$-complex vector bundle $E\to M$. The proof of Theorem
\ref{th.RR.G.beta.hamilton} is then completed. $\Box$
\medskip
{\em Proof of Lemma \ref{lem.induction.slice} :}
\medskip
Through the identification
$\ensuremath{\hbox{\bf T}}(G\times_{G_{\sigma}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}) \cong
G\times_{G_{\sigma}}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\oplus
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}})$, the vector fiels $\ensuremath{\mathcal{H}}^{\sigma}$ and
$\ensuremath{\mathcal{H}}^{^{G}}$ satisfy the relation $\ensuremath{\mathcal{H}}^{^{G}}_{[g,y]}\cong
\ensuremath{\mathcal{H}}^{\sigma}_{y},\ [g,y]\in \ensuremath{\mathcal{U}}^{^{G,\beta}}$. The symbol
$\sigma_{[g,y;X+v]}$ of $\ensuremath{\hbox{\rm Thom}}_{G,[\beta]}^{\mu}(M)$ at $[g,y;X+v]\in
G\times_{G_{\sigma}}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\oplus
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}})$ acts on
$\wedge^{\bullet}_{\widetilde{J}}\ensuremath{\hbox{\bf T}}_{[g,y]}\ensuremath{\mathcal{U}}^{^{G,\beta}} \cong
\wedge^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}\otimes
\wedge^{\bullet}_{J_{\sigma}}\ensuremath{\hbox{\bf T}}_{y}\ensuremath{\mathcal{U}}^{^{\sigma,\beta}}$ as the
product
$$
\sigma_{[g,y;X+v]}=Cl(X)\odot Cl_{y}(v-\ensuremath{\mathcal{H}}^{\sigma}_{y})\ .
$$
Now we see that $[g,y;X+v]\to Cl(X)\odot Cl_{y}(v-\ensuremath{\mathcal{H}}^{\sigma}_{y})$
is homotopic, as $G$-transversally elliptic symbol, to
$\widetilde{\sigma}: [g,y;X+v]\to Cl(0)\odot
Cl_{y}(v-\ensuremath{\mathcal{H}}^{\sigma}_{y})$, and $\widetilde{\sigma}$ is, by
definition, the image of
$\ensuremath{\hbox{\rm Thom}}_{G_{\sigma},[\beta]}^{\mu}(\ensuremath{\mathcal{Y}}_{\sigma})
\wedge^{\bullet}_{\ensuremath{\mathbb{C}}}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\sigma}$ by $i_{*}$. The proof of
Lemma \ref{lem.induction.slice} is then completed. $\Box$
\medskip
\subsection{The singular case.}\label{non-regulier}
In this section, we do not assume that $0$ is a regular value of
$\mu_{_G}$, and we use the `shifting trick' to
compute $[RR^{^G}(M,L)]^G$ in term of reduced manifolds
of the type $\mu_{_G}^{-1}(a)/G_{a}$, for every $\mu_{_G}$-moment
bundle $L$. We know from Theorem \ref{QR=RQ-regulier} that $[RR^{^G}(M,L)]^G=0$
if $0\notin \mu_{_G}(M)$ since every moment bundle is
strictly positive (see Lemma \ref{L-a-positif}).
So, we assume for the rest of this section that $0\in \mu_{_G}(M)$.
Let $\ensuremath{\mathcal{O}}_a$ be the coadjoint orbit through $a\in\ensuremath{\mathfrak{g}}^*
$. It has a canonical symplectic 2-form and the moment map $\ensuremath{\mathcal{O}}_a\to\ensuremath{\mathfrak{g}}^*$
for the $G$-action is the inclusion. We denote by $\overline{\ensuremath{\mathcal{O}}_a}$
the coadjoint orbit $\ensuremath{\mathcal{O}}_a$ with the opposite symplectic form. The
product $M\times\overline{\ensuremath{\mathcal{O}}_a}$ is a symplectic manifold with a
Hamiltonian moment map
\begin{eqnarray*}
\mu_a :M\times\overline{\ensuremath{\mathcal{O}}_a}&\longrightarrow&\ensuremath{\mathfrak{g}}^*\\
(m,\xi)&\longmapsto& \mu_{_G}(m)-\xi\ .
\end{eqnarray*}
On the symplectic manifold $M\times\overline{\ensuremath{\mathcal{O}}_a}$ we have a
quantization map $RR^{^G}(M\times\overline{\ensuremath{\mathcal{O}}_a},-)$ with the
following property: for any $G$-vector bundles $E$ and $F$ over
$M$ and $\ensuremath{\mathcal{O}}_a$ respectively, we have
$RR^{^G}(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_a^*(E)\otimes(\pi_a')^*(F))=
RR^{^G}(M,E)\cdot RR^{^G}(\overline{\ensuremath{\mathcal{O}}_a},F)$ in $R(G)$. Here
we denote by $\pi_a:M\times\overline{\ensuremath{\mathcal{O}}_a}\to M$ the projection to
the first factor and $\pi_a'$ the projection to the second factor.
Since $RR^{^G}(\overline{\ensuremath{\mathcal{O}}_a},\ensuremath{\mathbb{C}})=1$ we have
\begin{equation}\label{shift-a}
RR^{^G}(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_a^*(L))= RR^{^G}(M,L)\ .
\end{equation}
We can now compute $[RR^{^G}(M,L)]^G$ by localizing the character
$RR^{^G}(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_a^*(L))$ with the
moment map $\mu_a$. We need the following Lemma which was proved
by Tian-Zhang \cite{Tian-Zhang97} for the prequantum line bundles.
\begin{lem}\label{L-a-positif}
Let $L$ be a $\mu_{_G}$-moment bundle over $M$. There exists
$\epsilon>0$ such that for any $\vert a\vert<\epsilon$, the vector
bundle $\pi_a^*(L)$ is $\mu_a$-positive. For $a=0$, the bundle
$L=\pi_0^*(L)$ is $\mu_{_{G}}$-strictly positive.
\end{lem}
Let $RR^{^G}_0(M\times\overline{\ensuremath{\mathcal{O}}_a},-)$ be the Riemann-Roch
character localized near $\mu_a^{-1}(0)\simeq \mu_{_G}^{-1}(\ensuremath{\mathcal{O}}_a)$.
Theorem \ref{QR=RQ-regulier}, Equality \ref{shift-a}, and Lemma \ref{L-a-positif}
show that
\begin{equation}\label{shift-a-0}
[RR^{^G}(M,L)]^G=[RR^{^G}_0(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_a^*(L))]^G\ ,
\end{equation}
for any moment bundle $L$ if $a\in\mu_{_G}(M)$ is close enough to $0$.
There exists a unique open face $\tau$ of the Weyl chamber $\ensuremath{\mathfrak{h}}_+$ such that
$\mu_{_G}(M)\cap\tau$ is dense in $\mu_{_G}(M)\cap\ensuremath{\mathfrak{h}}_+$.
The face $\tau$ is called the principal face of $(M,\mu_{_G})$ \cite{L-M-T-W}.
All points in the open face $\tau$ have the same connected
centralizer $G_{\tau}$. Let $A_{\tau}$ be the identity component of
the center of $G_{\tau}$ and $[G_{\tau},G_{\tau}]$ its semisimple part. Note
that we have an identification between
the Lie algebra $\ensuremath{\mathfrak{a}}_{\tau}$ of $A_{\tau}$ and the
linear span of the face
$\tau$. The Principal-cross-section Theorem \cite{L-M-T-W} tells us that
$Y_{\tau}:=\mu_{_G}^{-1}(\tau)$ is a symplectic $G_{\tau}$-manifold,
with a trivial action of $[G_{\tau},G_{\tau}]$. So, the restriction
of $\mu_{_G}$ on $\ensuremath{\mathcal{Y}}_{\tau}$ is a moment map
$\mu_{\tau}: \ensuremath{\mathcal{Y}}_{\tau}\to\ensuremath{\mathfrak{a}}_{\tau}$ for the Hamiltonian action of
the torus $A_{\tau}$. We decompose the torus $A_{\tau}$
in a product of two subtorus $A_{\tau}=A_{\tau}^1\times A_{\tau}^2$
where $A_{\tau}^{1}$ is the identity component of the principal
stabilizer for the action of $A_{\tau}$ on $\ensuremath{\mathcal{Y}}_{\tau}$.
We take now $a$ with value in $\tau\cap\mu_{_G}(M)$.
For generic values $a\in \tau\cap\mu_{_G}(M)$,
$\mu_{_G}^{-1}(a)=\mu_{\tau}^{-1}(a)$ is
a smooth manifold of $M$ with a locally free action of $A_{\tau}^{2}$, hence
the quotient $\ensuremath{\mathcal{M}}_a:=\mu_{_G}^{-1}(a)/G_a=
\mu_{\tau}^{-1}(a)/(A_{\tau}^2)$
is a symplectic orbifold. We denote by $RR(\ensuremath{\mathcal{M}}_{a},-)$ the quantization
map defined by the choice of a compatible almost complex structure.
If $L$ is a $\mu_{_{G}}$-moment bundle on $M$, $L\vert_{\ensuremath{\mathcal{Y}}_{\tau}}$
is a $\mu_{\tau}$-moment bundle: the action of
$A_{\tau}^{1}[G_{\tau},G_{\tau}]$ on $L\vert_{\ensuremath{\mathcal{Y}}_{\tau}}$ is trivial.
Then the quotient $L\vert_{\mu_{\tau}^{-1}(a)}/G_{a}=
L\vert_{\mu_{\tau}^{-1}(a)}/(A_{\tau}^2)$ is an orbifold line bundle
over $\ensuremath{\mathcal{M}}_a$ for generic $a$.
We compare now the Riemann-Roch character $RR^{^{G_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},-)$
localized near $\mu_{\tau}^{-1}(a)$ by the moment map $\mu_{\tau}-a$
and the Riemann-Roch character $RR^{^G}_0(M\times\overline{\ensuremath{\mathcal{O}}_a},-)$
localized near $\mu_{a}^{-1}(0)=G\cdot(\mu_{\tau}^{-1}(a)\times\{a\})$.
All we need is contained in the following
\begin{prop}\label{RR-a}
Let $E$ be a G-vector bundle over $M$, and take $a\in\tau$. We have
$RR^{^G}_0(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_{a}^{*}E)=
{\rm Ind}^{^G}_{_{G_{\tau}}}\left(RR^{^{G_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},E\vert_{\ensuremath{\mathcal{Y}}_{\tau}})\right)$,
in particular
$[RR^{^G}_0(M\times\overline{\ensuremath{\mathcal{O}}_a},\pi_{a}^{*}E)]^G=
[RR^{^{G_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},E\vert_{\ensuremath{\mathcal{Y}}_{\tau}})]^{G_{\tau}}$.
\end{prop}
If $L$ is a $\mu_{_{G}}$-moment bundle, the action of
$A_{\tau}^{1}[G_{\tau},G_{\tau}]$ on $L\vert_{\ensuremath{\mathcal{Y}}_{\tau}}$ is trivial, then
$[RR^{^{G_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},L\vert_{\ensuremath{\mathcal{Y}}_{\tau}})]^{G_{\tau}}=
[RR^{^{A^2_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},L\vert_{\ensuremath{\mathcal{Y}}_{\tau}})]^{A^{2}_{\tau}}$.
Finally, for every generic
value $a\in\tau\cap\mu_{_G}(M)$, the quotient
$L_{a}:=L\vert_{\mu_{\tau}^{-1}(a)}/A^{2}_{\tau}$ is an orbifold line
bundle over $\ensuremath{\mathcal{M}}_{a}$, so from subsection \ref{subsec.RR.O.hamilton}
we get $[RR^{^{A^{2}_{\tau}}}_0(\ensuremath{\mathcal{Y}}_{\tau},L\vert_{\ensuremath{\mathcal{Y}}_{\tau}})]^{A^{2}_{\tau}}=
RR(\ensuremath{\mathcal{M}}_{a},L_{a})$.
With this last equality, Proposition \ref{L-a-positif}, and
equality (\ref{shift-a-0}) we have proved the central result of
this section
\begin{prop}\label{RR-0-singulier}
Suppose that $0\in\mu_{_{G}}(M)$. If $L$ is a $\mu_{_{G}}$-moment bundle,
there exist $\epsilon>0$, such that
$$
[RR^{^{G}}(M,L)]^{G}= RR(\ensuremath{\mathcal{M}}_{a},L_{a})\ ,
$$
for every generic value $a\in \tau\cap\mu_{_{G}}(M)$ with $\vert
a\vert<\epsilon$.
\end{prop}
\subsubsection{Proof of Lemma \ref{L-a-positif}}
Let $L$ be a $\mu_{_{G}}$-moment bundle over $M$, where
$\mu_{_{G}}:M\to\ensuremath{\mathfrak{g}}^{*}$ is a Hamiltonian moment map.
Recall that the Lie
algebra $\ensuremath{\mathfrak{g}}$ is identified to $\ensuremath{\mathfrak{g}}^{*}$ trough
an invariant scalar product $(-,-)$. Let $H$ be a maximal
torus of $G$ with Lie algebra $\ensuremath{\mathfrak{h}}$.
\begin{lem}\label{critere-L-positif} For $\beta\in \ensuremath{\mathfrak{h}}$ and
$m\in M^{\beta}\cap\mu_{_{G}}^{-1}(\gamma)$,
the weight $\alpha$ for the action of $\ensuremath{\mathbb{T}}_{\beta}$ on $L_{m}$
satifies $(\alpha,\beta)=(\gamma,\beta)$.
\end{lem}
{\em Proof }: Let $N$ be the connected component of $M^{\beta}$
containing $m$, and let $m'$ be a point of $N^{H}$. Since $N$ is
connected, $\alpha$ is also the weight for the action of
$\ensuremath{\mathbb{T}}_{\beta}$ on $L_{m'}$, and $\mu_{_{G}}(m')$ is the
weight for the action of $H$ on $L_{m'}$: then
$(\alpha,X)=(\mu_{_{G}}(m'),X)$ for every $X\in
Lie(\ensuremath{\mathbb{T}}_{\beta})$. But the map $x\to(\mu_{_{G}}(x),\beta)$
is constant on $N$, then
$(\gamma,\beta)=(\mu_{_{G}}(m),\beta)=
(\mu_{_{G}}(m'),\beta)=(\alpha,\beta)$. $\Box$
\medskip
The element $a$ is taken in $\ensuremath{\mathfrak{h}}$. The critical set of the function
$||\mu_{a}||^2:M\times\ensuremath{\mathcal{O}}_{a}\to\ensuremath{\mathbb{R}}$ admits the following
decomposition $\ensuremath{\hbox{\rm Cr}}(||\mu_{_a}||^2)= G\cdot(\ensuremath{\hbox{\rm Cr}}(||\mu_{_a}||^2)\cap
(M\times \{a\}))= G\cdot
\Big(\left(\ensuremath{\hbox{\rm Cr}}(||\mu_{_{G_{a}}}-a||^2)\cap\mu_{_{G}}^{-1}(\ensuremath{\mathfrak{g}}_{a})\right)
\times \{a\}\Big)$,
where $\mu_{_{G_{a}}}:M\to\ensuremath{\mathfrak{g}}_{a}$ is the moment map for the action
of $G_{a}$. Let $\ensuremath{\mathcal{B}}_{a}$ the finite subset of $\ensuremath{\mathfrak{h}}$ defined
by $\ensuremath{\mathcal{B}}_{a}=\{\beta\in\ensuremath{\mathfrak{h}},\ M^{\beta}\cap\mu_{_{G}}^{-1}(\beta
+a)\neq \emptyset\}$. Finally we have the decomposition
$$
\ensuremath{\hbox{\rm Cr}}(||\mu_{_a}||^2)=\bigcup_{\beta\in\ensuremath{\mathcal{B}}_{a}}
G\cdot\left(M^{\beta}\cap\mu_{_{G}}^{-1}(\beta
+a)\times\{a\}\right)\ .
$$
Using Lemma \ref{critere-L-positif}, we see that $\pi_{a}^{*}L$ is
$\mu_{a}$-positive if and only if
\begin{equation}\label{L-a-bis}
(\beta + a,\beta)\geq 0\quad {\rm for\ every}\quad
\beta\in\ensuremath{\mathcal{B}}_{a}\ .
\end{equation}
We first see that it is trivially true
if $a=0$: in this case $L$ is strictly positive.
Let $\mu_{_{H}}:M\to\ensuremath{\mathfrak{h}}$ be the moment map for the maximal torus $H$.
Consider the finite set $\ensuremath{\mathcal{B}}_{H,a}$ which parametrizes the critical
set of $||\mu_{_H}-a||^2$: $\ensuremath{\mathcal{B}}_{H,a}=
\{\beta\in\ensuremath{\mathfrak{h}},\ M^{\beta}\cap\mu_{_{H}}^{-1}(\beta +a)\neq
\emptyset\}$. We have obviously the inclusion $\ensuremath{\mathcal{B}}_{a}\subset
\ensuremath{\mathcal{B}}_{H,a}$, so it suffices to show (\ref{L-a-bis}) for
$\ensuremath{\mathcal{B}}_{H,a}$.
To finish our proof we use now a characterisation of the set
$\ensuremath{\mathcal{B}}_{H,a}$ we gave in \cite{pep1}. There exists a finite collection
$\ensuremath{\mathcal{B}}$ of affine subspaces of $\ensuremath{\mathfrak{h}}$ such that
$$
\ensuremath{\mathcal{B}}_{H,a}\subset\{P_{\Delta}(a)-a,\Delta\in \ensuremath{\mathcal{B}}\}
$$
for every $a\in\ensuremath{\mathfrak{h}}$. Here $P_{\Delta}:\ensuremath{\mathfrak{h}}\to\ensuremath{\mathfrak{h}}$ is the
orthogonal projection on $\Delta$. It is now easy to
compute the sign of $(P_{\Delta}(a),P_{\Delta}(a)-a)$ for
all $\Delta\in\ensuremath{\mathcal{B}}$. A simple computation gives
$(P_{\Delta}(a),P_{\Delta}(a)-a)=\vert P_{\Delta}(0)\vert^{2}
-(a,P_{\Delta}(0))$. Hence, either $0\in \Delta$ and then
$(P_{\Delta}(a),P_{\Delta}(a)-a)$ is equal to $0$ for all $a\in\ensuremath{\mathfrak{h}}$
, or $0\notin \Delta$ and then $(P_{\Delta}(a),P_{\Delta}(a)-a)>0$ if
$\vert a\vert<\vert P_{\Delta}(0)\vert$. We can take $\epsilon=
\inf_{0\notin \Delta}\vert P_{\Delta}(0)\vert$ in
Lemma \ref{L-a-positif}. $\Box$
\subsubsection{Proof of Proposition \ref{RR-a}}
Since the point $a$ takes value in $\tau$ we identify the
coadjoint orbit $\ensuremath{\mathcal{O}}_{a}$ with $G/G_{\tau}$. Let $\ensuremath{\mathcal{H}}^{a}$ be the
Hamiltonian vector field of the function
$\frac{1}{2}\parallel\mu_{a}\parallel^{2}:M\times G/G_{\tau}\to\ensuremath{\mathbb{R}}$.
To simplify the notations,
$\ensuremath{\mathcal{Y}}_{\tau}$ will denote a small neighborhood
of $\mu^{-1}_{_{G}}(a)$ in the symplectic slice
$\mu^{-1}_{_{G}}(\tau)$ such that
the open subset $\ensuremath{\mathcal{U}}:=(G\times_{G_{\tau}}\ensuremath{\mathcal{Y}}_{\tau})\times
G/G_{\tau}$ is then a neighborhood of
$\mu_{a}^{-1}(0)=G\cdot(\mu_{\tau}^{-1}(a)\times\{\bar{e}\})$ which
satisfies $\overline{\ensuremath{\mathcal{U}}}\cap\{ \ensuremath{\mathcal{H}}^{a}=0\}=\mu_{a}^{-1}(0)$.
Following Definition \ref{def.thom.beta.f}, the localized Riemann-Roch
character $RR^{^G}_{0}(M\times G/G_{\tau},-)$ is computed by means of
the Thom class $\ensuremath{\hbox{\rm Thom}}_{G,[0]}^{\mu_{a}}(M\times G/G_{\tau})\in
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}})$. On the other hand, the localized Riemann-Roch
character $RR^{^{G_{\tau}}}_{0}(\ensuremath{\mathcal{Y}}_{\tau},-)$ is computed by means of
the Thom class $\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau})\in
K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}\ensuremath{\mathcal{Y}}_{\tau})$.
Proposition \ref{RR-a} will follow from a simple relation
between $\ensuremath{\hbox{\rm Thom}}_{G,[0]}^{\mu_{a}}(M\times G/G_{\tau})$ and
$\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau})$.
First, one considers the isomorphism
\begin{eqnarray}\label{def-U-prime}
\phi:\ensuremath{\mathcal{U}}&\to&\ensuremath{\mathcal{U}}'\\
([g;y],[h])&\to&[g;[g^{-1}h],y]\ ,\nonumber
\end{eqnarray}
with $\ensuremath{\mathcal{U}}':=G\times_{G_{\tau}}(G/G_{\tau}\times\ensuremath{\mathcal{Y}}_{\tau})$, and
let $\phi^{*}:K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}')\to K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}})$ be
the induced isomorphism. After one consider the inclusion
$i:G_{\tau}\ensuremath{\hookrightarrow} G$ which induces an isomorphism
$i_{*}:K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}(G/G_{\tau}\times\ensuremath{\mathcal{Y}}_{\tau}))\to
K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}')$ (see subsection \ref{subsec.induction.def}). Let
$j:\ensuremath{\mathcal{Y}}_{\tau}\ensuremath{\hookrightarrow} G/G_{\tau}\times\ensuremath{\mathcal{Y}}_{\tau}$ be the
$G_{\tau}$-invariant inclusion map defined by $j(y):=(\bar{e},y)$. We
have then a pushforward map $j_{!}:
K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}\ensuremath{\mathcal{Y}}_{\tau})\to
K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}(G/G_{\tau}\times\ensuremath{\mathcal{Y}}_{\tau}))$.
Finally we have produced a map $\Theta:=\phi^{*}\circ i_{*}\circ
j_{!}$ from $K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}\ensuremath{\mathcal{Y}}_{\tau})$ to
$K_{G}(\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}})$, such that $\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{U}}}^{G}(\Theta(\sigma))=
{\rm Ind}^{^G}_{_{G_{\tau}}}(\ensuremath{\hbox{\rm Index}}_{\ensuremath{\mathcal{Y}}_{\tau}}^{G_{\tau}}(\sigma))$ for every
$\sigma\in K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}\ensuremath{\mathcal{Y}}_{\tau})$.
Proposition \ref{RR-a} is an immediate consequence of the following
\begin{lem}
We have the equality
$$
\Theta\left(\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau})\right)=
\ensuremath{\hbox{\rm Thom}}_{G,[0]}^{\mu_{a}}(M\times G/G_{\tau})\ .
$$
\end{lem}
{\em Proof }: Let $\mu_{a}':=\mu_{a}\circ\phi^{-1}$ be the moment map
on $\ensuremath{\mathcal{U}}'$, and let $\ensuremath{\mathcal{H}}^{',a}$ be the Hamiltonian vector field
of $\parallel\mu_{a}'\parallel$. For the tangent manifold $\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}'$
we have the decomposition
$$
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{U}}'\simeq G\times_{G_{\tau}}\left(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}\oplus
G\times_{G_{\tau}}(\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}})\oplus
\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}_{\tau}\right)\ .
$$
A small computation gives $\ensuremath{\mathcal{H}}^{',a}(m)=pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}(ha)+
R(m) + \ensuremath{\mathcal{H}}^{\tau}_{a}(y)+ S(m)$
for $m=[g;y,[h]]\in \ensuremath{\mathcal{U}}'$, where $R(m)\in
\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}$ and $S(m)\in \ensuremath{\hbox{\bf T}}_{y}\ensuremath{\mathcal{Y}}_{\tau}$ vanishes
when $m\in G\times_{G_{\tau}}(\{\bar{e}\}\times\ensuremath{\mathcal{Y}}_{\tau})$, i.e.
$[h]=\bar{e}$. Here $\ensuremath{\mathcal{H}}^{\tau}_{a}$ is the Hamiltonian vector field
of the function $\frac{1}{2}\parallel\mu_{\tau}-a\parallel^{2}:
\ensuremath{\mathcal{Y}}_{\tau}\to\ensuremath{\mathbb{R}}$.
The transversally elliptic symbol $\sigma_{1}:=
(\phi^{-1})^{*}(\ensuremath{\hbox{\rm Thom}}_{G,[0]}^{\mu_{a}}(M\times G/G_{\tau}))$
is equal to the exterior product
$$
\sigma_{1}(m, \xi_{1}+\xi_{2}+ v)=
Cl(\xi_{1}-pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}(ha))\odot
Cl(\xi_{2}-R(m))\odot
Cl(v-\ensuremath{\mathcal{H}}^{\tau}_{a}- S(m))\ ,
$$
with $\xi_{1}\in\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}$,
$\xi_{2}\in\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}$, $v\in\ensuremath{\hbox{\bf T}}\ensuremath{\mathcal{Y}}_{\tau}$.
Now we simplify the symbol $\sigma_{1}$ whithout changing its
$K$-theoretic class. Since $\ensuremath{\hbox{\rm Char}}(\sigma_{1})\cap\ensuremath{\hbox{\bf T}}_{G}\ensuremath{\mathcal{U}}'
= G\times_{G_{\tau}}(\{\bar{e}\}\times\ensuremath{\mathcal{Y}}_{\tau})$, we can transform
$\sigma_{1}$ through the
$G_{\tau}$-invariant diffeomorphism $h=e^{X}$ from a neighborhood
of $0$ in $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}$ to a neighborhood of $\bar{e}$ in
$G/G_{\tau}$. This gives $\sigma_{2}\in
K_{G}(\ensuremath{\hbox{\bf T}}_{G}(G\times_{G_{\tau}}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}\times
\ensuremath{\mathcal{Y}}_{\tau})))$ defined by
\begin{eqnarray*}
\lefteqn{\sigma_{2}([g,X,y], \xi_{1}+\xi_{2}+ v)
=}\\
& &
Cl(\xi_{1}-pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}(e^{X}a))\odot
Cl(\xi_{2}-R(m))\odot
Cl(v-\ensuremath{\mathcal{H}}^{\tau}_{a}- S(m))\ .
\end{eqnarray*}
Now trivial homotopies link $\sigma_{2}$
with the symbol $\sigma_{3}$, where we have removed the terms
$R(m)$ and $S(m)$, and where we have replaced
$pr_{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}}(e^{X}a)=[X,a]+o([X,a])$ by the term
$[X,a]$:
$$
\sigma_{3}([g,X,y], \xi_{1}+\xi_{2}+ v)=
Cl(\xi_{1}-[X,a])\odot
Cl(\xi_{2})\odot
Cl(v-\ensuremath{\mathcal{H}}^{\tau}_{a})\ .
$$
Now, we get $\sigma_{3}=i_{*}( \sigma_{4})$ where the symbol
$\sigma_{4}\in K_{G_{\tau}}(\ensuremath{\hbox{\bf T}}_{G_{\tau}}(\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}\times
\ensuremath{\mathcal{Y}}_{\tau}))$ is defined by
$$
\sigma_{4}(X,y;\xi_{2}+ v)=
Cl(-[X,a])\odot Cl(\xi_{2})\odot Cl(v-\ensuremath{\mathcal{H}}^{\tau}_{a})\ .
$$
So $\sigma_{4}$ is equal to the exterior product of $(y,v)\to
Cl(v-\ensuremath{\mathcal{H}}^{\tau}_{a})$, which is
$\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau})$, with the
transversally elliptic symbol on $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}$:
$(X,\xi_{2})\to Cl(-[X,a])\odot Cl(\xi_{2})$. As in
Lemma \ref{lem.indice.wedge.V.inverse}, we see that the
$K$-theoretic class of this former symbol is equal to $k_{!}(\ensuremath{\mathbb{C}})$
where $k:\{0\}\ensuremath{\hookrightarrow}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\tau}$. This shows that
$$
\sigma_{4}=k_{!}(\ensuremath{\mathbb{C}})\odot
\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau})=
j_{!}(\ensuremath{\hbox{\rm Thom}}_{G_{\tau},[0]}^{\mu_{\tau}-a}(\ensuremath{\mathcal{Y}}_{\tau}))\ .
$$
$\Box$
\section{Appendix A: G=SU(2)}
We restrict our attention to an action of $G=SU(2)$ on a compact
manifold $M$. We suppose that $M$ is endowed with a $G$-invariant
almost complex structure $J$ and an abstract moment map $f:M\to\ensuremath{\mathfrak{g}}$.
In this situation, the decomposition $RR^{^{G}}(M,-)=
\sum_{\beta\in\ensuremath{\mathcal{B}}_{_{G}}}RR_{\beta}^{^{G}}(M,-)$ becomes simple.
Let $S^{1}$ be the maximal torus of $SU(2)$, and $f_{_{S^{1}}}:M\to
\ensuremath{\mathbb{R}}$ the induced moment map for the $S^{1}$-action. The critical set
$\{\ensuremath{\mathcal{H}}^{^G}=0\}$ has a particularly simple expression: $
\{\ensuremath{\mathcal{H}}^{^G}=0\}=f^{-1}(0)\cup G.M^{S^{1}}_+$,
where $M^{S^1}_{+}$ is the union of the connected components
$F\subset M^{S^1}$ with $f_{S^1}(F)>0$. Note that the critical set
$\{\ensuremath{\mathcal{H}}^{^{S^1}}=0\}$ is equal to
$f_{_{S^{1}}}^{-1}(0)\cup M^{S^{1}}$,
\medskip
\underline{The non-symplectic case}
\medskip
Here the induction formula of Theorem \ref{th.induction.G.H}, and
Proposition \ref{prop.loc.RR.G.beta} gives
\begin{equation}\label{exemple1}
RR^{^{G}}(M,E)=RR^{^{G}}_{0}(M,E)+
{\rm Hol}^{^G}_{_{S^1}}\Big(\Theta(E)(t).(1-t^{-2})\Big)
\end{equation}
where $\Theta(E)\in R^{-\infty}(S^1)$ is determined by
\begin{equation}\label{exemple1.1}
\Theta(E)= (-1)^{r_{\ensuremath{\mathcal{N}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR^{^{S^1}}(M^{S^1}_{+},E\vert_{M^{S^1}_{+}}\otimes\det
\ensuremath{\mathcal{N}}^{+}\otimes S^k((\ensuremath{\mathcal{N}}\otimes\ensuremath{\mathbb{C}})^{+}))\ .
\end{equation}
Here $\ensuremath{\mathcal{N}}\to M^{S^1}_{+}$ is the normal bundle of
$M^{S^1}_{+}$ in $M$.
\medskip
\underline{The Hamiltonian case}
\medskip
Here we suppose that $(M,\omega)$ is a symplectic manifold, with
moment map $\mu$ and a $\omega$-compatible almost complex structure $J$.
Let $\ensuremath{\mathcal{Y}}=\mu^{-1}(\ensuremath{\mathbb{R}}_{>0})$ be the symplectic slice associated
to the interior of the Weyl chamber $\ensuremath{\mathbb{R}}_{>0}\subset Lie(S^1)$.
The induction formula of Theorem \ref{th.RR.G.beta.hamilton}
gives
\begin{equation}\label{exemple2}
RR^{^{G}}(M,E)=RR^{^{G}}_{0}(M,E)+
{\rm Hol}^{^G}_{_{S^1}}\Big(\widetilde{\Theta}(E)\Big)
\end{equation}
where $\widetilde{\Theta}(E)\in R^{-\infty}(S^1)$ is determined by
\begin{equation}\label{exemple2.2}
\widetilde{\Theta}(E)= (-1)^{r_{\widetilde{\ensuremath{\mathcal{N}}}}}\sum_{k\in\ensuremath{\mathbb{N}}}
RR^{^{S^1}}(M^{S^1}_{+},E\vert_{M^{S^1}_{+}}\otimes\det
\widetilde{\ensuremath{\mathcal{N}}}^{+}\otimes S^k((\widetilde{\ensuremath{\mathcal{N}}}\otimes\ensuremath{\mathbb{C}})^{+}))\ .
\end{equation}
Here $\widetilde{\ensuremath{\mathcal{N}}}\to M^{S^1}_{+}$ is the normal bundle of
$M^{S^1}_{+}$ in $\ensuremath{\mathcal{Y}}$.
\medskip
Recall that the irreducible characters $\phi_{n}$ of $G=SU(2)$ are
labelled by $\ensuremath{\mathbb{Z}}_{\geq 0}$, and are completely determined by the
relation $\phi_{n}={\rm Hol}^{^G}_{_{S^1}}(t^n)$ in $R(G)$
(See Lemma \ref{lem.Weyl}). Hence the component
${\rm Hol}^{^G}_{_{S^1}}\Big(\Theta(E)(t).(1-t^{-2})\Big)$
of (\ref{exemple1}) does not contain the trivial character $\phi_{0}$
if $\Theta(E)=\sum_{n\in \ensuremath{\mathbb{Z}}}a_{n}t^n$ with
\begin{equation}\label{plus.grand.3}
a_{n}\neq 0\Longrightarrow n\geq 3\ .
\end{equation}
(\ref{exemple1.1}) tells us that
(\ref{plus.grand.3}) is satisfied if the weights for the
action of $S^1$ in the fibers of the complex vector bundle
$E\vert_{M^{S^1}_{+}}\otimes\det\ensuremath{\mathcal{N}}^{+}$ are all
bigger than $3$.
\medskip
The conditions are weaker in the `Hamiltonian' situation. The term
\break ${\rm Hol}^{^G}_{_{S^1}}(\widetilde{\Theta}(E))$
of (\ref{exemple2}) does not contain the trivial character $\phi_{0}$
if $\widetilde{\Theta}(E)=\sum_{n\in \ensuremath{\mathbb{Z}}}a_{n}t^n$ with
\begin{equation}\label{plus.grand.1}
a_{n}\neq 0\Longrightarrow n\geq 1\ ,
\end{equation}
and this condition is fulfilled if the weights for the
action of $S^1$ in the fibers of the complex vector bundle
$E\vert_{M^{S^1}_{+}}\otimes\det\widetilde{\ensuremath{\mathcal{N}}}^{+}$
are all bigger than $1$. Here we have another important difference:
the vector bundle $\widetilde{\ensuremath{\mathcal{N}}}^{+}\to M^{S^1}_{+}$ is not equal to
the zero bundle if $0\in\mu(M)$ (see Lemma \ref{lem.N.beta.plus}).
We see finally that, in the Hamiltonian case, the condition
`{\em $E$ is $\mu$-positive}' implies
$$
0\in\mu(M)\Longrightarrow \left[ RR^{^{G}}(M,E)\right]^{G}=
\left[ RR^{^{G}}_{0}(M,E)\right]^G\ .
$$
\section{Appendix B: Induction map and multiplicities}
Let $G$ be a compact connected Lie group, with maximal torus $H$, and
$\ensuremath{\mathfrak{h}}_{+}^{*}\subset \ensuremath{\mathfrak{h}}^{*}=(\ensuremath{\mathfrak{g}}^{*})^{H}$ some choice of
positive Weyl chamber. We denote by $\ensuremath{\mathfrak{R}}_{+}$ the associated system of
positive roots, and we label the irreducible representations of $G$
by the set $\Lambda^{*}_{+}=\Lambda^{*}\cap \ensuremath{\mathfrak{h}}^{*}_{+}$ of dominant
weights. For any weights $\alpha\in \Lambda^{*}$ we denote by
$H\to \ensuremath{\mathbb{C}}^{*},\ h\mapsto h^{\alpha}$ the corresponding
character : $(\exp(X))^{\alpha}=e^{\imath\langle \alpha,X\rangle}$
for $X\in \ensuremath{\mathfrak{h}}$.
Let $W$ be the Weyl group of $(G,H)$, and ${\rm L}^{2}(H)$ be the vector space of
square integrable complex functions on $H$. For $f\in{\rm L}^{2}(H)$, we consider
$J(f)$ \break $=\sum_{w\in W}(-1)^{w}\, w.f$,
where $W\to \{1,-1\},\ w\to (-1)^{w}$, is the signature operator and
$w.f\in {\rm L}^{2}(H)$ is defined by $w.f(h)=f(w^{-1}.h),\ h\in H$
(see Section 7.4 of \cite{Bourbaki.Lie.9}). The map
$\frac{1}{\vert W\vert}J$ is the orthogonal projection from ${\rm L}^{2}(H)$
to the space of $W$-anti-invariant elements of ${\rm L}^{2}(H)$.
Let $\rho\in\ensuremath{\mathfrak{h}}^{*}$ be the half sum of the positive roots. The
function $H\to \ensuremath{\mathbb{C}}^{*},\ h\mapsto h^{\rho}$ is well defined as an
element of ${\rm L}^{2}(H)$ (even if $\rho$ is not a weight). The Weyl's character
formula can be written in the following way. For any dominant weight
$\lambda\in \Lambda^{*}_{+}$, the restriction
$\chi_{_{\lambda}}^{_{G}}\vert_{H}$
of the irreducible $G$-character $\chi_{_{\lambda}}^{_G}$ satisfies
\begin{equation}\label{eq.Weyl.1}
J(h^{\rho}).\chi_{_{\lambda}}^{_G}\vert_{H}= J(h^{\lambda+\rho})\quad {\rm
in}\quad {\rm L}^{2}(H)\ .
\end{equation}
For our purpose we give an expression of the character $\chi_{_{\lambda}}^{_G}$
through the induction map ${\rm Ind}^{^G}_{_H}:\ensuremath{\mathcal{C}^{-\infty}}(H)\to\ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$ (see
(\ref{eq:fonction.induction})).
Consider the affine action of the Weyl group on
the set of weights : $w\circ \lambda =w.(\lambda+\rho)-\rho$ for $w\in
W$ and $\lambda\in \Lambda^{*}$.
\begin{lem}\label{lem.Weyl}
1) For any dominant weight $\lambda\in \Lambda^{*}_{+}$, the
character $\chi_{_{\lambda}}^{_G}$ is determined by the relation
$\chi_{_{\lambda}}^{_G}= {\rm Ind}^{^G}_{_H}\Big(h^{\lambda}\prod_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}
(1-h^{\alpha})\Big)$ in $\ensuremath{\mathcal{C}^{-\infty}}(G)^{G}$.
\noindent 2) For $\lambda\in\Lambda^{*}$ and $w\in W$, we have
${\rm Ind}^{^G}_{_H}(h^{w\circ\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))=$ \break
$(-1)^{w}{\rm Ind}^{^G}_{_H}(h^{\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))$.
\noindent 3) For any weight $\lambda$, the following statements are
equivalent :
a) $\ {\rm Ind}^{^G}_{_H}(h^{\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))=0$,
b) $\ W\circ \lambda\cap \Lambda^{*}_{+}=\emptyset$,
c) The element $\lambda+\rho$ is not a regular element of $\ensuremath{\mathfrak{h}}^{*}$.
\end{lem}
{\em Proof of 1) :}To prove it, we need the
following relations proved in \cite{Bourbaki.Lie.9}[section 7.4] :
i) $\ \overline{J(h^{\rho})}=h^{-\rho}\prod_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha})$,
\quad
ii) $\ J(h^{\rho}).\overline{J(h^{\rho})}=\prod_{\alpha\in\ensuremath{\mathfrak{R}}}(1-h^{\alpha})$.
Let $dg$ and $dt$ be respectively the normalized Haar measures on $G$ and $H$.
For any $f\in\ensuremath{\mathcal{C}^{\infty}}(G)^{G}$ we have
\begin{eqnarray*}
\int_{G}\chi^{_G}_{_{\lambda}}(g)\, f(g)\, dg &=&
\frac{1}{\vert W\vert}\int_{H}\chi_{_{\lambda}}^{_G}\vert_{H}(h)\,
\Pi_{\alpha\in\ensuremath{\mathfrak{R}}}(1-h^{\alpha})\, f\vert_{H}(h)\, dh
\hspace{2cm} [1] \\
&=&
\frac{1}{\vert W\vert}\int_{H}J(h^{\lambda+\rho})\,
\overline{J(h^{\rho})}\, f\vert_{H}(h)\, dh
\hspace{3cm} [2] \\
&=&
\int_{H}h^{\lambda+\rho}\,\overline{J(h^{\rho})}\, f\vert_{H}(h)\,dh
\hspace{4cm} [3] \\
&=&
\int_{H} h^{\lambda}\,\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha})\,
f\vert_{H}(h)\, dh \ .
\hspace{3cm} [4]
\end{eqnarray*}
The first equality is the Weyl integration formula. The equality
$[2]$ comes from ii) and (\ref{eq.Weyl.1}). Since $\frac{1}{\vert
W\vert} J$ is the orthogonal projection on
${\rm L}^{2}(H)^{W-anti-invariant}$ and $h\mapsto \overline{J(h^{\rho})}\,
f\vert_{H}(h)$ is $W$-anti-invariant we obtain the third equality.
The equality $[4]$ comes from i).
{\em Proof of 2) : } From $i)$, wee see that
$h^{w\circ\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha})=
h^{w(\lambda+\rho)}\overline{J(h^{\rho})}=$ \break
$(-1)^{w}\, w^{-1}.( h^{\lambda+\rho}\overline{J(h^{\rho})})= (-1)^{w}\,
w^{-1}.(h^{\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))$, hence
the relation $2)$ is proved since ${\rm Ind}^{^G}_{_H}$ is $W$-invariant.
{\em Proof of 3) :} The implication $a)\Longrightarrow b)$ is an
immediate consequence of $1)$ and $2)$. Proposition 3 in section 7.4 of
\cite{Bourbaki.Lie.9} tells us that $\{J(h^{\lambda'+\rho}),\
\lambda'\in \Lambda^{*}_{+}\}$ is an orthogonal basis of the Hilbert
space ${\rm L}^{2}(H)^{W-anti-invariant}$. For $\lambda\in \Lambda^{*}$ and
$\lambda'\in\Lambda^{*}_{+}$ we have
$<J(h^{\lambda+\rho}),J(h^{\lambda'+\rho})>_{{\rm L}^{2}}=$
$\vert W\vert<J(h^{\lambda+\rho}),h^{\lambda'+\rho}>_{{\rm L}^{2}}=
\vert W\vert\sum_{w\in W}(-1)^w\int_{T}t^{w\circ\lambda-\lambda'}dt$.
Thus, the condition
$W\circ \lambda\cap \Lambda^{*}_{+}=\emptyset$ is equivalent to
$J(h^{\lambda+\rho})=0$. But the equality $[2]$ gives
${\rm Ind}^{^G}_{_H}(h^{\lambda}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))=$ \break
$\frac{1}{\vert W\vert}
{\rm Ind}^{^G}_{_H}(J(h^{\lambda+\rho})h^{-\rho}\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))$,
hence $J(h^{\lambda+\rho})=0$ implies the point $a)$. We have proved
that $b)\Longrightarrow a)$. Finally we see that
$J(h^{\lambda+\rho})=0 \Longleftrightarrow \exists w\in W,
w.(\lambda+\rho)=\lambda+\rho \Longleftrightarrow \lambda+\rho\ {\rm
is\ not\ a\ regular\ value\ of\ } \ensuremath{\mathfrak{h}}^{*}$. We have proved
that $b)\Longleftrightarrow c)$. $\Box$
\bigskip
From the previous Lemma, we see that $v\mapsto
{\rm Ind}^{^G}_{_H}(v(h)\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha}))$ is the holomorphic
induction map
\begin{equation}\label{eq.holomorphe.G.H}
{\rm Hol}^{^G}_{_H}:R(H)\to R(G)\ .
\end{equation}
We keep the same notation for the extended map
${\rm Hol}^{^G}_{_H}:R^{-\infty}(H)\to R^{-\infty}(G)$. Note that the choice of a
positive Weyl chamber $\ensuremath{\mathfrak{h}}^{*}_{+}$ determines a complex structure on
$\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}$, and $\Pi_{\alpha\in\ensuremath{\mathfrak{R}}_{+}}(1-h^{\alpha})$ is the
trace of the virtual $H$-representation
$\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\, \in R(H)$. Then the map ${\rm Hol}^{^G}_{_H}$
will be defined simply by the relation
${\rm Hol}^{^G}_{_H}(v)={\rm Ind}^{^G}_{_H}(v\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$.
\begin{rem}\label{wedge.C-wedge.R}
The relations i) and ii) used in the proof of the past lemma show
that $\sum_{w\in W}w. \prod_{\alpha>0}(1-h^{\alpha})=
\sum_{w\in W}w.(\overline{J(h^{\rho})}h^{\rho})=
\overline{J(h^{\rho})}.J(h^{\rho})=
\prod_{\alpha}(1-h^{\alpha})$. In other words
$\sum_{w\in W}w.\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}=
(\wedge_{\ensuremath{\mathbb{R}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})\otimes\ensuremath{\mathbb{C}}=
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\,
\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\overline{\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}}$
in $R(H)$. These equalities give
\begin{equation}\label{eq.Weyl.Hol}
{\rm Ind}^{^G}_{_H}\Big((\sum_w w.\phi)\, \wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}}\Big)=
{\rm Ind}^{^G}_{_H}(\phi\,\wedge_{\ensuremath{\mathbb{R}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})
\end{equation}
since ${\rm Ind}^{^G}_{_H}$ is $W$-invariant. The Weyl integration formula is usually
state as the relation $f=\frac{1}{\vert W\vert}
{\rm Ind}^{^G}_{_H}(f\vert_{H}\,\wedge_{\ensuremath{\mathbb{R}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$ for any
$f\in \ensuremath{\mathcal{C}^{\infty}}(G)^G$. But $f\vert_{H}$ is $W$-invariant, so
(\ref{eq.Weyl.Hol}) gives
$\frac{1}{ \vert W\vert}
{\rm Ind}^{^G}_{_H}(f\vert_{H}\,\wedge_{\ensuremath{\mathbb{R}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})=
{\rm Ind}^{^G}_{_H}(f\vert_{H}\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{h}})$.
Finally, for any $\phi\in R(G)$, the Weyl integration formula is
equivalent to the following equality in $R(G)$:
$$
\phi={\rm Hol}^{^G}_{_H}(\phi\vert_{H}) .
$$
\end{rem}
\begin{rem}\label{hol.egale.1}
A weight $\lambda$ satisfies ${\rm Hol}^{^G}_{_H}(h^{\lambda})=\pm 1$ if and only if
$0\in\,W\circ \lambda\cap\Lambda^*_{+}$, that is $\lambda=
-(\rho -w.\rho)$ for some $w\in W$. But a small computation shows that
$\rho -w.\rho=\sum_{\alpha>0,w^{-1}.\alpha<0}\alpha$, hence
$\langle \rho -w.\rho,X\rangle\geq 0$ for any $X\in\ensuremath{\mathfrak{h}}_{+}$.
Finally the equality ${\rm Hol}^{^G}_{_H}(h^{\lambda})=\pm 1$ implies that
$\langle \lambda,X\rangle\leq 0$ for any $X\in\ensuremath{\mathfrak{h}}_{+}$.
\end{rem}
\medskip
Consider now the stabiliser $G_{\beta}$ of the non-zero element
$\beta\in\ensuremath{\mathfrak{h}}_{+}$. The subgroup $H$ is also a maximal torus of $G_{\beta}$.
The Weyl group $W_{\beta}$ of $(G_{\beta},H)$ is identified with
$\{w\in W,\ w.\beta=\beta\}$. We consider a Weyl chamber
$\ensuremath{\mathfrak{h}}^{*}_{+,\beta}\subset\ensuremath{\mathfrak{h}}^{*}$ for $G_{\beta}$ that contains
the Weyl chamber $\ensuremath{\mathfrak{h}}^{*}_{+}$ of $G$. The irreducible representations
$\chi_{_{\lambda}}^{_{G_{\beta}}},\ \lambda\in\Lambda_{+,\beta}^{*}$ of
$G_{\beta}$ are labelled by the set $\Lambda_{+,\beta}^{*}=
\Lambda^{*}\cap\ensuremath{\mathfrak{h}}_{+,\beta}^{*}$ of dominant weights.
We have a unique `holomorphic' induction map ${\rm Hol}^{^G}_{_{G_{\beta}}}:R(G_{\beta})\to
R(G)$ such that ${\rm Hol}^{^G}_{_H}={\rm Hol}^{^G}_{_{G_{\beta}}}\circ{\rm Hol}^{^{G_{\beta}}}_{_H}$. This map is defined
precisely by the equation\footnote{We take on $\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}$
the complex structure defined by $\beta$.}
\begin{equation}\label{eq:Hol-G-beta}
{\rm Hol}^{^G}_{_{G_{\beta}}}(v)={\rm Ind}^{^G}_{_{G_{\beta}}}\left(v\,\wedge_{\ensuremath{\mathbb{C}}}^{\bullet}\ensuremath{\mathfrak{g}}/\ensuremath{\mathfrak{g}}_{\beta}\right)\ ,
\end{equation}
for every $v\in R(G_{\beta})$.
\bigskip
We finish this appendix with some general remarks about
$P$-transversally elliptic symbols on a compact manifold $M$, when a
subgroup $\ensuremath{\mathbb{T}}$ in the center of $P$ acts trivially on $M$.
More precisely, let $H$ be a compact maximal torus in $P$, $\ensuremath{\mathfrak{h}}_{+}$
be a choice of a positive Weyl chamber in the Lie algebra $\ensuremath{\mathfrak{h}}$ of
$H$, and let $\beta\in \ensuremath{\mathfrak{h}}_{+}$ be a non-zero element in the center
of the Lie algebra $\ensuremath{\mathfrak{p}}$ of $P$\footnote{The Lie group $P$ is
supposed connected then $\beta\in(\ensuremath{\mathfrak{p}})^{P}$.}. We suppose here that
the subtorus $\ensuremath{\mathbb{T}}\subset H$, which is equal to the closure of
$\{\exp(t.\beta),\ t\in\ensuremath{\mathbb{R}}\}\ $, acts trivially on $M$.
Every $P$-equivariant complex vector bundle $E\to M$ can be decomposed
relatively to the $\ensuremath{\mathbb{T}}$-action:
$E=\oplus_{a\in\hat{\ensuremath{\mathbb{T}}}}E^a\otimes\ensuremath{\mathbb{C}}_{a}$, where
$E^{a}:=\hom_{\ensuremath{\mathbb{T}}}(E,\ensuremath{\mathbb{C}}^*_{a})$\footnote{The torus $\ensuremath{\mathbb{T}}$ acts on
the complex line $\ensuremath{\mathbb{C}}_{a}$ with the representation $t\to t^a$.} is a
$P$-complex vector bundle with a trivial action of $\ensuremath{\mathbb{T}}$. Then,
each $P$-equivariant symbol $\sigma:p^{*}(E_{1})\to p^{*}(E_{2})$
where $E_{1},E_{2}$ are $P$-equivariant complex vector bundles over
$M$, and where $p:\ensuremath{\hbox{\bf T}} M\to M$ is the canonical projection, admits a
finite $P\times\ensuremath{\mathbb{T}}$-equivariant decomposition
\begin{equation}\label{eq:sigma-a}
\sigma=\sum_{a\in\hat{\ensuremath{\mathbb{T}}}}\sigma^a\otimes\ensuremath{\mathbb{C}}_{a}.
\end{equation}
Here $\sigma^a:p^{*}(E^a_{1})\to p^{*}(E^a_{2})$ is a $P$-equivariant
symbol, trivial for the $\ensuremath{\mathbb{T}}$-action.
Let us consider the inclusion map $i:\ensuremath{\mathbb{T}}\ensuremath{\hookrightarrow} H$,
with the induced maps $i:Lie(\ensuremath{\mathbb{T}}) \to \ensuremath{\mathfrak{h}}$ at the level of Lie algebra and
$i^{*}:\ensuremath{\mathfrak{h}}^{*}\to Lie(\ensuremath{\mathbb{T}}) ^{*}$. Note that $i^{*}(\lambda)$ is a
weight for $\ensuremath{\mathbb{T}}$ if $\lambda$ is a weight for $H$.
\begin{lem}\label{lem.multiplicites.tore}
Let $M$ be a $P$-manifold with the same properties as above.
Let $\sigma:p^{*}(E_{1})\to p^{*}(E_{2})$ be a $P$-equivariant
transversally elliptic symbol over $M$ and denote by
$m_{\lambda}(\sigma),\, \lambda\in \Lambda^{*}_{P,+}$, the
multiplicities of its index : $\ensuremath{\hbox{\rm Index}}_{M}^P(\sigma)=
\sum_{\lambda\in \Lambda^{*}_{P,+}}
m_{\lambda}(\sigma)\chi^{_P}_{_{\lambda}}$. Then, if
$m_{\lambda}(\sigma)\neq 0$, the weight $a=i^{*}(\lambda)$
occurs in the decomposition (\ref{eq:sigma-a}).
\end{lem}
\begin{coro}\label{coro.multiplicites.tore}
Suppose that the weights $a\in \hat{\ensuremath{\mathbb{T}}}$ which occur in the
decomposition (\ref{eq:sigma-a}) satisfy
$\langle a,\beta\rangle\geq \eta$ for some fixed $\eta\in\ensuremath{\mathbb{R}}$.
Then, for the multiplicities, we get
$$
m_{\lambda}(\sigma)\neq 0\Longrightarrow
\langle \lambda,\beta\rangle\geq \eta\ .
$$
In particular, $\ensuremath{\hbox{\rm Index}}_{M}^P(\sigma)$ does not contain the
trivial representation when $\eta>0$.
\end{coro}
\begin{rem}The previous Lemma and Corollary remain true if $M$ is a
$P$-invariant open subset of a compact $P$-manifold.
\end{rem}
For the Corollary, we have just to notice
that\footnote{We use the same notations for
$\beta\in Lie(\ensuremath{\mathbb{T}}) $ and $i(\beta)\in\ensuremath{\mathfrak{h}}$.}
$\langle\lambda,\beta\rangle
=\langle a,\beta\rangle$ for
$a=i^{*}(\lambda)$. Then, if we have
$\langle a,\beta\rangle\geq \eta$ for all $\ensuremath{\mathbb{T}}$-weights
occurring in $\sigma$, we get $\langle\lambda,\beta\rangle\geq \eta $
for every $\lambda$ such that $m_{\lambda}(\sigma)\neq 0$.
\medskip
{\em Proof of Lemma \ref{lem.multiplicites.tore}}:
Let $P'$ be a Lie subgroup of $P$ such that
$r:\ensuremath{\mathbb{T}}\times P'\to P,\ r(t,g)=t.g $, is a finite covering of $P$.
The map $r$ induces $r^{*}:K_{P}(\ensuremath{\hbox{\bf T}}_{P}M)\to K_{\ensuremath{\mathbb{T}}\times
P'}(\ensuremath{\hbox{\bf T}}_{P'}M)$\footnote{Note that $\ensuremath{\hbox{\bf T}}_{P'}M=\ensuremath{\hbox{\bf T}}_{P}M$ because $\ensuremath{\mathbb{T}}$
acts trivially on $M$.} and an injective map $r^{*}:R^{-\infty}(P)\to
R^{-\infty}(\ensuremath{\mathbb{T}}\times P')$, such that
$\ensuremath{\hbox{\rm Index}}_{M}^{\ensuremath{\mathbb{T}}\times P'}(r^{*}\sigma)=r^{*}(\ensuremath{\hbox{\rm Index}}_{M}^P(\sigma))$.
The decomposition (\ref{eq:sigma-a}) can be read through the
identification $K_{\ensuremath{\mathbb{T}}\times P'}(\ensuremath{\hbox{\bf T}}_{P'}M)=$\break
$K_{P'}(\ensuremath{\hbox{\bf T}}_{P'}M)\otimes R(\ensuremath{\mathbb{T}})$: we have
$r^*\sigma=\sum_{a\in\hat{\ensuremath{\mathbb{T}}}}\sigma^a\otimes\ensuremath{\mathbb{C}}_{a}$
with $\sigma^a\in K_{P'}(\ensuremath{\hbox{\bf T}}_{P'}M)$. Hence
\begin{equation}\label{decomposition-1}
\ensuremath{\hbox{\rm Index}}_{M}^{\ensuremath{\mathbb{T}}\times P'}(r^{*}\sigma)(t,g)=\sum_{a\in\hat{\ensuremath{\mathbb{T}}}}
\ensuremath{\hbox{\rm Index}}_{M}^{P'}(\sigma^a)(g).\, t^a\ ,\quad (t,g)\in \ensuremath{\mathbb{T}}\times P'\ .
\end{equation}
The irreducible characters $\chi^{_P}_{_{\lambda}}$ satisfy
$r^{*}\chi^{_P}_{_{\lambda}}(t,g)=\chi^{_P}_{_{\lambda}}\vert_{P'}(g).\,
t^{i^{*}(\lambda)}$. If we start from the
decomposition $\ensuremath{\hbox{\rm Index}}_{M}^P(\sigma)=\sum_{\lambda\in \Lambda^{*}_{P,+}}
m_{\lambda}(\sigma)\chi^{_P}_{_{\lambda}}$ relative to the
irreducible characters of $P$, we get
\begin{equation}\label{decomposition-2}
r^{*}\left(\ensuremath{\hbox{\rm Index}}_{M}^{\ensuremath{\mathbb{T}}\times P'}(\sigma)\right)(t,g)=
\sum_{a\in\hat{\ensuremath{\mathbb{T}}}}\left(\sum_{i^{*}(\lambda)=a}
m_{\lambda}(\sigma)\chi^{_P}_{_{\lambda}}\vert_{P'}(g)\right) .\,
t^a\ ,
\end{equation}
for any $(t,g)\in \ensuremath{\mathbb{T}}\times P'$.
If we compare (\ref{decomposition-1}) and
(\ref{decomposition-2}), we get $\ensuremath{\hbox{\rm Index}}_{M}^{P'}(\sigma^a)=
\sum_{i^{*}(\lambda)=a}m_{\lambda}(\sigma)
\chi^{_P}_{_{\lambda}}\vert_{P'}$. The map $r^{*}:R^{-\infty}(P)\to
R^{-\infty}(\ensuremath{\mathbb{T}}\times P')$ is injective, so $\sum_{i^{*}(\lambda)=a}
m_{s\lambda}(\sigma)\chi^{_P}_{_{\lambda}}\vert_{P'}=0$ if and only
if $m_{\lambda}(\sigma)=0$ for every $\lambda$ satisfying
$i^{*}(\lambda)=a$. Hence if the multiplicity
$m_{\lambda}(\sigma)$ is non zero, the element $a=i^{*}(\lambda)$
is a weight for the action of $\ensuremath{\mathbb{T}}$ on
$\sigma:p^{*}(E_{1})\to p^{*}(E_{2})$. $\Box$
\bigskip
{\small
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Q: SCNProgram read from depth buffer I am investigating some advanced rendering in Scenekit, and what SCNTechnique and SCNProgram can be used for.
I would like to implement a multi-pass render like so:
*
*First; render the full scene using the in-built PBR shader,
*Second; render a single node with a SCNProgram which accesses both the colour and depth output from the first pass.
I know I can do something like the following with a SCNTechnique:
...
<key>passes</key>
<dict>
<key>pass_scene</key>
<dict>
<key>draw</key>
<string>DRAW_SCENE</string>
<key>excludeCategoryMask</key>
<string>The bit mask of the node to exclude</string>
<key>outputs</key>
<dict>
<key>color</key>
<string>COLOR</string>
<key>depth</key>
<string>DEPTH</string>
</dict>
</dict>
<key>pass_object</key>
<dict>
<key>draw</key>
<string>DRAW_SCENE</string>
<key>includeCategoryMask</key>
<string>The bit mask of the node to exclude</string>
<key>inputs</key>
<dict>
<key>color</key>
<string>COLOR</string>
<key>depth</key>
<string>DEPTH</string>
</dict>
<key>outputs</key>
<dict>
<key>color</key>
<string>COLOR</string>
</dict>
<key>metalVertexShader</key>
<string>myCustomVertShader</string>
<key>metalFragmentShader</key>
<string> myCustomFragShader </string>
</dict>
</dict>
<key>sequence</key>
<array>
<string>pass_scene</string>
<string>pass_object</string>
</array>
...
But I would like to use a SCNProgram for the object render, instead of specifying metalVertexShader & metalFragmentShader in the SCNTechnique.
I can't figure out how to pass through the colour and depth buffers from the first pass of the SCNTechnique into the SCNProgram for a second pass.
Does anyone know how to go about this? The documentation is difficult to piece together, and hard to debug issues.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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{"url":"https:\/\/guide.coap.online\/copt\/en-doc\/amplinterface.html","text":"# AMPL Interface\uf0c1\n\nAMPL is an algebraic modeling language for describing large-scale complex mathematical problems, it was hooked to many commercial and open-source mathematical optimizers, with various data interfaces and extensions, and received high popularity among both industries and institutes, see Who uses AMPL? for more information. The solver coptampl uses Cardinal Optimizer to solve linear programming, convex quadratic programming, convex quadratic constrainted programming and mixed integer programming problems. Normally coptampl is invoked by AMPL\u2019s solve command, which gives the invocation:\n\ncoptampl stub -AMPL\n\n\nin which stub.nl is an AMPL generic output file (possibly written by 'ampl -obstub' or 'ampl -ogstub'). After solving the problem, coptampl writes a stub.sol file for use by AMPL\u2019s solve and solution commands. When you run AMPL, this all happens automatically if you give the AMPL commands:\n\nampl: option solver coptampl;\nampl: solve;\n\n\n## Installation Guide\uf0c1\n\nTo use coptampl in AMPL, you must have a valid AMPL license and make sure that you have installed Cardinal Optimizer and setup its license properly, see How to install Cardinal Optimizer for details. Be sure to check if it satisfies the following requirements for different operating systems.\n\n### Windows\uf0c1\n\nOn Windows platform, the coptampl.exe utility and the copt.dll dynamic library contained in the Cardinal Optimizer must appear somewhere in your user or system PATH environment variable (or in the current directory).\n\nTo test if your setting meets the above requirements, you can check it by executing commands below in command prompt:\n\ncoptampl -v\n\n\nAnd you are expected to see output similar to the following on screen:\n\nAMPL\/x-COPT Optimizer [6.0.0] (windows-x86), driver(20220526), MP(20220526)\n\nIf the commands failed, then you should recheck your settings.\n\n### Linux\uf0c1\n\nOn Linux platform, the coptampl utility must appears somewhere in your $PATH environment variable, while the libcopt.so shared library must appears somewhere in your $LD_LIBRARY_PATH environment variable.\n\nSimilarly, to test if your setting meets the above requirements, just execute commands below in shell:\n\ncoptampl -v\n\n\nAnd you are expected to see output similar to the following on screen:\n\nAMPL\/x-COPT Optimizer [6.0.0] (linux-x86), driver(20220526), MP(20220526)\n\n### MacOS\uf0c1\n\nOn MacOS platform, the coptampl utility must appears somewhere in your $PATH environment variable, while the libcopt.dylib dynamic library must appears somewhere in your $DYLD_LIBRARY_PATH environment variable.\n\nYou can execute commands below in shell to see if your settings meets the above requirements:\n\ncoptampl -v\n\n\nAnd you are expected to see output similar to the following on screen:\n\nAMPL\/x-COPT Optimizer [6.0.0] (macos-x86), driver(20220526), MP(20220526)\n\n## Solver Options and Exit Codes\uf0c1\n\nThe coptampl utility offers some options to customize its behavior. Uers can control it by setting the environment variable copt_options or use AMPL\u2019s option command. To see all available options, please invoke:\n\ncoptampl -=\n\n\nThe supported parameters and their interpretation for current version are shown in Table 4:\n\n Parameter Interpretation barhomogeneous whether to use homogeneous self-dual form in barrier bariterlimit iteration limit of barrier method barthreads number of threads used by barrier basis whether to use or return basis status bestbound whether to return best bound by suffix conflictanalysis whether to perform conflict analysis crossoverthreads number of threads used by crossover cutlevel level of cutting-planes generation divingheurlevel level of diving heuristics dualize whether to dualize a problem before solving it dualperturb whether to allow the objective function perturbation dualprice specifies the dual simplex pricing algorithm dualtol the tolerance for dual solutions and reduced cost feastol the feasibility tolerance heurlevel level of heuristics iisfind whether to compute IIS and return result iismethod specify the IIS method inttol the integrality tolerance for variables logging whether to print solving logs logfile name of log file exportfile name of model file to be exported lpmethod method to solve the LP problem matrixtol input matrix coefficient tolerance mipstart whether to use initial values for MIP problem miptasks number of MIP tasks in parallel nodecutrounds rounds of cutting-planes generation of tree node nodelimit node limit of the optimization objno objective number to solve count whether to count the number of solutions stub name prefix for alternative MIP solutions written presolve level of presolving before solving a problem relgap the relative gap of optimization absgap the absolute gap of optimization return_mipgap whether to return absolute\/relative gap by suffix rootcutlevel level of cutting-planes generation of root node rootcutrounds rounds of cutting-planes generation of root node roundingheurlevel level of rounding heuristics scaling whether to perform scaling before solving a problem simplexthreads number of threads used by dual simplex sos whether to use \u2018.sosno\u2019 and \u2018.ref\u2019 suffix sos2 whether to use SOS2 to represent piecewise linear terms strongbranching level of strong branching submipheurlevel level of Sub-MIP heuristics threads number of threads to use timelimit time limit of the optimization treecutlevel level of cutting-planes generation of search tree wantsol whether to generate \u2018.sol\u2019 file\n\nPlease refer to COPT Parameters for details.\n\nAMPL uses suffix to store or pass model and solution information, and also some extension features, such as support for SOS constraints. Currently, coptampl support suffix information as shown in Suffix supported by coptampl :\n\n Suffix Interpretation absmipgap absolute gap for MIP problem bestbound best bound for MIP problem iis store IIS status of variables or constraints nsol number of pool solutions written ref weight of variable in SOS constraint relmipgap relative gap for MIP problem sos store type of SOS constraint sosno type of SOS constraint sosref store variable weight in SOS constraint sstatus basis status of variables and constraints\n\nUsers who want to know how to use SOS constraints in AMPL, please refer to resources in AMPL\u2019s website: How to use SOS constraints in AMPL .\n\nWhen solving finished, coptampl will display a status message and return exit code to AMPL. The exit code can be displayed by:\n\nampl: display solve_result_num;\n\n\nIf no solution was found or something unexpected happened, coptampl will return non-zero code to AMPL from Table 6:\n\n Exit Code Interpretation 0 optimal solution 200 infeasible 300 unbounded 301 infeasible or unbounded 600 user interrupted\n\n## Example Usage\uf0c1\n\nThe following section will illustrate the use of AMPL by a well-known example called \u201cDiet problem\u201d, which finds a mix of foods that satisfies requirements on the amounts of various vitamins, see AMPL book for details.\n\nSuppose the following kinds of foods are available for the following prices per unit, see Table 7:\n\n Food Price BEEF 3.19 CHK 2.59 FISH 2.29 HAM 2.89 MCH 1.89 MTL 1.99 SPG 1.99 TUR 2.49\n\nThese foods provide the following percentages, per unit, of the minimum daily requirements for vitamins A, C, B1 and B2, see Table 8:\n\n A C B1 B2 BEEF 60% 20% 10% 15% CHK 8 0 20 20 FISH 8 10 15 10 HAM 40 40 35 10 MCH 15 35 15 15 MTL 70 30 15 15 SPG 25 50 25 15 TUR 60 20 15 10\n\nThe problem is to find the cheapest combination that meets a week\u2019s requirements, that is, at least 700% of the daily requirements for each nutrient.\n\nTo summarize, the mathematical form for the above problem can be modeled as shown in Eq. 6:\n\n(6)$\\begin{split}\\textrm{Minimize: } & \\\\ & \\sum_{j \\in J} cost_j \\cdot buy_j \\\\ \\textrm{Subject to: } & \\\\ & n\\_min_i \\leq \\sum_{j \\in J} amt_{i, j} \\cdot buy_j \\leq n\\_max_i \\,\\,\\, \\forall i \\in I \\\\ & f\\_min_j \\leq buy_j \\leq f\\_max_j \\,\\,\\, \\forall j \\in J\\end{split}$\n\nThe AMPL model for above problem is shown in diet.mod, see Listing 7:\n\nListing 7 diet.mod\n 1# The code is adopted from:\n2#\n3# https:\/\/github.com\/Pyomo\/pyomo\/blob\/master\/examples\/pyomo\/amplbook2\/diet.mod\n4#\n5# with some modification by developer of the Cardinal Optimizer\n6\n7set NUTR;\n8set FOOD;\n9\n10param cost {FOOD} > 0;\n11param f_min {FOOD} >= 0;\n12param f_max {j in FOOD} >= f_min[j];\n13\n14param n_min {NUTR} >= 0;\n15param n_max {i in NUTR} >= n_min[i];\n16\n17param amt {NUTR, FOOD} >= 0;\n18\n19var Buy {j in FOOD} >= f_min[j], <= f_max[j];\n20\n21minimize Total_Cost:\n22 sum {j in FOOD} cost[j] * Buy[j];\n23\n24subject to Diet {i in NUTR}:\n25 n_min[i] <= sum {j in FOOD} amt[i, j] * Buy[j] <= n_max[i];\n\n\nThe data file for above problem is shown in diet.dat, see Listing 8:\n\nListing 8 diet.dat\n 1# The data is adopted from:\n2#\n3# https:\/\/github.com\/Pyomo\/pyomo\/blob\/master\/examples\/pyomo\/amplbook2\/diet.dat\n4#\n5# with some modification by developer of the Cardinal Optimizer\n6\n7data;\n8\n9set NUTR := A B1 B2 C ;\n10set FOOD := BEEF CHK FISH HAM MCH MTL SPG TUR ;\n11\n12param: cost f_min f_max :=\n13 BEEF 3.19 0 100\n14 CHK 2.59 0 100\n15 FISH 2.29 0 100\n16 HAM 2.89 0 100\n17 MCH 1.89 0 100\n18 MTL 1.99 0 100\n19 SPG 1.99 0 100\n20 TUR 2.49 0 100 ;\n21\n22param: n_min n_max :=\n23 A 700 10000\n24 C 700 10000\n25 B1 700 10000\n26 B2 700 10000 ;\n27\n28param amt (tr):\n29 A C B1 B2 :=\n30 BEEF 60 20 10 15\n31 CHK 8 0 20 20\n32 FISH 8 10 15 10\n33 HAM 40 40 35 10\n34 MCH 15 35 15 15\n35 MTL 70 30 15 15\n36 SPG 25 50 25 15\n37 TUR 60 20 15 10 ;\n\n\nTo solve the problem with coptampl in AMPL, just type commands in command prompt on Windows or shell on Linux and MacOS:\n\nampl: model diet.mod;\nampl: data diet.dat;\nampl: option solver coptampl;\nampl: option copt_options 'logging 1';\nampl: solve;\n\n\ncoptampl solve it quickly and display solving log and status message on screen:\n\nx-COPT 5.0.1: optimal solution; objective 88.2\n1 simplex iterations\n\n\nSo coptampl claimed it found the optimal solution, and the minimal cost is 88.2 units. You can further check the solution by:\n\nampl: display Buy;\n\n\nAnd you will get:\n\nBuy [*] :=\nBEEF 0\nCHK 0\nFISH 0\nHAM 0\nMCH 46.6667\nMTL 0\nSPG 0\nTUR 0\n;\n\n\nSo if we buy 46.667 units of MCH, we will have a minimal cost of 88.2 units.","date":"2023-03-25 11:51:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34589990973472595, \"perplexity\": 9562.257857444285}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945323.37\/warc\/CC-MAIN-20230325095252-20230325125252-00335.warc.gz\"}"}
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{"url":"https:\/\/www.physicsforums.com\/threads\/proof-by-induction.741607\/","text":"# Proof by induction\n\n1. Mar 4, 2014\n\n### mikky05v\n\n1. The problem statement, all variables and given\/known data\nI have two problems that I can't figure out how to turn into summations so I can solve them.\n\n1. Prove by induction ($\\forall$n)P(n), P(n) is 3|(4$^{n}$-1)\nI believe the | means divides.\n\n2. Prove by induction ($\\forall$n)P(n), P(n) is n! $\\leq$n$^{n}$\n\n2. Mar 5, 2014\n\n### maajdl\n\nThe symbol \"|\" probably means \"divides\" in this context:\n\nhttp:\/\/en.wikipedia.org\/wiki\/List_of_mathematical_symbols\n\nWhat are the meaning of these \"P(n)\"?\n\nThere you can read what a proof by induction means:\n\nhttp:\/\/en.wikipedia.org\/wiki\/Mathematical_induction\n\nFor example, for problem 1, the first step is to check the statement for some special value of n (typically a small value).\nTake n = 1, for example, and check that 3 divides (4^n-1), which is quite obvious.\nThen try to prove that if the statement is true for any special value n, then it must also be true for the next value n+1 .\nThis can be done in two lines of algebra.\n\nHave fun.\n\n3. Mar 5, 2014\n\n### micromass\n\nStaff Emeritus\nTurn into summations? What do you mean with that?\n\n4. Mar 5, 2014\n\n### mikky05v\n\nSo that i can solve them with a classic proof by induction where you solve p (1) and show thats true then solve p (k) and p (k+1) using the induction hypothesis. I dontt know how to do this properly without a summstion or how to turn those statements into summations in the form\n$$\\sum_{i=1}^{n}i=n$$\ndoes that ake sense?\n\n5. Mar 5, 2014\n\n### LCKurtz\n\nNo. Your problem has nothing to do with summations.\n\n6. Mar 5, 2014\n\n### micromass\n\nStaff Emeritus\nPerhaps the problem is that you'e only seen induction before when you had to do stuff with summations. But summations are not necessary in order for a proof by induction to work.\n\nLet's prove (1). You need to prove $P(n)$ holds for all naturals $n$. If you prove this by induction, then you need to prove two things:\n\n(a) Prove that $P(1)$ holds.\n(b) Prove that if $P(k)$ holds for some natural $k$, then $P(k+1)$ holds.\n\nCan you prove (a) first?\n\n7. Mar 5, 2014\n\n### mikky05v\n\nHmmm. Ok yes p (1) works, and icanpull out a p (k) and a p (k+1) but everything i know how to do stops there.\n\n1. P (1) 3|4^1-1=3 and yes 3|3\n$$p (k)= 3|(4^k-11)$$\n$$p (k+1)= 3|(4^{k+1}-1)$$\nnow what?\n\nLast edited: Mar 5, 2014\n8. Mar 5, 2014\n\n### mikky05v\n\nOk so\n$$p (k)= 3|(4^k-1)$$\nand i need to get to\n$$p (k+1)=3|(4^{k+1}-1)$$\nSo p (k) is for some m\n$$4^k -1=3m$$\nAnd p (k+1) is for some p\n$$4^{k+1}-1=3p$$\nNow manipulating p (k) i can add 1 to bothsides and thenmultiply both sides by 4 and then subtract 1\n$$4^{k+1}-1=4(3m+1)-1$$\nI feel like this is a good direction but i dont know what to do to prove 3|rhs\n\n9. Mar 5, 2014\n\n### micromass\n\nStaff Emeritus\nGood. Now prove that $4(3m + 1) - 1$ can be written as $3a$ for some integer $a$.\n\n10. Mar 5, 2014\n\n### mikky05v\n\nAlright i think i have a solution for both of them now. Thank you all very much for your help","date":"2017-12-13 17:55:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6624273061752319, \"perplexity\": 1135.7713429389537}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948529738.38\/warc\/CC-MAIN-20171213162804-20171213182804-00796.warc.gz\"}"}
| null | null |
Conversations est un client XMPP libre pour Android.
Il supporte l'échange de messages chiffrés. Le code source est fourni sous la licence GNU GPL et est géré sur GitHub. L'application est disponible via F-Droid et le Google Play.
Histoire
La version 2.8.0, publiée fin avril 2020 introduit la gestion des appels audio et video via Jingle.
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|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,532
|
\section{Introduction}
One of the big challenges of the past decade has been the construction of renormalizable quantum field theories (QFTs) on {non-commutative} space-time, usually plagued by the infamous {UV/IR mixing} problem~\cite{Minwalla:1999,Douglas:2001,Rivasseau:2007a}.
Only in very special cases has this task been successful so far, namely in the case of scalar field theories formulated in even dimensional $\theta$-deformed Euclidean space: In fact, three such renormalizable models are known today\footnote{The other two renormalizable scalar models were introduced in references~\cite{Rivasseau:2008a,Grosse:2008df} and developed further in~\cite{Tanasa:2008a,Blaschke:2008b,Tanasa:2008b,Tanasa:2008c}.}, the first of which being the so-called Grosse-Wulkenhaar model~\cite{Grosse:2003,Grosse:2004b,Rivasseau:2005a}. In that model, the action is supplemented by an oscillator-like term which modifies the propagator in such a way that it becomes Langmann-Szabo invariant~\cite{Szabo:2002} (just like the modified action, in fact).
Motivated by the success of the Grosse-Wulkenhaar model, some {non-commutative} gauge field models including similar oscillator-like terms in the action were put forward in Refs.~\cite{Blaschke:2007b,Grosse:2007,Wulkenhaar:2007}. The current work mainly deals with loop calculations based on the previous letter~\cite{Blaschke:2007b}, aiming to clarify whether the action we proposed there can lead to a renormalizable {non-commutative} gauge theory. In doing so, we find effective one-loop contributions to the action showing great similarities to the so-called ``induced gauge theories'' with oscillator terms which were put forward in~\cite{Grosse:2007,Wulkenhaar:2007}. These findings lead us to the conclusion, that (even though we do not claim to have solved the many problems concerning {non-commutative} gauge field theories in general, as was discussed in~\cite{Blaschke:2009c}) the implementation of a Grosse-Wulkenhaar oscillator term into gauge theories seems to be a promising ansatz for a renormalizable {non-commutative} gauge theory. However, it also seems to be necessary to do this in a gauge invariant manner, i.e. to consider an action derived by (or motivated from) the ``induced gauge theories''.
Our paper is organized as follows: Section~\ref{sec:action_and_properties} briefly reviews the action and its symmetries followed by Section~\ref{sec:1loop} where some one-loop results are presented.
We finally compare the effective action with the induced gauge theory of~\cite{Grosse:2007,Wulkenhaar:2007} in Section~\ref{sec:conclusion} and discuss the consequences.
\section{The {non-commutative} gauge field model with oscillator term}
\label{sec:action_and_properties}
In the simplest case of $\th$-deformed space-time, one considers a constant commutator $\starco{x^\mu}{x^\nu} \equiv x^{\mu} \star
x^{\nu} -x^{\nu} \star x^{\mu} = {\rm i} \th^{\mu \nu}$, i.e. a Heisenberg algebra for the space-time coordinates, realized through the so-called Groenewold-Moyal $\star$-product~\cite{Groenewold:1946,Moyal:1949}. Considering four dimensional Euclidean space, in Ref.~\cite{Blaschke:2007b} we proposed the following {non-commutative} $U_\star(1)$ gauge field action\footnote{Some other candidates for {non-commutative} gauge field theories without oscillator terms were in fact presented in Refs.~\cite{Slavnov:2003,Blaschke:2008a,Blaschke:2009a,Vilar:2009,Blaschke:2009b}. However, these will not be discussed here.} at
tree-level:
\begin{align}\label{new-action}
\Gamma^{(0)}&=S_{\text{inv}}+S_{\text{m}}+S_{\text{gf}}\,,\nonumber\\
S_{\text{inv}}&=\inv{4}\int d^4x\, F_{\mu\nu}\star F_{\mu\nu}\,,\nonumber\\
S_{\text{m}}&=\frac{\Omega^2}{4}\int d^4x\left(\inv{2}\staraco{\tilde{x}_\mu}{A_\nu}\star\staraco{\tilde{x}_\mu}{A_\nu}+\staraco{\tilde{x}_\mu}{\bar{c}}\star\staraco{\tilde{x}_\mu}{c} \right)=\nonumber\\
&=\frac{\Omega^2}{8}\int d^4x\left(\tilde{x}_\mu\star\mathcal{C}_\mu\right)\,,\nonumber\\
S_{\text{gf}}&=\int d^4x\Bigg[B\star\partial_\mu A_\mu-\inv{2}B\star B-\bar{c}\star\partial_\mu sA_\mu-\frac{\Omega^2}{8}\,\widetilde{c}_\mu\star s\,\mathcal{C}_\mu\Bigg]\,,
\end{align}
with
\begin{align}
F_{\mu\nu}&=\partial_\mu A_\nu-\partial_\nu A_\mu-{\rm i}g\starco{A_\mu}{A_\nu}\,,\nonumber\\
\mathcal{C}_\mu&=\Big(\staraco{\staraco{\tilde{x}_\mu}{A_\nu}}{A_\nu}+\starco{\staraco{\tilde{x}_\mu}{\bar{c}}}{c}+\starco{\bar{c}}{\staraco{\tilde{x}_\mu}{c}}\Big)\,,\nonumber\\
\tilde{x}_\mu&=\left(\th^{-1}\right)_{\mu\nu}x_\nu\,.
\end{align}
The field $A_\mu$ denotes the {non-commutative} generalization of a $U(1)$ gauge field (hence the notation $U_\star(1)$ for the deformed algebra), and $B$ is the multiplier field implementing a non-linear gauge fixing\footnote{Notice, that in the limit $\Omega\to0$ this becomes a Feynman gauge.}
\begin{align}
\var{\Gamma^{(0)}}{B}&=\partial_\mu A_\mu-B+\frac{\Omega^2}{8}\Big(\starco{\staraco{\tilde{x}_\mu}{c}}{\widetilde{c}_\mu}-\staraco{\tilde{x}_\mu}{\starco{\widetilde{c}_\mu}{c}}\Big)=0.
\end{align}
Furthermore, we have an additional multiplier field $\widetilde{c}_\mu$ imposing on-shell BRST invariance of the expression $\mathcal{C}_\mu$, $\Omega$ is a constant parameter and $c$, $\bar{c}$ are the ghost/antighost, respectively. The action (\ref{new-action}) is invariant under the BRST transformations given by
\begin{align}\label{BRST}
&sA_\mu=D_\mu c=\partial_\mu c-{\rm i}g\starco{A_\mu}{c}, && s\bar{c}=B,\nonumber\\
&sc={\rm i}g{c}\star{c}, && sB=0,\nonumber\\
&s\widetilde{c}_\mu=\tilde{x}_\mu, && s^2\varphi=0\ \forall\ \varphi\in\left\{A_\mu,B,c,\bar{c},\widetilde{c}_\mu\right\}.
\end{align}
Using these transformations, the action can be written succinctly as
\begin{align}
\Gamma^{(0)} = \int d^4x \left(
\frac 14 F_{\mu\nu}\star F_{\mu\nu} + s\left( \frac{\Omega^2}8 \tilde c_\mu \star \mathcal{C}_\mu +
\bar c\star \partial_\mu A_\mu - \frac 12 \bar c \star B \right) \right)\,.
\end{align}
The properties of the appearing fields are summarized in Table 1, where $g_\sharp$ denotes the ghost number.
\begin{table}[!ht]
\caption{Properties of the fields.}
\label{tab:field_prop}
\centering
\begin{tabular}{l c c c c c}
\hline
\hline
Field & $A_\mu$ & $c$ & $\bar c$ & $\tilde c_\mu$ & $B$ \\[2pt]
\hline
$g_\sharp$ & 0 & 1 & -1 & -1 & 0 \\
Mass dim. & 1 & 0 & 2 & 1 & 2 \\
Statistics & b & f & f & f & b \\
\hline
\hline
\end{tabular}
\end{table}
\noindent
By introducing external sources $\rho_\mu$ and $\sigma$ for the non-linear BRST transformations, one derives the Slavnov-Taylor identity
\begin{align}\label{Slavnov_Taylor_identities}
\text{S}(\Gamma_{\text{tot}})=\int d^4x
\,\Bigg(\dif{\Gamma_{\text{tot}}}{\rho_\mu}\star\dif{\Gamma_{\text{tot}}}{A_\mu}+\dif{\Gamma_{\text{tot}}}{\sigma}\star\dif{\Gamma_{\text{tot}}}{c}+B\star\dif{\Gamma_{\text{tot}}}{\bar{c}}
+\tilde{x}_\mu\star\dif{\Gamma_{\text{tot}}}{\tilde{c}_\mu}\Bigg)=0\,,
\end{align}
with
\begin{align}
\Gamma_{\text{tot}}&=\Gamma^{(0)}+\Gamma_{\text{ext}}\,,\nonumber\\
\Gamma_{\text{ext}}&=\int d^4x\left(\rho_\mu \star sA_\mu+ \sigma \star sc\right)\,.
\end{align}
Equation (\ref{Slavnov_Taylor_identities}) describes the symmetry content with respect to \eqref{BRST}. When taking its functional derivative with respect to $A_\rho$ and $c$ and then setting all fields to zero one arrives at
\begin{align}\label{transversality_relation}
\partial^z_\mu\vvar{\Gamma_{\text{tot}}}{A_\rho(y)}{A_\mu(z)}
=\int d^4x\,\Bigg(\tilde{x}_\mu\frac{\delta^3\Gamma_{\text{tot}}}{\d c(z)\d A_\rho(y)\d\tilde{c}_\mu(x)}\Bigg)\neq 0\,.
\end{align}
Usually one would expect to obtain the transversality condition for the one-particle irreducible (1PI) two-point graph. However, in this case the oscillator term breaks gauge invariance and hence transversality, as can bee seen from the equation above: Instead one has a Ward identity relating the 1PI two-point graph to a three-point graph involving the new field $\widetilde{c}_\mu$. Graphically we can express this relation as depicted in Figure~\ref{almost_transversality_relation}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.8]{almost_transversality_relation.eps}
\caption{Ward identity replacing transversality}
\label{almost_transversality_relation}
\end{figure}
The bilinear parts of the action \eqref{new-action} lead to the following gauge field and ghost propagators in momentum space:
\begin{align}\label{newprop}
G^{A}_{\mu\nu}(p,q)&=(2\pi)^4K_M(p,q)\d_{\mu\nu},\nonumber\\
G^{\bar{c} c}(p,q)&=(2\pi)^4K_M(p,q)\,,
\end{align}
where $K_M(p,q)$ denotes the so-called Mehler kernel
\begin{align}\label{Mehler-long-short}
K_M(p,q)&=\frac{\omega^3}{8\pi^2}\int\limits_{0}^{\infty}d\alpha\inv{\sinh^2(\alpha)}\exp\left(-\tfrac{\omega}{4}\coth\left(\tfrac{\alpha}{2}\right)\left(p-q\right)^2-\tfrac{\omega}{4}\tanh\left(\tfrac{\alpha}{2}\right)\left(p+q\right)^2\right),
\end{align}
which is going to be responsible for improving the IR behaviour of the model. In the limit $\omega=\frac{\th}{\Omega}\to\infty$ these propagators
reduce to the usual ones, i.e.
\begin{align*}
\lim\limits_{\Omega\to0}K_M(p,q)=\inv{p^2}\d^4\left(p-q\right)\,.
\end{align*}
This can be easily verified by integrating over one of the arguments in the Mehler kernel:
\begin{align}
\lim\limits_{\Omega\to0}\int d^4qK_M(p,q)=\lim\limits_{\Omega\to0}\inv{p^2}\left(1-{\rm e}^{-\frac{\omega}{2}p^2}\right)=\inv{p^2}\,.
\end{align}
In the following sections we will discuss some one-loop properties of this model, but first some comments are in order:
In contrast to the usual situation, the multiplier field $B$ appears in a $\widetilde{c} Bc$ vertex (in addition to the $BB$ and $BA$ propagators). However, since the new multiplier $\widetilde{c}_\mu$ does not propagate, it is impossible to build a graph including these additional Feynman rules, but with only external gauge field legs. In fact, these are the types of graphs we are going to concentrate on in the following, and hence the Feynman rules including the fields $B$ and $\widetilde{c}$ are ignored for now.
\section{One-loop computations}\label{sec:1loop}
\subsection{Power counting}\label{sec:powercounting}
Before starting the explicit calculations, we may derive estimations for the ``worst case'', i.e. the superficial degree of ultraviolet divergence, which via {UV/IR mixing} is directly related to the degree of {non-commutative} IR divergence.
We take into account the powers of internal momenta $k$ each Feynman rule contributes and also that each loop integral over 4-dimensional space increases the degree by 4. For example, the gauge boson propagator usually behaves like $1/k^2$ for large $k$ and therefore reduces the degree of divergence by 2, whereas each ghost vertex contributes one power of $k$ to the numerator of a graph, hence increasing the degree by one. However, in the present model all propagators, being essentially Mehler kernels, depend on two momenta which additionally need to be integrated in loop calculations. Taking into account these considerations for all Feynman rules we arrive at
\begin{align}
d_{\gamma}=4L-6I_A-6I_c-5I_{AB}-4I_B+V_c+V_{3A}+V_{\widetilde{c} cA}\,,
\end{align}
where the $I$ and $V$ denote the number of the various types of internal lines and vertices, respectively. The number of loop integrals $L$ is given by
\begin{align*}
L=2I_A+2I_c+2I_{B}+2I_{AB}-(V_c+V_{3A}+V_{4A}+V_{\widetilde{c} cA}+V_{\widetilde{c} c2A}+V_{\widetilde{c} Bc}+V_{\widetilde{c}\bc2c} -1)\,.
\end{align*}
Furthermore, we take into account the relations
\begin{align}\label{eq:power-counting-relations}
E_{c/\bar c}+2I_c&=2V_c+V_{\widetilde{c} cA}+V_{\widetilde{c} c2A}+V_{\widetilde{c} Bc}+3V_{\widetilde{c}\bc2c}\,,\nonumber\\
E_A+2I_A+I_{AB}&=V_c+3V_{3A}+4V_{4A}+V_{\widetilde{c} cA}+2V_{\widetilde{c} c2A}\,,\nonumber\\
E_B+2I_{B}+I_{AB}&=V_{\widetilde{c} Bc}\,,\nonumber\\
E_{\tilde c}&=V_{\widetilde{c} cA}+V_{\widetilde{c} c2A}+V_{\widetilde{c} Bc}+V_{\widetilde{c}\bc2c}\,,
\end{align}
between the various Feynman rules describing how they (and how many) can be connected to one another. The $E_{c/\bar{c}}$, $E_A$, $E_{\widetilde{c}}$ and $E_B$ denote the number of external lines of the respective fields. Using these relations one can eliminate all internal lines and vertices from the power counting formula and arrive at
\begin{align}\label{power-counting_final}
d_\gamma=4-E_A-E_{c/\bar c}-E_{\tilde c}-2E_B\,.
\end{align}
For the gauge boson self-energy we therefore expect the degree of divergence (UV and {non-commutative} IR) to be at worst quadratically. Gauge invariance usually reduces the degree of UV divergence to be merely logarithmic. However, due to the Ward identity \eqref{transversality_relation} whose right hand side is non-zero, the UV divergence will in fact be worse in our case, namely quadratic.
\subsection{Tadpole graphs}\label{sec:tadpoles}
The two possible one-point functions (tadpoles) of this model at one-loop level are depicted in Figure~\ref{fig:tadpoles}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.65]{tadpoles.eps}
\caption{tadpole graphs}
\label{fig:tadpoles}
\end{figure}
According to the Feynman rules given in \eqref{Mehler-long-short} and in Appendix~\ref{app:vertices}, the sum of tadpole graphs is given by
\begin{align}
\Pi_\mu(p)&=2{\rm i}g\int d^4k \int d^4k'\d^4\left(p+k'-k\right)\sin\left(\frac{k\tilde{p}}{2}\right)K_M(k,k')\left[2k_\mu+3k'_\mu\right]\,,
\end{align}
where the abbreviation $\tilde{p}_\mu\equiv\th_{\mu\nu}p_\nu$ has been introduced\footnote{Concerning notation, notice that while in $x$-space we use $\tilde{x}_\mu\equiv(\th^{-1})_{\mu\nu}x_\nu$, in momentum space we have $\tilde{p}_\mu\equiv\th_{\mu\nu}p_\nu$ (and likewise for all other momenta such as $\k_\mu$ or $\tilde{q}_\mu$).}. In the following, we furthermore assume that the deformation
matrix $( \th_{\mu\nu} )$ has the simple block-diagonal form
\begin{align}
( \th_{\mu\nu} )
=\th\left(\begin{array}{cccc}
0&1&0&0\\
-1&0&0&0\\
0&0&0&1\\
0&0&-1&0
\end{array}
\right) \, , \qquad {\rm with} \ \; \th \in \mathbb{R} \, .
\end{align}
Hence one has the identity $\tilde{p}^2=\th^2p^2$ which will significantly simplify calculations.
Making use of
\begin{align}
\sin\left(\frac{k\tilde{p}}{2}\right)=\sum_{\eta=\pm1}\frac{\eta}{2{\rm i}}\exp\left(\frac{{\rm i}\eta}{2}k\tilde{p}\right)\,,
\end{align}
and considering ``short'' and ``long'' variables defined by $u=k-k'$ and $v=k+k'$ one arrives at
\begin{align}
\Pi^\varepsilon_\mu(p)&=\frac{g\omega^3}{2^{8}\pi^2}\sum\limits_{\eta=\pm1}\int d^4v\left[5v_\mu-p_\mu\right]\int\limits_{\varepsilon}^{\infty}d\a\frac{\eta {\rm e}^{\frac{{\rm i}\eta}{4}v\tilde{p}}}{\sinh^2\a}\exp\left(-\tfrac{\omega}{4}\left[\coth\left(\tfrac{\a}{2}\right)p^2+\tanh\left(\tfrac{\a}{2}\right)v^2\right]\right)\nonumber\\
&=\frac{5{\rm i}g\tilde{p}_\mu}{64}\int\limits_{\varepsilon}^{\infty}d\a\frac{\cosh\left(\frac{\a}{2}\right)}{\sinh^5\left(\frac{\a}{2}\right)}\exp\left[-\inv{4}\coth\left(\frac{\a}{2}\right)\left(\omega+\frac{\th^2}{4\omega}\right)p^2\right]\,,
\end{align}
regularizing the integrals by introducing a UV cutoff $\varepsilon = 1/\Lambda^2$.
Now we consider the following expansion:
\begin{align}\label{expansion-tadpole}
\int \frac{d^4p}{(2\pi)^4}\,\Pi^\varepsilon_\mu(p)&\left[A_\mu(0)+p_\nu\left(\partial^p_\nu A_\mu(p)\big|_{p=0}\right)+\frac{p_\nu p_\rho}{2}\left(\partial^p_\nu\partial^p_\rho A_\mu(p)\big|_{p=0}\right)+\right.\nonumber\\
&\left. \;+\frac{p_\nu p_\rho p_\sigma}{6}\left(\partial^p_\nu\partial^p_\rho\partial^p_\sigma A_\mu(p)\big|_{p=0}\right)+\ldots\right]\,.
\end{align}
All terms of even order (i.e. of order 0,2,4,\ldots) are zero for symmetry reasons. Of the other terms, we now show that only the first two, namely orders 1 and 3, diverge in the limit $\varepsilon\to 0$:
\begin{itemize}
\item \emph{order 1:}
\begin{align}
\int \frac{d^4p}{(2\pi)^4}\,p_\nu\Pi^\varepsilon_\mu(p)&=
\int\limits_{\varepsilon}^{\infty}d\a\frac{5{\rm i}g\th_{\mu\nu}}{32\pi^2\omega^3\left(1+\frac{\Omega^2}{4}\right)^3\sinh^2(\a)}\nonumber\\
&=\frac{5{\rm i}g\th_{\mu\nu}}{32\pi^2\omega^3\left(1+\frac{\Omega^2}{4}\right)^3}\left[\inv{\varepsilon}-1+\mathcal{O}(\varepsilon)\right].
\end{align}
With the external field, we obtain a counter term of the form
\begin{align}
\label{tadpole-order1}
\left(\partial^p_\nu A_\mu(p)\big|_{p=0}\right)& \int \frac{d^4p}{(2\pi)^4}\,p_\nu\Pi^\varepsilon_\mu(p) =\nonumber\\
&=
\frac{5 g \Omega^2}{32\pi^2 \omega\left(1+\frac{\Omega^2}{4}\right)^3}
\left[ \inv{\varepsilon}-1+\mathcal{O}(\varepsilon) \right] \int d^4x \, \tilde{x}_\mu A_\mu(x) \,.
\end{align}
\item \emph{order 3:}
\begin{align}
\int \frac{d^4p}{(2\pi)^4}\, \frac{p_\a p_\b p_\gamma}{6} \Pi^\varepsilon_\mu(p) & =
\frac {-5{\rm i}g \left(\d_{\a\b}\th_{\mu\gamma}+\d_{\b\gamma}\th_{\mu\a}+\d_{\a\gamma}\th_{\mu\b}\right) } {24\pi^2\omega^4 \left(1+\frac{\Omega^2}{4}\right)^4}
\left[ \ln\varepsilon + \mathcal{O}(0) \right],
\end{align}
and with the external field we get the counter term
\begin{align}
\label{tadpole-order3}
\nonumber
\left(\partial^p_\a \partial^p_\b \partial^p_\gamma A_\mu(p)\big|_{p=0}\right)
&\int \frac{d^4p}{(2\pi)^4}\, \frac{p_\a p_\b p_\gamma}{6} \Pi^\varepsilon_\mu(p) =\\
&
= \frac{ 5g } { 8\pi^2 } \frac { \Omega^4 } { \left(1+\frac{\Omega^2}{4}\right)^4}
\left[ \ln\varepsilon + \mathcal{O}(0) \right] \int d^4x\, \tilde{x}_\mu {\tilde{x}}^2 A_\mu (x)
\,.
\end{align}
\item \emph{order 5 and higher:}\\
These orders are \emph{finite}. The contribution to order $5 + 2n$, $n\ge 0$ is proportional to
\begin{align}
\int\limits_{0}^{\infty} d\a \frac{ \sinh^n \frac \a 2 } { \cosh^{n+4} \frac \a 2 } = \frac 4 { (n+1) (n+3) }\,.
\end{align}
\end{itemize}
Notice, that all tadpole contributions would vanish in the limit $\Omega\to0$ as expected. When keeping $\Omega\neq0$, the two divergent terms can be removed by renormalization, i.e. by considering the appropriate counter terms given in Eqns.~(\ref{tadpole-order1}) and (\ref{tadpole-order3}), respectively. Remarkably, these terms are present in the induced action calculated in Refs.~\cite{Grosse:2007,Wulkenhaar:2007}. The fact that these graphs do not vanish also means that we need to find the correct vacuum for $\Omega\neq0$ by solving the equations of motion, which at the classical level read
{\allowdisplaybreaks
\begin{subequations}\label{eom-osci}
\begin{align}
\var{\Gamma^{(0)}}{A_\nu}&=\left(-\Delta_4+\Omega^2\tilde{x}^2\right)A_\nu+{\rm i}g\starco{A_\mu}{F_{\mu\nu}}+{\rm i}g\partial_\mu\starco{A_\mu}{A_\nu}+{\rm i}g\staraco{\partial_\nu\bar{c}}{c}+\nonumber\\*
&\quad\;\,+\partial_\nu(\partial A)-\partial_\nu B+\frac{\Omega^2}{8}\Big(\staraco{\starco{D_\nu c}{\widetilde{c}_\mu}}{\tilde{x}_\mu}+\starco{\staraco{D_\nu c}{\tilde{x}_\mu}}{\widetilde{c}_\mu}\Big)-\nonumber\\*
&\quad\;\,-{\rm i}g\frac{\Omega^2}{8}\Big(\staraco{c}{\staraco{\tilde{x}_\mu}{\staraco{A_\nu}{\widetilde{c}_\mu}}}+\staraco{c}{\staraco{\widetilde{c}_\mu}{\staraco{\tilde{x}_\mu}{A_\nu}}}\Big)=0,\\
\var{\Gamma^{(0)}}{B}&=\partial_\mu A_\mu-B+\frac{\Omega^2}{8}\Big(\starco{\staraco{\tilde{x}_\mu}{c}}{\widetilde{c}_\mu}-\staraco{\tilde{x}_\mu}{\starco{\widetilde{c}_\mu}{c}}\Big)=0,\\
\var{\Gamma^{(0)}}{\bar{c}}&=\left(-\Delta_4+\Omega^2\tilde{x}^2\right)c-{\rm i}g\frac{\Omega^2}{8}\Big(\staraco{\staraco{\tilde{x}_\mu}{c\star c}}{\widetilde{c}_\mu}+\staraco{\tilde{x}_\mu}{\staraco{\widetilde{c}_\mu}{c\star c}}\Big)\nonumber\\*
&\quad+{\rm i}g\partial_\mu\starco{A_\mu}{c}=0,\\
\var{\Gamma^{(0)}}{c}&=\left(\Delta_4-\Omega^2\tilde{x}^2\right)\bar{c}+\frac{\Omega^2}{8}\Big(\staraco{\widetilde{c}_\mu}{\staraco{\tilde{x}_\mu}{B}}+\staraco{\tilde{x}_\mu}{\staraco{\widetilde{c}_\mu}{B}}\Big)-\nonumber\\*
&\quad\;\,-{\rm i}g\starco{A_\mu}{\partial_\mu\bar{c}}-\frac{\Omega^2}{8}D_\nu\Big(\staraco{\tilde{x}_\mu}{\staraco{A_\nu}{\widetilde{c}_\mu}}+\staraco{\staraco{\tilde{x}_\mu}{A_\nu}}{\widetilde{c}_\mu}\Big)+\nonumber\\*
&\quad\;\,+{\rm i}g\frac{\Omega^2}{8}\Big(\starco{c}{\starco{\widetilde{c}_\mu}{\staraco{\tilde{x}_\mu}{\bar{c}}}}-\starco{c}{\staraco{\tilde{x}_\mu}{\starco{\bar{c}}{\widetilde{c}_\mu}}}\Big)=0,\\
\var{\Gamma^{(0)}}{\widetilde{c}_\mu}&=-\frac{\Omega^2}{8}s\,\mathcal{C}_\mu=0.
\end{align}
\end{subequations}
Finding solutions to these equations is the task of a work in progress\footnote{In fact, some work in this respect has been done in Refs.~\cite{Wallet:2007b,Wallet:2008a} for the induced gauge theory case without BRST ghosts.}.
}
\subsection{Two point functions at one-loop level}
\label{sec:one-loop-vac-pol}
In this section we analyze the divergence structure of the gauge boson self-energy at one-loop level. The relevant graphs are depicted in Figure~\ref{fig:photonself}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.65]{two_point_one_loop_graphs.eps}
\caption{Gauge boson self-energy --- amputated graphs}
\label{fig:photonself}
\end{figure}
Explicitly, the sum of these graphs is computed using the same techniques as in the previous section (except for the expansion which is not necessary here). Once more we use long and short variables such as $u=k-k'$ and $v=k+k'$ and including their symmetry factors arrive at the following expressions for the three separate graphs:
{\allowdisplaybreaks
\begin{subequations}
\begin{align}
\Pi^a_{\mu\nu}&=-\frac{3g^2\d_{\mu\nu}}{8}\int d^4v \,K_M(p-p',v)\Big[\sin\left(\tfrac{(v+p')\tilde{p}}{4}\right)\sin\left(\tfrac{(v+p)\tilde{p}'}{4}\right)\nonumber\\*
&\quad\hspace{5.3cm}+\sin\left(\tfrac{(v-p')\tilde{p}}{4}\right)\sin\left(\tfrac{(v-p)\tilde{p}'}{4}\right)\Big]\,,
\\
\Pi^b_{\mu\nu}&=\frac{g^2}{8}\int d^4u\,d^4v\,K_M(u,v)K_M(u+p-p',v+p+p')\sin\left(\tfrac{(v+u)\tilde{p}}{4}\right)\sin\left(\tfrac{(v-u)\tilde{p}'}{4}\right)\nonumber\\*
&\quad\qquad\times\Big[\tfrac{5}{2}(v_\mu v_\nu-u_\mu u_\nu)+\tfrac{3}{2}(u_\mu v_\nu-v_\mu u_\nu)+\tinv{2}p'_\mu(v+u)_\nu+\tinv{2}(v-u)_\mu p_\nu\nonumber\\*
&\quad\qquad\qquad+2p_\mu(v-u)_\mu+2(v+u)_\mu p'_\nu+2p_\mu p'_\nu-4p'_\mu p_\nu \nonumber\\*
&\quad\qquad\qquad+\d_{\mu\nu}\left(\tfrac{v^2-u^2}{2}+\tfrac{p'(v+u)}{2}+\tfrac{(v-u)p}{2}+5pp'\right)\Big]\,,
\\
\Pi^c_{\mu\nu}&=\frac{-g^2}{16}\int d^4u\,d^4v\,K_M(u,v)K_M(u+p-p',v+p+p')\,(v+u)_\mu(v-u+2p')_\nu\nonumber\\*
&\quad\hspace{2.7cm}\times\sin\left(\tfrac{(v+u)\tilde{p}}{4}\right)\sin\left(\tfrac{(v-u)\tilde{p}'}{4}\right)\,,
\end{align}
\end{subequations}
where $p$ and $p'$ denote the external momenta.
In the limit $\Omega\to0$ one would expect a result $\Pi_{\mu\nu}(p,p')=\Pi_{\mu\nu}(p)\d^4(p-p')$ where the transversality property $p_\mu\Pi_{\mu\nu}=0$ holds. However, due to $\Omega\neq0$ these properties are not fulfilled, i.e. transversality is broken and one cannot split off a delta function. In order to reveal the divergence structure of the general result without the ``smeared out delta function'', we additionally integrate over $p'$. Finally, noticing that the parameter integrals entering from the Mehler kernels \eqref{Mehler-long-short}, are dominated by the region of small $\alpha$, one also needs to approximate for $\alpha\ll1$ in order to extract the UV and IR divergent terms. We hence arrive at
}
\begin{align}\label{eq:one-loop_divergences}
\Pi_{\mu\nu}^{\text{div}}(p)&=
\frac{g^2 \d _{\mu\nu} \left(1-\tfrac{3}{4}\Omega^2\right)}{4\pi^2 \omega\,\varepsilon \left(1+\tfrac{\Omega^2}{4}\right)^3}+
\frac{3 g^2 \delta_{\mu\nu} \Omega^2}{8\pi^2 \tilde{p}^2 \left(1+\tfrac{\Omega^2}{4}\right)^2}+
\frac{2 g^2 \tilde{p}_\mu \tilde{p}_\nu }{\pi ^2 (\tilde{p}^2)^2 \left(1+\tfrac{\Omega^2}{4}\right)^2}\nonumber\\
&\quad+\text{logarithmic UV divergence}\,.
\end{align}
In the limit $\Omega\to0$ (i.e. $\omega\to\infty$) this expression reduces to the usual transversal term
\begin{align}
\lim\limits_{\Omega\to0}\Pi_{\mu\nu}^{\text{div}}(p)&=\frac{2g^2}{\pi^2} \frac{\tilde{p}_\mu\tilde{p}_\nu}{(\tilde{p}^2)^2}+\text{logarithmic UV divergence}\,,
\end{align}
which is quadratically IR divergent\footnote{In fact, this term is consistent with previous results~\cite{Blaschke:2005b,Hayakawa:1999b,Ruiz:2000} calculated in the ``na{\"i}ve'' model, i.e. without any additional non-local terms in the action.} in the external momentum $p$ and logarithmically UV divergent.
Notice that the general result \eqref{eq:one-loop_divergences}, on the other hand, not only breaks transversality due to the first two terms, but also has an ultraviolet divergence parameterized by $\varepsilon$, whose degree of divergence is higher compared to the (commutative) gauge model without oscillator term. Both properties are due to the term $S_{\text{m}}$ in the action which breaks gauge invariance (cf. Eqn. (\ref{transversality_relation})).
\subsection{Vertex corrections at one-loop level}
\label{sec:vertex-corr}
The calculation of the vertex corrections generally proceeds along the lines of the previous section. Due to the vast amount of terms, it is however useful to use a computer: In fact we ``taught'' Wolfram Mathematica$^{\text{\textregistered}}$ to perform exactly the same steps that we would have done by hand.
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.8]{3A-vertex-corr.eps}
\caption{One loop corrections to the 3A-vertex.}
\label{fig:1loop_3A_all}
\end{figure}
Hence, computing the graphs depicted in \figref{fig:1loop_3A_all} (and approximating for momentum conservation as in the previous subsections), one eventually finds a linear IR divergence of the form:
\begin{align}\label{eq:3Avert}
\Gamma^{\text{3A,IR}}_{\mu\nu\rho}(p_1,p_2,p_3)=\frac{-8{\rm i}g^3}{\pi^2\left(4+\Omega^2\right)^3}\sum\limits_{i=1}^{3}\bigg[&
\frac{16\tilde{p}_{i,\mu}\tilde{p}_{i,\nu}\tilde{p}_{i,\rho}}{\tilde{p}_i^4}+\frac{3\Omega^2}{\tilde{p}_i^2}
\left(\delta_{\mu\nu}\tilde{p}_{i,\rho}+\d_{\mu\rho}\tilde{p}_{i,\nu}+\d_{\nu\rho}\tilde{p}_{i,\mu}\right)\bigg],
\end{align}
where $p_3=-p_1-p_2$. Once more, this expression is not transversal due to the non-vanishing oscillator term parametrized by $\Omega$. However, in the limit $\Omega\to0$ transverality is recovered, and \eqref{eq:3Avert} reduces to the well-known expression~\cite{Matusis:2000jf, Armoni:2000xr,Ruiz:2000}
\begin{align}
\lim\limits_{\Omega\to0}V^{\text{1loop}}_{\mu\nu\rho}(p_1,p_2,p_3)=\frac{-2{\rm i}g^3}{\pi^2}\sum\limits_{i=1}^{3}\bigg[&
\frac{\tilde{p}_{i,\mu}\tilde{p}_{i,\nu}\tilde{p}_{i,\rho}}{\tilde{p}_i^4}\bigg].
\end{align}
In the ultraviolet, the graphs of \figref{fig:1loop_3A_all} diverge only logarithmically.
Additionally, one has of course also one-loop corrections to the 4A-vertex $\Gamma^{\text{4A,IR}}_{\mu\nu\rho\sigma}$. However, these show only a logarithmic divergence, as expected from the power counting (\ref{power-counting_final}).
\section{Discussion}
\label{sec:conclusion}
As already mentioned, the occurring UV counter terms of Section~\ref{sec:1loop} are present in the induced gauge action, e.g.~\cite{Grosse:2007,Wulkenhaar:2007}.
Let us compare the expressions in more detail here. The induced (Euclidean) gauge action (in the notation of~\cite{Grosse:2007}) is given by
\begin{align}
\label{result2}
\nonumber
\Gamma_{I}= \int d^4x\, \bigg\{&
\frac{3}{\th_I} \left(1-\rho_I^2\right) \left(\tilde \mu_I^2-\rho_I^2\right)\left(\tilde X_\nu \star \tilde X_\nu -\tilde{x}^2\right)
\\
&
+ \frac{3}{2}\left(1-\rho_I^2\right)^2 \left( \left(\tilde X_\mu\star \tilde X_\mu\right)^{\star 2}-\left(\tilde{x}^2\right)^2 \right)
+ \frac{\rho_I^4}{4} F_{\mu\nu}\star F_{\mu\nu}
\bigg\}\,,
\end{align}
where
\begin{align}
\rho_I = \frac{1-\Omega_I^2}{1+\Omega_I^2},\qquad
\tilde \mu_I^2 = \frac{\mu_I^2\th_I}{1+\Omega_I^2}\,.
\end{align}
The parameter $\mu_I$ denotes the mass of the scalar field\footnote{The index $I$ in all variables of \eqref{result2} merely indicate that they belong to the ``induced'' action and need not be equal to the according ones in our present model.}.
Through its coupling to $A_\mu$, the scalar field ``induced'' the effective one-loop action \eqref{result2} above. The so-called covariant coordinates $\tilde X_\mu$ are furthermore defined as
\begin{align}
\tilde X_\mu&=\tilde{x}_\mu+A_\mu\,.
\end{align}
The first expression in the induced action (\ref{result2}) can be written as
\begin{align}\label{eq:induced-first-term}
\frac{3}{\th_I} \left(1-\rho_I^2\right) \left(\tilde \mu_I^2-\rho_I^2\right)\left(\tilde X_\nu \star \tilde X_\nu -\tilde{x}^2\right)
= \frac{3}{\th_I} \left(1-\rho_I^2\right) \left(\tilde \mu_I^2-\rho_I^2\right)\left(2 \tilde{x}_\nu A_\nu + A_\nu \star A_\nu\right)\,.
\end{align}
The first term of \eqref{eq:induced-first-term} has to be compared with the first expression in (\ref{tadpole-order1}), whereas the second one corresponds to
the first term of the self energy (\ref{eq:one-loop_divergences}). However, the exact coefficients do not match. But since (\ref{tadpole-order1}) and
(\ref{eq:one-loop_divergences}) only take one-loop effects into account this cannot be expected.
Due to technical difficulties, we did not calculate the logarithmic UV divergences in all cases. Therefore, we
can only compare the term proportional to $\tilde{x}^2 (\tilde{x} A)$ given in Eq.(\ref{tadpole-order3}).
The respective term in the induced action --- stemming from the $(\tilde X_\mu \star \tilde X_\mu)^2$ term --- reads
\begin{align}
6\left(1-\rho_I^2\right)^2 \tilde{x}^2 \left(\tilde{x} A\right)\,.
\end{align}
In conclusion, one can state that the induced gauge theory action of Refs.~\cite{Grosse:2007,Wulkenhaar:2007} seems to be the more fundamental one when considering {non-commutative} gauge theories with Grosse-Wulkenhaar oscillator terms.
The UV counter terms we have encountered here can be nicely accommodated.
However, these models exhibit non-trivial vacuum configurations (which are discussed in e.g.~\cite{Grosse:2007jy, Wallet:2008a, Wohlgenannt:2008mn}) due to non-vanishing tadpoles, and it is hence not (yet) clear, how to do higher order loop calculations. Especially, a (highly desirable) general proof of renormalizability will be very involved.
\subsection*{Acknowledgements}
The authors are indebted to R.~Sedmik for providing valuable assistance with Wolfram Mathematica$^{\text{\textregistered}}$.\\
The work of D.~N.~Blaschke, E. Kronberger and M. Wohlgenannt was supported by the ``Fonds zur F\"orderung der Wissenschaftlichen Forschung'' (FWF) under contracts P20507-N16 and P21610-N16.
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{
"redpajama_set_name": "RedPajamaArXiv"
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Hyemeyohsts Wolf Storm (Lame Deer, Montana 1935) és un escriptor xeiene. Estudià a la Universitat de Montana i ha escrit llibres força polèmics per a posar en ficció una interpretació de la història sense autorització tribal. Ha escrit Seven arrows (1972), Song of Heyoehkah (1981), i Lighteningbold (1994).
Enllaços externs
Bibliografia
Persones de Montana
Amerindis estatunidencs
Escriptors amerindis estatunidencs
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/biz.libretexts.org\/Courses\/Lumen_Learning\/Book%3A_Retail_Management_(Lumen)\/12%3A_Retail_Pricing_and_Sales_Strategies\/12.01%3A_Price_Elasticity_Calculations","text":"# 12.1: Price Elasticity Calculations\n\n\nLearning Objectives\n\n\u2022 Calculate the price elasticity of a product based on the given situation\n\nRemember, elasticity\u00a0measures the responsiveness of one variable to changes in another variable. In the last section we looked at\u00a0price elasticity of demand, or how much a change in price affects the quantity demanded. In this section we will look at both\u00a0elasticity of demand and elasticity of supply.\u00a0Supply can also be elastic, since a change in price will influence the quantity supplied.\n\nIn order to measure elasticity, we need to calculate percentage change, also known as a\u00a0growth\u00a0rate.\u00a0The formula for computing a growth rate is straightforward:\n\nPercentage\u00a0change =\u200b\u200b\u200b Change\u00a0in\u00a0quantity\u200b\u200b \/ Quantity\n\nSuppose that a\u00a0job pays $10 per hour. At some point, the individual doing the job is given a$2-per-hour raise. The percentage change (or growth rate) in pay is:\n\n$2 \/$10 = 0.20\u00a0or\u00a020%.\n\nNow, recall that we defined elasticity as the percentage change in something divided by the percentage change in something else. Let\u2019s take the price elasticity of demand as an example. The price elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price:\n\nPrice\u00a0elasticity\u00a0of\u00a0demand =\u200b Percentage\u00a0change\u00a0in quantity demanded \/\u00a0Percentage change in price\n\nThere are two general methods\u00a0for calculating elasticities: the point elasticity approach and the midpoint (or arc) elasticity approach. Elasticity looks at the percentage change in quantity demanded divided by the percentage change in price, but which quantity and which price should be the denominator in the percentage calculation? The point approach uses the initial price and initial quantity to measure percent change. This makes the math easier, but the more accurate approach is the midpoint approach, which uses the average price and average quantity over the price and quantity change. (These are the price and quantity halfway between the initial point and the final point.\u00a0Let\u2019s compare the two approaches.)\n\nSuppose the quantity demanded of a product\u00a0was 100\u00a0at one point on the demand curve, and then it moved to\u00a0103 at another point. The growth rate, or percentage change in quantity demanded, would be the change in quantity demanded (103\u2212100) divided by the average\u00a0of the two quantities demanded\u00a0(103+100) \/ 2\u200b.\n\nIn other words,\u00a0the growth rate:\n\n103\u2212100 \/ ((103+100) \/ 2)\n\n= 3 \/ 101.5\n\n= 0.0296\n\n= 2.96% growth\n\nNote that if we used the point approach, the calculation would be:\n\n(103\u2013100) \/ 100\n\n= 3%\u00a0growth\n\nThis produces\u00a0nearly the same result as the slightly more complicated midpoint method (3% vs. 2.96%). If you need a rough approximation, use the point\u00a0method. If you need accuracy, use the midpoint\u00a0method.\n\nIf:\n\n% change in quantity > % change in price > 1 = Elastic demand\n\n% change in quantity > % change in price < 1 = Inelastic demand","date":"2022-08-16 16:59:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8495480418205261, \"perplexity\": 699.8413417017048}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572408.31\/warc\/CC-MAIN-20220816151008-20220816181008-00771.warc.gz\"}"}
| null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/936212\/question-about-the-central-limit-theorem","text":"# Question about the Central Limit Theorem\n\nThe version of the CLT in my book states that if $X_1,...,X_n$ is a random sample, with mean $\\mu$ and standard deviation $\\sigma^2$, then $W=\\frac{\\bar X-\\mu}{\\sigma\/\\sqrt n}$~$N(0,1)$ as $n\\rightarrow\\infty$. I also hear this phrased as \"the sample mean is approximately normally distributed for large $n.$\" I'm having trouble justifying this second characterization (which I guess is just a corollary of the CLT). I know that $\\mu_{\\bar X}=\\mu$ and $\\sigma^2_{\\bar X}$, and I know that if $X$~$N(\\mu,\\sigma^2)$, then $Z=\\frac{X-\\mu}{\\sigma}$~$N(0,1).$ It seems like the second characterization is using the converse of this result. Is the converse also true? That is, if $Z$~$N(0,1)$, does that imply $\\sigma Z+\\mu$~$N(\\mu,\\sigma^2)?$\n\n\u2022 For your final question, yes. If $Z$~$N(0,1)$ then $\\sigma Z+\\mu$~$N(\\mu,\\sigma^2)$. It is just a change in location and scale parameters \u2013\u00a0Henry Sep 18 '14 at 8:02\n\nWhen $n$ is large $$W=\\frac{\\bar{X}-\\mu}{\\sigma\/\\sqrt{n}}\\sim N(0,1)$$ approximately, or equivalently, $$\\bar{X}= \\mu+\\frac{\\sigma}{\\sqrt{n}}W\\sim N(\\mu,\\sigma^2\/n)$$ approximately. That is, the sample mean $\\bar{X}$ is approximately normal distributed with mean $\\mu$ and variance $\\sigma^2\/n$ when $n$ is large.","date":"2019-08-23 02:13:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9605005383491516, \"perplexity\": 135.44931026035638}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027317817.76\/warc\/CC-MAIN-20190823020039-20190823042039-00238.warc.gz\"}"}
| null | null |
Air France-KLM reports fall in profits for third quarter
Air France-KLM has reported disappointing third quarter financial results, with net profits dropping €420 million to €366 million for the three months to September 30th.
The European airline group reported a quarterly operating profit of €900 million, which was again down by €165 million compared to the same period last year.
Fuel costs were up by €135 million.
There was, however, a two per cent gain in revenue to €7.7 billion.
Pointing to a "challenging" global economy, Air France-KLM chief executive, Benjamin Smith, said: "Air France-KLM Group's performance in the third quarter showed resilience amid geopolitical uncertainties and softening macro-economic environment.
"Operationally, we achieved a solid performance during the summer peak travel period.
"Air France and KLM ranked in the top European legacy carriers in terms of punctuality.
"Based on long-haul forward bookings on average ahead of last year and renewed commitment to a strict cost discipline, we are confident that we can deliver on our annual objectives of reduced unit cost and stable leverage ratio.
"All the group's employees are mobilised to ensure the success of our strategic ambitions, which we will further outline on the occasion of the upcoming investor day next week."
The group was impacted by a stronger dollar and €100 million in costs from the phasing out of the Airbus A380.
Future Investment Initiative comes to a close in Riyadh
TUI to return to Sharm El Sheikh for summer 2020
Air France-KLM orders 60 Airbus A220 planes
Air France-KLM places new Airbus A350-900 order
Air France-KLM Group signs huge Airbus A220 deal
Virgin Atlantic expands bluebiz loyalty scheme in partnership with Air France-KLM
European Commission gives approval to Air France-KLM investment in Virgin Atlantic
Smith appointed chief executive at Air France-KLM
Accor drops plans for Air France-KLM stake
Air France-KLM completes acquisition of Virgin Atlantic stake
|
{
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|
José Maria Chaves dos Reis (ur. 21 listopada 1962 w Oeiras do Pará) – brazylijski duchowny katolicki, biskup Abaetetuby od 2013.
Życiorys
21 listopada 1996 otrzymał święcenia kapłańskie i został inkardynowany do prałatury terytorialnej Cametá. Po święceniach został rektorem niższego seminarium. W latach 2000-2006 pracował przy katedrze - początkowo jako wikariusz, a od 2002 jako proboszcz. W 2006 został rektorem wyższego seminarium oraz wikariuszem generalnym prałatury (od 2013 diecezji).
3 lipca 2013 papież Franciszek mianował go biskupem diecezji Abaetetuba. Sakry udzielił mu 5 października 2013 biskup Jesús María Cizaurre Berdonces.
Bibliografia
[dostęp 2013-07-03]
Informacja o nominacji w Biuletynie watykańskim z dnia 3 lipca 2013
Brazylijscy biskupi katoliccy
Urodzeni w 1962
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 6,181
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|
{
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Coin Story
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HomeNewsTSB Gaming Receives $2M from Square Enix, True Global Ventures, B Cryptos
TSB Gaming Receives $2M from Square Enix, True Global Ventures, B Cryptos
9 months ago Abdulhay Mahmoud News 0
The Sandbox (TSB) announced that, the third quarter of 2019 had been a particularly successful one for the game, according to a press release by TSB Gaming's parent company Animoca Brands.
Between July and September 2019 TSB Gaming raised US$2.01 million, consisting of 83% in cash and 17% in Bitcoin and Tether cryptocurrency at the time of the transaction, through the issue of "SAND" utility tokens and simple agreement for future equity (SAFE) convertible securities. SAND will be utilised in the upcoming blockchain version of The Sandbox user generated content game platform (the "Game").
The investment was made by Square Enix Co., Ltd. ("Square Enix"), B Cryptos Inc. ("B Cryptos"), Mindfulness Venture Fund I, L.P. ("Mindfulness Capital"), and True Global Ventures 4 Plus Fund GP ("TGV"), among others (the "Investors").
Sebastien Borget, COO of Pixowl and of TSB Gaming, said:
"The Sandbox is a pioneer in the space of blockchain gaming, NFTs, and true digital ownership, and we are thrilled to have Square Enix with us as a strategic investor, supporting our ambitious vision of building the Metaverse where Creators can make games on their LANDs and trade ASSETS on our marketplace in a totally decentralized manner."
Investors Include Big Names
Borget said big news for the platform is the involvement of a major game developer like Square Enix.
Square Enix itself has in fact recognized the potential of blockchain in gaming. In a letter from a new year, Yosuke Matsuda, president of the firm, said the technology will become a key growth factor for Square Enix in the future:
"Meanwhile, games using blockchain are no longer in their infancy and are gradually coming to represent a more significant presence. Rather than treating blockchain gaming as an opportunity for speculative investment, we believe establishing whether it is capable of bringing something new to our customers' gaming experiences will be the key to growth."
Square Enix developed major hits in the gaming industry such as the latest addition in the Tomb Raider series and the long-awaited remake of Final Fantasy VII that is slated for release in April.
TSB is a virtual world built on the blockchain of the Ethereum to allow players to create, own and monetize their own voxel gaming experiences.
As the namesake suggests it is possible to use SAND tokens in the game. They can be bought by players and creators, earned through gameplay and exchanged with Ether (ETH).
Successful Presales of Virtual Spaces of the Game
The game of mobile creation had over 40 million downloads and brought in more than 1,300 ETH during the first and second presales of virtual spaces of the game, called LAND. The third presale, starting on March 31, is expected to do similarly well in the second half of 2020, before the game launches.
Abdulhay Mahmoud 414 Articles
Abdulhay Mahmoud is a creative writer with over 15 years of experience in journalism, translation, and investor relations. He has B.A in English and Literature from a reputable University. He recently became a contributor at Cryptolydian.com to fulfill his thirst in reporting digital coins and blockchain-related news, an interest was built over the years.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,115
|
<?php
/**
* MySQL DB. All data is stored in data_pdo_mysql database
* Create an empty MySQL database and set the dbname, username
* and password below
*
* This class will create the table with sample data
* automatically on first `get` or `get($id)` request
*/
use Luracast\Restler\RestException;
class DB_PDO_MySQL
{
private $db;
function __construct()
{
try {
//Make sure you are using UTF-8
$options = array(PDO::MYSQL_ATTR_INIT_COMMAND => 'SET NAMES utf8');
//Update the dbname username and password to suit your server
$this->db = new PDO(
'mysql:host=localhost;dbname=data_pdo_mysql',
'username',
'password',
$options
);
$this->db->setAttribute(PDO::ATTR_DEFAULT_FETCH_MODE,
PDO::FETCH_ASSOC);
//If you are using older version of PHP and having issues with Unicode
//uncomment the following line
//$this->db->exec("SET NAMES utf8");
} catch (PDOException $e) {
throw new RestException(501, 'MySQL: ' . $e->getMessage());
}
}
function get($id, $installTableOnFailure = FALSE)
{
$this->db->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);
try {
$sql = $this->db->prepare('SELECT * FROM authors WHERE id = :id');
$sql->execute(array(':id' => $id));
return $this->id2int($sql->fetch());
} catch (PDOException $e) {
if (!$installTableOnFailure && $e->getCode() == '42S02') {
//SQLSTATE[42S02]: Base table or view not found: 1146 Table 'authors' doesn't exist
$this->install();
return $this->get($id, TRUE);
}
throw new RestException(501, 'MySQL: ' . $e->getMessage());
}
}
function getAll($installTableOnFailure = FALSE)
{
$this->db->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);
try {
$stmt = $this->db->query('SELECT * FROM authors');
return $this->id2int($stmt->fetchAll());
} catch (PDOException $e) {
if (!$installTableOnFailure && $e->getCode() == '42S02') {
//SQLSTATE[42S02]: Base table or view not found: 1146 Table 'authors' doesn't exist
$this->install();
return $this->getAll(TRUE);
}
throw new RestException(501, 'MySQL: ' . $e->getMessage());
}
}
function insert($rec)
{
$sql = $this->db->prepare("INSERT INTO authors (name, email) VALUES (:name, :email)");
if (!$sql->execute(array(':name' => $rec['name'], ':email' => $rec['email'])))
return FALSE;
return $this->get($this->db->lastInsertId());
}
function update($id, $rec)
{
$sql = $this->db->prepare("UPDATE authors SET name = :name, email = :email WHERE id = :id");
if (!$sql->execute(array(':id' => $id, ':name' => $rec['name'], ':email' => $rec['email'])))
return FALSE;
return $this->get($id);
}
function delete($id)
{
$r = $this->get($id);
if (!$r || !$this->db->prepare('DELETE FROM authors WHERE id = ?')->execute(array($id)))
return FALSE;
return $r;
}
private function id2int($r)
{
if (is_array($r)) {
if (isset($r['id'])) {
$r['id'] = intval($r['id']);
} else {
foreach ($r as &$r0) {
$r0['id'] = intval($r0['id']);
}
}
}
return $r;
}
private function install()
{
$this->db->exec(
"CREATE TABLE authors (
id INT AUTO_INCREMENT PRIMARY KEY ,
name TEXT NOT NULL ,
email TEXT NOT NULL
) DEFAULT CHARSET=utf8;"
);
$this->db->exec(
"INSERT INTO authors (name, email) VALUES ('Jac Wright', 'jacwright@gmail.com');
INSERT INTO authors (name, email) VALUES ('Arul Kumaran', 'arul@luracast.com');"
);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,047
|
Q: Input-dependent error: sendto() error code 22 (Invalid argument) depending on input size I'm having issues implementing a C UDP socket program. The code below works perfectly with any input shorter than 56 characters, but if I feed it 56 characters or more, sendto complains that I gave it invalid arguments (error code 22).
For instance, this will send correctly:
./talkerDemo localhost qqqqqwwwwweeeeeqqqqqwwwwweeeeeqqqqqwwwwweeeeeqqqqqwwwww
But this won't:
./talkerDemo localhost qqqqqwwwwweeeeeqqqqqwwwwweeeeeqqqqqwwwwweeeeeqqqqqwwwwwH
What gives?
/*
** talker.c
** Adapted from http://beej.us/guide/bgnet/html/single/bgnet.html#datagram
*/
#include <stdio.h>
#include <stdlib.h>
#include <unistd.h>
#include <errno.h>
#include <string.h>
#include <sys/types.h>
#include <sys/socket.h>
#include <netinet/in.h>
#include <arpa/inet.h>
#include <netdb.h>
#define SERVERPORT "4242" // the port users will be connecting to
int main(int argc, char *argv[])
{
int sockfd;
struct addrinfo hints, *servinfo, *p;
int rv;
int numbytes;
if (argc != 3) {
fprintf(stderr,"usage: talker hostname message\n");
exit(1);
}
memset(&hints, 0, sizeof hints);
hints.ai_family = AF_UNSPEC;
hints.ai_socktype = SOCK_DGRAM;
if ((rv = getaddrinfo(argv[1], SERVERPORT, &hints, &servinfo)) != 0) {
fprintf(stderr, "getaddrinfo: %s\n", gai_strerror(rv));
return 1;
}
// loop through all the results and make a socket
for(p = servinfo; p != NULL; p = p->ai_next) {
if ((sockfd = socket(p->ai_family, p->ai_socktype,
p->ai_protocol)) == -1) {
perror("talker: socket");
continue;
}
break;
}
if (p == NULL) {
fprintf(stderr, "talker: failed to create socket\n");
return 2;
}
//============================================================
// !!!!!!! Eror occurs here:
if ((numbytes = sendto(sockfd, argv[2], strlen(argv[2]), 0,
p->ai_addr, p->ai_addrlen)) == -1) {
perror("talker: sendto");
exit(1);
}
//============================================================
freeaddrinfo(servinfo);
printf("talker: sent %d bytes to %s\n", numbytes, argv[1]);
close(sockfd);
return 0;
}
EDIT
This was the version of the code I was actually running. Before posting the question, I had gone back to the original (above) to see if the problem occurred with that implementation too - and I seemed it was. But it turns out I was being thick and was using the wrong binary... derp
/*
** UDPTalker.hpp -- a datagram sockets "server"
** Adapted from http://beej.us/guide/bgnet/html/single/bgnet.html#datagram
*/
#ifndef UDPTALKER_H
#define UDPTALKER_H
#include <stdio.h>
#include <stdlib.h>
#include <unistd.h>
#include <errno.h>
#include <string.h>
#include <sys/types.h>
#include <sys/socket.h>
#include <netinet/in.h>
#include <arpa/inet.h>
#include <netdb.h>
#include <iostream>
#include <string>
#define UDPT_DEFAULT_PORT "4243"
#define UDPT_DEFAULT_HOST "localhost"
#define UDPT_MAXBUFLEN 2048
class UDPTalker {
int sockfd;
struct addrinfo hints, *servinfo, *p;
int rv;
int numbytes;
std::string host;
std::string port;
public:
//! Takes target hostname/ip and port as arguments. Defaults: ("localhost", "4243")
UDPTalker(std::string host = UDPT_DEFAULT_HOST, std::string port = UDPT_DEFAULT_PORT);
~UDPTalker();
void send(std::string msg);
};
#endif // UDPTALKER_H
Here's the corresponding .cpp:
// File UDPTalker.cpp
#include "UDPTalker.hpp"
UDPTalker::UDPTalker(std::string h, std::string port) : host(h), port(port) {
memset(&hints, 0, sizeof hints);
hints.ai_family = AF_UNSPEC;
hints.ai_socktype = SOCK_DGRAM;
if ((rv = getaddrinfo(host.c_str(), port.c_str(), &hints, &servinfo)) != 0) {
throw std::runtime_error(std::string("getaddrinfo: ").append(gai_strerror(rv)));
}
// loop through all the results and make a socket
for(p = servinfo; p != NULL; p = p->ai_next) {
if ((sockfd = socket(p->ai_family, p->ai_socktype,
p->ai_protocol)) == -1) {
perror("talker: socket");
continue;
}
break;
}
freeaddrinfo(servinfo);
if (p == NULL) {
throw std::runtime_error("talker: failed to create socket\n");
}
}
UDPTalker::~UDPTalker() {
close(sockfd);
}
void UDPTalker::send(std::string msg) {
if ((numbytes = sendto(sockfd, msg.c_str(), msg.size(), 0,
p->ai_addr, p->ai_addrlen)) == -1) {
perror("talker: sendto! ");
}
// printf("talker: sent %d bytes to %s\n", numbytes, host.c_str());
}
A: freeaddrinfo(servinfo); frees up the memory used by servinfo. This means that the pointer p is now pointing to empty memory, so when it gets passed to sendto it may have invalid content. My guess is that for some reason the extra byte in the input string was "rolling over" that memory location when the class' send method was being called. The fix was moving freeaddrinfo(servinfo); from the constructor to the destructor:
UDPTalker::UDPTalker(std::string h, std::string port) : host(h), port(port) {
memset(&hints, 0, sizeof hints);
hints.ai_family = AF_UNSPEC;
hints.ai_socktype = SOCK_DGRAM;
if ((rv = getaddrinfo(host.c_str(), port.c_str(), &hints, &servinfo)) != 0) {
throw std::runtime_error(std::string("getaddrinfo: ").append(gai_strerror(rv)));
}
// loop through all the results and make a socket
for(p = servinfo; p != NULL; p = p->ai_next) {
if ((sockfd = socket(p->ai_family, p->ai_socktype,
p->ai_protocol)) == -1) {
perror("talker: socket");
continue;
}
break;
}
if (p == NULL) {
throw std::runtime_error("talker: failed to create socket\n");
}
}
UDPTalker::~UDPTalker() {
freeaddrinfo(servinfo);
close(sockfd);
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,067
|
Charles Tillman (born 1932/1933) is an American politician who served on the Jackson, Mississippi City council for twelve years and as acting mayor after the death of Chokwe Lumumba, becoming the 5th consecutive African-American mayor of the city.
Biography
Tillman graduated with a B.A. in business education from Alcorn State University in 1958 and a M.A. in guidance and counseling from the University of Southern California in 1962. He did additional coursework at Atlanta University and took post-graduate classes at the University of Notre Dame and at Jackson State University. He began his career as a counselor at Rowan and Brinkley middle schools in Jackson before becoming a school principal and later president of the Jackson Public Schools Board. He was an active volunteer which led to his running for the Jackson City Council in 2005. In May 2005, he narrowly defeated Betty Dagner-Cook, 903 votes to Dagner-Cook's 845 votes in the Democratic primary for Ward 5 councilmember. On June 7, 2005, he defeated Independent candidate Joe Louis Sanders in the general election, 2,586 votes to Sanders' 384 votes.
After the death of mayor Chokwe Lumumba, Tillman was sworn in as acting mayor on February 25, 2014. He served as mayor until April 24, 2014 when Tony Yarber was sworn in after a special election.
In the 2021 election, seeking a 4th term on the City Council, he was defeated by Vernon Hartley, 936 votes to Tillman's 482.
Awards
In 1982, Tillman received the Governor's Distinguished Service Award for Outstanding Voluntary Community Service and the National Council of Negro Women's Appreciation Award for Outstanding and Dedicated Service.
Personal life
He has two children.
References
External links
Office of the Mayor
1930s births
Living people
Year of birth uncertain
African-American mayors in Mississippi
Mississippi Democrats
University of Southern California alumni
Alcorn State University alumni
Mississippi city council members
Mayors of Jackson, Mississippi
21st-century American politicians
21st-century African-American politicians
20th-century African-American people
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,738
|
\section{Equation of the Euler-Bernoulli beam}
We consider an Euler Bernoulli rod
under a distributed transversal force $g$ and axial force $P$.
The differential equation of equilibrium for the displacement
$w$ is given in \cite{Atanackovic:97} in the form
\begin{equation}
\frac{d^{2}}{dx^{2}}\Big( EI \frac{d^{2}w(x)}{dx^{2}} \Big)
+P \frac{d^{2} w(x)}{dx^{2}}=g(x)
\qquad x\in\left[0,l\right]. \label{1.2}%
\end{equation}
Here, $E$ is the modulus of elasticity, $I$ is the moment of
inertia, and $l$ is the length of the rod. In our analysis we will
allow for nonconstant, $x$-depended, even discontinuous
coefficients $I$ and $P$. When there is a discontinuity in $I$ at
some point $x_{0}$ the rod can be considered to consist of two
different, but connected, parts, i.e., $EI(x) =
EI_{1}+H(x-x_{0})(EI_{2}-EI_{1})$ where $I_1 \not= I_2$ are the
corresponding moments of inertia respectively and $H$ denotes the
Heaviside function.
Equation (\ref{1.2}) has been studied in
\cite{YS:01}, where the authors discuss possible jump discontinuities at
$x_{0}$ in the
displacement
\begin{equation}
\Delta := w(x_{0}+) - w(x_{0}-) \label{1.6}%
\end{equation}%
(as well as in the rotation $\theta := w'(x_{0}+)-w'(x_{0}-)$), where a
suffix $+$ or
$-$ in the function argument denotes the limit from the right or left.
A solution Ansatz of the form $w=w_{1}+H\left( x-x_{0}\right) \left(
w_{2}-w_{1}\right)$ is then used, where $w_1$ and $w_2$ solve the equation
to the left
and to the right of $x_0$. In course of justifying this could be called a
solution $\Delta$ was being forced to vanish in order to avoid ill-defined
products involving a Dirac delta.
Here, we investigate the corresponding mathematical issues left
open: first, we analyze the possibility to give a meaning to the
notion of 'distributional solution' in the context of the
distributional product hierarchy described in \cite[Chapter
II]{O:92} (see also the Appendix for a brief review); second, we
show that indeed $\Delta$ necessarily has to vanish then, which is
consistent with the calculations in \cite{YS:01}; more precisely,
if $w$ were to have a jump discontinuity then the model product $[a \cdot w'']$,
which is the most general in the distributional product hierarchy, can not exist.
Thus, in order to allow for solutions with
jump discontinuities in the displacement one is forced to go beyond
intrinsic distributional products and use, e.g.\ algebras of
generalized functions (cf.\ \cite{GKOS:01,O:92}). For example, there has been
active research on such issues for hyperbolic partial differential equations with
discontinuous coefficients, where in certain cases non-existence of
distributional solutions
has been proved (cf.\ \cite{Hoermann:03,HdH:01,HdH:01c,LO:91,O:88,O:89}).
For notational simplification and structural clarity we put $A=EI_{1}$,
$B=EI_{2}$
(hence $A\neq B$) and
\begin{equation}
a(x) = A + (B-A) H(x-x_{0}) =A H( x_{0}-x)
+B H(x-x_{0}). \label{1.12}
\end{equation}
Then the governing differential equation with boundary conditions for the
Euler-Bernoulli rod with jump discontinuities in the bending read
\begin{equation}
\frac{d^{2}}{dx^{2}}
\Big( a(x) \cdot\frac{d^{2}w(x)}{dx^{2}} \Big) + P(x)
\frac{d^{2}w(x)}{dx^{2}} =
g(x), \qquad x \in[0,1] \label{1.3}
\end{equation}
\begin{align}
w(0) & =0;\hspace{0.5cm}w(l)=0;\nonumber\\
\frac{d^{2}w}{dx^{2}}(0) & =0;\hspace{0.5cm}\frac{d^{2}w}{dx^{2}}(l)=0.
\label{1.4}
\end{align}
Mechanically, a global condition of equilibrium is expressed by equality of
the bending
moments
\begin{equation}
E I_1 w''(x_0 -) = E I_2 w''(x_0 +). \label{1.5}
\end{equation}
We may use the substitution $u = w''$ to lower the order of equation
(\ref{1.3})\ and boundary conditions (\ref{1.4})
\begin{equation}
\frac{d^{2}}{dx^{2}}\Big( a(x) \cdot u(x) \Big) + P(x)\, u(x) =
g(x) \qquad x \in[0,1] \label{1.1}
\end{equation}
with
\begin{equation}
u(0)=u(1)=0. \label{1.11}
\end{equation}
\begin{remark} Note that the above substitution is equivalent to imposing the
additional boundary problem $\frac{d^{2}}{dx^{2}}w(x)=u(x)$ with $w(0)=w(1)=0$,
which is uniquely solvabile once $u$ is determined by (\ref{1.1}-\ref{1.11}). In
the sequel we will thus only consider $u$.
\end{remark}
In equation (\ref{1.1}) the
product of the distributions $a$ and $u$ arises. There are several concepts
of partialy defined
products in the space of distributions. In the current paper we use the
so-called model product (cf.\ \cite{O:92}) to give a meaning to the
differential
equation.
\begin{remark} {\bf (Comparison with $L^2$-operator theory.)} The above boundary
value problem (\ref{1.1}-\ref{1.11}) can as
well be investigated in the classical functional analytic context of unbounded
operators on $L^2([0,1])$. Singularities of the coefficient functions
then have a significant influence on choices for an appropriate
domain. In course of the current paper, we follow an intrinsic distribution
theoretic view, which allows for a wider class of solutions, right-hand sides in
the differential equation, as well as variations in the solution concept itself.
To illustrate the situation in an unbounded operator approach we briefly sketch
the constructions for the case where $a$ is given by (\ref{1.12}) and $P$ is a
real constant. It is natural to implement the boundary conditions (\ref{1.11})
into the domain of the operator. Furthermore, we have to specify the meaning of
the formal expression $(a u)''$. Note that requiring that $u$ belongs to the
Sobolev space $H^2(]0,1[)$ makes $u''$ well-defined in $L^2([0,1])$ and gives
sense to the boundary conditions $u(0) = u(1) = 0$. Observe that under these
hypotheses $a u = A u_- + B u_+$, where $u_-$ (resp.\ $u_+$) vanishes to the
right (resp.\ left) of $x_0$ and is continuously differentiable on the left
(resp.\ right) up to $x_0$. Thus, by Schwartz' formula (\cite[Chapitre II,
\mathhexbox278 2]{Schwartz:66}), we have $(a u)'' = a u'' + (B u_+'(x_0)-A u_-'(x_0))
\cdot \de_{x_0} + (B u_+(x_0) - A u_-(x_0)) \cdot \de_{x_0}'$, which is in $L^2$
only for $u$ such that the coefficients of $\de_{x_0}$ and $\de_{x_0}'$ vanish.
Therefore, we define the operator $T u := a \cdot u''$ with domain
$$
D(T) := \{ u \in H^2(]0,1[) : u(0) = u(1) = 0, B u_-(x_0) - A u_+(x_0) = 0,
B u_+'(x_0)-A u_-'(x_0) = 0 \}.
$$
It is straightforward to check that $T$ is
symmetric, i.e., $D(T) \subseteq D(T^*)$ and $T^* \mid_{D(T)} = T$, where $T^*$
denotes the adjoint of $T$. In fact, one
can prove that $T$ is self-adjoint along the following lines: Let $\vphi\in
\Cinfc(]0,1[) \cap D(T)$ and $v\in D(T^*)$; interpreting $L^2$-inner products
$\inp{.}{.}$ in terms of distributional actions $\dis{.\,}{.}$ and
vice versa we obtain
$\dis{\ovl{T^*v}}{\vphi} = \inp{\vphi}{T^* v}=\inp{T\vphi}{v} = \inp{a \vphi''}{v}
= \dis{\ovl{(a v)''}}{\vphi}$, which implies that $(a v)'' \in L^2$, forcing
that $v$ belongs to $H^2$ and satisfies the conditions appearing in $D(T)$ at
$x_0$. Furthermore, integration by parts is then applicable with $u\in D(T)$
yielding $\inp{T u}{v} = \inp{u}{a \cdot v''} + B u'(1) v(1) - A u'(0) v(0)$;
since $u \mapsto \inp{T u}{v}$ has to be a continuous linear
functional with respect to the $L^2$-norm $v(0)$ and $v(1)$ have to vanish. Hence
$v$ is in $D(T)$ and $T^* = T$.
We observe that the original differential operator in Equation (\ref{1.1}) is of
the form $T + P I$, where $I$ denotes the identity operator. Therefore, questions
concerning uniqueness and existence of solutions to (\ref{1.1}-\ref{1.11}) when
$g\in L^2$ directly relate to spectral properties of $T$. One can view
corresponding results obtained in Section 3 below in this context.
\end{remark}
\section{Solution concept based on the model product}
We analyze the properties of a distributional solution $u$ to
problem (\ref{1.1}-\ref{1.11}) in detail when the product $a \cdot
u$ is interpreted as a 'model product'. Throughout this and the
following two sections we focus on regularity issues stemming from
the highest order terms in the equation. Therefore we make the
assumption that
\begin{center} $P$ is constant. \end{center}
We will remove this assumption and generalize our results in a final section
allowing
for jump discontinuities in $P$ as well.
\begin{definition}\label{d.1}
Let $\dot{\D}'([0,1]) := \{ v\in\D'(\R); \supp v \subseteq [0,1] \}$. We
call $u
\in \dot{\D}'([0,1])$ a \emph{solution} to (\ref{1.1}-\ref{1.11}) if the
following holds:
\begin{description}
\item{(A1)} The model product $[a \cdot u]$ of $u$ and $a$ (defined as in
\cite{O:92}, see also the Appendix) exists in $\D'(\R)$
\item{(A2)} The equation
\begin{equation}
( [a \cdot u])'' + P u = g \label{2.1}
\end{equation}
holds in $\mathcal{D}^{\prime}( \mathbb{R})$.
\end{description}
\end{definition}
\begin{remark}
\begin{trivlist}
\item{(i)} The boundary conditions (\ref{1.11}) are implemented into the
definition of
the space of prospective solutions $\dot{\D}'([0,1])$ in the following
sense: if $u \in
\dot{\D}'([0,1])$ happens to be a continuous function then $u(0) = u(1) =
0$.
\item{(ii)} Note that (A1) is equivalent to the existence of the model
products $[
H_{-}\cdot u] $ and $[ H_{+}\cdot u]$ where $H_{-}(x) = H(x_{0}-x)$ and
$H_{+} =
H(x-x_{0})$.
\end{trivlist}
\end{remark}
\begin{lemma}\label{T2}
\begin{trivlist}
\item{(i)} Let $u \in \dot{\D}'([0,1])$ satisfy (A1-2) then
$[H_{-}\cdot u]$ and $[ H_{+}\cdot u]$ belong to $\dot{\D}'([0,1])$.
\item{(ii)} $[ H_{-}\cdot\delta_{x_{0}}^{(k)}]$ and
$[H_{+}\cdot\delta_{x_{0}}^{(k)}]$
exist if and only if $k=0$, in which case we have
$[H_{-} \cdot\delta_{x_{0}}] = -\frac{\delta_{x_{0}}}{2}$,
$[ H_{+}\cdot\delta_{x_{0}} ] = \frac{\delta_{x_{0}}}{2}$. (Cf.\ similar
investigations in \cite[Lemma 4]{HdH:01})
\end{trivlist}
\end{lemma}
\begin{proof} Let $\vphi\in\D(\R)$ with $\int \vphi = 1$ and $\vphi_{\eps}(x) :=
\vphi(x/ \eps) / \eps$ be a model delta net (cf.\ \cite{O:92},(7.9)).
(i) By definition
$[H_{\pm}\cdot u] = \lim_{\eps\to 0}(H_{\pm} \ast \vphi_{\eps})
\cdot (u \ast \vphi_{\eps})$.
Let $w_{\eps} := (H_{\pm} \ast \vphi_{\eps}) \cdot (u \ast \vphi_{\eps})$ and
$\psi\in\D(\R)$ with $\supp \psi \cap [0,1] = \emptyset$ then
$\dis {w_{\eps}}{\psi} = \dis {H_{\pm} \ast \vphi_{\eps}}{(u \ast
\vphi_{\eps})\cdot\psi}$.
Since $\supp(u \ast \vphi_{\eps})\subseteq [0,1]+ \supp\vphi_{\eps} \subseteq
[-d_{\eps},1+d_{\eps}]$ for some $d_{\eps} \to 0$ (as $\eps \to 0$) we have
$(u \ast \vphi_{\eps})\cdot\psi = 0$ and thus $\dis {w_{\eps}}{\varphi} =
0$.
(ii) For any $\psi \in\D(\R)$
\begin{align*}
\dis{ [ H_{-}\cdot\delta_{x_{0}}^{(k)} ] }{\psi} & = \lim_{\eps\to 0}
\dis{ ( H(x_{0} - x) \ast
\vphi_{\eps})(\delta_{x_{0}}^{(k)} \ast\vphi_{\eps})}{\psi} \\
& = \lim_{\eps\to 0} \frac{1}{\eps^{k+1}}
\int\limits_{\R} \int\limits_{\frac{x-x_{0}}{\eps}}^{\infty}\vphi(t)
\vphi^{(k)}(\frac{x-x_{0}} { \eps}) \psi(x)\, dt dx\\
& = \lim_{\eps\to 0} \frac{1}{\eps^{k}} \int\limits_{\R}
\int\limits_{z}^{\infty}\vphi(t) \vphi^{(k)}(z) \psi(\eps z + x_{0}) \,
dt dz
\end{align*}
cannot be convergent for all $\psi$ as $\eps\to 0$ if $k \not= 0$. In case
$k=0$
we obtain the formula $[H_{-}\cdot\delta_{x_{0}}] = - \delta_{x_{0}}/2$ by
dominated convergence and the fact that $\int_{\R}\int_{z}^{\infty}\vphi(t)
\vphi(z)\, dz dt = - 1/2$.
The proof for $[ H_{+}\cdot\delta_{x_{0}} ]$ is similar.
\end{proof}
\begin{theorem}\label{T1}
Let $u \in \dot{\D}'([0,1])$ be a solution in the sense of Definition
\ref{d.1}.
Then $u$ is a locally integrable function.
\end{theorem}
\begin{proof}
\emph{Step 1:} Putting $\widetilde{u}_{-} = u \mid_{(0,x_{0})}$ and
$\widetilde{u}_{+}=u \mid_{(x_{0},1)}$ yields
\begin{align}
A\widetilde{u}_{-}'' + P\widetilde{u}_{-} &= g \mid_{(0,x_{0})}
\label{2.2} \\
B\widetilde{u}_{+}'' + P\widetilde{u}_{+} &= g \mid_{(x_{0},1)}.
\label{2.3}
\end{align}
Solving these two differential equations with constant coefficients we get
\begin{equation}\label{2.4}
\widetilde{u}_{-}=\widetilde{u}_{-h}+\widetilde{u}_{-p}\;
\qquad\text{and}\qquad
\widetilde{u}_{+} =\widetilde{u}_{+h}+\widetilde{u}_{+p},
\end{equation}
where
\[
\widetilde{u}_{-h}(x) = C_{1} e^{\sqrt{-P/A}x} +
C_{2}e^{-\sqrt{-P/A}x}, \quad
\widetilde{u}_{+h}(x) = D_{1}e^{\sqrt{-P/B}x} +
D_{2}e^{-\sqrt{-P/B}x}
\]
and
\[
\widetilde{u}_{-p}(x) = \frac{1}{2\sqrt{-P/A}}(\int_{0}^{x}g(\tau)
e^{\sqrt{-P/A}(x-\tau)}d\tau-
\int_{0}^{x}g(\tau)e^{-\sqrt{-P/A}(x-\tau)}d\tau)
\]
with a similar formula for $\widetilde{u}_{+p}(x)$ replacing $P/A$ by $P/B$
and
integration limits from $1$ to $x$.
Here, $\widetilde{u}_{-h}$, $\widetilde{u}_{+h}$ are smooth and
$\widetilde{u}_{-p}$, $\widetilde{u}_{+p}$ are absolutely continuous.
Therefore
$\widetilde{u}_{-}$ and $\widetilde{u}_{+}$ are absolutely continuous
functions on open subintervals $(0,x_{0})$ and $(x_{0},1)$ respectively.
Also, by
explicit formula, we see that $\widetilde{u}_{-}(x_{0}-) :=
\lim_{x\to x_{0}-}\widetilde{u}_{-}(x)$ and
$\widetilde{u}_{+}(x_{0}+) := \lim_{x\to x_{0}+}\widetilde{u}_{+}(x)$
exist.
\emph{Step 2:} Define $\widetilde{u}\in \ensuremath{L_{\text{loc}}^{1}}(\R) $ by
\begin{equation}\label{2.5}
\widetilde{u}(x) =
\begin{cases} 0 & -\infty<x\leq0 \\
\widetilde{u}_{-}(x) & 0<x<x_{0} \\
\widetilde{u}_{+} (x) & x_{0}<x<1 \\
0 & 1\leq x<\infty
\end{cases}.
\end{equation}
We have that $\widetilde{u}\in\dot{\D}'([0,1])$ and
$(u-\widetilde{u})\mid_{\R\backslash\{x_{0}\}} = 0$.
Therefore $\supp(u-\widetilde{u}) =\{x_{0}\}$, which implies that
\begin{equation}
u=\widetilde{u}+\sum_{k=0}^{N}c_{k}\delta_{x_{0}}^{(k)},\qquad
c_{k}\in\C,N\in\N_{0}. \label{2.6}
\end{equation}
By Lemma \ref{T2} and Assumption (A1) $N=0$ in (\ref{2.6}). Hence
\begin{equation}
u=\widetilde{u}+c_{0}\delta_{x_{0}}. \label{2.8}
\end{equation}
\emph{Step 3:} By Assumption (A2) we now obtain
\begin{equation}
( u\cdot a)''=g-Pu=g-P\widetilde{u}-cP\delta_{x_{0}}, \label{2.7}
\end{equation}
where $g-P\widetilde{u}\in \ensuremath{L_{\text{loc}}^{1}}(\mathbb{R}) $.
Let $w$ be a primitive function for $g-P\widetilde{u}$.
Then $w-c_{0}PH(x-x_{0})$ is one for $(u\cdot a)'$.
Therefore
\[
u\cdot a=W-c_{0}P(x-x_{0})_{+},
\]
where $x_{+}$ denote kink function, i.e.
$x_{+}=\left\{\begin{array}
[c]{c}
x\\
0
\end{array}
\begin{array}
[c]{c}
x>0\\
x\leq0
\end{array}
\right.$
and $W$ is primitive function for $w$.
Since $ W\in C^{1} $ and the kink function is absolutely continuous we have
that $u\cdot a$ is
absolutely continuous. But then (\ref{2.8}) and (\ref{1.12}) imply that
\[
u\cdot a=\widetilde{u}\cdot a+\frac{c_{0}}{2}\delta_{x_{0}}(B-A)
\]
which is absolutely continuous if and only if $c_{0}=0.$
This in turn yields $u=\widetilde{u}$ and therefore $u$ is locally
integrable
as $\widetilde{u}$ is.
\end{proof}
\section{Existence and uniqueness of an $L^1([0,1])$-solution}
As we have seen in the previous section, a distributional solution
in the sense of Definition \ref{d.1} necessarily is a locally
integrable function. In this case, we can interpret the product
$u\cdot a$ as a duality product (cf.\ \cite{O:92} or the Appendix).
We analyze this situation more closely.
\begin{proposition}\label{M1}
If $u\in L^1([0,1])$ is a solution to (\ref{1.1}-\ref{1.11}) then $u\in
C^{1}([0,1]\backslash\{x_{0}\})$
and $u$ has a jump at $x=x_{0}$.
\end{proposition}
\begin{proof}
If $u\in L^1([0,1])$ then the differential equation (\ref{1.1}) yields
$(u\cdot a)''\in L^1([0,1])$, hence $(u\cdot a)'\in
C_{\text{abs}}([0,1])$
and thus $u\cdot a\in C^{1}([0,1])$.
Therefore we also have that
$u\cdot a\mid_{[0,x_{0})}=Au\mid_{[0,x_{0})} \in C^{1}([0,x_{0}))$ in turn
$u\in C^{1}([0,x_{0}))$.
Similarly, $u\in C^{1} ((x_{0},1])$.
Furthermore $\lim_{x\to x_{0}-}u\cdot a(x)=A\cdot u(x_{0})$ and therefore
$u(x_{0}-)=\lim_{x\to x_{0}-}u(x)$ exists.
Similarly for $u(x_{0}+) =\lim_{x\to x_{0}+}u(x)$. But $a\cdot u$ is
continuous, so that
$\lim_{x\to x_{0}-}u\cdot a(x) =\lim_{x\to x_{0}+}u\cdot a(x)$ and thus
\begin{equation}\label{3.1}
Au(x_{0}-) = Bu(x_{0}+),
\end{equation}
which implies the global equilibrium condition (\ref{1.5}).
If $u\in C([0,1])$ then (\ref{3.1}) implies $A=B$,
which contradicts the assumption $I_{1}\neq I_{2}$.
This means that $u$ has to be discontinuous at $x_{0}$.
\end{proof}
\begin{remark}\label{AC2}
As a matter of fact we have $u'\in C_{\text{abs}}([0,1]\backslash
\{x_{0}\})$.
Indeed, since $(u\cdot a)'\in C_{\text{abs}}([0,1])$ reasoning as above
we obtain that $u'$ is absolutely continuous off $x_{0}$, so that
$u'(x_{0}-)$, $u'(x_{0}+)$ exist and obtain
\begin{equation}\label{3.11}
Au'(x_{0}-) = Bu'(x_{0}+).
\end{equation}
\end{remark}
Now we are in a position to construct a solution to (\ref{1.1}-\ref{1.11}).
\begin{lemma}\label{P_lemma} For any choice of $A > 0$, $B > 0$, and $0< x_0
<1$ there exists a strictly increasing sequence $(P_l)_{l\in\N}$
of positive real numbers $P_l$ such that the following holds:
\begin{enumerate}
\item[(i)] If $P < 0$
\item[or]
\item[(ii)] if $P >0$ and $P \not= P_l$ for all $l\in\N$
\end{enumerate}
then there is a unique solution to (\ref{2.2}) and (\ref{2.3})
with $\widetilde{u}_{-}(0)=0 $ and $\widetilde{u}_{+}(1)=0 $, which satisfies the
stability conditions
\begin{align}
A\widetilde{u}_{-}(x_0)=B\widetilde{u}_{+}(x_0) \label{sk} \\
A\widetilde{u}'_{-}(x_0)=B\widetilde{u}'_{+}(x_0). \label{sk1}
\end{align}
\end{lemma}
\begin{remark} \label{excases}
In case $P = P_{l}$ for some $l\in\N$ the solution is not unique or even may
fail to exist. Investigation of these cases seems possible in a direct way
without requiring further analytical tools.
\end{remark}
\begin{proof} Any solutions to (\ref{2.2}) and (\ref{2.3}) are given by
(\ref{2.4}).
\emph{Case $P < 0$:} The solution formulae (\ref{2.4}), adapted to the
boundary
conditions at $0$ and $1$, give
$$\widetilde{u}_{-}(x) = 2C_{1}\sinh \sqrt{-P/A}\,x + \widetilde{u}_{-p}(x) $$
and
$$\widetilde{u}_{+}(x) = 2D_{1}e^{\sqrt{-P/B}x}\sinh \sqrt{-P/B}\,(x-1) +
\widetilde{u}_{+p}(x), $$
where
\[
\widetilde{u}_{-p}(x) = \frac{1}{\sqrt{-P/A}}\int_{0}^{x}g(\tau)
\sinh \sqrt{-P/A}(x-\tau) \, d\tau
\]
and similarly to $\widetilde{u}_{+p}(x)$ (replacing $P/A$ with $P/B$ and
integration limits
from $1$ to $x$). The stability conditions (\ref{sk}-\ref{sk1}) are
equivalent to the
linear system $Hy=z$ with
\begin{equation}
H := \left[\begin{array}
[c]{cc}
2A\sinh\sqrt{-P/A}x_{0}
& -2Be^{\sqrt{-P/B}}\sinh\sqrt{-P/B}(x_{0}-1) \\
2A\sqrt{-P/A}\cosh\sqrt{-P/A}x_{0}
&-2B\sqrt{-P/B}e^{\sqrt{-P/B}}\cosh\sqrt{-P/B}(x_{0}-1)
\end{array}\right]
\end{equation}
and
\[
y := \left[\begin{array}[c]{c}
C_{1}\\
D_{1}
\end{array} \right],\qquad
z := \left[\begin{array}[c]{c}
Bu_{+p}(x_{0}) - Au_{-p}(x_{0})\\
Bu_{+p}'(x_{0}) - Au_{-p}'(x_{0})
\end{array}\right].
\]
Further we have
\begin{align*}
\det H & =4AB e^{\sqrt{-P/B}}
\left(-\sqrt{-P/B}\sinh
\sqrt{-P/A}x_{0}\cosh\sqrt{-P/B}(x_{0}-1)
\right. \nonumber\\
& \left. +\sqrt{-P/A}\sinh\sqrt{-P/B}
(x_{0}-1)\cosh\sqrt{-P/A}x_{0}
\right). \label{5.1}
\end{align*}
Since $$\sinh\sqrt{-A/P}x_{0}\cosh\sqrt{-B/P}(x_{0}-1) > 0 $$
and $$\sinh\sqrt{-B/P}(x_{0}-1)\cosh\sqrt{-A/P}x_{0} < 0 $$
we have $ \det H<0 $
hence unique solvability of the above linear system.
\emph{Case $P >0$:}
If $P>0$ then the solutions $\widetilde{u}_{-}$, $\widetilde{u}_{+}$ involve $\sin$ and
$\cos$
(instead of $\sinh$ and $\cosh$) and the determinant of the corresponding
linear system $H y = z$ reads
\begin{align*}
\det H & =4AB e^{\sqrt{-P/B}}
\left(-\sqrt{P/B}\sin
\sqrt{P/A}x_{0}\cos\sqrt{P/B}(x_{0}-1)
\right. \nonumber\\
& \left. +\sqrt{P/A}\sin\sqrt{P/B}
(x_{0}-1)\cos\sqrt{P/A}x_{0}
\right).
\end{align*}
When $\det H \not= 0$ we have the same situation as in the case $P<0$.
Observe that the
set $Z_0$ of values for $P$ such that any cosine factor occurring in the
above determinant vanishes is at most countable. Apart from these values, to
find
$P>0$ for which $\det H = 0 $ is equivalent to solving
$$ h(s) := \tan(s) + \nu \mu \tan(\mu s) = 0, $$
where $s=\sqrt{P/A} x_0 > 0$, $ \mu = \sqrt{A/B} (1/x_0 - 1) > 0$,
and $\nu = x_0 / (1 - x_0) > 0$. One observes that there is a countable
discrete set
of singularities of $h$, at which the limits from the left and right are
$+\infty$
and $-\infty$ respectively. Since $h$ is continuous otherwise, there is a
countable
set $Z_1$ of (positive) zeroes. To summarize, the union $Z_0 \cup Z_1$
makes up a
sequence $(P_l)_{l\in\N}$ with the required property.
\end{proof}
\begin{remark} We point out that the above proof of Lemma \ref{P_lemma}
does not give the
minimum set of values $P_l$ to be removed. In fact, only those elements in
$Z_0$ have to
occur in $(P_l)$ which make both cosine factors vanish. Note that the
latter can only
happen, when $\sqrt{B/A} x_0 / (1 - x_0)$ is a rational number of the form
$(2l+1)/(2k+1)$ with integers $k$, $l$.
\end{remark}
\begin{theorem}\label{T3}
Let $P$ satisfy one of the conditions (i)-(ii) in Lemma \ref{P_lemma}. Let
$\widetilde{u}_{-}$ and $\widetilde{u}_{+}$ be the solutions to (\ref{2.2}) and (\ref{2.3})
obtained in Lemma \ref{P_lemma} and define
\begin{equation}\label{3.2}
u_{-}(x)=\left\{\begin{array}
[c]{c}
\widetilde{u}_{-}(x) \\
0
\end{array}
\begin{array}
[c]{c}
x\in\left[ 0,x_{0}\right] \\
x\in\left[ x_{0},1\right]
\end{array},\right. \qquad
u_{+}(x)=\left\{\begin{array}
[c]{c}
0\\
\widetilde{u}_{+}(x)
\end{array}
\begin{array}
[c]{c}
x\in [0,x_{0}] \\
x\in [x_{0},1]
\end{array} \right. .
\end{equation}
Then
\begin{equation}\label{3.3}
u(x) = u_{-}(x) + u_{+}(x)
\end{equation}
is the unique solution to (\ref{1.1}-\ref{1.11}) in the sense of
Definition \ref{d.1},
belongs to $L^1([0,1])$ and satisfies the boundary conditions in the
classical
sense.
\end{theorem}
\begin{proof}
Since $a(x) = AH(x_{0}-x) + BH(x-x_{0})$ we have
$$
(u\cdot a)(x) = Au_{-}(x) + Bu_{+}(x)
$$
which we will differentiate twice.
Recall (\cite[Chapitre II, \mathhexbox278 2]{Schwartz:66}) that if a function $f$ is in
$C_{\text{abs}}([0,1]\backslash\{x_{0}\})$, such
that $\lim_{x\to x_{0}-} f(x) = f(x_{0}-)$ and $\lim_{x\to x_{0}+}f(x) =
f(x_{0}+)$
exist, then the distributional derivative $\frac{d}{dx}f$ satisfies
\begin{equation}\label{3.5}
\frac{d}{dx}f(x) = f'(x) + \big(f(x_{0}+)-
f(x_{0}-)\big)\cdot\delta_{x_{0}},
\end{equation}
where $f'$ denotes the (class of) function(s) in $L^1([0,1])$ equal to
the pointwise
derivative of $f$ almost everywhere in $[0,1]\setminus\{ x_{0} \}$.
Therefore
\[
\frac{d}{dx}(u\cdot a) =
A u_{-}'+Bu_{+}'+\big(B u(x_{0}+)- Au(x_{0}-)\big)\cdot\delta_{x_{0}}
\]
and
\begin{multline}\label{3.6}
\frac{d^{2}}{dx^{2}}(u\cdot a) = Au_{-}'' + Bu_{+}''+
\big(B u'(x_{0}+) - A u'(x_{0}-)\big)\cdot\delta_{x_{0}}\\
+ \big(B u(x_{0}+) - Au(x_{0}-)\big)\cdot\delta_{x_{0}}'.
\end{multline}
By construction we have that
$$
Au_{-}''(x)=\left\{\begin{array}
[c]{c}
(-Pu+g)(x)\\
0
\end{array}
\begin{array}
[c]{c}
x\in[0,x_{0}] \\
x\in[x_{0},1]
\end{array}\right., \;
B u_{+}''(x) =\left\{\begin{array}
[c]{c}
0\\
(-Pu+g)(x)
\end{array}
\begin{array}
[c]{c}
x\in[0,x_{0}] \\
x\in[x_{0},1]
\end{array}\right..$$
Thus (\ref{3.1}) and (\ref{3.11}) imply that
\[
\frac{d^{2}}{dx^{2}}(u\cdot a) = -Pu + g.
\]
Note that $u$ is continuous near the boundaries $x=0$ and $x=1$, thus
the conditions (\ref{1.11}) follow by construction.
For uniqueness, we first observe that any solution $u$ has to be in
$\ensuremath{L_{\text{loc}}^{1}}$ by
Theorem \ref{T1}. Furthermore, due to Proposition \ref{M1} it also has to
satisfy the
stability conditions (\ref{3.1}-\ref{3.11}). Hence Lemma \ref{P_lemma}
implies
uniqueness.
\end{proof}
\section{Generalization to discontinuous axial force}
We extended the analysis of the previous section to investigate solvability
of the same type of differential equation
\[
\frac{d^{2}}{dx^{2}}[a(x)\cdot u(x)]+ Pu(x) = g(x)
\]
with boundary condition $u(0) = u(1) = 0$, where the force $P$ now is a jump
function of the form
\[
P = P_{1}+ H(x-x_{0})(P_{1}-P_{2})
\]
with real numbers $P_1$, $P_2$.
As with constant $P$ we obtain that any solution to the
differential equation (in a sense similar to Definition \ref{d.1})
necessarily is a locally integrable function and
continuously differentiable off $x_{0}$ with a jump at $x=x_0$.
\begin{remark}\label{r2}
Note that condition (A1) in Definition \ref{d.1} implies that the
model product $[P \cdot u]$ exists. Therefore we will now require $u$ to be a
solution to the differential equation in the sense of Definition \ref{d.1} with
(\ref{2.1}) replaced by
\[
([a \cdot u])'' + [P \cdot u] = g.
\]
\end{remark}
\begin{theorem}
(i) Let $u\in \dot{\D}'([0,1])$ be a solution in the sense of Remark
\ref{r2}.
Then $u$ is a locally integrable function.
(ii) If $u\in \ensuremath{L_{\text{loc}}^{1}}$ is a solution then $u\in
C^{1}([0,1]\backslash\{x_{0}\})$
and $u$ has a jump discontinuity at $x=x_{0}$. Furthermore, equations
(\ref{3.1}) and
(\ref{3.11}) hold.
\end{theorem}
\begin{proof}
\emph{Step 1:} As in the proof of Theorem \ref{T1} we can set
$\widetilde{u}_{-}=u\left|_{( 0,x_{0}) }\right. $
and $\widetilde{u}_{+}=u\left|_{( x_{0},1)}\right.$.
Then we have
\begin{align}
A\widetilde{u}_{-}'' + P_{1}\widetilde{u}_{-} &
=g\left|_{(0,x_{0})}\right.\label{6.6}\\
B\widetilde{u}_{+}'' + P_{2}\widetilde{u}_{+} &
=g\left|_{(x_{0},1)}\right.\label{6.7}
\end{align}
Solving these two differential equations with constant coefficients we get
\begin{equation}\label{6.3}
\widetilde{u}_{-}=\widetilde{u}_{-h} + \widetilde{u}_{-p}
\hspace{0.5cm}\text{and}\hspace{0.5cm}
\widetilde{u}_{+}=\widetilde{u}_{+h}+\widetilde{u}_{+p}\
\end{equation}
where
\begin{equation*}
\widetilde{u}_{-h}(x) =C_{1}e^{\sqrt{-P_{1}/A}x} +
C_{2}e^{-\sqrt{-P_{1}/A}x},\;
\widetilde{u}_{+h}(x) =D_{1}e^{\sqrt{-P_{2}/B}x} +
D_{2}e^{-\sqrt{-P_{2}/B}x}
\end{equation*}
and
\[
\widetilde{u}_{-p}(x) = \frac{1}{2\sqrt{-P_{1}/A}}
\big( \int_{0}^{x}g(\tau)e^{\sqrt{-P_{1}/A}(x-\tau)}d\tau
-\int_{0}^{x}g(\tau)e^{-\sqrt {-P_{1}/A}(x-\tau)}d\tau\big),
\]
with a similar formula for $\widetilde{u}_{+p}(x)$ replacing $P_{1}/A$ by
$P_{2}/B$ and
integration limits from $1$ to $x$. Again,
$\widetilde{u}_{-h},\widetilde{u}_{+h}$ are smooth and
$ \widetilde{u}_{-p}$,$\widetilde{u}_{+p}$ are absolutely continuous.
Therefore $\widetilde{u}_{-}$ and $\widetilde{u}_{+}$
are absolutely continuous on open subintervals $(0,x_0)$ and $(x_0,1)$.
Also, there exist $\widetilde{u}_{-}(x_{0-}) = \lim_{x\rightarrow
x_{0-}}\widetilde{u}_{-}(x)$
and $\widetilde{u}_{+}(x_{0+}) = \lim_{x\rightarrow x_{0+}}\widetilde{u}_{+}(x)$.
\emph{Step 2:} Precisely as in Step 2 of the proof of Theorem \ref{T1} we
obtain
\begin{equation}\label{6.2}
u=\widetilde{u}+c\delta_{x_{0}}.
\end{equation}
\emph{Step 3:} Equation (\ref{6.2}) and $P=P_{1}+ H(x-x_{0})(
P_{1}-P_{2})$
leads to $ Pu=P_{1}\widetilde{u}_{-}+ P_{2}\widetilde{u}_{+} +
\frac{c}{2}\delta_{x_{0}}(P_{2}-P_{1})$.
Since $P_{1}\widetilde{u}_{-}+P_{2}\widetilde{u}_{+}\in \ensuremath{L_{\text{loc}}^{1}}(\R)$
we have that $w=\int(g-P_{1}\widetilde{u}_{-}-P_{2}\widetilde{u}_{+})$ is
absolutely continuous and its primitive function $W$ is $C^{1}$.
The differential equation $( u\cdot a)'' = g - Pu $
implies
\[
u\cdot a=W-\frac{c}{2}\left( P_{2}-P_{1}\right) \left( x-x_{0}\right)
_{+},
\]
which is absolutely continuous. The same arguments as in Theorem \ref{T1}
yield that $c=0$ and hence $u=\widetilde{u}$ is locally integrable. This proves
(i).
For part (ii) we may reason as in the proof of Proposition \ref{M1} we get
that
second part of theorem is valid.
\end{proof}
The construction of a solution rests on the following lemma which corresponds to
Lemma \ref{P_lemma}.
\begin{lemma}\label{P1_lemma} For any choice of $A > 0$, $B > 0$, and $0< x_0
<1$ there exist one-dimensional submanifolds $\cal{M}$ and $\cal{N}$ of $\R^2$
such that the following holds:
\begin{enumerate}
\item[(i)] If $P_1 < 0$ and $P_2 < 0$
\item[or]
\item[(ii)] if $P_1 > 0$, $P_2 > 0$ and $(P_1,P_2) \not\in \cal{M}$
\item[or]
\item[(iii)] if $P_1 > 0$, $P_2 < 0$, and $(P_1,P_2) \not\in \cal{N}$
\end{enumerate}
then there is a unique solution to (\ref{6.6}) and (\ref{6.7}) with
$\widetilde{u}_{-}(0)=0 $ and $\widetilde{u}_{+}(1)=0 $, which satisfies the
stability conditions
\begin{align}
A\widetilde{u}_{-}(x_0)=B\widetilde{u}_{+}(x_0) \label{ask} \\
A\widetilde{u}'_{-}(x_0)=B\widetilde{u}'_{+}(x_0). \label{ask1}
\end{align}
\end{lemma}
\begin{remark}
Similarly as in Remark \ref{excases} for the cases where $(P_1,P_2)$ belongs to
$\cal{M}$ or $\cal{N}$ the solution is not unique or may
fail to exist, explicit investigation of which could be carried out
along the lines of the following proof.
\end{remark}
\begin{proof}
As in the proof of Lemma \ref{P_lemma} any solution to (\ref{6.6}) and (\ref{6.7}) is given
by (\ref{6.3}).
\emph{In case $P_1 < 0$ and $P_2 < 0$} the solution
formulae (\ref{6.3}) with boundary conditions at $0$ and $1$, and stability conditions (\ref{ask})
and (\ref{ask1}) lead to a linear ($2$x$2$) system $ Hy=z $ with $y$ and $z$ as in Lemma
\ref{P_lemma} and
\begin{align*}
\det H & =4AB e^{\sqrt{-P_2/B}}
\left(-\sqrt{-P_2/B}\sinh
\sqrt{-P_1/A}x_{0}\cosh\sqrt{-P_2/B}(x_{0}-1)
\right. \nonumber\\
& \left. +\sqrt{-P_1/A}\sinh\sqrt{-P_2/B}
(x_{0}-1)\cosh\sqrt{-P_1/A}x_{0}
\right) < 0.
\end{align*}
Therefore we have a unique solution.
\vspace{0.2cm}
\noindent
\parbox[c]{.6\linewidth}{\ \hphantom{bla}\emph{Case $P_1 > 0$, $P_2 > 0$}:
$\det H$ now reads
\begin{align*}
\det H & =4AB e^{\sqrt{P_2/B}}
\left(-\sqrt{P_2/B}\sin
\sqrt{P_1/A}x_{0}\cos\sqrt{P_2/B}(x_{0}-1)
\right. \nonumber\\
& \left. + \sqrt{P_1/A}\sin\sqrt{P_2/B}
(x_{0}-1)\cos\sqrt{P_1/A}x_{0}
\right).
\end{align*}
Whenever this is nonzero we have a unique solution.
To see where it vanishes let $ s =
\sqrt{P_1/A}x_{0} $, $ t = \sqrt{P_2/B}(x_{0}-1)$, $\nu = x_0/(1-x_0)$ and
analyze the function
\begin{equation*}
f(s,t)=\nu t \sin s\cos t + s\sin t\cos s.
\end{equation*}
By direct inspection one deduces that $\grad f$ is
nonzero when $f=0$
which yields that the zero set ${\cal M} '=\{ (s,t)\in{\R^2}; f(s,t)=0 \}$ is a
one-dimensional
submanifold of $\R^2$.
We set ${\cal M} = \{ (P_1,P_2)\in{\R^2}; (s,t)\in\cal M ' \} $.}
\parbox{.35\linewidth}{\includegraphics[width=5cm,height=5cm]{M.eps}\\
{\small Figure: $\cal{M}'$ is a union of infinitely
many closed concentric curves (plot for the case $\nu =
6$).}}
\vspace{0.1cm}
\emph{Case $P_1 > 0$, $P_2 < 0$}: we have
\begin{align*}
\det H & =4AB e^{\sqrt{P_2/B}}
\left(\sqrt{-P_2/B}\sinh
\sqrt{-P_1/A}x_{0}\cos\sqrt{P_2/B}(x_{0}-1)
\right. \nonumber\\
& \left. + \sqrt{P_1/A}\sin\sqrt{P_2/B}
(x_{0}-1)\cosh\sqrt{P_1/A}x_{0}
\right).
\end{align*}
and as above one can show that the zero set is a one-dimensional
submanifold $\cal N$ of $\R^2$. In the complement $\det H \not= 0$, solution exists and is unique.
\end{proof}
\begin{theorem} Let $P_1$ and $P_2$ satisfy one of the conditions (i)-(iii) in
Lemma \ref{P1_lemma}.
Let $\widetilde{u}_{-}(x)$ and $\widetilde{u}_{+}(x)$ be the solutions to (\ref{6.6}) and
(\ref{6.7}) obtained in Lemma \ref{P1_lemma} and define
\begin{equation} \label{6.5}
u_{-}(x)=\left\{\begin{array}
[c]{c}
\widetilde{u}_{-}(x)\\
0
\end{array}
\begin{array}
[c]{c}
x\in [0,x_{0}] \\
x\in [x_{0},1]
\end{array}\right.
\hspace{0.5cm}
u_{+}(x)=\left\{\begin{array}
[c]{c}
0\\
\widetilde{u}_{+}(x)
\end{array}
\begin{array}
[c]{c}
x\in [0,x_{0}] \\
x\in [x_{0},1]
\end{array}\right. .
\end{equation}
Then
\begin{equation}\label{6.4}
u(x) = u_{-}(x) + u_{+}(x)
\end{equation}
is a solution to (\ref{1.1}-\ref{1.11}) with
$P=P_{1}+H(x-x_{0})(P_{1}-P_{2})$.
\end{theorem}
\begin{proof}
Proceeding as in the proof of Theorem \ref{T3} we arrive at (\ref{3.6}). By
construction we have
$$Au_{-}'' = \left\{\begin{array}
[c]{c}
-P_{1}u+g\\
0
\end{array}
\begin{array}
[c]{c}
x\in\left[ 0,x_{0}\right] \\
x\in\left[ x_{0},1\right]
\end{array}
\right.
\qquad \text{and}\qquad
Bu_{+}''=\left\{\begin{array}
[c]{c}
0\\
-P_{2}u+g
\end{array}
\begin{array}
[c]{c}
x\in[0,x_{0}]\\
x\in[x_{0},1]
\end{array}
\right. .
$$
Employing (\ref{ask}) and (\ref{ask1}) we get
$\frac{d^{2}}{dx^{2}}(u\cdot a) = -(P_{1}+(P_{2}-P_{1})H(x-x_{0}))u + g$.
\end{proof}
\section{Approximation by regularization}
In this section we investigate the possibility to approximate the solution
to (\ref{1.1}), (\ref{1.11}) using some regularization of the coefficient
$a(x)$. Throughout this section we will assume that $P$ is constant and such
that the solution to (\ref{1.1}), (\ref{1.11}) is unique.
Let $Qu:=(au)''+ Pu$ and consider the equation $Qu = g\in L^1([0,1])$.
Suppose that $a_{\eps}$ is a smooth regularization of the jump disconuity in $a$ such that
$a_\eps \in C^2([0,1])$ with $\lim_{\eps\to 0} a_{\eps}^{(j)} = a^{(j)}$ uniformly on compact subsets of $[0,1]\setminus\{x_0\}$ for derivative orders $j = 0, 1,2$. Let $Q_{\eps}u := (au_{\eps})''+ Pu$.
\begin{proposition} Let $g\in L^1([0,1])$.\\[1mm]
(i) If $u_{\eps} \in \dot{\D}'([0,1])$ denotes the solution to
$$
Q_{\eps}u = g, \qquad u(0)=0, u(1)=0,
$$
then $u_{\eps}$ belongs to the space $AC^2([0,1])$ of continuously diifferentiable functions whose derivates are loccaly integrable.\\[1mm]
(ii) Let $u$ be the solution to $Qu = g$. Then $u_{\eps}\to u$ uniformly on compact subsets of $[0,1]\setminus\{x_0\}$.
\end{proposition}
\begin{proof}
(i):
Note that $a_{\eps} Q_{\eps} u_{\eps} = a_{\eps}g$ is
equivalent to the Sturm
Liouville problem $L u_{\eps} := (pu_{\eps}')' + q u_{\eps} = a_{\eps} g$
with $p := a_{\eps}^2$ and
$ q := a_{\eps}(a_{\eps}'' + P)$. If $Q_{\eps} u_{\eps} = a_{\eps} g$ then
$(a_{\eps}^2 u_{\eps}')' = a_{\eps} g - a_{\eps}(a_{\eps}'' + P)\in L^1([0,1]) $
since $a_{\eps} g\in L^1([0,1])$ and $a_{\eps}$ is $C^2$. Therefore
$a_{\eps}^2 u_{\eps}'$ is absolutely continuous. Since $a_{\eps}$ is bounded from below away from zero this
implies that $u_{\eps}'$ is absolutely continuous as well. Thus
$u_{\eps}\in AC^2([0,1])$.\\[1mm]
(ii): Let $v_{\eps}:= u_{\eps}-u$. Then we obtain
\begin{equation} \label{Qeps_eq_feps}
Q_{\eps}v_{\eps} = ([(a_{\eps}-a)\cdot u])'' : = f_{\eps},
\qquad v_{\eps}(0)=v_{\eps}(1)=0.
\end{equation}
By assumption and using that $a_{\eps} - a$ is $C^2$ off $x_0$ we have that $f_{\eps}\to 0$ in $\ensuremath{L_{\text{loc}}^{1}}([0,1]\setminus\{x_0\})$.
Integrating twice in (\ref{Qeps_eq_feps}) gives
\begin{equation}
(a_{\eps}v_{\eps})(x) = \int_0^x (x-y)f_{\eps}(y)dy + P \int_0^x
(x-y)v_{\eps} dy + a_{\eps}(0)v_{\eps}(0) +
(a_{\eps}v_{\eps})'(0)\, x.
\end{equation}
Let $K$ be a compact subset of $[0,x_0)$.
Since $u$ and $u_{\eps}$ are $AC^2$ on $K$ (as noted in Remark \ref{AC2} and the prooof above) and both belong to $\dot{\D}'([0,1])$ we have that $v_{\eps}(0) = v'_{\eps}(0)=0$. Furthermore, by
$|(a_{\eps}v_{\eps})(x)|\geq A |v_{\eps}| > 0$ we obtain
$$
|v_{\eps}(x)|\leq \frac{1+P}{A}\int_0^x |x-y||f_{\eps}(y)|dy + \frac{1+P}{A}\int_0^x
|x-y||v_{\eps}(y)|dy.
$$
Applying Gronwall's inequality we get
$$
|v_{\eps}(x)| \leq \frac{1+P}{A}\, e^{\frac{(1+P) x}{A}}\int_0^x |x-y||f_{\eps}(y)|dy
\leq\frac{1+P}{A}\, e^{\frac{1+P}{A}}\, \|f_{\eps}\|_{L^1(K)} .
$$
As noted above, we have $f_{\eps} \to 0$ in $\ensuremath{L_{\text{loc}}^{1}}([0,1]\setminus\{x_0\})$, thus $v_{\eps} \to 0$ uniformly on $K$.
The reasoning in case of a compact subset contained in $(x_0,1]$ is similar. Since an arbitrary compact subset of $[0,1]\setminus\{x_0\}$ is the disjoint union of two compact subsets in either part of $[0,1]\setminus\{x_0\}$ the assertion is proved.
\end{proof}
We illustrate the convergence in an example: we put $A=1$, $B =2$, $P=1$, $x_0 = 1/2$,
and design $a_{\eps}$ as the $C^2$ function, which is defined by a fifth order odd polynomial in $[-\eps,\eps]$ and equal to $a$ otherwise. As right-hand side we choose $g(x) = - \cos(11 x) / \sqrt{|x - 2/3|}$. Note that $g \in L^1([0,1]) \setminus L^2([0,1])$. The following plots show the regularized coefficients and corresponding solutions for parameter values $\eps = 1/10, 1/30, 1/100$.
\vspace{0.5cm}
\parbox{.4\linewidth}{\includegraphics[width=6cm,height=4cm]{regcoff.eps}}
\hfill
\parbox{.5\linewidth}{\includegraphics[width=6cm,height=4cm]{regsolcos.eps}}
\vspace{0.1cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,092
|
Q: sigaction() sa_flags and POSIX.1-2001 Base Spec This is an excerpt from SIGACTION(2):
POSIX.1-1990 specified only SA_NOCLDSTOP. POSIX.1-2001 added SA_NOCLDSTOP, SA_NOCLDWAIT,
SA_NODEFER, SA_ONSTACK, SA_RESETHAND, SA_RESTART, and SA_SIGINFO. Use of these latter values
in sa_flags may be less portable in applications intended for older UNIX implementations.
And from FEATURE_TEST_MACROS(7):
_POSIX_C_SOURCE
· (Since glibc 2.3.3) The value 200112L or greater additionally exposes definitions
corresponding to the POSIX.1-2001 base specification (excluding the XSI extension).
This value also causes C95 (since glibc 2.12) and C99 (since glibc 2.10) features to
be exposed (in other words, the equivalent of defining _ISOC99_SOURCE).
But in reality, the flags added in 2001 spec is not exposed when the value 200112L is used. So, the following code does not compile. I tried glibc and uclibc.
#define _POSIX_C_SOURCE 200112L
#include <string.h>
#include <signal.h>
#include <unistd.h>
static int ret = 1;
void handle_signal (const int s) {
ret = 0;
}
int main (const int argc, const char **args) {
struct sigaction sa;
memset(&sa, 0, sizeof(sa));
sa.sa_flags = SA_RESETHAND;
sa.sa_sigaction = handle_signal;
sigaction(SIGINT, &sa, NULL);
sigaction(SIGTERM, &sa, NULL);
pause();
return ret;
}
What am I missing here? Are the flags not 2001 base spec? Are they an extension? Or is this just a bug in Linux libc implementations?
A: In the GNU C library, SA_RESETHAND is exposed by __USE_XOPEN_EXTENDED or __USE_XOPEN2K8. As a result, setting _POSIX_C_SOURCE to 200112 isn't sufficient; instead, you need
#define _XOPEN_SOURCE 600
(which enables __USE_XOPEN_EXTENDED) or
#define _POSIX_C_SOURCE 200809L
(which enables __USE_XOPEN2K8).
As I understand it, this is POSIX-conformant: before Issue 7, SA_RESETHAND, SA_RESTART, SA_SIGINFO, SA_NOCLDWAIT, and SA_NODEFER were part of the XSI option. They were moved to Base in Issue 7 (2008).
man 2 sigaction comes from the Linux man-pages project, not from the GNU C library, so discrepancies do happen.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,216
|
\section{Introduction and Results}
This paper pertains to the functions $N(T, \chi)$ and $N_{K}(T)$, respectively the number of zeroes $\rho = \beta + i\gamma$ of $L(s, \chi)$ and of $\zeta_{K}(s)$ in the region $0<\beta <1$ and $|\gamma|\leq T$. The purpose of this paper is to prove the following two theorems.
\begin{theorem}\label{McCurleyThm}
Let $T\geq 1$ and $\chi$ be a primitive nonprincipal character modulo $k$. Then
\begin{equation}\label{Thm11}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq 0.317\log kT + 6.401.
\end{equation}
In addition, if the right side of (\ref{Thm11}) is written as $C_{1}\log kT + C_{2}$, one may use the values of $C_{1}$ and $C_{2}$ contained in Table \ref{PrTable}.
\end{theorem}
\begin{theorem}\label{ZetaThm}
Let $T\geq 1$ and $K$ be a number field with degree $n_{K} = [K: \mathbb{Q}]$ and absolute discriminant $d_{K}$. Then
\begin{equation}\label{Thm12}
\bigg|N_{K}(T) - \frac{T}{\pi}\log\left\{ d_{K} \left( \frac{T}{2\pi e}\right)^{n_{K}}\right\}\bigg|\leq 0.317\left\{ \log d_{K} + n_{K}\log T\right\} + 6.333n_{K} + 3.482.
\end{equation}
In addition, if the right side of (\ref{Thm12}) is written as $D_{1}\left\{ \log d_{K} + n_{K}\log T\right\} + D_{2} n_{K} + D_{3}$, one may use the values of $D_{1}, D_{2}$ and $D_{3}$ contained in Table \ref{PrTable2}.
\end{theorem}
Theorem \ref{McCurleyThm} and Table \ref{PrTable} improve on a result due to McCurley \cite[Thm 2.1]{McCurley}; Theorem \ref{ZetaThm} and Table \ref{PrTable2} improve on a result due to Kadiri and Ng \cite[Thm 1]{NgKad2012}. The values of $C_{1}$ and $D_{1}$ given above are less than half of the corresponding values in \cite{McCurley} and \cite{NgKad2012}. The improvement is due to Backlund's trick --- explained in \S \ref{BT} --- and some minor optimisation.
Explicit expressions for $C_{1}$ and $C_{2}$ and for $D_{1}, D_{2}$ and $D_{3}$ are contained in (\ref{3.15}) and (\ref{C2f1}) and in (\ref{FinalD}) and (\ref{FinalD2}). These contain a parameter $\eta$ which, when varied, gives rise to Tables \ref{PrTable} and \ref{PrTable2}. The values in the right sides of (\ref{Thm11}) and (\ref{Thm12}) correspond to $\eta = \frac{1}{4}$ in the tables. Note that some minor improvement in the lower order terms is possible if $T\geq T_{0}>1$; Tables \ref{PrTable} and \ref{PrTable2} give this improvement when $T\geq 10$.
\begin{table}[h]
\caption{$C_{1}$ and $C_{2}$ in Theorem \ref{McCurleyThm} and in \cite{McCurley} for various values of $\eta$ }
\label{PrTable}
\centering
\begin{tabular}{c c c c c c }
\hline\hline
$\qquad\eta\qquad$ & \multicolumn{2}{c} \textrm{McCurley \cite{McCurley}\qquad} & \multicolumn{2}{c}{When $T\geq 1$\qquad\qquad} & When $T\geq 10$\\
& $C_{1}$ & $C_{2}$ & $C_{1}$ & $C_{2}$ & $C_{2}$ \\[0.5 ex] \hline
0.05 &0.506 &16.989 &0.248 & 9.339 &8.666 \\
0.10 & 0.552 & 13.202 & 0.265 & 8.015 & 7.311 \\
0.15 & 0.597 &11.067 & 0.282 &7.280 & 6.549 \\
0.20 & 0.643 &9.606 &0.300 & 6.778 &6.021 \\
0.25 & 0.689 & 8.509 & 0.317 & 6.401 & 5.616 \\
0.30 & 0.735 &7.641 &0.334 &6.101 &5.288 \\
0.35 & 0.781 & 6.929 & 0.351 & 5.852 & 5.011 \\
0.40 & 0.827 & 6.330 & 0.369 & 5.640 & 4.770 \\
0.45 & 0.873 & 5.817 & 0.386 & 5.456 & 4.556 \\
0.50 & 0.919 & 5.370 & 0.403 & 5.294 & 4.363 \\ \hline\hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{$D_{1}, D_{2}$ and $D_{3}$ in Theorem \ref{ZetaThm} and in \cite{NgKad2012} for various values of $\eta$ }
\label{PrTable2}
\centering
\begin{tabular}{c c c c c c c c c c}
\hline\hline
$\eta$ & \multicolumn{3}{c} \textrm{Kadiri and Ng \cite{NgKad2012}} & & \multicolumn{2}{c}{When $T\geq 1$} & \multicolumn{2}{c} {When $T\geq 10$}\\
& $D_{1}$ & $D_{2}$ & $D_{3}$ & $D_{1}$ & $D_{2}$ & $D_{3}$ & $D_{2}$ & $D_{3}$ \\[0.5 ex] \hline
0.05 &0.506& 16.95& 7.663& 0.248& 9.270& 3.005& 8.637& 2.069 \\
0.10 & 0.552& 13.163& 7.663& 0.265& 7.947& 3.121& 7.288& 2.083 \\
0.15 & 0.597& 11.029& 7.663& 0.282& 7.211& 3.239& 6.526& 2.099 \\
0.20 & 0.643& 9.567& 7.663& 0.300& 6.710& 3.359& 5.997& 2.116 \\
0.25 & 0.689& 8.471& 7.663& 0.317& 6.333& 3.482& 5.593& 2.134\\
0.30 & 0.735& 7.603& 7.663& 0.334& 6.032& 3.607& 5.265& 2.153 \\
0.35 & 0.781& 6.891& 7.663& 0.351& 5.784& 3.733& 4.987& 2.173\\
0.40 &0.827& 6.292& 7.663& 0.369& 5.572& 3.860& 4.746& 2.193 \\
0.45 &0.873& 5.778& 7.663& 0.386& 5.388& 3.988& 4.532& 2.215 \\
0.50 & 0.919& 5.331& 7.663& 0.403& 5.225& 4.116& 4.339& 2.238 \\ \hline\hline
\end{tabular}
\end{table}
Explicit estimation of the error terms of the zero-counting function for $L(s, \chi)$ is done in \S \ref{Ds}. Backlund's trick is modified to suit Dirichlet $L$-functions in \S \ref{BT}. Theorem \ref{McCurleyThm} is proved in \S \ref{LemSec}. Theorem \ref{ZetaThm} is proved in \S \ref{zetasec}.
The Riemann zeta-function, $\zeta(s)$, is both a Dirichlet $L$-function (albeit to the principal character) and a Dedekind zeta-function. The error term in the zero counting function for $\zeta(s)$ has been improved, most recently, by the author \cite{TrudS}. One can estimate the error term in the case of $\zeta(s)$ more efficiently owing to explicit bounds on $\zeta(1+it)$, for $t\gg 1$. It would be of interest to see whether such bounds for $L(1+ it, \chi)$ and $\zeta_{K}(1+ it)$ could be produced relatively easily --- this would lead to an improvement of the results in this paper.
\section{Estimating $N(T, \chi)$}\label{Ds}
Let $\chi$ be a primitive nonprincipal character modulo $k$, and let $L(s, \chi)$ be the Dirichlet $L$-series attached to $\chi$. Let $a= (1-\chi(-1))/2$ so that $a$ is $0$ or $1$ according as $\chi$ is an even or an odd character. Then the function
\begin{equation}\label{xie}
\xi(s, \chi) = \left(\frac{k}{\pi}\right)^{(s+a)/2} \Gamma\left(\frac{s+a}{2}\right)L(s, \chi),
\end{equation}
is entire and satisfies the functional equation
\begin{equation}\label{xif}
\xi(1-s, \overline{\chi}) = \frac{i^{a}k^{1/2}}{\tau(\chi)}\xi(s, \chi),
\end{equation}
where $\tau(\chi) = \sum_{n=1}^{k}\chi(n)\exp(2\pi in/k)$.
Let $N(T, \chi)$ denote the number of zeroes $\rho = \beta + i\gamma$ of $L(s, \chi)$ for which $0< \beta < 1$ and $|\gamma|\leq T$. For any $\sigma_{1}>1$ form the rectangle $R$ having vertices at $\sigma_{1} \pm iT$ and $1-\sigma_{1} \pm iT$, and let $\mathcal{C}$ denote the portion of the boundary of the rectangle in the region $\sigma \geq \frac{1}{2}$. From Cauchy's theorem and (\ref{xif}) one deduces that
\begin{equation*}\label{N1}
N(T, \chi) = \frac{1}{\pi} \Delta_{\mathcal{C}} \arg \xi(s, \chi).
\end{equation*}
Thus
\begin{equation}\label{N2}
\begin{split}
N(T, \chi) &= \frac{1}{\pi}\left\{ \Delta_{\mathcal{C}}\arg\left(\frac{k}{\pi}\right)^{(s+a)/2} + \Delta_{\mathcal{C}}\arg\Gamma\left(\frac{s+a}{2}\right) + \Delta_{\mathcal{C}}\arg L(s, \chi)\right\}\\
&= \frac{T}{\pi} \log\frac{k}{\pi} + \frac{2}{\pi} \Im\log\Gamma\left(\frac{1}{4} + \frac{a}{2} + i\frac{T}{2}\right) + \frac{1}{\pi}\Delta_{\mathcal{C}}\arg L(s, \chi).
\end{split}
\end{equation}
To evaluate the second term on the right-side of (\ref{N2}) one needs an explicit version of Stirling's formula. Such a version is provided in \cite[p.\ 294]{Olver}, to wit
\begin{equation}\label{Stirling}
\log\Gamma(z) = (z-\frac{1}{2})\log z - z + \frac{1}{2} \log 2\pi + \frac{\theta}{6|z|},
\end{equation}
which is valid for $|\arg z| \leq \frac{\pi}{2}$, and in which $\theta$ denotes a complex number satisfying $|\theta| \leq 1$.
Using (\ref{Stirling}) one obtains
\begin{equation}\label{Storl}
\begin{split}
\Im\log\Gamma\left(\frac{1}{4} + \frac{a}{2} + i\frac{T}{2}\right) = & \frac{T}{2} \log\frac{T}{2e} + \frac{T}{4} \log\left( 1+ \frac{(2a+1)^{2}}{4T^{2}}\right) \\
&+ \frac{2a-1}{4}\tan^{-1}\left(\frac{2T}{2a +1}\right) + \frac{\theta}{3|\frac{1}{2} + a+ iT|}.
\end{split}
\end{equation}
Denote the last three terms in (\ref{Storl}) by $g(a, T)$. Using elementary calculus one can show that $|g(0,T)|\leq g(1, T)$ and that $g(1, T)$ is decreasing for $T\geq 1$.
This, together with (\ref{N2}) and (\ref{Storl}), shows that
\begin{equation}\label{3.11}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq \frac{1}{\pi}\big| \Delta_{\mathcal{C}}\arg L(s, \chi)\big| + \frac{2}{\pi}g(1, T).
\end{equation}
All that remains is to estimate $\Delta_{\mathcal{C}}\arg L(s, \chi)$. Write $\mathcal{C}$ as the union of three straight lines, viz.\ let $\mathcal{C} = \mathcal{C}_{1} \cup \mathcal{C}_{2} \cup \mathcal{C}_{3}$, where $\mathcal{C}_{1}$ connects $\frac{1}{2} - iT$ to $\sigma_{1} - iT$; $\mathcal{C}_{2}$ connects $\sigma_{1} - iT$ to $\sigma_{1} + iT$; and $\mathcal{C}_{3}$ connects $\sigma_{1} + iT$ to $\frac{1}{2} + iT$. Since $L(\overline{s}, \chi) = \overline{L(s, \overline{\chi})}$ a bound for the integral on $\mathcal{C}_{3}$ will serve as a bound for that on $\mathcal{C}_{1}$. Estimating the contribution along $\mathcal{C}_{2}$ poses no difficulty since
\begin{equation*}
|\arg L(\sigma_{1} + it, \chi)| \leq |\log L(\sigma_{1} + it, \chi)| \leq \log\zeta(\sigma_{1}).
\end{equation*}
To estimate $\Delta_{\mathcal{C}_{3}}\arg L(s, \chi)$ define
\begin{equation}\label{wwbw}
f(s) = \frac{1}{2} \{L(s+ iT, \chi)^{N} + L(s-iT, \overline{\chi})^{N}\},
\end{equation}
for some positive integer $N$, to be determined later. Thus $f(\sigma) = \Re L(\sigma + iT, \chi)^{N}$. Suppose that there are $n$ zeroes of $\Re L(\sigma + iT, \chi)^{N}$ for $\sigma\in [\frac{1}{2}, \sigma_{1}]$. These zeroes partition the segment into $n+1$ intervals. On each interval $\arg L(\sigma + iT, \chi)^{N}$ can increase by at most $\pi$. Thus
\begin{equation*}
|\Delta_{\mathcal{C}_{3}} \arg L(s, \chi)| = \frac{1}{N} |\Delta_{\mathcal{C}_{3}} \arg L(s, \chi)^{N}| \leq \frac{(n+1)\pi}{N},
\end{equation*}
whence (\ref{3.11}) may be written as
\begin{equation}\label{stp}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq \frac{2}{\pi}\left\{\log\zeta(\sigma_{1}) + g(1,T)\right\} + \frac{2(n+1)}{N}.
\end{equation}
One may estimate $n$ with Jensen's Formula.
\begin{Lem}[Jensen's Formula]\label{JL}
Let $f(z)$ be holomorphic for $|z-a|\leq R$ and non-vanishing at $z=a$. Let the zeroes of $f(z)$ inside the circle be $z_{k}$, where $1\leq k\leq n$, and let $|z_{k} - a| = r_{k}$. Then
\begin{equation}\label{JF1}
\log\frac{R^{n}}{|r_{1} r_{2} \cdots r_{n}|} = \frac{1}{2\pi}\int_{0}^{2\pi} \log f(a+ Re^{i\phi})\, d\phi - \log |f(a)|.
\end{equation}
\end{Lem}
This is done in \S \ref{LemSec}.
\section{Backlund's Trick}\label{BT}
For a complex-valued function $F(s)$, and for $\delta>0$ define $\Delta_{+}\arg F(s)$ to be the change in argument of $F(s)$ as $\sigma$ varies from $\frac{1}{2}$ to $\frac{1}{2} + \delta$, and define $\Delta_{-}\arg F(s)$ to be the change in argument of $F(s)$ as $\sigma$ varies from $\frac{1}{2}$ to $\frac{1}{2} - \delta$.
Backlund's trick is to show that if there are zeroes of $\Re F(\sigma + iT)^{N}$ on the line $\sigma \in[\frac{1}{2}, \sigma_{1}]$, then there are zeroes on the line $\sigma \in[1-\sigma_{1}, \frac{1}{2}]$. This device was introduced by Backlund in \cite{Backlund1918} for the Riemann zeta-function.
Following Backlund's approach one can prove the following general lemma.
\begin{Lem}\label{BL}
Let $N$ be a positive integer and let $T\geq T_{0}\geq 1$. Suppose that there is an upper bound $E$ that satisfies
\begin{equation*}
| \Delta_{+}\arg F(s) + \Delta_{-}\arg F(s)| \leq E,
\end{equation*}
where $E = E(\delta, T_{0})$. Suppose further that there exists an $n\geq 3 + \lfloor NE/\pi\rfloor$ for which
\begin{equation}\label{C3gt}
n\pi \leq|\Delta_{\mathcal{C}_{3}} \arg F(s)^{N}| < (n+1)\pi.
\end{equation}
Then there are at least $n$ distinct zeroes of $\Re F(\sigma + iT)^{N}$, denoted by $\rho_{\nu}= a_{\nu} + iT$ (where $1\leq \nu \leq n$ and $\frac{1}{2} \leq a_{n}< a_{n-1} < \cdots \leq \sigma_{1}$), such that the bound $|\Delta\arg F(s)^{N}| \geq \nu\pi$ is achieved for the first time when $\sigma$ passes over $a_{\nu}$ from above.
In addition there are at least $n-2 - \lfloor N E/\pi\rfloor$ distinct zeroes $\rho_{\nu}' = a_{\nu}' + iT$ (where $1\leq \nu \leq n-2$ and $1-\sigma_{1} \leq a_{1}' < a_{2}' < \cdots \leq \frac{1}{2}$).
Moreover
\begin{equation}\label{jenpre}
a_{\nu} \geq 1-a_{\nu}', \quad \textrm{for } \nu = 1, 2, \ldots, n-2 - \lfloor NE/\pi\rfloor ,
\end{equation}
and, if $\eta$ is defined by $\sigma_{1} = \frac{1}{2} + \sqrt{2}(\eta + \frac{1}{2})$, then
\begin{equation}\label{finalae}
\prod_{\nu = 1}^{n} |1 + \eta - a_{\nu}| \prod_{\nu = 1}^{n -2 - \lfloor NE/\pi\rfloor } |1 + \eta - a_{\nu}'|\leq (\tfrac{1}{2} + \eta)^{2n-2 - \lfloor NE/\pi\rfloor }.
\end{equation}
\end{Lem}
\begin{proof}
It follows from (\ref{C3gt}) that $|\arg F(s)^{N}|$ must increase as $\sigma$ varies from $\sigma_{1}$ to $\frac{1}{2}$. This increase may only occur if $\sigma$ has passed over a zero of $\Re F(s)^{N}$, irrespective of its multiplicity. In particular as $\sigma$ moves along $\mathcal{C}_{3}$
\begin{equation*}
|\Delta\arg F(s)^{N}| \geq \pi, 2\pi,\ldots, n\pi.
\end{equation*}
Let $\rho_{\nu}= a_{\nu} + it$ denote the distinct zeroes of $\Re F(s)^{N}$ the passing over of which produces, for the first time, the bound $|\Delta\arg F(s)^{N}| \geq \nu\pi$. It follows that there must be $n$ such points, and that $\frac{1}{2} \leq a_{n} < a_{n-1}< \ldots < a_{2} < a_{1} \leq \sigma_{1}$. Also if $\frac{1}{2} + \delta \geq a_{\nu}$ then
\begin{equation}\label{gmcd}
|\Delta_{+}\arg F(s)^{N}| \geq (n-\nu)\pi.
\end{equation}
For (\ref{gmcd}) is true when $\nu = n$ and so, by the definition of $\rho_{\nu}$, it is true for all $1\leq \nu\leq n$.
By the hypothesis in Lemma \ref{BL},
\begin{equation}\label{sas3e}
| \Delta_{+}\arg F(s)^{N} + \Delta_{-}\arg F(s)^{N}| \leq NE.
\end{equation}
When $\frac{1}{2} + \delta \geq a_{\nu},$ (\ref{gmcd}) and (\ref{sas3e}) show that
\begin{equation}\label{jargo}
|\Delta_{-}\arg F(s)^{N}| \geq (n-\nu - NE/\pi)\pi,
\end{equation}
for $1 \leq \nu \leq n-2-\lfloor NE/\pi\rfloor $. When $\frac{1}{2} + \delta = a_{\nu}$ and $\nu= n-2-\lfloor NE/\pi\rfloor$, it follows from (\ref{jargo}) that $|\Delta_{-}\arg F(s)^{N}| \geq \pi$. The increase in the argument is only possible if there is a zero of $\Re F(s)^{N}$ the real part of which is greater than $\frac{1}{2} - \delta = 1- a_{n-2-\lfloor NE/\pi\rfloor}$. Label this zero $\rho_{n-2-\lfloor NE/\pi\rfloor}' = a_{n-2-\lfloor NE/\pi\rfloor}' + iT$. Repeat the procedure when $\nu = n-3 -\lfloor NE/\pi\rfloor, \ldots, 2, 1$, whence (\ref{jenpre}) follows. This produces a positive number of zeroes in $[1-\sigma_{1}, \frac{1}{2}]$ provided that $n\geq 3 + \lfloor NE/\pi\rfloor$.
For zeroes $\rho_{\nu}$ lying to the left of $1+ \eta$ one has
\begin{equation*}
|1+ \eta - a_{\nu}||1+ \eta - a_{\nu}'| \leq (1+ \eta - a_{\nu})(\eta + a_{\nu}),
\end{equation*}
by (\ref{jenpre}). This is a decreasing function for $a_{\nu}\in[\frac{1}{2}, 1+ \eta]$ and so, for these zeroes
\begin{equation}\label{lbt}
|1+ \eta - a_{\nu}||1+ \eta - a_{\nu}'| \leq (\tfrac{1}{2} + \eta)^{2}.
\end{equation}
For zeroes lying to the right of $1+ \eta$ one has
\begin{equation*}
|1+ \eta - a_{\nu}||1+ \eta - a_{\nu}'| \leq (a_{\nu} - 1 - \eta)(\eta + a_{\nu}).
\end{equation*}
This is increasing with $a_{\nu}$ and so, for these zeroes
\begin{equation}\label{rbt}
|1+ \eta - a_{\nu}||1+ \eta - a_{\nu}'| \leq \sigma_{1}^{2} - \sigma_{1} - \eta(1+ \eta).
\end{equation}
The bounds in (\ref{lbt}) and (\ref{rbt}) are equal\footnote{McCurley does not use Backlund's trick. Accordingly, his upper bounds in place of (\ref{lbt}) and (\ref{rbt}) are $\frac{1}{2} + \eta$ and $\sigma_{1} - 1 - \eta$. These are equal at $\sigma_{1} = \frac{3}{2} + 2\eta$, which is his choice of $\sigma_{1}$.} when $\sigma_{1} = \frac{1}{2} + \sqrt{2}(\eta + \frac{1}{2})$. Thus (\ref{finalae}) holds for $\sigma_{1} = \frac{1}{2} + \sqrt{2}(\eta + \frac{1}{2})$. For the unpaired zeroes
one may use the bound $|1+\eta - a_{\nu}| \leq \frac{1}{2} + \eta$, whence (\ref{finalae}) follows.
\end{proof}
\subsection{Applying Backlund's Trick}
Apply Jensen's formula on the function $F(s)$, with $a = 1+\eta$ and $R = r(\frac{1}{2}+ \eta)$, where $r>1$. Assume that the hypotheses of Lemma \ref{BL} hold.
If $1+ \eta - r(\frac{1}{2} + \eta) \leq 1-\sigma_{1}$ then all of the $2n-1-\lfloor NE/\pi\rfloor$ zeroes of $\Re F(\sigma+ iT)^{N}$ are included in the contour. Thus the left side of (\ref{JF1}) is
\begin{equation}\label{MayDay}
\begin{split}
&\log\frac{\{r(\frac{1}{2}+ \eta)\}^{2n-2-\lfloor NE/\pi\rfloor}}{|1+\eta - a_{1}| \cdots |1+ \eta - a_{n}||1+ \eta - a'_{1}|\cdots |1+ \eta - a_{n-2-\lfloor NE/\pi\rfloor}'|} \\
&\geq (2n -2-\lfloor NE/\pi\rfloor)\log r,
\end{split}
\end{equation}
by (\ref{finalae}).
If the contour does not enclose all of the $2n-2-[NE/\pi]$ zeroes of $\Re F(\sigma + iT)^{N}$, then the following argument, thoughtfully provided by Professor D.R.\ Heath-Brown, allows one still to make a saving.
To a zero at $x+ it$, with $\frac{1}{2} \leq x \leq 1+ \eta$ one may associate a zero at $x' + it$ where, by (\ref{jenpre}), $1-x \leq x' \leq \frac{1}{2}$. Thus, for an intermediate radius, zeroes to the right of $\frac{1}{2}$ yet still close to $\frac{1}{2}$ will have their pairs included in the contour. Let $X$ satisfy $1 + \eta - (\frac{1}{2} + \eta)/r < X < \min\{1+ \eta, r(\frac{1}{2} + \eta) - \eta\}$. Since $r>1$, this guarantees that $X>\frac{1}{2}$. For a zero at $x+ it$ consider two cases: $x \geq X$ and $x <X$.
In the former, there is no guarantee that the paired zero $x'+ it$ is included in the contour. Thus the zero at $x+ it$ is counted in Jensen's formula with weight
\begin{equation}\label{1stHBc}
\log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - x} \geq \log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - X}.
\end{equation}
Now, when $x<X$, the paired zero at $x'$ is included in the contour, since $1+ \eta - r(\frac{1}{2} + \eta) <1-X < 1-x \leq x'$. Thus, in Jensen's formula, the contribution is
\begin{equation}\label{uofx}
\begin{split}
\log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - x} + \log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - x'} &\geq \log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - x} + \log \frac{r(\frac{1}{2} + \eta)}{\eta + x}\\
&= \log \frac{r^{2} (\frac{1}{2} + \eta)^{2}}{(1+ \eta - x)(\eta + x)}.
\end{split}
\end{equation}
The function appearing in the denominator of (\ref{uofx}) is decreasing for $x\geq \frac{1}{2}.$ Thus the zeroes at $x+ it$ and $x' + it$ contribute at least $2\log r$.
Suppose now that there are $n$ zeroes in $[\frac{1}{2}, \sigma_{1}]$, and that there are $k$ zeroes the real parts of which are at least $X$. The contribution of all the zeroes ensnared by the integral in Jensen's formula is at least
\begin{equation*}
k\log \frac{r(\frac{1}{2} + \eta)}{1+ \eta - X} + 2(n-k)\log r = k\log\frac{(\frac{1}{2} + \eta)}{r(1+ \eta - X)} + 2n\log r \geq 2n\log r,
\end{equation*}
which implies (\ref{MayDay})
\subsection{Calculation of $E$ in Lemma \ref{BL}}
From (\ref{xie}) and (\ref{xif}) it follows that
\begin{equation*}\label{args}
\Delta_{+}\arg \xi(s, \chi) = - \Delta_{-}\arg \xi(s, \chi).
\end{equation*}
Since $\arg (\pi/k)^{-\frac{s+a}{2}} = -\frac{t}{2}\log (\pi/k)$ then $\Delta_{\pm} (\pi/k)^{-\frac{s+a}{2}} =0$, whence
\begin{equation*}\label{zetaarg}
| \Delta_{+}\arg L(s, \chi) + \Delta_{-}\arg L(s, \chi)| = | \Delta_{+}\arg\Gamma(\tfrac{s+a}{2}) + \Delta_{-}\arg\Gamma(\tfrac{s+a}{2})|.
\end{equation*}
Using
(\ref{Stirling}) one may write
\begin{equation}\label{llde}
\bigg| \Delta_{+}\arg \Gamma \left(\frac{s+a}{2}\right) + \Delta_{-}\arg\Gamma\left(\frac{s+a}{2}\right)\bigg| \leq G(a, \delta, t),
\end{equation}
where
\begin{equation}\label{Gdef}
\begin{split}
G(a, \delta, t) = &\frac{1}{2}(a-\frac{1}{2} + \delta) \tan^{-1}\frac{a+ \frac{1}{2} + \delta}{t} + \frac{1}{2}(a-\frac{1}{2} - \delta) \tan^{-1}\frac{a+ \frac{1}{2} - \delta}{t}\\ & - (a-\frac{1}{2}) \tan^{-1} \frac{a+ \frac{1}{2}}{t}
-\frac{t}{4}\log\left[1+ \frac{2\delta^{2}\{t^{2} - (\frac{1}{2} + a)^{2}\} + \delta^{4}}{\left\{t^{2} + (\frac{1}{2} + a)^{2}\right\}^{2}}\right]\\
&+\frac{1}{3}\left\{\frac{1}{|\frac{1}{2} + \delta +a + it|} + \frac{1}{|\frac{1}{2} - \delta +a + it|} + \frac{2}{|\frac{1}{2} +a + it|}\right\}.
\end{split}
\end{equation}
One can show that $G(a, \delta, t)$ is decreasing in $t$ and increasing in $\delta$, and that $G(1, \delta, t) \leq G(0, \delta, t)$. Therefore, since, in Lemma \ref{BL}, one takes $\sigma_{1} = \frac{1}{2} + \sqrt{2}(\frac{1}{2} + \eta)$ it follows that $\delta = \sqrt{2}(\frac{1}{2} + \eta)$, whence one may take
\begin{equation}\label{Etake}
E = G(0, \sqrt{2}(\tfrac{1}{2} + \eta), t_{0}),
\end{equation}
for $t\geq t_{0}$.
\section{Proof of Theorem 1}\label{LemSec}
First, suppose that $|\Delta_{\mathcal{C}_{3}} \arg L(s, \chi)^{N}| < 3 + \lfloor NE/\pi\rfloor$. Thus (\ref{3.11}) becomes
\begin{equation}\label{polk}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq \frac{2}{\pi}\left\{\log\zeta(\sigma_{1}) + g(1,T) + E\right\} + \frac{6}{N}.
\end{equation}
Now suppose that $|\Delta_{\mathcal{C}_{3}} \arg L(s, \chi)^{N}| \geq 3 + \lfloor NE/\pi\rfloor$, whence Lemma \ref{BL} may be applied.
To apply Jensen's formula to the function $f(s)$, defined in (\ref{wwbw}), it is necessary to show that $f(1+\eta)$ is non-zero: this is easy to do upon invoking an observation due to Rosser \cite{Rossers}. Write $L(1+\eta + iT, \chi) = Ke^{i\psi}$, where $K> 0$. Choose a sequence of $N$'s tending to infinity for which $N\psi$ tends to zero modulo $2\pi$. Thus
\begin{equation}\label{Rost}
\frac{f(1+\eta)}{|L(1+\eta+iT, \chi)|^{N}} \rightarrow 1.
\end{equation}
Since $\chi$ is a primitive nonprincipal character then $f(s)$ is holomorphic on the circle. It follows from (\ref{JF1}) and (\ref{MayDay}) that
\begin{equation}\label{jenco1}
n \leq \frac{1}{4\pi\log r} J - \frac{1}{2\log r} \log |f(1+\eta)| +1 + \frac{NE}{2\pi},
\end{equation}
where
\begin{equation*}\label{71}
J = \int_{-\frac{\pi}{2}}^{\frac{3\pi}{2}} \log|f(1+\eta + r(\frac{1}{2} + \eta)e^{i\phi})|\, d\phi.
\end{equation*}
Write $J = J_{1} + J_{2}$ where the respective ranges of integration of $J_{1}$ and $J_{2}$ are $\phi\in[-\pi/2, \pi/2]$ and $\phi\in[\pi/2, 3\pi/2]$. For $\sigma>1$
\begin{equation}\label{ie1}
\frac{\zeta(2\sigma)}{\zeta(\sigma)} \leq |L(s, \chi)| \leq \zeta(\sigma),
\end{equation}
which shows that
\begin{equation}\label{J1f}
J_{1} \leq N\int_{-\pi/2}^{\pi/2}\log\zeta(1+\eta + r(\tfrac{1}{2}+ \eta)\cos\phi)\, d\phi.
\end{equation}
On $J_{2}$ use
\begin{equation*}
\log |f(s)| \leq N\log |L(s + iT, \chi)|,
\end{equation*}
and the convexity bound \cite[Thm 3]{Rademacher}
\begin{equation}\label{rconv}
|L(s, \chi)| \leq \left(\frac{k|s+1|}{2\pi}\right)^{(1+\eta - \sigma)/2} \zeta(1+\eta),
\end{equation}
valid for $-\eta \leq \sigma \leq 1+\eta$, where $0<\eta \leq \frac{1}{2}$, to show that
\begin{equation}\label{J2}
J_{2} \leq \pi N \log \zeta(1+\eta) + N\frac{r(\frac{1}{2} + \eta)}{2} \int_{\pi/2}^{3\pi/2}(-\cos\phi) \log\left\{\frac{kTw(T, \phi, \eta, r)}{2\pi}\right\}\, d\phi,
\end{equation}
where
\begin{equation}\label{wder}
\begin{split}
w(& T, \phi, \eta,r)^{2} = \\
&1+ \frac{2r(\frac{1}{2} + \eta)\sin\phi}{T} + \frac{r^{2}(\frac{1}{2} + \eta)^{2} + (2+ \eta)^{2} + 2r(\frac{1}{2} + \eta)(2+\eta)\cos\phi}{T^{2}}.
\end{split}
\end{equation}
For $\phi\in[\pi/2, \pi]$, the function $w(T, \phi, \eta, r)$ is decreasing in $T$; for $\phi\in[\pi, 3\pi/2]$ it is bounded above by $w^{*}(T, \phi, \eta,r)$ where
\begin{equation}\label{wder2}
w^{*}(T, \phi, \eta,r)^{2} = 1 + \frac{r^{2}(\frac{1}{2} + \eta)^{2} + (2+ \eta)^{2} + 2r(\frac{1}{2} + \eta)(2+\eta)\cos\phi}{T^{2}},
\end{equation}
which is decreasing in $T$.
To bound $n$ using (\ref{jenco1}) it remains to bound $-\log|f(1+\eta)|$. This is done by using (\ref{Rost}) and (\ref{ie1}) to show that
\begin{equation*}
-\log |f(1+\eta)| \rightarrow -N \log |L(1+\eta + iT)| \leq -N \log[\zeta(2+2\eta)/\zeta(1+\eta)].
\end{equation*}
This, together with (\ref{stp}), (\ref{polk}), (\ref{jenco1}), (\ref{J1f}), (\ref{J2}) and sending $N\rightarrow \infty$, shows that, when $T\geq T_{0}$
\begin{equation}\label{3.14}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq \frac{r(\frac{1}{2} + \eta)}{2\pi\log r} \log kT + C_{2},
\end{equation}
where
\begin{equation*}
\begin{split}
C_{2} = \frac{2}{\pi}& \left\{\log\zeta(\tfrac{1}{2} + \sqrt{2}(\tfrac{1}{2} + \eta)) + g(1, T) + \frac{E}{2}\right\} + \frac{3}{2\log r} \log\zeta(1+\eta)\\ &- \frac{\log \zeta(2+2\eta)}{\log r} + \frac{1}{2\pi\log r}\int_{-\pi/2}^{\pi/2} \log\zeta(1+ \eta + r(\tfrac{1}{2}+ \eta)\cos\phi)\, d\phi\\
& +\frac{r(\frac{1}{2} + \eta)}{4\pi\log r}\bigg\{ -2\log 2\pi + \int_{\pi/2}^{\pi} (-\cos\phi)\log w(T_{0}, \phi, \eta, r)\, d\phi \\
&\qquad\qquad\qquad\qquad\qquad\qquad+ \int_{\pi}^{3\pi/2} (-\cos\phi)\log w^{*}(T_{0}, \phi, \eta, r)\, d\phi\bigg\}.
\end{split}
\end{equation*}
\subsection{A small improvement}\label{Asi}
Consider that what is really sought is a number $p$ satisfying $-\eta \leq p <0$ for which one can bound $L(p+it, \chi)$, provided that $1+ \eta - r(\frac{1}{2} + \eta)\geq p$. Indeed the restriction that $p\geq-\eta$ can be relaxed by adapting the convexity bound, but, as will be shown soon, this is unnecessary.
The convexity bound (\ref{rconv}) becomes the rather ungainly
\begin{equation*}
|L(s, \chi)| \leq \left\{\left(\frac{k|1+ s|}{2\pi}\right)^{(1/2 - p)(1 + \eta - \sigma)} \zeta(1 - p)^{1 + \eta - \sigma} \zeta(1+ \eta)^{\sigma - p}\right\}^{1/(1 + \eta - p)},
\end{equation*}
valid for $-\eta \leq p \leq \sigma \leq 1 + \eta$. Such an alternation only changes $J_{2}$, whence the coefficient of $\log kT$ in (\ref{3.14}) becomes
\begin{equation*}
\frac{r(\frac{1}{2} + \eta)(\frac{1}{2} - p)}{\pi(1+ \eta -p)\log r}.
\end{equation*}
This is minimised when $r = (1 + \eta - p)/(1/2 + \eta)$, whence (\ref{3.14}) becomes
\begin{equation}\label{3.15}
\bigg|N(T, \chi) - \frac{T}{\pi}\log\frac{kT}{2\pi e}\bigg| \leq \frac{\frac{1}{2} - p}{\pi\log \left(\frac{1+ \eta - p}{1/2 + \eta}\right)} \log kT + C_{2},
\end{equation}
where
\begin{equation}\label{C2f1}
\begin{split}
&C_{2} = \frac{2}{\pi} \left\{\log\zeta(\tfrac{1}{2} + \sqrt{2}(\tfrac{1}{2} + \eta)) + g(1, T) + \frac{G(0, \sqrt{2}(\frac{1}{2} + \eta), T_{0})}{2}\right\} \\
&+ \frac{1}{\log\left(\frac{1+ \eta - p}{1/2 + \eta}\right)}\bigg\{ \frac{3}{2}\log\zeta(1+\eta)
- \log \zeta(2+2\eta) +\frac{1}{\pi}\log \frac{\zeta(1-p)}{\zeta(1+ \eta)}\\
& + \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} \log\zeta(1+ \eta + (1+ \eta - p)\cos\phi)\, d\phi
+\frac{\frac{1}{2} - p}{2\pi}\bigg(-2\log 2\pi \\
&+ \int_{\pi/2}^{\pi} (-\cos\phi)\log w(T_{0}, \phi, \eta, r)\, d\phi + \int_{\pi}^{3\pi/2} (-\cos\phi)\log w^{*}(T_{0}, \phi, \eta, r)\, d\phi\bigg)\bigg\},
\end{split}
\end{equation}
in which $g(1, T)$, $G(a, \delta, T_{0})$, $w$ and $w^{*}$ are defined in (\ref{Storl}), (\ref{Gdef}), (\ref{wder}) and (\ref{wder2}).
The coefficient of $\log kT$ in (\ref{3.15}) is minimal when $p=0$ and $r = \frac{1 + \eta}{1/2 + \eta}$. One cannot choose $p=0$ nor should one choose $p$ to be too small a negative number lest the term $\log \zeta(1-p)/\zeta(1+ \eta)$ become too large. Choosing $p=-\eta/7$ ensures that $C_{2}$ in (\ref{3.15}) is always smaller than the corresponding term in McCurley's proof. Theorem \ref{McCurleyThm} follows upon taking $T_{0} = 1$ and $T_{0} = 10$. One could prove different bounds were one interested in `large' values of $kT$. In this instance the term $C_{2}$ is not so important, whence one could choose a smaller value of $p$.
\section{The Dedekind zeta-function}\label{zetasec}
This section employs the notation of \S\S \ref{Ds}-\ref{BT}. Consider a number field $K$ with degree $n_{K} = [K: \mathbb{Q}]$ and absolute discriminant $d_{K}$. In addition let $r_{1}$ and $r_{2}$ be the number of real and complex embeddings in $K$, whence $n_{K} = r_{1} + 2r_{2}$. Define the Dedekind zeta-function to be
\begin{equation*}
\zeta_{K}(s) = \sum_{\mathfrak{a} \subset \mathcal{O}_{K}} \frac{1}{(\mathbb{N}\mathfrak{a})^{s}},
\end{equation*}
where $\mathfrak{a}$ runs over the non-zero ideals. The completed zeta-function
\begin{equation}\label{feded}
\xi_{K}(s) = s(s-1) \left(\frac{d_{K}}{\pi^{n_{K}}2^{2r_{2}}}\right)^{s/2} \Gamma(s/2)^{r_{1}} \Gamma(s)^{r_{2}} \zeta_{K}(s)
\end{equation}
satisfies the functional equation
\begin{equation}\label{feeded}
\xi_{K}(s) = \xi_{K}(1-s).
\end{equation}
Let $a(s) = (s-1)\zeta_{K}(s)$ and let
\begin{equation}\label{fdefded}
f(\sigma) = \frac{1}{2}\left\{a(s+ iT)^{N} + a(s-iT)^{N}\right\}.
\end{equation}
It follows from (\ref{feded}) and (\ref{feeded}) that
\begin{equation}\label{DedEE}
\bigg|\Delta_{+} \arg a(s) + \Delta_{-} \arg a(s)\bigg| \leq F(\delta, t)+ n_{K} G(0, \delta, t),
\end{equation}
where $F(\delta, t) = 2\tan^{-1}\frac{1}{2t} - \tan^{-1} \frac{1/2 + \delta}{t} - \tan^{-1}\frac{1/2 - \delta}{t}$, and $G(0, \delta, t)$ is defined in (\ref{Gdef}).
Thus, following the arguments in \S\S \ref{Ds}-\ref{LemSec}, one arrives at
\begin{equation}\label{NDed}
\bigg|N_{K}(T) - \frac{T}{\pi}\log\left\{ d_{K} \left( \frac{T}{2\pi e}\right)^{n_{K}}\right\}\bigg| \leq \frac{2(n+1)}{N} + \frac{2n_{K}}{\pi}\left\{ |g(0, T)| + \log \zeta(\sigma_{1})\right\} + 2,
\end{equation}
where $n$ is bounded above by (\ref{jenco1}), in which $f(s)$ is defined in (\ref{fdefded}). Using the right inequality in
\begin{equation}\label{dedin}
\frac{\zeta_{K}(2\sigma)}{\zeta_{K}(\sigma)}\leq |\zeta_{K}(s)|\leq \left\{\zeta(\sigma)\right\}^{n_{K}},
\end{equation}
one can show that the corresponding estimate for $J_{1}$ is
\begin{equation}\label{JDedFin}
J_{1}/N \leq \pi\log T + \int_{-\pi/2}^{\pi/2} \left\{\log \tilde{w}(T, \phi, \eta, r) + n_{K} \log \zeta( 1 + \eta + r(\tfrac{1}{2} + \eta)\cos \phi)\right\}\, d\phi
\end{equation}
where
\begin{equation}\label{wdert}
\tilde{w}(T, \phi, \eta,r)^{2} =1+ \frac{2r(\frac{1}{2} + \eta)\sin\phi}{T} + \frac{r^{2}(\frac{1}{2} + \eta)^{2} + \eta^{2} + 2r\eta(\frac{1}{2} + \eta)\cos\phi}{T^{2}}.
\end{equation}
For $\phi\in[0, \pi/2]$, the function $\tilde{w}(T, \phi, \eta, r)$ is decreasing in $T$; for $\phi\in[-\pi/2, 0]$ it is bounded above by $\tilde{w}^{*}(T, \phi, \eta,r)$ where
\begin{equation}\label{wder2t}
\tilde{w}^{*}(T, \phi, \eta,r)^{2} =1+ \frac{r^{2}(\frac{1}{2} + \eta)^{2} + \eta^{2} + 2r\eta(\frac{1}{2} + \eta)\cos\phi}{T^{2}}.
\end{equation}
which is decreasing in $T$.
The integral $J_{2}$ is estimated using the following convexity result.
\begin{Lem}\label{convLem}
Let $-\eta\leq p<0$. For $p \leq 1+ \eta - r(\frac{1}{2} + \eta)$ the following bound holds
\begin{equation*}
\begin{split}
|a(s)|^{1 + \eta - p} \leq \left(\frac{1 -p}{1 +p}\right)^{1+ \eta - \sigma} &\zeta_{K}(1+ \eta)^{\sigma - p} \zeta_{K}(1-p)^{1 + \eta - \sigma} |1+ s|^{1 + \eta - p}\\
&\times \left\{d_{K} \left(\frac{|1 + s|}{2\pi}\right)^{n_{K}}\right\}^{(1 + \eta - \sigma)(1/2 - p)}.
\end{split}
\end{equation*}
\end{Lem}
\begin{proof}
See \cite[\S 7]{Rademacher}. When $p = -\eta$ the bound reduces to that in \cite[Thm 4]{Rademacher}.
\end{proof}
Using this it is straightforward to show that
\begin{equation}\label{FinJ2}
\begin{split}
J_{2}/N &\leq \frac{2r(\frac{1}{2} + \eta)}{1 + \eta - p} \left\{ \log \frac{\zeta_{K}(1 -p)}{\zeta_{K}(1 + \eta)} + \log \frac{1 - p}{1 +p} + (1/2 - p)\log \frac{d_{K}}{(2\pi)^{n_{K}}}\right\} \\
& + \pi \log\zeta_{K}(1 + \eta) + \log T\left(\pi + \frac{2rn_{K}(\frac{1}{2} + \eta)(\frac{1}{2} - p)}{1 + \eta - p}\right)\\
& + \int_{\pi/2}^{3\pi/2} \log w(T_{0}, \phi, \eta, r)\left(1 + \frac{n_{K}r(\frac{1}{2} + \eta)(\frac{1}{2} - p)(-\cos\phi)}{1 + \eta - p}\right)\, d\phi\\
\end{split}
\end{equation}
The quotient of Dedekind zeta-functions can be dispatched easily enough using
\begin{equation*}
-\frac{\zeta_{K}'}{\zeta_{K}} (\sigma) \leq n_{K}\left\{ -\frac{\zeta'}{\zeta}(\sigma)\right\}
\end{equation*}
to show that
\begin{equation*}
\log\frac{\zeta_{K}(1-p)}{\zeta_{K}(1 + \eta)} = \int_{1-p}^{1 + \eta} -\frac{\zeta_{K}'}{\zeta_{K}}(\sigma)\, d\sigma \leq n_{K}\int_{1-p}^{1 + \eta} -\frac{\zeta'}{\zeta}(\sigma)\, d\sigma \leq n_{K} \log\frac{\zeta(1-p)}{\zeta(1 + \eta)}.
\end{equation*}
Finally the term $-\log|f(1 + \eta)|$ is estimated as in the Dirichlet $L$-function case --- cf.\ (\ref{Rost}). This shows that
\begin{equation*}\label{RosDed}
\log|f(1+ \eta)| \geq N\log\frac{\zeta_{K}(2 + 2\eta)}{\zeta_{K}(1 + \eta)} + \frac{N}{2}\log(\eta^{2} + T^{2}) + o(1).
\end{equation*}
This, together with (\ref{NDed}), (\ref{JDedFin}), (\ref{wdert}), (\ref{wder2t}) and (\ref{FinJ2}) and sending $N\rightarrow\infty,$ shows that, when $T\geq T_{0}$,
\begin{equation}\label{FinalD}
\begin{split}
\bigg|N_{K}(T) - \frac{T}{\pi}\log\left\{ d_{K} \left( \frac{T}{2\pi e}\right)^{n_{K}}\right\}\bigg| &\leq \frac{r(\frac{1}{2} + \eta)(\frac{1}{2} - p)}{\pi\log r(1 + \eta - p)}\left\{ \log d_{K} + n_{K}\log T\right\}\\
& + \left(C_{2} -\frac{2}{\pi}\left[g(1, T) - |g(0, T)|\right]\right)n_{K} + D_{3},
\end{split}
\end{equation}
where $C_{2}$ is given in (\ref{C2f1}) and
\begin{equation}\label{FinalD2}
\begin{split}
D_{3} &= 2 + \frac{r(\frac{1}{2} + \eta)}{\pi\log r (1 + \eta - p)} \log \left(\frac{1 -p}{1 +p}\right) + \frac{1}{\pi}F( \sqrt{2}(\tfrac{1}{2} + \eta), T_{0}) \\
&+ \frac{1}{2\pi \log r}\bigg( \int_{-\pi/2}^{0} \log \tilde{w}^{*}(T_{0}, \phi, \eta, r) \, d\phi + \int_{0}^{\pi/2} \log \tilde{w}(T_{0}, \phi, \eta, r) \, d\phi\\
&\qquad\qquad\qquad+ \int_{\pi/2}^{\pi} \log w(T_{0}, \phi, \eta, r) \, d\phi + \int_{\pi}^{3\pi/2} \log w^{*}(T_{0}, \phi, \eta, r) \, d\phi\bigg)
\end{split}
\end{equation}
If one chooses $p = -\eta/7$, to ensure that the lower order terms in (\ref{FinalD}) are smaller than those in \cite{NgKad2012}, one arrives at Theorem \ref{ZetaThm}. One may choose a smaller value of $p$ if one is less concerned about the term $D_{2}$.
\section*{Acknowledgements}
I should like to thank Professor Heath-Brown and Professors Ng and Kadiri for their advice. I should also like to thank Professor Giuseppe Molteni and the referee for some constructive remarks.
\bibliographystyle{plain}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Cynthia massachusettensis är en fjärilsart som beskrevs av Gunder 1925. Cynthia massachusettensis ingår i släktet Cynthia och familjen praktfjärilar. Inga underarter finns listade i Catalogue of Life.
Källor
Praktfjärilar
massachusettensis
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: Contravariance/Covariance, why can't cast this? Let's face it, I am still having some difficulties to understand the constrains when it's time to use covariance and contravariance in generics.
I wonder, why if I have this:
public interface IFasterListCov<out T>
{}
public interface IFasterListCon<in T>
{}
public class FasterList<T> : IList<T>, IFasterListCov<T>, IFasterListCon<T>
The third cast fails:
public void QueryNodes<T>() where T:INode
{
//somehow I can convert IFasterListCon<INode> to IFasterListCon<T>
IFasterListCon<INode> nodes = (IFasterListCon<INode>)_nodesDB[type];
//I guess this works because _nodesDB[type] is indeed a FasterList<T> object
//note: I am wrong, I can cast whatever INode implementation, not just T, which made me very confused :P
IFasterListCon<T> nodesT = (IFasterListCon<T>)nodes;
//I can't cast IFasterListCon<T> back to FasterList<T>
FasterList<T> nodeI = nodesT as FasterList<T>; //null
}
Dictionary<Type, IFasterListCov<INode>> _nodesDB;
to be clear _nodesDB[type] is a FasterList<T> declared through IFasterListCov<INode>
A: In the scenario where you're calling QueryNodes<MyNode>, in order for your last cast to get a non-null value, the actual instance that you get with _nodesDB[type] must be a FasterList<MyNode>. It's not good enough for it to be FasterList<SomeOtherMostlyCompatibleNode>.
The runtime is very strict about types, it keeps track of the actual runtime types of everything involved, it's not good enough for the data types to be similar, or for you to only have MyNode objects populating your FasterList<SomeOtherMostlyCompatibleNode>, or anything else. If the types are not exactly what they should be, you need to do some sort of programmatic conversion, not just cast.
A: MyType : IMyType does not make Generic<IMyType> and Generic<MyType> related in any way.
In your particular case it is likely that nodesT is FasterList<Node> which is not FasterList<INode>.
Note that this conversion work for interface which support variance (co/contra) when you can specify in/out as you see in successful conversion to interface. See one of many questions for details - i.e. Generic Class Covariance.
There is also excellent answer about List co-variance - C# variance problem: Assigning List<Derived> as List<Base> which shows that List<Derived> and List<Base> can't be cast between each other:
List<Giraffes> giraffes = new List<Giraffes>();
List<Animals> animals = new List<Animals>() {new Lion()};
(giraffes as List<Animals>).Add(new Lion()); // What? Lion added to Girafes
Giraffe g = (animals as List<Giraffes>)[0] ; // What? Lion here?
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: Generating Slidify slides with knit to HTML on RStudio Version 0.99.467 After writing up a Slidify content when pressing knit HTML button on RStudio IDE version 0.99.467 it only renders HTML file and NOT the slides? I have used default YAML and a YAML used by Ramnath from his github repository examples with the same result. Is there a change or what am I missing?
A: Ok, this steps fixed my issue.
Restart R (or alternatively restart RStudio).
install.packages('stringr')
devtools::install_github('muschellij2/slidify')
library(slidify)
slidify("index.Rmd")
full discussion is found here
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"redpajama_set_name": "RedPajamaStackExchange"
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Antoni Roca i Rosell (Barcelona, 1951) és professor d'història de la ciència i de la tècnica a la Universitat Politècnica de Catalunya i membre del Centre de Recerca per a la Història de la Tècnica (ETSEIB). Des de 2005 és el coordinador de la Càtedra UNESCO de Tècnica i Cultura de la mateixa Universitat. Des del 1993 fins al 2009 va ser president de la Societat Catalana d'Història de la Ciència i de la Tècnica, filial de l'Institut d'Estudis Catalans.
S'interessa en el procés de difusió i d'assimilació de les idees científiques, principalment la física, i de l'enginyeria i les tecnologies a Catalunya i a Espanya, en el context de l'anomenada civilització europea, temàtiques sobre les quals ha publicat un centenar d'articles i capítols de llibre així com una quinzena de llibres com a autor o coordinador.
Referències
Enllaços externs
Professors de la Universitat Politècnica de Catalunya
Científics catalans del sud contemporanis
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Q: Using log scale on vertical axis with values less than 1 I have some data in a Google spreadsheet that need to be plotted. It needs to be plotted in log scale, however, all of it is less than 1. When I select "Log scale" on the vertical axis, all the columns start at 1 and go downwards, like this:
This doesn't look very good, since I want to compare the three values and show that "test3" is much higher than the other two, but the column for "test3" is smaller, so visually it seems that "test3" is the smallest value (at first glance, anyway). Is it possible to have them the opposite way, that is, start at the bottom of the chart (at whatever minimum value is set for the vertical axis), and go up to the given value?
A: Answer:
Unfortunately, at present this isn't possible to do.
More Information
I did a bit of digging for this and I found that there's a workaround described on Web Apps Stack Exchange, but I couldn't get this to work for log graphs - the workaround assumes that the values used are greater than 1 and there's no concern with using 0 as a base point for an axis.
In this case, the only thing you will be able to do is to download the chart as an image and manually edit it in a image editing software to get the axes to flip.
Feature Request:
You can however let Google know that this is a feature that is important for access to their APIs, and that you would like to request they implement it.
Google's Issue Tracker is a place for developers to report issues and make feature requests for their development services, I'd urge you to make a feature request there. The best component to file this under would be the Sheets Editor component, as it would be a new Feature. Alternatively, you can suggest this by following the Help > Report a problem menu item from the Sheets UI.
There is a glimmer of hope for this! A feature request was made already for this, but ratrher than for Charts and Sheets, for use within Data Studio. This feature request you can see here, to which you can click the star (☆) in the top left to let Google know more people want this kind of feature to be implemented.
References:
*
*How to reverse the y-axis in a Google Sheets graph? - Web Applications Stack Exchange
*Reverse only the left Y axis or Right Y Axis - Issue Tracker
*Sheets Editor Feature Request Form - Google Issue Tracker
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\section{1. Introduction}
\begin{table}[!b]
\caption{Networks Terminology and Notation}
\centering
\small
\begin{tabular}{| l | l | l |}
\hline
Term & Notation & Description \\
\hline
\hline
adjacency matrix & $A$ & A square matrix whose elements $A_{ij}$ have a value\\
& & different from 0 if there is an edge from some node $i$\\
& & to some node $j$. $A_{ij} = 1$ if the link is a simple \\
& & connection (unweighted graph). $A_{ij} = w_{ij}$ when the \\
& & link is assigned some kind of weight (weighted graphs).\\
& & If the graph is undirected (links connect nodes\\
& & symmetrically), $A$ is symmetric.\\
& &\\
degree & $k_i$& The number of nodes a node $i$ is connected to\\
& &\\
in-degree & $k_i^{\text{in}}$ &In a directed network, the number of incoming edges to\\
& & a node $i$\\
& &\\
out-degree & $k_i^{\text{out}}$ & In a directed network, the number of outgoing edges \\
& & emanating from a node $i$ \\
& &\\
weight & $w_{ij}$ & In a weighted network, weight assigned to an edge from\\
& & some node $i$ to some node $j$\\
& &\\
strength & $s_i = \sum^{k_i}_{j = 1} w_{ij}$ & The sum of weights attached to ties belonging to some\\
& & node $i$ \\
& &\\
Erd\H{o}s-R\'{e}nyi & $G(n, p)$ & A random graph of $n$ nodes and edges generated by \\
random graph& & connecting a pair of nodes with some probability $p$ \\
& & independently of all other edges\\
& &\\
Call Detail Records &CDRs & Digital records of the attributes of a certain instance of a \\
& & telecommunication transaction (such as the start time or \\
& & duration of a call), but not the content.\\
\hline
\end{tabular}
\end{table}
Humans interact with each other both online and in-person, forming and dissolving social ties throughout our lives. The flexible architecture of networks or graphs make them a useful paradigm for modeling these complex relationships at the individual, group, and population levels. Social network nodes typically represent individuals, and edges the connections between individuals, such as friendships, sexual contacts, or cell phone calls. Social networks have been shown to have a direct impact on public health \cite{Chris,Chris2,Fowl,Fowl2,Good}. For example, a recent study examined the social networks of households in Malegaon, India, finding that households that refuse to have their children vaccinated against polio have a disproportionate number of social ties to other vaccine-reluctant and vaccine-refusing households \cite{Polio}. Several studies have now successfully modeled the spread of epidemics through various populations, finding that different network structures have an effect on the potential efficacy of an intervention \cite{Banerjee, Valente, VanderWeele}. Studies have also leveraged network properties to target highly connected individuals in public health interventions \cite{Kim2}. The structure of connections in contact networks have also been shown to affect statistical power in cluster randomized trials \cite{Banerjee, Staples}. Additionally, new classes of connectivity-informed study designs for cluster randomized trials have been proposed recently, and these designs appear to simultaneously improve public health impact and detect intervention effects \cite{Banerjee, Kim, Harling}.
There is also accumulating evidence that the habits of our friends influence our own behavior, such as the uptake of smoking or lifestyle choices that can lead to obesity \cite{Chris, Chris2, Fowl, Fowl2}. Moreover, electronic billing records have been used to study patient-physician interaction networks to learn about structural properties of these networks and how these properties are associated with the quality and cost of health care \cite{Landon, Kim, Sima}.
Network structure can be studied at different scales ranging from local to global. Microscopic (local) structures include one or a few nodes, macroscopic (global) structures involve most to all nodes, and mesoscopic structures lie between the microscopic and macroscopic scales. It has been shown that the different structures are not independent \cite{Fort}. Specifically, several microscopic mechanisms are known to give rise to microsopic, mesoscopic, and macroscopic structure \cite{Bianconi, Fort, Kumpula}. For example, triadic closure, the process of getting to know a friend of a friend, can generate network communities \cite{Fort, Kumpula, Porter}. The term community here refers to a group of nodes that are densely connected to one another but only sparsely connected to the rest of the network. Community structure is of particular interest because most social networks have meaningful community structure that is related to their function. Communities also arise from humans forming tightly-knit groups through shared interests and similar characteristics, and they play an important role in the spread of disease and information \cite{Chris, Chris2, Fort}.
Social network data has traditionally been collected from surveys, mostly capturing small, static network snapshots at one point in time \cite{Faust}.
Dozens of different metrics have been created to quantify and study the structure of these simple networks. However, with the recent availability of increasingly rich, complex network data, limitations of these metrics have become increasingly clear. For example, betweenness centrality, the number or proportion of all pairwise shortest paths in a network that pass through a specified node, is used quite broadly but becomes much more computationally demanding as the size of the network increases and, even more importantly, it is unclear how meaningful this metric is in very large social networks. Another example of a widely used metric is the clustering coefficient, which is defined as the fraction of paths of length two in the network that are closed, i.e., groups of three nodes where ``the friend of my friend is also my friend'' \cite{Watts}.
The clustering coefficient has subsequently been extend to weighted and directed networks \cite{Saramaki, Tore}.
For the classic Erd\H{o}s-R\'{e}nyi random graph, the local clustering coefficient (the average clustering coefficient taken across all nodes in the network) asymptotically tends to $p$ where $p$ is the probability of forming a tie between any two nodes in the network \cite{Lecture}.
Most social networks are more clustered than corresponding random networks \cite{Newman3, Newman}. This observation is expected since people are more likely to become friends with others whom they meet through their current friends. While an expression has been derived for the mean of the local clustering coefficient, an expression for the variance has not been presented. Thus, classification of a given value for clustering as either high or low, and whether that value is statistically significant, is not currently possible and its value cannot be compared across networks.
The rest of this paper is organized as follows. In Section 2, we introduce the microscopic metric known as edge overlap and define extensions of edge overlap for weighted and directed networks. We then present two closed-form expressions for the mean and variance of each version of edge overlap for the Erd\H{o}s-R\'{e}nyi random graph and its weighted and directed counterparts. We then demonstrate the accuracy of our mean and variance approximations through simulation. Finally, we apply our results to empirical social network data and quantify the difference in the observed average overlap to the value expected for a corresponding random graph. We present the results of our data analysis in Section 3 and discuss our conclusions in Section 4. Supplementary material is contained in Appendices A, B, C and D.
\section{2. Methods}
\subsection{2.1 Overlap Extensions}
A central microscopic metric, which captures the effect of triadic closure and is related to the clustering coefficient, is edge overlap, the proportion of common friends two connected individuals share. In mathematical terms, the overlap between two connected individuals $i$ and $j$ is defined as
\begin{equation}\label{eq:1}
o_{ij} = \frac{n_{ij}}{(k_i -1) + (k_j -1) - n_{ij}}
\end{equation}
where $n_{ij}$ is the number of common neighbors of nodes $i$ and $j$, and $k_i$ $(k_j)$ denotes the degree, or number of connections, node $i$ $(j)$ has. Note that the tie between nodes $i$ and $j$ is not included in the calculation; overlap for the edge $(i, j)$ is defined only where $A_{ij} = 1$ and $k_1 + k_j >2$. Currently, edge overlap is only defined for simple networks in which edges are both unweighted and undirected \cite{Onnela}. Moreover, expressions for the mean and variance of edge overlap do not yet exist, making it hard to carry out statistical comparisons of this metric across networks, in particular networks of different sizes.
In a weighted network, each edge has a weight assigned to it. We define weighted overlap in Eq. \eqref{eq:2} as the proportion of total weight associated with ties to common friends nodes $i$ and $j$ share, and denote it $o^W_{ij}$:
\begin{eqnarray}\label{eq:2}
o^W_{ij} = \frac{\sum^{n_{ij}}_{k=1}(w_{ik} + w_{jk})}{s_i + s_j - 2w_{ij}}.
\end{eqnarray}
Here, $n_{ij}$ is the number of common neighbors of nodes $i$ and $j$, $w_{ij}$ denotes the weight associated with the tie between nodes $i$ and $j$, and $s_i$ $(s_j)$ denotes the strength of node $i$ $(j)$. According to the definition, the common friends of two connected individuals are first identified, the weights associated with these edges are summed together, and this sum is then divided by the combined strengths of the two nodes excluding the tie that connects them. The last step is intended to ensure consistency with the original version of edge overlap, i.e., the weight of the tie between the two individuals being considered is not included in the calculation of $o^W_{ij} $. Also, the metric is only defined for $w_{ij} > 0$ and for $s_i + s_j > 2w_{ij}$.
In a directed network, each edge has a direction associated with it. Thus, ties between nodes can be reciprocated, meaning that there can be an edge pointing from node $i$ to $j$ and another edge pointing from $j$ to $i$. For directed networks, the concept of a `common friend' of two individuals is ambiguous due to the directionality associated with the ties. We define a common friend as a node that creates a directed path of length two between the two nodes either from $i$ to $j$, $j$ to $i$, or both. Defining a common friend in this manner allows information to flow between $i$ and $j$ via a neighbor of both $i$ and $j$. To illustrate this, let $i$ and $j$ be the two connected individuals of interest, and $k$ a potential common friend. If there is a directed edge from $i$ to $k$ and a directed edge from $k$ to $j$, then there is a path a length two from $i$ to $j$ through $k$, and $k$ is considered a common friend. Using this criterion, we define directed overlap in Eq. \eqref{eq:3} as the proportion of paths of length two between two connected individuals, and denote it $o^D_{ij}$:
\begin{eqnarray}\label{eq:3}
o^D_{ij} = \frac{ \sum^{n}_{k = 1} (A_{ik}A_{kj} + A_{jk}A_{ki}) }{\text{min}(k_i^{\text{in}}, k_j^{\text{out}}) + \text{min}(k_j^{\text{in}}, k_i^{\text{out}} ) - 1}.
\end{eqnarray}
Here, $A_{ij}$ is the $(i,j)$ element of the directed adjacency matrix, $k_i^{\text{in}}$ $(k_j^{\text{in}})$ denotes the in-degree of node $i$ $(j)$, $k_i^{\text{out}}$ $(k_j^{\text{out}})$ denotes the out-degree of node $i$ $(j)$, and min$(\cdot,\cdot)$ the minimum of the two arguments. We consider each edge separately, even in the case of unreciprocated edges, and again, the tie between nodes $i$ and $j$ is not included in the calculation. The metric is only defined if $\text{min}(k_i^{\text{in}}, k_j^{\text{out}}) + \text{min}(k_j^{\text{in}}, k_i^{\text{out}}) > 1$.
\begin{figure}[t]
\vspace{-25pt}
\centering
\includegraphics[scale = 0.6]{networks_diagram.pdf}
\vspace{-75pt}
\captionsetup{width=\textwidth}
\caption{Schematics of edge overlap for (a) an unweighted network, (b) weighted network, and (c) directed network. Nodes are labeled with letters and weights are labeled with numbers.}
\end{figure}
\subsection{2.2 Erd\H{o}s-R\'{e}nyi Random Graph Models}
With the extensions of edge overlap defined above, one can easily compute the mean overlap (simple or weighted or directed) across all edges in the network. However, in order to make meaningful comparisons, such as to learn whether the observed value is small or large for the given network, or whether it represents a statistically significant deviation from what might be expected to occur at random, one needs to consider suitable null models and derive both the expected value and the variance of overlap under these null models. The Erd\H{o}s-R\'{e}nyi random graph model, often denoted $G(n,p)$, is the simplest model for generating random graphs \cite{Erdos}. In this model, graphs are created by considering ${n \choose 2}$ distinct pairs of $n$ nodes and connecting each pair with probability $p$ independently of all other dyads (node pairs). The random process can therefore be thought of as a series of Bernoulli trials or coin flips. Suppose we have a coin that lands on heads with probability $p$. If the coin flip results in heads, the two nodes are connected, otherwise, they are not. Note that here the number of edges is not fixed, but rather the probability of creating an edge.
\par The weighted random graph (WRG) is the weighted counterpart of the canonical Erd\H{o}s-R\'{e}nyi random graph \cite{Diego}. In this case, a network of $n$ nodes is generated by selecting each pair of nodes and carrying out a series of independent Bernoulli trials for each pair with success probability $p$. This process is continued until the first failure is encountered, and every success preceding the failure adds a unit weight to the tie. Note that if the first Bernoulli trial is a failure, the two nodes will not be connected. We can again relate this to the tossing of a coin. If the coin lands on heads with probability $p$, the weight associated with an edge is given by the number of heads flipped until the first tails appears, and therefore tie weights are distributed according to the geometric distribution. This process is repeated for every distinct pair of nodes in the network.
\par The directed random graph is the directed version of the Erd\H{o}s-R\'{e}nyi random graph, and it is generated in a very similar manner as its canonical counterpart. For two nodes $i$ and $j$, in a network of $n$ nodes, an edge pointing from $i$ to $j$ is created with probability $p$ and, likewise, an edge pointing from $j$ to $i$ is also connected independently with probability $p$ \cite{Erdos, Erdos2, Ballobas}. In this case, in the coin analog of the model, we flip a coin twice for each pair of nodes, one flip for each direction. This process is repeated for every pair of nodes in the network.
\subsection{2.3 Erd\H{o}s-R\'{e}nyi Overlap}
In order to perform inference about overlap, i.e., to compare point estimates of overlap across networks, we need to know the mean and variance of each version of overlap under the null model in question. To fix our notation, we will let uppercase letters stand for random variables: $K_i$ denotes the degree of node $i$, $N_{ij}$ the number of common neighbors of nodes $i$ and $j$, $S_i$ the strength of node $i$, $W_{ij}$ the weight of the edge connecting nodes $i$ and $j$, $K^{in}_i$ the in-degree of node $i$, $K^{out}_i$ the out-degree of node $i$, and $A_{ij}$ the adjacency matrix element, where a nonzero (positive) value represents the existence of an edge between nodes $i$ and $j$ (binary in the case of unweighted graphs).
For the Erd\H{o}s-R\'{e}nyi random graph, a given node is connected to each of the remaining $n-1$ nodes with probability $p$, and its resulting degree can thus be viewed as a sum of independent Bernoulli trials. Therefore, as is well known, $K_i \sim$ binomial$(n-1, p)$, which can be approximated by a Poisson$(np)$ distribution for large $n$. For any pair of (connected) nodes, the probability of both nodes being connected to the same neighboring node, meaning that they have a common neighbor, is $p^2$ as each edge occurs independently of any others. Moreover, the total number of possible common friends two nodes can have is $n - 2$. Thus, $N_{ij} \sim$ binomial$(n-2, p^2)$, which can similarly be approximated by a Poisson$(np^2)$ random variable for large $n$. With these definitions, the numerator of edge overlap is a Poisson random variable, and the denominator is the difference of two Poisson random variables, known as a Skellam random variable \cite{Skellam}. In this case, the denominator is a Skellam$(2np - 2 - np^2)$ random variable. We can now view overlap as a random variable as in Eqn. \eqref{eq:unweightedrv}.
\begin{equation}\label{eq:unweightedrv}
O_{ij} = \frac{N_{ij}}{(K_i -1) + (K_j -1) - N_{ij}}
\end{equation}
Edge overlap is a ratio of two dependent random variables since the maximum number of possible common friends is bounded by the min$(K_i, K_j)$. This dependency increases the difficulty of deriving exact expressions for the mean and variance of overlap. However, despite this dependence, we can approximate both the mean and variance in two different ways. The first approach observes the weakness of the dependence between the numerator and denominator and simply ignores it, defining the ratio as a function of independent random variables. Approximations for the mean and variance of the ratio are then derived using Taylor expansions of the function about the means of the random variables \cite{Kendall, Johnson}. This results in Eqs. \eqref{eq:unmean} and \eqref{eq:unvar} (for details, see Appendix A.1.).
\begin{eqnarray}
\mathbb{E}[O_{ij}] &=& \frac{p}{2-p}\label{eq:unmean} \\[15pt]
\text{Var}(O_{ij}) &=& \frac{np^2}{(2np - 2 - np^2)^2} + \frac{n^2p^4(2np - 2 + np^2)}{(2np - 2 - np^2)^4}.\label{eq:unvar}
\end{eqnarray}
Our second approach incorporates results from \cite{Oxford}, where the local clustering coefficient for an Erd\H{o}s-R\'{e}nyi random graph is also written as a ratio of dependent random variables with the intention of deriving its distribution. The dependency is eliminated by replacing the random variable in the denominator with its expectation, and this approximation turns the denominator into a constant. Thus, the distribution of the clustering coefficient is approximated by a scaled version of the random variable in the numerator. It is subsequently shown that this is a good approximation for the actual distribution. We adopt the same approach here, and approximate the distribution of edge overlap by replacing the denominator with its expectation. We then derive the mean and variance of $O_{ij}$ using the distributional properties of the numerator. This results in the expressions in Eqs. \eqref{eq:unmean2} and \eqref{eq:unvar2} (for details, see Appendix B.1.):
\begin{eqnarray}
\mathbb{E}[O_{ij}] &=& \frac{p}{2-p}\label{eq:unmean2} \\[15pt]
\text{Var}(O_{ij}) &=& \frac{np^2}{(2np - 2 - np^2)^2}.\label{eq:unvar2}
\end{eqnarray}
Note that the expressions for the mean, Eqs. \eqref{eq:unmean} and \eqref{eq:unmean2}, are equivalent, but the expressions for the variance, Eqs. \eqref{eq:unvar} and \eqref{eq:unvar2} differ, with the expression for Eq. \eqref{eq:unvar2} corresponding to the first term of Eq. \eqref{eq:unvar}.
We use the same two approaches for the weighted and directed cases. For the weighted Erd\H{o}s-R\'{e}nyi random graph (WRG), we first define the distributions of $W_{ij}$ and $S_i$. Given how WRGs are constructed (as given above), the tie weights follow a geometric distribution, such that if an edge is placed between a pair of nodes with probability $p$, tie weight distribution will be $W_{ij} \sim$ geometric$(1-p)$. It then follows that node strength $S_i$ is a sum of geometric random variables, i.e., is the sum of the weights of the ties that are adjacent to the given node, leading to $S_i \sim$ negative binomial$(n-1, 1-p)$ \cite{Diego}.
For the first approach, the numerator can be written as $\sum^{N_{ij}}_{k=0} (W_{ik} + W_{jk})$, where $N_{ij}$ is again the number of common neighbors of nodes $i$ and $j$, and is distributed as in the unweighted Erd\H{o}s-R\'{e}nyi random graph. Thus, the numerator is a sum of geometric random variables, where the number of summed variables is itself a random variable. Moreover, we must have $W_{ik} > 0$ and $W_{jk} >0 $ since a common neighbor of two nodes can only exist if both nodes are attached to the node in question (the common neighbor). To address this constraint, each of the random variables must first be transformed into zero-truncated geometric random variables, and their mean and variance altered correspondingly. We can now write weighted overlap as a random variable as in Eqn. \eqref{eq:weightedrv}.
\begin{eqnarray}\label{eq:weightedrv}
O^W_{ij} = \frac{\sum^{N_{ij}}_{k=1}(W_{ik} + W_{jk})}{S_i + S_j - 2W_{ij}}.
\end{eqnarray}
Now hierarchical models can be used to find the mean and variance of the numerator, and these results combined with the mean and variance values of the denominator can be used to derive the expressions in Eqs. \eqref{eq:wmean} and \eqref{eq:wvar} (for details, see Appendix A.2.):
\begin{eqnarray}
\mathbb{E}[O^W_{ij}] &=& p \label{eq:wmean}\\[15pt]
\text{Var}(O^W_{ij}) &=& \frac{p+1}{n}.\label{eq:wvar}
\end{eqnarray}
The second approach again replaces the denominator with its expectation. The mean and variance derivations are then straightforward and result in the expressions in Eqs. \eqref{eq:wmean2} and \eqref{eq:wvar2}. Again, the expressions for the mean are equivalent for both approaches, and the variance expressions are quite similar (for details, see Appendix B.2.):
\begin{eqnarray}
\mathbb{E}[O^W_{ij}] &=& p \label{eq:wmean2}\\[15pt]
\text{Var}(O^W_{ij}) &=& \frac{np^2(p+2)}{2(np-1)^2}.\label{eq:wvar2}
\end{eqnarray}
The derivations for the directed Erd\H{o}s-R\'{e}nyi random graph are more complicated and do not have a closed form due to the minimum expressions in the denominator. Focusing on the numerator, each of the $A_{ik}A_{kj}$ and $A_{jk}A_{ki}$ terms is equal to one if and only if both adjacency matrix values are equal to 1, which happens with probability $p^2$ since each edge is independent. Thus, each of the terms is a Bernoulli$(p^2)$ random variable, and the numerator consists of a sum of 2$n$ independent Bernoulli random variables, meaning it is a binomial$(2n, p^2)$ random variable, which we will again approximate with a Poisson$(2np^2)$ random variable. The denominator includes the minimum of two identically distributed random variables $K^{in}_i$ and $K^{out}_i$. Due to the definition given above, the in and out degrees of nodes $i$ and $j$ cannot equal 0, making them zero-truncated binomial$(n-1, p)$ random variables, which will also be approximated as zero-truncated Poisson$(np)$ random variables since $n$ is assumed to be large. We can now write directed overlap as a random variable as in Eqn. \eqref{eq:directedrv}.
\begin{eqnarray}\label{eq:directedrv}
O^D_{ij} = \frac{ \sum^{n}_{k = 1} (A_{ik}A_{kj} + A_{jk}A_{ki}) }{\text{min}(K_i^{\text{in}}, K_j^{\text{out}}) + \text{min}(K_j^{\text{in}}, K_i^{\text{out}} ) - 1}.
\end{eqnarray}
The mean and variance of the denominator can now be calculated and used to derive the expressions in Eqs. \eqref{eq:dmean} and \eqref{eq:dvar} \cite{Kendall} (for details, see Appendix A.3.):
\begin{equation}
\mathbb{E}[O^D_{ij}] = \frac{np^2}{e^{-2np}\sum^{(n-1)}_{k=1}\left[\sum^{(n-1)}_{j=k} \frac{(np)^j}{j!} \right]^2 - 0.5} \label{eq:dmean}\\[15pt]
\end{equation}
\begin{equation}
\text{Var}(O^D_{ij}) = \frac{2n^2p^4}{(2e^{-2np}\sum^{(n-1)}_{k=1}\left[\sum^{(n-1)}_{j=k} \frac{(np)^j}{j!} \right]^2 - 1)^2} + \frac{\frac{32n^3p^5e^{np}}{e^{np} - 1} \left[ 1 - \frac{np}{e^{np}-1}\right]}{(2e^{-2np}\sum^{(n-1)}_{k=1}\left[\sum_{j=k}^{(n-1)} \frac{(np)^j}{j!} \right]^2 - 1)^2}.\label{eq:dvar}
\end{equation}
The second approach again replaces the denominator with its expectation, and the mean and variance derivations result in the expressions in Eqs. \eqref{eq:dmean2} and \eqref{eq:dvar2} (for details, see Appendix B.3.). Again, the expressions for the mean are equivalent for both approaches, but note that the expression for the variance using the second approach in Eq. \eqref{eq:dvar2} is equivalent to the first term of the variance resulting from the first approach in Eq. \eqref{eq:dvar}.
\begin{eqnarray}
\mathbb{E}[O^D_{ij}] &=& \frac{np^2}{e^{-2np}\sum^{(n-1)}_{k=1}\left[\sum^{(n-1)}_{j=k} \frac{(np)^j}{j!} \right]^2 - 0.5} \label{eq:dmean2}\\[15pt]
\text{Var}(O^D_{ij}) &=& \frac{2n^2p^4}{(2e^{-2np}\sum^{(n-1)}_{k=1}\left[\sum^{(n-1)}_{j=k} \frac{(np)^j}{j!} \right]^2 - 1)^2} \label{eq:dvar2}
\end{eqnarray}
\subsection{2.4 Simulation Studies}
We conducted simulation studies to evaluate the accuracy of the proposed mean and variance expressions for each version of Erd\H{o}s-R\'{e}nyi edge overlap. We simulated 5,000 realizations of networks with $n =1,000$ nodes for various values of $p \in (0,1)$. The mean and variance of edge overlap was calculated for each network realization, and those values subsequently averaged over all simulations. We considered values of $p > 1/n$, such that the resulting average degree $np > 1$, which ensures (asymptotically) that the graphs have non-vanishing largest connected components.
Figure \ref{fig:sims} displays the simulation results and accuracy of our approximations. The top row contains the results for the mean unweighted overlap (Figure \ref{fig:sims}a), mean weighted overlap (Figure \ref{fig:sims}b) and mean directed overlap (Figure \ref{fig:sims}c). In each plot, the red dots represent the simulated results, black lines represent the theoretical values using the first approach and blue lines the second approach. Note that each expression for average overlap is equivalent for the two approaches, making only the black lines visible. The bottom row of panels shows the results for the variance of unweighted overlap (Figure \ref{fig:sims}d), weighted overlap (Figure \ref{fig:sims}e) and directed overlap (Figure \ref{fig:sims}f). In each plot, black lines represent the theoretical values using the first approach, blue lines the second approach, and the red dots the simulated values.
For each version of overlap, our theoretical approximations of the mean closely match the simulations, with the unweighted case being the best fit for all values of $np$. The approximations of the variance overall are not as accurate, where the accuracy of the fit depends on the value of $np$. In the unweighted case (Figure \ref{fig:sims}d), both theoretical approaches match the simulated values for average degree $np \geq 10$ until about $np = 100$. The first approximation then deviates from the simulated values, followed by the second approach deviating from them when $np \approx 300$. In the weighted case (Figure \ref{fig:sims}e) the first approximation is more accurate than the second for average degree less than or equal to about 30. The approaches are then equally precise until the average degree is approximately 170; after this point, the second approximation is closer to the simulated values. In the directed case (Figure \ref{fig:sims}f) the two approximations are equivalent and closely match the simulated values until the average degree reaches about 10. After that point, approach two is more accurate. Furthermore, in all cases, both approximations systematically overestimate variability. We stress that this overestimation leads to inflated standard errors and thus to conservative hypothesis tests, which is preferable over the opposite situation, i.e., having deflated standard errors and anti-conservative tests.
\subsection{2.5 Data Analysis}
As an application of our derivations to analysis of empirical social networks, we used social network data collected in 2006 from 75 villages housed in 5 districts in rural southern Karnataka, India, all within 3 hours driving distance from Bangalore (Figure \ref{fig:india}) \cite{Banerjee}. The data were collected as part of a study that examined how participation in a microfinance program diffuses through social networks. First, a baseline survey was conducted in all 75 villages. The survey consisted of a village questionnaire, a full census that collected data on all households in the villages, and a detailed follow-up survey fielded to a subsample of individuals. The village questionnaire collected data on village leadership, the presence of pre-existing non-governmental organizations (NGOs) and savings self-help groups and various geographical features of the area. The household census gathered demographic information, GPS coordinates of each household and data on a variety of amenities for every household in each village (roof type, latrine type, and access to electric power). The individual surveys were administered to a random sample of villagers in each village and were stratified by religion and geographic sub-location. Over half of the households in each stratification were sampled, yielding a sample of about 46$\%$ of all households per village. The individual questionnaire asked for information including age, sub-caste, education, language, native home, and occupation of the person. Additionally, the survey included social network data along 12 dimensions: friends or relatives who visit the respondent's home, friends or relatives the respondent visits, any kin in the village, nonrelatives with whom the respondent socializes, those who the respondent receives medical advice from, who the respondent goes to pray with, from whom the respondent would borrow money, to whom the respondent would lend money, from whom the respondent would borrow or to whom the respondent would lend material goods,
\begin{figure}[!t]
\centering
\vspace{5pt}
\includegraphics[scale = 0.4]{sims-all.png}
\captionsetup{width=\textwidth}
\caption{Simulation results for the mean (top row) and variance (bottom row) of each type of Erd\H{o}s-R\'{e}nyi overlap. The first column corresponds to the unweighted Erd\H{o}s-R\'{e}nyi overlap, the second column to the weighted Erd\H{o}s-R\'{e}nyi overlap and the third to the directed Erd\H{o}s-R\'{e}nyi overlap case. The top row plots (a), (b) and (c) plot the average overlap on the $y$-axis and average degree ($np$) on the $x$-axis. The red dots represent values from the simulations, and the black line represents the theoretical outcome using approach 1 and the blue line represents the theoretical outcome using approach 2. Note that the blue lines are completly covered by the black lines since the values for average overlap are the same for both approaches. The bottom row plots (d), (e) and (f) plot the variance of edge overlap on the $y$-axis and average degree ($np$) on the $x$-axis. In each plot, the red dots represent values from the simulations, the black line represents the theoretical outcome using approach 1 and the blue line represents the theoretical outcome using approach 2.}
\label{fig:sims}
\end{figure}
\noindent from whom the respondent gets advice, and to whom the respondent gives advice.
\begin{figure}[!th]
\caption{A map of the districts of Karnataka, India. The five districts colored in green house all of the villages included in the data set. The districts included are Bangalore, Bangalore Rural, Kolar, Ramanagara and Chikballapura \cite{IndiaData, IndiaR}.}
\label{fig:india}
\centering
\vspace{5pt}
\includegraphics[scale = 0.2]{karn_map.png}
\captionsetup{width=\textwidth}
\end{figure}
The median pairwise distance between villages was 46km and the number of cross-village ties was minimal, allowing the villages to be regarded as independent networks. The villages were linguistically homogeneous but had variability in caste. Each village contained anywhere from 354 to 1775 residents, with a total population of 69,441 people in the 75 villages combined. The number of edges across all social networks totaled 2,361,745 which included 480 self-loops and 6,402 isolated dyads. The average degree was 6.79 (standard deviation of 4.03), and the average number of connected components was 17.99 per village. Among the respondents for whom covariate data was collected via the individual surveys, 55.4\% were female and 44.6\% were male. The mean age was 39 years with a range of 10 to 99 years. Four different castes were represented: scheduled caste, scheduled tribe, general caste, and OBC (``other backward castes''), with a majority of respondents members of the general and OBC castes ($\approx$ 69.5\%) \cite{Banerjee}.
\begin{table}[htb]
\centering
\vspace{5pt}
\begin{tabular}{|ccl|}
\hline
Label & \hspace{0.75cm} & Type of social interaction \\
\hline
\hline
1 && The respondent borrows money from this individual\\
2 && The respondent gives advice to this individual \\
3 && The respondent helps this individual make a decision \\
4 && The respondent borrows kerosene or rice from this individual\\
5 && The respondent lends kerosene or rice to this individual\\
6 && The respondent lends money to this individual\\
7 && The respondent obtains medical advice from this individual\\
8 && The respondent engages socially with this individual \\
9 && The respondent is related to this individual\\
10 && The respondent goes to temple with this individual\\
11 && The respondent has visited this individual's home\\
12 && The respondent has been invited to this individual's home\\
\hline
\end{tabular}
\caption{The types of social interactions recorded for individuals in each village. }
\label{table:relations}
\end{table}
We first calculated the average unweighted overlap for each type of social relationship (labeled 1-12, see Table \ref{table:relations}) for each village by treating all ties as undirected and by removing all self-loops since they do not contribute to edge overlap (Figure \ref{fig:full_raw}). Then we standardized each average overlap by subtracting the expected mean and dividing by the standard deviation under the null; the results from the unweighted Erd\H{o}s-R\'{e}nyi overlap derivations using the first approach discussed above (Figure \ref{fig:full_stand} in Appendix C). We stratified edges according to the availability of nodal attributes (since not all villagers completed an individual survey), sex, caste and age. Here we detail our results from stratifying by sex with Figures \ref{fig:sex_raw} and \ref{fig:sex_stand} showing raw and standardized overlap for female-female (F/F), male-male (M/M) and male-female (M/F) ties. For details and figures of stratification by attribute availability, age and caste, see Appendix C.
We next collapsed the twelve unweighted networks into one weighted network. Specifically, the weight of a tie between two individuals corresponds to the number of types of social relationships they are engaged in with each other. For example, if individual $i$ borrows money from, gives advice to and goes to temple with individual $j$, the weight of the (undirected) tie between $i$ and $j$ would be equal to 3. Similar to the unweighted networks, we stratified the weighted networks by nodal attributes, including the presence or absence of attribute information, sex, caste and age. Figure \ref{fig:sex_w} shows the distributions of raw and standardized weighted overlap for F/F, M/M and M/F ties. See Appendix C for figures stratified by attribute availability, caste and age.
\section{3. Results}
Here we detail our observations of the figures in the previous section where overlap is stratified by sex. For explanations about the figures detailing stratification by attribute information, caste and age, see Appendix D. In Figure \ref{fig:sex_raw}, the median average unweighted overlap is the largest for F/F ties, followed by M/F ties and then M/M ties. There is a clear separation in the values of average overlap between F/F and M/M ties with no overlap in values for interaction types 1, 2, 3, 4, 5, 6, 8, and 11. This suggests that women in these villages tend to form `cliques', tighter friendship circles where most individuals interact with each other more regularly and intensely than others in the same setting, much more than men for every type of social interaction. This kind of social development is quite common among females and has been studied in the social sciences \cite{Alison1, Alison2}. However, this trend could also be due to the significant difference in the average degree for males and females across the villages (Figure \ref{fig:sex_degrees} in Appendix C). The degrees of two attached nodes directly effects the value of overlap; it is easier for pairs of nodes with smaller degrees to have a higher value of overlap due to the smaller number of neighbors they need to have in common. The values of average overlap
\begin{figure}[!h]
\vspace{5pt}
\includegraphics[scale = 0.5]{overlap_gender_box_inclusive.pdf}
\captionsetup{width=\textwidth}
\caption{Distribution of average unweighted overlap for each village for each type of social interaction stratified by sex. A female individual is labeled with an `F' and a male individual is labeled with an `M'. We stratified the edges by sex, and labeled an edge between two female individuals as `F/F', an edge between two male individuals as `M/M', and an edge between a female individual and a male individual as `M/F'. The y-axis represents the proportion of average edge overlap and the x-axis represents the type of social interaction.}
\label{fig:sex_raw}
\end{figure}
\begin{figure}[!b]
\includegraphics[scale = 0.5]{overlap_gender_box_inclusive_stand.pdf}
\captionsetup{width=\textwidth}
\caption{Distribution of standardized unweighted overlap for each village for each type of social interaction stratified by sex. A female individual is labeled with an `F' and a male individual is labeled with an `M'. We stratified the edges by sex, and labeled an edge between two female individuals as `F/F', an edge between two male individuals as `M/M', and an edge between a female individual and a male individual as `M/F'. The y-axis represents the standardized value, also known as the Z-score, and the x-axis represents the type of social interaction.}
\label{fig:sex_stand}
\end{figure}
\noindent for the M/F ties are closer to the values for F/F ties than M/M ties and their distributions tend to have smaller variance compared to the other types of ties. This suggests that individuals who have mixed-sex social ties typically have more friends in common than individuals who are part of a M/M social tie. Interestingly, when the average overlap values are standardized, which effectively adjusts for differences in average degree, M/F and M/M ties have much more similar values and are still well below the F/F ties values. The only exceptions are for interaction types 9 and 10 where the F/F and M/M ties have comparable values. All values are significantly higher than expected under the null, which is not surprising.
\par Figure \ref{fig:sex_w} shows that when ties are aggregated across interaction types, the values of average weighted overlap for F/F and M/F ties are very similar. The distribution for F/F ties has larger values and more variation, but its median is almost equivalent to that of the M/F ties distribution. It can also be seen that the values for average weighted overlap are much smaller for M/M ties; in fact there is no overlap in values between the M/M ties and the F/F and M/F ties. This again points to females having the tendency to create social `cliques' more often than males. This trend is also seen when all values are standardized (Figure \ref{fig:sex_w}b). Again, all values are significantly higher than expected for each type of tie, as we would expect from Figure \ref{fig:sex_stand} above.
\begin{figure}[!t]
\begin{subfigure}{.45\textwidth}
\centering
\vspace{5pt}
\includegraphics[width=1.05\textwidth]{woverlap_gender_box_inclusive.pdf}
\caption*{(a)}
\label{fig:sex_raw_w}
\end{subfigure}
\qquad
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{woverlap_stand_gender_box_inclusive.pdf}
\caption*{(b)}
\label{fig:sex_stand_w}
\end{subfigure}
\caption{Distribution of average weighted overlap (a) and standardized weighted overlap (b) stratified by sex. A female individual is labeled with an `F' and a male individual is labeled with an `M'. We stratified the edges by sex, and labeled an edge between two female individuals as `F/F', an edge between two male individuals as `M/M', and an edge between a female individual and a male individual as `M/F'. The y-axis in (a) represents the proportion of average weighted edge overlap, and the y-axis in (b) represents the standardized value, also known as the Z-score.}
\label{fig:sex_w}
\end{figure}
\section{4. Conclusions and Discussion}
In this paper we introduced extensions of edge overlap for weighted and directed networks. We also used the classic Erd\H{o}s-R\'{e}nyi random graph and its weighted and directed counterparts to define a null model and derive approximations for the expected mean and variance of edge overlap for each type of graph. Edge overlap can be standardized using these approximations allowing its comparison across networks of different size. We used these approximations in a data analysis of the social networks of 75 villages in rural India. We found that overall, the average proportion of overlap was much higher than expected under the null for each type of social interaction, especially when the social activity was going to temple together.
We also found that there is a marked difference in the amount of overlap between female-female ties and male-male ties, with female-female ties consistently achieving much higher values of overlap. This could be a consequence of two types of mechanisms; the average degrees of males versus females and the tendency of women forming friendship `cliques' with other women much more frequently than men forming the same types of friendship circles with other men. We found that in this case, men have a significantly higher degree than women across all networks. Whichever mechanism is at work here, this structural information could lead to an alternative method of eliciting social network data to optimize diffusion or intervention strategies based on the type of tie.
While our work generalizes a central microscopic network metric, making it more broadly applicable, there are limitations to our work. The Erd\H{o}s-R\'{e}nyi random graph model is a simple and somewhat naive null model in the context of social networks. This model does not preserve the degree distribution and is relatively easy to reject. An alternative would be to derive these expressions for the configuration model, which does preserve the degree distribution. However, deriving the mean and variance under the configuration model null model would be considerably more difficult. Another limitation with our mean and variance approximations is the ignoring of the correlations that are present among the random variables in the overlap expressions. In each version of overlap, the number of common neighbors is constrained by the degree of the edge-sharing nodes, making the numerator dependent upon the denominator. While our approximations are quite precise for the majority of values of mean degree, they could be improved if the correlation were also included in the approximations.
\section{Acknowledgments}
We thank Banerjee et. al. for making the India data set publicly available.
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\bibliographystyle{abbrv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 416
|
\section{Introduction}
\label{sec:intro}
In this paper we introduce a new tool -- the Riemann-Hilbert method -- into the study of minimal surfaces in the real
Euclidean space $\mathbb{R}^n$, and null curves in the complex Euclidean space $\mathbb{C}^n$, for any $n\ge3$,
and we obtain several applications. A special case of this technique is already available for $n=3$ (cf.\ \cite{AF2}),
but the general case treated here, especially for $n>3$, is more subtle and requires a novel approach.
Our first aim is to delve into the analysis of global geometric properties of minimal surfaces in $\mathbb{R}^n$
bounded by Jordan curves. The classical Plateau problem amounts to finding a minimal surface spanning
a given contour. In 1931, Douglas \cite{Do} and Rad\'o \cite{Ra} independently solved this problem
for any Jordan curve in $\mathbb{R}^n$. A major topic in global theory is the study of geometry of {\em complete minimal surfaces},
that is, minimal surfaces which are complete in the intrinsic distance.
The analysis of the asymptotic behavior, the conformal structure, and the influence of topological embeddedness are
central questions in this field; see \cite{MP1,MP2} for recent surveys.
By the isoperimetric inequality, minimal surfaces in $\mathbb{R}^n$ spanning rectifiable Jordan curves are not complete.
Our first main result provides complete minimal surfaces with (nonrectifiable) Jordan boundaries
in $\mathbb{R}^n$ for $n\ge 3$ which are normalized by any given bordered Riemann surface.
\begin{theorem}\label{th:Jordan}
Let $M$ be a compact bordered Riemann surface.
Every conformal minimal immersion $F\colon M\to\mathbb{R}^n$ $(n\geq 3)$ of class $\mathscr{C}^1(M)$
can be approximated arbitrarily closely in the $\mathscr{C}^0(M)$ topology by a continuous map $\wt F\colon M\to\mathbb{R}^n$
such that $\wt F|_{M\setminus bM}\colon M\setminus bM\to\mathbb{R}^n$
is a conformal complete minimal immersion, $\wt F|_{bM}\colon bM\to\mathbb{R}^n$ is a topological embedding, and the flux
of $\wt F$ equals the one of $F$. In particular, $\wt F(bM)$ consists of finitely many Jordan curves.
If $n\ge 5$ there exist embeddings $\wt F \colon M\hra\mathbb{R}^n$ with these properties.
\end{theorem}
Theorem \ref{th:Jordan} shows that every finite collection of smooth Jordan curves in $\mathbb{R}^n$ spanning a connected minimal
surface can be approximated in the $\mathscr{C}^0$ topology by families of Jordan curves spanning {\em complete} connected minimal
surfaces; hence it can be viewed as an {\em approximate solution of the Plateau problem by complete minimal surfaces.}
Recall that a {\em compact bordered Riemann surface} is a compact connected surface $M$, endowed with a
complex (equivalently, a conformal) structure, whose boundary $bM\neq\emptyset$ consists of finitely many smooth Jordan curves.
The interior $\mathring M=M\setminus bM$ of such $M$ is an (open) {\em bordered Riemann surface}.
A {\em conformal minimal immersion} $F\colon M\to\mathbb{R}^n$ is an immersion
which is angle preserving and harmonic; every such map parametrizes a minimal surface in $\mathbb{R}^n$
(see Sec.\ \ref{sec:prelim}). The {\em flux} of $F$
is the group homomorphism $\mathrm{Flux} (F)\colon H_1(M,\mathbb{Z})\to \mathbb{R}^n$ given on any closed curve
$\gamma\subset M$ by $\mathrm{Flux}(F)(\gamma)=\int_\gamma \Im (\partial F)$,
where $\partial F$ is the complex differential of $F$ (see (\ref{eq:flux}) below)
and $\Im$ denotes the imaginary part.
Theorem \ref{th:Jordan} generalizes pioneering results of Mart\'{i}n and Nadirashvili \cite{MN} who dealt with
immersed minimal discs in $\mathbb{R}^3$. Their method relies on a refinement of Nadirashvili's construction of a complete
bounded minimal disc in $\mathbb{R}^3$ \cite{Na} and is based on a recursive application of classical Runge's approximation
theorem. By using the same technique, Alarc\'on \cite{A1} constructed compact complete minimal immersions in
$\mathbb{R}^3$ with arbitrary finite topology; i.e., continuous maps
$F\colon \overline \Omega\to\mathbb{R}^3$ such that $F|_\Omega\colon\Omega\to\mathbb{R}^3$ is a conformal complete minimal immersion,
where $\Omega$ is a relatively compact domain in an open Riemann surface. However, neither the conformal structure
of $\Omega$, nor the topology of its boundary, can be controlled by this method; in particular, it can not be ensured that
$F(\overline\Omega\setminus\Omega)\subset \mathbb{R}^3$ consists of Jordan curves.
Indeed, the use of Runge's theorem in Nadirashvili's technique does not enable one to control the placement in $\mathbb{R}^3$
of the entire surface, and hence one must cut away pieces of the surface in order to keep it suitably bounded; this surgery
causes the aforementioned problems. By a different technique, relying on Runge-Mergelyan type theorems
(cf.\ \cite{AL1,AL-No}), Alarc\'on and L\'opez obtained analogous results for nonorientable minimal surfaces in $\mathbb{R}^3$ \cite{AL-Ritore},
null holomorphic curves in $\mathbb{C}^3$, and complex curves in $\mathbb{C}^2$ \cite{AL-Israel}. (Recall that a null curve in $\mathbb{C}^n$, $n\geq 3$,
is a complex curve whose real and imaginary parts are minimal surfaces in $\mathbb{R}^n$.)
Their technique still does not suffice to control the conformal structure of the surface or the topology of its boundary.
By introducing the Riemann-Hilbert method into the picture, Alarc\'on and Forstneri\v c \cite{AF2}
recently constructed complete bounded minimal surfaces in $\mathbb{R}^3$, and null curves in $\mathbb{C}^3$, normalized by
any given bordered Riemann surface. The principal advantage of the Riemann-Hilbert method over Runge's theorem
in this problem is that it enables one to work on a fixed bordered Riemann surface, controlling its global
placement in $\mathbb{R}^n$ or $\mathbb{C}^n$ at all stages of the construction.
The main novelty of Theorem \ref{th:Jordan} is that {\em we prescribe both the complex structure}
(any bordered Riemann surface) {\em and the asymptotic behavior}
(bounded by Jordan curves) of complete bounded minimal surfaces; furthermore, we obtain
results in any dimension $n\ge 3$. This is achieved by developing the Riemann-Hilbert technique,
first used in \cite{AF2} for $n=3$, in any dimension $n\ge 3$, and by further improving its implementation
in the recursive process. Theorem \ref{th:Jordan} is new even in the case $n=3$.
Furthermore, for $n>3$ this seems to be the first known approximation result by complete bounded minimal surfaces,
even if one does not take care of the conformal structure of the surface and the asymptotic behavior of its boundary.
Previous results in this line are known for complex curves in $\mathbb{C}^n$, $n\geq 2$,
and null curves in $\mathbb{C}^n$, $n\geq 3$; cf.\ \cite{AF,AF2,AL-CY}.
The Riemann-Hilbert method developed in this paper also allows us to establish essentially optimal results
concerning {\em proper complete minimal surfaces in convex domains}; see Theorems \ref{th:proper} and \ref{th:topology} below.
These results, and the methods used in their proof, will hopefully provide a step towards the more ambitious goal
of finding optimal geometric conditions on a domain $D\subset\mathbb{R}^n$ for $n\ge 3$ which guarantee that $D$ admits plenty
of proper (possibly also complete) conformal minimal immersions from any given bordered Riemann surface.
An explicit motivation comes from the paper \cite{DF2007} on proper holomorphic images
of bordered Riemann surfaces in complex manifolds endowed with an exhaustion function whose Levi form
has at least two positive eigenvalues at every point.
We shall say that a domain $\mathscr{D} \subset\mathbb{R}^n$ is {\em smoothly bounded} if it is
bounded and its boundary $b\mathscr{D}=\overline \mathscr{D}\setminus \mathscr{D}$ is smooth (at least of class $\mathscr{C}^2$).
\begin{theorem}\label{th:proper}
Let $\mathscr{D}\subset\mathbb{R}^n$ $(n\ge 3)$ be a bounded strictly convex domain with $\mathscr{C}^2$ smooth boundary,
let $M$ be a compact bordered Riemann surface, and let $F\colon M\to \overline\mathscr{D}$ be a conformal minimal immersion
of class $\mathscr{C}^1(M)$. Then the following assertions hold:
\begin{enumerate}[\rm (a)]
\item If $F(M)\subset \mathscr{D}$ then $F$ can be approximated uniformly on compacts in $\mathring M = M\setminus bM$
by continuous maps $\wt F\colon M\to \overline \mathscr{D}$ such that
$\wt F|_{\mathring M}\colon \mathring M \to\mathscr{D}$ is a conformal complete proper minimal immersion with
$\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
\vspace{1mm}
\item If $F(bM)\subset b\mathscr{D}$ then $F$ can be approximated in the $\mathscr{C}^0(M)$ topology
by continuous maps $\wt F\colon M\to\overline \mathscr{D}$ such that
$\wt F|_{\mathring M}\colon \mathring M \to\mathscr{D}$ is a conformal complete
proper minimal immersion.
\end{enumerate}
In either case, the frontier $\wt F(bM)\subset b\mathscr{D}$ consists of finitely many closed curves.
If $n\geq 5$ then the approximation can be achieved by maps $\wt F$ which are embeddings on $\mathring M$.
\end{theorem}
Theorem \ref{th:proper} is proved in Sec.\ \ref{sec:topology}.
It may be viewed as a version of Theorem \ref{th:Jordan} in which we additionally ensure that
the boundary curves of a minimal surface are contained in the boundary of the domain, at the cost of losing
topological embeddedness of these curves. A partial result in this direction
can be found in \cite{A2} where the first named author constructed compact complete proper minimal immersions of
surfaces with arbitrary finite topology into smoothly bounded strictly convex domains in $\mathbb{R}^3$,
but without control of the conformal structure on the surface or the flux of the immersion.
In part {\rm (a)} of Theorem \ref{th:proper} it is clearly impossible to ensure approximation in the $\mathscr{C}^0(M)$ topology;
however, a nontrivial upper bound for the maximum norm $\|\wt F-F\|_{0,M}$,
depending on the placement of the boundary $F(bM)\subset\mathscr{D}$, is provided by Theorem \ref{th:proper2}.
In part (b) the flux can be changed by an arbitrarily small amount.
Unlike in Theorem \ref{th:Jordan}, we are unable to guarantee that the boundary $\wt F(bM)\subset b\mathscr{D}$
consists of Jordan curves; see Remark \ref{rem:Jordan}. Hence the following remains an open problem.
\begin{problem}
Let $\mathscr{D}$ be a smoothly bounded, strictly convex domain in $\mathbb{R}^n$ for some $n\ge 3$.
Does there exist a complete proper minimal surface in $\mathscr{D}$ bounded by finitely
many Jordan curves in the boundary $b\mathscr{D}$ of $\mathscr{D}$? What is the answer if $\mathscr{D}$ is the unit ball of $\mathbb{R}^n$?
\end{problem}
Theorem \ref{th:proper} fails in general for weakly convex domains.
For instance, no polyhedral region of $\mathbb{R}^3$ admits a complete proper minimal disc that is
continuous up to the boundary \cite{AN,Na2}; it is easily seen that such a disc
would violate Bourgain's theorem on the radial variation of bounded analytic functions \cite{Bourgain}.
(See also \cite{G2} for the case of complex discs in a bidisc of $\mathbb{C}^2$.)
Our next result shows that the situation is rather different if we do not insist on continuity up to the boundary.
\begin{theorem}\label{th:topology}
Let $D$ be a convex domain in $\mathbb{R}^n$ for some $n\ge 3$.
\begin{enumerate}[\rm (a)]
\item If $M$ is a compact bordered Riemann surface and $F\colon M\to D$ is a conformal minimal immersion
of class $\mathscr{C}^1(M)$, then $F$ can be approximated uniformly on compacts in $\mathring M$
by conformal complete proper minimal immersions $\wt F\colon \mathring M\to D$ with $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
If $n\geq 5$ then $\wt F$ can be chosen an embedding.
\vspace{1mm}
\item Every open orientable smooth surface $S$ carries a full complete proper minimal immersion
$S\to D$ (embedding if $n\ge 5$) with arbitrary flux.
\end{enumerate}
\end{theorem}
Recall that a minimal surface in $\mathbb{R}^n$ is said to be {\em full} if it is not contained in any affine hyperplane.
In part (b), the flux is meant with respect to the conformal structure induced on the surface
$S$ by the Euclidean metric of $\mathbb{R}^n$ via the immersion $S\to\mathbb{R}^n$.
Theorem \ref{th:topology} is proved in Sec.\ \ref{sec:topology}. The case $D=\mathbb{R}^n$ was already established in \cite{AFL,AL1}
where conformal complete proper minimal immersions $\mathcal{R}\to\mathbb{R}^n$ (embeddings if $n\geq 5$) with arbitrary flux are constructed
for every open Riemann surface $\mathcal{R}$. However, since the existence of a nonconstant positive harmonic function
is a nontrivial condition on an open Riemann surface, it is clearly impossible to prescribe the conformal type of a
full minimal surface in any either convex or smoothly bounded domain $D$ different from $\mathbb{R}^n$.
Ferrer, Mart\'in, and Meeks \cite{FMM} proved Theorem \ref{th:topology} {\rm (b)} for $n=3$
but without the control of the flux, whereas the cases $n\in\{3,4,6\}$ and vanishing flux follow from the results in \cite{AF1,AL-CY}.
If one is merely interested in the existence of proper minimal surfaces (without approximation),
then it suffices to assume that the domain $D\subset \mathbb{R}^n$ admits a smooth strongly convex boundary point $p\in bD$.
Indeed, by using the approximation statement in Theorem \ref{th:proper2} one can find proper conformal
minimal immersions into $D$ with boundaries in a small neighbourhood of $p$ in $bD$,
thereby proving the following corollary (see Sec.\ \ref{sec:topology}).
\begin{corollary}\label{co:bdddomains}
If $D\subset\mathbb{R}^n$ $(n\ge 3)$ is a domain with a $\mathscr{C}^2$ smooth strictly convex boundary point then the following hold.
\begin{enumerate}[\rm (a)]
\item
Every compact bordered Riemann surface $M$ admits a continuous map $\wt F\colon M \to \overline D$
such that $\wt F(\mathring M)\subset D$ and $\wt F|_{\mathring M}\colon \mathring M \to D$ is a conformal
full complete proper minimal immersion. If $n\geq 5$ then $\wt F$ can be taken to be an embedding on $\mathring M$.
\vspace{1mm}
\item
Every open orientable smooth surface $S$ carries a full complete proper minimal immersion $S\to D$
(embedding if $n\ge 5$) with arbitrary flux.
\end{enumerate}
\end{corollary}
Note that every smoothly bounded relatively compact domain in $\mathbb{R}^n$ admits a strictly convex boundary point,
so Corollary \ref{co:bdddomains} applies to such domains.
To the best of our knowledge, Theorems \ref{th:Jordan}, \ref{th:proper}, and \ref{th:topology},
and Corollary \ref{co:bdddomains} provide the first examples of complete bounded {\em embedded}
minimal surfaces in $\mathbb{R}^5$ with controlled topology; furthermore, they solve the {\em conformal Calabi-Yau problem}
for {\em embedded bordered Riemann surfaces in convex domains of $\mathbb{R}^n$} for any $n\geq 5$.
Recall that the {\em Calabi-Yau problem} deals with the existence and geometric properties of
complete bounded minimal surfaces; see for instance \cite{AL-CY,AF2} and the references therein for
the state of the art of this subject. Regarding the {\em embedded Calabi-Yau problem}, it is still unknown whether there
exist complete bounded embedded minimal surfaces in $\mathbb{R}^3$. By results of Colding and Minicozzi \cite{CM} and
Meeks, P\'erez, and Ros \cite{MPR}, there is no such surface with finite genus and at most countably many ends.
Recently Alarc\'on and L\'opez \cite{AL-JEMS} and Globevnik \cite{Gl,G3} constructed complete bounded
embedded complex curves (hence minimal surfaces) in $\mathbb{C}^2\equiv\mathbb{R}^4$;
however, their method does not provide any information on the topology and the conformal structure of their examples.
Globevnik actually constructed a holomorphic function $f$ on the unit ball of $\mathbb{C}^n$ for any $n\ge 2$ \cite{Gl}
and, more generally, on any pseudoconvex domain $D\subset \mathbb{C}^n$ for $n\ge 2$ \cite{G3},
such that every divergent curve in $D$ on which $f$ is bounded has infinite length.
It follows that every level set $\{f=c\}$ of such a
function is a complete closed complex hypersurface in $D$.
By applying the Riemann-Hilbert technique, developed in Sec.\ \ref{sec:RH} below, it is straightforward to
extend all main results of the paper \cite{DF2014} to null hulls of
compact sets in $\mathbb{C}^n$ and minimal hulls of compact sets in $\mathbb{R}^n$ for any $n> 3$.
As pointed out in \cite{DF2014}, the only reason for restricting to $n=3$ was that a
Riemann-Hilbert lemma for null curves in dimension $n>3$ was not available at that time.
We postpone this to a subsequent publication.
The proofs of our results depend in an essential way on a new tool that we obtain in this paper,
namely the Riemann-Hilbert boundary value problem for null curves in $\mathbb{C}^n$ and minimal surfaces in $\mathbb{R}^n$
for any $n\ge 3$ (cf.\ Theorems \ref{th:RH} and \ref{th:RHCMI}), generalizing the one developed in
\cite{AF2} in dimension $n=3$. We also use a number of other recent ideas and techniques:
gluing holomorphic sprays on Cartan pairs (cf.\ \cite{DF2007} and \cite{F2011}),
the Mergelyan approximation theorem for conformal minimal immersions in $\mathbb{R}^n$ (cf.\ \cite{AL1,AFL}),
the general position theorem for minimal surfaces in $\mathbb{R}^n$ for $n\ge 5$ (cf.\ \cite{AFL}), the method of exposing boundary points
on a bordered Riemann surface (Forstneri\v c and Wold \cite{FW0}), and the circle of ideas around the construction of
compact complete minimal immersions in $\mathbb{R}^3$ (cf.\ \cite{AL-Israel} and Mart\'in and Nadirashvili \cite{MN}) and complete
bounded minimal surfaces in $\mathbb{R}^3$ normalized by bordered Riemann surfaces (cf.\ \cite{AF2} and also \cite{AF}).
Our methods easily adapt to give results analogous to Theorems \ref{th:Jordan}, \ref{th:proper}, and \ref{th:topology}
in the context of complex curves in $\mathbb{C}^n$, $n\geq 2$, and holomorphic null curves in $\mathbb{C}^n$, $n\geq 3$.
Indeed, all tools used in the proof are available for these families of curves:
the Riemann-Hilbert method for holomorphic null curves in arbitrary dimension is provided in this paper
(see Theorem \ref{th:RH}), while Runge-Mergelyan type theorems for null curves are proved in \cite{AF1,AL1}.
For example, by following the proof of Theorem \ref{th:Jordan} one can show the following result.
\begin{theorem}
\label{th:further}
Every bordered Riemann surface $M$ admits a continuous map $F\colon M\to \mathbb{C}^n$, $n\ge 2$, such that
$F|_{\mathring M}\colon \mathring M \to\mathbb{C}^n$ is a complete holomorphic immersion
and $F(bM)$ is a finite union of Jordan curves. If $n\ge 3$ then there is a topological
embedding $F\colon M\hra \mathbb{C}^n$ with these properties.
\end{theorem}
A similar result can be established for null curves $F\colon M\to\mathbb{C}^3$, $n\ge 3$; recall that the general position of
null curves in $\mathbb{C}^n$ is embedded for any $n\ge 3$ (cf.\ \cite[Theorem 2.4]{AF1}).
\subsection*{Outline of the paper}
In Sec.\ \ref{sec:prelim} we introduce the notation and preliminaries.
In Sec.\ \ref{sec:RH} we develop the Riemann-Hilbert method for null curves
(Lemmas \ref{lem:RH3}, \ref{lem:RH} and Theorem \ref{th:RH})
and minimal surfaces (Theorem \ref{th:RHCMI}) in arbitrary dimension $n\ge 3$.
Sec.\ \ref{sec:Jordan} is devoted to the proof of Theorem \ref{th:Jordan}.
The main technical part is contained in Lemma \ref{lem:Jordan2}, asserting that any conformal minimal immersion
$M\to\mathbb{R}^n$ of a compact bordered Riemann surface can be approximated in the $\mathscr{C}^0(M)$ topology
by conformal minimal immersions $M\to\mathbb{R}^n$ whose boundary distance from a fixed interior point is as big as desired.
In Sec.\ \ref{sec:Jordan} we also prove that the general position of the boundary curves $bM\to\mathbb{R}^n$
of a conformal minimal immersion $M\to\mathbb{R}^n$ is embedded for $n\geq 3$; see Theorem \ref{th:gp}.
Lemma \ref{lem:Jordan2} is also exploited in the proof of Theorem \ref{th:proper2} in Sec.\ \ref{sec:proper}.
The latter result is the key to the proof of Theorems \ref{th:proper}, \ref{th:topology}
and Corollary \ref{co:bdddomains} given in Sec.\ \ref{sec:topology}.
\section{Notation and preliminaries}\label{sec:prelim}
We denote by $\langle\cdot,\cdot\rangle$, $\|\cdot\|$, and $\mathrm{dist}(\cdot,\cdot)$ the Euclidean scalar product, norm, and distance in $\mathbb{R}^n$, $n\in\mathbb{N}$. Given a vector $\mathbf{x}\in\mathbb{R}^n\setminus\{0\}$ we denote by
$\langle\mathbf{x}\rangle^\bot=\{\mathbf{w}\in\mathbb{R}^n\colon \langle\mathbf{w},\mathbf{x}\rangle=0\}$ its orthogonal complement.
If $K$ is a compact topological space and $f\colon K\to\mathbb{R}^n$ is a continuous function,
we denote by $\|f\|_{0,K}$ the maximum norm of $f$.
Set $\mathbb D=\{\zeta\in\mathbb{C}\colon |\zeta|<1\}$ and $\mathbb{T}= b{\mathbb D}=\{\zeta\in\mathbb{C}\colon |\zeta|=1\}$.
As usual we will identify $\mathbb{C}^n\equiv \mathbb{R}^{2n}$. We shall write $\imath=\sqrt{-1}$ . By $\Re(\mathbf{z})$ and $\Im(\mathbf{z})$
we denote the real and the imaginary part of a point $\mathbf{z}\in\mathbb{C}^n$. Let $\mathbf{z}=(z_1,\ldots,z_n)$ be complex
coordinates on $\mathbb{C}^n$. Denote by $\Theta$ the holomorphic bilinear form on $\mathbb{C}^n$ given by
\begin{equation}\label{eq:bilinear}
\Theta(\mathbf{z},\mathbf{w})= \sum_{j=1}^n z_j w_j.
\end{equation}
Let $\mathfrak{A}=\mathfrak{A}^{n-1}\subset \mathbb{C}^n$ denote the {\em null quadric}
\begin{equation}\label{eq:Agot}
\mathfrak{A}^{n-1} =\{\mathbf{z}=(z_1,\ldots,z_n) \in\mathbb{C}^n\colon \Theta(\mathbf{z},\mathbf{z})=z_1^2+\ldots+z_n^2=0\}.
\end{equation}
This is a conical algebraic subvariety of $\mathbb{C}^n$ that is not contained in any hyperplane of $\mathbb{C}^n$
and is nonsingular except at the origin. We also write $\mathfrak{A}^{n-1}_*=\mathfrak{A}^{n-1}\setminus\{0\}$.
In the sequel we shall omit the superscript when the dimension is clear from the context.
Let us recall the basic facts concerning minimal surfaces (see e.g.\ Osserman \cite{Osserman}).
Let $M$ be an open Riemann surface, and let $\theta$ a nowhere vanishing holomorphic $1$-form on $M$
(such exists by the Oka-Grauert principle, cf.\ Theorem 5.3.1 in \cite[p.\ 190]{F2011}).
The exterior derivative on $M$ splits into the sum $d=\di+\dibar$ of the $(1,0)$-part $\di$ and the $(0,1)$-part $\dibar$.
An immersion $F=(F_1,\ldots,F_n)\colon M\to\mathbb{R}^n$ ($n\geq 3$) is {\em conformal} (angle preserving)
if and only if its Hopf differential $\sum_{j=1}^n (\partial F_j)^2$ vanishes everywhere on $M$,
that is to say, if $\partial F/\theta\in\mathfrak{A}$ (\ref{eq:Agot}). A conformal immersion $F\colon M\to\mathbb{R}^n$ is
minimal if and only if it is harmonic, and in this case $\Phi:=\partial F$ is a $\mathbb{C}^n$-valued holomorphic $1$-form
vanishing nowhere on $M$. Given a base point $p_0\in M$, $F$ can be written in the form
\begin{equation}\label{eq:FPhi}
F(p)=F(p_0)+\Re\Big(\int_{p_0}^p \Phi\Big), \quad p\in M.
\end{equation}
This is called the {\em Weierstrass representation} of $F$.
Conversely, if an $n$-dimensional holomorphic 1-form $\Phi$ on $M$ has vanishing real periods
(i.e., its real part $\Re(\Phi)$ is exact) and satisfies $\Phi/\theta\colon M\to \mathfrak{A}_*$, then the map
$F\colon M\to\mathbb{R}^n$ given by \eqref{eq:FPhi} is a conformal minimal immersion.
Let $H_1(M;\mathbb{Z})$ denote the first homology group of $M$ with integer coefficients.
The {\em flux map} of a conformal minimal immersion $F\colon M \to\mathbb{R}^n$ is the group homomorphism
$\mathrm{Flux}(F)\colon H_1(M;\mathbb{Z})\to\mathbb{R}^n$ given by
\begin{equation} \label{eq:flux}
\mathrm{Flux}(F)(\gamma)=\Im\Big(\int_\gamma \partial F\Big) \quad \text{for every closed curve }\gamma\subset M.
\end{equation}
Since the $1$-form $\di F$ on $M$ is holomorphic and therefore closed, the integral on the right hand side is
independent of the choice of a path in a given homology class.
Next we introduce the mapping spaces that will be used in the paper.
If $M$ is an open Riemann surface then $\CMI(M,\mathbb{R}^n)$ denotes the set of all conformal minimal immersions $M\to \mathbb{R}^n$.
Assume now that $M$ is a {\em compact bordered Riemann surface}, i.e., a compact connected Riemann surface with smooth boundary
$\emptyset \ne bM \subset M$ and interior $\mathring M=M\setminus bM$. For any $r\in \mathbb{Z}_+$ we denote by $\mathscr{A}^r(M)$ the space
of all functions $M\to \mathbb{C}$ of class $\mathscr{C}^r(M)$ that are holomorphic in $\mathring M$. We write $\mathscr{A}^0(M)=\mathscr{A}(M)$.
It is classical that every compact bordered Riemann surface $M$ can be considered as a smoothly bounded compact domain in
an open Riemann surface $\wt M$ and, by Mergelyan's theorem, each function in $\mathscr{A}^r(M)$ can be approximated in the
$\mathscr{C}^r(M)$ topology by holomorphic functions on a neighborhood of $M$.
For any $r\in \mathbb{N}$ we denote by $\CMI^r(M,\mathbb{R}^n)$ the set of all conformal minimal immersions $M\to\mathbb{R}^n$
of class $\mathscr{C}^r(M)$. More precisely, an immersion $F\colon M\to \mathbb{R}^n$ of class $\mathscr{C}^r$ belongs to
$\CMI^r(M,\mathbb{R}^n)$ if and only if $\di F$ is a $(1,0)$-form of class $\mathscr{C}^{r-1}(M)$
which is holomorphic in the interior $\mathring M=M\setminus \partial M$ and has
range in the punctured null quadric $\mathfrak{A}_*$ (\ref{eq:Agot}).
For $r=0$ we define $\CMI^0(M,\mathbb{R}^n)$ as the class of all continuous maps $F\colon M\to \mathbb{R}^n$ such that
$F\colon\mathring M\to\mathbb{R}^n$ is a conformal minimal immersion.
By the local Mergelyan theorem for conformal minimal immersions \cite[Theorem 3.1 (a)]{AFL},
every $F\in \CMI^r(M,\mathbb{R}^n)$ for $r\ge 1$ can be approximated in the $\mathscr{C}^r(M)$ topology by
conformal minimal immersions on an open neighborhood of $M$ in the ambient surface $\wt M$.
If $M$ is Runge in $\wt M$ then every such $F$ can also be approximated in the $\mathscr{C}^r(M)$ topology
by conformal minimal immersions $\wt M\to\mathbb{R}^n$ (cf.\ \cite[Theorem 5.3]{AFL}).
We say that a holomorphic map $f\colon M\to \mathfrak{A}_*$ is {\em nondegenerate} if the image $f(M)\subset\mathfrak{A}_*$
is not contained in any complex hyperplane of $\mathbb{C}^n$. Clearly nondegenerate implies {\em nonflat},
where the latter condition means that $f(M)$ is not contained in a (complex) ray of the null cone $\mathfrak{A}$.
If $f$ is nonflat then the linear span of the tangent spaces $T_{f(p)}\mathfrak{A}$ over all
points $p\in M$ equals $\mathbb{C}^n$ (cf.\ \cite[Lemma 2.3]{AFL}).
The latter condition implies the existence of a dominating
and period dominating holomorphic spray of maps $f_w\colon M\to \mathfrak{A}_*$, with the parameter
$w$ in a ball in some $\mathbb{C}^N$ and with the core map $f_0=f$ (see \cite[Lemma 5.1]{AF1}
or \cite[Lemma 3.2]{AFL}). This will be used in the proof of Theorems \ref{th:RH} and \ref{th:RHCMI}.
A conformal minimal immersion $F\colon M\to \mathbb{R}^n$ is said to be {\em nondegenerate} if the map
$f=\partial F/\theta \colon M\to \mathfrak{A}_*$ is nondegenerate, and is said to be {\em full} if $F(M)$
is not contained in a hyperplane of $\mathbb{R}^n$. Nondegenerate conformal minimal immersions $M\to\mathbb{R}^n$ are full,
but the converse is true only in the case $n=3$ (see \cite{Osserman}).
If $M$ is an open Riemann surface, we denote by $\CMI_*(M,\mathbb{R}^n)\subset \mathrm{CMI}(M,\mathbb{R}^n)$
the subset consisting of all immersions which are nondegenerate on every connected component of $M$.
The analogous notation is used for compact bordered Riemann surfaces:
$\CMI^r_*(M,\mathbb{R}^n)$ denotes the space of all $F\in \CMI^r(M,\mathbb{R}^n)$ which are nondegenerate
on every connected component of $M$. By Theorem 3.1 in \cite{AFL}, $\CMI_*^r(M,\mathbb{R}^n)$ is a dense subset of
$\CMI^1(M,\mathbb{R}^n)$ in the $\mathscr{C}^1(M)$ topology for every $r\in\mathbb{N}$.
If $M$ is an open Riemann surface and $F\in\CMI(M,\mathbb{R}^n)$, we denote by $\mathrm{dist}_F(\cdot,\cdot)$
the intrinsic distance in $M$ induced by the Euclidean metric of $\mathbb{R}^n$ via $F$; i.e.
\[
\mathrm{dist}_F(p,q)=\inf\{\mathrm{length}\, F(\gamma) \colon \gamma\subset M \text{ arc connecting $p$ and $q$}\},
\]
where $\mathrm{length}$ denotes the Euclidean length in $\mathbb{R}^n$.
Likewise we define $\mathrm{dist}_F$ on $M$ when $M$ is a compact bordered Riemann surface.
If $M$ is open, the immersion $F\colon M\to\mathbb{R}^n$ is said to be {\em complete} if $\mathrm{dist}_F$
is a complete metric on $M$;
equivalently, if the image $F(\gamma)$ of any divergent curve $\gamma\subset M$
(i.e., a curve which eventually leaves any compact subset of $M$) is a curve of
infinite length in $\mathbb{R}^n$.
\section{Riemann-Hilbert problem for null curves in $\mathbb{C}^n$}\label{sec:RH}
In this section we find approximate solutions of a general Riemann-Hilbert boundary value problem for null curves
and for confomal minimal immersions.
Let $n\in\mathbb{N}$, $n\geq 3$. Recall that $\mathfrak{A}^{n-1}\subset \mathbb{C}^n$ is the null quadric
(\ref{eq:Agot}) and $\mathfrak{A}^{n-1}_*=\mathfrak{A}^{n-1}\setminus\{0\}$.
We shall drop the superscript when the dimension $n$ is clear from the context.
We begin with the following essentially optimal result in dimension $n=3$ in which there is no restriction
on the type of null discs attached at boundary points of the central null disc.
Lemma \ref{lem:RH3} generalizes \cite[Lemma 3.1]{AF2} which pertains to the case of
linear null discs of the form $\xi\mapsto r(\zeta)\,\xi \mathbf{u}$ in a constant null direction
$\mathbf{u}\in \mathfrak{A}_*$. The corresponding result for ordinary holomorphic discs can be found in several
sources, see e.g.\ \cite[Lemma 3.1]{DF2012}.
\begin{lemma} \label{lem:RH3}
Let $F\colon\overline{\mathbb D}\to\mathbb{C}^3$ be a null holomorphic disc of class $\mathscr{A}^1(\mathbb D)$.
Assume that $I$ is a proper closed segment in the circle $\mathbb{T}=b\mathbb D$,
$r\colon \mathbb{T} \to \mathbb{R}_+ := [0,1]$ is a continuous function supported on $I$ (the {\em size function}),
and $\sigma \colon I \times\overline{\mathbb D}\to\mathbb{C}^3$ is a map of class $\mathscr{C}^1$ such that
for every $\zeta\in I$ the map $\overline{\mathbb D} \ni \xi \mapsto \sigma(\zeta,\xi)$ is an immersed holomorphic null disc
with $\sigma(\zeta,0)=0$. Let $\varkappa\colon \mathbb{T} \times\overline{\mathbb D}\to\mathbb{C}^3$ be given by
\begin{equation}\label{eq:varkappa}
\varkappa(\zeta,\xi)=F(\zeta) + \sigma\bigl(\zeta,r(\zeta)\, \xi\bigr)
\end{equation}
where we take $\sigma\bigl(\zeta,r(\zeta)\, \xi\bigr)=0$ for $\zeta\in\mathbb{T}\setminus I$.
Given numbers $\epsilon>0$, $0<\rho_0<1$ and an open neighborhood $U$ of $I$ in $\overline{\mathbb D}$,
there exist a number $\rho'\in [\rho_0,1)$ and a null holomorphic immersion
$G\colon\overline{\mathbb D}\to\mathbb{C}^n$ such that $G(0)=F(0)$ and the following conditions hold:
\begin{enumerate}[\it i)]
\item $\mathrm{dist}(G(\zeta),\varkappa(\zeta,\mathbb{T}))<\epsilon$ for all $\zeta\in \mathbb{T}$,
\item $\mathrm{dist}(G(\rho\zeta),\varkappa(\zeta,\overline{\mathbb D}))<\epsilon$ for all $\zeta\in \mathbb{T}$ and all
$\rho\in [\rho',1)$, and
\item $G$ is $\epsilon$-close to $F$ in the $\mathscr{C}^1$ topology on
$(\overline{\mathbb D}\setminus U) \cup \rho'\overline{\mathbb D}$.
\end{enumerate}
Moreover, given an upper semicontinuous function $\phi \colon\mathbb{C}^3\to \mathbb{R}\cup\{-\infty\}$,
we may achieve in addition that
\begin{equation}
\label{eq:small-increase}
\int_{I} \phi \bigl( G(\E^{\imath t})\bigr) \, \frac{dt}{2\pi} \le
\int^{2\pi}_0 \!\! \int_{I} \phi \bigl(\varkappa(\E^{\imath t},\E^{\imath s})\bigr)
\frac{dt}{2\pi} \frac{ds}{2\pi} + \epsilon
\end{equation}
\end{lemma}
\begin{proof}
Let $\pi\colon\mathbb{C}^2\to\mathbb{C}^3$ be the homogeneous quadratic map defined by
\begin{equation}\label{eq:pi}
\pi(u,v)=\left(u^2-v^2,2uv,-\imath(u^2+v^2)\right), \quad (u,v)\in\mathbb{C}^2.
\end{equation}
Note that $\pi$ is a two-sheeted parametrization of the null quadric
$\mathfrak{A}\subset \mathbb{C}^3$ (\ref{eq:Agot}), commonly called the
{\em spinor parametrization} of $\mathfrak{A}$, and $\pi$ is branched only at the point $(0,0)\in\mathbb{C}^2$.
In particular, $\pi\colon \mathbb{C}^2_* =\mathbb{C}^2\setminus\{0\} \to \mathfrak{A}_*$ is a doubly sheeted
holomorphic covering projection.
Set $\mu(\zeta,\xi) = \sigma\bigl(\zeta,r(\zeta)\, \xi\bigr)$ and extend it by zero to points $\zeta\in\mathbb{T}\setminus I$.
The conditions on $\sigma$ imply that the partial derivative
\[
\sigma_2(\zeta,\xi):= \frac{\di \sigma}{\di \xi}(\zeta,\xi) \in\mathfrak{A}_*, \quad (\zeta,\xi)\in I\times \overline{\mathbb D}
\]
has values in $\mathfrak{A}_*$. Note that
$ \frac{\di}{\di \xi} \mu(\zeta,\xi)=r(\zeta) \, \sigma_2(\zeta,r(\zeta)\,\xi)$.
Since $I\times \overline{\mathbb D}$ is simply connected, there is a
lifting $\varsigma\colon I\times \overline{\mathbb D} \to \mathbb{C}^2_*$ such that $\pi\circ\varsigma=\sigma_2$. Set
\[
\eta(\zeta,\xi) = \sqrt{r(\zeta)}\, \varsigma \bigl(\zeta,r(\zeta)\, \xi\bigr),
\quad (\zeta,\xi)\in \mathbb{T}\times\overline{\mathbb D}.
\]
Then $\eta(\zeta,\xi)$ is holomorphic in $\xi\in\mathbb D$ for every fixed $\zeta\in\mathbb{T}$, and we have that
\[
\pi\bigl(\eta(\zeta,\xi)\bigr) = r(\zeta) \, \sigma_2(\zeta,r(\zeta)\,\xi)
= \frac{\di}{\di \xi} \mu(\zeta,\xi).
\]
We can approximate $\eta$
as closely as desired in the sup norm on $\mathbb{T}\times\overline{\mathbb D}$ by a rational map
\begin{equation}\label{eq:tilde-eta}
\wt \eta(\zeta,\xi) = \sum_{j=0}^l B_j(\zeta)\, \xi^j
\end{equation}
where every $B_j(\zeta)$ is a $\mathbb{C}^2$-valued Laurent polynomial with the only pole at $\zeta=0$. Set
\begin{equation}\label{eq:tilde-mu}
\wt \mu(\zeta,z) = \int_0^z \pi\bigl( \wt\eta(\zeta,\xi)\bigr) \,d\xi = \sum_{k=1}^m A_k(\zeta) \, z^k
\end{equation}
where $\pi$ is the projection (\ref{eq:pi}), $m=2l+1$,
and $A_k(\zeta)$ are $\mathbb{C}^3$-valued Laurent polynomials with the only pole at $\zeta=0$.
Then $\wt\mu$ is uniformly close to $\mu$ on $\mathbb{T}\times\overline{\mathbb D}$, and it suffices to prove
the lemma with $\mu$ replaced by $\wt \mu$.
To simplify the notation we now drop the tildes and assume that the functions $\eta$ and $\mu$
are given by (\ref{eq:tilde-eta}) and (\ref{eq:tilde-mu}), respectively. In particular, we have
\[
\mu_2(\zeta,\xi):= \frac{\di \mu}{\di \xi}(\zeta,\xi) =\pi(\eta(\zeta,\xi)).
\]
\begin{lemma}\label{lem:estimate}
Let $\mu(\zeta,\xi)= \sum_{k=1}^m A_k(\zeta) \, \xi^k$
where every $A_k(\zeta)$ is a Laurent polynomial with the only pole at $\zeta=0$.
Write $\mu_2(\zeta,\xi)=\frac{\di \mu}{\di \xi}(\zeta,\xi)$. Then
\begin{equation}\label{eq:estimate}
\lim_{N\to\infty} \sup_{|z|\le 1,\, |c|=1} \left |
\int_0^z c N\zeta^{N-1} \mu_2(\zeta,c\, \zeta^N)\, d\zeta - \mu(z,c z^N) \right | =0.
\end{equation}
\end{lemma}
\begin{proof}
We have
$
\mu_2(\zeta,\xi) = \frac{\di \mu}{\di \xi}(\zeta,\xi) = \sum_{k=1}^m A_k(\zeta) \, k \xi ^{k-1}
$
and hence
\[
c N\zeta^{N-1} \mu_2(\zeta,c\, \zeta^N) = \sum_{k=1}^m c^k A_k(\zeta) kN\zeta^{kN-1}.
\]
If $N\in\mathbb{N}$ is chosen big enough then $A_k(\zeta)\zeta^N$ vanishes at $\zeta=0$
for every $k=1,\ldots,m$. For such $N$ integration by parts gives
\begin{equation}\label{eq:IntN}
\int_0^z A_k(\zeta) kN\zeta^{kN-1} \,d\zeta = \int_0^z A_k(\zeta)\,d\zeta^{kN}
= A_k(z) z^{kN} - \int_0^z A'_k(\zeta) \zeta^{kN} \,d\zeta.
\end{equation}
Since $A'_k(\zeta)=\sum_{|j| \le m_k} A'_{k,j}\zeta^j$ for some integer $m_k\in\mathbb{N}$, we have
\[
\int_0^z A'_k(\zeta) \zeta^{kN} d\zeta
= \sum_{|j| \le m_k} \int_0^z A'_{k,j}\zeta^{j+kN}\,d\zeta
= \sum_{|j| \le m_k} \frac{A'_{k,j} z^{j+kN+1}}{j+kN+1}.
\]
The right hand side converges to zero uniformly on $\overline{\mathbb D}= \{|z|\le 1\}$ when
$N\to\infty$. Multiplying the equation (\ref{eq:IntN}) by $c^k\in \mathbb{T}$,
summing over $k=1,\ldots,m$ and observing that
\[
\sum_{k=1}^m c^k A_k(z) z^{kN} = \sum_{k=1}^m A_k(z) (cz^N)^k = \mu(z,c z^N)
\]
we get (\ref{eq:estimate}). This proves Lemma \ref{lem:estimate}.
\end{proof}
For every point $c=\E^{\imath\phi}\in\mathbb{T}$ with $\phi\in [0,2\pi)$ we set
$\sqrt c = \E^{\imath \phi/2}$. Consider the sequence of maps
$g_N\colon\mathbb{C}\times\mathbb{T} \to \mathbb{C}^2$ $(N\in\mathbb{N})$ given by
\begin{equation}\label{eq:gN}
g_N(\zeta,c) = \sqrt{c} \, \sqrt{2N+1}\, \zeta^N \eta(\zeta,c\, \zeta^{2N+1}).
\end{equation}
Note that $g_N$ is a holomorphic polynomial in $\zeta\in \mathbb{C}$ for every sufficiently big $N$, say
$N\ge N_0$. Since the projection $\pi$ (\ref{eq:pi}) is homogeneous quadratic, we have that
\[
\pi(g_N(\zeta,c)) = c (2N+1)\zeta^{2N} \pi\bigl( \eta(\zeta,c\, \zeta^{2N+1})\bigr)
= c (2N+1)\zeta^{2N} \mu_2(\zeta,c\, \zeta^{2N+1})
\]
and hence
\[
\int_0^z \pi(g_N(\zeta,c)) \,d\zeta = \int_0^z c (2N+1)\zeta^{2N} \mu_2(\zeta,c\, \zeta^{2N+1})\, d\zeta.
\]
By Lemma \ref{lem:estimate} we have
\begin{equation}\label{eq:estimate2}
\lim_{N\to\infty} \sup_{|z|\le 1,\, c\in\mathbb{T}} \left |
\int_0^z \pi\bigl(g_N(\zeta,c)\bigr) d\zeta - \mu(z,c z^{2N+1}) \right | =0.
\end{equation}
The derivative $F' \colon \overline{\mathbb D}\to \mathfrak{A}^2_*$ of the given null disc $F$
lifts to a continuous map $h=(u,v)\colon \overline{\mathbb D}\to\mathbb{C}^2_*$ that is holomorphic on $\mathbb D$.
With $g_N$ as in (\ref{eq:gN}) we consider the sequence of maps $h_N\colon \overline{\mathbb D}\times \mathbb{T} \to\mathbb{C}^2$
given for $N\ge N_0$ by
\begin{equation}\label{eq:hN}
h_N(\zeta,c) = h(\zeta) + g_N(\zeta,c),\quad \zeta \in \overline{\mathbb D},\ c\in\mathbb{T}.
\end{equation}
A general position argument shows that for a generic choice of $F$
we have $h_N(\overline{\mathbb D}\times \mathbb{T})\subset \mathbb{C}^2_*$ for all sufficiently big $N\in \mathbb{N}$ (see the proof of Lemma 3.1 in
\cite{AF2} for the details). Assume that this is the case. Consider the holomorphic null discs
\[
F_N(z,c) = F(0) + \int_0^z \pi(h_N(\zeta,c))\,d\zeta,\quad z\in \overline{\mathbb D},\ c\in \mathbb{T}.
\]
Since $\pi$ is a homogeneous quadratic map, we have
\begin{equation}\label{eq:pi-hN}
\pi(h_N(\zeta,c)) = \pi(h(\zeta)) + \pi(g_N(\zeta,c)) + R_N(\zeta,c)
\end{equation}
where each component of the remainder term $R_N(\zeta,c)$ is a linear combination with constant
coefficients of terms $g_{N,j}(\zeta,c) u(\zeta)$ and $g_{N,j}(\zeta,c) v(\zeta)$ for $j=1,2$.
(Here we write $g_N=(g_{N,1},g_{N,2})$.) We claim that
\begin{equation}\label{eq:estimate3}
\lim_{N\to\infty} \sup_{|z|\le 1,\, c\in\mathbb{T}} \left |\int_0^z R_N(\zeta,c) \, d\zeta \right | =0.
\end{equation}
To see this, set
\[
C_1= \sup_{ |\zeta| \le1, |z|\le 1} |\zeta^{N_0}\eta(\zeta,z)|,\quad
C_2=\max\bigl\{ \sup_{ |\zeta| \le1} |u(\zeta)|,\, \sup_{ |\zeta| \le1}|v(\zeta)| \bigr\}.
\]
Then $\sup_{ |\zeta| \le1} |\zeta^{N_0} \eta(\zeta,c \zeta^{2N+1})| \le C_1$ for $N\in\mathbb{N}$.
Given $z\in\overline{\mathbb D}$, $c\in\mathbb{T}$, $j\in\{1,2\}$ and $N\ge N_0$ we then have
\begin{eqnarray}\label{eq:est-gN}
\left| \int_0^z g_{N,j}(\zeta,c) u(\zeta) d\zeta \right| &\le &
\int_0^{|z|} \sqrt{2N+1}\, |\zeta|^{N-N_0}
|\zeta^{N_0}\eta(\zeta,c\, \zeta^{2N+1})| \cdotp |u(\zeta)| \, d|\zeta| \cr
&\le & C_1C_2 \int_0^{|z|} \sqrt{2N+1}\, |\zeta|^{N-N_0} \, d|\zeta| \cr
&\le & C_1C_2 \frac{\sqrt{2N+1}}{N-N_0+1}.
\end{eqnarray}
Clearly the right hand side converges to zero as $N\to+\infty$. The same estimate holds
with $u(\zeta)$ replaced by $v(\zeta)$. Since $R_N(\zeta,c)$
is a linear combination of finitely many such terms whose number is independent of $N$,
(\ref{eq:estimate3}) follows.
Since $\pi(h(\zeta))= F'(\zeta)$, we get by integrating the equation (\ref{eq:pi-hN}) and
using the estimates (\ref{eq:estimate2}), (\ref{eq:estimate3}) that
\begin{equation}\label{eq:FN}
F_N(z,c) = F(z) + \mu(z,c z^{2N+1}) + E_N(z,c) = \varkappa(z,c z^{2N+1}) + E_N(z,c)
\end{equation}
where
\begin{equation}\label{eq:est-EN}
\lim_{N\to\infty} \sup_{|z|\le 1,\, c\in\mathbb{T}} |E_N(z,c)|=0.
\end{equation}
It is easily seen that for every $c\in\mathbb{T}$ and for all sufficiently big $N\in \mathbb{N}$ the null disc
$G=F_N(\cdotp,c)$ satisfies conditions {\em i)} -- {\em iii)} in Lemma \ref{lem:RH3}; a suitable choice of the constant
$c\in\mathbb{T}$ ensures that it also satisfies condition (\ref{eq:small-increase}).
(See the proof of \cite[Lemma 3.1]{AF2} and of \cite[Lemma 3.1]{DF2012} for the details.)
\end{proof}
We now proceed to the case $n>3$. This requires some additional preparations.
Let $\mathbf{u}, \mathbf{v}, \mathbf{w}\in \mathfrak{A}_*$ be linearly independent null vectors such that
\begin{equation}\label{eq:nondeg}
c:=\Theta(\mathbf{u},\mathbf{v})\ne 0,\quad b:=\Theta(\mathbf{u},\mathbf{w})\ne 0, \quad a:=\Theta(\mathbf{v},\mathbf{w})\ne 0,
\end{equation}
where $\Theta$ is the complex bilinear form on $\mathbb{C}^n$ given in \eqref{eq:bilinear}.
Denote by $\mathfrak{A}_{(\mathbf{u},\mathbf{v},\mathbf{w})}$ the intersection of $\mathfrak{A}$ with the complex
$3$-dimensional subspace $\mathscr{L}(\mathbf{u},\mathbf{v},\mathbf{w})$ of $\mathbb{C}^n$ spanned by the vectors $\mathbf{u},\mathbf{v},\mathbf{w}$.
Condition (\ref{eq:nondeg}) ensures that $\mathfrak{A}_{(\mathbf{u},\mathbf{v},\mathbf{w})}$
is biholomorphic (in fact, linearly equivalent) to the $2$-dimensional null quadric $\mathfrak{A}^2\subset\mathbb{C}^3$.
Indeed, a calculation shows that $\alpha \mathbf{u}+\beta \mathbf{v}+ \gamma \mathbf{w}\in \mathfrak{A}$
for some $(\alpha,\beta,\gamma)\in \mathbb{C}^3$ if and only if
\[
\alpha\beta\, \Theta(\mathbf{u},\mathbf{v})+ \alpha\gamma\, \Theta(\mathbf{u},\mathbf{w})+\beta\gamma\, \Theta(\mathbf{v},\mathbf{w})=0.
\]
Using the notation (\ref{eq:nondeg}), the above equation is equivalent to
\[
\left(\frac \alpha a-\imath \frac \beta b\right)^2+
\left(\frac \beta b-\imath \frac\gamma c\right)^2
+ \left(\frac\gamma c-\imath \frac \alpha a\right)^2=0.
\]
This is the equation of the null quadric $\mathfrak{A}^2\subset\mathbb{C}^3$ (\ref{eq:Agot}) in the coordinates
\[
z_1=\frac \alpha a-\imath \frac \beta b,
\quad z_2=\frac \beta b-\imath \frac\gamma c,
\quad z_3=\frac\gamma c-\imath \frac \alpha a.
\]
Note that
\[
\mathbf{z}=(z_1,z_2,z_3)= (\alpha,\beta,\gamma)\cdotp A(a,b,c)
\]
where $A(a,b,c)$ is the following nonsingular $3\times 3$ matrix with holomorphic coefficients:
\begin{equation}\label{eq:A}
A(a,b,c) =\left(\begin{matrix} 1/a & 0 & -\imath/a \cr
-\imath/b & 1/b & 0 \cr
0 & -\imath/c & 1/c \cr
\end{matrix}
\right).
\end{equation}
We are using row vectors and matrix product on the right for the convenience of notation.
Let $\pi\colon\mathbb{C}^2\to\mathbb{C}^3$ be the homogeneous quadratic map
given by (\ref{eq:pi}). Note that
\[
\pi(1,0)= (1,0,-\imath),\quad \pi(0,1)=(-1,0,-\imath).
\]
Recall that the restriction $\pi\colon \mathbb{C}^2_*=\mathbb{C}^2\setminus\{0\} \to \mathfrak{A}^2_*$
is a doubly sheeted holomorphic covering projection. We have
\begin{eqnarray*}
\pi\left(\frac 1 {\sqrt a},0\right) &=& \left(\frac 1 a,0,-\frac\imath a\right)
= (1,0,0) \cdotp A(a,b,c), \cr
\pi\left(\imath\sqrt{\frac\imath{2b}},-\sqrt{\frac\imath{2b}}\right) &=&
\left(-\frac\imath b,\frac 1 b,0\right) = (0,1,0) \cdotp A(a,b,c).
\end{eqnarray*}
The choice of $\sqrt{ }$ is fine on any simply connected subset in the domain space.
Pick a holomorphically varying family of linear automorphisms $\phi_{(a,b)}$ of $\mathbb{C}^2$ such that
\[
\phi_{(a,b)}(0,0)=(0,0), \quad \phi_{(a,b)}(1,0)=\left(\frac 1 {\sqrt a},0\right), \quad
\phi_{(a,b)}(0,1)=\left(\imath \sqrt{\frac\imath{2b}},-\sqrt{\frac\imath{2b}}\right).
\]
This is achieved by taking $\phi_{(a,b)}(s,t)= (s,t) \cdotp B(a,b)$ where $B$ is the $2\times 2$ matrix
\begin{equation}\label{eq:B}
B(a,b) =\left(\begin{matrix} \frac{1}{\sqrt a} & 0 \cr
\imath \sqrt{\frac\imath{2b}} & -\sqrt{\frac\imath{2b}} \cr
\end{matrix} \right).
\end{equation}
The map
\[
\mathbb{C}^2\ni (s,t) \mapsto
\bigl(\alpha(s,t), \beta(s,t),\gamma(s,t)\bigr)=
\pi\bigl((s,t)\cdotp B(a,b)\bigr) \cdotp A(a,b,c)^{-1} \in \mathbb{C}^3
\]
is homogeneous quadratic in $(s,t)$ and depends holomorphically on $(a,b,c)$,
and hence on the triple $(\mathbf{u},\mathbf{v},\mathbf{w})$ of null vectors satisfying (\ref{eq:nondeg}).
By the construction the associated map
\begin{equation}\label{eq:psi}
\mathbb{C}^2\ni (s,t) \mapsto
\psi_{(\mathbf{u},\mathbf{v},\mathbf{w})}(s,t) =\alpha(s,t) \mathbf{u}+\beta(s,t) \mathbf{v}+ \gamma (s,t)\mathbf{w}
\end{equation}
is a holomorphically varying parametrization of the quadric $\mathfrak{A}_{(\mathbf{u},\mathbf{v},\mathbf{w})}$ satisfying
\begin{equation}\label{eq:normalization}
\psi_{(\mathbf{u},\mathbf{v},\mathbf{w})}(\mathbf{e}_1) =\mathbf{u}, \quad \psi_{(\mathbf{u},\mathbf{v},\mathbf{w})}(\mathbf{e}_2) =\mathbf{v}
\end{equation}
where $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$.
Note that $\psi_{(\mathbf{u},\mathbf{v},\mathbf{w})}$ is well defined on the set of triples $(\mathbf{u},\mathbf{v},\mathbf{w})\in (\mathfrak{A}^{n-1})^3$
satisfying condition (\ref{eq:nondeg}), except for the indeterminacies caused by the square roots
in the entries of the matrix $B$ (\ref{eq:B}).
These reflect the fact that $\pi$ is a doubly sheeted quadratic map, so
we have four different choices providing the normalization (\ref{eq:normalization}).
In the sequel we shall hold fixed a pair of null vectors $\mathbf{u},\mathbf{v}\in \mathfrak{A}_*$, subject to the condition
$c=\Theta(\mathbf{u},\mathbf{v})\ne 0$, and will assume that $\mathbf{w}=f(\zeta)$ where
$f\colon \overline{\mathbb D}\to \mathfrak{A}_*$ is a holomorphic map such that the triple
of null vectors $(\mathbf{u},\mathbf{v},f(\zeta))$ satisfies (\ref{eq:nondeg})
for every $\zeta\in \overline{\mathbb D}$.
The following lemma provides an approximate solution of the Riemann-Hilbert problem
for null holomorphic discs in $\mathbb{C}^n$ for any $n\ge 3$ under the condition that the null discs
attached at boundary points of the (arbitrary) center null disc $F$ have a constant
direction vector $\mathbf{u}\in \mathfrak{A}_*$. (For $n=3$ this result is subsumed by Lemma \ref{lem:RH3}.)
At this time we are unable to prove the exact analogue of Lemma \ref{lem:RH3}
in dimensions $n>3$, but the present version is entirely sufficient for the
applications in this paper.
\begin{lemma} \label{lem:RH}
Fix an integer $n\ge 3$ and let $\mathfrak{A}=\mathfrak{A}^{n-1}$ be the null quadric (\ref{eq:Agot}).
Assume that $\mathbf{u},\mathbf{v}\in\mathfrak{A}_*=\mathfrak{A}\setminus\{0\}$ are null vectors such that
$\Theta(\mathbf{u},\mathbf{v})\ne0$. Let $F\colon\overline{\mathbb D}\to\mathbb{C}^n$ be a null holomorphic
immersion of class $\mathscr{A}^1(\mathbb D)$ whose derivative $f=F'\colon \overline{\mathbb D}\to \mathfrak{A}_*$
satisfies the following nondegeneracy condition:
\begin{equation}\label{eq:nondeg2}
\Theta(\mathbf{u},f(\zeta))\ne 0\ \ \text{and} \ \ \Theta(\mathbf{v},f(\zeta))\ne 0
\quad \forall \zeta\in\overline{\mathbb D}.
\end{equation}
Let $r\colon \mathbb{T}=b\mathbb D \to \mathbb{R}_+ := [0,+\infty)$ be a continuous function (the {\em size function}),
and let $\sigma \colon \mathbb{T} \times\overline{\mathbb D}\to\mathbb{C}$ be a function of class $\mathscr{C}^1$ such that
for every $\zeta\in \mathbb{T}$ the function $\overline{\mathbb D}\ni \xi \mapsto \sigma(\zeta,\xi)$ is holomorphic on $\mathbb D$,
$\sigma(\zeta,0)=0$, the partial derivative $\frac{\di \sigma}{\di \xi}$ is nonvanishing on $\mathbb{T}\times \overline{\mathbb D}$,
and the winding number of the function $\mathbb{T} \ni \zeta \mapsto \frac{\di \sigma}{\di \xi}(\zeta,0) \in \mathbb{C}_*$
equals zero. Set $\mu(\zeta,\xi) = r(\zeta) \sigma(\zeta,\xi)$ and let
$\varkappa\colon \mathbb{T} \times\overline{\mathbb D}\to\mathbb{C}^n$ be given by
\begin{equation}\label{eq:varkappa2}
\varkappa(\zeta,\xi)=F(\zeta) + \mu(\zeta,\xi)\, \mathbf{u} = F(\zeta)+ r(\zeta)\, \sigma(\zeta,\xi)\, \mathbf{u}.
\end{equation}
Given numbers $\epsilon>0$ and $0<\rho_0<1$, there exist a number $\rho'\in [\rho_0,1)$
and a null holomorphic disc $G\colon\overline{\mathbb D}\to\mathbb{C}^n$
such that $G(0)=F(0)$ and the following conditions hold:
\begin{enumerate}[\it i)]
\item $\mathrm{dist}(G(\zeta),\varkappa(\zeta,\mathbb{T}))<\epsilon$ for all $\zeta\in \mathbb{T}$,
\item $\mathrm{dist}(G(\rho\zeta),\varkappa(\zeta,\overline{\mathbb D}))<\epsilon$ for all $\zeta\in \mathbb{T}$ and all
$\rho\in [\rho',1)$, and
\item $G$ is $\epsilon$-close to $F$ in the $\mathscr{C}^1$ topology on $\{\zeta\in\mathbb{C}\colon |\zeta|\leq \rho'\}$.
\end{enumerate}
Furthermore, if $I$ is a compact arc in $\mathbb{T}$ such that the function $r$ vanishes on $\mathbb{T}\setminus I$
and $U$ is an open neighborhood of $I$ in $\overline{\mathbb D}$, then in addition to the above
\begin{enumerate}[\it i)]
\item[\it iv)] one can choose $G$ to be $\epsilon$-close to $F$ in the $\mathscr{C}^1$ topology
on $\overline{\mathbb D}\setminus U$.
\end{enumerate}
Moreover, given an upper semicontinuous function $\phi \colon\mathbb{C}^n\to \mathbb{R}\cup\{-\infty\}$
and a closed arc $I\subset \mathbb{T}$, we may achieve in addition that
\begin{equation}
\label{small-increase2}
\int_{I} \phi \bigl( G(\E^{\imath t})\bigr) \, \frac{dt}{2\pi} \le
\int^{2\pi}_0 \!\! \int_{I} \phi \bigl(\varkappa(\E^{\imath t},\E^{\imath s})\bigr)
\frac{dt}{2\pi} \frac{ds}{2\pi} + \epsilon
\end{equation}
\end{lemma}
\begin{remark}
For every $\zeta\in\mathbb{T}$ the map $\overline{\mathbb D}\ni \xi \mapsto \sigma(\zeta,\xi)\mathbf{u}$ in the lemma is an
immersed holomorphic disc directed by the null vector $\mathbf{u}\in\mathfrak{A}_*$; the nonnegative function
$r(\zeta)\ge 0$ is used to rescale it. If the support of $r$ is contained in a proper subarc $I$ of the circle
$\mathbb{T}$ then it suffices to assume that $\sigma(\zeta,\xi)$ is defined for $\zeta\in I$, and in this
case the winding number condition on the function $ \frac{\di \sigma}{\di \xi}(\zeta,\xi)\ne 0 $ is irrelevant.
Conditions (\ref{eq:small-increase}) and (\ref{small-increase2}) are not used in this paper, but
they will be used in the envisioned applications to minimal hulls.
\end{remark}
\begin{proof}
Write $\mathfrak{A}=\mathfrak{A}^{n-1}$ and fix a pair of null vectors
$\mathbf{u},\mathbf{v}\in\mathfrak{A}_*$ as in the lemma. Given a vector $\mathbf{w}\in \mathfrak{A}_*$ such that the triple
$(\mathbf{u},\mathbf{v},\mathbf{w})$ satisfies condition (\ref{eq:nondeg}), we denote by
$\psi_\mathbf{w} \colon \mathbb{C}^2\to \mathfrak{A}$ the map $\psi_{(\mathbf{u},\mathbf{v},\mathbf{w})}$ (\ref{eq:psi});
hence by (\ref{eq:normalization}) we have that
\[
\psi_\mathbf{w} (\mathbf{e}_1) = \mathbf{u},\quad \psi_\mathbf{w} (\mathbf{e}_2) =\mathbf{v}.
\]
Recall that $f=F'\colon \overline{\mathbb D} \to \mathfrak{A}_*$ is a map of class $\mathscr{A}(\mathbb D)$, and
condition (\ref{eq:nondeg2}) implies that the triple of null vectors $(\mathbf{u},\mathbf{v},f(\zeta))$
satisfies condition (\ref{eq:nondeg}) for every $\zeta\in \overline{\mathbb D}$.
Due to simple connectivity of the disc the coefficients of the matrix function $B$
(\ref{eq:B}) are well defined functions of class $\mathscr{A}(\mathbb D)$, and there is a holomorphic map
$h=(u,v)\colon \overline{\mathbb D}\to \mathbb{C}^2_*$ satisfying
\begin{equation}\label{eq:h}
\psi_{f(\zeta)}(h(\zeta))= f(\zeta),\quad \zeta\in\overline{\mathbb D}.
\end{equation}
The conditions on the functions $r\ge 0$ and $\sigma$ ensure that the partial derivative
\[
\mu_2(\zeta,\xi):=
\frac{\di \mu}{\di \xi}(\zeta,\xi) = r(\zeta)\, \frac{\di \sigma}{\di \xi}(\zeta,\xi)
\]
admits a continuous square root
\[
\eta(\zeta,\xi) = \sqrt{\mu_2(\zeta,\xi)}, \qquad (\zeta,\xi)\in \mathbb{T}\times\overline{\mathbb D}
\]
that is holomorphic in $\xi\in\mathbb D$ for every fixed $\zeta\in\mathbb{T}$. We can approximate $\eta$
as closely as desired in the sup norm on $\mathbb{T}\times\overline{\mathbb D}$ by a rational function of the form
\begin{equation}\label{eq:tilde-eta2}
\tilde\eta(\zeta,\xi) = \sum_{j=0}^l B_j(\zeta)\, \xi^j
\end{equation}
where every $B_j(\zeta)$ is a Laurent polynomial with the only pole at $\zeta=0$. Set
\begin{equation}\label{eq:tilde-mu2}
\tilde\mu(\zeta,z) = \int_0^z \tilde\eta(\zeta,\xi)^2 \,d\xi
= \sum_{k=1}^m A_k(\zeta) \, z^k
\end{equation}
where $m=2l+1$ and $A_k(\zeta)$ are Laurent polynomials with the only pole at $\zeta=0$.
Then $\tilde\mu$ is uniformly close to $\mu$ on $\mathbb{T}\times\overline{\mathbb D}$, and it suffices to prove
the lemma with $\mu$ replaced by $\tilde \mu$.
We now drop the tildes and assume that the functions $\eta$ and $\mu$
are given by (\ref{eq:tilde-eta2}) and (\ref{eq:tilde-mu2}), respectively.
For every point $c=\E^{\imath\phi}\in\mathbb{T}$ with $\phi\in [0,2\pi)$ we set
$\sqrt c = \E^{\imath \phi/2}$ and consider the sequence of functions
\[
g_N(\zeta,c) = \sqrt{c} \, \sqrt{2N+1}\, \zeta^N \eta(\zeta,c\, \zeta^{2N+1}).
\]
(Formally this coincides with (\ref{eq:gN}), except that $g_N$ is now scalar-valued.)
Note that $g_N$ is a holomorphic polynomial in $\zeta\in \mathbb{C}$ for every sufficiently big $N$ and
\[
\int_0^z g_N(\zeta,c)^2 \,d\zeta = \int_0^z c (2N+1)\zeta^{2N} \mu_2(\zeta,c\, \zeta^{2N+1})\, d\zeta
\]
where $\mu_2(\zeta,\xi)=\frac{\di \mu}{\di \xi}(\zeta,\xi)$.
By Lemma \ref{lem:estimate} we have
\begin{equation}\label{eq:estimate-gN}
\lim_{N\to\infty} \sup_{|z|\le 1,\, c\in\mathbb{T}} \left |
\int_0^z g_N(\zeta,c)^2 \,d\zeta - \mu(z,c z^{2N+1}) \right | =0.
\end{equation}
For every sufficiently big $N \in \mathbb{N}$ we define the map
$h_N=(u_N,v_N)\colon \overline{\mathbb D}\times\mathbb{T}\to\mathbb{C}^2$ by
\[
h_N(\zeta,c)= h(\zeta) + g_N(\zeta,c) \,\mathbf{e}_1 = \bigl( u(\zeta) + g_N(\zeta,c), v(\zeta) \bigr).
\]
By general position, moving $f$ and hence $h$ slightly, we can ensure that
$h_N(\zeta,c)\ne (0,0)$ for all $(\zeta,c)\in \overline{\mathbb D}\times\mathbb{T}$ and all sufficiently big $N\in \mathbb{N}$.
Indeed, we claim that this holds as long as $v$ has no zeros on $\mathbb{T}\times \mathbb{T}$. Under this
assumption we have that $v(\zeta)\ne 0$ in an annulus $\rho_1 \le |\zeta|\le 1$
for some $\rho_1 <1$, whence $h_N(\zeta,c) \ne (0,0)$ for such $\zeta$ and for any $N\in \mathbb{N}$
and $c\in\mathbb{T}$. Since $h_N \to h$ uniformly on
$\{|\zeta|\le \rho_1\}\times\mathbb{T}$ as $N\to +\infty$
and $h$ does not assume the value $(0,0)$, the same is true for $h_N$
for all sufficiently big $N\in \mathbb{N}$.
By the definition of $\psi_{f(\zeta)}$ the map
\[
f_N(\zeta,c) := \psi_{f(\zeta)}\bigl(h_N(\zeta,c)\bigr) \in \mathbb{C}^n,
\quad (\zeta,c)\in \overline{\mathbb D}\times\mathbb{T}
\]
has range in the punctured null quadric $\mathfrak{A}_*$, and hence the map
\[
\overline{\mathbb D}\ni z\mapsto F_N(z,c) = F(0)+ \int_0^z f_N(\zeta,c)\, d\zeta \in \mathbb{C}^n
\]
is an immersed holomorphic null disc for every $c\in\mathbb{T}$ and for all sufficiently big $N\in \mathbb{N}$.
We claim that if $N\in\mathbb{N}$ is chosen big enough then the null disc
$G=F_N(\cdot,c)$ satisfies Lemma \ref{lem:RH}
for a suitable choice of the constant $c=c_N \in\mathbb{T}$.
Indeed, since all maps in the definition of $\psi_{f(\zeta)}$ are
either linear (given by a product with the matrices $A^{-1}$ and $B$)
or homogeneous quadratic (the projection $\pi$ given by (\ref{eq:pi})), we infer that
\begin{equation}\label{eq:psi-f}
\psi_{f(\zeta)} \left( g_N(\zeta,c) \mathbf{e}_1\right) =
g_N(\zeta,c)^2 \, \mathbf{u}, \quad (\zeta,c)\in \overline{\mathbb D}\times\mathbb{T}.
\end{equation}
As we also have $\psi_{f(\zeta)}(h(\zeta))=f(\zeta)$ by (\ref{eq:h}), we get for
$(\zeta,c)\in \overline{\mathbb D}\times\mathbb{T}$ that
\begin{equation}\label{eq:fN}
f_N(\zeta,c) = \psi_{f(\zeta)}(h_N(\zeta,c)) = f(\zeta) + g_N(\zeta,c)^2 \, \mathbf{u} + R_N(\zeta,c).
\end{equation}
In order to estimate the remainder $R_N(\zeta,c)$
we observe that the terms in $f_N(\zeta,c)$ are of three different kinds as follows:
\begin{itemize}
\item[\rm (a)] Terms which contain $u^2$, $v^2$ or $uv$ (where $h=(u,v)$);
the sum of all such terms equals $f(\zeta)$ in view of (\ref{eq:h}).
\item[\rm (b)] Terms which do not contain any component $u,v$ of $h$;
the sum of all such terms equals $g_N(\zeta,c)^2 \, \mathbf{u}$ in view of (\ref{eq:psi-f}).
\item[\rm (c)] Terms which contain exactly one component $u,v$ of $h$
and exactly one copy of the function $g_N(\zeta,c)$.
All such terms are placed in the remainder $R_N$.
\end{itemize}
The terms described above are multiplied by various elements of the matrices $A^{-1}$ (\ref{eq:A}) and
$B$ (\ref{eq:B}); those are functions in $\mathscr{A}(\mathbb D)$ depending on $f$ and $h$
but not depending on $N$. By integrating the equation (\ref{eq:fN})
we see that the map $F_N(z,c)$ is of the form (\ref{eq:FN}) and the remainder
$E_N(z,c)$ satisfies (\ref{eq:est-EN}) as is seen from the estimates (\ref{eq:estimate-gN}) and (\ref{eq:est-gN}).
\end{proof}
By using Lemmas \ref{lem:RH3} and \ref{lem:RH} we can prove the following result which gives an approximate
solution to the Riemann-Hilbert problem for bordered Riemann surfaces as null curves
in $\mathbb{C}^n$. The case $n=3$ corresponds to \cite[Theorem 3.4]{AF2}.
The proof given there extends directly to the present situation, replacing
\cite[Lemma 3.1]{AF2} with Lemma \ref{lem:RH} above.
\begin{theorem}[{\bf Riemann-Hilbert problem for null curves in $\mathbb{C}^n$}]\label{th:RH}
Fix an integer $n\ge 3$ and let $\mathfrak{A}=\mathfrak{A}^{n-1}\subset\mathbb{C}^n$ denote the null quadric (\ref{eq:Agot}).
Let $M$ be a compact bordered Riemann surface with boundary $bM\ne\emptyset$, and let $I_1,\ldots,I_k$ be a finite collection of pairwise disjoint compact subarcs of $bM$ which are not connected components of $bM$. Choose a thin annular neighborhood
$A\subset M$ of $bM$ and a smooth retraction $\rho\colon A\to bM$. Assume that
\begin{itemize}
\item $F\colon M \to\mathbb{C}^n$ is a null holomorphic immersion of class $\mathscr{A}^1(M)$,
\item $\mathbf{u}_1,\ldots,\mathbf{u}_k \in \mathfrak{A}_*=\mathfrak{A}\setminus\{0\}$ are null vectors (the {\em direction vectors}),
\item $r \colon bM \to \mathbb{R}_+$ is a continuous nonnegative function supported on $I:=\bigcup_{i=1}^k I_i$,
\item $\sigma \colon I \times\overline{\mathbb D}\to\mathbb{C}$ is a function of class $\mathscr{C}^1$ such that
for every $\zeta\in I$ the function $\overline{\mathbb D}\ni \xi \mapsto \sigma(\zeta,\xi)$ is holomorphic on $\mathbb D$,
$\sigma(\zeta,0)=0$, and the partial derivative $\frac{\di \sigma}{\di \xi}$ is nowhere vanishing on $I \times \overline{\mathbb D}$.
\end{itemize}
Consider the continuous map $\varkappa\colon bM \times\overline{\mathbb D}\to\mathbb{C}^n$ given by
\[
\varkappa(\zeta ,\xi)=\left\{\begin{array}{ll}
F(\zeta ); & \zeta \in bM\setminus I \\
F(\zeta ) + r(\zeta) \sigma(\zeta,\xi) \mathbf{u}_i; & \zeta \in I_i,\; i\in\{1,\ldots,k\}.
\end{array}\right.
\]
Given a number $\epsilon>0$ there exists an arbitrarily small open neighborhood $\Omega\subset M$ of
$I=\bigcup_{i=1}^k I_i$ and a null holomorphic immersion $G\colon M\to\mathbb{C}^n$ of class $\mathscr{A}^1(M)$
satisfying the following properties:
\begin{enumerate}[\it i)]
\item $\mathrm{dist}(G(\zeta ),\varkappa(\zeta ,\mathbb{T}))<\epsilon$ for all $\zeta \in bM$.
\item $\mathrm{dist}(G(\zeta ),\varkappa(\rho(\zeta ),\overline{\mathbb D}))<\epsilon$ for all $\zeta \in \Omega$.
\item $G$ is $\epsilon$-close to $F$ in the $\mathscr{C}^1$ topology on $M\setminus \Omega$.
\end{enumerate}
\end{theorem}
\begin{proof}
Recall that $\Theta$ is the bilinear form (\ref{eq:bilinear}) on $\mathbb{C}^n$.
For each index $i\in \{1,\ldots,k\}$ we choose a null vector $\mathbf{v}_i\in \mathfrak{A}_*$
such that $\Theta(\mathbf{u}_i,\mathbf{v}_i)\ne 0$. Pick a holomorphic $1$-form $\theta$ without zeros on $M$.
Then $dF=f\theta$ where $f\colon M\to \mathfrak{A}_*$ is a map of class $\mathscr{A}(M)$.
Deforming $F$ slightly if needed we may assume that $f$ is nondegenerate
(see Sec.\ \ref{sec:prelim}). By a small deformation of the pairs of null vectors $(\mathbf{u}_i,\mathbf{v}_i)$
we may also assume that the following conditions hold on each of the arcs $I_i$ (cf.\ (\ref{eq:nondeg2})):
\[
\Theta(\mathbf{u}_i,f(\zeta ))\ne 0\quad \text{and}\quad \Theta(\mathbf{v}_i,f(\zeta ))\ne 0 \quad \text{for all } \zeta\ \in I_i.
\]
By continuity there is a neighborhood $U_i\subset A\subset M$ of the arc $I_i$ such that
the same conditions hold for all $\zeta \in U_i$.
By \cite[Lemma 5.1]{AF1} there exists a {\em spray of maps} $f_w\colon M\to \mathfrak{A}_*$ of class $\mathscr{A}(M)$,
depending holomorphically on a parameter $w$ in a ball $B\subset \mathbb{C}^N$
for some big integer $N$ and satisfying the following properties:
\begin{itemize}
\item[\rm (a)] the spray is {\em dominating}, i.e., the partial differential
$
\frac{\di}{\di w}\big|_{w=0} f_w(\zeta ) \colon \mathbb{C}^N\to \mathbb{C}^n
$
is surjective for every $\zeta \in M$.
\item[\rm (b)] The spray is {\em period dominating} in the following sense.
Let $C_1,\ldots, C_l$ be smooth closed curves in $\mathring M$ which form the basis
of the homology group $H_1(M;\mathbb{Z})$. Then the period map
$\mathcal{P}=(\mathcal{P}_1,\ldots,\mathcal{P}_l)\colon B \to (\mathbb{C}^n)^l$ with the components
\[
\mathcal{P}_j(w) = \int_{C_j} f_w\theta \in \mathbb{C}^n, \quad w\in B,\ j=1,\ldots,l
\]
has the property that $\frac{\di}{\di w}\big|_{w=0} \mathcal{P}(w): \mathbb{C}^N\to (\mathbb{C}^n)^l$ is surjective.
\end{itemize}
For each $i=1,\ldots, l$ we pick a compact, smoothly bounded, simply connected domain
$D_i$ in $M$ (a disc) such that $D_i\subset U_i$, $D_i$ contains a neighborhood of the arc $I_i$
in $M$, and $D_i\cap D_j=\emptyset$ for $1\le i\ne j\le l$.
Since the curves $C_j$ lie in the interior of $M$, we can choose the $D_i$'s small enough such that
$\bigcup_{j=1}^l C_j \cap \bigcup_{i=1}^k D_i=\emptyset$. For every $i=1,\ldots,l$
the function $\sigma(\zeta,\xi)$ can be extended to $(\zeta,\xi) \in bD_i\times \overline{\mathbb D}$ such that conditions of
Lemma \ref{lem:RH} are fulfilled on the disc $D_i$,
and the function $r$ extends to $bD_i$ such that it vanishes on $bD_i\setminus I_i$.
Under these conditions we can apply Lemma \ref{lem:RH} on each disc $D_i$
to approximate the restricted spray $f_w|_{D_i}$ as closely as desired,
uniformly on $D_i\setminus V_i$ for a small neighborhood $V_i\subset D_i$ of the arc $I_i$,
by a holomorphic spray $g_{i,w}\colon D_i\to\mathfrak{A}_*$ of class $\mathscr{A}(D_i)$ such that the integrals
$G_{i,w}(\zeta ) = \int^\zeta g_{i,w}\theta$ $(\zeta \in D_i,\ w\in B)$
with suitably chosen initial values at some point $p_i\in D_i$ satisfy the conclusion of Lemma \ref{lem:RH}.
(To be precise, we need a parametric version of Lemma \ref{lem:RH} with a holomorphic dependence
on the parameter $w$ of the spray. It is clear that the proof of Lemma \ref{lem:RH}
gives this without any changes.)
Assuming that these approximations are close enough, the domination property (a) of the
spray $f_{w}$ allow us to glue the sprays $f_w$ and $g_{i,w}$ for $i=1,\ldots, l$
into a new holomorphic spray $\tilde g_w\colon M\to \mathfrak{A}_*$ which approximates
$f_w$ very closely on $M\setminus \bigcup_{i=1}^l V_i$. (The parameter ball $B\subset\mathbb{C}^N$
shrinks a little. For the details and references regarding this gluing see \cite[Theorem 3.4]{AF2}.)
The period domination property of $f_w$ (condition (b) above) implies
that there exists $w_0\in B$ close to $0$ such that the map
$g=\tilde g_{w_0}\colon M\to \mathfrak{A}_*$ has vanishing periods over the curves $C_j$.
The holomorphic map $G\colon M\to \mathbb{C}^n$, defined by
$G(\zeta ) = F(p)+\int_p^\zeta g\theta$ $(\zeta \in M$) for a fixed initial point $p\in M$,
is then a null holomorphic immersion satisfying Theorem \ref{th:RH} provided that all
approximations were sufficiently close.
\end{proof}
By adapting Theorem \ref{th:RH} to conformal minimal immersions we obtain the following.
\begin{theorem}[\bf Riemann-Hilbert problem for conformal minimal immersions in $\mathbb{R}^n$]
\label{th:RHCMI}
Assume that $n\ge 3$ and the data $M$, $I_1,\ldots,I_k\subset bM$, $I=\bigcup_{i=1}^k I_i$, $r \colon bM \to \mathbb{R}_+$,
$\sigma\colon I \times \overline{\mathbb D}\to\mathbb{C}$, $A\subset M$ and $\rho\colon A\to bM$ are as in Theorem \ref{th:RH}.
Let $F\in\CMI^1(M,\mathbb{R}^n)$. For each $i=1,\ldots, k$ let $\mathbf{u}_i,\mathbf{v}_i\in \mathbb{R}^n$ be a pair of orthogonal vectors
satisfying $\|\mathbf{u}_i\|=\|\mathbf{v}_i\|>0$. Consider the continuous map
$\varkappa\colon bM \times\overline{\mathbb D}\to\mathbb{R}^n$ given by
\[
\varkappa(\zeta ,\xi)=\left\{\begin{array}{ll}
F(\zeta ), & \zeta \in bM\setminus I; \\
F(\zeta ) + r(\zeta )\bigl( \Re \sigma(\zeta,\xi) \mathbf{u}_i+ \Im \sigma(\zeta,\xi) \mathbf{v}_i \bigr), &
\zeta \in I_i,\; i\in\{1,\ldots,k\}.
\end{array}\right.
\]
Given a number $\epsilon>0$ there exist an arbitrarily small open neighborhood $\Omega\subset M$ of
$I=\bigcup_{i=1}^k I_i$ and a conformal minimal immersion $G\in\CMI^1(M,\mathbb{R}^n)$
satisfying the following properties:
\begin{enumerate}[\it i)]
\item $\mathrm{dist}(G(\zeta ),\varkappa(\zeta ,\mathbb{T}))<\epsilon$ for all $\zeta \in bM$.
\item $\mathrm{dist}(G(\zeta ),\varkappa(\rho(\zeta ),\overline{\mathbb D}))<\epsilon$ for all $\zeta \in \Omega$.
\item $G$ is $\epsilon$-close to $F$ in the $\mathscr{C}^1$ norm on $M\setminus \Omega$.
\item $\mathrm{Flux} (G)=\mathrm{Flux} (F)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The conditions on $\mathbf{u}_i,\mathbf{v}_i\in\mathbb{R}^n$ imply that
$\wt \mathbf{u}_i=\mathbf{u}_i-\imath \mathbf{v}_i\in\mathfrak{A}_*$ is a null vector for every $i=1,\ldots,k$.
Pick a holomorphic $1$-form $\theta$ without zeros on $M$. Then
$\di F=f\theta$ where $f\colon M\to \mathfrak{A}_*$ is a holomorphic map of class $\mathscr{A}(M)$.
We apply the proof of Theorem \ref{th:RH} to the map $f$
and the null vectors $\wt \mathbf{u}_i$, but with the following difference.
At the very last step of the proof we can argue that we obtain a holomorphic
map $g=\tilde g_{w_0}\colon M\to \mathfrak{A}_*$ in the new spray
(for some value of the parameter $w_0\in \mathbb{C}^N$ close to $0$) such that
\[
\int_{C_j} g\theta = \int_{C_j} f\theta = \int_{C_j}\di F; \quad j=1,\ldots,l.
\]
Since $\int_{C_j}2\Re (\di F) = \int_{C_j} dF=0$, the real periods of $g$ vanish
and the imaginary periods equal those of $f$. Hence the map $G\colon M\to\mathbb{R}^n$,
given by $G(\zeta )=F(p_0)+\int_{p_0}^\zeta \Re(g\theta)$
for some fixed $p_0\in M$, is a conformal minimal immersion with $\mathrm{Flux}(G)=\mathrm{Flux}(F)$
(property {\it iv)}). Properties {\it i)--iii)} of $G$ follow from
the corresponding properties of the map $\int^\zeta g\theta$
on each disc $D_i\subset M$ constructed in the proof of Theorem \ref{th:RH}.
(In fact, with a correct choice of initial values we have $G(\zeta ) =\int^\zeta \Re(g\theta)$
for $\zeta \in D_i$, $i=1,\ldots,k$.)
\end{proof}
\section{Complete minimal surfaces bounded by Jordan curves}\label{sec:Jordan}
In this section we prove Theorem \ref{th:Jordan}. The key in the proof is the following lemma
which asserts that every conformal minimal immersion $M\to\mathbb{R}^n$ of a compact bordered Riemann surface
can be approximated as close as desired in the $\mathscr{C}^0(M)$ topology by conformal minimal immersions
with arbitrarily large intrinsic diameter.
\begin{lemma}\label{lem:Jordan2}
Let $M$ be a compact bordered Riemann surface, let $n\geq 3$ be a natural number, and let $G\in\CMI_*^1(M,\mathbb{R}^n)$.
Given a point $p_0\in \mathring M$ and a number $\lambda>0$, we can approximate $G$ arbitrarily closely
in the $\mathscr{C}^0(M)$ topology by a conformal minimal immersion $\wh G\in\CMI_*^1(M,\mathbb{R}^n)$ such that
$\mathrm{dist}_{\wh G}(p_0,bM)>\lambda$ and $\mathrm{Flux}(\wh G)=\mathrm{Flux}(G)$.
\end{lemma}
The notation $\CMI_*^1(M)$ has been introduced in Sec.\ \ref{sec:prelim}. Since conformal minimal immersions are
harmonic, the approximation in the above lemma takes place in the $\mathscr{C}^r$ topology on compact subsets of $\mathring M$
for all $r\in\mathbb{Z}_+$. However, if $\lambda>\mathrm{dist}_G(p_0,bM)$ then the approximation in the $\mathscr{C}^1(M)$ topology is clearly impossible.
Lemma \ref{lem:Jordan2} will follow by a standard recursive application of the following result.
\begin{lemma}\label{lem:Jordan1}
Let $M$ be a compact bordered Riemann surface and let $n\geq 3$.
Consider $F\in\CMI^1(M,\mathbb{R}^n)$, a smooth map $\mathfrak{Y}\colon bM\to\mathbb{R}^n$, and a number $\delta>0$ such that
\begin{equation}\label{eq:lemJordan1}
\|F-\mathfrak{Y}\|_{0,bM}<\delta.
\end{equation}
Fix a point $p_0\in \mathring M$ and choose a number $d>0$ such that
\begin{equation}\label{eq:d}
0<d<\mathrm{dist}_F(p_0,bM).
\end{equation}
Then for each $\eta>0$ the map $F$ can be approximated uniformly on compacts in $\mathring M$ by nondegenerate
conformal minimal immersions $\wh F\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying the following properties:
\begin{enumerate}[\rm (a)]
\item $\|\wh F-\mathfrak{Y}\|_{0,bM}<\sqrt{\delta^2+\eta^2}$.
\item $\mathrm{dist}_{\wh F}(p_0,bM)> d+\eta$.
\item $\mathrm{Flux}(\wh F)=\mathrm{Flux}(F)$.
\end{enumerate}
\end{lemma}
The key idea in the proof of Lemma \ref{lem:Jordan1} is to push the $F$-image of each point $p\in bM$
a distance approximately $\eta$ in a direction approximately orthogonal to $F(p)-\mathfrak{Y}(p)\in\mathbb{R}^n$.
Conditions {\rm (a)} and {\rm (b)} will then follow from \eqref{eq:lemJordan1}, \eqref{eq:d}, and Pythagoras' Theorem.
The main improvement of Lemmas \ref{lem:Jordan2} and \ref{lem:Jordan1} with respect to previous related results in the literature
is that we do not change the conformal structure on the source bordered Riemann surface $M$. This particular point is the key that
allows us to ensure that the complete minimal surfaces constructed in Theorem \ref{th:Jordan} are bounded by Jordan curves.
\begin{proof}[Proof of Lemma \ref{lem:Jordan1}]
We assume that $M$ is a smoothly bounded compact domain in an open Riemann surface $\wt M$. Furthermore,
in view of the Mergelyan theorem for conformal minimal immersions into $\mathbb{R}^n$
(Theorems 3.1 and 5.3 in \cite{AFL}) we may also assume that $F$ extends to $\wt M$
as a conformal minimal immersion in $\CMI_*(\wt M,\mathbb{R}^n)$.
Fix a number $\epsilon>0$ and a compact set $K\Subset M$.
To prove the lemma, we will construct a nondegenerate conformal minimal immersion
$\wh F\in\CMI_*^1(M,\mathbb{R}^n)$ which is $\epsilon$-close to $F$ in the $\mathscr{C}^1(K)$ norm and
satisfies conditions {\rm (a)}, {\rm (b)}, and {\rm (c)}.
Enlarging $K$ if necessary we assume that $K$ is a smoothly bounded compact domain in $\mathring M$
which is a strong deformation retract of $M$, that $p_0\in\mathring K$, and (see \eqref{eq:d}) that
\begin{equation}\label{eq:>tau}
\mathrm{dist}_F(p_0,bK)>d.
\end{equation}
By general position we may also assume that
\begin{equation}\label{eq:neq}
\text{the map $F-\mathfrak{Y}$ does not vanish anywhere on $bM$.}
\end{equation}
Denote by $\alpha_1,\ldots,\alpha_k$ the connected components of $bM$
and recall that every $\alpha_i$ is a smooth Jordan curve in $\wt M$.
Fix a number $\epsilon_0>0$ which will be specified later.
By \eqref{eq:lemJordan1} and the continuity of $F$ and $\mathfrak{Y}$ there exist a natural number $l\geq 3$ and compact connected
subarcs $\{\alpha_{i,j}\subset\alpha_i\colon (i,j)\in \mathtt{I}:=\{1,\ldots,k\}\times\mathbb{Z}_l\}$ (here $\mathbb{Z}_l=\mathbb{Z}/l\mathbb{Z}$) such that
\begin{equation}\label{eq:alpha}
\bigcup_{j\in\mathbb{Z}_l}\alpha_{i,j}=\alpha_i, \quad i\in\{1,\ldots,k\},
\end{equation}
and, for every $(i,j)\in \mathtt{I}$,
the arcs $\alpha_{i,j}$ and $\alpha_{i,j+1}$ have a common endpoint $p_{i,j}$ and are otherwise disjoint,
$\alpha_{i,j}\cap\alpha_{i,a}=\emptyset$ for all $a\in\mathbb{Z}_l\setminus\{j-1,j,j+1\}$,
\begin{equation}\label{eq:Y-Y}
\text{$\|\mathfrak{Y}(p)-\mathfrak{Y}(q)\|<\epsilon_0$ for all $\{p,q\}\subset \alpha_{i,j}$,}
\end{equation}
and
\begin{equation}\label{eq:F-Yalpha}
\text{$\|F(p)-\mathfrak{Y}(q)\|<\delta$ and $\|F(p)-F(q)\|<\epsilon_0$ for all $\{p,q\}\subset \alpha_{i,j}$}.
\end{equation}
For every $(i,j)\in \mathtt{I}$ we denote by
\begin{equation}\label{eq:piij}
\pi_{i,j}\colon\mathbb{R}^n\to {\rm span}\bigl\{F(p_{i,j})-\mathfrak{Y}(p_{i,j})\bigl\}\, \subset\mathbb{R}^n
\end{equation}
the orthogonal projection onto the affine real line ${\rm span}\{F(p_{i,j})-\mathfrak{Y}(p_{i,j})\}$; cf.\ \eqref{eq:neq}.
The first main step in the proof consists of perturbing $F$ near the points $\{p_{i,j}\colon (i,j)\in \mathtt{I}\}$ in order to find a conformal
minimal immersion in $\CMI_*^1(M,\mathbb{R}^n)$ which is close to $F$ in the $\mathscr{C}^1(K)$ topology and the distance between $p_0$ and
$\{p_{i,j}\colon (i,j)\in \mathtt{I}\}$ in the induced metric is large in a suitable way. This deformation procedure is enclosed in the following result.
\begin{lemma}\label{lem:step1}
Given a number $\epsilon_1>0$, there exists a nondegenerate conformal minimal immersion
$F_0\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying the following properties:
\begin{enumerate}[\rm ({P}1)]
\item $F_0$ is $\epsilon_1$-close to $F$ in the $\mathscr{C}^1(K)$ topology.
\item $\|F_0(p)-\mathfrak{Y}(q)\|<\delta$ and $\|F_0(p)-F(q)\|<\epsilon_0$ for all $\{p,q\}\subset\alpha_{i,j}$,
for all $(i,j)\in \mathtt{I}$.
\item $\mathrm{Flux}(F_0)=\mathrm{Flux}(F)$.
\item For every $(i,j)\in I$ there exists a small open neighborhood $U_{i,j}$ of $p_{i,j}$ in $M$, with $\overline U_{i,j}\cap K=\emptyset$, fulfilling the following condition: If $\gamma\subset M$ is an arc with initial point in $K$ and final point in $\overline U_{i,j}$,
and if $\{J_{a,b}\}_{(a,b)\in \mathtt{I}}$ is any partition of $\gamma$ by Borel measurable subsets, then
\[
\sum_{(a,b)\in \mathtt{I}} \mathrm{length} \, \pi_{a,b}(F_0(J_{a,b})) >\eta,
\]
where $\eta>0$ is the real number given in the statement of Lemma \ref{lem:Jordan1} and $\pi_{a,b}$ are the
projections in \eqref{eq:piij}, $(a,b)\in \mathtt{I}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Choose a family of pairwise disjoint Jordan arcs $\{\gamma_{i,j}\subset \wt M\colon (i,j)\in \mathtt{I}\}$ such that
each $\gamma_{i,j}$ contains $p_{i,j}$ as an endpoint, is attached transversely to $M$ at $p_{i,j}$,
and is otherwise disjoint from $M$. The set
\begin{equation}\label{eq:S}
S:=M\cup\bigcup_{(i,j)\in \mathtt{I}} \gamma_{i,j} \subset\wt M
\end{equation}
is {\em admissible} in $\wt M$ the sense of \cite[Def.\ 5.1]{AFL}.
Take a smooth map $u\colon S\to\mathbb{R}^n$ satisfying the following properties:
\begin{enumerate}[\rm (i)]
\item $u=F$ on a neighborhood of $M$.
\item $\|u(x)-\mathfrak{Y}(q)\|<\delta$ and $\|u(x)-F(q)\|<\epsilon_0$ for all
$(x,q)\in (\gamma_{i,j-1}\cup \alpha_{i,j}\cup \gamma_{i,j})\times \alpha_{i,j}$, for all $(i,j)\in \mathtt{I}$.
\item If $\{J_{a,b}\}_{(a,b)\in \mathtt{I}}$ is a partition of $\gamma_{i,j}$ by Borel measurable subsets, then
\[
\sum_{(a,b)\in \mathtt{I}} \mathrm{length}\, \pi_{a,b}(u(J_{a,b})) > 2\eta.
\]
\end{enumerate}
Notice that condition \eqref{eq:F-Yalpha} allows one to choose a map $u$ satisfying {\rm (i)} and {\rm (ii)}.
To ensure also {\rm (iii)}, one can simply choose $u$ over each arc $\gamma_{i,j}$ to be highly oscillating in the direction
of $F(p_{a,b})-\mathfrak{Y}(p_{a,b})$ for all $(a,b)\in\mathtt{I}$, but with sufficiently small extrinsic diameter so that {\rm (ii)}
remains to hold. (Recall \eqref{eq:neq} and \eqref{eq:piij} and take into account that $\mathtt{I}$ is finite.)
Let $\theta$ be a nowhere vanishing holomorphic $1$-form on $\wt M$ (such exists by the Oka-Grauert principle;
cf.\ Theorem 5.3.1 in \cite[p.\ 190]{F2011}). It is then easy to find a smooth function $f\colon S\to\mathfrak{A}_*^{n-1}$,
holomorphic in a neighborhood of $M$,
such that the pair $(u,f\theta)$ is a {\em generalized conformal minimal immersion} on the set $S$ (\ref{eq:S})
in the sense of \cite[Def.\ 5.2]{AFL}.
Fix a number $\epsilon_2>0$ which will be specified later.
Since $M$ is a strong deformation retract of $S$ (\ref{eq:S}) and taking {\rm (i)} into account,
the Mergelyan theorem for conformal minimal immersions into $\mathbb{R}^n$ \cite[Theorem 5.3]{AFL} furnishes a
conformal minimal immersion $G\in\CMI_*^1(\wt M,\mathbb{R}^n)$ such that
\begin{enumerate}[\rm ({P}1)]
\item[\rm (iv)] $G$ is $\epsilon_2$-close to $u$ in the $\mathscr{C}^0(S)$ and the $\mathscr{C}^1(M)$ topologies, and
\item[\rm (v)] $\mathrm{Flux}(G)=\mathrm{Flux}(F)$.
\end{enumerate}
Let $V\subset \wt M$ be a small open neighborhood of $S$. For every $(i,j)\in \mathtt{I}$ let $q_{i,j}$ denote
the endpoint of $\gamma_{i,j}$ different from $p_{i,j}$. If $\epsilon_2>0$ is small enough,
then properties {\rm (ii)} and {\rm (iii)} guarantee the existence of small open neighborhoods
$W'_{i,j}\Subset W_{i,j}$ of $p_{i,j}$ and $V_{i,j}$ of $\gamma_{i,j}$ in $V \setminus K$, $(i,j)\in \mathtt{I}$,
satisfying the following conditions:
\begin{enumerate}[\rm ({P}1)]
\item[\rm (vi)] $V_{i,j}\cap M\Subset W'_{i,j}\Subset W_{i,j}\Subset V\setminus K$.
\item[\rm (vii)] $\|G(x)-\mathfrak{Y}(q)\|<\delta$ and $\|G(x)-F(q)\|<\epsilon_0$ for all
$(x,q)\in (W_{i,j-1}\cup V_{i,j-1}\cup \alpha_{i,j}\cup W_{i,j}\cup V_{i,j})\times \alpha_{i,j}$, for all $(i,j)\in \mathtt{I}$.
\item[\rm (viii)] If $\gamma'_{i,j}\subset W_{i,j}\cup V_{i,j}$ is an arc with initial point in $W_{i,j}$ and final point $q_{i,j}$,
and if $\{J_{a,b}\}_{(a,b)\in \mathtt{I}}$ is a partition of $\gamma'_{i,j}$ by Borel measurable subsets, then
\[
\sum_{(a,b)\in \mathtt{I}} \mathrm{length}\, \pi_{a,b}(G(J_{a,b})) >2\eta.
\]
\end{enumerate}
Without loss of generality we assume in addition that the compact sets $\overline W_{i,j}\cup\overline V_{i,j}$, $(i,j)\in \mathtt{I}$,
are pairwise disjoint.
By \cite[Theorem 2.3]{FW0} (see also \cite[Theorem 8.8.1]{F2011}) there exists a smooth diffeomorphism $\phi\colon M\to\phi (M)\subset V$ satisfying the following properties:
\begin{enumerate}[\rm (i)]
\item[\rm (ix)] $\phi \colon \mathring M \to \phi(\mathring M)$ is a biholomorphism.
\item[\rm (x)] $\phi$ is as close as desired to the identity in the $\mathscr{C}^1$ topology on $M \setminus \bigcup_{(i,j)\in \mathtt{I}} W'_{i,j}$.
\item[\rm (xi)] $\phi(p_{i,j}) = q_{i,j}\in b\,\phi(M)$ and $\phi(M \cap W'_{i,j}) \subset W_{i,j}\cup V_{i,j}$ for all $(i,j)\in \mathtt{I}$.
\end{enumerate}
Let us check that, if $\epsilon_2>0$ is sufficiently small and if the approximation in {\rm (x)} is close enough,
then the conformal minimal immersion
\begin{equation}\label{eq:G}
F_0:=G\circ \phi\in \CMI_*^1(M,\mathbb{R}^n)
\end{equation}
satisfies the conclusion of Lemma \ref{lem:step1}.
Indeed, property {\rm (vi)} gives that
\begin{equation}\label{eq:K}
K\Subset M \setminus \bigcup_{(i,j)\in \mathtt{I}} W_{i,j}\Subset M \setminus \bigcup_{(i,j)\in \mathtt{I}} W'_{i,j}.
\end{equation}
Therefore {\rm (i)}, {\rm (iv)}, and {\rm (x)} ensure that $F_0$ is $\epsilon_1$-close to $F$ in the $\mathscr{C}^1(K)$ topology
provided that $\epsilon_2$ is small enough, thereby proving {\rm (P1)}.
In order to check {\rm (P2)} fix $(i,j)\in\mathtt{I}$ and take $\{p,q\}\in\alpha_{i,j}$. If $p\in \alpha_{i,j} \setminus (W'_{i,j-1}\cup W'_{i,j})$,
then $F_0(p)\approx G(p)$ by \eqref{eq:G} and {\rm (x)}, hence {\rm (vii)} gives that $\|F_0(p)-\mathfrak{Y}(q)\|<\delta$ and
$\|F_0(p)-F(q)\|<\epsilon_0$. If $p\in W'_{i,j-1}\cup W'_{i,j}$, then {\rm (xi)} gives that
$\phi(p)\in W_{i,j-1}\cup V_{i,j-1}\cup W_{i,j}\cup V_{i,j}$, and so \eqref{eq:G} and {\rm (vii)} imply that
$\|F_0(p)-\mathfrak{Y}(q)\|<\delta$ and $\|F_0(p)-F(q)\|<\epsilon_0$ as well. This proves {\rm (P2)}.
Property {\rm (P3)} is directly implied by {\rm (v)} and \eqref{eq:G}.
Finally, in order to check {\rm (P4)} fix $(i,j)\in\mathtt{I}$, let $\gamma\subset M$ be an arc with the initial point in $K$
and the final point $p_{i,j}$, and let $\{J_{a,b}\}_{(a,b)\in\mathtt{I}}$ be a partition of $\gamma$ by Borel measurable subsets.
Properties \eqref{eq:K} and {\rm (x)} give that $\phi(K)\Subset V \setminus \bigcup_{(i,j)\in \mathtt{I}} W_{i,j}$.
Properties {\rm (vi)}, {\rm (x)}, and {\rm (xi)} guarantee that $\phi(\gamma)$ has a connected subarc contained in
$W_{i,j}\cup V_{i,j}$ with the initial point in $W_{i,j}$ and the final point $q_{i,j}$. Therefore, \eqref{eq:G} and
condition {\rm (viii)} trivially implies the existence of neighborhoods $U_{i,j}$ of $p_{i,j}$ satisfying {\rm (P4)}.
This proves Lemma \ref{lem:step1}.
\end{proof}
We continue with the proof of Lemma \ref{lem:Jordan1}.
Let $F_0\in\CMI_*^1(M,\mathbb{R}^n)$ and $\{U_{i,j}\colon (i,j)\in \mathtt{I}\}$ be furnished by Lemma \ref{lem:step1} for a given
number $\epsilon_1>0$ that will be specified later. Up to a shrinking, we may assume that the sets $\overline U_{i,j}$,
$(i,j)\in \mathtt{I}$, are simply connected, smoothly bounded, and pairwise disjoint.
Roughly speaking, $F_0$ meets conditions {\rm (a)} and {\rm (c)} in Lemma \ref{lem:Jordan1} (cf.\ properties {\rm (P2)}
and {\rm (P3)}), but it satisfies condition {\rm (b)} only on the sets $bM\cap \overline U_{i,j}$, $(i,j)\in \mathtt{I}$ (cf.\ {\rm (P4)}).
To conclude the proof we now perturb $F_0$ near the points of $bM$ where it does not meet {\rm (b)}
(i.e.,\ outside $\bigcup_{(i,j)\in \mathtt{I}} U_{i,j}$), preserving what has already been achieved so far.
It is at this stage where the Riemann-Hilbert problem for minimal surfaces in $\mathbb{R}^n$ (see Theorem \ref{th:RHCMI})
will be exploited.
Let $\epsilon_3>0$ be a positive number that will be specified later.
Take an annular neighborhood $A\subset M\setminus K$ of $bM$ and a smooth retraction $\rho\colon A\to bM$.
In view of condition {\rm (P2)} we may choose a family of pairwise disjoint, smoothly bounded closed disc
$\overline D_{i,j}$ in $M\setminus K$, $(i,j)\in\mathtt{I}$, satisfying
\begin{equation}\label{eq:F0D}
\text{$\|F_0(p)-\mathfrak{Y}(q)\|<\delta$\ for all $(p,q)\in \overline D_{i,j}\times\alpha_{i,j}$}
\end{equation}
and the following properties:
\begin{enumerate}[\it i)]
\item $\bigcup_{(i,j)\in \mathtt{I}} \overline D_{i,j}\subset A$.
\item $\overline D_{i,j}\cap bM$ is a compact connected Jordan arc in $\alpha_{i,j}\setminus\{p_{i,j-1},p_{i,j}\}$ with an endpoint
in $U_{i,j-1}$ and the other endpoint in $U_{i,j}$.
\item $\rho(\overline D_{i,j})\subset \alpha_{i,j}\setminus\{p_{i,j-1},p_{i,j}\}$ and $\|F_0(\rho(x))-F_0(x)\|<\epsilon_3$ for all
$x\in\overline D_{i,j}$ and all $(i,j)\in \mathtt{I}$.
\end{enumerate}
For each $(i,j)\in \mathtt{I}$ we also choose a pair of compact connected Jordan arcs
$\beta_{i,j}\Subset I_{i,j}\Subset \overline D_{i,j}\cap\alpha_{i,j}$ with an endpoint in $U_{i,j-1}$ and the other endpoint in
$U_{i,j}$, and a pair of vectors $\mathbf{u}_{i,j}$, $\mathbf{v}_{i,j}\in \mathbb{R}^n$, such that
\begin{equation}\label{eq:uijvij}
\|\mathbf{u}_{i,j}\|=1 = \|\mathbf{v}_{i,j}\|; \quad \text{$\mathbf{u}_{i,j}$, $\mathbf{v}_{i,j}$, and $F(p_{i,j})-\mathfrak{Y}(p_{i,j})$ are pairwise orthogonal}.
\end{equation}
Let $\mu\colon bM\to\mathbb{R}_+$ be a continuous function such that
\begin{equation}\label{eq:mu}
0\leq\mu\leq\eta,\quad \text{$\mu=\eta$\ \ on $\bigcup_{(i,j)\in I} \beta_{i,j}$,\quad \text{$\mu=0$\;
on $bM\setminus \bigcup_{(i,j)\in \mathtt{I}} I_{i,j}$}}.
\end{equation}
Consider the continuous map $\varkappa\colon bM \times\overline{\mathbb D}\to\mathbb{R}^n$ given by
\begin{equation}\label{eq:vark}
\varkappa(x,\xi)=\left\{\begin{array}{ll}
F_0(x), & x\in bM\setminus \bigcup_{(i,j)\in \mathtt{I}} I_{i,j}; \\
F_0(x) + \mu(x)\,(\Re \xi \mathbf{u}_{i,j} + \Im \xi \mathbf{v}_{i,j}), & x\in I_{i,j},\; (i,j)\in\mathtt{I}.
\end{array}\right.
\end{equation}
In this setting, Theorem \ref{th:RHCMI} provides for every $(i,j)\in \mathtt{I}$ an arbitrarily small
open neighborhood $\Omega_{i,j}\subset \overline D_{i,j}$ of
$I_{i,j}$ in $M$ and a nondegenerate conformal minimal immersion $\wh F\in \CMI_*^1(M,\mathbb{R}^n)$
satisfying the following properties:
\begin{enumerate}[\rm ({P}1)]
\item[\rm (P5)] $\mathrm{dist}(\wh F(x),\varkappa(x,\mathbb{T}))<\epsilon_3$ for all $x\in bM$.
\item[\rm (P6)] $\mathrm{dist}(\wh F(x),\varkappa(\rho(x),\overline{\mathbb D}))<\epsilon_3$ for all
$x\in \Omega:=\bigcup_{(i,j)\in\mathtt{I}}\Omega_{i,j}$.
\item[\rm (P7)] $\wh F$ is $\epsilon_3$-close to $F_0$ in the $\mathscr{C}^1$ topology on $M\setminus \Omega$.
\item[\rm (P8)] $\mathrm{Flux} (\wh F)=\mathrm{Flux} (F_0)$.
\end{enumerate}
Recall that $\pi_{i,j}$ is the projection \eqref{eq:piij}.
Note that {\rm (P6)}, \eqref{eq:vark}, and property {\it iii)} ensure that
\begin{enumerate}[\rm ({P}1)]
\item[\rm (P9)] $\pi_{i,j}\circ\wh F$ is $2\epsilon_3$-close to $\pi_{i,j}\circ F_0$ in the $\mathscr{C}^0(\Omega_{i,j})$ topology
for all $(i,j)\in\mathtt{I}$.
\end{enumerate}
Let us check that $\wh F$ satisfies the conclusion of Lemma \ref{lem:Jordan1}
provided that the positive numbers $\epsilon_0$, $\epsilon_1$, and $\epsilon_3$ are chosen sufficiently small.
Notice that properties {\rm (P1)} and {\rm (P7)} imply that
\begin{equation}\label{eq:close}
\text{$\wh F$ is $(\epsilon_1+\epsilon_3)$-close to $F$ in the $\mathscr{C}^1(K)$ topology},
\end{equation}
and hence $\wh F$ and $F$ are $\epsilon$-close in $\mathscr{C}^1(K)$ provided that $\epsilon_1+\epsilon_3<\epsilon$;
take into account that $\Omega\subset\bigcup_{(i,j)\in I} \overline D_{i,j}\subset A\subset M\setminus K$.
Let us now check property {\rm (a)} in Lemma \ref{lem:Jordan1}.
Fix a point $p\in bM$. If $p\in bM\setminus \Omega$ then
by {\rm (P7)} we have $\|\wh F(p)- F_0(p)\|<\epsilon_3$,
and hence {\rm (P2)} ensures that $\|\wh F(p)-\mathfrak{Y}(p)\|<\sqrt{\delta^2+\eta^2}$
provided that $\epsilon_3>0$ is small enough.
Assume now that $p\in bM\cap\Omega$; then $p\in bM\cap \Omega_{i,j}$ for some $(i,j)\in \mathtt{I}$.
In view of {\rm (P5)}, \eqref{eq:mu}, and \eqref{eq:vark}, we have that
\begin{equation}\label{eq:whFp}
\big\|\wh F(p)- \big(F_0(p) + \mu(p)\,(\Re \xi \mathbf{u}_{i,j} + \Im \xi \mathbf{v}_{i,j})\big)\big\|<\epsilon_3\quad \text{for some $\xi\in\mathbb{T}$.}
\end{equation}
On the other hand, taking into account \eqref{eq:uijvij}, we obtain
\begin{multline*}
\big\|\big(F_0(p) + \mu(p)\,(\Re \xi \mathbf{u}_{i,j} + \Im \xi \mathbf{v}_{i,j})\big) -\mathfrak{Y}(p)\big\| \leq
\\
\|F_0(p)-F(p_{i,j})\| + \sqrt{\|F(p_{i,j})-\mathfrak{Y}(p_{i,j})\|^2 + \mu(p)^2}
+ \|\mathfrak{Y}(p_{i,j})-\mathfrak{Y}(p)\| \stackrel{\text{{\rm (P2)}},\eqref{eq:Y-Y}}{<}
\\
\sqrt{\|F(p_{i,j})-\mathfrak{Y}(p_{i,j})\|^2 + \mu(p)^2}+2\epsilon_0.
\end{multline*}
Together with \eqref{eq:whFp}, \eqref{eq:mu}, and \eqref{eq:lemJordan1} we get
\[
\|\wh F(p)-\mathfrak{Y}(p)\|<\sqrt{\|F(p_{i,j})-\mathfrak{Y}(p_{i,j})\|^2 + \mu(p)^2}+2\epsilon_0+\epsilon_3<\sqrt{\delta^2+\eta^2},
\]
where the latter inequality holds provided that $\epsilon_0$ and $\epsilon_3$ are chosen small enough from the beginning.
This proves property {\rm (a)} in Lemma \ref{lem:Jordan1}.
Let us now verify property {\rm (b)}. Recall that $p_0\in \mathring K$. If $\epsilon_1$ and $\epsilon_3$ are small enough,
then \eqref{eq:>tau} and \eqref{eq:close} ensure that
\begin{equation}\label{eq:distwhF}
\mathrm{dist}_{\wh F}(p_0,bK)>d.
\end{equation}
We now estimate $\mathrm{dist}_{\wh F}(bK,bM)$. Properties {\rm (P4)}, {\rm (P7)}, and {\rm (P9)} guarantee the following:
\begin{claim}\label{cla:ep3}
If $\epsilon_3>0$ is chosen small enough, then for every arc $\gamma\subset M\setminus \mathring K$ with the
initial point in $bK$ and the final point in $\bigcup_{(i,j)\in\mathtt{I}}\overline U_{i,j}$,
and for any partition $\{J_{a,b}\}_{(a,b)\in\mathtt{I}}$ of $\gamma$ by Borel measurable subsets satisfying
$\gamma\cap\overline \Omega_{a,b}\subset J_{a,b}$ for all $(a,b)\in\mathtt{I}$,
we have
\[
\mathrm{length}\, \wh F(\gamma) \geq \sum_{(a,b)\in\mathtt{I}}\mathrm{length} \, \pi_{a,b}(\wh F(J_{a,b})) >\eta.
\]
\end{claim}
Consider now an arc $\gamma\subset M\setminus \bigcup_{(i,j)\in\mathtt{I}} \overline U_{i,j}$ with the initial point in $bK$,
the final point in $bM$, and otherwise disjoint from $K$.
Then there exist $(i,j)\in\mathtt{I}$ and a subarc $\hat\gamma$ of $\gamma$ with the endpoints
$q\in M\setminus\Omega$ and $p\in \beta_{i,j}$
satisfying $\hat\gamma\subset \overline\Omega_{i,j}\setminus (\overline U_{i,j-1}\cup \overline U_{i,j})$.
In view of \eqref{eq:uijvij}, \eqref{eq:mu} and \eqref{eq:whFp} there exists $\xi\in\mathbb{T}$ such that
\begin{equation}\label{eq:whF-F0}
\|\wh F(p)- F_0(p)\|> \mu(p)\|\Re \xi \mathbf{u}_{i,j} + \Im \xi \mathbf{v}_{i,j}\|-\epsilon_3=\eta-\epsilon_3.
\end{equation}
On the other hand, we have
\begin{eqnarray*}
\mathrm{length}\, \wh F(\gamma) & \geq & \mathrm{length}\, \wh F(\hat \gamma) \ge \|\wh F(q)-\wh F(p)\|
\\
& \geq & \|\wh F(p)-F_0(p)\|
\\
& & - \|\wh F(q)-F_0(q)\| - \|F_0(q)-F_0(\rho(q))\| - \|F_0(\rho(q))-F_0(p)\|
\\
& > & \|\wh F(p)-F_0(p)\| -\epsilon_0-2\epsilon_3,
\end{eqnarray*}
where in the last inequality we used {\rm (P2)}, {\it iii)}, and {\rm (P7)}. Combining this inequality and \eqref{eq:whF-F0}
we get that $\mathrm{length}\, \wh F(\gamma)>\eta-\epsilon_0-3\epsilon_3$. Together with Claim \ref{cla:ep3} we obtain that
$\mathrm{dist}_{\wh F}(bK,bM)>\eta-\epsilon_0-3\epsilon_3$ and, taking into account \eqref{eq:distwhF}, $\mathrm{dist}_{\wh F}(p_0,bM)>d+\eta$
provided that $\epsilon_0$ and $\epsilon_3$ are small enough. This shows property {\rm (b)}.
Finally, condition {\rm (c)} is trivially implied by {\rm (P3)} (cf.\ Lemma \ref{lem:step1})
and {\rm (P8)}, thereby concluding the proof of Lemma \ref{lem:Jordan1}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:Jordan2}]
Let $\epsilon>0$. We shall find $\wh G\in\CMI_*^1(M,\mathbb{R}^n)$ which is $\epsilon$-close to $G$ in the $\mathscr{C}^0(M)$ topology and
satisfies $\mathrm{dist}_{\wh G}(p_0,bM)>\lambda$ and $\mathrm{Flux}(\wh G)=\mathrm{Flux}(G)$.
Choose numbers $d_0$ and $\delta_0$ such that $0<d_0<\mathrm{dist}_G(p_0,bM)$
and $0<\delta_0<\epsilon$. Set
\[
c:=\frac{\sqrt{6(\epsilon^2-\delta_0^2)}}{\pi}>0.
\]
Consider the following sequences defined recursively:
\[
d_j:=d_{j-1}+\frac{c}{j}>0,\qquad \delta_j:=\sqrt{\delta_{j-1}^2+\frac{c^2}{j^2}}>0,\quad j\in\mathbb{N}.
\]
Observe that
\begin{equation}\label{eq:limits}
\{d_j\}_{j\in\mathbb{Z}_+}\nearrow +\infty,\qquad \{\delta_j\}_{j\in\mathbb{Z}_+}\nearrow \epsilon.
\end{equation}
We claim that there exists a sequence $G_j\in\CMI_*^1(M,\mathbb{R}^n)$
$(j\in\mathbb{Z}_+)$ of conformal minimal immersions enjoying the following properties:
\begin{enumerate}[\rm (a$_{j}$)]
\item $\|G_j-G\|_{0,bM}<\delta_j$.
\item $\mathrm{dist}_{G_j}(p_0,bM)>d_j$.
\item $\mathrm{Flux}(G_j)=\mathrm{Flux}(G)$.
\end{enumerate}
We proceed by induction, beginning with the immersion $G_0:=G$. For the inductive step we assume the existence of
$G_j\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying {\rm (a$_j$)}, {\rm (b$_j$)}, and {\rm (c$_j$)} for some $j\in\mathbb{Z}_+$.
Applying Lemma \ref{lem:Jordan1} to the data
\[
F=G_j,\quad \mathfrak{Y}=G|_{bM},\quad \delta=\delta_j,\quad p_0,\quad \eta=\frac{c}{j+1},\quad d=d_j,
\]
we obtain a conformal minimal immersion $G_{j+1}\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying
{\rm (a$_{j+1}$)}, {\rm (b$_{j+1}$)}, and {\rm (c$_{j+1}$)}, hence closing the induction step.
By {\rm (a$_j$)}, the Maximum Principle, and the latter assertion in \eqref{eq:limits}, $G_j$ is $\epsilon$-close to $G$ in the
$\mathscr{C}^0(M)$ topology for all $j\in \mathbb{Z}_+$. On the other hand, {\rm (b$_j$)} and the former assertion in \eqref{eq:limits} ensure that
$\mathrm{dist}_{G_j}(p_0,bM)>d_j>\lambda$ for any large enough $j\in\mathbb{Z}_+$. In view of {\rm (c$_j$)}, to conclude the proof it suffices
to choose $\wh G:=G_j$ for a sufficiently large $j\in\mathbb{N}$.
\end{proof}
Another important point in the proof of Theorem \ref{th:Jordan} is that the general position of
conformal minimal immersions $M\to\mathbb{R}^n$ is embedded if $n\geq 5$ (cf.\ \cite[Theorem 1.1]{AFL}).
Moreover, it is easy to derive from the proof in \cite{AFL} that the general position of the boundary curves
of conformal minimal immersions $M\to\mathbb{R}^n$ is also embedded for any $n\geq 3$.
The following is the precise result that will be used in the proof of Theorem \ref{th:Jordan}.
\begin{theorem}\label{th:gp}
Let $M$ be a compact bordered Riemann surface and let $n\geq 3$ and $r\ge 1$ be natural numbers.
\begin{enumerate}[\rm (a)]
\item Every conformal minimal immersion $F\in \CMI^r(M,\mathbb{R}^n)$ can be approximated in the $\mathscr{C}^r(M)$ topology
by nondegenerate conformal minimal immersions $\wt F\in \CMI_*^r(M,\mathbb{R}^n)$ such that
$\wt F|_{bM}\colon bM\to\mathbb{R}^n$ is an embedding and $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
\item If $n\geq 5$ then every nondegenerate conformal minimal immersion $F\in \CMI_*^r(M,\mathbb{R}^n)$ can be approximated in the
$\mathscr{C}^r(M)$ topology by nondegenerate conformal minimal embeddings $\wt F\in \CMI_*^r(M,\mathbb{R}^n)$ satisfying
$\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
\end{enumerate}
\end{theorem}
As already said, assertion {\rm (b)} in this theorem is proved in \cite[Theorem 4.1]{AFL}.
\begin{proof}[Proof of {\rm (a)}]
By \cite[Theorem 3.1 (a)]{AFL} we may assume that $F\in \CMI_*^r(M,\mathbb{R}^n)$ is nondegenerate.
We consider the {\em difference map} $\delta F\colon M\times M\to \mathbb{R}^n$, defined by
\[
\delta F(x,y)=F(y)-F(x), \qquad x,y\in M.
\]
Clearly $F$ is injective if and only if $(\delta F)^{-1}(0)= D_M:=\{(x,x): x\in M\}$,
the diagonal of $M\times M$. Since $F$ is an immersion, it is locally injective, and hence there is an open
neighborhood $U\subset M\times M$ of $D_M$
such that $\delta F$ does not vanish anywhere on $\overline U\setminus D_M$.
In this setting, the construction in \cite[Sec.\ 4]{AFL} furnishes a neighborhood $\Omega \subset \mathbb{R}^N$
of the origin in a Euclidean space and a real analytic map $H\colon \Omega \times M \to \mathbb{R}^n$ satisfying
the following conditions:
\begin{enumerate}[\rm (i)]
\item $H(0,\cdotp)=F$.
\item $H(\xi,\cdotp)\in \CMI_*^r(M)$ and $\mathrm{Flux}(H(\xi,\cdotp))=\mathrm{Flux}(F)$ for every $\xi \in \Omega$.
\item The difference map $\delta H\colon \Omega \times M\times M \to \mathbb{R}^n$, defined by
\[
\delta H(\xi,x,y) = H(\xi,y)-H(\xi,x), \qquad \xi\in \Omega, \ \ x,y\in M,
\]
is a submersive family on $(M\times M) \setminus U$, in the sense that the partial differential
\[
\di_\xi|_{\xi=0} \, \delta H(\xi,x,y) \colon \mathbb{R}^N \to \mathbb{R}^n
\]
is surjective for every $(x,y)\in (M\times M) \setminus U$.
\end{enumerate}
Set $\psi_H:=H|_{\Omega\times bM}$ and $\delta \psi_H:=(\delta H)|_{\Omega\times bM\times bM}$.
From {\rm (iii)} and the compactness of $(bM\times bM)\setminus U$ we obtain that the partial differential
$\di_\xi (\delta\psi_H(\xi,x,y))\colon\mathbb{R}^N\to\mathbb{R}^n$ is surjective for all $\xi$ in a neighborhood
$\Omega'\subset\Omega$ of $0\in\mathbb{R}^N$, for every $(x,y)\in (bM\times bM)\setminus U$.
This implies that the map $\delta \psi_H\colon \Omega'\times (bM\times bM)\setminus U\to\mathbb{R}^n$
is transverse to any submanifold of $\mathbb{R}^n$, in particular, to the origin $0\in\mathbb{R}^n$.
The standard transversality argument due to Abraham \cite{Abraham}
(see also \cite[Sec.\ 7.8]{F2011}) ensures that for a generic choice of $\xi\in\Omega'$ the difference map
$\delta \psi_H(\xi,\cdotp,\cdotp)$ is transverse to $0\in\mathbb{R}^n$ on $(bM\times bM)\setminus U$.
Since $n\geq 3$ and $(bM\times bM)\setminus U$ is of real dimension 2, it follows that
\begin{equation}\label{eq:psiH}
\text{$\delta \psi_H(\xi,\cdotp,\cdotp)$ does not vanish anywhere on $(bM\times bM)\setminus U$.}
\end{equation}
If we choose $\xi={\xi_0}$ close enough to $0\in\mathbb{R}^N$ and such that \eqref{eq:psiH} holds,
then the conformal minimal immersion $\wt F=H({\xi_0},\cdotp)\in\CMI_*^r(M,\mathbb{R}^n)$ satisfies the conclusion of
Theorem \ref{th:gp} and is arbitrarily close to $F$ in the $\mathscr{C}^r(M)$ topology.
(See the proof of \cite[Theorem 4.1]{AFL} for further details.)
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:Jordan}]
We may assume that $M$ is a smoothly bounded compact domain in an open Riemann surface
$\wt M$. By \cite[Theorem 3.1 (a)]{AFL} we can assume that $F$ is nondegenerate, $F\in\CMI_*^1(M,\mathbb{R}^n)$.
By Theorem \ref{th:gp} above we may also assume that $F|_{bM}$ is an embedding and,
if $n\geq 5$, that $F\colon M\to\mathbb{R}^n$ is an embedding.
Choose a compact domain $M_0\subset\mathring M$, a point $p_0\in \mathring M_0$, and set $F_0:=F$. Let $\theta$ be a
holomorphic $1$-form in $\wt M$ vanishing nowhere on $M$ and denote by ${\rm d}\colon M\times M\to \mathbb{R}$ the distance function
on the Riemannian surface $(M,|\theta|^2)$.
Pick a number $\epsilon_0>0$.
By applying Lemma \ref{lem:Jordan2} and Theorem \ref{th:gp} we shall inductively construct a sequence
$\{\Xi_j=(M_j,\epsilon_j,F_j)\}_{j\in\mathbb{N}}$, where $M_j$ is a compact domain in $\mathring M$,
$\epsilon_j>0$, and $F_j\in \CMI_*^1(M,\mathbb{R}^n)$, satisfying
the following properties for all $j\in\mathbb{N}$:
\begin{enumerate}[\rm (1$_{j}$)]
\item $M_{j-1} \Subset \mathring M_{j}$ and $\max\{{\rm d}(p,bM)\colon p\in bM_j\}<1/j$.
\item $\max\big\{\|F_j-F_{j-1}\|_{0,M}, \big\|(\partial F_j-\partial F_{j-1})/\theta\big\|_{0,M_{j-1}}\big\}<\epsilon_j$.
\item $\mathrm{dist}_{F_j}(p_0,bM_k)>k$ for all $k\in\{0,\ldots,j\}$.
\item $\mathrm{Flux}(F_j)=\mathrm{Flux}(F)$.
\item $F_j|_{bM}$ is an embedding and, if $n\geq 5$, $F_j$ is an embedding.
\item $\epsilon_j<\min\left\{ \epsilon_{j-1}/2 , \tau_j,\varsigma_j\right\},$
where the numbers $\tau_j$ and $\varsigma_j$ are defined as follows:
\end{enumerate}
\begin{equation}\label{eq:tauj}
\tau_j=\frac1{2^j}\min_{k\in\{0,\ldots,j-1\}} \min_{p\in M} \left\|\frac{\partial F_k}{\theta}(p) \right\|,
\end{equation}
\begin{equation}\label{eq:varsigmaj}
\varsigma_j=\left\{
\begin{array}{ll}
\displaystyle \frac1{2j^2} \inf\left\{ \|F_{j-1}(p)-F_{j-1}(q)\|\colon p,q\in bM,\, {\rm d}(p,q)>\frac1{j}\right\} & \text{if $n\in\{3,4\}$}
\\
&
\\
\displaystyle \frac1{2j^2} \inf\left\{ \|F_{j-1}(p)-F_{j-1}(q)\|\colon p,q\in M,\, {\rm d}(p,q)>\frac1{j}\right\} & \text{if $n\geq 5$.}
\end{array}
\right.
\end{equation}
Notice that $\Xi_0=(M_0,\epsilon_0,F_0)$ meets conditions {\rm (3$_0$)}, {\rm (4$_0$)}, and {\rm (5$_0$)}, whereas {\rm (1$_0$)},
{\rm (2$_0$)}, and {\rm (6$_0$)} are void. Let $j\in\mathbb{N}$ and assume inductively the existence of triples $\Xi_0,\ldots, \Xi_{j-1}$
satisfying the above conditions. Since $F_0,\ldots,F_{j-1}$ are immersions, the number $\tau_j$ \eqref{eq:tauj} is positive.
Moreover, {\rm (5$_{j-1}$)} ensures that the number $\varsigma_j$ \eqref{eq:varsigmaj} is positive as well.
Therefore there exists $\epsilon_j>0$ satisfying {\rm (6$_j$)}. Lemma \ref{lem:Jordan2} ensures that $F_{j-1}$ can be approximated
in the $\mathscr{C}^0(M)$ topology (and hence also in the $\mathscr{C}^1(M_{j-1})$ topology) by a conformal minimal immersion
$F_j\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying $\mathrm{dist}_{F_j}(p_0,bM)>j$ and {\rm (4$_j$)}.
Taking into account {\rm (3$_{j-1}$)}, we may choose an $F_j$ with these properties and a compact region
$M_j\subset \mathring M$ satisfying also {\rm (1$_j$)}, {\rm (2$_j$)}, and {\rm (3$_j$)}.
Furthermore, in view of Theorem \ref{th:gp}, we may also assume that $F_j$ meets condition {\rm (5$_j$)}.
This concludes the inductive step and hence the construction of the sequence $\{\Xi_j\}_{j\in\mathbb{N}}$.
By properties {\rm (1$_j$)} and {\rm (6$_j$)}, which hold for all $j\in\mathbb{N}$, we have $\bigcup_{j=1}^\infty M_j=M$
and the sequence $\{F_j\}_{j\in\mathbb{N}}$ converges uniformly on $M$ to a continuous map
\[
\wt F:=\lim_{j\to\infty} F_j\colon M\to\mathbb{R}^n
\]
which is $\epsilon_0$-close to $F$ in the $\mathscr{C}^0(M)$ topology and
whose restriction to $\mathring M$ is conformal and harmonic.
To finish the proof, it remains to show that
\begin{enumerate}[\rm (a)]
\item $\wt F|_{\mathring M}\colon \mathring M\to\mathbb{R}^n$ is a complete immersion,
\item $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$,
\item $\wt F|_{bM}\colon bM\to\mathbb{R}^n$ is injective, and
\item if $n\geq 5$ then $\wt F\colon M\to\mathbb{R}^n$ is injective.
\end{enumerate}
Indeed, take a point $p\in \mathring M$. From $\bigcup_{j=1}^\infty M_j=M$
and {\rm (1$_j$)} we see that there exists a number $j_0\in\mathbb{N}$ such that $p\in M_{j}$ for all $j\geq j_0$. We have
\begin{eqnarray*}
\left\| \frac{\partial \wt F}{\theta}(p) \right\| & \geq &
\left\| \frac{\partial F_{j_0}}{\theta} (p) \right\| - \sum_{j>j_0}
\left\| \frac{\partial F_j}{\theta}(p) - \frac{\partial F_{j-1}}{\theta}(p) \right\|
\\
& \stackrel{\text{\rm (2$_j$), (6$_j$)}}{>} & \left\| \frac{\partial F_{j_0}}{\theta}(p) \right\| -
\sum_{j>j_0} \tau_j
\\
& \stackrel{\text{\eqref{eq:tauj}}}{\geq} & \left\| \frac{\partial F_{j_0}}{\theta}(p) \right\|
\biggl(1-\sum_{j>j_0}\frac1{2^j}\biggr)
\\
& > & \frac12 \left\| \frac{\partial F_{j_0}}{\theta}(p) \right\| \;>\; 0.
\end{eqnarray*}
Since this holds for each point $p\in \mathring M$, $\wt F|_{\mathring M}$ is an immersion.
As $\wt F$ is a uniform limit on $M$ of conformal harmonic immersions,
it is a conformal harmonic immersion on $\mathring M$, and hence $\wt F\in\CMI_*^0(M,\mathbb{R}^n)$.
Property {\rm (3$_j$)} says that for every $k\in \mathbb{N}$ and all $j\ge k$ we have
$\mathrm{dist}_{F_j}(p_0,bM_k)>k$. Property {\rm (2$_j$)} ensures that the sequence $F_j$ converges to
$\wt F$ in the $\mathscr{C}^1(M_k)$ topology, and hence in the limit we get that
$\mathrm{dist}_{\wt F}(p_0,bM_k) \ge k$. Since this holds for all $k\in\mathbb{N}$, we see that
$\wt F|_{\mathring M}$ is complete, thereby proving {\rm (a)}.
Property {\rm (b)} is a trivial consequence of {\rm (4$_j$)}, $j\in\mathbb{N}$.
In order to check properties {\rm (c)} and {\rm (d)}, pick a pair of distinct points $p,q\in M$.
If $n\in\{3,4\}$, assume that $\{p,q\}\subset bM$.
Choose $j_0\in\mathbb{N}$ such that ${\rm d}(p,q)>\frac1{j}$ for all $j\geq j_0$. Given $j>j_0$ we have
\begin{eqnarray*}
\|F_{j-1}(p)-F_{j-1}(q)\| & \leq & \|F_j(p)-F_{j-1}(p)\|+\|F_j(q)-F_{j-1}(q)\|
\\
& & +\|F_j(p)-F_j(q)\|
\\
& \stackrel{\text{\rm (2$_j$), (6$_j$)}}{<} & 2\varsigma_j+\|F_j(p)-F_j(q)\|
\\
& \stackrel{\text{\eqref{eq:varsigmaj}}}{<} & \frac1{j^2}\|F_{j-1}(p)-F_{j-1}(q)\| + \|F_j(p)-F_j(q)\|.
\end{eqnarray*}
Therefore,
\[
\|F_j(p)-F_j(q)\| > \left(1-\frac1{j^2}\right) \|F_{j-1}(p)-F_{j-1}(q)\|, \quad j>j_0,
\]
and hence
\[
\|F_{j_0+k}(p)-F_{j_0+k}(q)\| > \|F_{j_0}(p)-F_{j_0}(q)\|
\prod_{j=j_0+1}^{j_0+k} \left(1-\frac1{j^2}\right), \quad k\in\mathbb{N}.
\]
Taking limits in the above inequality as $k$ goes to infinity, we conclude that
\[
\|\wt F(p)-\wt F(q)\|\geq \frac12\|F_{j_0}(p)-F_{j_0}(q)\| > 0,
\]
where the latter inequality in ensured by {\rm (5$_{j_0}$)}.
This completes the proof of Theorem \ref{th:Jordan}.
\end{proof}
\section{Complete proper minimal surfaces in convex domains} \label{sec:proper}
In this section we prove a technical result, Theorem \ref{th:proper2}, which will be used
in the following section to prove Theorems \ref{th:proper}, \ref{th:topology} and Corollary \ref{co:bdddomains}.
Assume that $\mathscr{D}$ is a smoothly bounded, strictly convex domain in $\mathbb{R}^n$, $n\geq 3$.
We denote by $\nu_\mathscr{D}$ the inner normal to $b\mathscr{D}$ and by $\kappa_\mathscr{D}^{\rm max}$ and $\kappa_\mathscr{D}^{\rm min}$ the maximum
and the minimum of the principal curvatures of points in $b\mathscr{D}$ with respect to $\nu_\mathscr{D}$. Obviously,
$0<\kappa_\mathscr{D}^{\rm min}\leq\kappa_\mathscr{D}^{\rm max}$. For any real number $-\infty<t<1/\kappa_\mathscr{D}^{\rm max}$ we denote by
$\mathscr{D}_t$ the smoothly bounded, strictly convex domain bounded by $b\mathscr{D}_t=\{p+t\nu_\mathscr{D}(p)\colon p\in b\mathscr{D}\}$.
Clearly, $t_1<t_2<1/\kappa_\mathscr{D}^{\rm max}$ implies $\mathscr{D}_{t_2}\Subset\mathscr{D}_{t_1}$, and
\begin{equation}\label{eq:curvature}
\frac1{\kappa_{\mathscr{D}_t}^{\rm max}}=\frac1{\kappa_{\mathscr{D}}^{\rm max}}-t,\quad \frac1{\kappa_{\mathscr{D}_t}^{\rm min}}
=\frac1{\kappa_{\mathscr{D}}^{\rm min}}-t,\quad \text{for all }t<\frac1{\kappa_\mathscr{D}^{\rm max}}.
\end{equation}
By the classical Minkowski theorem, every convex domain in $\mathbb{R}^n$ can be exhausted by an increasing sequence of
smoothly bounded, strictly convex domains.
\begin{theorem}\label{th:proper2}
Let $n\geq 3$ be a natural number, let $\mathscr{L}\Subset\mathscr{D}\Subset\mathbb{R}^n$ be smoothly bounded (of class at least $\mathscr{C}^2$)
strictly convex domains, and let $\eta>0$ be such that $\mathscr{D}\subset\mathscr{L}_{-\eta}$.
Let $M$ be a compact bordered Riemann surface and let $F\in \CMI^1(M,\mathbb{R}^n)$ be a conformal minimal immersion with
$F(bM)\subset \mathscr{D}\setminus\overline\mathscr{L}$. Given a number $\mu >0$ and a compact set $K\subset \mathring M$
there exists a continuous map $\wt F\colon M\to\overline\mathscr{D}$ satisfying the following conditions:
\begin{enumerate}[\rm (i)]
\item $\wt F|_{\mathring M}\colon\mathring M\to \mathscr{D}$ is a complete proper conformal minimal immersion.
\item $\wt F(bM)\subset b\mathscr{D}$ is a finite family of closed curves.
\item $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
\item $\|\wt F-F\|_{0,M}< \sqrt{2 \eta^2+ {2 \eta}/{\kappa_\mathscr{L}^{\rm min}}}$.
\item $\|\wt F-F\|_{1,K}<\mu$.
\end{enumerate}
If $n\geq 5$ then we can choose $\wt F$ to be an embedding on $\mathring M$.
\end{theorem}
Unfortunately we are unable to ensure that the frontier
$\wt F(b M)\subset b\mathscr{D}$ consists of Jordan curves even when $n\geq 5$;
the reason is explained in Remark \ref{rem:Jordan} at the end of the section.
The proof Theorem \ref{th:proper2} uses an inductive procedure in which we alternately apply
the following two types of deformations to a conformal minimal immersion $F\colon M\to \mathbb{R}^n$:
\begin{itemize}
\item[\rm (i)] Push the boundary $F(bM)$ closer to $b\mathscr{D}$
while keeping the resulting immersion suitably close to $F$ in the
$\mathscr{C}^0(M)$ sense, depending on how far is $F(bM)$ from $b\mathscr{D}$.
This deformation is provided by Lemma \ref{lem:proper} below.
\item[\rm (ii)] Increase the interior boundary distance of the immersion by a prescribed
(arbitrarily big) amount by a deformation which is arbitrarily small in the $\mathscr{C}^0(M)$
sense. Such deformation is provided by Lemma \ref{lem:Jordan2} in Sec.\ \ref{sec:Jordan}.
\end{itemize}
The resulting sequence of conformal minimal immersions $F_k\colon M\to \mathscr{D}$
$(k\in\mathbb{Z}_+)$ converges uniformly on $M$ to a continuous map $\wt F\colon M\to\overline\mathscr{D}$
satisfying Theorem \ref{th:proper2}.
We begin with technical preparations.
\begin{lemma}\label{lem:proper}
Let $n\ge 3$, let $\mathscr{L}\Subset\mathscr{D}$ be smoothly bounded, strictly convex domains in $\mathbb{R}^n$,
and let $\eta>0$ be such that $\mathscr{D}\subset \mathscr{L}_{-\eta}$.
Let $M$ be a compact bordered Riemann surface and let $F\in\CMI_*^1(M,\mathbb{R}^n)$.
Assume that for a compact set $K\subset \mathring M$ we have
\begin{equation}\label{eq:lemproper}
F(M\setminus \mathring K)\subset \mathscr{D}\setminus\overline\mathscr{L}.
\end{equation}
Given a number $\delta$
satisfying $0<\delta<1/\kappa_\mathscr{D}^{\rm max}$, $F$ can be approximated as closely as desired
in the $\mathscr{C}^1(K)$ topology by a conformal minimal immersion $\wt F\in\CMI_*^1(M,\mathbb{R}^n)$
enjoying the following properties:
\begin{enumerate}[\rm (a)]
\item $\|\wt F-F\|_{0,M}< \sqrt{2 \eta^2+ {2 \eta}/{\kappa_\mathscr{L}^{\rm min}}}$.
\item $\wt F(M\setminus \mathring K)\subset \mathscr{D}\setminus\overline\mathscr{L}$.
\item $\wt F(bM)\subset \mathscr{D}\setminus\overline\mathscr{D}_\delta$.
\item $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$.
\end{enumerate}
\end{lemma}
\begin{proof}
The main idea is to perturb $F$ near $bM$ in such a way that the image of each point $p\in bM$ is moved into
the convex shell $\mathscr{D}\setminus\overline\mathscr{D}_\delta$ by pushing it in a direction orthogonal to the inner unit normal $\nu_\mathscr{L}(p)$ of $b\mathscr{L}$ at $p$.
By Pythagoras' theorem and basic theory of convex domains, condition \eqref{eq:lemproper} ensures that it will be enough to push each point a distance
smaller than $\sqrt{\eta^2+ {2\eta}/{\kappa_\mathscr{L}^{\rm min}}}$, allowing us to guarantee condition {\rm (a)}.
We may assume that $M$ is a smoothly bounded domain in an open Riemann surface $\wt M$.
Without loss of generality we may also assume that $\delta>0$ is small enough so that $\overline\mathscr{L}\subset \mathscr{D}_\delta$.
In view of \eqref{eq:lemproper} we may choose a constant $\varsigma>0$ such that
\begin{equation}\label{eq:eta0}
\overline\mathscr{L}_{-\varsigma}\subset\mathscr{D}_{\delta},\quad F(M\setminus \mathring K)\subset \mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma}.
\end{equation}
Pick another constant $c>0$ to be specified later. For every point $\mathbf{x}\in b\mathscr{L}$ set
\begin{equation} \label{eq:cball}
B_\mathbf{x}:=b\mathscr{L} \cap \mathbb{B}_{\mathbf{x}}(c),
\end{equation}
where $\mathbb{B}_{\mathbf{x}}(c)$ denotes the open euclidean ball in $\mathbb{R}^n$ centered at $\mathbf{x}$ with radius $c>0$. Set
\begin{equation}\label{eq:Ox}
O_\mathbf{x}=\mathscr{D}\cap\{\mathbf{y}-t\nu_\mathscr{L}(\mathbf{y}) \colon \mathbf{y}\in B_\mathbf{x},\; t>\varsigma\}\subset \mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma}.
\end{equation}
Assume that $c>0$ is small enough so that $B_\mathbf{x}$ is a topological open ball and
\begin{equation}\label{eq:Ox1}
O_\mathbf{x} \subset \wt O_\mathbf{x}:= \{\mathbf{y}\in\mathscr{D} \colon \langle\mathbf{y}-\mathbf{x},\nu_\mathscr{L}(\mathbf{x})\rangle
<-\varsigma/2\} \subset \mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma/2} \quad \forall \mathbf{x}\in b\mathscr{L}
\end{equation}
\begin{figure}[ht]
\begin{center}
\scalebox{0.28}{\includegraphics{D-L3}}
\end{center}
\caption{The convex domains $\mathscr{L}$ and $\mathscr{D}$.}
\label{fig:D-L}
\end{figure}
(see Figure \ref{fig:D-L}). Observe that \eqref{eq:Ox1} holds in the limit case $c=0$.
Set $\mathcal{O}:=\{O_\mathbf{x}\colon \mathbf{x}\in b\mathscr{L}\}$ and notice that $\mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma}=\bigcup_{\mathbf{x}\in b\mathscr{L}}O_\mathbf{x}$.
Denote by $\alpha_1,\ldots,\alpha_k$ the connected boundary curves of $M$. Since $\mathcal{O}$ is an open covering of the compact set
$F(bM)\subset\mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma}$ (cf. \eqref{eq:eta0}), there exist a natural number $l\geq 3$ and
compact connected subarcs $\{\alpha_{i,j}\colon (i,j) \in \mathtt{J}:=\{1,\ldots,k\}\times \mathbb{Z}_l\}$ satisfying
\[
\bigcup_{j\in\mathbb{Z}_l} \alpha_{i,j}=\alpha_i\quad \text{for all $i\in\{1,\ldots,k\}$},
\]
and, for every $(i,j)\in\mathtt{J}$, $\alpha_{i,j}$ and $\alpha_{i,j+1}$ have a common endpoint $p_{i,j}$ and are otherwise disjoint,
$\alpha_{i,j}\cap\alpha_{i,a}=\emptyset$ for all $a\in\mathbb{Z}_l\setminus\{j-1,j,j+1\}$, and
\begin{equation}\label{eq:alphaU}
F(\alpha_{i,j})\subset O_{i,j}:=O_{\mathbf{x}_{i,j}}\in\mathcal{O}\quad \text{for some $\mathbf{x}_{i,j}\in b\mathscr{L}$}.
\end{equation}
\begin{lemma}\label{lem:arcs}
(Notation as in Lemma \ref{lem:proper}.) Let $\varsigma>0$ be
such that (\ref{eq:eta0}) holds.
Given a number $\epsilon_1>0$, there exists $F_0\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying the following properties:
\begin{enumerate}[\rm ({P}1)]
\item $F_0$ is $\epsilon_1$-close to $F$ in the $\mathscr{C}^1(K)$ topology.
\item $F_0(p_{i,j})\in \mathscr{D}\setminus\overline\mathscr{D}_{\delta/2}$ for all $(i,j)\in\mathtt{J}$.
\item $F_0(\alpha_{i,j})\subset O_{i,j}$ for all $(i,j)\in\mathtt{J}$.
\item $F_0(M\setminus \mathring K)\subset \mathscr{D}\setminus \overline\mathscr{L}_{-\varsigma}$.
\item $\mathrm{Flux}(F_0)=\mathrm{Flux} (F)$.
\end{enumerate}
\end{lemma}
\begin{proof}
For each $(i,j)\in\mathtt{J}$ we choose an arc $\gamma_{i,j}\subset \wt M$ with the endpoint $p_{i,j}\in bM$
and otherwise disjoint from $M$ such that the arcs $\gamma_{i,j}$, $(i,j)\in\mathtt{J}$, are pairwise disjoint and
\[
S:=M\cup\bigcup_{(i,j)\in\mathtt{J}} \gamma_{i,j}\subset\wt M
\]
is an {\em admissible set} in the sense of \cite[Def.\ 5.1]{AFL}.
(This implies that the Mergelyan approximation theorem holds on $S$.)
Let $v\colon S\to\mathscr{D}\subset \mathbb{R}^n$ be a smooth map satisfying the following conditions:
\begin{enumerate}[\rm (i)]
\item $v=F$ on a neighborhood of $M$.
\item $v(\gamma_{i,j})\subset O_{i,j}\cap O_{i,j+1}$ and $v(q_{i,j})\in \mathscr{D}\setminus \overline\mathscr{D}_{\delta/2}$, where $q_{i,j}$
is the endpoint of $\gamma_{i,j}$ different from $p_{i,j}$, for all $(i,j)\in\mathtt{J}$.
(Observe that
$O_{i,j}\cap O_{i,j+1}=\mathscr{D}\cap\{\mathbf{y}-t\nu_\mathscr{L}(\mathbf{y}) \colon \mathbf{y}\in B_{\mathbf{x}_{i,j}}\cap B_{\mathbf{x}_{i,j+1}},\; t>\varsigma\}\neq\emptyset$.
See \eqref{eq:Ox}, \eqref{eq:alphaU}, and Figure \ref{fig:D-L}).
\end{enumerate}
Pick a number $\epsilon_2>0$ which will be specified later. As in the proof of Lemma \ref{lem:step1}, we may use the Mergelyan theorem
for conformal minimal immersions \cite[Theorem 5.3]{AFL} to obtain $G\in\CMI_*(\wt M,\mathbb{R}^n)$ satisfying the following properties:
\begin{enumerate}[\rm (i)]
\item[\rm (iii)] $G$ is $\epsilon_2$-close to $v$ in the $\mathscr{C}^1(M)$ and the $\mathscr{C}^0(S)$ topologies.
\item[\rm (iv)] $\mathrm{Flux}(G)=\mathrm{Flux}(F)$.
\end{enumerate}
Let $V\subset \wt M$ be a small open neighborhood of $S$. If $\epsilon_2>0$ is small enough,
then properties {\rm (i)}, {\rm (ii)}, and {\rm (iii)} ensure the existence of small open neighborhoods
$U_{i,j}$ of $\alpha_{i,j}$, $V_{i,j}$ of $\gamma_{i,j}$, and $W'_{i,j}\Subset W_{i,j}$ of $p_{i,j}$ in
$V \setminus K$, $(i,j)\in \mathtt{J}$, satisfying the following conditions:
\begin{enumerate}[\rm (i)]
\item[\rm (v)] $V_{i,j}\cap M\Subset W'_{i,j}\Subset W_{i,j}\Subset U_{i,j}\cap U_{i,j+1}\Subset V\setminus K$.
\item[\rm (vi)] $G(V_{i,j-1}\cup U_{i,j}\cup V_{i,j})\subset O_{i,j}$. (Take into account \eqref{eq:alphaU}.)
\item[\rm (vii)] $G(q_{i,j})\in \mathscr{D}\setminus\mathscr{D}_{\delta/2}$.
\end{enumerate}
Without loss of generality we assume in addition that the compact sets $\overline W_{i,j}\cup\overline V_{i,j}$, $(i,j)\in \mathtt{J}$,
are pairwise disjoint.
By \cite[Theorem 2.3]{FW0} (see also \cite[Theorem 8.8.1, p.\ 365]{F2011}), there exists a smooth
diffeomorphism $\phi\colon M\to\phi (M)\subset V$ satisfying the following properties:
\begin{enumerate}[\rm (i)]
\item[\rm (viii)] $\phi \colon \mathring M \to \phi(\mathring M)$ is a biholomorphism.
\item[\rm (ix)] $\phi$ is as close as desired to the identity in the $\mathscr{C}^1$ topology on $M \setminus \bigcup_{(i,j)\in \mathtt{J}} W'_{i,j}$.
\item[\rm (x)] $\phi(p_{i,j}) = q_{i,j}\in b\,\phi(M)$ and $\phi(M \cap W'_{i,j}) \subset W_{i,j}\cup V_{i,j}$ for all $(i,j)\in \mathtt{J}$.
\end{enumerate}
Set $F_0:=G\circ\phi\in\CMI_*^1(M,\mathbb{R}^n)$.
If $\epsilon_2>0$ is small enough and the approximation in {\rm (ix)} is close enough, then $F_0$ satisfies the conclusion of the lemma.
Indeed, {\rm (P1)} is ensured by {\rm (i)}, {\rm (iii)}, and {\rm (ix)};
{\rm (P2)} by {\rm (vii)} and {\rm (x)}; {\rm (P3)} by {\rm (ix)}, {\rm (x)}, {\rm (v)}, and {\rm (vi)}; {\rm (P4)} by {\rm (ix)},
{\rm (i)}, {\rm (iii)}, \eqref{eq:eta0}, {\rm (x)}, {\rm (vi)}, and \eqref{eq:Ox}; and {\rm (P5)} by {\rm (iv)} and the definition of $F_0$.
\end{proof}
We continue with the proof of Lemma \ref{lem:proper}.
Let $F_0\in\CMI_*^1(M,\mathbb{R}^n)$ be provided by Lemma \ref{lem:arcs} for some $\epsilon_1>0$ which will be specified later.
In view of {\rm (P2)} and {\rm (P3)}, each arc $\alpha_{i,j}$ contains a proper connected compact subarc
$I_{i,j}\Subset\alpha_{i,j}$ such that
\begin{equation}\label{eq:IO}
F_0(\overline{\alpha_{i,j}\setminus I_{i,j}})\subset (\mathscr{D}\setminus\overline\mathscr{D}_{\delta/2})\cap O_{i,j}.
\end{equation}
Here $O_{i,j}:=O_{\mathbf{x}_{i,j}}\in\mathcal{O}$ for a certain point $\mathbf{x}_{i,j}\in b\mathscr{L}$, cf.\ \eqref{eq:alphaU}.
Pick an annular neighborhood $A\subset M\setminus K$ of $bM$ and a smooth retraction $\rho\colon A\to bM$.
Choose pairwise disjoint, smoothly bounded closed disc $\overline D_{i,j}$ in $A$, $(i,j)\in\mathtt{J}$, such that
\begin{equation}\label{eq:Dij2}
I_{i,j}\Subset \overline D_{i,j}\cap\alpha_{i,j}, \quad
\rho(\overline D_{i,j})\subset \overline D_{i,j}\cap\alpha_{i,j},
\quad\text{and}\quad F_0(\overline D_{i,j})\subset O_{i,j}.
\end{equation}
Set $I:=\bigcup_{(i,j)\in\mathtt{J}} I_{i,j}$. Choose pairs of unitary orthogonal vectors
$\{\mathbf{u}_{i,j},\mathbf{v}_{i,j}\}\subset\langle \nu_\mathscr{L}(\mathbf{x}_{i,j})\rangle^\bot$, where $\mathbf{x}_{i,j}\in b\mathscr{L}$ were given in
\eqref{eq:alphaU}, $(i,j)\in \mathtt{J}$, and consider a continuous map
$\varkappa\colon bM \times\overline{\mathbb D}\to\mathbb{R}^n$ of the form
\[
\varkappa(x,\xi)=\left\{\begin{array}{ll}
F_0(x), & x\in bM\setminus I \\
F_0(x) + r(x)\,(\Re \sigma(x,\xi) \mathbf{u}_{i,j} + \Im \sigma(x,\xi) \mathbf{v}_{i,j}), & x\in I_{i,j},\; (i,j)\in\mathtt{J},
\end{array}\right.
\]
where $r\colon bM\to\mathbb{R}_+$ and $\sigma\colon I\times\overline\mathbb D\to\mathbb{C}$ are functions as in Theorem \ref{th:RH} such that
\begin{equation}\label{eq:convexdiscs}
\varkappa(bM\times \mathbb{T})\subset \mathscr{D}\setminus\overline\mathscr{D}_\delta.
\end{equation}
Such functions clearly exist; one can for instance take $r$ and $\sigma$ so that $\varkappa(x,\overline\mathbb D)$ is the planar
convex disc $\overline\mathscr{D}_{\delta/2}\cap (F_0(x)+\mathrm{span}\{\mathbf{u}_{i,j},\mathbf{v}_{i,j}\})$ for all $x\in bM$.
From properties {\rm (P3)}, \eqref{eq:Ox1}, \eqref{eq:IO}, and \eqref{eq:convexdiscs}
we infer that
\begin{equation}\label{eq:varDT}
\varkappa(bM\times \overline\mathbb D)\subset \mathscr{D}\setminus\overline\mathscr{L}_{-\varsigma/2}.
\end{equation}
Fix a number $\epsilon_3>0$ which will be specified later. Theorem \ref{th:RHCMI} furnishes arbitrarily small open neighborhoods
$\Omega_{i,j}\subset \overline D_{i,j}$ of $I_{i,j}$ in $M$, $(i,j)\in \mathtt{J}$, and a conformal minimal immersion
$\wt F\in \CMI_*^1(M,\mathbb{R}^n)$ satisfying the following properties:
\begin{enumerate}[\rm ({P}1)]
\item[\rm (P6)] $\mathrm{dist}(\wt F(x),\varkappa(x,\mathbb{T}))<\epsilon_3$ for all $x\in bM$.
\item[\rm (P7)] $\mathrm{dist}(\wt F(x),\varkappa(\rho(x),\overline{\mathbb D}))<\epsilon_3$ for all $x\in \Omega:=\bigcup_{(i,j)\in\mathtt{J}}\Omega_{i,j}\subset M\setminus K$.
\item[\rm (P8)] $\wt F$ is $\epsilon_3$-close to $F_0$ in the $\mathscr{C}^1$ topology on $M\setminus \Omega$.
\item[\rm (P9)] $\mathrm{Flux} (\wt F)=\mathrm{Flux} (F_0)$.
\end{enumerate}
Let us check that the immersion $\wt F$ satisfies the conclusion of Lemma \ref{lem:proper} provided that
the positive numbers $\epsilon_1$ and $\epsilon_3$ are small enough.
First of all, properties {\rm (P8)} and {\rm (P1)} ensure that $\wt F$ is as close to $F$ in the $\mathscr{C}^1(K)$ topology as desired
if $\epsilon_1$ and $\epsilon_3$ are small enough (observe that $K\subset M\setminus \Omega$).
Pick a point $p\in bM$ and let $(i,j)\in\mathtt{J}$ with $p\in\alpha_{i,j}$.
In view of {\rm (P3)}, \eqref{eq:varDT}, \eqref{eq:Ox1}, and {\rm (P6)}, we have
\begin{equation}\label{eq:wtFp}
\wt F(p)\in \wt O_{\mathbf{x}_{i,j}} \setminus\overline\mathscr{D}_\delta \subset \mathscr{D}.
\end{equation}
This proves condition {\rm (c)} in the lemma. Since $\mathscr{L}_{-\eta}$ is smoothly bounded and strictly convex, it is contained in the
Euclidean ball in $\mathbb{R}^n$ centered at $\mathbf{x}_{i,j}+\frac1{\kappa_\mathscr{L}^{\rm min}}\nu_\mathscr{L}(\mathbf{x}_{i,j})$ with radius
$1/\kappa^{\rm min}_{\mathscr{L}_{-\eta}}=\eta + 1/{\kappa_\mathscr{L}^{\rm min}}$; cf.\ \eqref{eq:curvature}. Therefore,
taking into account that $\mathscr{D}\subset \mathscr{L}_{-\eta}$ and \eqref{eq:wtFp}, Pythagoras' theorem ensures that
\[
\|\wt F(p)- (\mathbf{x}_{i,j}-t\nu_\mathscr{L}(\mathbf{x}_{i,j}))\|<\sqrt{2\eta^2+\big(\frac2{\kappa_\mathscr{L}^{\rm min}}
-\varsigma\big)\eta-\frac{\varsigma}{\kappa_\mathscr{L}^{\rm min}}} \quad \text{for all $t\in \big(\frac{\varsigma}2,\eta\big)$.}
\]
Since $F(p)$ lies in the convex domain $\wt O_{\mathbf{x}_{i,j}}\subset\mathscr{L}_{-\eta}$ (see \eqref{eq:alphaU} and \eqref{eq:Ox1}),
we have that $t_p:=\langle F(p)-\mathbf{x}_{i,j}, -\nu_\mathscr{L}(\mathbf{x}_{i,j})\rangle \in (\varsigma/2,\eta)$.
Together with \eqref{eq:alphaU} and \eqref{eq:Ox}, and taking into account \eqref{eq:curvature}, basic trigonometry gives
\[
\|F(p)-(\mathbf{x}_{i,j}-t_p\nu_\mathscr{L}(\mathbf{x}_{i,j}))\|<c (\eta \kappa_{\mathscr{L}}^{\rm max}+1) \sqrt{1-c^2 (\kappa_{\mathscr{L}}^{\rm max})^2/4},
\]
where $c>0$ is the constant given in \eqref{eq:cball}.
The last two inequalities ensure $\|\wt F(p)-F(p)\|< \sqrt{2\eta^2+2\eta/\kappa_\mathscr{L}^{\rm min}}$, proving {\rm (a)},
provided that $c>0$ is chosen small enough.
In order to check {\rm (b)}, notice that, if $\epsilon_3>0$ is sufficiently small, {\rm (P8)} and {\rm (P4)} give that
$\wt F(M\setminus (K\cup\Omega))\subset \mathscr{D}\setminus \overline\mathscr{L}_{-\varsigma}$. On the other hand, {\rm (P7)}
and \eqref{eq:varDT} guarantee that $\wt F(\Omega)\subset \mathscr{D}\setminus \overline\mathscr{L}_{-\varsigma/2}$ as well.
Finally, {\rm (P5)} and {\rm (P9)} trivially imply {\rm (d)}. This concludes the proof of Lemma \ref{lem:proper}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:proper2}]
By the Mergelyan theorem for conformal minimal immersions \cite{AFL}, we may assume that $F\in\CMI_*^1(M,\mathbb{R}^n)$.
Moreover, if $n\geq 5$, we may also assume that $F$ is an embedding; see Theorem \ref{th:gp}.
Let $\mu >0$ and $\eta>0$ be as in the theorem. Since $F(bM)\subset \mathscr{D}\setminus\overline\mathscr{L}$, there exist
a number $\epsilon >0$ and a smoothly bounded compact domain $K_0\subset \mathring M$ which is a
strong deformation retract of $M$ such that $K\subset K_0$, $\overline \mathscr{L}_{-\epsilon}\subset \mathscr{D}$, and
\[
F(M\setminus \mathring K_0)\subset \mathscr{D}\setminus\overline\mathscr{L}_{-\epsilon}.
\]
Since $\epsilon>0$, it follows from \eqref{eq:curvature} that
$\sqrt{2(\eta-\epsilon)^2+{2(\eta-\epsilon)}/{\kappa_{\mathscr{L}_{-\epsilon}}^{\rm min}}}< \sqrt{2\eta^2+{2\eta}/{\kappa_\mathscr{L}^{\rm min}}}$;
we therefore may choose a sequence $-1/\kappa_\mathscr{D}^{\rm max}>\delta_1>\delta_2>\cdots >\lim_{j\to\infty}\delta_j=0$ satisfying
$\overline\mathscr{L}_{-\epsilon}\subset \mathscr{D}_{\delta_1}$ and
\begin{equation}\label{eq:sumetas}
\sqrt{2(\eta-\epsilon)^2+\frac{2(\eta-\epsilon)}{\kappa_{\mathscr{L}_{-\epsilon}}^{\rm min}}}+
\sum_{j\geq 1} \sqrt{2\delta_j^2+\frac{2\delta_j}{\kappa_\mathscr{D}^{\rm min}}} <
\sqrt{2\eta^2+\frac{2\eta}{\kappa_\mathscr{L}^{\rm min}}}.
\end{equation}
Set $F_0:=F$, $\delta_0:=\eta-\epsilon$, $\mathscr{B}^0:=\mathscr{L}_{-\epsilon}$, and $\mathscr{B}^j:=\mathscr{D}_{\delta_j}$ for all $j\in\mathbb{N}$.
Fix a point $p_0\in \mathring K$ and a number $\epsilon_0$ with $0<\epsilon_0<\mu$.
By recursively applying Lemma \ref{lem:proper}, Lemma \ref{lem:Jordan2}, and Theorem \ref{th:gp} we may
construct a sequence $\{\Xi_j=(K_j, F_j, \epsilon_j)\}_{j\in\mathbb{N}}$, where
$K_j$ is a smoothly bounded compact domain in $\mathring M$ which is a strong deformation retract of $M$
and we have $\bigcup_{j\in\mathbb{N}} K_j=\mathring M$, $F_j\in\CMI_*^1(M,\mathbb{R}^n)$, and $\epsilon_j>0$,
satisfying the following conditions for all $j\in\mathbb{N}$:
\begin{enumerate}[\rm (a{$_j$})]
\item $K_{j-1}\subset \mathring K_j$.
\item $F_j$ is $\epsilon_j$-close to $F_{j-1}$ in the $\mathscr{C}^1(K_{j-1})$ topology.
\item $\|F_j-F_{j-1}\|_{0,M}<\sqrt{2\delta_{j-1}^2+{2\delta_{j-1}}/{\kappa_{\mathscr{B}^{j-1}}^{\rm min}}}$.
\item $F_j(M\setminus \mathring K_{j-1})\subset \mathscr{D}\setminus \overline{\mathscr{B}^{j-1}}$.
\item $F_j(M\setminus \mathring K_j)\subset \mathscr{D}\setminus \overline{\mathscr{B}^j}$.
\item $\mathrm{Flux}(F_j)=\mathrm{Flux} (F)$.
\item $\mathrm{dist}_{F_j}(p_0, bK_i)>i$ for all $i\in\{0,\ldots,j\}$.
\item If $n\geq 5$, then $F_j$ is an embedding.
\item If $n\ge 5$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j,\varsigma_j\right\}$,
where the number $\tau_j$ is defined by (\ref{eq:tauj}) and
\[
\varsigma_j= \frac1{2j^2} \inf\left\{ \|F_{j-1}(p)-F_{j-1}(q)\| : p,q\in M,\ {\rm d}(p,q)>\frac1{j}\right\}.
\]
If $n=3,4$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j\right\}$.
\end{enumerate}
Notice that $\Xi_0=(F_0,K_0,\epsilon_0)$ satisfies {\rm (e$_0$)}, {\rm (f$_0$)}, {\rm (g$_0$)}, and {\rm (h$_0$)},
whereas the other conditions are void for $j=0$. Let $j\in\mathbb{N}$ and assume the existence of triples $\Xi_0,\ldots, \Xi_{j-1}$
enjoying these conditions. Fix $\epsilon_j>0$ such that (${\rm i}_j$) holds. Applying Lemma \ref{lem:proper}
to the data
\[
\mathscr{L}=\mathscr{B}^{j-1},\quad \mathscr{D},\quad \eta=\delta_{j-1},\quad M,\quad F=F_{j-1},\quad K=K_{j-1},\quad \delta=\delta_j,
\]
we obtain $F_j\in\CMI_*^1(M,\mathbb{R}^n)$ satisfying {\rm (b$_j$)}, {\rm (c$_j$)}, {\rm (d$_j$)}, {\rm (f$_j$)},
$F_j(bM)\subset \mathscr{D}\setminus\overline{\mathscr{B}^j}$, and $\mathrm{dist}_{F_j}(p_0, bK_i)>i$ for all $i\in\{0,\ldots,j-1\}$.
Therefore we may choose $K_j\Subset M$ fulfilling conditions {\rm (a$_j$)} and {\rm (e$_j$)}.
We now apply Lemma \ref{lem:Jordan2} to approximate $F_j$ uniformly on $M$
by a conformal minimal immersion $\wh F_j\in \CMI_*^1(M,\mathbb{R}^n)$ such that $\wh F_j(M) \subset \mathscr{D}$,
$\mathrm{Flux}(\wh F_j)=\mathrm{Flux} (F)$, and $\mathrm{dist}_{\wh F_j}(p_0,bM) > j$.
Assuming as we may that the approximation is close enough,
$\wh F_j$ satisfies all the properties of $F_j$ that we have verified so far.
Replacing $F_j$ by $\wh F_j$ and enlarging the set $K_j$ if necessary we may assume that
condition {\rm (g$_j$)} holds as well.
Finally, Theorem \ref{th:gp} enables us to ensure condition {\rm (h$_j$)}, thereby closing the induction.
Properties $({\rm c}_j)$ and \eqref{eq:sumetas} guarantee that the sequence $\{F_j\}_{j\in\mathbb{N}}$ converges in the
$\mathscr{C}^0(M)$ topology to a continuous map $\wt F= \lim_{j\to+\infty} F_j\colon M\to\mathbb{R}^n$ satisfying
Theorem \ref{th:proper2} {\rm (iv)}; take into account that $\mathscr{B}^j=\mathscr{D}_{\delta_j}$ and so
$\kappa_{\mathscr{B}^j}^{\rm min}>\kappa_{\mathscr{D}}^{\rm min}$ for all $j\in\mathbb{N}$ (cf.\ \eqref{eq:curvature}).
From conditions {\rm (b$_j$)} and {\rm (i$_j$)} we obtain that
\begin{equation}\label{eq:properness}
\text{$\wt F$ is $\epsilon_j$-close to $F_j$ in the $\mathscr{C}^1(K_j)$ topology for all $j\in\mathbb{Z}_+$.}
\end{equation}
In particular, $\wt F$ is $\epsilon_0$-close to $F_0=F$ in the $\mathscr{C}^1(K)$ topology;
since $\epsilon_0<\mu$, property {\rm (v)} in Theorem \ref{th:proper2} holds.
Furthermore, as in the proof of Theorem \ref{th:Jordan}, we see that conditions
{\rm (g$_j$)}, {\rm (h$_j$)}, {\rm (i$_j$)}, and \eqref{eq:properness} ensure that $\wt F|_{\mathring M}\colon \mathring M\to\mathbb{R}^n$
is a complete minimal immersion which is an embedding if $n\geq 5$.
Finally, property {\rm (f$_j$)} give that $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F)$, whereas {\rm (d$_j$)} and \eqref{eq:properness} guarantee
that $\wt F(\mathring M)\subset\mathscr{D}$ and $\wt F|_{\mathring M}\colon \mathring M\to \mathscr{D}$ is a proper map; recall that
$\mathscr{B}^j=\mathscr{D}_{\delta_j}$ for all $j\in\mathbb{N}$ and that $\{\delta_j\}_{j\in\mathbb{N}}\searrow 0$.
Since $\wt F$ is continuous on $M$, it follows that $\wt F(bM)\subset b\mathscr{D}$ is a finite
family of curves.
\end{proof}
\begin{remark}\label{rem:Jordan}
Our method does not ensure that the map $\wt F|_{bM}\colon bM\to\mathbb{R}^n$ in Theorem \ref{th:proper2} is an embedding.
The reason is that, at each step in the recursive procedure, we can only assert that $F_j$ is
$\sqrt{2\delta_{j-1}^2+{2\delta_{j-1}}/{\kappa_{\mathscr{D}}^{\rm min}}}$-close to $F_{j-1}$ in the
$\mathscr{C}^0(bM)$ topology (cf.\ {\rm (c$_j$)}), and the number $\delta_{j-1}$ is given a priori in the construction of
$F_{j-1}$ (or in other words, $F_{j-1}$ depends on $\delta_{j-1}$). To guarantee embeddedness of $\wt F(bM)$ a more
accurate approximation, depending on the geometry of $F_{j-1}(bM)$, would be required.
\qed\end{remark}
\section{Proof of Theorems \ref{th:proper}, \ref{th:topology} and Corollary \ref{co:bdddomains}} \label{sec:topology}
\subsection{Proof of Theorem \ref{th:proper}}
Part (a) is a direct consequence of Theorem \ref{th:proper2};
indeed for $F$ and $\mathscr{D}$ as in Theorem \ref{th:proper} {\rm (a)}, just take any smoothly bounded,
strictly convex domain $\mathscr{L}\Subset \mathscr{D}$ with $F(bM)\cap \overline\mathscr{L}=\emptyset$ and apply Theorem \ref{th:proper2}.
We now prove part (b).
Let $F\colon M\to\overline \mathscr{D}$ be as in Theorem \ref{th:proper} {\rm (b)} and let $\epsilon>0$. Up to a translation we may
assume without loss of generality that the origin $0\in\mathbb{R}^n$ lies in $\mathscr{D}$. Set $d:=\max\{\|x\|\colon x\in b\mathscr{D}\}>0$.
Fix $\lambda\in(0,\min\{1,1/2d\kappa_{\mathscr{D}}^{\rm max}\})$ to be specified later. Set $F_0:=(1-\lambda) F\in\CMI^1(M,\mathbb{R}^n)$
and observe that $F_0(bM)\subset \mathscr{D}\setminus\overline \mathscr{D}_{2\lambda d}$ and
\begin{equation}\label{eq:co1}
\|F_0-F\|_{0,M}\leq \lambda d.
\end{equation}
Theorem \ref{th:proper2} applied to the data $\mathscr{L}=\mathscr{D}_{2\lambda d},$ $\mathscr{D}$, $\eta=2\lambda d$, and $F=F_0$,
furnishes a continuous map $\wt F\colon M\to \overline\mathscr{D}$ such that $\wt F(\mathring M)\subset \mathscr{D}$,
$\wt F|_{\mathring M}\colon \mathring M\to\mathscr{D}$ is a conformal complete proper minimal immersion
(embedding if $n\geq 5$), $\mathrm{Flux}(\wt F)=\mathrm{Flux}(F_0)$, and
\[
\big\|\wt F-F_0\big\|_{0,M}<\sqrt{8\lambda^2 d^2+{4\lambda d}\left(\frac1{\kappa_\mathscr{D}^{\rm min}}-2\lambda d\right)}
\]
(take into account \eqref{eq:curvature}). Together with \eqref{eq:co1} we obtain that $\|\wt F-F\|_{0,M}<\epsilon$
provided that $\lambda>0$ is chosen sufficiently small. This shows that the flux can be changed by an arbitrarily small amount
when passing from $F$ to $\wt F$. This completes the proof of Theorem \ref{th:proper}.
\subsection{Proof of Theorem \ref{th:topology}}
Let $D\subset \mathbb{R}^n$ $(n\geq 3)$ be a convex domain. Take an exhaustion
$\mathscr{B}^0\Subset\mathscr{B}^1\Subset\cdots \Subset \cup_{j\in\mathbb{Z}_+} \mathscr{B}^j=D$ of $D$ by smoothly bounded, strictly convex domains
$\mathscr{B}^j\subset\mathbb{R}^n$. Choose a sequence $\{\lambda_j\}_{j\in\mathbb{Z}_+}\searrow 0$ with $0<\lambda_j<1/\kappa_{\mathscr{B}^j}^{\rm max}$,
and denote by $\delta_j$ the Hausdorff distance between
$\overline{\mathscr{B}^j_{\lambda_j}}$ and $\overline{\mathscr{B}^{j+1}}$ for all $j\in\mathbb{Z}_+$. It follows that $\delta_j>\lambda_j$ and
$\mathscr{B}^{j+1}\subset\mathscr{B}^j_{-\delta_j+\lambda_j}$ for all $j\in\mathbb{Z}_+$. (Observe that possibly $\mathscr{B}^j_{-\delta_j+\lambda_j}\nsubseteq D$.)
\medskip
\noindent{\em Proof of part (a)}. Let $M$ be a compact bordered Riemann surface. Let $F_0\in\CMI_*^1(M,\mathbb{R}^n)$ be an immersion
(embedding if $n\geq 5$) satisfying $F_0(bM)\subset \mathscr{B}^1\setminus\overline{\mathscr{B}^0}$. Choose $K_0$ any smoothly bounded
compact domain in $\mathring M$ which is a strong deformation retract of $M$ and with
$F_0(M\setminus \mathring K_0)\subset \mathscr{B}^0\setminus\overline{\mathscr{B}^0_{\lambda_0}}$, and any number $\epsilon_0>0$.
As in the proof of Theorem \ref{th:proper2}, we may recursively apply Lemma \ref{lem:proper}, Lemma \ref{lem:Jordan2},
and Theorem \ref{th:gp} in order to construct a sequence $\{\Xi_j=(K_j, F_j, \epsilon_j)\}_{j\in\mathbb{N}}$, where
$K_j$ is a smoothly bounded compact domain in $\mathring M$ which is a strong deformation retract of $M$ and we have
$\bigcup_{j\in\mathbb{N}} K_j=\mathring M$, $F_j\in\CMI_*^1(M,\mathbb{R}^n)$, and $\epsilon_j>0$,
satisfying the following conditions for all $j\in\mathbb{N}$:
\begin{enumerate}[\rm (a{$_j$})]
\item $K_{j-1}\subset \mathring K_j$.
\item $\|F_j-F_{j-1}\|_{1,K_{j-1}}<\epsilon_j$.
\item $\|F_j-F_{j-1}\|_{0,M}<\sqrt{2\delta_{j-1}^2+{2\delta_{j-1}}/{\kappa_{\mathscr{B}_{\lambda_{j-1}}^{j-1}}^{\rm min}}}$.
\item $F_j(M\setminus \mathring K_{j-1})\subset \mathscr{B}^j\setminus \overline{\mathscr{B}_{\lambda_{j-1}}^{j-1}}$.
\item $F_j(M\setminus \mathring K_j)\subset \mathscr{B}^j\setminus \overline{\mathscr{B}_{\lambda_j}^j}$.
\item $\mathrm{Flux}(F_j)=\mathrm{Flux} (F_0)$.
\item $\mathrm{dist}_{F_j}(p_0, bK_i)>i$ for all $i\in\{0,\ldots,j\}$.
\item If $n\geq 5$, then $F_j$ is an embedding.
\item If $n\ge 5$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j,\varsigma_j\right\}$,
where the number $\tau_j$ is defined by (\ref{eq:tauj}) and
\[
\varsigma_j= \frac1{2j^2} \inf\left\{ \|F_{j-1}(p)-F_{j-1}(q)\| : p,q\in M,\ {\rm d}(p,q)>\frac1{j}\right\}.
\]
If $n=3,4$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j\right\}$.
\end{enumerate}
(Property {\rm (c$_j$)} is useless in this proof and can be ruled out. In fact, unlikely in Theorem \ref{th:proper2}, it does not enable us
to ensure that the sequence $\{F_j\}_{j\in\mathbb{Z}_+}$ converges up to $bM$; see Remark \ref{rem:continuous} for a more detailed explanation.)
In this case, to pass from $F_{j-1}$ to $F_j$ in the inductive step we apply Lemma \ref{lem:proper} to the data
\[
\mathscr{L}=\mathscr{B}_{\lambda_{j-1}}^{j-1},\quad \mathscr{D}=\mathscr{B}^j,\quad \eta=\delta_{j-1},\quad F=F_{j-1},\quad \delta=\lambda_j.
\]
As in the proof of Theorem \ref{th:proper2}, and taking into account that $D=\cup_{j\in\mathbb{Z}_+}\mathscr{B}^j$, these properties ensure that
$\{F_j\}_{j\in\mathbb{N}}$ converges uniformly on compact subsets of $\mathring M$ to a conformal complete proper minimal immersion
(embedding if $n\geq 5$) $\wt F\colon \mathring M\to D$. Furthermore, since $F_0$ is full then $\wt F$ is also full provided the
$\epsilon_j$'s are chosen small enough at each step. This concludes the proof of part {\rm (a)}.
\medskip
\noindent{\em Proof of part (b)}.
Let $\wt M$ be an open Riemann surface and let $\mathfrak{p}\colon H_1(\wt M;\mathbb{Z})$ be a group homomorphism.
Exhaust $\wt M$ by an increasing sequence $M_0\subset M_1\subset\cdots\subset \bigcup_{j=0}^\infty M_j=\wt M$ of
compact smoothly bounded connected Runge regions such that $M_0$ is a disc and the Euler characteristic of
$M_j\setminus\mathring M_{j-1}$ satisfies $\chi(M_j\setminus\mathring M_{j-1})\in\{0,-1\}$ for all $j\in\mathbb{N}$.
Set $K_0:=M_0$ and let $F_0\in\CMI_*^1(K_0,\mathbb{R}^n)$ be an immersion (embedding if $n\geq 5$) satisfying
$F_0(K_0)\subset \mathscr{B}^0\setminus\overline{\mathscr{B}^0_{\lambda_0}}$. Fix $\epsilon_0>0$ and a point $p_0\in \mathring M$.
We shall construct a sequence $\{\Xi_j=(K_j, F_j, \epsilon_j)\}_{j\in\mathbb{Z}_+}$ where $K_j\subset M_j$ is a smoothly bounded
compact Runge domain which is a strong deformation retract of $M_j$, $F_j\in\CMI_*^1(K_j,\mathbb{R}^n)$, and $\epsilon_j>0$,
satisfying the following conditions:
\begin{enumerate}[\rm (a{$_j$})]
\item[\rm (a{$_j$})] $K_{j-1}\subset \mathring K_j$.
\item[\rm (b{$_j$})] $\|F_j-F_{j-1}\|_{1,K_{j-1}}<\epsilon_j$.
\item[\rm (d{$_j$})] $F_j(K_j\setminus \mathring K_{j-1})\subset \mathscr{B}^j\setminus \overline{\mathscr{B}_{\lambda_{j-1}}^{j-1}}$.
\item[\rm (e{$_j$})] $F_j(b K_j)\subset \mathscr{B}^j\setminus \overline{\mathscr{B}_{\lambda_j}^j}$.
\item[\rm (f{$_j$})] $\mathrm{Flux}(F_j)=\mathfrak{p}|_{H_1(K_j;\mathbb{Z})}$.
\item[\rm (g{$_j$})] $\mathrm{dist}_{F_j}(p_0, bK_i)>i$ for all $i\in\{0,\ldots,j\}$.
\item[\rm (h{$_j$})] If $n\geq 5$, then $F_j$ is an embedding.
\item[\rm (i{$_j$})] If $n\ge 5$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j,\varsigma_j\right\}$,
where the number $\tau_j$ is defined by (\ref{eq:tauj}) and
\[
\varsigma_j= \frac1{2j^2} \inf\left\{ \|F_{j-1}(p)-F_{j-1}(q)\| : p,q\in K_{j-1},\ {\rm d}(p,q)>\frac1{j}\right\}.
\]
If $n=3,4$ then $\epsilon_j<\min\left\{ \epsilon_{j-1}/2, \tau_j\right\}$.
\end{enumerate}
(Observe that there is no property {\rm (c{$_j$})} in the above list; this is not a misprint, we labeled the properties in this way in
order to emphasize that, under our current assumptions, a condition similar to {\rm (c{$_j$})} in the proof of part {\rm (a)} is not expected.)
The triple $\Xi_0=(K_0,F_0,\epsilon_0)$ meets the above conditions for $j=0$, except for {\rm (a$_0$)}, {\rm (b$_0$)}, {\rm (d$_0$)},
and {\rm (i$_0$)} which are void. For the inductive step, assume that we have triples $\Xi_0,\ldots,\Xi_{j-1}$ satisfying the required
properties for some $j\in\mathbb{N}$ and let us construct $\Xi_j$. Fix $\epsilon_j>0$ to be specified later. Let us distinguish cases depending
on whether the Euler characteristic $\chi(M_j\setminus\mathring M_{j-1})$ equals $0$ or $-1$.
\smallskip
\noindent{\em Case 1:} $\chi(M_j\setminus\mathring M_{j-1})=0$. In this case there
is no change of topology when passing from $M_{j-1}$ to $M_j$. Therefore, $K_{j-1}$ is a strong deformation retract of $M_j$.
By the Mergelyan Theorem for conformal minimal immersions \cite[Theorem 5.3]{AFL}
we may find a smoothly bounded compact region $K_j\subset M_j$ and may approximate $F_{j-1}$ by a map
$\wt F_j\in\CMI_*^1(K_j,\mathbb{R}^n)$ such that the triple $\wt\Xi_j=(K_j,\wt F_j,\epsilon_j)$ satisfies {\rm (a$_j$)}, {\rm (b$_j$)},
{\rm (d$_j$)}, {\rm (f$_j$)}, and {\rm (g$_j$)} for $i=0,\ldots,j-1$. (For {\rm (d$_j$)} take into account {\rm (e$_{j-1}$)}.)
Applying Lemma \ref{lem:proper} to the data
\[
M=K_j,\quad \mathscr{L}=\mathscr{B}_{\lambda_{j-1}}^{j-1},\quad \mathscr{D}=\mathscr{B}^j,\quad \eta=\delta_j,\quad F=\wt F_j,\quad \delta=\lambda_j,
\]
we obtain $F_j\in\CMI_*^1(K_j,\mathbb{R}^n)$ such that the triple $\Xi_j=(K_j,F_j,\epsilon_j)$ meets condition {\rm (e$_j$)} in addition to
the above properties. Finally, by Lemma \ref{lem:Jordan2} and Theorem \ref{th:gp}, we may also assume that {\rm (g$_j$)},
{\rm (h$_j$)}, and {\rm (i$_j$)} are also satisfied.
\smallskip
\noindent{\em Case 2:} $\chi(M_j\setminus\mathring M_{j-1})=-1$. In this case there exists a smooth arc
$\gamma\subset \mathring M_j\setminus \mathring K_{j-1}$ with both endpoints in $bK_{j-1}$ and otherwise disjoint with
$K_{j-1}$ such that $\chi(M_j\setminus(\mathring K_{j-1}\cup\gamma))=0$. We may also assume that
$S=K_{j-1}\cup\gamma\Subset M_j$ is an admissible subset in the sense of \cite[Def.\ 5.1]{AFL}.
Extend $F_{j-1}$ to a generalized conformal minimal immersion $(F_{j-1},f\theta)$ on $S$,
in the sense of \cite[Def.\ 5.2]{AFL}, such that
\begin{equation}\label{eq:tildeF}
\text{$F_{j-1}(\gamma)\subset \mathscr{B}^{j-1}\setminus \overline{\mathscr{B}_{\lambda_{j-1}}^{j-1}}$\; and\;
$\int_\alpha \Im(f\theta)=\mathfrak{p}(\alpha)$ for every closed curve $\alpha\subset S$;}
\end{equation}
take into account {\rm (e$_{j-1}$)} and {\rm (f$_{j-1}$)}.
By the Mergelyan theorem for conformal minimal immersions \cite[Theorem 5.3]{AFL} we may approximate $F_{j-1}$ on $S$ by maps $\wt F_{j-1}\in\CMI_*^1(M_j,\mathbb{R}^n)$.
If the approximation is close enough then, taking into account Theorem \ref{th:gp},
there exists a smoothly bounded compact Runge region $L_{j-1}\Subset M_j$ which is is a strong deformation retract of $M_j$
and satisfies $S\Subset L_{j-1}$ and
$\chi(M_j\setminus \mathring{L}_{j-1})=0$, such that the triple $\wt\Xi_{j-1}=(L_{j-1},\wt F_{j-1},\epsilon_{j})$ satisfies
\begin{enumerate}[\rm (a{$_{j}$})]
\item[\rm ($\wt{\rm a}${$_{j}$})] $K_{j-1}\Subset \mathring{L}_{j-1}\subset M_j$.
\item[\rm ($\wt{\rm c}${$_{j}$})] $\|\wt F_{j-1}-F_{j-1}\|_{1, K_{j-1}}<\epsilon_j/2$.
\item[\rm ($\wt{\rm d}${$_{j}$})] $\wt F_{j-1}(L_{j-1}\setminus \mathring K_{j-1})\subset \mathscr{B}^{j-1}\setminus \overline{\mathscr{B}_{\lambda_{j-1}}^{j-1}}$.
(Take into account {\rm (e$_{j-1}$)} and \eqref{eq:tildeF}.)
\item[\rm ($\wt{\rm f}${$_{j}$})] $\mathrm{Flux}(\wt F_{j-1})=\mathfrak{p}|_{H_1(L_{j-1};\mathbb{Z})}$.
(Take into account {\rm (f$_{j-1}$)} and \eqref{eq:tildeF}.)
\item[\rm ($\wt{\rm g}${$_{j}$})] $\mathrm{dist}_{\wt F_{j-1}}(p_0, bK_i)>i$ for all $i\in\{0,\ldots,j-1\}$.
(Take into account {\rm (g$_{j-1}$)}.)
In particular, $\mathrm{dist}_{\wt F_{j-1}}(p_0, bL_{j-1})>j-1$; see {\rm ($\wt{\rm a}$$_j$)}.
\item[\rm ($\wt{\rm h}${$_{j}$})] If $n\geq 5$ then $\wt F_{j-1}$ is an embedding.
\end{enumerate}
This reduces the proof to Case 1, closing the induction and concluding the construction of the sequence $\{\Xi_j\}_{j\in\mathbb{Z}_+}$.
Set $\mathcal{R}:=\bigcup_{j\in\mathbb{Z}_+} K_j$. Since $\bigcup_{j\in\mathbb{Z}_+} M_j=\wt M$ and $K_j$ is a strong deformation retract of $M_j$
for all $j\in\mathbb{Z}_+$, property {\rm (a$_j$)} ensures that $\mathcal{R}\subset \wt M$ is an open domain homeomorphic to $\wt M$.
Given $\epsilon>0$, properties {\rm (b$_j$)} and {\rm (i$_j$)} guarantee that we may choose the numbers $\epsilon_j>0$ small enough
in the inductive construction so that the sequence $\{F_j\}_{j\in\mathbb{Z}_+}$ converges uniformly on compact subsets of $\mathcal{R}$ to a
conformal minimal immersion $\wt F:=\lim_{j\to\infty} F_j\colon \mathcal{R}\to \mathbb{R}^n$ which is $\epsilon$-close to $F$ in the
$\mathscr{C}^1(M)$ topology; recall that $K_0=M$. Further, if the $\epsilon_j$'s are chosen sufficiently small, conditions
{\rm (d$_j$)}, {\rm (f$_j$)}, {\rm (g$_j$)}, and {\rm (h$_j$)} guarantee that $\wt F(\mathcal{R})\subset D$, $\wt F\colon\mathcal{R}\to D$
is a proper map, $\mathrm{Flux}(\wt F)=\mathfrak{p}$, $\wt F$ is complete, and, if $n\geq 5$, $\wt F$ is an embedding.
(Recall that $\cup_{j\in\mathbb{Z}_+}\mathscr{B}^j=D$.) See the proof of Theorem \ref{th:proper2} for details on how to check these properties.
This concludes the proof of part {\rm (b)}.
\begin{remark}\label{rem:continuous}
Our method does not ensure that $\wt F\colon \mathring M\to D$ in Theorem \ref{th:topology} {\rm (a)}
extends continuously up to $bM$. The reason is that, at each step in the recursive process, we only have that $F_j$ is
$\sqrt{2\delta_{j-1}^2+{2\delta_{j-1}}/{\kappa_{\mathscr{B}_{\lambda_{j-1}}^{j-1}}^{\rm min}}}$-close to $F_{j-1}$
in the $\mathscr{C}^0(bM)$ topology (see {\rm (c$_j$)}). Since the domains $\mathscr{B}^j$'s need not be parallel to each other,
this sequence is not necessarily Cauchy (in fact neither
$\{\delta_j\}_{j\in\mathbb{Z}_+}$ nor $\{1/\kappa_{\mathscr{B}_{\lambda_j}^{j-1}}^{\rm min}\}_{j\in\mathbb{Z}_+}$ need to
be bounded sequences in general) and so we do not get convergence of the sequence $\{F_j\}_{j\in\mathbb{Z}_+}$ up to $bM$.
\qed \end{remark}
\subsection{Proof of Corollary \ref{co:bdddomains}}
Let $D$ be a domain in $\mathbb{R}^n$ with a smooth strictly convex boundary point $x\in bD$, that is to say, $bD$ is smooth and
has positive principal curvatures with respect to the inner normal in a neighborhood of $x$.
There exist a number $r>0$ and a smoothly bounded, strictly convex domain $\mathscr{D}\subset D$ such that $x\in b\mathscr{D}$ and
$U:=b\mathscr{D}\cap \mathbb{B}(x,r)\subset bD$, where $\mathbb{B}(x,r)\subset\mathbb{R}^n$ denotes the Euclidean ball centered at $x$ with radius $r$.
Fix a number $\lambda\in(0,1/\kappa_{\mathscr{D}}^{\rm max})$ to be specified later.
\medskip \noindent {\em Proof of part (a).}
Let $M$ be a compact bordered Riemann surface and let $F\in\CMI^1_*(M,\mathbb{R}^n)$ be a conformal minimal immersion satisfying
$F(M)\subset \mathscr{D}\setminus\overline\mathscr{D}_\lambda$ and
\begin{equation}\label{eq:co2}
\|F(p)-x\|<\lambda\quad \text{for all $p\in M$.}
\end{equation}
Theorem \ref{th:proper2}, applied to the data $\mathscr{L}=\mathscr{D}_\lambda,$ $\mathscr{D}$, $\eta=\lambda$, and $F$, furnishes a continuous map
$\wt F\colon M\to \overline\mathscr{D}$ such that $\wt F(\mathring M)\subset \mathscr{D}\subset D$,
$\wt F|_{\mathring M}\colon \mathring M\to\mathscr{D}$ is a conformal complete proper minimal immersion (embedding if $n\geq 5$),
and
\[
\big\|\wt F-F\big\|_{0,M}<\sqrt{2\lambda^2 +{2\lambda}\left(\frac1{\kappa_\mathscr{D}^{\rm min}}-\lambda\right)}
\]
(take into account \eqref{eq:curvature}). In view of \eqref{eq:co2} we get that $\|\wt F(p)-x\|<r$ for all $p\in M$, provided that
$\lambda>0$ is chosen small enough. Since $\wt F(bM)\subset b\mathscr{D}$, we obtain that $\wt F(bM)\subset U\subset bD$ and hence
$\wt F|_{\mathring M}\colon \mathring M\to D$ is proper. Finally, since $F$ is full, $\wt F$ is also full provided the
approximation is close enough. This completes the proof of part (a).
\medskip \noindent {\em Proof of part (b).}
Pick a number $r'$ with $0<r'<r$ and a decreasing sequence
$\{\lambda_j\}_{j\in\mathbb{Z}_+}\searrow 0$ with $0<\lambda_j<\min\{r',1/2\kappa_{\mathscr{D}}^{\rm max}\}$ for all $j\in\mathbb{Z}_+$.
These constants will be specified later. Set $\mathscr{B}^j=\mathscr{D}_{\lambda_j}$ and $\delta_j=\lambda_j-\lambda_{j+1}$ for all $j\in\mathbb{Z}_+$.
Let $\wt M$, $\mathfrak{p}$, and $\{M_j\}_{j\in\mathbb{Z}_+}$ be as in the proof of Theorem \ref{th:topology} {\rm (b)}.
Let $F_0\colon M_0\to \mathbb{R}^n$ be a nondegenerate conformal minimal immersion with
$F_0(M_0)\subset \mathbb{B}(x,r')\cap(\mathscr{B}^0\setminus\overline{\mathscr{B}^0_{\lambda_0}})$.
As in the proof of Theorem \ref{th:topology} {\rm (b)} we may recursively construct a sequence
$\{\Xi_j=(K_j,F_j,\epsilon_j)\}_{j\in\mathbb{Z}_+}$ satisfying conditions {\rm (a$_j$)}, {\rm (b$_j$)}, {\rm (d$_j$)},{\rm (e$_j$)},
{\rm (f$_j$)}, {\rm (g$_j$)}, {\rm (h$_j$)}, and {\rm (i$_j$)} there, and also
\begin{enumerate}[\rm (a$_j$)]
\item[\rm (c$_j$)] $F_j(K_j)\subset \mathbb{B}\Big(x,r'+\sum_{i=1}^j\sqrt{2\delta_{i-1}^2+{2\delta_{i-1}}/{\kappa_{\mathscr{B}^{i-1}_{\lambda_{i-1}}}^{\rm min}}}\Big)$ for all $j\in\mathbb{Z}_+$.
\end{enumerate}
Indeed, this extra condition is directly granted by Lemma \ref{lem:proper} {\rm (a)} when passing from $\Xi_{j-1}$ to $\Xi_j$;
in case $\chi(M_j\setminus \mathring M_{j-1})=-1$ we take the arc $\gamma$ so that
$F_{j-1}(\gamma)\subset (\mathscr{B}^{j-1}\setminus\overline{\mathscr{B}^{j-1}_{\lambda_{j-1}}}) \cap \mathbb{B}\Big(x,r'+\sum_{i=1}^{j-1}\sqrt{2\delta_{i-1}^2+{2\delta_{i-1}}/{\kappa_{\mathscr{B}^{i-1}_{\lambda_{i-1}}}^{\rm min}}}\Big)$,
which is possible in view of {\rm (c$_{j-1}$)} and {\rm (e$_{j-1}$)}.
Taking into account that $\delta_j=\lambda_j-\lambda_{j+1}$ and
$1/\kappa_{\mathscr{B}^j_{\lambda_j}}^{\rm min}=-2\lambda_j+1/\kappa_{\mathscr{D}}^{\rm min}$ (cf.\ \eqref{eq:curvature}),
the above properties ensure that the sequence $\{F_j\}_{j\in\mathbb{Z}_+}$ converges to a conformal complete proper nondegenerate
minimal immersion (embedding if $n\geq 5$) $\wt F\colon \mathcal{R}\to\mathscr{D}$, where $\mathcal{R}=\cup_{j\in\mathbb{Z}_+} M_j$ is homeomorphic to
$\wt M$, satisfying $\mathrm{Flux}(\wt F)=\mathfrak{p}$ and $\wt F(\mathcal{R})\subset b\mathscr{D}\cap\mathbb{B}(x,r)\subset bD$, provided that $r'$ and the $\lambda_j$'s
are chosen small enough. This concludes the proof.
\subsection*{Acknowledgements}
A. Alarc\'on is supported by the Ram\'on y Cajal program of the Spanish Ministry of Economy and Competitiveness.
A.\ Alarc\'{o}n and F.\ J.\ L\'opez are partially supported by MCYT-FEDER grant MTM2011-22547 and
Junta de Andaluc\'ia grant P09-FQM-5088.
B.\ Drinovec Drnov\v sek and F.\ Forstneri\v c are partially supported by the research program P1-0291 and the grant J1-5432
from ARRS, Republic of Slovenia.
Part of this work was made when F.\ Forstneri\v c visited the institute IEMath-Granada with support by the GENIL-SSV 2014 program.
Part of this work was made when B.\ Drinovec Drnov\v sek visited University of Oslo. She would like
to thank the Department of Mathematics for the hospitality and partial financial support.
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\section{Introduction}\label{sec:Introduction}
Improvements in the spatial and temporal resolution of solar observations have led to a recent deluge in reported magnetohydrodynamic (MHD) wave motions (e.g. see Nakariakov \& Verwichte \citealp{Nakariakov2005}; De Moortel \citealp{Ineke2005}; Banerjee et al. \citealp{Banerjee2007}; Ruderman \& Erd\'elyi \citealp{Misha2009}; Goossens et al. \citealp{Goossens2011}; McLaughlin et al. \citealp{McLaughlinREVIEW} for a recent list). Here, we focus on observations of transverse motions in the solar atmosphere (e.g. Tomczyk et al. \citealp{Tomczyk2007}; De Pontieu et al. \citealp{Bart2007}; Cirtain et al. \citealp{Cirtain2007}; Erd\'elyi \& Taroyan \citealp{Robertus2008}; He et al. \citealp{He2009a}; \citealp{He2009b}; Liu et al. \citealp{Liu2009}; \citealp{Liu2011}; Morton et al. \citealp{Morton2011}; Okamoto et al. \citealp{Okamoto2011}). These transverse motions have been called Alfv\'en waves by some authors, although this is subject to debate and they are alternatively interpreted as kink waves (e.g. see arguments by Erd\'elyi \& Fedun \citealp{RobertusViktor}; Van Doorsselaere et al. \citealp{Van2008}). The dispute rests not with the observations themselves, but with the appropriate interpretation: MHD wave modes of an overdense cylinder versus MHD waves of a homogeneous plasma. These arguments and others, e.g. whether or not a stable waveguide actually exists in the solar atmosphere, are not the focus of this current paper.
Tomczyk et al. (\citealp{Tomczyk2007}) utilised the {\emph{CoMP/Coronal Multi-channel Polarimeter}} instrument to report on ubiquitous, small-amplitude, transverse disturbances, propagating along magnetic field lines. The authors do not report on how the waves are generated, but do speculate the waves may originate from within the chromospheric network that forms the footpoints of the observed loops. De Pontieu et al. (\citealp{Bart2007}) used {\emph{Hinode/SOT}} measurements in an attempt to reveal Alfv\'en/transversal waves in the chromosphere with strong amplitudes ($10-30\:$km/s) and periods $100-500\:$seconds. Ca II H-line images also reveal a plethora of dynamic, jet-like extrusions called chromosphere spicules, or {\emph{type II spicules}}. These spicules undergo a swaying/oscillatory motion perpendicular to their own axis, which the authors described as Alfv\'enic motions. Again, this interpretation is disputed by other authors (e.g. He et al. {\citealp{He2009b}}; Verth et al. {\citealp{Verth2011}}) who interpreted these spicule oscillations as kink waves, due to the fact that spicules are overdense in comparison with the ambient plasma.
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The Chelsea School District offers several educational options for students. If you are interested in specific enrollment information regarding any of these programs, please contact your child's building principal.
VLAC is available to students in Genesee, Lapeer, Livingston, Oakland, Washtenaw and Wayne counties whose parents have chosen to guide their education using home-based learning. Students who enroll in VLAC will be provided, at no cost, the equipment (including computer, printer/scanner and internet access) and materials needed to participate in virtual learning using the Calvert curriculum. Each student will also receive a tailored program of learning designed to meet their specific educational needs and goals, and will receive instructional oversight by highly qualified Michigan teachers.
The South & West Washtenaw Consortium delivers Career & Technical Education to Chelsea, Dexter, Lincoln, Manchester, Milan and Saline school district students. All CTE courses are designed to equip students with entry level skills needed to gain employment, and also prepare them for Post-Secondary training and education. Many courses offer certifications necessary to enter the job market, in addition to Advanced Placement college credit, which articulate to local community colleges.
W.A.V.E Washtenaw is a free, public high school program for students in Washtenaw County school districts. W.A.V.E Washtenaw provides support through an online mentor, a designated team leader and subject matter experts working together to design and support an academic program that meets the Michigan State standards and prepares each student for further education and career skills.
ECA is a public, early/middle college program located on the campus of Eastern Michigan University. It exists in partnership with local school districts, including Washtenaw Intermediate School District. The program gives students an opportunity to earn college credits while still in high school and offers strong, academically focused students a chance to enroll in advanced, college-level coursework. It also provides an option for students who are either struggling or don't feel connected to their school an alternative.
Section 21f of the State School Aid Act of 2013 entitles public school students in grades 5-12 to take up to two online courses per academic term as requested by the pupil, with the consent of the pupil's parent or legal guardian. MVU is the primary option in the Chelsea School District. Please see links below to browse the online catalog and refer to FAQs for more in-depth information.
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{"url":"http:\/\/lmcaraig.com\/understanding-image-histograms-with-opencv\/","text":"#### Lou Marvin Caraig\n\nSoftware Development Engineer at source{d}\n\n28 December 2018 - 213 days ago\n\n# Understanding image histograms with OpenCV\n\n\u2022 What is an image histogram?\n\u2022 Calculating the histogram\n\u2022 Grayscale histogram\n\u2022 Color histogram\n\u2022 Multi-dimensional histogram\n\u2022 2D histogram\n\u2022 3D histogram\n\u2022 Histogram equalization\n\u2022 Theory and derivation of the formula\n\u2022 Example\n\u2022 Histogram equalization for grayscale images\n\u2022 Histogram equalization for colored images\n\u2022 Conclusions\n\nIn this post, I want to explore what is an image histogram, how it is useful to understand an image and how it can be calculated using OpenCV which is de facto the standard tool for computer vision. In this post, I'm going to use OpenCV 3 with Python 3.6.\n\n## What is an image histogram?\n\nA histogram is a graph or a plot that represents the distribution of the pixel intensities in an image. In this post, we're going to focus on the RGB color space (see here if you want an explanation about the difference between some color spaces such as RGB and Lab), hence the intensity of a pixel is in the range $[0, 255]$. When plotting the histogram we have the pixel intensity in the X-axis and the frequency in the Y-axis. As with any other histogram, we can decide how many bins to use.\n\nA histogram can be calculated both for the grayscale image and for the colored image. In the first case we have a single channel, hence a single histogram. In the second case we have three channels, hence three histograms.\n\nCalculating the histogram of an image is very useful as it gives an intuition regarding some properties of the image such as the tonal range, the contrast, and the brightness.\n\n## Calculating the histogram\n\nLet's now see how we can calculate the histogram of any given image using OpenCV and display them using matplotlib.\n\nOpenCV provides the function cv2.calcHist to calculate the histogram of an image. The signature is the following:\n\ncv2.calcHist(images, channels, mask, bins, ranges)\n\nwhere:\n\n1. images - is the image we want to calculate the histogram of wrapped as a list, so if our image is in variable image we will pass [image],\n2. channels - is the index of the channels to consider wrapped as a list ([0] for grayscale images as there's only one channel and [0], [1] or [2] for color images, if we want to consider the channel blue, green or red respectively),\n3. mask - is a mask to be applied on the image if we want to consider only a specific region (we're gonna ignore this in this post),\n4. bins - is a list containing the number of bins to use for each channel,\n5. ranges - is the range of the possible pixel values which is $[0, 256]$ in case of RGB color space (where $256$ is not inclusive).\n\nThe returned value hist is a numpy.ndarray with shape (n_bins, 1) where hist[i][0] is the number of pixels having an intensity value in the range of the i-th bin.\n\nWe can simplify this interface by wrapping it with a function that in addition to calculate the histogram it also draws it (at the moment we're going to fix the number of bins to $256$):\n\nimport cv2 from matplotlib\nimport pyplot as plt\n\ndef draw_image_histogram(image, channels, color='k'):\nhist = cv2.calcHist([image], channels, None, [256], [0, 256])\nplt.plot(hist, color=color)\nplt.xlim([0, 256])\n\nwhere image can be loaded using cv2.imread('path\/to\/image').\n\nLet's now see the histograms of these three sample images:\n\n### Grayscale histogram\n\nLet's start by considering the histogram of the grayscale version of the above sample images. We can write the following helper function to display using matplotlib the histogram of the grayscale version of an image:\n\nimport cv2 from matplotlib\nimport pyplot as plt\n\ndef show_grayscale_histogram(image):\ngrayscale_image = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)\ndraw_image_histogram(grayscale_image, [0])\nplt.show()\n\nIf we execute this function for the sample images we obtain the following histograms:\n\nLet's now analyze these plots and see what kind of information we can extract from them. From the first one, we can infer that all the pixels of the corresponding image have a low intensity as their almost all in the $[0, 60]$ range approximately. From the second one, we can see that the distribution of the pixel intensities is still more skewed over the darker side as the median value is around $80$, but the variance is much larger. Then from the last one, we can infer that the corresponding image is much lighter overall but also have some dark regions.\n\nI think that even if the histograms weren't given in the same order of the sample images, anyone could be able to guess which histogram belongs to each image by simply doing the above analysis.\n\nHere are the grayscale images with the corresponding histograms:\n\n### Color histogram\n\nLet's now move onto the histograms of the colored sample images. Even in this case, we can write the following helper function to display using matplotlib the histogram an image:\n\nimport cv2\nfrom matplotlib import pyplot as plt\n\ndef show_color_histogram(image):\nfor i, col in enumerate(['b', 'g', 'r']):\ndraw_image_histogram(image, [i], color=col)\nplt.show()\n\nIf we execute this function for the sample images we obtain the following histograms:\n\nThe plots are in the same order as the sample images. As we could have expected from the first plot we can see that all the channels have low intensities corresponding to very dark red, green and blue. We also have to consider that the color black, which is given by $(0, 0, 0)$ in RGB, is abundant in the corresponding image and that may explain why all the channels have peaks in the lower part of the X-axis. Anyway, this can't be much appreciated in this type of visualization given that we're plotting the three channels independently from each other. Later we will see how we can observe the distribution of the combination of the channels' values by using multi-dimensional histograms.\n\nFrom the second plot, we can observe that there's a dark red peak that may correspond to the rocks and the mountains while both the green and the blue channel have a wider range of values. From the last plot, if we exclude the peaks of all the channels in the interval $[0, 30]$, we can observe the opposite of what we saw in the first plot. All the three channels have high intensities and if we consider that $(255, 255, 255)$ in RGB corresponds to white, then by looking at the image it's clear why the histogram has this distribution.\n\nHere are the sample images with the corresponding histograms:\n\n### Multi-dimensional histogram\n\nAs anticipated above we're now going to take a look at multi-dimensional histograms.\n\n#### 2D histogram\n\nThe idea is that instead of examining each channel separately we analyze them together in groups of 2 (RG, GB, BR) or all together. In the case of 2D histograms, the resulting plot will have the pixel intensities of a channel on the X-axis, the pixel intensities of another channel on the Y-axis, and the frequency is given by the color of the plot. Let's look at an example as it is easier to show it than explaining it.\n\nHere's the code that plots each 2D color histograms one for each pair combinations.\n\nimport cv2\nfrom matplotlib import pyplot as plt\nfrom matplotlib import ticker\n\ndef show_image_histogram_2d(image, bins=32, tick_spacing=5):\nfig, axes = plt.subplots(1, 3, figsize=(12, 5))\nchannels_mapping = {0: 'B', 1: 'G', 2: 'R'}\nfor i, channels in enumerate([[0, 1], [0, 2], [1, 2]]):\nhist = cv2.calcHist(\n[image], channels, None, [bins] * 2, [0, 256] * 2)\n\nchannel_x = channels_mapping[channels[0]]\nchannel_y = channels_mapping[channels[1]]\n\nax = axes[i]\nax.set_xlim([0, bins - 1])\nax.set_ylim([0, bins - 1])\n\nax.set_xlabel(f'Channel {channel_x}')\nax.set_ylabel(f'Channel {channel_y}')\nax.set_title(f'2D Color Histogram for {channel_x} and '\nf'{channel_y}')\n\nax.yaxis.set_major_locator(\nticker.MultipleLocator(tick_spacing))\nax.xaxis.set_major_locator(\nticker.MultipleLocator(tick_spacing))\n\nim = ax.imshow(hist)\n\nfig.colorbar(im, ax=axes.ravel().tolist(),\norientation='orizontal')\nfig.suptitle(f'2D Color Histograms with {bins} bins',\nfontsize=16)\nplt.show()\n\nLet's now take a look at what this function does. Most of the code is just to have a nicer plot, actually, the logic is really simple. On line 4 we initialize the figure by configuring 3 subplots all in the same row. Each subplot will contain a 2D histogram. On line 7 we simply define the mapping between the channel index and the corresponding channel name. Keep in mind that the color space of the images loaded using OpenCV is BGR and not RGB, that's why index 0 correspond to B and not R and so on.\n\nOn line 8 we start to loop over each pair combinations of channels. The first combination is [0, 1] which correspond to the pair BG, the second one is [0, 2] which correspond to the pair BR, the third one is [1, 2] which correspond to the pair GR.\n\nOn line 9 we can see that we call the same function cv2.calcHist that we used for 1D histograms. In order to consider 2D histogram the channels attribute now contains two values instead of a single one, the bins attribute defines how many bins to use for each channel and the value of ranges attribute must be repeated ([0, 256] becomes [0, 256, 0, 256]). The returned value hist is a numpy.ndarray with shape (bins, bins) where hist[i][j] is the number of pixels having an intensity value in the range of the $i$-th bin for the channel in the X-axis and in the range of the $j$-th bin for the channel in the Y-axis.\n\nOn lines 12\u201324 we simply set the axis limits, tick spacing and the labels to show and on line 26 we draw the plot. After the for-loop on lines 28\u201330 we add the horizontal colorbar and we finally show the plot.\n\nHere's the generated plot for the sample image with mid-tones.\n\nYou may have noticed that we used only $32$ bins instead of $256$ as previously. That's because using $256$ bins means having $256 * 256 = 65536$ separate pixel counts which are not very practical and also a waste of resource. Here's the 2D color histogram for the same sample image using $256$ bins.\n\n#### 3D histogram\n\nWe can easily extend what we've done for the 2D histogram to calculate 3D histogram. Essentially the call to cv2.calcHist becomes as follows:\n\ncv2.calcHist([image], [0, 1, 2], None, [bins] * 3, [0, 256] * 3)\n\nEven in this case the returned value hist is a numpy.ndarray with shape (bins, bins, bins). Unfortunately, we cannot easily visualize this histogram.\n\n## Histogram equalization\n\nThe histogram equalization process is an image processing method to adjust the contrast of an image by modifying the image's histogram. The intuition behind this process is that histograms with large peaks correspond to images with low contrast where the background and the foreground are both dark or both light. Hence histogram equalization stretches the peak across the whole range of values leading to an improvement in the global contrast of an image.\n\nIt is usually applied to grayscale images and it tends to produce unrealistic effects, but it is highly used where high contrast is needed such as in medical or satellite images.\n\nOpenCV provides the function cv2.equalizeHist to equalize the histogram of an image. The signature is the following:\n\ncv2.equalizeHist(image)\n\nwhere:\n\n1.image - is the grayscale image that we want to equalize (we're going to see later how we can extend to a colored image).\n\nThis function returns the image with the equalized histogram.\n\n### Theory and derivation of the formula\n\nIf you're not interested in the theoretical part, feel free to skip this section and its subsection. We're now going to show which formula is being applied when doing histogram equalization and how to derive it.\n\nLet $n$ be the number of pixels of the given image and $[p_0,p_k]$ the range of the gray levels. Let's also define the histogram of this image as $H(p)$ where $H(p_i)$ is the number of pixels having gray level $p_i$. Our goal is to have a histogram $G(q)$, where $G$ is defined similarly to $H$, which is uniform over all the levels in the range $[q_0,q_k]$. Hence we need to find a monotonic transformation function $q=\\mathcal{T}(p)$.\n\nGiven that the total sum of bins doesn't change and thanks to the monotonicity we can imply:\n\n$\\sum_{i_0}^k G(q_i) = \\sum_{i_0}^k H(q_i)$\n\nAdditionally given that $G$ is uniform, we have:\n\n$G(q) = \\frac{n}{q_k - q_0}$\n\nSince the exactly uniform histogram can be obtained only in the continuous space, let's change the first formula as follows:\n\n\\begin{aligned} \\int_{q_0}^q G(s) \\,ds &= \\int_{p_0}^p H(s) \\,ds \\\\ \\int_{q_0}^q \\frac{n}{q_k - q_0} \\,ds &= \\int_{p_0}^p H(s) \\,ds \\\\ \\frac{n}{q_k - q_0} \\int_{q_0}^q \\,ds &= \\int_{p_0}^p H(s) \\,ds \\\\ \\frac{n (q - q_0)}{q_k - q_0} &= \\int_{p_0}^p H(s) \\,ds \\\\ q &= \\frac{q_k - q_0}{n} \\int_{p_0}^p H(s) \\,ds + q_0 \\\\ \\end{aligned}\n\nWe have found our function $\\mathcal{T}$:\n\n$q = \\mathcal{T} = \\frac{q_k - q_0}{n} \\int_{p_0}^p H(s) \\,ds + q_0$\n\nIn the discrete space, this corresponds to:\n\n$q = \\mathcal{T} = \\left(\\frac{q_k - q_0}{n} \\sum_{i=p_0}^p H(i)\\right) + q_0$\n\nThis may seem a complicated formula, but essentially the function $\\mathcal{T}$ is simply a lookup table: if in the original histogram the bin count at level $p$ is equal to $v$, then in the equalized histogram that bin is \"moved\" at level $q$. This mechanism is applied to all levels $p$.\n\n#### Example\n\nLet's see how the histogram equalization works by doing it \"by hand\" on a very simple fake image. Let our sample grayscale image be defined as follows:\n\n$\\begin{bmatrix} 0 & 32 & 96 & 64 \\\\ 32 & 32 & 32 & 32 \\\\ 0 & 128 & 0 & 32 \\\\ 128 & 128 & 255 & 64 \\end{bmatrix}$\n\nWe want to equalize the image so that the whole range $[0, 255]$ is used. Hence we have $q_0=0$, $q_k=255$, $p_0=0$, and $n=16$. We can then calculate the following table:\n\n$\\begin{matrix} p & H(p) & \\sum_{i=0}^p H(i) & q \\\\ & & & \\\\ \\hline & & & \\\\ 0 & 3 & 3 & 48 \\\\ 32 & 6 & 9 & 143 \\\\ 64 & 2 & 11 & 175 \\\\ 96 & 1 & 12 & 191 \\\\ 128 & 3 & 15 & 239 \\\\ 255 & 1 & 16 & 255 \\end{matrix}$\n\nwhere $q$ is simply:\n\n$q = \\frac{255}{16}\\sum_{i=0}^p H(i)$\n\nWe can now plot both the original and the equalized histogram. We can notice that cumulative equalized histogram is very close to the optimal one.\n\n### Histogram equalization for grayscale images\n\nLet's now see how we can easily equalize a grayscale image and show it. Here's the code:\n\nimport cv2\nfrom matplotlib import pyplot as plt\n\ndef show_grayscale_equalized(image):\ngrayscale_image = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)\neq_grayscale_image = cv2.equalizeHist(grayscale_image)\nplt.imshow(eq_grayscale_image, cmap='gray')\nplt.show()\n\nAnd here's the comparison between the original grayscale sample image and the equalized one:\n\nAs we can see the global contrast has been improved by the equalization process and the histogram has been stretched.\n\n### Histogram equalization for colored images\n\nIt's possible to extend the histogram equalization and apply it to colored images. The most naive approach consists in applying the same process to all the three RGB channels separately and rejoining them together. The problem is that this process changes the relative distributions of the color and may consequently yield to dramatic changes in the image's color balance. Here's the code:\n\nimport cv2\nfrom matplotlib import pyplot as plt\ndef show_rgb_equalized(image):\nchannels = cv2.split(image)\neq_channels = []\nfor ch, color in zip(channels, ['B', 'G', 'R']):\neq_channels.append(cv2.equalizeHist(ch))\neq_image = cv2.merge(eq_channels)\neq_image = cv2.cvtColor(eq_image, cv2.COLOR_BGR2RGB)\nplt.imshow(eq_image)\nplt.show()\n\nAnd here's the result of applying the algorithm to a colored sample image:\n\nAn alternative is to first convert the image to the HSL or HSV color space and then apply the histogram equalization only on the lightness or value channel by leaving the hue and the saturation of the image unchanged. Here's the code that applies the histogram equalization on the value channel of the HSV color space:\n\nimport cv2\nfrom matplotlib import pyplot as plt\n\ndef show_hsv_equalized(image):\nH, S, V = cv2.split(cv2.cvtColor(image, cv2.COLOR_BGR2HSV))\neq_V = cv2.equalizeHist(V)\neq_image = cv2.cvtColor(cv2.merge([H, S, eq_V]),\ncv2.COLOR_HSV2RGB)\nplt.imshow(eq_image)\nplt.show()\n\nAnd here's the result of applying the algorithm on the same sample image:\n\n## Conclusions\n\nIn this post we saw how to extract and visualize the histogram of an image, both grayscale and colored, and what kind of intuitions we can infer from that. We also discussed how histogram equalization works and how it can be extended to colored images.\n\nThere are also other algorithms for histogram equalization that are more robust that haven't been considered here such as AHE (Adaptive Histogram Equalization) and CLAHE (Contrast Limited Adaptive Histogram Equalization).\n\nImage histograms are simple, but largely used in image processing. One interesting application is the usage of image histograms to build an image search engine based on the similarity between them such explained in this blog post.","date":"2019-09-21 23:57:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 47, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8571241497993469, \"perplexity\": 866.7506972757238}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514574710.66\/warc\/CC-MAIN-20190921231814-20190922013814-00399.warc.gz\"}"}
| null | null |
CRISPR Patents Disallowed, But Where Are the Journalists?
Posted in Deception, Europe, Patents at 11:29 pm by Dr. Roy Schestowitz
Journalists and corporate pressure groups are not the same thing
The Shark Tank patent meme
Summary: The narrative surrounding last week's decision against CRISPR patents may have been virtually monopolised by the litigation think tanks and law firms; it certainly feels like no journalism is left to rebut them, fact-check, and introspect
THE European Patent Office (EPO) has spent more than half a decade lowering patent quality. Ask EPO examiners. They'll say so off the record (for fear of retribution they won't say so openly and publicly).
It didn't start with Battistelli, but it became far more profound under Battistelli and it continues to get worse with António Campinos, who intervenes in legal cases (more recently in support of software patents in Europe).
In the U.S. Patent and Trademark Office (USPTO) and even more so in American courtrooms we've seen Alice and Mayo routinely used — via 35 U.S.C. § 101 — to squash patents on naturally-repeating/recurring phenomena and maths. That's just common sense, isn't it?
"Last year the opposition/s folks at the EPO came to the conclusion that CRISPR patents are impermissible and days ago the EPO judges doubled down on that decision."Here in Europe it's possible that later this year EPO judges will reject a lot of software patents by deciding on a 'simulation' patent; it won't be a European 'Alice' because the level of EPO tribunals is nowhere near that of SCOTUS in the US. But that can be a start…
Last year the opposition/s folks at the EPO came to the conclusion that CRISPR patents are impermissible and days ago the EPO judges doubled down on that decision. That's a pretty big deal with wide-ranging ramifications. Having surveyed all the EPO-CRISPR news I was able to find, it seems clear to me that no journalism exists anymore in the area of patents. It's all just pressure groups and press releases of large corporations. It's grotesque; prove me wrong/show otherwise…
Before this decision, roughly a week beforehand, Jamie Atkins and Claire Weston (Kilburn & Strode LLP) published "CRISPR priority appeal – What you need to know ahead of the hearing". This links to their site (of course) and says: "Arguably the highest profile appeal at the EPO in recent years (T 0844/18) will be heard before the Board of Appeal in Munich on 13-17 January 2020…"
One can guess whose side Kilburn & Strode LLP takes; they're a litigation giant, so the more patents, the merrier (to them anyway).
Back in December we saw some shameless self-promotion by Sterne, Kessler, Goldstein & Fox P.L.L.C., under the giveaway headline: "Prosecuting Bioinformatics Patent Applications in Europe" (the only "bio" they care about it "bio-litigation").
Yesterday's IAM blog post said: "This week, all eyes are on the EPO Boards of Appeal's oral proceedings in T0844/18, the Broad Institute's appeal against the revocation of its key patent on CRISPR-Cas9."
Life Sciences Intellectual Property Review, a site whose sole purpose seems to be advocacy of such patents, wrote a lot about it throughout the week [1, 2, 3].
Except a couple of blog posts, which were cited here a day ago, I've found nothing else but this mainstream media piece behind a paywall. It said: "The saga over who owns the valuable patents that cover CRISPR/Cas9 gene editing took a new turn this week with a court decision that goes against the Broad Institute of Harvard and MIT and two local startups."
There are paid press releases, such as this one: (Yahoo, Business Wire)
ERS Genomics Limited, which was formed to provide broad access to the foundational CRISPR/Cas9 intellectual property held by Dr. Emmanuelle Charpentier, announced that yesterday the European Patent Office's (EPO) Technical Board of Appeal (Board) upheld the earlier EPO Opposition Division ruling that the Broad Institute patent EP2771468 is not formally entitled to a number of its asserted priority dates and all its claims are therefore not novel and fully revoked.
EP2771468 is viewed as the Broad Institute's foundational CRISPR-Cas9 patent in Europe. In January of 2018 the Opposition Division found that all claims of the Broad Institute's patent were invalid because the Broad Institute was not entitled to its two earliest priority dates and thus the claims lacked novelty in light of prior art. The Broad Institute appealed the Opposition Division's decision, which the Board affirmed yesterday. Thus, all claims of the Broad Institute's patent remain fully revoked with no option left to overturn this decision.
So there's panic among patent zealots looking to 'own' all lives; they panic because even EPO judges explain that patents on life are fake, bogus, invalid patents.
GenomeWeb, a patent maximalism site focused on that domain (their track record suggests so always), said this:
The European Patent Office (EPO) Board of Appeal has upheld the revocation of a Broad Institute CRISPR patent in Europe.
The Science Board wrote:
An important patent on CRISPR technology held by the Broad Institute has been revoked by the European Patent Office (EPO).
On January 16, the EPO's Technical Board of Appeal upheld a previous ruling in 2018 that the Broad Institute's EP2771468 patent was not novel and therefore should be revoked. EP2771468 is seen as the institute's foundational CRISPR-Cas9 patent in Europe, noted a release from ERS Genomics.
The Broad Institute had appealed the earlier decision, saying the issue related to the "current interpretation of rules that dictate what happens when the names of inventors differ across international applications. This interpretation affects many other European patents that rely on U.S. provisional patent applications, and is inconsistent with treaties designed to harmonize the international patent process, including that of the United States and Europe," according to background information about the patent.
Notice how nobody bothers explaining why such patents are so troubling and problematic. These monopolists are shedding a tear this weekend because there were planning to turn organisms into 'private property' and it's not quite working as they hoped.
How long will it take for a real article to explain all this? We'll watch closely… █
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Links 19/1/2020: Wine 5.0 RC6, Alpine 3.11.3
Posted in News Roundup at 4:54 am by Dr. Roy Schestowitz
Desktop/Laptop
Why Windows 7 Users Should Switch to Linux
Now, if you think that Linux is intimidating, confusing, or (and I mean this nicely) for nerds: I hear you. At first glance it can often look that way.
But Linux suffers from poor marketing rather than poor product. With no multi-million dollar marketing campaign out there promoting one specific flavour of Linux it's left to enthusiasts to 'sell' each system, usually on its technical merits — which is what makes things seem rather niche.
However it's 2020 and Linux-based operating systems (called 'distros') like Ubuntu and Linux Mint are very user-friendly. They let you continue to do pretty much all of that you currently do on Windows 7 just as easily, but with safety first.
Should I upgrade from Windows 7 to Linux?
After 10 years of service, Microsoft has finally decided to cease support for Windows 7. This means that the OS will no longer receive any type of updates and that anyone still using it will be exposed to online threats.
Given this change, many users are left wondering what they should do. Most think that they are limited to just three choices…
Windows 7 Support Ended. Here's You Should Do.
As per the information available, there are 200+ million devices that are still running Windows 7. The number might increase, as, it is difficult to estimate the offline devices which might be still running Windows 7.
CentOS 8 (1911) derived from RedHat Linux 8.1 Enterprise released
With the release of RedHat Linux 8.1 Enterprise, we knew that it was only a matter of time for CentOS V8 (1911) to be released. Now that it is finally here let's have a look at what this new update has in store for us.
Since RedHat Linux 8.1 Enterprise powers this version of CentOS, it's going to accompany all the features and improvements that came with the latest RHLE. If you've not been keeping up with RHLE, you need not worry as we're going to discuss all the changes that were implemented with this update.
Enterprise Insights: Red Hat And The Public Cloud
Open source projects are the epicenter of technology innovation today. Docker and Kubernetes are revolutionizing cloud-native computing, along with data-focused projects like Mongo and Redis and many others. Even as open source projects drive innovation, however, sponsoring companies face a growing existential threat from hyper-scale cloud providers.
Red Hat is the recognized leader in enterprise open source support. It's a successful public company with a track record of growth, so it was somewhat puzzling to understand why the Red Hat board decided to sell to IBM this past year.
Red Hat Shows Off Their vDPA Kernel Patches For Better Ethernet Within VMs
Red Hat engineers have been developing virtual data path acceleration (vDPA) as a standard data plane that is more flexible than VirtIO full hardware offloading. The goal is providing wire-speed Ethernet interfaces to virtual machines in an open manner.
This patch series was sent out on Thursday by Red Hat's Jason Wang. This implements the vDPA bus for the Linux kernel as well as providing a vDPA device simulator and supporting vDPA-based transport within VirtIO.
What is the latest kernel release for my version of Red Hat Enterprise Linux?
I read an interesting question on the Red Hat Learning Community forums recently. What is the latest kernel version for my version of Red Hat Enterprise Linux (RHEL)? In this post we'll see how you can find out.
Some users, trying to be helpful, gave a specific version of the kernel package. Unfortunately, that might only be valid at the time of writing. A better approach would be to understand where to get that information about the latest kernel version for a given version of RHEL.
When Red Hat releases a major or minor update to RHEL, they ship it with a specific branch of the kernel version. This page in the customer portal shows the kernel version "branch" associated with a release of RHEL (e.g. RHEL7.6).
dnf-automatic – Install Security Updates Automatically in CentOS 8
Security updates play a crucial role in safeguarding your Linux system against cyber-attacks and breaches which can have a devastating effect on your critical files, databases and other resources on your system.
You can manually apply security patches on your CentOS 8 system, but it is much easier as a system administrator to configure automatic updates. This will give you the confidence that your system will be periodically checking for any security patches or updates and applying them.
Linux 5.4.13
I'm announcing the release of the 5.4.13 kernel.
All users of the 5.4 kernel series must upgrade.
The updated 5.4.y git tree can be found at:
git://git.kernel.org/pub/scm/linux/kernel/git/stable/linux-stable.git linux-5.4.y
and can be browsed at the normal kernel.org git web browser:
https://git.kernel.org/?p=linux/kernel/git/stable/linux-s…
Linux 4.19.97
Linux 4.14.166
Btrfs Async Discard Support Looks To Be Ready For Linux 5.6
After months of work by Facebook engineers, it looks like the new async discard support for Btrfs is ready for the upcoming Linux 5.6 cycle as a win for this Linux file-system on solid-state storage making use of TRIM/DISCARD functionality.
Btrfs has been handling its DISCARD functionality synchronously during the transaction commit, but that can lead to performance issues depending upon the amount of TRIM'ing required and how the drive behaves. With this async discard support, the work is punted off the transaction commit.
BTRFS Guide | The Best Desktop File System
This video goes over BTRFS in-depth, and in this guide, you will learn basic commands, structure, snapshots, and raid capabilities.
Ryzen 9 3900X vs. Ryzen 9 3950X vs. Core i9 9900KS In Nearly 150 Benchmarks
This week our AMD Ryzen 9 3950X review sample finally arrived and so we've begun putting it through the paces of many different benchmarks. The first of these Linux tests with the Ryzen 9 3950X is looking at the performance up against the Ryzen 9 3900X and Intel Core i9 9900KS in 149 different tests.
The Ryzen 9 3950X is AMD's new $749 USD processor just below the Threadripper 3960X. The 3950X offers sixteen cores / 32 threads with a 3.5GHz base clock and 4.7GHz maximum turbo clock. Over the 3900X at $499 is four extra cores / eight threads, a 300MHz lowering of the base clock, 100MHz higher maximum boost clock, and larger L1 and L2 caches while both of these Zen 2 processors share the same TDP of 105 Watts.
Ryzen CPUs On Linux Finally See CCD Temperatures, Current + Voltage Reporting
The 10 Best Presentation Software for Linux in 2020
Presentation always plays a vital role in decision making, and in closing any kind of deal. It provides graphical descriptions and clears the situation. In ancient times, we used papers for presentation. With the revolution of our modern technology, we shifted to screens from the papers and developed a lot of tools for making our work easy. If any company's core system in Linux, then they should go through the whole post and find out the best presentation software for Linux.
Our world has many types of operating systems for our personal computers and laptops. Among them, Linux is one of the most popular ones because it's free and has a lot of open-source tools. With that, a user can customize his/ her's operating system at his/ her will. But getting the right presentation software for Linux distributions can be quite tough. Don't fear, and we will discuss the presentation software in our content today. I hope you will get the right match for your work.
5 Best GUI Clients for PostgreSQL on Ubuntu
Written in C, PostgreSQL which is also known as Postgres is one of the most popular relational database management systems. macOS server has it as default database and is also available for other operating systems such as Windows, FreeBCD, OpenBCD and Linux. As PostgreSQL is one of the most used database management systems in the world, it is used as the backbone of many small to large applications and software's.
Even though I feel working in command-line is best way to learn anything in the world of application and software development, there are some limitations while working with databases in command-line. It requires great experience of working in command-line or it could get really messy for newbies as well as for professionals.
XMPP – Fun with Clients
As I already wrote in my last blog post there's much development in XMPP, not only on the server side, but also on the client side. It's surely not exaggerated to say that Conversations on Android is the de-facto standard client-wise. So, if you have an Android phone, that's the client you want to try&use. As I don't have Android, I can't comment on it. The situation on Linux is good as well: there are such clients as Gajim, which is an old player in the "market" and is available on other platforms as well, but there is with Dino a new/modern client as well that you may want to try out.
The situation for macOS and iOS users are not that good as for Windows, Linux or Android users. But in the end all clients have their pro and cons… I'll try to summarize a few clients on Linux, macOS and iOS…
Shift on Stack: api_port failure
How To Git Commit With Message
Windscribe VPN on openSUSE
With all the talk of VPN (Virtual Private Network) services to keep you safe and my general lack of interest in the subject, I was talking to Eric Adams, my co-host on the DLN Xtend podcast about the subject. He was telling me that he was hesitant to recommend any service so he gave me some option to try out. The one I chose, after doing a little reading was Windscribe.
I am new to the VPN game so I want to be careful in saying, I am recommending this as the perfect solution but rather demonstrating how I set it up and how I am using it on my openSUSE Tumbleweed system. Much in the same way Eric informed me about it.
How to Install Minecraft on Ubuntu 18.04
How to Remove Top Bar Application Menu in Ubuntu 19.10, 18.04
How to add podcasts to a mobile device with Linux
How to transfer a file from a remote computer to a local machine on Linux/Ubuntu
How to Make Linux Mint Look Like Windows 7
50 Useful and Productive cURL Command in Linux
The cURL utility is a simple yet powerful command-line tool for transferring files to/from remote locations. Its full form stands for 'Client URL'. It has cemented its position as one of the best tools for remote data transfer over the internet. cURL offers a robust collection of commands that enable many advanced functionalities. Additionally, most curl command in Linux works exceptionally well for headless agents and/or automated scripts. To help you get started with cURL, our editors have compiled this thoughtfully curated introductory guide. Although it's meant as a starting point for beginning Linux users, seasoned users can use it as a reference guide.
The cURL utility supports a wide variety of protocols and features. We've outlined the essential commands with appropriate examples and suggest readers try them interactively for gaining first-hand experience on them. As with any Linux tool, your expertise with cURL will only grow when you continue to use it in everyday life.
How To Install Selenium Chrome On Centos 7
How To Limit User's Access To The Linux System
How to Get Started with Raspberry Pi
How to Install Arduino IDE on Ubuntu 18.04
How to Preserve Directory Path with CP Command
How to Set or Change Timezone in Ubuntu Linux [Beginner's Tip]
Install Steam on OpenSUSE to Play Games
What is Kanban and How to use Kanban in Linux
in 1997, directly after University, I started as an IT specialist and have been working in this area ever since in different roles. During these more than 20 years of being part of and later also leading IT related projects, our teams were using several methods and supporting software to plan our projects in the best possible way. Not all were equally successful. Currently our teams are working in a Scrum approach which is part of the Agile methodology. To support this way of working we use among others a Kanban board to plan and monitor our work. Kanban is nothing new but seems extremely popular at the moment. It is not only a great approach for large and complex projects, but also on a smaller scale for your study and your personal projects. In this article, which will be a part of a series on productivity apps, I want to explain three topics: What is Kanban, Why should you use Kanban to be more productive and What are the best Kanban apps for Linux.
What's your favorite Linux terminal trick?
The beginning of a new year is always a great time to evaluate new ways to become more efficient. Many people try out new productivity tools or figure out how to optimize their most mundane processes. One area to assess is the terminal. Especially in the world of open source, there are tons of ways to make life at the terminal more efficient (and fun!) with shortcuts and commands.
We asked our writers about their favorite terminal trick. They shared their time-saving tips and even a fun terminal Easter egg. Will you adopt one of these keyboard shortcuts or command line hacks? Do you have a favorite you'd like to share? Tell us about it by taking our poll or leaving a comment.
Wine or Emulation
Wine Announcement
The Wine development release 5.0-rc6 is now available.
Barring any last minute issue, this is expected to be the last
release candidate before the final 5.0.
What's new in this release (see below for details):
- Bug fixes only, we are in code freeze.
The source is available from the following locations:
https://dl.winehq.org/wine/source/5.0/wine-5.0-rc6.tar.xz
http://mirrors.ibiblio.org/wine/source/5.0/wine-5.0-rc6.tar.xz
Binary packages for various distributions will be available from:
https://www.winehq.org/download
You will find documentation on https://www.winehq.org/documentation
You can also get the current source directly from the git
repository. Check https://www.winehq.org/git for details.
Wine is available thanks to the work of many people. See the file
AUTHORS in the distribution for the complete list.
Wine 5.0-RC6 Released With Another 21 Fixes
We'll likely see the Wine 5.0 stable release next week or the following week, but for now Wine 5.0-RC6 is available as the newest weekly release candidate.
Given the code freeze that's been in place for over the past month, there are no new features but only bug fixes at this stage. Wine 5.0-RC6 ships with 21 known bug fixes in total.
Some of the fixes in Wine 5.0-RC6 are for Brothers In Arms – Hell's Highway, Tomb Raider, The Witcher Enhanced Edition, Serious Sam Classic, and other games. There are also fixes for applications like 7-Zip, Acrobat Reader, and Pale Moon.
The sixth Release Candidate for Wine 5.0 is out now
The Wine team have released a sixth and perhaps final Release Candidate for the upcoming stable release of Wine 5.0. What's going to be one of the biggest releases ever, with some truly massive feature improvements since Wine 4.0 back in January last year.
Going by how many Release Candidates they've done before (7 for 4.0 and 6 for 3.0 and 6 for 2.0), the final stable Wine 5.0 release could well be next week.
For this sixth Release Candidate of Wine 5.0, they noted 21 bug fixes. As always, some might have been fixed in older versions that have been retested recently. From the recent fixes you should see a better experience with The Witcher Enhanced Edition, Dark Messiah of Might and Magic, Serious Sam Classic, Far Cry 5 and probably more too.
Desktop Environments/WMs
K Desktop Environment/KDE SC/Qt
So you want to make a KDE video…
KDE is running a competition in search of the next great promotional video for KDE's Plasma desktop and KDE's applications.
The prizes are two fantastic TUXEDO computers, one per category, which will undoubtedly boost your film rendering capacity. There are also 12 goodie packages for runner-ups, and who doesn't need more Linux shirts, caps and stickers?
Although we have already received some interesting entries, we feel it may be time to help video artists out there with ideas from the judges themselves.
Below, Julian Schraner, Ivana Isadora Devčić, and Paul Brown from the Promo team and Farid Abdelnour from the Kdenlive team give their views on what a KDE promotional video should look like, where to find resources, and which pitfalls may hurt your film if you fall for them.
Learning about our users
In a product like Plasma, knowing the kind of things our existing users care about and use sheds light on what needs polishing or improving. At the moment, the input we have is either the one from the loudest most involved people or outright bug reports, which lead to a confirmation bias.
What do our users like about Plasma? On which hardware do people use Plasma? Are we testing Plasma on the same kind of hardware Plasma is being used for?
Some time ago, Volker Krause started up the KUserFeedback framework with two main features. First, allowing to send information about application's usage depending on certain users' preferences and include mechanisms to ask users for feedback explicitly. This has been deployed into several products already, like GammaRay and Qt Creator, but we never adopted it in KDE software.
The first step has been to allow our users to tune how much information Plasma products should be telling KDE about the systems they run on.
GNOME Desktop/GTK
Doing Things That Scale
I used to have an Arch GNU/Linux setup with tons of tweaks and customizations. These days I just run vanilla Fedora. It's not perfect, but for actually getting things done it's way better than what I had before. I'm also much happier knowing that if something goes seriously wrong I can reinstall and get to a usable system in half an hour, as opposed to several hours of tedious work for setting up Arch. Plus, this is a setup I can actually install for friends and relatives, because it does a decent job at getting people to update when I'm not around.
Until relatively recently I always set a custom monospace font in my editor and terminal when setting up a new machine. At some point I realized that I wouldn't have to do that if the default was nicer, so I just opened an issue. A discussion ensued, a better default was agreed upon, and voilà — my problem was solved. One less thing to do after every install. And of course, everyone else now gets a nicer default font too!
I also used to use ZSH with a configuration framework and various plugins to get autocompletion, git status, a fancy prompt etc. A few years ago I switched to fish. It gives me most of what I used to get from my custom ZSH thing, but it does so out of the box, no configuration needed. Of course ideally we'd have all of these things in the default shell so everyone gets these features for free, but that's hard to do unfortunately (if you're interested in making it happen I'd love to talk!).
Years ago I used to maintain my own extension set to the Faenza icon theme, because Faenza didn't cover every app I was using. Eventually I realized that trying to draw a consistent icon for every single third party app was impossible. The more icons I added, the more those few apps that didn't have custom icons stuck out. Nowadays when I see an app with a poor icon I file an issue asking if the developer would like help with a nicer one. This has worked out great in most cases, and now I probably have more consistent app icons on my system than back when I used a custom theme. And of course, everyone gets to enjoy the nicer icons, not only me.
TROMjaro Updates Deliver Lighter, Better Manjaro
TROMjaro is based on Arch Linux but is much simpler to install and maintain. That makes it a good computing platform to investigate.
While it is easier to install and use than most other Arch Linux-based distributions, users already familiar with how the Linux OS works will have a less challenging experience than newcomers to Linux migrating from macOS and Microsoft Windows.
The TROMjaro desktop has its own unique look and feel, and its own approach to handling software.
Existing TROMjaro users do not have to do anything special to update their systems if they installed an earlier ISO. Just update the TROM repository.
Alpine 3.11.3 released
The Alpine Linux project is pleased to announce the immediate availability of version 3.11.3 of its Alpine Linux operating system.
Screenshots/Screencasts
VIDEO: Linux Lite 4.8 Features and Desktop Tour
Linux Lite eases Windows 7 users transition to Linux much more comfortable by offering simple software like Team Viewer, VLC, Firefox, TimeShift backup utility, and a full Microsoft Office compatible office suite in LibreOffice.
Why Pop!_OS is NOT just an Ubuntu clone (interview with Jeremy Soller)
There are many myths in the Linux community, with one of them being that Pop!_OS is just an Ubuntu clone.
Astra Linux Common Edition 2.12.22 overview | Ensures the protection of confidential information.
In this video, I am going to show an overview of Astra Linux Common Edition 2.12.22 and some of the applications pre-installed.
Arch Family
Tuxedo's New Manjaro Linux Laptops Will Include Massive Customization
The Manjaro/Tuxedo Computers partnership will also offer some intense customization options, Forbes adds.
"Want your company logo laser-etched on the lid? OK. Want to swap out the Manjaro logo with your logo on the Super key? Sure, no problem. Want to show off your knowledge of fictional alien races? Why not get a 100% Klingon keyboard?"
Fedora Family
Fedora's CoreOS Released and Out for Preview (Stable & Testing Version)!
Fedora's CoreOS Released Now: The team Fedora officially announced that the CoreOS stable and testing version are released for users. CoreOS is designed for handling the running containerized workloads securely and at scale. Fedora's CoreOS combines the OCI support, SELinux security for atomic host, provisioning tools and automatic update model of container Linux with the packaging technology.
Linux Container Fedora CoreOS Released For Public Use: Download Now!
As per the official blog, Fedora CoreOS does not give guaranteed stability, which is challenging to achieve along with the incremental and evolving development required by Fedora CoreOS.
However, Fedora CoreOS is still under active development. The CoreOS team promises to provide tools and work over the time to manage the impact of any regressions or breaking changes from automatic updates.
Future of CoreOS Container Linux and Fedora Atomic Host
CoreOS Container Linux will be maintained for a few more months, as mentioned in the latest blog by Fedora CoreOS team, whose end-of-life date will be announced later this month.
Fedora Atomic Host has already reached end-of-life, and the users are highly recommended to migrate to Fedora CoreOS as soon as possible.
Upcoming Enhancements
Fedora CoreOS also serves as the upstream to Red Hat CoreOS. It aims to provide the best container host to run containerized workloads securely and at scale.
Debian Is Making The Process Easier To Bisect Itself Using Their Wayback Machine
For a decade now snapshot.debian.org has been around for accessing old Debian packages and to find packages by dates and version numbers. Only now though is a guide materializing for leveraging this Debian "wayback machine" in order to help in bisecting regressions for the distribution that span multiple/unknown packages.
The bisecting is intended for Debian Sid users of the latest bleeding-edge packages and to helping track down what specific package versions may have introduced a regression. This Debian snapshot archive offers a JSON-based API to query changed packages based upon dates and from there with leveraging Git can make the bisecting manageable.
Stremio
There is a new tool available for Sparkers: Stremio
What is Stremio?
Stremio is a one-stop hub for video content aggregation. Discover, organize and watch video from all kind of sources on any device that you own.
Movies, TV shows, series, live television or web channels like YouTube and Twitch.tv – you can find all this on Stremio.
Debian 10, the clean install
Events have ended my upgrade procrastination. Last week my hard drive started having many errors. Fortunately it lasted long enough for me to copy all of its contents to my USB backup drive. (My /home/brad directory is automatically backed up daily, but I also have separate partitions for downloaded files, PDFs, Linux CD images, and archived photos from my digital camera…and those only get backed up now and then.) Then a quick trip to the store for a new SATA hard drive.
I suppose I could have copied my old root partition over to the new drive. But I've been running 32-bit Debian 8 ("Jessie"), which is now two versions behind. And I've been noticing more and more applications that I want to run are only being distributed for 64-bit Linux. So I decided to do a clean install of 64-bit Debian 10 ("Buster"), with my preferred MATE desktop (now a standard option with Debian).
gnu Linux Debian – top 1000 packages by install – popularity contest
remember: only the installs are counted where the user said yes during setup to: "do you want to participate in popularity contest?" (guess that many Linux users are privacy sensitive and a lot of them probably say "no")
Canonical/Ubuntu Family
[Ubuntu] Design and Web team summary – 17 January 2020
The second iteration of this year is the last one before our mid-cycle sprint next week.
Here's a short summary of the work the squads in the Web & Design team completed in the last 2-week iteration.
5 key steps to take your IoT device to market
IoT businesses are notoriously difficult to get off the ground. No matter how good your product is or how good your team is, some of the biggest problems you will face are just in getting to market and maintaining your devices once they're in the field. The webinar will take a look at how Canonical's Brand Store product allows you to get to market while catering for long term problems and the need to keep your product up to date in the future.
More specifically, this webinar will look at the common problems we see organisations facing on their way to getting an IoT device to market, and cover five key steps to solve these problems. Along the way we will dig a little into serval case studies Canonical has done with various customers and partners to show you what has already been achieved with these solutions.
E-con ships 5MP cam for Nano Dev Kit, which gets rev'd with second CSI link
E-Con has launched a $69, 5-megapixel "e-CAM50_CUNANO" camera for the Jetson Nano Dev Kit with a MIPI-CSI2 interface and Linux driver. Meanwhile, Nvidia launched a revised "B01" version of the kit with a second CSI connector and support for the Xavier NX.
E-Con Systems has followed its 3.4-megapixel e-CAM30_CUNANO camera for the Jetson Nano Dev Kit with a 5-megapixel model. Like the 3.4MP model, the e-CAM50_CUNANO uses a MIPI-CSI2 interface, but is built around OnSemi's 1/2.5″ AR0521, a 2.2µm pixel CMOS image sensor with integrated ISP.
Pumpkin SBCs feature new MediaTek SoCs and mainline Linux/Android SDK
MediaTek has launched two "Pumpkin i300 EVK" SBCs with a quad -A35 MediaTek i300 and unveiled a Pumpkin i500 EVK based on the octa-core -A73/-A53 i500. The SBCs feature a BayLibre-developed, MediaTek Rich IoT SDK v20.0 with Yocto and Android 10.
At CES last week, MediaTek teased an OLogic built Pumpkin i500 EVK AI Vision SBC built around its powerful octa-core Cortex-A73 and -A53 MediaTek i500. Due in February, the SBC will join two recently launched Pumpkin i300 EVK SBCs built around different variations of the quad-core, Cortex-A35 based MediaTek i300 (MT8362) SoC.
A beginner tries PCB assembly
I wrote last year about my experience with making my first PCB using JLCPCB. I've now got 5 of the boards in production around my house, and another couple assembled on my desk for testing. I also did a much simpler board to mount a GPS module on my MapleBoard – basically just with a suitable DIP connector and mount point for the GPS module. At that point I ended up having to pay for shipping; not being in a hurry I went for the cheapest option which mean the total process took 2 weeks from order until it arrived. Still not bad for under $8!
Just before Christmas I discovered that JLCPCB had expanded their SMT assembly option to beyond the Chinese market, and were offering coupons off (but even without that had much, much lower assembly/setup fees than anywhere else I'd seen). Despite being part of LCSC the parts library can be a bit limited (partly it seems there's nothing complex to assemble such as connectors), with a set of "basic" components without setup fee and then "extended" options which have a $3 setup fee (because they're not permanently loaded, AIUI).
Digitizing a analog water meter
Sadly, my meter is really dirt under the glass and i couldn't manage to clean it. This will cause problems down the road.
The initial idea was easy, add a webcam on top of the meter and read the number on the upper half it. But I soon realized that the project won't be that simple. The number shows only the use of 1m^3 (1000 liters), this means that I would have a change only every couple of days, which is useless and boring. So, I had to read the analog gauges, which show the fraction in 0.0001, 0.001, 0.01 and 0.1 m^3. This discovery blocked me, and I was like "this is way to complicated".
I have no idea how I found or what reminded me of OpenCV, but that was the solution. OpenCV is an awesome tool for computer vision, it has many features like Facial recognition, Gesture recognition … and also shape recognition. What's a analog gauge? It's just a circle with an triangular arrow indicating the value.
NVIDIA Jetson Nano Developer Kit-B01 Gets an Extra Camera Connector
Launched in March 2019, NVIDIA Jetson Nano developer kit offered an AI development platform for an affordable $99.
Fake cases — make sure yours is the real deal
We've had some reports of people finding cases that pretend to be official Raspberry Pi products online — these are fakes, they're violating our trademark, they're not made very well, and they're costing you and us money that would otherwise go to fund the Raspberry Pi Foundation's charitable work. (Reminder, for those who are new to this stuff: we're a not-for-profit, which means that every penny we makes goes to support our work in education, and that none of us gets to own a yacht.)
Mobile Systems/Mobile Applications
Gesture Plus by Jawomo lets you customize Android 10′s gesture
LG V50 Android 10 update starts rolling out in South Korea
More Nokia phones on Android 10
Download: Xiaomi Mi A2 receives official Android 10 update
Leaked Galaxy S20 spec sheet details huge screens, big batteries, Android 10, more
Huawei P Smart 2019 Starts Receiving Android 10-Based EMUI Firmware Update
Today's Tablet Sales: iPads, Amazon Fire, Android Tablets Best Deals [Updated]
Android Circuit: New Samsung Galaxy S20 Leaks, Huawei's Big Google Problem, Radical OnePlus 8 Lite Revealed
Android secrets: Here are the best Google features you didn't know about
5 Android apps you shouldn't miss this week
Android's version of AirDrop, 'Fast Share,' has been renamed to 'Nearby Sharing'
Concept Android Auto app alerts you of pedestrians before you see them
Android Auto's support for Google Assistant routines has been broken since 2018
This concept Android Auto app uses nearby phones to detect pedestrians
How to get started with Android apps on your Chromebook
Google already testing Stadia on non-Pixel Android phones
Chrome for Android prepares Google Assistant integration for voice search
Blockbuster Android update will bring the feature millions have been waiting for
Which Android Tablet Should You Get?
Google app's new Labs lets you disable Doodles in the homescreen widget
7 things your Android phone can do that people with iPhones only dream
How to enable RCS messaging in Google Messages on Android phones
It's time for Google to build a video editor for Chromebooks and Android
CES 2020: 8 exciting Android announcements you might have missed
TicPods 2 Pro review: Still a solid Android alternative to Apple AirPods [Video]
The Note 10 is the best Android phone but this is why I use a Pixel
Xbox Console Streaming Preview for Android Goes Global
Fleeceware – The Free Android Apps That Secretly Cost $240 Per Year
These were the most downloaded apps on IOS and Android in 2019
Warning: If your Android phone has THESE apps
Android alert: 25 apps that could be stealing from you, have YOU installed one?
The stable Android 10 update now rolling out for the LG V50 ThinQ in Korea
LG V50 ThinQ gets stable Android 10 update in South Korea, global rollout to follow
LG V50 ThinQ Is Starting To Get Android 10, Surprising Absolutely Everyone
Android 10 stable has started to roll out to LG V50 phones
Honor 8X EMUI 10 (Android 10) stable update starts rolling out
EMUI 10 now rolling out on Huawei P30 Lite
Samsung Galaxy A60 and M40 may also get Android 10 a little bit ahead of schedule
Asus ROG Phone 2 Android 10 update imminent as it gets WiFi & Bluetooth certification
Google Pixel 4 running Android 11 (Android R) caught on Geekbench, Developer Preview coming soon?
Firefox for Android migration is about to begin
Firefox Nightly for Android to get a major update next week
Chrome for Android tests recommending tabs you should close, even if you won't listen
Chrome for Android tests reminding you to clean up your old tabs, since you clearly can't handle them yourself
The best sci-fi Android games for a space adventure
10 best auto chess and auto battle games for Android!
Android Circuit: Samsung Galaxy S20 Ultra Leaks, Huawei Fights Google, OnePlus Confirms Key Feature
Google Assistant routines haven't worked on Android Auto for over a year, still no fix in sight (Update: Google acknowledges)
Android is finally receiving a feature Google should have introduced years ago
Android R spotted running on Google Pixel 4
The Beauty Of OLED Screens And Why All Android Devices Should Use Them
Is Huawei getting ready to launch its Android rival?
How to reset your Android device and erase your personal data
What is Android TV? Everything You Need To Know To Get Started
PinePhone, the $149 Linux Phone, Has Started Shipping for the Brave of Heart
The long-anticipated PinePhone Linux-powered smartphone has finally started shipping to customers who were brave enough to purchase the first batch.
The PinePhone Linux phone has been available for pre-order since mid-November 2019, and those who bought one paid as low as $149 USD for the device, which doesn't ship with an operating system preinstalled. After a bit of delay, the PinePhone is now finally shipping to customers, starting today, January 17th.
"We're now ready and I am happy to confirm that PinePhones will begin shipping on January 17th, 2020," said PINE64′s Lukasz Erecinski. "The dispatch process will take a couple of days, however, so your unit may ship anytime between the 17th and 25th. At any rate, you'll have your PinePhone soon."
The $150 PinePhone feels like it's from an alternate universe
In our current mobile world, there are essentially two worlds: Android and iOS. But for the oddballs who aren't satisfied with either option, Pine64 has started shipping the PinePhone, a Linux-based, kinda-DIY smartphone that sells for just $150.
Originally a crowdsourced startup, Pine64 has made single-board computers and the PineBook laptop in the past, so it isn't new to developing alternative hardware for Linux. As ArsTechnica wrote, the company's efforts are not an attempt to create a third mainstream category for smartphones, but the PinePhone still looks to be an interesting concept.
This Linux smartphone is now shipping for $150
Computer and developer-board maker Pine64 has started shipping the first edition of its much-anticipated – at least in the open-source community – PinePhone, after pre-orders sold out. Dubbed "Brave Heart", the device is indeed designed only for the keener hobbyists.
Shipping at only $149.99, Brave Heart is a fully open-source smartphone running Linux, which the company claims was developed "with the community for the community", which means with developers and early adopters, and for developers and early adopters; and in this case, preferably for those who have extensive Linux experience.
In a departure from Android and iOS, Pine's new project provides a platform for customers to develop Linux-on-phone projects. It does not come with a pre-installed OS, but supports all major Linux phone projects such as Ubuntu Touch, Sailfish OS and Plasma Mobile.
Although buyers get to choose their OS, therefore, it will be up to them to upload the platform to the Pine Phone – meaning the device is not designed for the average Joe.
Motorola One Zoom – The Lumia is dead, long live One
Here we go. I've had the Motorola One Zoom for about a month now, and I have to say it's a good phone. In fact, let's do it systematically. Pros: excellent camera, good sounds, fast performance, splendid battery time, configurable operating system that is steadily improving. Cons: the phone is super heavy and cumbersome to hold, you need a lot of patience getting everything configured and tweaked, especially if you care about privacy.
But did I accomplish my mission? Yes I did. I have found a worthy successor to my Lumia. One Zoom matches and even supersedes some of the aspects of my previous phone, which makes me happy, as my degrees of freedom and my sense of nerdy control have not changed (for the worse). I really disliked Android many years ago, but it has evolved into a pretty solid system, and even someone like me, with almost zero interest in the mobile nonsense, can appreciate the improvements and advancements. You have the tools to change anything you like, and while the defaults are silly and lax, you can make Android work as you please.
Now, Android will most likely never be as clean and elegant as Windows Phone. But the super-rich app ecosystem does compensate for that. At the end of the day, it is a small compromise, here and there, but overall, 95% of things are just as good as they were, or even better. I'd say that qualifies as a pleasant and successful transition from one platform to another. It was a little bit traumatic due to the SIM card mistake and the manual data transfer, but I might even say that I'm cautiously, mildly enjoying it.
So there we are. An end of an era, and a start of a new one. Motorola One Zoom seems like a really nifty mid-range phone. I do wonder what high-end devices offer, but that's a story for another time. Meanwhile, if you want a capable, fast phone with lots of juice and a splendid camera with a real optical zoom, this is a really clever choice. I'm happy with my decision. 9.5/10. Bye Lumia, welcome One.
Free, Libre, and Open Source Software
7 things I learned from starting an open source project
I'm currently involved—heavily involved—in Enarx, an open source (of course!) project to support running sensitive workloads on untrusted hosts. I've had involvement in various open source projects over the years, but this is the first for which I'm one of the founders. We're at the stage now where we've got a fair amount of code, quite a lot of documentation, a logo, and (important!) stickers. The project will hopefully be included in a Linux Foundation group—the Confidential Computing Consortium—so things are going very well indeed.
I thought it might be useful to reflect on some of the things we did to get things going. To be clear, Enarx is a particular type of project, one that we believe has commercial and enterprise applications. It's also not mature yet, and we'll have hurdles and challenges along the way. What's more, the route we've taken won't be right for all projects, but hopefully, there's enough here to give a few pointers to other projects or people considering starting one up.
Open source: A matter of license and lock-in
Recently, a few bits of newsworthy information hit the open source landscape. Separately, these pieces of news were not that glaring, but when you put them together something a bit more ominous comes into focus–something I never would have thought to be an issue within the open source community.
Before I get into this, I want to preface this by saying I am not usually one to cry foul, wolf, or squirrel! I prefer to let those pundits who make a living at gleaning the important bits out of the big bowl of alphabet soup and draw their own conclusions. But this time, I think it's important I chime in.
Yes, at this very moment I am donning my tin foil hat. Why? Because I think it's necessary. And with me sporting that shiny chapeau, understand every word you are about to read is conjecture.
Why Did Red Hat Drop Its Support for Docker's Runtime Engine?
[KDE] Itinerary extraction in Nextcloud Hub
Nextcloud announced their latest release and among the many new features is itinerary extraction from emails. That's using KDE's extraction engine, the same that powers similar features in KMail as well.
One open source chat tool to rule them all
Last year, I brought you 19 days of new (to you) productivity tools for 2019. This year, I'm taking a different approach: building an environment that will allow you to be more productive in the new year, using tools you may or may not already be using.
Instant messaging and chat have become a staple of the online world. And if you are like me, you probably have about five or six different apps running to talk to your friends, co-workers, and others. It really is a pain to keep up with it all. Thankfully, you can use one app (OK, two apps) to consolidate a lot of those chats into a single point.
Finance goes agile as open source checks the security box
"At Northwestern Mutual, we've finally gotten past that curve," said Sean Corkum (pictured, right), senior engineer at Northwestern Mutual. "Now we're trying to make it even easier for our internal developers to participate in open source … and contribute more to the community."
Top NLP Open Source Projects For Developers In 2020
Kiwi TCMS: Project roadmap 2020
Hello testers, the Kiwi TCMS team sat down together last week and talked about what we feel is important for us during the upcoming year. This blog post outlines our roadmap for 2020!
Let's Talk With Neal Gompa of Fedora @ openSUSE Conference
In this episode of Let's Talk, we sat down with Neal Gompa of the Fedora community at openSUSE Conference
FOSSCOMM 2019 aftermath
FOSSCOMM (Free and Open Source Software Communities Meeting) is a Greek conference aiming at free-software and open-source enthusiasts, developers, and communities. This year was held at Lamia from October 11 to October 13.
It is a tradition for me to attend this conference. Usually, I have presentations and of course, booths to inform the attendees about the projects I represent.
This year the structure of the conference was kind of different. Usually, the conference starts on Friday with a "beer event". Now it started with registration and a presentation. Personally, I made my plan to leave Thessaloniki by bus. It took me about 4 hours on the road. So when I arrived, I went to my hotel and then waited for Pantelis to go to University and set up our booths.
Google announces end of support dates for Chrome Apps on Windows, Mac, Linux and Chrome OS
The end of support for Chrome apps has been a long time coming — Google announced more than two years ago that it was going to start winding things down.
The Chrome Web Store has already been stripped of the App section on Windows, macOS and Linux, and now Google has announced that it is to be pulled from Chrome OS too. The company has also revealed the dates on which support will be dropped completely for all platforms.
Linux disk resizing on Chromebooks pushed back to Chrome OS 81
The good news is that plans have been in the works since March of last year to allow you to reclaim some of that space by shrinking or resizing the Linux storage. The bad news is that after being pushed back twice since the feature is being put off again; this time until Chrome OS 81.
You'd think this would be a relatively simple thing to implement but in reality, it's not. That's because the Chrome OS filesystem has evolved in the past year and due to expected support for a particular file type for older Linux kernels never worked out.
I'd rather the Chromium team take their time for a well designed and implemented solution so as not to break any functionality. Plus there's the challenge of having enough free storage to restore a container backup.
The Pros And Cons Of Using Tor Browser
The Tor Browser sends all traffic data through the Tor Network. Some also call it the anonymity network. It does so by connecting to a server with a list of Tor nodes and bouncing the connection through three different proxies.
There's an entry node, a middle relay, and an exit node that are selected at random. The connection between these different nodes is encrypted, and the IP addresses aren't public. It makes the whole process safe and anonymous.
The Tor Browser is easy to download and install and doesn't cost a penny to boot. Volunteers created and run the Tor Project, so donations are welcome. Moreover, the code is open-source, which means anyone can review it. It ensures that the code stays secure, and no one can use it for any hidden exploitative money-making purposes.
FSF
GNU Projects
GNU Guile 3.0.0 now available
GNU Guile, a programming and extension language for the GNU Project, is now available as version 3.0.0. According to the team, this is the first release in the stable 3.0 release series.
The major new feature in this version is just-in-time (JIT) native code generation, which helps speed up performance. In this release, microbenchmark performance is twice as good as the 2.2 release, and some individual benchmarks have seen improvements up to 32 times as fast, according to the project maintainers.
Other new features include support for interleaved definitions and expressions in lexical contexts, native support for structured exceptions, and improved support for R6RS and R7RS Scheme standards.
Experimental Support For C++20 Coroutines Has Landed In GCC 10
As of this morning experimental support for C++20 coroutines has been merged into the GCC 10 compiler!
Coroutines allow a function to have its execution stopped/suspended and then to be resumed later. Coroutines is one of the big features of C++20. Sample syntax and more details on C++ coroutines can be found at cppreference.com.
Coroutines support for GCC has been under development for months and now as a late addition to GCC 10 is the experimental implementation.
GNU Binutils 2.34 Branched – Bringing With It "debuginfod" HTTP Server Support
With GNU Binutils 2.34 comes debuginfod support, which is the HTTP server catching our eye while the debuginfod server is distributed as part of the latest elfutils package. This isn't for a general purpose web server thankfully but is an HTTP server for distributing ELF/DWARF debugging information and source code. With debuginfod enabled, Binutils' readelf and objdump utilities can query the HTTP server(s) for debug files that cannot otherwise be found. Enabling this option requires building Binutils using –with-debuginfod.
Licensing / Legal
Fugue open sources Regula to evaluate Terraform for security misconfigurations and compliance violations
Regula rules are written in Rego, the open source policy language employed by the Open Policy Agent project and can be integrated into CI/CD pipelines to prevent cloud infrastructure deployments that may violate security and compliance best practices.
"Developers design, build, and modify their own cloud infrastructure environments, and they increasingly own the security and compliance of that infrastructure," said Josh Stella, co-founder and CTO of Fugue.
"Fugue builds solutions that empower engineers operating in secure and regulated cloud environments, and Regula quickly and easily checks that their Terraform scripts don't violate policy—before they deploy infrastructure."
Programming/Development
Announcing git-cinnabar 0.5.3
Git-cinnabar is a git remote helper to interact with mercurial repositories. It allows to clone, pull and push from/to mercurial remote repositories, using git.
Steve Kemp: Announce: github2mr
myrepos is an excellent tool for applying git operations to multiple repositories, and I use it extensively.
I've written several scripts to dump remote repository-lists into a suitable configuration format, and hopefully I've done that for the last time.
Perl / Raku
Term::ANSIColor 5.01
This is the module included in Perl core that provides support for ANSI color escape sequences.
This release adds support for the NO_COLOR environment variable (thanks, Andrea Telatin) and fixes an error in the example of uncolor() in the documentation (thanks, Joe Smith). It also documents that color aliases are expanded during alias definition, so while you can define an alias in terms of another alias, they don't remain linked during future changes.
JavaScript destructuring like Python kwargs with defaults
I'm sure it's been blogged about a buncha times before but, I couldn't find it, and I had to search too hard to find an example of this.
Create the input text box with tkinter
In the previous post, I have written a python program to create the database, earning table as well as input the first row of data into the earning table. In this chapter, I will create a simple UI to accept the user's input so we do not need to hardcoded the values into the SQL query. I will leave the SQL commit code to the next chapter, we will only create a simple input's UI in this chapter first.
A description box and the earning box of the Earning Input user interface
As you can see I will create the above simple UI with tkinter which can then be further upgraded in the future to include more stuff.
Start using 2FA and API tokens on PyPI
To increase the security of PyPI downloads, we have added two-factor authentication (2FA) as a login security option, and API tokens for uploading packages. This is thanks to a grant from the Open Technology Fund, coordinated by the Packaging Working Group of the Python Software Foundation.
If you maintain or own a project on the Python Package Index, you should start using these features. Click "help" on PyPI for instructions. (These features are also available on Test PyPI.)
How to Build RESTful APIs with Python and Flask
For some time now I have been working with Python but I just got to try out Flask recently, so I felt it would be nice to write about it. In this aritcle I'll discuss about Flask and how you can use it to build RESTfull APIs.
Flask is a Python-based microframework that enables you to quickly build web applications; the "micro" in microframework simply means Flask aims to keep the core simple but extensible.
Reading Binary Data with Python
When you deal with external binary data in Python, there are a couple of ways to get that data into a data structure. You can use the ctypes module to define the data structure or you can use the struct python module.
You will see both methods used when you explore tool repositories on the web. This article shows you how to use each one to read an IPv4 header off the network. It's up to you to decide which method you prefer; either way will work fine.
Python 3.7.5 : Django security issues – part 001.
Django like any website development and framework implementation requires security settings and configurations.
Today I will present some aspects of this topic and then I will come back with other information.
How to display flash messages in Django templates
Sometimes we need to show the one-time notification, also known as the flash messages in our Django application. For this Django provides the messages framework. We are going to use the same here.
To show flash messages in the Django application, we will extend our previous project Hello World in Django 2.2. Clone the git repository, check out the master branch and set up the project on your local machine by following the instructions in the README file.
Baby Dinosaurs on Noah's Ark
In violation of the separation of church and state, American tax dollars are funneled to fundamentalist private schools teaching crackpot absurdities – such as a claim that Noah probably took two baby dinosaurs onto his ark.
Brecht in Berlin
One of Brecht's best-loved and most-performed works, The Caucasian Chalk Circle was written in Los Angeles in 1944 while the playwright was still in exile. He returned to Germany in 1947, but the play remained behind, a parting gift to the country that had offered him refuge from the Nazis, but which, with the Cold War heating up and the House Un-American Activities Committee in full lather, had turned against his socialist politics. With Brecht back in East Berlin, The Caucasian Chalk Circle received its premiere in May of 1948 in a student production at Carleton College in Northfield, Minnesota under the direction of the young theater professor Henry Goodman, a World War II veteran.
A native of Minneapolis, Goodman had been stationed in Berlin during the occupation and had seen Brecht's work onstage in the devastated city. From America Brecht had tried to stop such productions. In September of 1945 he wrote to the Soviet cultural official Mikhail Apletin requesting that all further stagings of his work be forbidden until his own repatriation to Germany. Brecht was appalled that a production of his Threepenny Opera—utterly inappropriate given the circumstances, he thought—had been mounted in August of 1945, just a few months after the end of the war, and likely seen by Goodman. In the new post-Nazi, post-War world, Brecht felt all his plays needed at least some revision. The only work he believed ready for immediate presentation was Fear and Misery of the Third Reich.
Why College Should Be Free
Obviously, I am making an analogy with free college. And obviously, people who don't support free college (and probably many who do) are going to reject this analogy. Here are some possible counterarguments: [...]
Health/Nutrition
Evidence Shows Whooping Cough Is Evolving Into a 'Superbug', Scientists Warn
In new world-first research, a team of Australian scientists has discovered how B. pertussis strains are adapting to the current acellular vaccine (ACV) used in Australia, which is similar to the ACVs used for whooping cough in other countries around the world.
"We found the whooping cough strains were evolving to improve their survival, regardless of whether a person was vaccinated or not," explains microbiologist Laurence Luu from UNSW.
Federal Regulators: Newark Beth Israel Put Patients in "Immediate Jeopardy"
Newark Beth Israel Medical Center's heart and lung transplant program was putting patients in "immediate jeopardy" before the hospital began to implement corrective measures, according to federal regulators.
In a pair of reports sent to the Newark, New Jersey, hospital on Dec. 12, the Centers for Medicare and Medicaid Services found that the transplant program repeatedly failed to fix mistakes. Spurred by ProPublica articles, the CMS investigation uncovered a series of incidents in which the hospital identified areas for improvement following botched surgeries but didn't carry out its own recommendations, allowing "subsequent adverse events to occur."
Integrity/Availability
Boeing discovers new software problem in 737 Max
The news comes following the release of internal documents showing employees knew about problems with pilot training for the 737 MAX and tried to conceal them from regulators. In the documents released by US lawmakers, Boeing employees said the aircraft was "designed by clowns, who in turn are supervised by monkeys," in an apparent reference to regulators.
A Georgia election server was vulnerable to Shellshock and may have been hacked
Forensic evidence shows signs that a Georgia election server may have been hacked ahead of the 2016 and 2018 elections by someone who exploited Shellshock, a critical flaw that gives attackers full control over vulnerable systems, a computer security expert said in a court filing on Thursday.
Shellshock came to light in September 2014 and was immediately identified as one of the most severe vulnerabilities to be disclosed in years. The reasons: it (a) was easy to exploit, (b) gave attackers the ability to remotely run commands and code of their choice, and (c) opened most Linux and Unix systems to attack. As a result, the flaw received widespread news coverage for months.
Micro Focus AD Bridge 2.0: Extending security policies and access controls to cloud-based Linux
With AD Bridge 2.0, organizations can leverage existing infrastructure authentication, security as well as policy, in order to simplify the migration of on-premises Linux Active Directory to the cloud, resulting in fully secured and managed Linux virtual machines in the cloud.
Pseudo-Open Source
Openwashing
Facebook Releases Open Source Speech Recognition Platform
Facebook has announced that it will be making its wav2letter@anywhere online speech recognition framework more readily available as an open source platform. The framework was developed by Facebook AI Research (FAIR), which claims that it has created the fastest open source automatic speech recognition (ASR) platform currently on the market.
Microsoft opens up Rust-inspired Project Verona programming language on GitHub
[Attacker] stole [sic] over 10,000 hospital files
Jason Corden-Bowen, of the CPS, said: "Moonie had no right to access confidential patient and staff records. He admitted his earlier wrongdoing and accepted a police caution yet he went ahead to reoffend knowing fully well it was not just against hospital procedures but it was wrong and illegal.
Privacy/Surveillance
Dear Reuters: This Is NOT How You Report On Dishonest, Disingenuous Talking Points From US Officials Regarding Encryption
Attorney General William Barr and his DOJ and FBI have really ramped up their bullshit campaign against the public being able to use encryption. President Trump himself weighed in himself with some ignorant statements, suggesting that Apple owes him some sort of quid pro quo, because his policies may have helped them on trade:
FBI used Graykey to unlock an iPhone 11 Pro, which was previously thought to be the most secure iPhone
A recent article by Thomas Brewster at Forbes highlights the fact that the FBI is able to unlock iPhones using a product called Graykey. Specifically, a product called Graykey was used in a case against Baris Ali Koch to unlock Koch's iPhone – an iPhone 11 Pro. Graykey works by bypassing the timeout functionality in iOS and allows for brute forcing of the passcode or password.
Police forces around the world continue to deploy facial recognition systems, despite no evidence of their utility
Last month, this blog wrote about governments around the world continuing to trial facial recognition systems, and the growing concerns this is provoking. There's one area in particular where facial recognition systems are deployed: law enforcement. That's hardly a surprise, since the legal system can only operate if it identifies alleged criminals that need to be arrested, tried and punished. But it also emphasizes how facial recognition is seen by many as a natural tool for controlling populations. The most famous example is in China, where facial recognition systems are widely deployed, especially in the turkic-speaking region of Xinjiang. The situation is getting so bad that even Chinese citizens are becoming concerned.
Report: Adult Site Leaks Extremely Sensitive Data of Cam Models
There are at least 875,000 keys, which represent different file types, including videos, marketing materials, photographs, clips and screenshots of video chats, and zip files. Within each zip folder – and there is apparently one zip folder per model – there are often multiple additional files (e.g. photographs and scans of documents), and many additional items that we chose not to investigate.
The folders included could be up to 15-20 years old, but are also as recent as the last few weeks. Even for older files, given the nature of the data, it is still relevant and of equal impact as newly added files.
2019 End of Year Campaign Wrap Up – Thanks for Helping Tor Take Back the Internet
Our commitment to privacy extends to our donors. We execute our fundraising in a way that is very different than other nonprofits. We never share your information with third parties. We never receive potential new donor information from outside sources. We do not track the behavior of our donors when you open our emails. We allow donors to choose what information you share with us. You can be more anonymous by sending a money order to our physical address or utilizing cryptocurrency to protect your personal information. Privacy is limited by requirements of the most popular donation methods, PayPal or credit card, but we are committed to offering privacy-preserving methods of making donations.
America's Most Powerful Woman Called Mark Zuckerberg a Crook. Facebook Has No Rebuttal
"They don't care about the impact on children. They don't care about truth," she said, adding that Facebook's only interest is turning a profit. Facebook has "schmoozed" the administration—a likely reference to CEO Mark Zuckerberg's secretive dinners with President Trump—because "all they want is their tax cuts and no antitrust action against them."
"I think that what they have said very blatantly, very clearly, that they intend to be accomplices for misleading the American people with money from god knows where. They didn't even check on the money from Russia in the last election. They never even thought they should," Pelosi said, before returning to the topic at hand, the Senate's looming impeachment trial.
The Secretive Company That Might End Privacy as We Know It
Clearview has shrouded itself in secrecy, avoiding debate about its boundary-pushing technology. When I began looking into the company in November, its website was a bare page showing a nonexistent Manhattan address as its place of business. The company's one employee listed on LinkedIn, a sales manager named "John Good," turned out to be Mr. Ton-That, using a fake name. For a month, people affiliated with the company would not return my emails or phone calls.
While the company was dodging me, it was also monitoring me. At my request, a number of police officers had run my photo through the Clearview app. They soon received phone calls from company representatives asking if they were talking to the media — a sign that Clearview has the ability and, in this case, the appetite to monitor whom law enforcement is searching for.
Defence/Aggression
A Conflict Fueled By Global Warming
One of the world's worst humanitarian crises is unfolding on the shores of Lake Chad, according to the United Nations. In Nigeria, Niger, Cameroon and Chad, people are suffering from more than just extreme poverty. Boko Haram and other violent Islamist groups combined with corrupt governments and the absence of any functional state administration are making their lives hell — not to mention diseases, natural disasters and overpopulation. All these problems are now being made worse by climate change.
Gun-Rights Activists Gear Up for Show of Force in Virginia
Police are scouring the internet for clues about plans for mayhem, workers are putting up chain-link holding pens around Virginia's picturesque Capitol Square, and one lawmaker even plans to hide in a safe house in advance of what's expected to be an unprecedented show of force by gun-rights activists.
City Of Dallas Shuts Down Business Of Man Who Called Cops Over 100 Times In 20 Months To Deal With Criminals Near His Car Wash
Let's talk about nuisance abatement laws. These are laws cities can use to shut down businesses that appear to draw more than their fair share of the criminal element.
Riots in Lebanon's Capital Leave More Than 150 Injured
Police fired volleys of tear gas and rubber bullets in Lebanon's capital Saturday to disperse thousands of protesters amid some of the worst rioting since demonstrations against the country's ruling elite erupted three months ago. More than 150 people were injured.
Eminem Attacked for Rapping About 2017 Manchester Arena Bombing
The latest album by rapper Eminem is facing backlash over lyrics that seemingly make light of the 2017 bombing of an Ariana Grande concert in Manchester, England.
As Iran and Iraq simmer, giants of Shiite world vie for influence
"Lots of Iraqis believe in Iran," says Hisham al-Hashemi, a security analyst with the European Institute of Peace. "They believe in it as an Islamic Revolution, and the right for this revolution to cross borders everywhere."
War in Syria drives out Christian community
The recent history of the Christians of the Khabur Valley is a nightmare in a loop. They are descendants of those who fled Anatolia during the 1915 genocide. Iraq was a first stop for many, but life there was far from easy either. After a massacre in northern Iraq in 1933, many crossed to French-controlled Syria, where they settled along the Khabur River.
Once a thriving community of 15,000 individuals in the Khabur Valley, local NGOs say there are a "few hundred" left in the area following the IS offensive in 2015.
"It was IS back in 2015 and it's very much the same people right now," said Lunan. Videos continue to emerge of Turkish-backed Islamists committing atrocities against civilians.
Michigan's 'green ooze' may be 'tip of iceberg' of state's toxic sites
The discovery of toxic PFAS chemicals at a contaminated site in Madison Heights could triple the cost of an already expensive cleanup effort, a state official told lawmakers Wednesday.
And taxpayers could get stuck with the bill to clean up the green ooze site, since it's unclear if its imprisoned owner can pay for it.
"Is it $2 million? Is it $20 million? I don't know, but it's not in the hundreds of thousands. It's in the millions," said Tracy Kecskemeti, southeast district coordinator for the Michigan Department of Environment, Great Lakes and Energy (EGLE).
Brazil's indigenous communities resist Bolsonaro
Meanwhile, in southern Brazil, the Kaingang are working to reforest their land with araucaria, an indigenous tree which produces a fruit they can eat and sell, and that supports indigenous wildlife. For Marcio Kokoj of the Kaingang Guarani Indigenous Environmental Association, the Bolsonaro government feels like a throwback to Brazil's military dictatorship that murdered thousands of indigenous people, drove thousands more from their lands, and tortured many and enslaved others between 1964 and 1985.
"The greatest concern today is with the attacks on the demarcations of our lands, we therefore need to make self-demarcation," Kokoj said. "If it depends on Bolsonaro, he will open up areas for large-scale agricultural production, multinationals, mining. That worries us a lot."
Self-demarcation is when indigenous communities draw the boundaries of their own territory, often expelling illegal occupants, such as loggers.
No one will be untouched by a warming planet, scientists say
Climate change is costing cities as they try to adapt and mitigate. Farmers are facing increasing challenges, which can lead to consumers paying more for food. Extreme weather disasters are on the rise in Canada and costing insurance companies, leading to higher premiums. People are dying in heat waves which are set to become more frequent as the planet warms; and hurricanes are stalling, meaning more people are in danger for longer periods of time.
With Passage of NAFTA 2.0, Congress Boosts Fossil Fuel Polluters in Mexico
NAFTA 2.0 cleared another hurdle on January 16 as the U.S. Senate approved the trade deal with bipartisan support.
This Is Our Last Decade to Get Climate Right
The Intergovernmental Panel on Climate Change warned in October 2018 that we have 12 years to keep global temperatures increases at 1.5 degrees Celsius or less. Now, we have more like 10 years.
Activists Find Evidence of Formosa Plant in Texas Still Releasing Plastic Pollution Despite $50 Million Settlement
Their suit against Formosa Plastics Corp. USA resulted in a $50-million-dollar settlement and a range of conditions in an agreement known as a consent decree. Key among the conditions was the company's promise to halt releasing the nurdles it manufactures into local waterways leading to the Texas Gulf Coast by January 15.
How One Utah Community Fought the Fracking Industry — and Won
'Pete Takes Money From Fossil Fuel Billionaires': Climate Activists Disrupt Buttigieg Rally in New Hampshire
"We are really concerned about candidates who have taken money from fossil-fuel executives. So that includes Joe Biden as well as Pete Buttigieg."
Big Oil sponsors Croatia's EU presidency
The six-month Croatian presidency of the Council of the EU has signed a deal with the country's national oil company INA to supply its fuel.
The part state-owned firm is now designated as the EU council presidency's "official gasoline supplier" – and comes during the launch of the European Commission's European Green Deal, a seminal policy that seeks to cut carbon emissions and fossil fuel consumption across the European Union over the next three decades.
"INA is proud to be official fuel supplier during Croatian's presidency of the Council of the European Union in the first half of 2020. The company will provide fuel for official vehicles," it said when asked for a comment.
The deal, described as an in-kind contribution, was confirmed to EUobserver by the secretariat of the Croatian Presidency (EU2020HR) whose spokesperson, in an email, also noted it had signed sponsorship deals with six other Croat companies.
"The EU2020HR has so far signed seven sponsorship agreements with exclusively Croatian companies. When we complete the list of sponsors full information will be posted on the EU2020HR website," said the secretariat on Wednesday (15 January).
Wildlife/Nature
Pacific Ocean's Warm "Blob" Killed a Million Seabirds in One Year
For this new study, published in the journal PLOS One Wednesday, researchers from the University of Washington, the United States Geological Survey, and other institutions determined that approximately 62,000 murre carcasses had washed ashore from central California through Alaska between the summer of 2015 and spring of 2016.
In some locations, the number of carcasses was 1,000 times more than normal. In total, the researchers estimate that one million murres died during the time period they studied, making the event the largest mass death of seabirds in recorded history.
Victoria Police rejects social media campaign claiming arson caused fires
Victoria Police say the fires engulfing the eastern part of Victoria are not being treated as the work of arsonists, as claimed by a widespread social media campaign.
In the past week, a Twitter hashtag #ArsonEmergency – mimicking the popular #BushfireEmergency hashtag – attracted thousands of tweets linking the fires to arsonists and casting doubt on the role of climate change in exacerbating the bushfires' severity.
But authorities have moved to dispel those claims.
"Police are aware of a number of posts circulating in relation to the current bushfire situation, however currently there is no intelligence to indicate that the fires in East Gippsland and north-east Victoria have been caused by arson or any other suspicious behaviour," a police spokeswoman said.
WHEN IT COMES TO THE AUSTRALIAN BUSHFIRES, RUPERT MURDOCH IS AN ARSONIST
In times of crisis, social media can become a hotbed of misinformation. Often it's accidental—people sharing faulty information they think is true, based on unconfirmed initial reports, or comical trolling, like the now-nearly-decade-old "shark on the highway" photoshop that gets shared during seemingly every hurricane landfall. But there is increasingly another sort of misinformation, and it's much more sinister. It's the deliberate injection of lies, disinformation, employing a combination of high profile trolls and bot armies to wage a public disinformation campaign.
The bushfires currently engulfing Australia offer an unwelcome example, one that I'm watching from front row seats here in the country. Little did I know when I planned my sabbatical in Sydney to study the impact of climate change on extreme weather events in Australia, that I would be here in the country to witness what is arguably the most profound example yet of that phenomenon. As I write this commentary, I'm looking out at smoke-tinged skies while the faint smell of smoke wafts in through an open window. Living through its consequences, even as we study it, is a recurring irony of climate science in the 21st century.
Australia's Fires Show How Wealth Inequality Compounds Climate Disasters
This is not the first time that Australia has been devastatingly burned. More than ten years ago, in February 2009, fires in Australia killed 173 people, injured thousands more and destroyed 2,000 homes. The day of the February 2009 blaze, which became known as Black Saturday, constituted the country's most deadly wildfire event in history. The fires were sudden and moved with alarming speed.
Of Course There Is a Class War, David Brooks
Though absolutely wrongheaded and naive, his latest New York Times column is very useful.
What Is the Earned Income Tax Credit (EITC), and Do I Qualify for It?
ProPublica has covered how budget cuts at the IRS have made it harder for the agency to ensure the billionaires of the world pay up, but the cuts haven't affected everyone equally.
In short: Wealthy taxpayers haven't faced as much scrutiny.
AstroTurf/Lobbying/Politics
Airbus Asks Court To Dismiss Chuck Yeager's Lawsuit, Pointing Out It Doesn't Allege Anything Actionable
You may recall that last summer we wrote about how American aviation legend Chuck Yeager decided to sue Airbus when the company mentioned the fact that Yeager broke the sound barrier in marketing material. Yeager's lawyer is Lincoln Bandlow, who has spent much of the past few years as a copyright troll after a formerly respectable career in which he once touted himself a free-speech fighter. His complaint, however, served mostly as comedic material. There were claims of trademark infringement and violation of Yeager's publicity rights. Neither made much sense, as repeating a historical fact, even in marketing material, does not constitute either violation and is clearly protected speech. It was only a matter of time before Airbus responded and now we have that response.
Marin to meet with Google, Apple CEOs at Davos conference
During the 22-24 January conference, Marin also meet with leaders of major tech firms including Google CEO Sundar Pichai, Apple CEO Tim Cook and Microsoft President Brad Smith to discuss "new technologies and data policy, corporate social responsibility, sustainable development and job creation," the statement says. Google has a major data centre in Hamina, while Microsoft shut down its product development operations in Finland in 2016.
Harry, Meghan to Give Up Public Funds, 'Highness' Titles Under Deal
Goodbye, your royal highnesses. Hello, life as — almost — ordinary civilians.
Bernie Sanders Denounces Trump's Effort to Divide Democrats
After President Donald Trump earlier in the day accused Democrats in Congress of "rigging the election again against Bernie Sanders," the 2020 presidential candidate shot back Friday evening to say the president's effort to divide Democrats would be unsuccessful.
"Let's Be Clear About Who Is Rigging What": Bernie Sanders Denounces Trump Effort to Divide Democrats
2020 candidate says only reason for impeachment trial is president's effort to "use the power of the federal government for his own political benefit."
Yes Minister Fan Fiction
I have been rather unwell this last week with atrial fibrillation, and at 5am last Sunday morning had the paramedics out and puzzling over the ECG results. This particularly severe episode was a result of being out in the cold and storm for hours on the AUOB march, and I felt so guilty at being a self-inflicted drain on the NHS that I declined their offer to take me into hospital and decided to recover at home.
How an Anti-Sexist Candidate Got Smeared as Sexist
The recent scandal alleging that Bernie Sanders told Elizabeth Warren a woman couldn't beat Trump captured attention for days. The manufactured narrative shows how the media repeats cynical, bad-faith attacks until they get seen as fact.
Censorship/Free Speech
Iranian Tech Users Are Getting Knocked Off the Web by Ambiguous Sanctions
Between targeted killings, retaliatory air strikes, and the shooting of a civilian passenger plane, the last few weeks have been marked by tragedy as tensions rise between the U.S. and Iranian governments. In the wake of these events, Iranians within the country and in the broader diaspora have suffered further from actions by both administrations—including violence and lethal force against protesters and internet shutdowns in Iran, as well as detention, surveillance and device seizure at the U.S. border and exacerbating economic conditions from U.S. sanctions. And to make matters worse, American tech companies are acting on sanctions through an overbroad lens, making it much harder for Iranian people to be able to share their stories with each other and with the broader world.
The Office of Foreign Assets Control (OFAC) of the U.S. Department of the Treasury administers and enforces economic and trade sanctions that target foreign countries, groups, and individuals. Some of these sanctions impact the export to Iran (or use by residents of the country) of certain types of technology, although trying to parse precisely which types are affected appears to have left some companies puzzled.
'King Of Bullshit News' Sees His Bullshit Libel Lawsuit Tossed For A Second Time
Michael Leidig, the owner of Central European News, wasn't thrilled BuzzFeed called him the "King of Bullshit News" in a 2015 article. The BuzzFeed investigation dug into CEN's publishing business and found the company did nothing more than generate a steady stream of salacious and rarely plausible "news" stories, which were then picked up by other "news" agencies (Mirror, Sun, etc.) less concerned with accurate reporting than with racking up page views.
Joe Biden Can't Tell The Difference Between The 1st Amendment & Section 230; Still Thinks Video Games Cause Violence
Joe Biden is the latest Democratic candidate for President interviewed by the NY Times editorial board, and if you're interested in tech policy, well, it's a doozy. Biden seems confused, misinformed, or simply wrong about a lot of issues from free speech to Section 230 to copyright to video games. It's really bad. We already knew he was on an anti 230 kick when he gave a confused quote on it late last year, but for the NY Times he goes even further:
Turkey restores access to Wikipedia after 991 days
Internet rights group Turkey Blocks first reported the block on April 29 2017, establishing technical evidence of a nationwide restriction in the absence of a court order or official notice. After the initial blocking, authorities explained that the entirety of Wikipedia was restricted because attempts by state officials to amend specific articles via the platform's own editorial process and legal demands had both been unsuccessful.
The site uses TLS encryption to enforce HTTPS security, an industry standard that provides user confidentiality and makes it impossible for governments to censor individual web pages. Hence, network operators blocked the entire platform in order to comply with the order. The sites were blocked with a combination of SNI filtering and DNS poisoning at the ISP level, with each telecom operator responsible for technical implementation of restrictions.
National Archives Censors Anti-Trump Messages from 2017 Women's March Photo to Avoid 'Political Controversy'
A massive, blown-up photo of the 2017 Women's March on Washington, D.C., currently on display in the National Archives, has been altered to remove anti-Trump messages, in order to avoid "political controversy."
As reported in the Washington Post, the photograph, taken by Mario Tama for Getty Images, sits near the entrance to a National Archives exhibit on the women's suffrage movement. But upon closer inspection, it's clear that the exhibited photo has been quietly changed, with any mentions of President Donald Trump, as well as references to parts of female anatomy like "vagina" and "pussy," blurred out from the protestors' signs.
Librarians, Advocacy Groups Take Action Against Missouri's Proposed Censorship Legislation
As news broke about HB 2044, legislation proposed in the state of Missouri which would open wide the doors to rampant censorship of library material and prosecution of librarians, librarians and library advocacy groups got to work.
"Public libraries already have procedures in place to assist patrons in protecting their own children while not infringing upon the rights of other patrons or restricting materials. The Missouri Library Association will always stand against censorship and for the freedom to read, and therefore opposes Missouri House Bill 2044," reads the statement from Missouri Library Association president Cynthia Dudenhoffer. "Public libraries exist to provide equitable access to information to all of its users, as it is key to having an informed populace. Missouri Library Association will always oppose legislation that infringes on these rights."
Freedom of Information / Freedom of the Press
Cambodia: Drop Charges Against 2 Journalists
Former RFA reporters Uon Chhin and Yeang Sothearin speak to the media outside of the Phnom Penh Municipal Court, Aug. 29, 2019.
Obama Freed Chelsea Manning Three Years Ago. Why Is She Still in Jail?
The country that Manning stepped into in 2017 was very different from the one she essentially left in 2010. For one, Donald Trump was president, and his administration was already outwardly hostile toward her—only days after he was inaugurated, Trump called Manning an "ungrateful traitor" in a tweet.
Impeachment trial security crackdown will limit Capitol press access
"There is no additional safety or security brought by bringing such a device into reporter work space and gives the impression that it is being done mostly to protect Senators from the bright light of the public knowing what they are doing in one of the country's most important moments," the Standing Committee of Correspondents wrote in a letter Tuesday to Senate Majority Leader Mitch McConnell and Minority Leader Charles E. Schumer.
EU could force mobile phone manufacturers to standardise chargers
"To reduce electronic waste and make consumers' life easier, MEPs want binding measures for chargers to fit all mobile phones and other portable devices," claims the briefing.
The parliament plans to vote on the measures in the next month.
Civil Rights/Policing
Hunger Striker Nearing Death in ICE Custody: "I Just Want Freedom"
Doctors and advocates for incarcerated immigrants are demanding that Immigration and Customs Enforcement (ICE) release five South Asian asylum seekers on hunger strike at a remote ICE jail in Louisiana, where at least three strikers are being force-hydrated and at least two are being force-fed after refusing to eat and drink for over 75 days.
Russia: Persecution of 'Undesirable' Activists
Biden Prioritizes Israel While Overlooking Daily Anti-Semitism in US
When, on January 2, Democratic presidential candidate Joe Biden responded to a questionnaire from the Jewish Telegraphic Agency (JTA) about anti-Semitism, Zionism, and his relationship to Jews and Jewish culture in the U.S., there was a noticeable absence in his response: practically any mention of American Jews.
Episode 63 – The Global Crisis of Child Labor Exploitation – Along The Line Podcast
Tens of Thousands of People Lost Driver's Licenses Over Unpaid Parking Tickets. Now, They're Getting Them Back.
Some 55,000 motorists will regain their right to drive this year after Illinois Gov. J.B. Pritzker signed legislation Friday that ends the practice of suspending licenses over unpaid parking tickets.
The law, known as the License to Work Act, goes into effect in July.
Why China's move to bar Human Rights Watch chief from Hong Kong was contrary to the city's Basic Law
Once again, the Chinese authorities are trying to accuse others for their own failures there – the people of Hong Kong are the loudest and strongest voices demanding democratic freedoms and the rule of law. The current situation is a direct result of Beijing's increasing encroachment on Hong Kong's freedoms and its rigid disregard of the legitimate grievances of the Hong Kong people.
British Police Failed To Stop Muslim Grooming Gang Due To Fears Over "Community Tensions"
The police operation identified 57 girls who had been exploited, including one who died after being injected with heroin by her abuser, but the case was shut down in 2005 and "very few" offenders were brought to justice.
Thousands Gather for Women's March Rallies Across the U.S.
Thousands gathered in cities across the country Saturday as part of the nationwide Women's March rallies focused on issues such as climate change, pay equity, reproductive rights and immigration.
DaBaby Shockingly Caught on Video Assaulting Hotel Worker
Fresh off a recent arrest, along with other altercations, rapper DaBaby has been caught on video assaulting a hotel worker.
FBI: Saudi government 'almost certainly' helps its citizens escape prosecution in US for serious crimes
Some of the cases date back 30 years, suggesting the Saudi government had spent decades subverting the U.S. criminal justice system and leaving untold numbers of victims without any recourse.
In April, a story co-published by The Oregonian/OregonLive and ProPublica showed how the FBI, Homeland Security and other agencies have been aware of the Saudi machinations since at least 2008 yet never intervened.
Internet Policy/Net Neutrality
Verizon Kills Cable Contracts As TV Sector Finally Starts Listening To Cord Cutters. Kind Of.
Remember cord cutting? The trend that cable and broadcast execs and countless sector analysts spent years claiming either wasn't real, didn't matter, or would most certainly end once Millennials started procreating? It set records in 2019, and despite some wishful thinking among cable TV executives, there's no real sign that the trend is going anywhere thanks to the continued rise of new streaming services.
ICANN Needs To Ask More Questions About the Sale of .ORG
Over 21,000 people, 660 organizations, and now six Members of Congress have asked ICANN, the organization that regulates the Internet's domain name system, to halt the $1.135 billion deal that would hand control over PIR, the .ORG domain registry, to private equity. There are crucial reasons this sale is facing significant backlash from the nonprofit and NGO communities who make the .ORG domain their online home, and perhaps none of them are more concerning than the speed of the deal and the dangerous lack of transparency that's accompanied it.
Less than three months have passed from the announcement of the sale—which took the nonprofit community by surprise—to the final weeks in which ICANN is expected to make its decision, giving those affected almost no chance to have a voice, much less stop it. The process so far, including information that the buyer, Ethos Capital, provided to ICANN in late December, raises more questions than it answers. U.S. lawmakers are correct that "the Ethos Capital takeover of the .ORG domain fails the public interest test in numerous ways."
2019 Income Sources
Trad pub certainly sells a bunch through Amazon. Traditional publishers put even more effort into diversification than I do. I'm going to assume that they're at least as successful as I am, and roughly one-third of my trad pub income comes through Amazon. This means Amazon backs roughly 42% of my income.
Book Review: 3D Printing and Intellectual Property
One of the subjects that IP laws have had to grapple with in the 21st century is that of 3D printing technology. 3D printers manufacture physical objects based on input from a digital file. The range of 3D printable products includes medical products, aesthetic products, plane parts, toys, buildings, guns etc. With the opportunities offered by 3D printing technology comes the worry from the IP community that the technology is susceptible to abuse and undue exploitation of IP rights. In 3D Printing and Intellectual Property, Lucas S. Osborn explains the technical aspects of 3D printing in plain language before going on to address how various aspects of IP laws (copyright, trademarks, patent and design laws) confronts the issues raised by 3D printing technology. A primer chapter on the foundations of IP law provides a much-needed context for the issues at the root of IP law concerns about 3D printing.
This book will be relevant to a variety of readers: lawyers, lawmakers, policymakers and judges. For its plain, easy-to-read language and its coverage of technical aspects of the 3D printing technology, this book will also be relevant to technologists and artists. Having said that, legal practitioners and scholars in the US, Europe and many countries in the Global North might find the book more relevant than practitioners from the Global South and the African continent, as it focuses on the former jurisdictions.
Proportionality clause in draft German patent reform bill falls short of not only eBay v. MercExchange but also the EU's definition
The debate over a draft patent reform bill presented by the German Federal Ministry of Justice and Consumer Protection is raging. I outlined some fundamental issues on Wednesday. Meanwhile, the ministry has published its draft bill.
On his (strongly recommended) Comparative Patent Remedies blog, Professor Thomas Cotter indicated my reaction to the draft bill made more sense to him after reading the explanations provided by the ministry. The rationale provided for the measure, and the legislative intent expressed in the "Eckpunktepapier," are very clear that they don't mean to change anything about where the case law already stands under the current statutory framework. Yesterday, CIPLITEC hosted a speech by Presiding Judge Ulrike Voss ("Voß" in German) of the Dusseldorf Higher Regional Court (not to be confused for Andreas Voß of the Karlsruhe Higher Regional Court, formerly Mannheim Regional Court). The last question an attendee had for the appellate judge was about what the reform bill would change, and essentially she said this:
There's the statute, and there's the rationale given by the ministry. As a judge, she'll be sure to read the rationale part as well, but the statute will be the law to apply. The rationale says that there's no intent to change anything, and the language of the statute is consistent with existing case law. Yet the courts may ask themselves whether the fact that the law was amended means that some change of the situation was intended anyway, contrary to the official statement by the ministry. She certainly predicted that the statute would encourage defendants to argue proportionality more frequently than before.
The judge went on to say that there might be a bit more of an opening for exceptional cases in which the defendant would, on the basis of proportionality, be granted not only a transitional (workaround/"use-up") period as stated by the Federal Court of Justice in the Heat Exchanger decision, but in some cases might be able to elude injunctive relief on a permanent basis. She did, however, note that "it would take a lot" to get to that point, and in response to a follow-up question agreed that the standard for that would be clearly more demanding than the one for a transitional period. In my opinion, while Heat Exchanger addressed the hypothetical possibility of a use-up period and stopped short of suggesting a permanent denial of injunctive relief, it didn't preclude that from being the outcome in some extreme cases either. So the door was never closed, but the judge has a point that it may be slightly more open now.
Shooting from the Hip and Curing a Premature Appeal
When I first wrote about Amgen v. Amneal, I focused on the patent law issue regarding Markush groups in the patent claims. Here, I want to circle back to the question appellate jurisdiction. I'll be teaching the 2nd semester of Civil Procedure at Mizzou this spring and appellate jurisdiction is the the first topic up for grabs.
Most patent infringement litigation reaches the Federal Circuit under the "final judgment rule" which creates appellate jurisdiction over cases "appealed from a final decision of a district court." 28 U.S.C. 1295(a)(1). Final decisions "are decisions that end litigation on the merits and leave nothing for the court to do but execute the judgment." Slip op.; Coopers & Lybrand v. Livesay, 437 U.S. 463, 467 (1978). Thus, in a patent case involving infringement claims and invalidity defenses, both must be resolved before appeal. (Note here, that there are some, albeit limited, options for interlocutory appeals prior to final judgment).
Too Short Analysis: The Federal Circuit did not cite any precedent or statute to support its conclusion that the oral-argument waiver can cure the lack of final judgment. The procedure principles here fall within the umbrella of "cumulative finality." Most courts have accepted the notion that a premature appeal can be cured by a subsequent entry of final judgment by the district court. However, some courts have taken an alternative approach.
Without eBay factor #2, German patent reform movement is left with nothing but Kremlinology, spin, and self-delusion: licensing vs. injunction
Thanks to both those favoring and those opposing patent injunction reform in Germany who gave me feedback. What I hear for the most part is that people appreciate my relentless pursuit of harsh analysis. The injunction-related part of the German patent reform bill is redundant boilerplate of the kind that's not really going to move the needle. Someone has to tell it like it is. The only good news so far is that Ingmar Jung, the rapporteur of the CDU/CSU (= the chancellor's party) group in the German Federal Parliament, told Handelsblatt that they're going to "scrutinize and discuss" the proposal.
I'm quoted in that same article, saying that the proposed reform, a draft version of which was revealed earlier this week, fails to deliver meaningful progress as a prevailing patentee simply needs to make a licensing offer that may be excessive, but just not excessive enough for the court to categorize as unreasonable based on a superficial analysis. In that case, the "infringer" will be considered an "unwilling licensee," and the injunction will come down regardless of the damage it may do.
While I agree with at least one of the thought leaders of the reform movement who says that he sorely misses (as do I) a reference to the commercial value of an invention underlying an injunction patent relative to the accused product, even that perspective isn't comprehensive enough. At the heart of the single biggest issue there's the total absence, from the statute as well as the government's rationale, of the second eBay v. MercExchange factor: the requirement for an injunction that monetary relief (= a damages award) be "inadequate" to make the patentee whole. That includes, but is not limited to, the intrinsic value of the invention at issue and its relevance to the accused product.
Obstruction Ahead! The IPEC decision in Adolf Nissen Elektrobau v Horizont Group
As an early Christmas gift, on 18 December 2019, His Honour Judge Hacon handed down a judgment in the matter of Adolf Nissen Elektrobau v Horizont Group. This case concerned the field of electric road traffic signs. Horizont's UK patent (the 'Patent') claimed a board to be mounted on to vehicles which had an arrangement of lights which included flashing directional arrows and a warning cross which was shown in a different colour in a constant light. The benefits of the invention was said to be causing "increased levels of attention" to road users due to the "unusual – and thus unexpected – form and colour". Oddly the Patent specification described mobile electric road traffic devices that were known in the art as those that include directional arrows and warning crosses both of which flash and are in a yellow colour. This, however, was not in line with UK road traffic regulations which require that the crosses are constant (not flashing) and are red. It was noted in the decision that yellow flashing arrows and crosses are consistent with German road traffic regulations.
Adolf Nissen ('Nissen') sought to revoke Horizont's Patent for lack of inventive step over three pieces of prior art: DE007 (a German Utility Model), a US Patent ('Pederson') and the prior use of one of Nissen's traffic boards. The court held that Horizont's Patent was invalid as it lacked inventive step over DE007 and one of Nissen's traffic boards. DE007 described a panel with flashing lights which could be "interconnected to represent a leftward or rightward arrow or a diagonal cross". Horizont argued, among other things, that DE007 differed from the Patent as it did not display a constant (i.e. not flashing) red cross. It was held that it would be obvious to adapt DE007 to show a constant red cross as well as flashing yellow arrows as these were conventional road traffic signals in the UK. The skilled person would have known how to achieve this as part of his common general knowledge ('CGK') using LEDs of two colours in the spotlights. Further it was held that the any reservations the skilled person might have concerning safety or regulatory approval were not relevant to the assessment of obviousness. Turning to the other successful piece of prior art: Nissen's traffic board. This was supplied with a purely yellow LED display which contained a full library of possible signs, including a cross or flashing yellow arrows. As this yellow LED display was not in line with UK regulations, it was argued that the skilled person would not find this purely yellow LED display useful and therefore it would be obvious to incorporate red LEDs to allow the optional display of a red cross, so as to comply with UK regulations. This would make the device familiar to UK road users.
Help or headache? Counsel share data tips and pitfalls
Data is providing both headaches and solutions for corporates, according to in-house lawyers, who say that data mining has proven to be a useful means of tracking down infringers.
Cross-Licenses, Royalty Stacking, and Patent Disputes
While the majority of U.S. patent litigation is now NPE litigation (and has been since at least 2009), litigation between operating companies continues to occur at roughly the same rate as it has for the past 20 years. (So much for the idea that IPR or eBay destroyed the ability for patent owners to enforce their intellectual property.)
These two categories of litigation are often fundamentally different, occurring for completely different reasons. For every Smartphone War, there are dozens of cases that are fundamentally about getting a competitor to pay a little more for the license they were probably already discussing taking. That's true in cellular technology, but it's true in a lot of other areas as well, ranging from semiconductor manufacturing to automotive to digital video. It's relatively rare for an operating company to file a lawsuit without discussing the possibility of a license first—after all, patent litigation is expensive and if you can avoid it, why wouldn't you? (This is also why large companies employ entire teams of people whose sole job is to make deals to license patents—at least, once they're aware those patents exist and have a chance to evaluate infringement.)
But there are a couple of considerations that can make it hard to get to an agreement as to that license. These considerations can result in litigation being filed—and often quickly settled after the parties refocus on negotiations, realizing that they'd far rather have a deal than spend tens of millions on protracted patent litigation.
Another issue is royalty stacking. When you have a complex multi-component product it could potentially require licensing hundreds or thousands of patents. Smartphones might implicate more than 250,000 patents. And if each licensee asks for 1% of the sales price, you can very quickly wind up paying more for licenses than you'll make from the product—at which point you just don't make the product. That's the essence of the royalty stacking problem.
One analysis, examining the royalty stack problem for smartphones, arrived at an estimate of more than $120 in patent royalties for a $400 smartphone. Each licensee, individually, might only receive $0.50 or $1.00 or $9.00 per phone—but, when summed up, the licenses approach the cost of the hardware. That situation recurs in many consumer products—for example, consumer audio-visual devices may need to license Wi-Fi patents, A/V codecs, user interface patents, and other features, all of which can add up quickly on a $50 or $100 device.
Effectively, the royalty stacking problem is a species of tragedy of the commons. Licensors are incentivized to extract the highest possible royalty from the manufacturer, but if every licensor does so, they run the risk of the product becoming unprofitable and the manufacturer ceasing to manufacture it. Even worse, licensors generally don't have knowledge of the total royalty stack of the party they're negotiating with, meaning that their idea of a reasonable portion of the stack may be very different from what the licensee knows is possible. (And licensees are generally obligated to keep their licenses secret, meaning they can't share how much they're paying as a way of reducing this knowledge gap during negotiation.)
At the end of the day, the licensor may think they're asking for something reasonable, while the licensee knows the requested royalty is out of line with licenses for similar patents and paying that rate would drive the product into unprofitability, a net loss for all involved. Again, this can make it difficult to reach a settlement.
2020 is set to be a crucial year for Standard-Essential Patent litigation in Europe
The debate over standard-essential patents (SEPs) is typically distinguished as much by concerns over competition than issues of patent law per se. Erixon argues: '…SEP disputes are less concerned about the rights and boundaries of patents, and more about antitrust limits to market behavior.' At the European level EU institutions acknowledge the policy concern that the owners of patents on technological standards (SEPs) could block competitors from making use of standards, and thus obstruct the development of efficient and thriving ICT and Internet of Things (IoT) sectors, an acute point in the 5G era. The aim is to balance the role of monopolistic patent rights (granted at the non-EU EPO) in the context of the EU's overriding focus on competition (embedded in Articles 101 to 109 of the Treaty on the Functioning of the European Union (TFEU)).
The inherent fragmentation of the European patent system, which has significant EU and non-EU aspects, works against the regulatory centralisation typically favoured by the EU. Further fragmentation is caused by the self-regulation model of standard-setting, which occurs at independent standard organisations such as ETSI and ISO. Yet, the EU is still able to exert a great deal of influence – in particular, in disputes over SEPs national courts are obliged to follow the CJEU's guidance on what amounts to fair, reasonable and non-discriminatory licensing of standard-essential patents (FRAND). At the judicial level, the challenges of resolving this issue are compounded by tension between the CJEU's role in making prescriptive rulings on matters of intellectual property and the implementation of these rulings by national courts in complex domestic litigation.
Key questions concern the global nature of SEP disputes, the interpretation of the CJEU guidance in Huawei (2015), and assessing the effect of the non-discrimination component of the FRAND (does it mean that materially the same licence terms as offered to Samsung must be offered to Huawei in the circumstances of the Unwired case?). The question of royalty-rate calculation is also the subject of appeal. In a linked case, the UKSC will also decide a case involving Conversant, Huawei and ZTE concerning whether England & Wales is an acceptable forum to decide global patent licensing disputes regarding patents valid in other territories.
Huawei seeks to overturn the earlier Court of Appeal and High Court rulings that held that seeking a global FRAND licence (rather than a territory-by-territory one) is fair and reasonable approach in the context of SEPs. In this view, if an implementer of the SEP technology refuses a FRAND global licence, an injunction can be sought by the SEP holder and granted by the court (as it will not constitute abuse of dominance). The UKSC will consider whether Unwired Planet, as SEP holder, made use of a 'time advantage' to improve its negotiation position.
Will the decision be handed down before the UK's formal exit from the EU on 31st January 2020? What could the impact of a 'no deal' or 'bare-bones' exit mean for the UK's legal system post-transition at the end of 2020? Could the impact of Brexit mean UK courts choosing (in coming years) to depart from the CJEU's guidance in Huawei v ZTE and develop an alternative approach? Looking to the year ahead, 2020 ought to bring a measure clarity on at least some these issues.
$4,000 Cash Prize for Prior Art on Universal Cipher Patent
On January 17, 2019, Unified added a new PATROLL contest with a $4,000 cash prize for prior art submissions for US 7,721,222. The '222 patent generally relates to a non-English text generation system by which text can be generated in any language without a keyboard. The '222 patent is owned by Universal Cipher, LLC (an NPE) and has been widely asserted in district court. To protect innovation and deter future frivolous assertions, Unified is offering a $4,000 cash prize for the best prior art on this patent.
Learned societies turn against scholarship and join publishers for profit
Many researchers, including the authors of this post, are members of learned societies whose mission is to support research and advance scholarly values. At the end of last year, we saw these values betrayed when over 100 societies joined with the Association of American Publishers in signing the letter to the White House. It is unsurprising that corporate publishers would resist reform. For them, any change to the rules threatens their profits. But why would learned societies purporting to represent researchers sign on? One reason is that some scholarly societies have come to depend on the lucrative business of journal publishing to subsidize their operations. In the case of one of the signatories, the American Psychological Association, publishing income accounts for ~80% of their revenue. Such learned societies face a dilemma: Their stated mission is to support scholarship, but the funds to do it come from a publishing system that has worked against scholarly values for more than three decades. We are saddened that these societies have decided to side with the publishers against the public interest.
Red Hat and IBM Jointly File Another Amicus Brief In Google v. Oracle, Arguing APIs Are Not Copyrightable
Monday Red Hat and IBM jointly filed their own amicus brief with the U.S. Supreme Court in the "Google vs. Oracle" case, arguing that APIs cannot be copyrighted.
"That simple, yet powerful principle has been a cornerstone of technological and economic growth for over sixty years. When published (as has been common industry practice for over three decades) or lawfully reverse engineered, they have spurred innovation through competition, increased productivity and economic efficiency, and connected the world in a way that has benefited commercial enterprises and consumers alike."
OSI Files Amicus Brief in Supreme Court's Google v. Oracle
The Open Source Initiative is proud to join OSI affiliate members Creative Commons, Mozilla Foundation, Software Freedom Conservancy, and Wikimedia Foundation along with other small, medium and open source technology organizations in filing an amicus curiae ("friend of the court") brief in the Google v. Oracle case pending before the U.S. Supreme Court.
In Google v. Oracle, Oracle successfully convinced the appeals court that Google's reuse of a limited number of Java declarations in its creation of the Android operating system is a copyright infringement and that a jury finding it fair use was mistaken. The brief asks that the Court reverse this decision and confirm that, as has been the common understanding for decades, API interfaces are not copyrightable and that their reuse by others is a fair use under copyright law.
Ariana Grande Faces Copyright Infringement Lawsuit Over '7 Rings'
A rapper is suing pop star Ariana Grande for copyright infringement, claiming that she copied the chorus of "7 Rings" from one of his songs.
'Local' Pirate Sites Are Thriving Around the World
The Pirate Bay, YTS and Animeflv are well-known pirate brands that are hugely popular in many countries. The same can't be said for Mrpiracy, Filma24, and Cimaclub. However, the latter and many other 'local' sites dwarf these major brands on their 'own' turf.
'Casting Couch' Movie Company Orders Cloudflare to Unmask Tube Site Pirates
Adult movie company AMA Multimedia has obtained a DMCA subpoena from a Washington court to help it track down individuals who uploaded content to various 'tube' sites. The subpoena orders Cloudflare to hand over the identities of uploaders and potentially site operators too but given the way the content seems to be delivered, it remains a question whether the former will be possible.
Manga Scanlation Teams Don't Want War, They Want Accessible Content
Huge scanlation platform MangaDex recently revealed that legal pressures had, among things, restricted its ability to receive donations from users. Following our report, a server administrator connected to several other groups gave us additional insight into these anti-piracy efforts. Amid the hostilities, however, it appears that all the scanlation community really wants is to improve the chances of manga titles arriving in the West.
Pirated Copy of '1917′ Leaks in Massive Screener Dump
Six pirated movie screeners have leaked in the span of just a few hours. The screener dump includes a copy of Golden Globe winner 1917, one of the most prominent leaks thus far. Also notable is the involvement of the group TOPKEK, which hasn't released any screeners before.
Judges Reject EPO Patents on Life as Constitutional Complaints Against the EPO Pile Up in Germany
Posted in Europe, Law, Patents at 3:45 am by Dr. Roy Schestowitz
5 challenges and counting…
Summary: EPO judges throw out patents on life (CRISPR at least); there's now growing hope that they'll have the courage to do the same to patents on software
THERE HAS been mostly good news coming from the European Patent Office (EPO) in recent days. We hope there will be positive impact and perhaps an end to software patents in Europe.
"As fewer readers may know, there are currently quite a few constitutional challenges against the EPO."As most readers know/are aware of, Team Campinos/Battistelli is unscientific and perhaps anti-scientific. The sole goal is granting as many patents as possible, irrespective of what the science says and what scientists need. It's not in vain that examiners are protesting and it is not without reason.
As fewer readers may know, there are currently quite a few constitutional challenges against the EPO. Richard Gillespie wrote about "Constitutional complaints against the EPO in Germany" just under a day ago. There's a decent roundup right there, naming 2 BvR 2480/10, 2 BvR 421/13, 2 BvR 756/16, 2 BvR 786/16, and, 2 BvR 561/18:
Patent Attorneys like myself are not known for their love of excitement. For example, I like reading lists. One regrettably exciting item that appears to have slipped off the 'things to look out for in 2020' lists that I have seen is the outcome of the constitutional complaints against the EPO in Germany. The outcome of these complaints could have potentially explosive implications for patent practice in Europe and they have not received enough attention.
At present there, are five constitutional complaints relating to the European Patent Office (EPO) before the German Federal Constitutional Court (BVerfG), namely, 2 BvR 2480/10, 2 BvR 421/13, 2 BvR 756/16, 2 BvR 786/16, and, 2 BvR 561/18. At issues is the lack of sufficient legal remedies at the EPO against negative decisions of the Boards of Appeal. I believe there is a clear risk that the BVerfG will uphold at least some of the constitutional complaints relating to the EPO. Such an outcome would likely mean that the European Patent Convention (EPC) in its present form is incompatible with the German constitution.
My reasoning is as follows: according to these complaints there is a question (amongst others) on whether or not Articles 19(4) and 103(1) of the German constitution (i.e. the Basic Law of the Federal Republic of Germany) have been violated. Article 19(4) states that if any person's rights are violated by a public authority, they have recourse to the courts. Article 103 deals with the right to a fair trial.
As noted in by Vissel (GRUR Int. 2019, 25) it is instructive to note the submissions of the Federal Republic of Germany during the Travaux Préparatoires of the EPC (emphasis added):
"The delegation of the Federal Republic of Germany opposed this request [to delete para. (b) of Art. 135]. It pointed out that the application of a national procedure should be possible not only in cases in which the applicant suffered a loss of rights as a result of the omission of an act but also where the European Patent Office had given a negative decision. It was in precisely these cases that there was a constitutional problem in the Federal Republic of Germany. The Basic Law required that every administrative act should be capable of being examined by a court. The Boards of Appeal of the European Patent Office, although similar to courts of law, were not in fact courts proper so that the possibility of recourse to a German Court had to be maintained. It should, however, be borne in mind that the Federal Republic did not at present intend to avail itself of the option available under para. 1(b). However, even if this option were applied, there would be little danger of any delay in the procedure since it was unlikely that proceedings would be initiated before the German patent authorities and the German Court after the European procedure had been concluded."
Hence, the provision of Article 135(1)(b) EPC was drafted for a situation in which the Boards of Appeal of the EPO could no longer be seen as independent courts.
This was a situation that had occurred within the German Patent Office when appeals against decisions of the Office were conducted internally. There was a constitutional complaint against the internal appeals of the German Patent Office because of a lack of sufficient legal remedies at the German Patent Office. This complaint was upheld and it ultimately lead to the establishment of the German Federal Patent Court.
We assume readers are aware of the constitutional complaint against the UPC and we have repeatedly shown that the press does not properly cover this (if at all). Amplifying the EPO's lies is not journalism and here's a new example of it ("New EU Patent System On Course For End Of 2020, Says EPO"). The EPO lies and some people copy-paste the lies, just like so-called 'reporters' who publish "Trump says" pieces. From the outline:
Progress is being made towards the implementation of the EU's new patent system, but the UK's insistence on severing all ties with the European Court could spell the end for its participation.
Could or will? Will. Has. This is hardly news.
The EPO's management has meanwhile moved on to its new lie (warning: epo.org link), having published this piece in which patent maximalists from all around the world push software patents agenda under the guise of "emerging" and "HEY HI" (AI). The EPO attributes this propaganda to "IP5" and says:
The five largest intellectual property offices held the inaugural meeting of their joint Task Force on New Emerging Technologies and Artificial Intelligence this week in Berlin. Known as the "IP5", the five offices – which are the EPO, the Japan Patent Office (JPO), the Korean Intellectual Property Office (KIPO), the China National Intellectual Property Administration (CNIPA) and United States Patent and Trademark Office (USPTO) – together handle about 85% of the world's patent applications. The meeting was organised jointly by the EPO and KIPO.
Launched at the IP5 annual meeting last June in Incheon, South Korea, the new task force will explore the legal, technical and policy aspects of new technologies and AI, their impact on the patent system and on operations at our five offices. The aim is to pinpoint which areas can most benefit from joint IP5 responses, ranging from employing AI to improve the patent grant process, to applying the patentability requirements to inventions in the field of AI, and handling applications for inventions created by machines.
"This task force is the IP5 offices' first joint response to a changing global patenting landscape and evolving user needs in the field," said Christoph Ernst, the EPO's Vice-President for Legal and International Affairs, opening the event. He added: "New emerging technologies and AI touch upon almost every aspect of daily life and seem to question the traditional models for the generation and utilisation of knowledge flows and decision-making. This translates into considerable challenges in IP, and the task force is a chance for us to demonstrate that we, as the world's leading offices, are agile and responsive to change."
It's very clear that Campinos, Iancu and the others just want to grant as many patents as possible, no matter the legality of these. This includes software patents.
Having said that, this EPO agenda has just suffered a major setback because CRISPR patents turn out (again) to be fake patents. This can, by extension, doom many other European Patents on life and nature.
The EPO has just tweeted: "Heinz Müller, #patent expert at @ige_ipi, will talk about the #patent landscape of #CRISPR at this event in Zurich…"
Maybe the EPO did not get the memo, but around the very same time (maybe the same day) judges found the courage to say no to CRISPR patents.
A site advocating for such patents (pressure group of the "life science" monopolists) wrote:
In a dramatic reversal, a European Patent Office's (EPO) board of appeal has upheld the revocation of a Broad Institute CRISPR/Cas9 patent.
Yesterday, the board indicated that it would refer several key issues at the heart of the case to a higher panel, potentially triggering a lengthy delay.
But today the board has announced that, after consideration, it is already equipped to decide the case and agreed with the earlier Opposition Division ruling that the Broad's patent lacks a valid priority claim.
Daniel Lim, partner at Kirkland & Ellis, said the decision was "quite the change of heart" from the board.
"I can imagine that the stakes involved in this case and the level of interest and scrutiny have not made the Board's life easy," he said.
Yesterday's proceedings opened with the announcement that the board intended to refer at least three questions to the EPO's enlarged board of appeal.
This has also been covered by Rose Hughes (AstraZeneca), who said:
The Board of Appeal (3.3.08) finished hearing submissions on priority from the parties this morning. Proceedings were then adjourned until the afternoon whilst the Board conferred. The parties undoubtedly had a tense lunch. The Board was either going to decide on the issue of priority or refer the issue to the EBA for clarification. There was a strong feeling following the comments made by the Board of Appeal on Day 3 that a referral to the EBA was likely. However, news came soon after recommencement of the proceedings that the Board of Appeal was to dismiss the appeal. [In a classic fake news saga, Merpel watched with bemusement today the ongoing proliferation of reports that the Board of Appeal had referred the matter to the EBA].
The immediate impact of the referral would have been to prolong the dispute. Even if the EBA had accepted the referral (far from certain), any decision from the EBA would not have been the end of the matter. The EBA is there to provide clarity on points of law. After a EBA decision, the case would then have had to be sent back to the Board of Appeal. Those wishing for legal clarity will welcome the Board of Appeal's decision to settle the matter today.
On the other hand, a fact easily forgotten amidst the all the excitement over this week's appeal hearing, is that the patent in dispute, EP2771468, is far from being the Broad Institute only patent relating to CRISPR. Whilst today is the end of the road for EP2771468, there are 5 divisional applications in the same family as the patent in dispute: EP2784162, EP2896697, EP2940140, EP2921557, EP3144390.
The patent family of EP2771468 is also, of course, not the only family relating to CRISPR. There are many other patents relating to aspects of CRISPR technology, owned by the Broad Institute and other parties, most notably University California Berkeley.
Could this be the most courageous decision these judges have made in recent years? More importantly, will there be 'consequences' for it? Will they soon decide to rule out software patents ('simulation') as well? Let's hope so. █
Posted in IRC Logs at 2:50 am by Needs Sunlight
StartPage (System1) Found New Spin Allies. Some Have Been Offered StartPage Jobs. Some Might Already be Working for StartPage in Secret.
Posted in Deception, Search at 2:47 am by Dr. Roy Schestowitz
When you have critics and you pay people to discredit them, what does that make you?
From StartPage with love
Summary: Pro-StartPage voices appear to be paid (or have been promised pay) by StartPage; the key strategy of StartPage seems to be, attack and betray people's privacy while paying people in particular positions to pretend otherwise
IT HAS been a while since we last touched the StartPage saga and little has changed. StartPage is still owned/controlled by a surveillance company and it is trying to muzzle/squash/discredit its critics.
"At this moment in time we do know for a fact about the conflict (some are more upfront about this than others), but we just don't know the full extent of it."Based on our understanding, as well as evidence we have but cannot divulge at this time, StartPage made job offers to people who are in a potential position to relist the company in privacy sites. In a sense, they're trying to pay their way into re-acceptance, without disclosing pertinent details. It's possible that some of these people are already on StartPage's payroll because they refuse to answer very simple questions.
At this moment in time we do know for a fact about the conflict (some are more upfront about this than others), but we just don't know the full extent of it. This corrupts or at least erodes trust in groups which claim to advance privacy agenda. If they receive money from surveillance companies, what does that tell us about them? Maybe some time soon we'll be able to publicly name the culprits, too. █
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: nautilus-compare on 22.04 This answer for installing nautilus-compare on 20.04 didn't work for 22.04 because it has no release file for jammy.
But it seemed to work if I pointed it to the focal package instead:
sudo add-apt-repository ppa:boamaod/nautilus-compare
sudo sed -i 's/jammy/focal/g' /etc/apt/sources.list.d/boamaod-ubuntu-nautilus-compare-jammy.list
sudo apt-get install nautilus-compare
Is this the best way to install nautilus-compare on Ubuntu 22.04?
A: Here is how I installed nautilus-compare on 22.04:
Install direct from source is possible by doing the following:
Install the nautilus python3 bindings
sudo apt-get install python3-nautilus
Download the latest source
wget https://launchpad.net/~boamaod/+archive/ubuntu/nautilus-compare/+sourcefiles/nautilus-compare/1.0.0~focal1/nautilus-compare_1.0.0~focal1.tar.xz
tar -xvf nautilus-compare_1.0.0~focal1.tar.xz
Install the source files
cd nautilus-compare-1.0.0/src
sudo cp nautilus-compare.py /usr/share/nautilus-python/extensions/
sudo mkdir /usr/share/nautilus-compare
sudo cp utils.py /usr/share/nautilus-compare
sudo cp nautilus-compare-preferences.py /usr/share/nautilus-compare
Install any missing python3 modules
At this point, if the python source uses any modules you don't have installed, then you will need to install them too, for example using pip. I already had all the modules so didn't need to.
Then restart nautilus and the compare menu should now appear.
nautilus -q && nautilus &
A: Reach Out to the Developer
The best way is to request the developers to provide a version pf the software for Ubuntu 22.04.
However, sometimes this is not possible, such as when the developer has stopped working on or abandoned the software.
Try to compile from the Source
If you know how, you may compile the source in your Ubuntu 22.04. This will make sure that all the dependencies are met.
Use the "Focal" PPA
This is the worst choice. How bad things can get will depend on what other software are in that PPA and the dependencies of those and the the installed software. This kind of mixing and matching can result into unresolvable dependency problems. Installing apps from another version or another distro can break your installation of the OS. In the worst case scenario your Ubuntu 22.04 can become completely unbootable.
There may be a technical reason why this software is not available for Ubuntu 22.04. If so there may be unpredictable issues now or later when you install some other software in your Ubuntu 22.04.
Hope this helps
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Renaissance covers a fascinating century of warfare, from the close of the Medieval era to the eve of the Thirty Years War. The period starts with the French invasion of Italy in 1494 and the armies are still largely Medieval in weaponry and organization.
While firearms and artillery are present on the battlefield they have yet to secure a dominant tactical role. However, by the close of the sixteenth century, traditional missile weapons such as the longbow and crossbow had been supplanted by arquebuses, muskets and pistols, while the proportion of firearms to pike had greatly increased since 1500.
The pike, once the primary infantry weapon for many armies - and often a decisive battle-winner in the hands of professional Swiss and Landsknect mercenaries - was, by the close of the period retained primarily as a protection against cavalry, a role it continued to perform for another century until the invention of the bayonet made it redundant.
With Renaissance, you can refight the Italian Wars between France and Spain as well as the various Anglo-Scottish battles of the 16th century, or lead Ottoman armies against Persians, Mamelukes and Western Christendom! Experience the pike & shot warfare of the French Wars of Religion and the Dutch Revolt against Spanish rule. These conflicts involve a wide range of different troop types and weaponry, providing a diverse and rewarding gaming experience.
Renaissance covers the Italian Wars between France and Spain as well as the various Anglo-Scottish battles of the 16th century and includes the Ottoman armies against Persians, Mamelukes and Western Christendom. Also included are the French Wars of Religion and the Dutch Revolt against Spanish rule.
5 campaigns with another 115 scenarios specifically for the campaigns.
A wide sampling of battles from small to large actions.
60 unique maps ranging in size from 480 hexes to 76,800 hexes providing ample ground for scenario designers to create their own actions.
Turn-based play with mostly 15 minute turns.
Both 2D and 3D views of the battlefield with multiple zoom-levels are supported.
Units include infantry, cavalry, artillery, leaders, and supply wagons as well as specialized formations such as block.
Detailed weapons including the lance, javelin, musket, arquebus, crossbow, and pike are included..
Command, Fatigue, Supply, and Terrain rules all affect game play.
The Musket and Pike game engine provides multiple play options including play against the computer AI, Play by E-mail (PBEM), LAN & Internet "live" play, and two player hotseat.
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
The model we start with can be compared with a lifting principle, $F(\gamma)(\zeta)=f(\zeta)$, $\zeta \in \mathbf{R}^{n}$
where $\widehat{g}(\zeta)=f(\zeta)$ and $\widehat{E}=F$. When $\gamma$ polynomial in an ideal $(I)$,
we assume thus that we have existence of $F$ analytic, such that $E * \gamma(D) \delta=g$.
More generally we can assume existence of $E \in \mathcal{D}_{L^{1}}'$,such that $E(\gamma)=g$.
We will restrict our attention to symbols $f$ with $\log \mid f \mid \in L^{1}$ and for a very regular boundary
(\cite{Dahn13}), we argue that the model can be represented using polynomial $\gamma$.
By further using the moment problem, we can continue $F$ to $C$,
given that the measure to $(I)^{\bot}$ is of bounded variation. If further for instance the measure is reduced,
we can continue to $\mathcal{E}'$.
We consider it as necessary for a global model to have a representation of the complementary ideal.
Assume the analytic symbols have a decomposition $(I) \oplus (I)^{\bot}$, we use the moment problem
to generalize this orthogonality (\cite{Riesz56}).
Assume that we have a continuous mapping $(I) \ni \gamma_{1} \rightarrow \gamma_{2} \in (I^{\bot})$.
A global model is invariant for change of local coordinates. We discuss instead a model invariant
for change of pairs of coordinates $H(\gamma)=H_{1}H_{2}(\gamma)$. Note that if $H$ polynomial,
we do not necessarily have $H_{1},H_{2}$ polynomials. We discuss $F(\gamma_{1} \rightarrow \gamma_{2})(\zeta)$,
where $\gamma_{1}$ is hyperbolic and $\gamma_{2}$ partially hypoelliptic.
We assume an invariance principle for movements in $(I)$, so that any movement of $(I)$ has an exact
correspondent movement in $(I)^{\bot}$ and conversely. When the movement $V$ has an analytic representation, we have a continuous mapping $V \rightarrow \tilde{V}$ into
the domain for $\zeta$. The main result that we will discuss in this paper is:
\newtheorem*{res15}{Main result}
\begin{res15}
Assume $\gamma_{1} \in (I_{1})$ an analytic symbol and consider a continuation $(I_{1}) \ni \gamma_{1} \rightarrow \gamma_{\delta} \in (I_{1})^{\bot}$,
for a parameter $\delta$. We assume there is a set $\Delta$ (lineality), where $\gamma_{1}=\gamma_{\delta}$.
We then have, given $f(\zeta)$ analytic, a solution $F_{\delta}$, such that $F_{\delta}(\gamma_{\delta})(\zeta)=f(\zeta)$.
We assume the continuation such that
$\frac{d F_{\delta}}{d x_{1}}/\frac{d F_{\delta}}{d y_{1}}=\frac{d F_{\delta}}{d x_{\delta}}/\frac{d F_{\delta}}{d y_{\delta}}$, under
the following conditions on the boundary: The boundary is given by the first surfaces $S$ to $f(\zeta)$,
that can be reached in at least one of three ways.
\begin{itemize}
\item[1] $S$ can be reached by $\Delta$
\item[2] $S$ can be reached through a movement $V^{\bot} \leftarrow U^{\bot}$, such that
$\tilde{V^{\bot}}$ and $\tilde{V}$ do not have points in common outside $S$.
\item[3] $S$ can be reached through an algebraic trajectory.
\end{itemize}
\end{res15}
Note that the mixed problem usually is given as $F \gamma_{\delta}=f_{\delta}$. But if we assume $\gamma_{1}$
has a fundamental solution $E_{1}$ and ${}^{t} E_{1} {}^{t }\gamma_{\delta} F=F \gamma_{\delta} E_{1}$,
then ${}^{t} E_{1} {}^{t} \gamma_{\delta} F \gamma_{1} = f_{\delta}$, where $f_{\delta}={}^{t} E_{1} {}^{t} \gamma_{\delta} f$
\section{The invariance principle}
We start with (\cite{Riesz43}) a movement on the hyperboloid $L(Ux,Uy)=L(x,y)$, that is ${}^{t}U \sim U^{-1}$
with respect to a lorentz metric form $L$.
The reflection invariance is defined by $L(Ux,y)=L(x,y)$ implies $y=0$, in this case $U \sim I$.
Particularly, we define the light cone $L(x,y)=0$, that is $x \bot y$ with respect to $L$.
Parallel to this we discuss a en movement with respect to euclidean metric $E(V x,V y)=E(x,y)$ that is ${}^{t}V \sim V^{-1}$
with respect to $E$ and we have an axes for invariance $E(V x,y)=E(x,y)$ implies $y=0$ and we conclude $V \sim I$.
When the movements are reflections, they are involutive.
The idea behind the model
is that an hyperbolic movement exactly corresponds to an euclidean translation. In the non- euclidean plane
we do not have any proper translation, but given two points $p,q$ there is a unique hyperbolic rotation
that maps $p$ onto $q$ and for which ``the line'' through the points is a trajectory. In this manner
the rotation axes is
conjugated to the plane through the origo and the points $p,q$. (\cite{Riesz43}).
Consider now $\gamma_{1} \in (I_{1})$ where orthogonality is relative $L$ and simultaneously $\gamma_{2} \in (I_{1})^{\bot}$
with orthogonality relative $E$. Using Radon Nikodym's theorem, we will continuously continue $\gamma_{1} \rightarrow \gamma_{2}$.
Assume $< \gamma_{1},\gamma_{2}>=0$ and $U \gamma_{1} \rightarrow I \gamma_{1}$ through a closed sequence
(continuous), say $M$. We can argue that $M={}^{\circ} ( M^{\circ} )$, that is if the condition
$< (U-I) \gamma_{1},(V-I) \gamma_{2}>=0$ implies $V \rightarrow I$ and conversely, when $V=I$, we have that $U=I$ and we have a geometric invariance principle.
Assume $J : \gamma_{1} \rightarrow \gamma_{2}$ continuous and $VJ=JU$, then
using the relation above for $M$ we have motivated an invariance principle.
We are here assuming
$< U \gamma_{1}, V \gamma_{2}>=0$ and $< (U-I) \gamma_{1},\gamma_{2}>=< \gamma_{1},(V-I) \gamma_{2}>=0$ on $\Delta$.
Now for the particular movements, we have
$\tau J = J h$, $e J= J \tilde{e}$ where $\tau$ is translation, $h$ a hyperbolic movement and $e,\tilde{e}$
rotation. Characteristic for
parabolic movements is a constant (euclidean) distance to the light cone (rotation axes).
Characteristic for elliptic and hyperbolic movements is that the rotation axes $R$ are one sided, that is
$\mid x \mid < \mid y \mid$ or $\mid x \mid> \mid y \mid$. Thus if we consider $\eta(x)=y/x$ we have
that the axes $R_{1} \rightarrow \mid \eta \mid < 1$ (hyperbolic) and $R_{2} \rightarrow \mid \eta \mid > 1$ (elliptic)
and $R_{3} \rightarrow \mid \eta \mid=1$ (parabolic). We can as usual map $\mid \eta \mid < 1$
on a half plane, in this way we can consider movement as ``one sided''. In particular the hyperbolic and elliptic movements map
half planes on to half planes.
Elliptic rotation can be represented through $e^{i<a \cdot x,a^{*} \cdot x^{*}>}$, where $a$ is a
scalar vector and where $a^{*}$, is defined such that $< a \cdot x, a^{*} x^{*}>=< x,x^{*} >$,
that is elliptic rotation is immediately represented in euclidean geometry. Using the Fourier Borel
transform, we can further associate elliptic rotation to translation of the symbol.
Assume the invariance principle $JU=VJ$ and $x \rightarrow x^{*}$ according to Legendre (reciprocal polars)
and note that $< x,x^{*}>=< x^{*},x>$ implies a normal transform.
We look for a continuous mapping $V^{\bot} \rightarrow V^{*}$. Assume $V_{e}$ corresponds to
rotational movement and let $\tilde{V_{e}}$ denote the inverse mapping, that is $V_{e}f \rightarrow \tilde{V_{e}} \zeta$,
we then have $V_{e}^{\bot} \ni x \rightarrow x^{*} \in V_{h}$ using Legendre,
that is $(V_{h})^{*} \rightarrow (V_{e}^{\bot})$ (reflexivity). In the same manner
$(V_{e})^{*} \simeq (V_{h})^{\bot}$ and $(V_{p})^{*} \simeq (V_{p}^{\bot}) \simeq (V_{p})$.
Formally $(e^{i <x, \cdot > + v(x)} f)^{\vee}= \tau \widehat{f}(x^{*})$. Using Parseval we have equivalent
sets in $L^{2}$. If we define a regular approximation through $H(U \varphi) \equiv 0$, if we assume
$\frac{d}{d x}H \neq 0$ and $H$ analytic, we have continuous induced relations for the inverse movements $\tilde{V}$
For simplicity we will in this article consider
one movement at a time, when we apply Radon-Nikodym's theorem and since the movements are not
dependent on sign of $L$, is sufficient to consider positive linear functionals.
Assume $\Delta$ a domain for the movement $I$ and $\gamma_{1}=\gamma_{2}$ on $\Delta$. Assume $(U - I) \gamma_{1}=0$ with respect to $L$
and $(V-I) \gamma_{2}=0$ with respect to $E$ and $F(J U \gamma_{1})=F(e^{v} V \gamma_{2})$ with $v \in L^{1}$
where $v = 0$ on $\Delta$. Note that the last condition is dependent on the movement.
Assume $w(\eta)=w(\frac{y}{x})=v(x,y)$
and assume $x \frac{d \eta}{d x}=-\frac{d y}{d x} - \frac{x}{y}$. Further let $d u=\frac{1}{x} d x - \frac{1}{y} d y$
then we have that $-d \eta = d u^{*}$, the harmonic conjugate.
As $w=0$, over $\Delta $ we have for a trajectory to $\eta$, that analyticity is preserved under the condition on vanishing flux $\int_{S} d u^{*}=0$
(a transmission property).
Note (\cite{Bendixsson01})
that for $X=d x/ d t,Y=d y/d t$, $(x X + y Y)/(x Y - y X)$ is passing through $0$ precisely like $-x/y - Y/X$. We assume in
the discussion that
$Y/X$ is not affected by the movement (Lie's point transform). Note also, that if
$Y/X=\rho>1$, we have if $U \rightarrow I$, that $\rho>1$ (hyperbolic) for degenerate points,
why if we limit ourselves to elliptic approximations, we do not have degenerate points.
The invariance principle for movements, has a correspondent principle in operator space.
Note that if we start from the parametrices as
Fredholm operators, with a decomposition $N(E) \bigoplus D(E)$, we see that modulo $C^{\infty}$,
that parametrices to hypoelliptic operators have $N(E)=\{ 0 \}$. Orthogonality for the symbol space
induces a corresponding relation in the operator space. If for two analytic symbols $f_{1},f_{2}$, we assume
$f_{1} \prec f_{2}$, we note that a necessary condition for inclusion for the correspondent space of
operators is $f_{1} \prec \prec f_{2}$. Thus, we have existence of $N$, such that for the operator space
$ \frac{d}{d x} f_{1}^{N} \prec f_{2}^{N}$. In particular, we can write $\frac{d}{d x} \log f_{1}^{N} \rightarrow 0$
for some $N$ and simultaneously $f_{1}^{N} \prec f_{2}^{N}$. Note that for a polynomial, we have
always $\frac{d}{d x} P \prec P$. Further, if $\mbox{ rad }f_{2} \subset \mbox{ rad }f_{1}$,
we have $N_{1} \subset N_{2}$, for the corresponding zero space. Radon Nikodym's theorem can be used,
$I(f_{1})=I(g f_{2})$, for $g \in L^{1}$. The conclusion is that a necessary condition for inclusion
of the corresponding operator spaces, is that one symbol ideal is strictly weaker that the other and
a strict dominance of symbol ideals implies an inclusion of operator spaces.
It is for this sufficient to consider the phase space, $\log f_{1}^{N} \prec \log f_{2}^{N}$
and $\frac{d}{d x} \log f_{1} \rightarrow 0$ in the $\infty$ implies $(I_{2}) \subset (I_{1})$.
Note that (\cite{Schwartz66}) $\mathcal{D}$ is dense in $\mathcal{D}_{L^{p}}$ and in
$\dot{B}$, but not in $B$. We have that $(\dot{B})'=\mathcal{D}_{L^{1}}'$ and $(B)'=\mathcal{D}_{L^{\infty}}'$.
Further we have that $(\dot{B})'$ is the limit in $B'$ of $\mathcal{E}'$.
Concerning $f \in (I_{1}) \subset (I_{He})^{\bot}$, we assume $(I_{He})$ with a global pseudo base
and $< f, d \mu>=0$ continued to $< \tilde{f}, d \mu >=0$. Assume the
continuation $\tilde{f} \sim e^{g} f$ with $g \in L^{1}$, that is $f \bot e^{-g} d \mu$. If $dv$
corresponds to $(I_{Phe})$, such that $dv \sim e^{-g} d \mu$, then $e^{g} d v$ is downward bounded.
Another example is given by $d \mu$ of type $0$,
$e^{- g_{1}} d \mu$ of type $A_{1}$, further $e^{-g} d \mu$ of finite type. A final example
is given by $d \mu$ with a trivial kernel.
Assume $\gamma_{1}=\gamma_{2}$, on $\Delta$ and construct a neighbourhood $J U \gamma_{1}=V J \gamma_{1}=V \gamma_{2}$.
We have $U \gamma_{1}^{2}=(U \gamma_{1})^{2}$, but $V \gamma_{2}^{2} \neq (V \gamma_{2})^{2}$ and
$V (e^{v} \gamma_{2}) \sim J U \gamma_{1}$ and $J \gamma_{1}^{(2)}=e^{v_{2}} \gamma_{2}^{(2)}$,
we then have $v_{2} \neq v_{1}$, but $v_{1}=v_{2}$ on $\Delta$. For instance:
$\{ 0 \} = \Delta (\gamma_{2}^{(2)}) \subset \Delta(\gamma_{2})= \Delta(\gamma_{1})=\Delta(\gamma_{1}^{(2)})$.
Thus, $\{ e^{v_{2}} - 1 \} \subset \{ e^{v_{1}} - 1 \}$, that is $\{ v_{2}=0 \} \subset \{ v_{1}=0 \}$
and for the correspondent geometric ideal, we have $I_{1} \subset I_{2}$. When we assume $U$ linear
in $\gamma$, we do not assume $U$ simultaneously linear in $\zeta$, for $V$ we do not assume $V$ linear in
$\phi$ or $\zeta$. $V = J U J^{-1}$ and $I = I_{Hyp}$ implies $V I^{\bot}=J U I$.
The boundary is defined by first surfaces $S$, invariant for all movements
and $S_{0}=S \backslash \{ x_{0} \}$.
Existence of regular approximations is guaranteed
by $\Delta$. Transversals are associated to reflection axles in $S$, we consider one movement at a time.
$\Delta$ is represented by $U^{\bot}=I$ and the choice of movement determines the properties of the neighbourhood of $\Delta$.
When $\gamma$ preserves a constant value in the $\infty$, we have $\gamma(x,y) \sim P(\frac{1}{x},\frac{1}{y})$,
as $x,y \rightarrow \infty$, for a polynomial $P$ and we can determine $\bot$ as independent of $\mid x \mid,\mid y \mid \rightarrow \infty$.
In the case when the regular approximation does not have a reduced $\bot$ measure, that is
$\int g_{reg} d \mu=0$ where $d \mu$ is not reduced, we must take into account orbits among the
possible approximations.
Sufficient for this to occur, is that we do an adjustment with point support measure
in Cousin's continuation over the boundary (\cite{Dahn13}, cfr the last section in this article). Thus, for a reduced representation in
the moment problem for the measure, we can assume transversals without orbits or orientation of orbits.
Assume $f=e^{\phi}$, and consider the problem when $U$ preserves analyticity. If $\big[ U,I \big]=\big[ I,U \big]$
and $U$ is acting linearly in the phase, we have that $U(f_{1}f_{2})=e^{U \phi_{1} + U \phi_{2}}
=U(f_{1}) U(f_{2})$ (``point topology''). Note that if $U=I$ on $W$ planar in $\mathcal{O}_{AD}$ (\cite{AhlforsSario60}),
we have that $U$ linear.
When $U=I$ on $W \ni \infty$ planar, we must assume $(X,Y)_{T} \rightarrow (X,Y)_{1/T}$ continuous.
When $U \rightarrow V$, it is not sufficient to consider tangents and we assume $F_{T} \rightarrow F_{1/T}$
preserves continuity over the axes for invariance.
Note that when $U$ is linear over $\phi$, we can assume $U(\overline{\phi})=\overline{U(\phi)}$,
$U(i \phi)=i U(\phi)$ and $U(- \phi)=-U(\phi)$. In particular this can be assumed when $\phi$ defines a planar domain
$\phi_{1} \bot \phi_{2}$ in $\mathcal{O}_{AD}$ (or standard complexified). For $J \phi$, we have
for some iterate $\phi_{1}^{N} \bot \phi_{2}^{N}$, however we also have a $e^{v}$ for $v \in L^{1}$
according to Radon-Nikodym's theorem, that is we have a non planar domain.
Since $v$ is determined by the movement, we will argue that it is sufficient to consider a domain for $v$
on one side of a hyperplane.
\newtheorem*{res1}{Invariance principle}
\begin{res1}
By considering movements as functionals, we can uniquely relate movements on the hyperboloid to movements
in an euclidean metric. We assume $J : \gamma_{1} \rightarrow \gamma_{2}$ continuous and $\gamma_{1}=\gamma_{2}$
on a set $\Delta \neq \{ 0 \}$. If we assume $\gamma_{j}$ analytic and $\gamma_{2}$ reduced
with respect to $\gamma_{1}$, we can represent $F(J \gamma_{1})=F(e^{v} \gamma_{2})$ and $v \in L^{1}$
The boundary measure represents a very regular distribution $B$ in $\mathcal{D}_{L^{1}}'$, with kernel
$\widehat{B} \subset L^{1}$.
\end{res1}
That is we do not assume $\gamma_{j} \in L^{1}$, but $\frac{d \gamma_{2}}{d \gamma_{1}} \in L^{1}$ with
respect to the boundary measure. Thus, since $\gamma_{2}$ is reduced, $\frac{d v}{d \gamma_{1}}$ is in $L^{1}$
and we can conclude that $v \in L^{1}$ with respect to the boundary measure. The boundary distribution is
thus of real type (\cite{Martineau}) and it is sufficient to consider $\mathbf{R}^{n}$ for the definition of boundary
condition. If we assume $B(e^{v})=\widehat{B}(v) \in L^{1}$
implies $v \in L^{1}$, we have that $v \in L^{1}(\mathbf{R}^{n})$.
Concerning the boundary condition, we note that the condition on a very regular boundary $\delta_{0} - C^{\infty}$
does not imply analyticity. The proposition that $\mbox{ ker }E$ (parametrix) can be replaced by
$C^{\infty}$, is implied by homogeneous hypoellipticity.
\section{Movements}
Assume that $\pi$ is a plane through the origo and $K$ is the light cone. We define
$V_{R} : \quad \pi \cap K=\{ 0 \}$,$V_{L} : \quad \pi \cap K=\{ L \}$,$V_{T} : \quad \pi \cap K=\{ L_{1},L_{2} \}$.
Assume $R$ reflection points (invariant points). We then have $R \in V_{T}$ implies elliptic
movements, $R \in V_{L}$ implies parabolic movements, $R \in V_{R}$ implies hyperbolic movements.
Every movement on the hyperboloid, can be given as reflections with respect to a plane through a fixed point.
They are divided into direct movements: reflection with respect to a plane through an axes and indirect
movements: reflection with respect to a plane through a point. We can for instance assume the axes
$x=y$ and the point is 0.
The lineality $\Delta$ is defined in the domain $\Omega$ through translation invariance for an analytic symbol.
The corresponding movement in lorentz geometry is $I$, that is
$I \gamma=\gamma$. More precisely $\tau F(\gamma)(\zeta)=F(\gamma)(\zeta)$
or equivalently $F(U \gamma)=F(\gamma)=F(I \gamma)$. Assume $U \rightarrow I$ is a movement and
let $U \rightarrow \tilde{U}$ be the mapping $(I) \rightarrow \Omega$. Given $U \gamma$
analytical, we have that $\tilde{U} \zeta$ defines a neighbourhood of $\Delta$.
In the same manner, if we let $U^{\bot} \rightarrow I$,
with $U^{\bot} \rightarrow \tilde{U}^{\bot}$, this does not imply $U^{\bot} \gamma \in H_{m}$, but given
$U^{\bot} \gamma$ analytic, we have that $\tilde{U}^{\bot} \zeta$ is continuous. Note that without the condition
on analyticity, we do not have $\lim_{U \rightarrow I} \tilde{U} = \lim_{U^{\bot} \rightarrow I} \tilde{U}^{\bot}$
For this reason, we define $\Delta=\{ \zeta \quad I \gamma = \gamma \quad \gamma \in (I) \}$, where
$(I)=\{ \gamma \quad F(\gamma)(\zeta)=f(\zeta) \}$ for some $F$. $\Omega$ can be defined through
$\{ \zeta \quad U \gamma(\zeta) \mbox{ close to } \gamma(\zeta) \quad \zeta \in \Delta \}$.
In the same manner, we can define $\Delta^{\bot}=\{ \zeta \quad I \gamma=\lim_{U^{\bot} \rightarrow I} U^{\bot} \gamma \}$.
Thus, we can assume $U \gamma \subset \{ F(\gamma)=const \}=S$ first surfaces with compatibility conditions
$U^{\bot} \gamma \cap S_{0} = \emptyset$ where $S_{0}=S \backslash \{ \zeta_{0} \}$ and $\zeta_{0}$
a singular point, further that $U^{\bot} \gamma$ is analytic close to $S$.
The movement $U_{e}$ (elliptic), has $U^{\bot}_{e}$ as ``transversals'', $U_{p}$ (parabolic) has
$U^{\bot}_{p}$ of the same character as $U_{p}$. For $U_{h}^{\bot}$ (hyperbolic) we have
$U_{h} \gamma \subset S$ and $U_{h}^{\bot} \gamma$ approximates $S$.
Further by the Fourier dual $(U_{h})^{*} \simeq (U_{e})^{\bot}$. Correspondingly in euclidean metric
we have $(V_{\tau})^{*} \simeq (V_{e})^{\bot}$ where $\tau$ is translation and $e$ is rotation.
We assume here $(I)=(I_{Hyp}) \oplus (I_{Phe})$ are
analytic functions, that is we assume $(I_{Hyp}) \ni \gamma \rightarrow J \gamma \in (I_{Phe})$
preserves analyticity.
Thus, we have $U_{e} \rightarrow U_{h}^{\bot}$
where transversal lines can be traced by translation, $U_{e}^{\bot} \rightarrow U_{h}$ first surfaces
(multivalued) can be traced through planar movements.
However we do not have that $U \gamma \in (I_{Hyp})$ implies $U^{\bot} \gamma \in (I_{Hyp})$,
further we do not assume
$U^{\bot} \gamma \subset (I_{Hyp})^{\bot}$ or $V^{\bot} J \gamma \notin (I_{Phe})$. We define $J$
through $L(\gamma, U^{\bot} \gamma)=0$ implies $E(J \gamma, V^{\bot} J \gamma)=0$. Thus, $L(U \gamma,U^{\bot} \gamma) \equiv 0$
for every $\gamma$. We use $<,>$ to denote respective scalar products.
For a regular (reversible) movement (axes $\notin H_{m}$) we have that $\big[ < \big[ I,U \big] \gamma,\xi>=0 \Rightarrow \xi=0 \big]$
implies $U=I$. In particular, if $< (U - U^{\bot}) \gamma,\xi>=0$ implies $\xi=0$ then $U=U^{\bot}$
that is we have points in common. Through the compatibility condition, we can assume points in common
on $S$. Note further $\Delta$ is joint for $\gamma,J \gamma$.
Given that the joint points can be defined through $\lim U^{\bot} \gamma = \lim U \gamma$ (with orientation)
we see that ${}^{t} U U^{\bot} - I \equiv 0$ in limes. Given $U \gamma \in \Sigma=\{ F(U \gamma)=const \}$,
by reverting the orientation for $U^{\bot} \gamma$ the continuation $U \gamma - U^{\bot} \gamma$
can be taken in the sense of Cousin (\cite{Dahn13}). We can assume product topology for $U \gamma$,
the continuation using $J$ is continuous and in $L^{1}$ (with respect to the boundary measure).
If ${}^{t} U \gamma \neq U^{\bot} \gamma$ except for a discrete set we can regard $U^{\bot} \gamma$
as a continuation of $U \gamma$. Note that we have existence of $\gamma$ with $U \gamma = I \gamma$
implies existence of $\gamma'$ such that $U^{\bot} \gamma'=\gamma'$ with $\gamma \neq \gamma'$.
Thus $U^{\bot} U \gamma = I U \gamma$
and $U U^{\bot} \gamma ' = U I \gamma'$ that is on invariant points we have $\big[ U,I \big]=\big[ I,U \big]$
(for instance $U \gamma=\gamma'$).
Further $< U \gamma, U^{\bot} \gamma> \equiv 0$ iff $< \gamma, {}^{t} U^{\bot} U \gamma> \equiv 0$
why ${}^{t} U^{\bot} U \in {}^{\circ}(I)$ (annihilators). The movements are primarily considered
in $H'$, though using the moment problem, if the movement is analytic in a set $E_{0}$, it can be
continued to $C$, assuming the compatibility conditions above.
Assume orthogonality is defined by
$<f,g> = I_{f}(g)=0$. We then have that $N(I_{f})$ is defined for $g \in H$, if $f$ has regular
kernel. If $N(I \oplus I^{\bot})=\{ 0 \}=N(I) \cap N(I^{\bot})$, we can write
$V \cup V^{\bot}=\Omega$ and $V \cap V^{\bot}=\{ 0 \}$.
When two mirrors are used, we get a non-commutative group. Consider the reflection
$S=(S_{1},S_{2})$ through the diagonal
$\pm x=y$, $R$ through the real axes and $T$ through $0$. Note that the diagonal has two
generators $S_{1},S_{2}$ and
$R S_{1} \neq S_{1} R$. But we have that $R S_{1} = S_{2} R$ and $TR S_{1}= R S_{1} T$.
Thus ${}^{t} T=T$,${}^{t} R=R$ and
${}^{t} S_{1} = S_{2}$. For $z \rightarrow z^{*}$ we have that $TR S_{1}=RT S_{1} = R S_{1} T = T S_{2} R = S_{2} T R =
S_{2} R T$. That is $z^{*}=-i z$. Note that for harmonic conjugation that for a closed form
$\varphi$, if $\varphi^{*}=-i\varphi$ that $\varphi=\alpha d z$, where $\alpha$ is analytic
locally.
A joint boundary for $(I_{1}),(I_{2})$ is represented using first surfaces
$\Sigma=$ $\{ F(U \gamma_{1})(\zeta)=$ $F(\gamma_{1})(\zeta) \}$ for every movement $U$ and
$\Sigma'=\{ F(V \gamma_{2})(\zeta)=$ $F(\gamma_{2})(\zeta) \}$,
for every movement $V$ and $\Sigma \simeq \Sigma'$. Thus, where
$J_{\delta} : \gamma_{1} \rightarrow \gamma_{2}$ and $J_{\delta} U=V J_{\delta}$
we assume $F(J_{\delta} U \gamma)=const.$ for all $\delta \rightarrow 0$. Consider the compatibility
condition $F(V^{\bot} \gamma_{2}) \neq 0$, for some $V^{\bot}$ close to and at distance from the boundary.
$V^{\bot} \gamma_{2}=J U^{\bot} \gamma_{1}$ is defined so that $F(V^{\bot} \gamma_{2}) \rightarrow \Sigma'$
iff $F (J U^{\bot} \gamma_{1}) \rightarrow \Sigma'$ regular. This is interpreted as two-sided,
that is the invariance principle in this case is extended with a compatibility condition.
\newtheorem*{res2}{Involutive movements}
\begin{res2}
Any movement on the hyperboloid is involutive. In our mixed model, any involutive movement in euclidean
metrics, has a correspondent movement on the hyperboloid. Orthogonal movements, are not necessarily
involutive, but our invariance principle maps $U^{\bot} \rightarrow V^{\bot}$ uniquely.
\end{res2}
\subsection{The lifting operator}
Consider $L(Ux,y) \rightarrow E(V x.y)$, where $E$ denotes the euclidean metric, through
the invariance principle. Define $\{ V x \}^{\bot}=\{ y=Rx \quad y \bot V x \}$
or $< V x, y >=R(V x)$, where $R$ is defined using Radon-Nikodym's theorem
and is specific for the movement. We define $V^{o}=\{ R \quad R(Vx) = 0 \quad \}$, a closed set
that is generated by $V$. Conversely, if $L({}^{t} R_{U} U x,x)=0$ $\forall x$, which implies
$E({}^{t} R_{V} V x,x)=0$ $\forall x$, where $R_{U}U=U^{\bot}$ and $R_{U}: X' \rightarrow X'$.
Thus, $\big[ L_{x}, U^{\bot} \big] \sim \big[ E_{x}, V^{\bot} \big]$
and over the reflection axes
$J(U^{\bot})=(JU)^{\bot}$.
Using the condition that $f$ and $g$ have lineality in common, we see that we have a continuous mapping
$\{ f-c \} \rightarrow \{ g-c \}$, through a planar (transversal) mapping.
In particular, with the compatibility conditions, we have $sng f \rightarrow sng g$. Note that we assume $\log f,\log g \in L^{1}$.
A reflection axes is defined by $R_{1} : {}^{t} r \gamma_{1}=\gamma_{1}$. Given $J : \gamma_{1} \rightarrow \gamma_{2}$
and $J {}^{t} r_{1}= r_{2} J$, there is a correspondent set for invariance $r_{2} \gamma_{2}=\gamma_{2}$
that is defined by $R_{2}$. For $\gamma_{1}$, we assume symmetry with respect to $R_{1}$. The corresponding
proposition for $\gamma_{2}$, is dependent on $e^{v}$ and symmetry for $v$ with respect to $R_{2}$.
Using Radon-Nikodym's theorem $({}^{t} r_{1} -1) \gamma_{1}=0$ and $({}^{t} r_{2} -1) \gamma_{2}=0$,
we then have $({}^{t} r_{2} -1) J e^{v}= ({}^{t} r_{1} -1)$. Assume $y=y(x)$ and consider
$k(x,y)=\frac{y}{x}$.
Given $k > 1$ and $\frac{d y}{d x} - \frac{y}{x}>0$, this implies $1 < \frac{y}{x} < \frac{d y}{d x}$ (elliptic movement).
Given $k < 1$ and $\frac{d y}{d x} - \frac{y}{x} < 0$, this implies $\frac{d y}{d x} < \frac{y}{x} < 1$
(hyperbolic movement).
Assume $r_{2} d \gamma_{2} \sim d r_{2} \gamma_{2}$, we then have
$\frac{r_{2} d \gamma_{2}}{r d \gamma_{1}}=e^{v}$ or $\frac{r_{2} (X_{2},Y_{2})}{r (X_{1},Y_{1})} \sim e^{v}$
where $\frac{d \gamma_{2}}{d t}=(X_{2},Y_{2})$. Given $\frac{k(X_{2},Y_{2})}{k(X_{1},Y_{1})}=const$
and $X_{1} < Y_{1}$, we have that $X_{2} < Y_{2}$ and so on. Note that $\gamma_{2}$ can be seen
as a continuation of $\gamma_{1} \subset (I_{He})^{\bot}$. Given $J r \sim {}^{t} r_{2} e^{v}$
and $< r \gamma_{1},\gamma_{2}> = < \gamma_{1}, r_{2} \gamma_{2}>$ and $< r \gamma_{1}, \gamma_{2} > = < {}^{t} r_{2} \gamma_{1},\gamma_{2}>$
that is $r \sim {}^{t} r_{2}$. Assume $A$ an annihilator for $(I_{1})$ and $B$ an annihilator
for $(I_{2})$, both closed operators. For instance $A=r-1$ and $B={}^{t} r_{2} -1 $. We then have
$(I)=(\mbox{ ker }A)$ and $(J)=(\mbox{ ker }B)$. We have that if $N(I)=N(J)$, given an analytic representation
of $(I),(J)$, that $\mbox{ rad }(I) \sim \mbox{ rad }(J)$. For movements, if we have analytic representations of
both $r, {}^{t} r_{2}$ and $r-1,{}^{t} r_{2} -1$, then the geometric sets coincide.
\newtheorem*{res3}{Fundamental representation}
\begin{res3}
We assume given $f$ analytic that $\gamma_{j}$ are analytic and that the equation $F(\gamma_{j})(\zeta) \sim f(\zeta)$
can be solved in $\mathcal{D}_{L^{1}}'$. When we can restrict the model to polynomial $\gamma_{j}$,
$F$ can be constructed from $f$ and the parametrices to $\gamma_{j}$.
In particular, where we have isolated singularities, we can represent $F$ as a measure with compact support.
\end{res3}
A very regular boundary is characterized by (\cite{Parreau51}) singularities located in a locally finite set of
isolated points or segments of analytic curves. Particularly, when the regular approximations have isolated
singularities for some higher (finite) order derivative. When the model is considered in $\mathcal{D}_{L^{1}}'$,
the real Fourier transform can be represented $P(\zeta) \tilde{f}_{0}(\zeta)$ (\cite{Dahn13}), for a polynomial $P$
and $\tilde{f}_{0}$ very regular. The fundamental representation can be derived in several ways. The parametrices
to hypoelliptic differential operators are very regular in the sense that the Schwartz kernel is regular outside the
diagonal. Assume for instance $E$ a parametrix to $\gamma(D) \delta$, where $\gamma(\zeta)$ is a hypoelliptic polynomial.
Then $\big[ E, \gamma(D) I \big] \sim \gamma(D) \big[ E, I \big]$ has Fourier-Borel transform $F(\zeta)\gamma(\zeta)$,
where $F$ is $\widehat{\big[ E,I \big]}$ and $\big[ E,I \big]=\big[ I,E \big]$, in this setting, when the parametrix is two-sided.
Note also that $F(\gamma(D) \varphi)=E(\gamma(\zeta) \widehat{\varphi})$, when $\widehat{\varphi} \rightarrow 1$.
\subsection{Reflections}
All movements in $(I_{Phe})^{\bot}$ can be represented as reflections. Singularities for
$(I_{Phe})$ can all be related to the boundary. Every point on the boundary can be reached through
reflections in $(I_{Phe})^{\bot}$ emanating from an ``origo'' on the boundary. If the points are
reached through planar reflection, then analyticity is preserved.
We can represent points that can be reached through hyperbolic movements (translation), parabolic movements (scaling)
elliptic movements (rotation) in the same leaf.
In cylindrical domains (order 0), the translations are parallel with one axes.
Pseudo convex domains are locally cylindrical at the boundary.
Given $P,Q \in H_{m}$ (the web of the hyperboloid), we can prove existence of unique movement
$P \rightarrow Q$, that corresponds to a unique (euclidean) translation $P \rightarrow Q$.
If $\pi$ is a plane through $P,Q$ and $R$ a line $\bot \pi$ implies a unique existence of a
rotation axes $R \sim PQ$ (line). $R$ is not uniquely given by the movement,
we have $\infty^{1}$ possibilities in $V_{T}$, $\infty^{1}$ in $V_{R}$ and $\infty^{2}$ in
$V_{L}$ (\cite{Riesz43}). However, given that $R$ is a hyperbolic movement,
it has a unique representation as a (euclidean) translation determined by $P,Q$.
\subsection{Distance functions}
Concerning the invariance principle, assume $d_{\Gamma}=d_{x} \otimes d_{y}$, where $d_{\Gamma}$ denotes
the distance to $x=y$ in $\gamma_{1}$. We now have three cases $d_{x}/d_{y}=const$ ($d_{\Gamma} = const$)
$d_{x}<d_{y}$ and $d_{x}>d_{y}$. The corresponding distance function
in $\gamma_{2}$ can be induced according to
$(J^{-1})^{*}d_{\Gamma}$ ( pseudo distance.)
The normal is defined in $(I_{1})$ starting with the tangent.
In the case where $(I_{2})$, we have blow-up according to
$df \rightarrow f$, why we prefer to start with invariant points.
On first surfaces where all points are invariant, we consider transversals
on the form $V^{\bot} \gamma$.
Consider $(I_{1}) \subset (I_{2})^{\bot}$ and $d=(v_{1},v_{2})$, that is product topology.
We assume Runge's property for $d( \gamma_{1} - \gamma_{2}) \sim \inf (v_{1}(\gamma_{1}),v_{2}(e^{v} J \gamma_{1}))$
for $v \in L^{1}$. Assume for instance $e^{v} \gamma_{2}=z e^{v_{1}(z)} \sim e^{w(\frac{1}{z})}$
and we have existence of $\eta$, such that $w(\frac{1}{z}) \sim \frac{1}{\eta(z)}$, where
$\eta \in L^{1}$. When $f = e^{\phi}$, we obviously have $\big[ I, e^{v} \big](e^{\phi})=I(e^{\phi + v})$
$ = \big[ e^{v},I \big](e^{\phi})$.
Assume that distances are given by $(v_{1},v_{2})$, we then have that for invariant points that
$v_{2} \leq v_{1}$ for a hyperbolic movement, $v_{1} \leq v_{2}$ for an elliptic movement. Assume
$v_{1} \mid f \mid \leq 1/v_{2}$, where $v_{2}$ is algebraic and $v_{1}=const$, we then have that
$v_{2} < 1$ for $\widehat{f} \in C^{N}$ and $v_{2} > 1$ for $\widehat{f} \in \mathcal{D}_{L^{1}}'$
(finite order). As $v_{1} \neq const.$ we have $v_{2}^{2} \mid f \mid \leq v_{1}v_{2} \mid f \mid \leq C$,
when $v_{2} \leq v_{1}$ and $v_{1}^{2} \mid f \mid \leq v_{1} v_{2} \mid f \mid \leq C$ in the converse
case. When $\deg v_{2}^{2} = N$, we have $\widehat{f} \in C^{N}$. For instance, $v_{1}(x,y)=const$
we have $v_{x1}(z)=v_{x1}(\overline{z})$ and $v_{x2}(y-x)=v_{x2}(x-y)$, that is we assume $x$ fixed
and $v_{1}v_{2}=v_{2}v_{1}$.
We define $v$ through movements $J U \gamma_{1}=V \gamma_{2}$. Assume $(d_{1},d_{2})$
distances to the euclidean axes, that is $d_{1} < d_{2}$ on the reflection axes, implies $V$
translation, $d_{2} < d_{1}$ implies $V$ rotation and $d_{1}=d_{2}$ implies $V$ scaling.
Assume $E_{0}$ with product topology and $d \mu \bot E_{0}$. The Runge property implies that
$d \mu \bot C$. Thus, if we have separate invariance in $x$ and $y$, we have
invariance in $(x,y)$.
Concerning monotropy, we have a separate (pluri complex) condition that is represented by $\epsilon-$
translation. For instance if $v = \big[ v \otimes I + I \otimes v \big]$, it is sufficient that
$\big[ I,e^{v} \big](f)=\int \big[ I,e^{v} \big](x,y) f(y) d y=\int ( \int I(x,z) e^{v}(z,y) d z) f(y) d y=\int e^{v}(x,y) f(y) d y$.
Assume for $x$ fixed, $G(f)(x)=\int e^{v(x,y)} f(y) d y$ and for $y$ fixed,
${}^{t} G(f)(y)=\int e^{v(x,y)} f(x) d x$.
Thus $< G(\gamma_{1}), \phi>=< \gamma_{1}, {}^{t}G(\phi) >$. Given
$J U \gamma_{1}=V \gamma_{2}$
and $J \gamma_{1} = \gamma_{2}$, which implies $U \gamma_{1}=\gamma_{1}$ and this implies $V \gamma_{2}=\gamma_{2}$.
Given $\gamma_{1} \bot J^{-1} \gamma_{2}$ with $F(J \gamma_{1})=F(e^{v} \gamma_{2})$, thus if
$U \gamma_{1} \bot V \gamma_{2}$, we must have $v=const$ on all invariant sets.
If we consider $(\gamma_{1},\gamma_{2}) \in (I_{Hyp}) \times (I_{Phe})$,
we can consider an associated distance $d=d_{1} + i d_{2}$.
In the case with a regular complement and $\gamma_{1} \rightarrow \gamma_{2}$ with joint lineality,
if $d_{2,N}$ denotes the distance to the lineality for $\gamma_{2}^{N}$, we have $d_{2,N} \geq d_{1,N}$.
Thus in the case when $\Omega \backslash \Delta_{N} \downarrow \{ 0 \}$, we have that $d_{2,N} \uparrow$,
as $N \uparrow$.
Alternatively we can give
$(x , y^{N}) \in (I_{Hyp}) \times (I_{He})$. We assume $\frac{1}{d_{2}}$ a pseudo distance
and also a distance.
Given the condition $d_{1}/d_{2} \rightarrow 0$,
when $y \rightarrow 0$, we have that $\frac{d_{1}d_{2}}{d_{1}^{2} - d_{2}^{2}}=\frac{1}{\frac{d_{1}}{d_{2}} - \frac{d_{2}}{d_{1}}} \rightarrow 0$,
as $y \rightarrow 0$. Thus if $\tilde{d}^{2}=(d_{1} + i d_{2})^{2}$, we have that
$\frac{\mbox{ Im } \tilde{d}^{2}}{\mbox{ Re } \tilde{d}^{2}} \rightarrow 0$,
as $y \rightarrow 0$.
Assume $\pi$ a line (axes for reflection) $= \{ d_{1}(z)=0 \}$ and $d_{1}(t z)=t d_{1}(z)$,
$t>0$.
If $\pi$ is a plane, we can use a two-sided distance
function, for instance $\{ d_{1}=const \} \cup \{ d_{2}=const \}$, that corresponds to
$\overline{N(d_{1})} \cup \overline{N(d_{2})} \sim \overline{N(d_{1} d_{2})}$.
Concerning first surfaces to distance functions, given $d^{2}$ algebraic (limit of algebraic functions),
when $d^{2}$ is a distance, it is locally 1-1. Given Schwartz type topology, we can assume these first
surfaces have the same properties as the zero sets.
Assume $S_{1}=\{ \zeta \quad d_{1}=0 \}$ and $\Sigma_{1}=\{ \gamma_{1} \quad F(\gamma_{1})=const \}$,
then $\gamma_{1}$ is locally bounded, we have thus $d_{\Sigma} \leq c d_{\Gamma}(f)$, that is
$d_{\Gamma}=0$ implies $d_{\Sigma}=0$. Assume $d_{1,\Sigma} \rightarrow d_{2,\Sigma}$.
If we assume $J_{\delta}$ continuous, we have $d_{\Gamma}=0$ implies $d_{2,\Sigma}=0$.
conversely, if $J_{\delta}^{-1}$ is continuous, we do not necessarily have $d_{2,\Gamma}=0$.
\subsection{Singularities}
$\Delta$ is defined through translation in $\Omega$, which corresponds to $\Delta=\{ I \gamma=\gamma \}$.
We define a neighbourhood of $\Delta$ with respect to a lorentz movement $U \rightarrow I$
and $U \rightarrow \tilde{U} \zeta \in \Omega$. In the case where $U \gamma \rightarrow I \gamma_{0}$,
then we can regard a singularity as an isolated point on $\Delta$, given that $U$ preserves
analyticity (planar), the point is reached using $\Delta$ as a strict carrier for the limit,
that is it is of no consequence what neighbourhood we consider. We assume all through this article
that $f(\zeta)=F(\gamma)(\zeta)$ and $\log \mid f \mid \in L^{1}$, that is the singularities
are given by $\log \mid f \mid$ and are of finite order.
In the case with $U_{e}$ elliptic, we have that $\mid V_{e} f \mid=\mid f \mid$, that is
these changes of variables do not affect the singularities. In this case, the singularities on $\mid f \mid=1$
can be seen as isolated and are completely defined by transversals. In the case where $\mbox{ Re }f \bot \mbox{ Im }f$,
we can start with an arbitrary point on $\mid f \mid = 1$, and approximate the singularity
by elliptic rotation, that is $U_{e}$ such that $\mid V_{e} f \mid=1$ and $\tilde{V}_{e} \zeta = \zeta_{0}$.
Given $\mid f \mid^{2}$ analytic (does not imply $f$ analytic), we can assume $\Omega=\{ \zeta \quad \mid f \mid=1 \}$
connected ($\mbox{ Im }f \sim \mbox{ Re }f$).
Consider the movement as a functional and $V_{e} F(\gamma)=F({}^{t} V_{e} \gamma) \sim V_{e} f$.
Then $< V_{e} \gamma,\gamma>=1$ implies $< \gamma, {}^{t} V_{e} \gamma>=1$, that is we can
assume ${}^{t} V_{e}$ rotation (given a normal model).
Concerning parabolic movements $U_{p} f=F({}^{t} U_{p} \gamma)$ and $\frac{\widetilde{y}}{\widetilde{x}}=\frac{y}{x}=\rho$
and $y=y(x)$, ${}^{t}U_{p} y=\widetilde{y}$. We are considering $\Phi(\frac{y}{x})=(t x, t y)$ for $t$ real.
It is for this reason sufficient to consider singularities that are given by $\eta(x)=y(x)/x$,
where $\eta$ is algebraic iff $y$ algebraic. Note that $\{ \eta=\frac{d \eta}{d x}=0 \} \subset \{ y=x \frac{d \eta}{d x}=0 \}$
We have that $x \frac{d \eta}{d x}=0$ iff $\frac{d y}{d x} \sim -x/y$ (\cite{Dahn13})
Note if $(x,y) \rightarrow (x',y')$ with $\frac{dy}{dx}=\frac{dy'}{dx'}$ and $y'(x)=y( \alpha x)={}^{t}A y (x)$,
then if ${}^{t}A=A$, $ \frac{d {}^{t} A y}{d x}=A \frac{d y}{d x}$ implies assuming ${}^{t}A =A$,
$\frac{d {}^{t }A}{d x}=0$.
For translation $V_{h} f=F({}^{t} V_{h} \gamma)$, in the moment problem,
assume that $A$ defines regular approximations and $V_{h} \gamma$
defines a movement on the first surface to $f$. Then,
$\int A d (V_{h} \gamma)=\int A d \gamma$, that is $A$ is not dependent on $V_{h}$.
In this case the singularity is not
affected by $V_{h}$. If $F$ is linear and $F : const \rightarrow const,$, we can assume
$\{ F(V_{h} \gamma)=0 \} \sim \{ F(\gamma) - \lambda' \}$,
that is first surfaces to analytic functions, with regularity conditions as with (\cite{Nishino68}).
In particular, when $\lambda \rightarrow \lambda_{0}$, there are $V_{h}$ such that
$V_{h} \gamma \rightarrow \gamma_{0}$.
For $\tilde{V}_{h} \zeta$ we can compare with Abel's problem (\cite{Julia24}).
Given isolated singularities, we can assume $\tilde{V}_{h} \zeta=\zeta + \eta$,
for some $\eta$.
Concerning the two mirror model, we consider $A \rightarrow \gamma_{1} \rightarrow \gamma_{2} \rightarrow B$
where $A,B$ are situated on first surfaces to $f$. In the planar case, where $\mid \gamma_{1} \mid \leq 1$,
$\mid \gamma_{2} \mid \leq 1$ and $\gamma_{1} \bot \gamma_{2}$, then every $B$ can be reached,
independent of starting point $A$ on a first surface. Thus in the planar case,
every pair of points on the first surfaces, can be combined using a continuous path.
Consider $U \rightarrow V \in (I_{2})'$ and $(I_{1}) \subset (I_{2})^{\bot}$, that gives
a continuation $\phi$ of $\gamma_{1}$, that is $< \phi,\gamma_{2}>=0$. Further
$< V \phi, \gamma_{2} >=< \phi,{}^{t} V \gamma_{2}>$ defines
${}^{t} V \in (I_{2})'$ and ${}^{t} V \gamma_{2}=\gamma_{2}$ iff
$V \phi = \phi$
and the invariance principle is here considered in $\mathcal{D}_{L^{1}}'$. Assume
$J : U \rightarrow V$ and $F(J U \gamma)={}^{t} J F(\gamma)$.
A chain given by $U$ is mapped by $J$ onto a chain given by $V$.
Concerning algebraicity, assume $\mid f \mid < C \mid P \mid$ in $\infty$, where $P$ is polynomial,
then for $P$ to serve as a weight in $\mathcal{D}_{P}'$, it is necessary that $I \prec \prec P$.
We then have that $P_{1} \prec \prec P_{2}$ implies $\mathcal{D}_{P_{2}}' \subset \mathcal{D}_{P_{1}}'$
which corresponds to analytic functionals of finite type. In particular, for a reduced polynomial, we have
$\{ P < \lambda \} \subset \subset \Omega$ and we assume $(f/P)$ holomorphic outside a compact.
In the case where $W g=g / Q$, that is $QW g=g$,
where $Q$ is hypoelliptic, then $W$ corresponds to a
very regular distribution, that is $W \sim I$ modulo $C^{\infty}$.
\newtheorem*{res4}{Singularities}
\begin{res4}
We assume all singularities for our model are on first surfaces to the symbol $f$ and can
be reached through involutive movements, movements linked to movements orthogonal to involutive movements or algebraic
approximations.
\end{res4}
\section{Invariant sets}
Consider $F(\gamma_{1} \rightarrow \gamma_{2})$ with $\gamma_{1} \bot \gamma_{2}$. Assume
$\Phi : F(\gamma_{1})(\zeta) \rightarrow U_{1}$ and in the same manner
$\Phi : F(\gamma_{2})(\zeta) \rightarrow \zeta \in U_{2}$.
Assume $J \gamma_{1} = \gamma_{2}$. The problem is now under which conditions on $J$,
do we have that there existence $\Phi$, such that $F(J \gamma)(\zeta) \rightarrow \zeta$ continuous.
Sufficient for this is naturally that $F,J$ are analytic, why $\Phi$ is continuous. This is however not necessary.
Assume $\gamma_{1}$ has a desingularization $U_{1}=\cup_{1}^{N} S_{j}$, where $S_{j}$ are connected. We
can now define $J$ on the covering $J {}^{t} r_{j} \gamma_{1}={}^{t} r_{j} \gamma_{2}$ ($r_{j}$ is restriction).
Note that when $\gamma_{2} \mid_{S_{1}} = \gamma_{2} \mid_{S_{2}}$, we do not necessarily have
$\gamma_{2} \rightarrow \zeta$ uniquely. The condition ${}^{t} r_{j} J = J {}^{t} r_{j}$,
means that $J$ is not 1-1. For every $\gamma_{2}$ partially hypoelliptic, naturally there is a $\gamma_{1}$ hyperbolic,
such that $J \gamma_{1}=\gamma_{2}$
Monodromy $f(\zeta) \rightarrow \gamma$ is necessary in order to separate $\gamma_{1}$ from
$\gamma_{2}$. Assume $\gamma_{1} \rightarrow S_{j}$ and $\gamma_{2} \rightarrow \tilde{S}_{j}$ and
$J : \gamma_{1} \rightarrow \gamma_{2}$, if we have $id : S_{j} \rightarrow \tilde{S}_{j}$ continuous,
we can define $\gamma_{2}$ on $S_{j}$, that is on the covering to $\gamma_{1}$. More precisely,
if $\widetilde{\Omega}$ is a covering defined by $\gamma_{1}$, we consider this as a domain for $\gamma_{2}$,
that is $\{ \tilde{\gamma_{2}} \}=\cup (\tilde{S_{j}},\gamma_{2}(\tilde{S_{j}}))$ with analytic continuation and we write
$\gamma_{2}(\widetilde{\Omega})=\tilde{\gamma_{2}}(\Omega)$. Concerning localization, starting from
$F(\gamma) \rightarrow \gamma_{1} \rightarrow \gamma_{2} \rightarrow V$, if $V$ is accessible from
$\gamma_{2}$, it is not necessarily accessible from $\gamma_{1}$, further accessible from
$\gamma_{1} \rightarrow \gamma_{2}$ does not imply accessible from $\gamma_{2},\gamma_{1}$ or
$F(\gamma)$. Further, if $F(\gamma) \rightarrow \zeta$ is not continuous, there are possibly $\tilde{\gamma}$, such that
$F(\tilde{\gamma}) \rightarrow \zeta$ is continuous.
Assume $F=\widehat{E}f$, where $E$ is a very regular parametrix to $\gamma_{2}$ and $F_{1}(\gamma_{1})=f$, then we must have that
$F(\gamma_{2})=f$, modulo $-\infty$ action. Thus, if $f \rightarrow f_{0}$, we have $\widehat{E}(f) \rightarrow f_{0}$
modulo $-\infty$. $E$ can be chosen with one-sided support, assuming $\gamma_{2}$ algebraic. When $E$
is constructed using Fredholm operators, $\gamma_{2} \sim \widehat{E}^{-1}(\varphi)$, as $\varphi \rightarrow \delta$.
Note that $\{ \gamma_{2} \leq \lambda \} \subset \subset \Omega$, where $\lambda$ is a constant and $\Omega$ a domain
in $\mathbf{R}^{n}$.
Assume, for an analytic quotient, $(F_{2}/F_{1})(\gamma) \rightarrow 0$ over $\mid \gamma \mid=1$ with positive measure,
we then have $F_{2} \bot F_{1}$ on $\mid \gamma \mid > 1$. More precisely
$(F_{2} / F_{1})(r_{T}' \gamma)(\zeta)=(F_{2} / F_{1})(\gamma)(\zeta_{T})$ with $\mid r_{T}' \gamma \mid=1$.
Define $E = \{ \zeta_{T} \quad \mid \gamma \mid=1 \quad (F_{2} / F_{1}) \rightarrow 0 \}$.
Thus, if $(F_{2}/F_{1}) (\gamma) (\zeta) \rightarrow 0$ for large $\zeta_{T}$, that is $F_{1} \bot F_{2}$
with respect to $\zeta$ and $\mid \gamma \mid=1$, we then have $F_{1} \bot F_{2}$ with respect to $\gamma$.
Write $(F_{1} \bot F_{2})(\zeta)$, when the orthogonality is taken in $\zeta$, we then have
$(F_{1} \bot F_{2})(\zeta)$ implies $(F_{1} \bot F_{2})(\gamma)$. In the same manner, if $F$
linear in $\gamma$, we have $\Delta(F)(\zeta)$ defines $\Delta(F)(\gamma)$.
Assume $f(\zeta_{T})=F(r_{T}' \gamma)(\zeta)$, where $r_{T}$ is assumed closed and locally 1-1,
why $r_{T}'$ is locally surjektive. Define the continuation through $r_{T}'$ and
$< \gamma_{2}, r_{T}' \gamma_{1}>=0$ or $< \gamma_{2}(\zeta_{T}),\gamma_{1}>=0$, note that the proposition
that $r_{T}' \gamma_{1}=\gamma_{2}^{\bot}$ is equivalent with the proposition that $\gamma_{2}(\zeta_{T})$
is locally 1-1 at the boundary. Note that hyperbolicity assumes a Cartier boundary, while hypoellipticity
assumes a bijective ramifier.
We can define $r_{T}' (I) = (I)(\Omega_{T})$.
Concerning accessibility, if for instance $\gamma_{1}=r_{T}' \gamma_{2}$, with $r_{T}' \rightarrow r_{T}$
locally 1-1 and closed (continuous), we have that $r_{T}'$ is surjektive $\sim J^{-1}$, that is
$\forall \gamma_{1}$, we have existence of $\gamma_{2}$, such that $F(\gamma_{1} \rightarrow \gamma_{2})(\zeta)$
solves the lifting problem. We can assume that $J(\gamma_{1} \bot \gamma_{2})(\gamma)$ induces a continuous
mapping $\tilde{J}(\gamma_{1} \bot \gamma_{2})(\zeta)$, that involves a complementary set to $\Omega$.
Assume $\gamma_{2}=J \gamma_{1}$, where we assume $\tilde{J} : N(\gamma_{1}) \rightarrow N(\gamma_{2})$.
If we using Radon-Nikodym's theorem and let $F(e^{v} \gamma_{2})=F(J \gamma_{1})$, that is
$F \circ e^{v} \sim F \circ J$, where $v \in L^{1}$. Then the condition $log J \in L^{1}$ implies
algebraic singularities. $J$ is here defined as dependent on the movements, that is we write $v_{j}$ $j=1,2,3$.
Note that invariant sets are not necessarily preserved under iteration of symbols.
The sets $\Delta$ (lineality) and $\bot$ orthogonality can not be assumed independent for $(I_{1})^{\bot}$.
Consider for example $f=f_{1} + i f_{2}$ and
$\frac{f_{1} f_{2}}{f_{1}^{2} - f_{2}^{2}}=\frac{1}{\frac{f_{1}}{f_{2}} - \frac{f_{2}}{f_{1}}}=\frac{1}{v - \frac{1}{v}}$,
implies $\frac{f_{1}}{f_{2}} \rightarrow 0$, when the quotient above $\rightarrow -0$ and
$\frac{f_{2}}{f_{1}} \rightarrow 0 $, when the quotient above $\rightarrow +0$. Further let
$e^{\phi}=f_{1}/f_{2}$, we then have $\mbox{ sinh } \phi=\frac{1}{2} \big[ e^{\phi} - e^{-\phi} \big]$,
that has $\rightarrow \infty$ as $\phi \rightarrow \infty$ and $\rightarrow - \infty$ as
$\phi \rightarrow -\infty$. Further, $1/\mbox{ sinh }\phi = \mbox{ csch }\phi$ that has
$\rightarrow -0$, as $\phi \rightarrow -\infty$ and $\rightarrow +0$ as $\phi \rightarrow + \infty$.
For a hyperbolic symbol, we have that $f/Pr f$ is real (\cite{Garding87}). Let $f_{1}/f_{2} = p_{m}^{1}/p_{m}^{2}$, where $p_{m}$ are highest order terms.
We then have that $p_{m}(tx)=t^{m}p_{m}(x)$, why if we choose $x$ such that $p_{m}^{1}(x) \neq 0$
and $p_{m}^{2}(x) \neq 0$, then we must have $f_{1}/f_{2}(tx)=const$, as $t \rightarrow \infty$,
why it is characteristic for hyperbolic symbols, that the real and imaginary parts are not orthogonal.
Assume $f^{N}=e^{\phi_{N}}$, with $\frac{d \phi_{N}}{d x} \rightarrow 0$ in the $\infty$, that is $d \phi_{N} \bot d x$
in $\infty$.
The condition $\frac{1}{x} \phi_{N}(x) \rightarrow 0$ as $x \rightarrow 0$, implies that $\phi_{N} \rightarrow 0$
faster than $1/x$ goes to $\infty$, why the condition in $\infty$ implies a condition at the boundary. Assume $\phi_{N} + p_{N}=I$,
that is an algebraic complement. In this case we have $\phi_{N} \sim I - p_{N}$, that is invertible outside
constant surfaces to $p_{N}$, which gives possibility for two-sided limits. Constant surfaces for
$\phi_{N}$ are constant surfaces to polynomials. An algebraic transversal implies oriented first
surfaces (one-sided orthogonality). On the other side
if $d \phi_{N} + p_{N} d x=0$ then we have $\frac{d \phi_{N}}{d x}=-p_{N}(x)$ and when $p_{N}$ are
reduced, we can assume $p_{N}(\frac{1}{x}) \sim \frac{1}{q_{N}(x)}$, for polynomials $q_{N}$.
Note the example $F(x,y)=\phi(x)\psi(y)$, where $\phi=\phi_{1} + i \phi_{2}$ and $\psi=\psi_{1} + i \psi_{2}$
We then have $F_{1} \bot F_{2}$ if $\frac{\psi_{2}}{\psi_{1}} + \frac{\phi_{2}}{\phi_{1}} / 1 - \frac{\phi_{2} \psi_{2}}{\phi_{1} \psi_{1}}$
that is if we have separately $\bot$ we have $\bot$.
Given analyticity we have $\tilde{\gamma} \in (I)(\Omega)=I(\tilde{\Omega})=\tilde{I}(\Omega)$,
where we assume $\gamma \subset \tilde{\gamma}$ implies $\tilde{\Omega} \subset \Omega$.
Further, $F \in \mathcal{D}_{\tilde{I}}' \subset \mathcal{D}_{I}'$, thus given that the continuation is allowed,
we have that the restriction is well-defined. Assume invariance is defined by $\{ \gamma \quad F(\phi \gamma)=F(\gamma) \}$
for a transformation $\phi$. Define $\Omega_{(2)}=\{ \gamma \quad F(\phi \gamma^{2})=F(\gamma^{2}) \}$,
it is then significant if $\phi \gamma^{2}=\gamma^{2}$
and $V_{2}=\{ \gamma \quad F^{2}(\phi \gamma)=F^{2}(\gamma) \}$. Consider $\Omega'=\{ \gamma \quad d F(\phi \gamma)=d F(\gamma) \}$
that is $\{ {}^{t} \phi d F(\gamma) = d F(\gamma) \}$ and $\Omega''=\{ d F^{2}(\phi \gamma)=d F^{2}(\gamma) \}$.
Define $(J_{1})=\{ \gamma \quad {}^{t} \phi d F(\gamma)=d F(\gamma) \}$, for $F$ fixed.
It is significant if ${}^{t} \phi$ is linear, when we study $\Omega=\Omega_{1} \cap \Omega_{2}$.
$(J_{1}) \subset (J_{2})$ implies $\Omega_{2} \subset \Omega_{1}$ according to algebraic geometry
and $(J_{2})' \subset (J_{1})'$ according to functional analyse.
We can if $F^{2}$ is linear over $d \gamma$,
conclude $\phi d \gamma - d \gamma \in \mbox{ ker }F^{2}$. If $F(0)=0$, we have $\mbox{ ker }F \subset \mbox{ ker }F^{2}$.
Assume that $\Delta$ defines a geometric ideal $(I)$, that is $g \in (I)$ implies $\tau f - f = g$,
for some $f$.
Assume that $\Omega_{j}$ gives invariant sets, then we have that $\Omega_{1} \cap \Omega_{2} \rightarrow (I_{1}) + (I_{2})$.
Sufficient for a disjoint decomposition is that $(I_{j})$ are given by positive functions.
Consider $f \in (I_{1})$ and $f=f_{+} - f_{-}$. We have that $\{ 0 \} \subset \Omega_{1} \cap \Omega_{2}$
implies $(I_{1}) + (I_{2}) \subset (J)$, where $(J)$ is the ideal corresponding to a disjoint decomposition.
Assume $f_{2}/f_{1}=\varphi$ and $M(\phi) \sim \varphi$ (arithmetic mean), then the proposition that $\phi$ is harmonic corresponds to,
$\frac{d}{d z} \frac{f_{2}}{f_{1}}$ real and analytic. When $\frac{d}{d z} \frac{f_{2}}{f_{1}}$ real,
we have $\frac{f_{2}}{f_{1}} \frac{d}{d z} \log \frac{f_{1}}{f_{2}}=\frac{d}{d z} \log \log \frac{f_{1}}{f_{2}}$.
Thus, if $\log \varphi \in L^{1}$ and $\frac{d}{d z} \log \log \frac{f_{1}}{f_{2}}$
is real and analytic, then we have existence of $\phi$ harmonic, such that $M(\phi) \sim \varphi$
is constant. Define $F(e^{\varphi})=\widehat{F}(\varphi)$ and $(\mathcal{F} S)(\varphi)=S(e^{\varphi})= S \exp \varphi$
and $(\mathcal{F}^{-1}S)(\varphi)=S \log \varphi$. We then have
$\widehat{(\log \frac{f_{2}}{f_{1}})}=\mathcal{F} I \log \frac{f_{2}}{f_{1}}=\mathcal{F} \mathcal{F}^{-1} I \frac{f_{2}}{f_{1}} \sim \frac{f_{2}}{f_{1}}$
Thus the condition $\log \frac{f_{2}}{f_{1}} \in L^{1}$ using the Lelong transform,
can be interpreted so that $f_{1} \bot f_{2}$ (one-sided).
Note that in the model in this article, an algebraic continuation of an hyperbolic operator
is not hyperbolic (and vice versa). Note that if the ideals are given by distance functions $d_{1},d_{2}^{\bot}$,
such that $d_{2}^{\bot} / d_{1} \rightarrow 0$ in $\infty$ and $d_{2}^{\bot}/d_{1} \leq \mid \tilde{\gamma} \mid$,
where $\tilde{\gamma}=J \gamma$, which does not imply $d_{1} \leq \mid \gamma \mid$.
Existence of $N$, such that $P$ is hyperbolic in the direction $N$, implies that $P$ is not hypoelliptic.
Since the principal part $p_{m}$ to a
phe operator, is independent of some variables, we have that there exists $N$ where $P$ is hyperbolic
(in dependent variables). Thus the restriction of $(I_{Phe})$, can in this way be a subset of $(I_{Hyp})$.
More precisely, given $(I)=(I_{Hyp}) \oplus (I_{Phe})$, when $(I_{Hyp})$ is seen as an extension of
$(I_{Phe})$, there is a corresponding restriction to dependent variables for $p_{m}$, as the domain
for $(I_{Hyp})$. Consider $\frac{1}{t^{k}} f(t x) \rightarrow Pr f(x)$, as $t \rightarrow 0$,
where $k$ is the order of zero. Particularly $t^{-N} f(t \eta) = Pr f(0)$ $\forall N$ and
$t \eta \in \Delta$, that is an ``infinite zero''. For a reduced operator, we have thus
$t^{N}f(x/t) \nrightarrow Pr f(0)$, for $\forall N$ and $t \rightarrow \infty$.
Reduced operators are of real type (type 0) and modulo $C^{\infty}$ we can always assume real type.
Assume that $S_{1}^{\bot}=\{ $ normals to $S_{1} \}$ $=N(f_{1}^{\bot} + f_{2})$, where $S_{j}$ are first surfaces.
The condition $S_{2} \cap S_{1}^{\bot} \neq \emptyset$ implies $S_{2}^{\bot} \cap S_{1} \neq \emptyset$.
Further $(f_{2} + f_{1}^{\bot})^{\bot}=f_{2}^{\bot} + f_{1}^{\bot \bot}$. The mapping $S_{2}^{\bot} \cap S_{1} \rightarrow S_{2} \cap S_{1}^{\bot}$
is a mapping between layers.
Consider the first surfaces to the iterated symbols and $M$ a retraction neighbourhood of $S_{j}$.
If we assume that $S_{j}$ stratum, we assume an embedding $S_{j} \hookrightarrow M$,
that is given a stratification $\{ S_{j} \}$, we assume
$S_{1} \hookrightarrow M \rightarrow_{projection} S_{2}$, which gives a mapping between $S_{1}$ and $S_{2}$.
Assume $S_{1}^{2} \supset S_{1}$ and assume the same conditions for $S_{2}$, consider
$S_{1}^{2} \rightarrow S_{1} \hookrightarrow M \rightarrow S_{2}^{2} \rightarrow S_{2}$,
and assume the compatibility condition, $S_{2}^{2} \cap S_{2} = \emptyset$ or $S_{2}^{2} \supset S_{2}$.
The condition that $f_{2}/f_{1}$ is polynomial in the infinity, is necessary to come to the conclusion
that the intersection $\bot$ is discrete (for instance \cite{Julia23}). The condition is also necessary to conclude that
the condition $\bot$ is independent of $\zeta$ when $\mid \zeta \mid \geq R$, $R$ large. When $\Delta_{(2)} \subset \Delta_{(1)}$
(index refers to iteration) we have for the first surfaces that intersect $\Delta$, that
$\tilde{S}^{1} \subset \tilde{S}^{2}$.
In the two systems we assume the lineality is the same,
it is characteristic for $(I_{1})$ that $S_{1} \sim S^{2}_{1}$. According to the compatibility conditions,
assuming $S \rightarrow \tilde{S}$
we can assume $\tilde{S}^{1} \subset \tilde{S}^{2}$ or $\tilde{S}^{1} \cap \tilde{S}^{2} = \emptyset$
Assume $\gamma_{1} \rightarrow \gamma_{2}$ with the corresponding first surfaces $S_{j}$ and
$\tilde{S}_{j}$, where we assume $\gamma_{1} \rightarrow \gamma_{2}$ continuous and $\gamma_{j}$ analytic,
that is we do not necessarily have existence of $S_{j} \rightarrow \tilde{S}_{j}$ continuous.
For the problem of continuing a desingularization to $\gamma_{2}$, it is sufficient to assume $\gamma_{1} \rightarrow \gamma_{2}$
locally 1-1. Alternatively, if we define $S_{j}$ through a distance function $d_{1,j}$ and in the same
manner for $\tilde{S}_{j}$ and $d_{2,j}$, we can define $d_{1,j}d_{2,j}$ and a ramifier $r_{T}'$
from $S_{j}$ to $\tilde{S_{j}}$. Injectivity for $r_{T}'$
is regarded relative $d_{1}d_{2}=0$. We assume Schwartz type topology according to (\cite{Martineau})), inclusion
of ideals $I_{1,j} \subset I_{1,j+1}$ is defined through $d_{1,j+1}/d_{1,j} \rightarrow 0$.
This implies surjectivity for an approximating sequence.
Thus, the condition $\gamma_{1} \rightarrow \gamma_{2}$
continuous, linear and locally 1-1, is sufficient for existence of path $S_{j} \rightarrow \tilde{S}_{j}$.
\newtheorem*{res5}{Continuation by continuity}
\begin{res5}
A sufficient condition on $J : \gamma_{1} \rightarrow \gamma_{2}$ continuous, to induce a continuation
of the correspondent first surfaces, is that $J$ is continuous, linear and locally 1-1.
\end{res5}
Note that this property is not necessary for a global mixed model, that is $f(\zeta) \rightarrow \gamma_{1} \rightarrow \gamma_{2} \rightarrow V$,
for a given geometric set $V$
\section{The two mirror model}
We discuss the mixed problem in a two base (two mirror) model, that is we consider $f(\zeta)=F(\gamma)(\zeta)$,
where $\gamma \in (I) \bigoplus (I)^{\bot}$. Assume $J_{\delta} \gamma_{1} \rightarrow \gamma_{\delta} \in (I_{1})^{\bot}$,
for a parameter $\delta \rightarrow 0$,
then given $F(V \gamma_{\delta}) - F(V^{*} \gamma_{\delta}) \sim 0$, where $V$ is a given movement,
the limit $V \rightarrow I$ exists as two sided. Further, When $F \sim GH$ and $F(\gamma_{1},\gamma_{2}^{-1}) \sim
\int G(\gamma_{1},\mu) H(\mu,\gamma_{2}^{-1}) d \mu$, when $\mu \rightarrow \delta_{0}$, then the two
mirror model goes over into a one mirror model and we have $F \sim (G \otimes I)(I \otimes H)$. Assume
the boundary $\Gamma$ can be given by one single function $\eta=y(x)/x$ (order 0). Denote $\eta_{1}^{*} \sim V^{*} \eta_{1}$
reflection through $V=I$, then we have in the two mirror model, $(\eta_{1}^{*})^{*} \sim \eta_{2}^{-1}$.
Assume $< U \eta_{1},\eta_{2}>=< \eta_{1}, V \eta_{2}>=0$ and $\mu \eta_{1} = U^{*} \eta_{1} \sim V^{-1} \eta_{2}$
and we assume $\eta_{1}=\eta_{2}$, over $\mu$ and $\Delta$. Then $\mu \eta_{1} \rightarrow \eta_{1}$
and $\mu^{-1} \eta_{2} \rightarrow \eta_{2}$, that is ${}^{t} \mu = \mu$ is involutive. When
$\mu \rightarrow \delta_{0}$ we have $\eta_{1}(0)=\eta_{2}(0)$. Otherwise, we assume $\mbox{ supp }\mu=\Delta$.
When $F$ is symmetric over the path, we have that $V$ is involutive relative $F$.
In the system $(\eta_{1},\eta_{2})$, when we consider $\Delta \rightarrow \mu$,
then $ \eta_{1}=\eta_{2}$ can be regarded as an abstract light cone, that is $\eta_{1} \times \eta_{2} \sim \mu$.
Concerning the
boundary condition $F(\eta_{2}) \mid_{t=0}=F(\eta_{1})$ and $\frac{d F}{d t}(\eta_{2}) \mid_{t=0}= \frac{d F}{d t}(\eta_{1})$.
In particular, when $\frac{d F}{d t}= \frac{d F}{d x} \frac{d x}{d t} + \frac{d F}{d y} \frac{d y}{d t}= -Y_{2} X_{1} + Y_{1} X_{2}=0$,
we have $\frac{Y_{2}}{X_{2}}=\frac{Y_{1}}{X_{1}}$ according to the Lie's point transform.
For the lifting operator we use an involutive condition,
that is if $\sharp{F}$ is the continuation of $F$ according to $\gamma_{1} \rightarrow \gamma_{2}$,
we assume $\frac{d F}{d x} \frac{d \sharp{F}}{d y} - \frac{d F}{d y} \frac{d \sharp{F}}{d x}=0$
in a neighbourhood of $\Delta$, which implies Lie's condition as above.
The lineality for the composite kernel can be represented as $\big[ G,H \big] (\xi + i t \eta)=\big[ G,H \big]$
and $\int G(x,y)H(y,z + i t \eta) d y=\int G(x + i t \eta,y)H(y,z) d y$. Thus, if $\big[ G,H \big]=\big[ H,G \big]$,
then $\Delta(\big[ G,H \big])=\Delta(G) \cap \Delta({}^{t}H)$. If $G$ has $\Delta(G)=\{ 0 \}$, the same holds for
the composition, only assuming symmetry.
Further that if $G=G_{1}G_{2}$ with $G_{1}$ hypoelliptic and $N(G_{1}) \neq N(G_{2})$, we do not necessarily have that $G$ is hypoelliptic,
since $\frac{G'}{G}=\frac{G_{1} '}{G_{1}} + \frac{G_{2} '}{G_{2}}$, where only one term in
the right hand side is assumed $\rightarrow 0$.
\newtheorem*{res6}{Two mirror movements}
\begin{res6}
The two mirror model refers to an involutive movement, using two reflection points. A generalization
to the use of two bases $\gamma_{j}$, requires localizers $F_{j}$ with one-sided support $j=1,2$. A necessary
condition for the two mirror model, to give a normal model, is that one of the limits is independent of orientation.
\end{res6}
Given a reflection in the plane with reflection points on the axles, assume that
the distances from $B$ to $A$ is given by $d_{1},d_{2},d_{3}$. We then have
$d_{3} \rightarrow 0$, $z \rightarrow A$ and $\rightarrow \infty$
as $z \rightarrow A'$. Further $d_{2} \rightarrow 0$ as $z \rightarrow A'$, $\rightarrow \infty$
as $z \rightarrow B'$.
Finally $d_{1} \rightarrow \infty$ as $z \rightarrow B$ and $\rightarrow 0$ as $z \rightarrow B'$,
that is $\frac{1}{d_{1}} \rightarrow 0$
as $z \rightarrow B$ and $\rightarrow \infty$ as $z \rightarrow B'$.
Note that $\frac{1}{d}$ does not necessarily define a distance.
\subsection{Orientation of limits}
Given a simply connected domain $\Omega$ in the plane, then
we have that a simple Jordan curve $\Gamma$ divides $\Omega$ into two connected components.
Assume $\Gamma$ is a simple curve that is transported in a normal model along the transversal.
If $V_{1},V_{2}$ are two sets intersecting the transversal, considered as ``antipodal'',
we do not necessarily have existence of a unique plane between $V_{1},V_{2}$, that intersects
the transversal. However according to Nishino (\cite{Nishino68}), if $V_{1},V_{2}$ are first
surfaces with constant values $c_{1},c_{2}$ and if $c_{1} < c_{2}$, we can determine the intermediate set
as a first surface to a scalar $c$, $c_{1} < c < c_{2}$.
Assume $d_{j}$ distances to respective set and
consider $< d_{j}, \mid f \mid >$, then if we assume $d_{1} / d_{2} \rightarrow 0$
in $\infty$, we have a continuous injection between respective spaces, weighted with
the distances. The relation defines a topological inclusion between respective ideals.
More precisely, under the condition $d_{1}/d_{2} \rightarrow 0$ in $\infty$, we can form
$L_{d_{2}},L_{d_{1}}$,
that is $L_{d_{2}} \subset L_{d_{1}}$. Thus, $J \gamma_{1} \in L_{d_{2}}$ $\Rightarrow \gamma_{1} \in L_{d_{1}}$.
Further $\int d_{1} \mid J^{-1} f \mid \leq \int d_{2} \mid f \mid$.
Assume $\eta=y/x \in \dot{B}$, we then have existence of $F \in \mathcal{D}_{L^{1}}'$
such that $F \sim P(D) \tilde{f}_{0}$, where $\tilde{f}_{0}$ very regular. When we have
$\eta \in B$, we assume existence of $\eta_{1} \in \dot{B}$ (\cite{Dahn13}), such that $\eta \sim_{m} \eta_{1}$.
Note that when $\eta_{2}$ corresponds to $\gamma^{2}$, if we have $\eta \in \dot{B}$ or $1/\eta \in \dot{B}$,
we have $\eta_{2} \in \dot{B}$. Existence of limits in $\mathcal{D}_{L^{1}}'$ can be seen
as a topological type (A) condition (\cite{Nishino75}). Note that when we do not have existence of limits $F(\frac{1}{\eta_{2}})$
in $\mathcal{D}_{L^{1}}'$, we have existence of limits $F(\eta_{2}^{\bot})$ (annihilators).
Consider $F(\gamma_{1} \rightarrow \gamma_{2}) \rightarrow U_{2}$ and $F(\gamma_{1} \rightarrow \gamma_{3}) \rightarrow U_{3}$
given $U_{2}^{\bot} \supset \Delta (\gamma_{j})$, $j=1,2$. When $r_{T} \zeta$ is locally 1-1 and
closed, we have that $r_{T}' \gamma$ is surjective. Thus, for every $\gamma_{2}$ we have existence of
$\gamma_{1}$ such that $J \gamma_{1}=\gamma_{2}$, thus starting from $\Delta$ we can always find
a homogeneous symbol with $\Delta$. Further, when $F(\gamma_{1} \rightarrow \gamma_{2}) \rightarrow U$
and $F(\gamma_{1} \rightarrow \gamma_{3}) \rightarrow U$, then $\gamma_{2}=r_{T}' \gamma_{3}$,
that is we have existence of $\mu$, a continuous path between $\gamma_{2}$ and $\gamma_{3}$. Thus the fact that we have
existence of an approximation of $U$, does not exclude existence of a longer path with the same limit.
In applications, the paths may have different order of zero's, thus different quality properties.
Starting from the moment problem and $J \gamma_{1}=\gamma_{2}$, given $\gamma_{2}$ a polynomial, we
have that $\liminf \gamma_{2} \leq \gamma_{2} \leq \limsup \gamma_{2}$, for the restriction to lines. For instance when $f$ is
of type (A), as long as $F(\gamma_{1} \rightarrow \gamma_{2})$ preserves this inequality and with finite limits,
we can solve the moment problem. When the limits coincide, the solution is unique.
\newtheorem*{res7}{Twosided limits}
\begin{res7}
Given $\gamma_{1}$ from $A$ to $A'$ (reflection point), $\gamma_{2}$ from $B$ to $B'$ and $A' \sim B'$.
We refer to this as a two-sided limit. When $\gamma_{2}$ also gives a path from $B'$ to $B$, the composition
gives a path from $A$ to $B$.
\end{res7}
We can divide two mirror model into $RF(z)=F(\overline{z})$ and $SF(z)=F(i z)$, that is we assume the reflection
points on the real axes or the diagonal.
When we do not have symmetry according to $F(\overline{z})=\overline{F}(z)$,
we must consider $f(z,\overline{z})$. In the same
manner if $F(i \gamma)=i F(\gamma)$ or rather $(F + i F^{*})^{*}=-i (F + i F^{*})$ (pure), we can
refer to this as a transmission property, otherwise we must consider $F(z,iz)$ (or $F(w,w^{*})$).
Assume in the two-mirror model, that $L$ is the segment between $\gamma_{1}$ and $\gamma_{2}$ and
consider $F(\gamma_{1} \rightarrow \gamma_{2})=\int G(\gamma_{1},L) H(L,\gamma_{2}) d L =\big[ G,H \big](\gamma_{1},\gamma_{2})$.
Given $F$ invariant for change of orientation, this corresponds to one-sidedness (with respect to two
mirrors). If $L \bot \gamma_{1}$ and $L \bot \gamma_{2}$, that is defined for instance by the distance
functions, we then have existence of $L \sim \gamma_{1} \times \gamma_{2}$,
where $L$ is assumed to have points in common with $\gamma_{1},\gamma_{2}$.
\subsection{Wellposedness}
Every movement in $(I_{Phe})^{\bot}$ can be represented as reflection (\cite{Riesz43}). This means that we have that
$\gamma_{1} \in (I_{Phe})^{\bot}$ and $F(\gamma_{1})=const$, can be reached through $U \gamma$ with
$\gamma \in (I_{Phe})^{\bot}$ and given by reflection relative an axes $R$ (not unique), that is $U=U_{R}$.
Assume compatibility conditions, $F(U^{\bot} \gamma) \neq const$, for the
approximation and assume that there are points in common, but $U^{\bot} \gamma$ is not necessarily in $(I_{Phe})^{\bot}$. Given that $\gamma \in \Sigma(S)$, a first surface,
we have that the reflection axes is a part of $\Sigma(S)$. Thus, if $R^{\bot}$ is the
reflection axes to $U^{\bot}$ as (an euclidean) $\bot$ to $R$, it can be used as a regular approximation.
F. and M. Riesz theorem: assume $f(\frac{1}{z})$ analytic and bounded with
$\mid f(\frac{1}{z}) \mid <M$ for $\mid \frac{1}{z} \mid < 1$.
Let $E=\{ f(e^{i \theta})=\lim_{r \rightarrow 1} f(r e^{i \theta})=0 \}$.
Assume the measure for $E$ is $>0$, on $\mid z \mid=1$, we then have
$f \equiv 0$ for $ \mid \frac{1}{z} \mid < 1$ (\cite{Riesz56}).
Assume $\Sigma= \{ \gamma \quad F(\gamma)=const \}$ equipped with a norm $\Sigma_{\rho}=\{ \gamma \quad \rho(\gamma)=1 \}$.
Application on my model, gives that if the segment $\mu$ exists
between $\gamma_{1}$ and $\gamma_{2}$ in the boundary $\Gamma$, forming a set of positive measure,
and if two solutions $F_{1} \equiv F_{2}$ on $\Sigma_{\rho}$, we have that $F_{1} \equiv F_{2}$ on $B_{\rho}=\{ \gamma \quad \rho(\gamma) \leq 1 \}$.
In this application, the boundary is continued with a transversal between the first surfaces. The continuation
principle (\cite{Martineau}) gives a representation of $F$ as a distribution of the continuation.
Assume $J_{1}=J_{x} \otimes J_{y}$ and assume $< (J - J_{1}) \gamma, T \gamma >=0$, $\forall \gamma$, implies $T$ is the id-mapping,
we then have that $J=J_{1}$. Note for $\gamma$ analytic, we have $\gamma=0$ iff $\gamma \mid_{L} = 0$
for every line $L$, that is we can consider a pluri complex formulation.
Note in this case if $\{ J \gamma_{1} < \lambda \}$ is semi algebraic locally,
we have that for instance $\{ J_{x} \gamma_{1} < \lambda \}$ is semi algebraic.
\subsection{Localisation}
Assume $\eta(x)=y(x)/x$ exact, then the ideal given by $\mbox{ ker }y$, can be represented with a
global pseudo base. The same proposition holds if $y$ is reduced.
An ideal that is defined by $\tau_{z} \phi/\phi \sim_{m} 0$
with $\tau_{z}$ compact, can be given a global pseudo base and when $\eta>0$ over an ideal, then
$\eta$ is exact and the ideal has a global pseudo base.
\newtheorem*{res8}{Localisation problem}
\begin{res8}
Given $F(\gamma_{0})(\Omega)$, determine $\gamma_{2}$ analytic
on $V$, such that $F(\gamma_{0}) \rightarrow \gamma_{0} \rightarrow \gamma_{2}$ continuous, further $\gamma_{2} \rightarrow V$
continuous and $\gamma_{2}^{N} \in (I_{He})$, that is representation using reduced measures.
\end{res8}
Assume $\{ \Omega_{j} \}$ a covering to $\gamma_{0}$ and $\Omega$. Given a point $\zeta_{0} \in V$
and $V_{k}$ a component, if $\gamma_{0} \in (I_{He})^{\bot}$, then we can give $\gamma_{2}$ as a
continuation on components $\Omega_{j}$ to $V_{k}$ and $< \gamma_{0},\gamma_{2}^{N} >=0$. When we have
$d \mu_{2} \in \mathcal{E}^{'0}$, $\gamma_{0}$ can be extended with zero over $V_{k}$.
Assume $F \gamma_{2}=1$ (invertible), where $\gamma_{2}$ is a hypoelliptic symbol,
and $F$ has representation with trivial kernel.
Assume $\mbox{ ker }F e^{v}=\{ 0 \}$, where $F(e^{v} \phi)=0$
implies $\phi=0$ or $e^{v} \phi \in \mbox{ ker }F$ for $v \in L^{1}$. We assume
$F(J \gamma_{1})=F(e^{v_{1}} \gamma_{2})=F(e^{v_{1} + v_{2}} \gamma_{2}^{2})$.
Thus, if $\gamma_{2}^{2}$ is hypoelliptic, we have $e^{v_{1} + v_{2}} \rightarrow 0$ in $\infty$,
that is $Fe^{v_{1} + v_{2}} \rightarrow 0$ and $F \nrightarrow 0$ in $\infty$. Then
$\log \mid F \mid + v \leq \log \mid F \mid$, we have that the type for $v \leq 0$, that is of type $-\infty$.
Note that if $JU=VJ$, then $U,V$ do not behave algebraically similar, that is $U \gamma_{1}^{2} = \gamma_{1}^{2}$
iff $U \gamma_{1}=\gamma_{1}$, but when $J \gamma_{1}=\gamma_{2}$, we do not have $J \gamma_{1}^{2}=(J \gamma_{1})^{2}$,
that is $J$ is not algebraic.
Note that if $f(x)=\frac{1}{x}g(\frac{1}{x})$ and $f(\frac{1}{x})=\frac{1}{h(x)}$, we have that
$x=\frac{f(\frac{1}{x})}{g(x)}$, that is $f(\frac{1}{x})h(x)=1$ and $f(\frac{1}{x}) \frac{1}{g(x)}=x$.
\subsection{Algebraic approximations}
The mapping $F(\gamma) \rightarrow \gamma \rightarrow \gamma_{2} \rightarrow \zeta$, does not necessarily preserve the
order of $0$ for the localisation. For the localisation, existence of order $0$ regular approximations is
sufficient. Note that concerning $\tau \rightarrow V_{e}$ using a contact transform, we have that the order of $0$
is preserved, but not the shape of obstacles. We will assume the boundary locally of order $0$
for some movement, that is it can be defined by a orthogonal movement locally. Assume $F(U^{\bot} \gamma) \rightarrow c$
regular and $F(U \gamma) \equiv c$,
given that $\gamma$ is polynomial, we have through the lifting principle, existence of $({}^{t} U^{\bot} F)$ with an analytic representation.
It is sufficient, that ${}^{t} U^{\bot} F \in \mathcal{D}_{L^{1}}'$ with $({}^{t} U^{\bot} F)(\gamma)(\zeta)$
analytic, that is $F \sim F(\zeta,\vartheta)$ kernel in $\mathcal{D}_{L^{1}}'$.
Assume $\{ \zeta \quad F(\gamma_{1})(\zeta)=c_{1} \}=S_{1}$
and $(I_{c_{1}})=\{ \gamma_{1} \quad F(\gamma_{1})=c \}$. Assume $\frac{d F}{d x_{1}}=-Y_{1}=0$
in $\zeta$ with $S_{1} \cap \{ Y_{1}=0 \}$ algebraic and the same condition for $X_{1}$. Assume $S_{1} \rightarrow \tilde{S}_{1}$ a continuation
by continuity. Assume $J : \gamma_{1} \rightarrow \gamma_{2}=\gamma_{1}^{\bot}$, where $\gamma_{1}^{\bot}$
is closed, given that we have existence of $F^{-1}$ continuous, we have $F \rightarrow \gamma_{1} \rightarrow \gamma_{1}^{\bot}$.
As $\{ F-c \} \rightarrow 0$, we have $\gamma_{1}^{\bot} \rightarrow {}^{o}(I_{c})$ (annihilator). When $\gamma_{1}^{\bot}$
partially hypoelliptic, we have $(\gamma_{1}^{\bot})^{N}$ locally 1-1 (downward bounded) which implies
$\zeta \rightarrow \zeta_{0}$, that is the limit exists.
For instance assume $\gamma_{1} \in (I_{c})(S_{1})$ and $\gamma_{2}$ such that $\mbox{ supp } \gamma_{2} \cap S_{1} = \emptyset$
and $F(\gamma_{2})=const$. If $(I_{1}) \subset (I_{He})^{\bot}$
where $(I_{1})=\{ d_{1}=0 \}$.
Note that $d_{2}^{\bot} / d_{1} \rightarrow 0$ defines a continuation of $\gamma_{1}$, that is
$\tilde{(I_{c}})=\{ \gamma \quad F(\gamma)=c \quad \gamma \bot \gamma_{2} \}$ and the ``moment problem''
gives that $(I_{1})$ has a continuation to $(I_{1})^{\bot}$, assuming
algebraic singularities.
Assume $J_{\delta} : \gamma_{1} \rightarrow \gamma_{2}$, for a parameter $\delta \rightarrow 0$,
such that $({}^{t} J_{\delta} F)(\gamma_{1})=c$ over an involutive set, that is $J_{\delta} \gamma_{1} \in F^{-1}(c) \sim (I_{c})$.
Given type (A), it is sufficient to assume $J_{\delta} \gamma_{1}$ a polynomial. If $\gamma_{1}^{\bot} \in \mathcal{D}_{L^{1}}'$
which implies $\gamma_{1}^{\bot} \sim P \mu$ and $P \mu (\gamma_{1})=\mu ({}^{t} P \gamma_{1}) \sim$
algebraic continuation close to the boundary.
Assume the continuation is given by a movement,
such that $(U^{\bot} \gamma)$ and $(U \gamma)^{\bot}$ have points in common.
For instance $U^{\bot} \gamma \rightarrow \{ F(\gamma)=c \}=S_{1}$ regularly and $F(U \gamma)=c$
and $\tilde{U} \zeta \subset S_{1}$. Let $V^{\bot} \gamma=J_{\delta} U^{\bot} \gamma_{1}$, where we assume
$\{ V^{\bot} \gamma \} \cap \{ F(\gamma_{2})=c \}=\tilde{S}_{1}$ algebraic, this can be seen as a
compatibility condition (when $v \in L^{1}$).
Consider for example $F$ as an analytic function of $\gamma$, where the boundary $\Gamma$ is defined as $\bot$
movements, that is if $V \gamma$ is translation, and $V=JU$, we can assume
$U^{\bot} \gamma$ are parallel planar movements. Given a normal model, if the measure for
$V \gamma$ is positive and $F(V \gamma)=f(\zeta)$, we have wellposedness according to a previous argument.
Note $F(V^{\bot} \gamma)= const$, that is $U \gamma$ denotes a $\sim$ planar movement.
If $R_{1}$ is the reflection axes and $R_{2}$ the corresponding axes in euclidean geometry.
Assume $\tilde{R_{2}}$ the corresponding points to $\tilde{V}$, that is $V \gamma = \gamma$ implies
$\zeta \in \tilde{R_{2}}$, then if $V-I$ has an analytic representation, this is a line.
\section{Discussion on change of base}
Assume $f \in (I)$ and $g \in (I^{\bot})$, we define $H(g)=H(f)=0$ ($H=\tau-1$) on $\Delta$, that is
$H(f)=A_{f}(H)$ and $H(g)=B_{g}(H)$ which implies $A_{f}(H)=B_{g}({}^{t} e^{\varphi} H)$ as
``inverse functionals''. Thus $H(g)=H(e^{\varphi} f)$, where $\varphi \in L^{1}$ in our model, which
corresponds to equivalent zero's. Consider $F(f+g) \sim F_{2}((f - e^{\varphi} g) + i (g + i e^{\varphi} f))$
assuming $F_{1} \bot F_{2}$. Assume $R(f)=e^{\varphi} g$ and $R^{*}(g)=-e^{\varphi} f$ where $J$ is reflection through $\Delta$
and $F(J f)=F(e^{\varphi} g)$, that is $F \sim F_{2}((1-R)(f) + i (1-R^{*}(g)))$.
Starting with $V \subset \Omega$, and $\Omega \backslash V$ analytic, we can form $(I)(V) \otimes (I)(\Omega \backslash V)$
Denote by $(I^{\bot})=\{ g \quad fg = 0 \}$, an annihilator ideal. The corresponding geometric set
is $V \cup \Omega \backslash V$.
If we assume $f,g$ positive, we have
$N((1-f)(1-g))=N(1 - fg)$, that is $(fg)^{\bot}=f^{\bot} g^{\bot}$.
Assume instead $(I) \oplus (I^{\bot})$ with base elements $f,g$ and $F_{1}={}^{t}e^{\varphi}F_{2}$, where ${}^{t} \varphi \rightarrow \infty$.
We then have $F(\alpha f + \beta g)=F_{2}((1 + i e^{\varphi})(\alpha f + \beta g))=F_{2}(\alpha f - e^{\varphi} \beta g + i (\beta g + e^{\varphi} \alpha f))$
and $= F_{2}( \alpha f - e^{\varphi} \beta g) + i F_{2}(\beta g + e^{\varphi} \alpha f)$.
Further, if $\alpha=\beta=1$, $\frac{g + e^{\varphi} f}{f - e^{\varphi} g}=
\frac{\frac{g}{f} + e^{\varphi}}{1 - e^{\varphi} \frac{g}{f}}$, why if $\frac{g}{f} \rightarrow 0$ in $\infty$
we have that the quotient above $\rightarrow \infty$. In the same manner, if $f/g \rightarrow 0$ in $\infty$
faster than $e^{\varphi} \rightarrow \infty$, the quotient $\rightarrow -0$ in $\infty$.
Thus, we have that $F_{1} \bot F_{2}$ one-sided, given $f \bot g$ one-sided.
Assume $(I)^{\bot}=\{ g \quad <f,g>=0 \}$ on a set of positive measure, then we have that $\mbox{ supp }g=W$
is the set where $f$ is not a polynomial. That is, $\int_{W} fg d x=0$ implies $fg \equiv 0$ on $W$ or that the measure for $W$
is zero and if $W$ contains a connected set where $fg \equiv 0$, we have that this set has measure zero.
If $fg \equiv 0$ on a line $L$ $\subset \Delta$ , this implies $g$ a zero divisor. Assume $\tau'=\tau-1$
algebraic and $\tau'(fg)=\tau' f \tau' g$ with $\tau f = f$ and $\tau' (fg) \equiv 0$, independent of
$g$. Assume now $< \tau' f,g > \equiv 0$ and $< f,\tau' g> \neq 0$. Define
$\Sigma=\{ < \tau' f,g >=< f,\tau' g> \}$.
If $g$ is reduced, we thus have $\Sigma = \{ 0 \}$.
Assume $(J_{f})=\{ g \quad <\tau' f,g >=< f,\tau' g> \quad \mbox{ on } L \}$ where
$L$ has positive measure. We then have $\tau' - {}^{t} \tau'$ is not algebraic!
We have that $L \subset N(J_{f})$ and $I N (J_{f}) \sim rad (J_{f})$.
Assume $\gamma_{1} \rightarrow \gamma_{2}$ defines a broken ray according to $\gamma_{2}^{* -1} \simeq \gamma_{1}^{*}$
thus $\gamma_{1},\gamma_{2}$ must have a point in common. It is sufficient for this to assume $\gamma_{2}$
has two sided limits.
Assume $(\gamma_{2}^{-1})^{*} \simeq (\gamma_{2}^{*})^{-1}$ which means $\gamma_{1}^{*} . \gamma_{2}^{*} \sim 1$
(Legendre). We denote this $\gamma_{1}^{*} \mathcal{L} \gamma_{2}^{*}$, $\gamma_{1} \bot \gamma_{2}$,
$\gamma_{1} \mathcal{L} \gamma_{1}^{*}$, $\gamma_{2} \mathcal{L} \gamma_{2}^{*}$, $\gamma_{1}^{*} \mathcal{L} \gamma_{2}^{*}$,
that is we can define $\bot \rightarrow \mathcal{L}$ continuous. Note that $x . x^{*} + y . y^{*} \sim 1$
assumes $x \bot y$, that is when $\gamma_{1}^{*} \mathcal{L} \gamma_{2}^{*}$, then we must have $\gamma_{1} \bot \gamma_{2}$. When reflexivity is given by $\gamma_{1}^{**}=(x_{1}^{**},y_{1}^{**})$
we have that $x_{1}^{**} \bot y_{1}^{**}$.
Consider now, $\{ \zeta \quad d_{\Gamma}(Y/X)=const \}$ that describes parabolic movements.
When $d_{\Gamma} \sim \mid \cdot \mid$
we have that $d_{\Gamma}(Y/X)=const$ implies $Y/X \nrightarrow 0$, that is $Y/X =const$ is complementary to
$ Y/X \rightarrow 0$ and conversely if $Y/X \rightarrow 0$ we have that $d_{\Gamma}(Y/X) \neq const.$
Assume $\{ \gamma_{2} < \lambda \}=M$ with $\gamma_{2}^{\bot}(\gamma_{2})=0$ implies $\gamma_{2} \in M$
where $\gamma_{2} \in (I_{2})$ on a set $V$ and we assume $(I_{2})$ closed with $(I_{2})^{\bot \bot} \simeq (I_{2})$.
If we define $(I_{1}) \subset (I_{2})^{\bot}$ through $d_{1},d_{2}^{\bot}$, we have that $(I_{1})$
can be considered as closed in $(I_{2})^{\bot}$ for instance $d_{2}^{\bot}/d_{1} \rightarrow 0$ with
$d_{2}^{\bot}(z) \sim d_{2}(\frac{1}{z}) \rightarrow 0$, when $z \rightarrow \infty$.
Note that if we assume $d_{2}(\gamma) \rightarrow 0$ implies $\gamma \rightarrow \gamma_{0}$ where
$\gamma$ reduced, we must have $d_{2}(\zeta) \rightarrow 0$ implies $\zeta \rightarrow \zeta_{0}$.
Consider the problem $L(U \gamma,U \gamma)=L(\gamma,\gamma)$ and $L(U \gamma,U^{\bot} \gamma)=0$. An
axes $L(U \gamma,\gamma)=L(\gamma,\gamma)$ corresponds to an axes for $U^{\bot}$ and $L(U^{\bot} \gamma, \gamma)=L(\gamma,\gamma)$.
If the condition $L(U \gamma_{1}, \xi)=0$ implies $\xi=0$, we must have $U=id$, that is $\gamma_{1}$ on the reflection axes.
We assume $\xi=0$ implies $J \xi=0$ since $J$ is assumed to preserve orthogonality, that is
$L=0 \rightarrow_{J} E=0$. This condition is also assumed for the inverse $J^{-1}$. Thus, if
$E(V \gamma_{2} , J \xi)=0$ implies $J \xi =0$, we must have $V=id$, that is a reflection axes in $L$,
has a corresponding axes for invariant points in euclidean metrics. In particular
$H_{1}(\gamma_{2})=< \gamma_{2}, J^{-1} \gamma_{2}>$
where $H_{1}(V \gamma_{2})=< V \gamma_{2}, \gamma_{1}>=< J U \gamma_{1}, \gamma_{1}>=H(U \gamma_{1})$.
$H_{1}(J \gamma_{1})=0$ implies $H(\gamma_{1})=0$ and we can use that $H_{1}(J \gamma_{1})=H(e^{v} \gamma_{1})$
that is $H : (\gamma_{1},L) \rightarrow (\gamma_{2},E)$ (positive) linear functionals.
Given a reflection axes $R$ to $U$, there is a reflection axes $R^{\bot}$ to
$U^{\bot}$, such that $R \cap R^{\bot} \neq \emptyset$ (note that $R^{\bot}$ is not unique).
Particularly, if $\gamma(\zeta) \in \Sigma$, where $\Sigma=\{ F(\gamma)(\zeta)=const \}$,
we assume $R^{\bot} \bot R$ with points in common. We have that $R^{\bot}=R^{\bot}(R)$, $R^{\bot} \cap \Sigma = \{ (x_{0},y_{0}) \}$,
for some $R^{\bot}$. When $F$ is constant on $R$, we have $F \neq const$ on $R^{\bot}$.
Consider $V,V^{\bot} \in H'$ and use the topological isomorphism $H'(E) \rightarrow Exp(E^{*})$,
where $Exp$ are entire functions or regular with slow growth (\cite{Martineau}). Assume compatibility conditions,
such that $V^{\bot} \gamma_{2}$ regular for $F$ or $F(V^{\bot} \gamma_{2}) = const.$. Then since
$\widehat{V} \cdot 1 = 1 \cdot \widehat{V}$, we have $\big[ V^{\bot},I \big] = \big[ I , V^{\bot} \big]$,
over regular approximations. Further, we assume $JI=IJ$ on the set corresponding to $\Delta$. If
$R^{\bot}$ is an axis for invariance for $V^{\bot}$, we have an axis $\widehat{R}$ to $\widehat{V}$
and $R^{\bot} \simeq \widehat{R}$, further a line $\tilde{R^{\bot}}$ corresponding to regular approximations.
Assume for parabolic rotation $\tau_{p}$ that $F(\tau_{p} \gamma)=const$, then over $S=\{ F(\gamma)=const \}$,
$\frac{d}{d \gamma} {}^{t} \tau_{p} \equiv 0$. Note that the condition
$\frac{\phi_{2}'}{\phi_{1}'}= c \frac{\phi_{2}}{\phi_{1}}$
corresponding to elliptic rotation, that $\frac{\phi_{1}^{2}}{a^{1}} + \frac{\phi_{2}^{2}}{b^{2}}=1$,
for constants $a,b$. Define $F(x,y)=F_{1}(\frac{y}{x})$ and $\tau_{e}$ elliptic rotation,
we then have ${}^{t} \tau_{e} F(x,y)={}^{t} \tau_{p} F_{1}(\frac{y}{x})$
and $F_{1}(\frac{y}{x})=F(x (1,\frac{y}{x}))$ or ${}^{t} \tau_{e} F = {}^{t} \tau_{p} F_{1}$.
In particular we consider the domain $D(F)$, that is we can define $D(F)=xD(F_{1})$
$\frac{1}{z} F(\frac{1}{z})=f(z)$ for $0 < \mid z \mid < R$.
Consider $G(z,z^{\bot})=(G(z),G^{\bot}(z))$, where $G(z),G^{\bot}(z)$ are real. Using Tarski-Seidenberg's theorem, if
$\{ (z,z^{\bot}) \quad G < \lambda \}$ is semi algebraic, we have the same for
$\{ z \quad G < \lambda \}$ and in the same manner for $z^{\bot}$.
Assume $G G^{\bot}=G^{\bot}G$, then $G^{2} \sim G^{2} - G^{\bot 2} + 2 i G G^{\bot}$. For the real part,
assume $\{ \frac{G}{G^{\bot}} < \frac{G^{\bot}}{G} \}$ bounded, this corresponds to $G^{\bot 2} \prec \prec G^{2}$
in $\infty$. When the set is unbounded, we have that $G^{\bot} < \lambda$ implies $G < \lambda$ in
$\infty$. If $\lim_{z \rightarrow z_{0}} G=\lim_{z^{\bot} \rightarrow z_{0}} G$,
we have that $z_{0}$ is an isolated point. Note that it remains, given $U \gamma$ a movement,
to define $U \rightarrow \tilde{U}$ and $\gamma(\tilde{U} \zeta)=U \gamma(\zeta)$.
Consider $G(z) \rightarrow G(z^{\bot})=G^{\bot}(z)$ as a projection operator and assume
$\bot$, dependent of orientation, that is one-sided, then we have that $\{ G < \lambda \}$ semi algebraic
$\Rightarrow \{ G^{\bot} < \lambda \}$ semi algebraic, can be seen as a transmission property.
Assume $V J = J U$ and $J : p \rightarrow q$. Consider $U \rightarrow V \in (I_{2})'$,
that is $V$ is a functional over $(I_{2})$. In particular when $( V - I) \gamma=0$
and $U \gamma_{1}=\gamma_{1}$ implies $V \gamma_{2}=\gamma_{2}$. Given $(I_{2})$ is closed
we have that $(I_{2})=({}^{o} I_{2})^{o}$. When $(I_{2}) \subset (I_{He})^{\bot}$ with Runge's property,
we can regard $(I_{2})$ as closed in $(I_{He})^{\bot}$ (with respect to $\dot{B}$).
The moment problem, for $(I_{2})=E_{0}$ provides a continuation to $C$.
For instance, $\gamma^{\bot}=e^{v} \gamma$ with $e^{v} \rightarrow 0$ (or $\infty$) and
$F(\gamma,\gamma^{\bot}) \rightarrow 0$
where $\gamma^{\bot}/\gamma \rightarrow 0$ in $\dot{B}$, thus we have existence $F_{1}$,
such that $F(\gamma,\gamma^{\bot})=F_{1}(e^{v})$.
\section{Intermediate ideals}
Define $V_{1}=\{ x \in \Omega \quad F(\phi x)=F(x) \}$, for a fixed $F$, that is $\phi x - x \in \mbox{ ker }F$,
an ideal of holomorphy. We assume here $F$ linear in $x$, but it is not assumed linear in $\Omega$.
Assume $V_{1}$ is given by $\{ f-c_{1} \}$, where $f$ is analytic and $c_{1}$ scalars and in
the same manner for $V_{2}$. Using the theorem on intermediate values, we can assume existence
of $V$ between $V_{1}$ and $V_{2}$.
Assume $(J)$ an ideal such that $(J)=\mbox{ ker } \phi_{1}$, for instance $\phi_{1}=\phi-1$ and $(J) \oplus (J)^{\bot}=(E)$, with $\{ 0 \} = N(J) \cap N(J^{\bot})$,
that is $\phi_{1} + \phi_{1}^{\bot}=id$ (local identity).
Given that $S_{1} \rightarrow S_{2}$ algebraic, we have type (A) first surfaces, that is we have a
lifting principle over an algebraic polyhedra (\cite{Nishino75}).
Concerning existence of $(I_{0})$ such that $(I_{1}) \subset (I_{0}) \subset (I_{2})^{\bot}$.
Given that the inclusion is continuous (closed) and injective, the existence can be derived
using the theorem of intermediate values.
A sufficient condition (and necessary) for inclusion between weighted space in $\mathcal{D}_{L^{1}}'$
is that quotient of the corresponding weights goes to $0$ in the $\infty$. If in this setting
$(I_{1})$ has the weight $\rho_{1}$,
$(I_{0})$ has the weight $\rho_{0}$ and $(I_{2})^{\bot}$ has the weight $\rho_{2}$.
Then the condition for the inclusion we are looking for is
$\rho_{2}/\rho_{0} \rightarrow 0$,$\rho_{0}/\rho_{1} \rightarrow 0$ and $\rho_{2}/\rho_{1} \rightarrow 0$.
For the hyperboloid we have given a desingularization $\mu$, that given $F$ meromorphic
then $F \circ \mu$ is holomorphic, we have a lifting principle over a desingularization.
Consider $(x,y) \rightarrow \frac{y}{x} = \eta(x)$. We denote the diagonal $V=\{ (x,y) \quad y(x) = x \}$.
Given $y=y(x)$ analytic $\in (I)$ and $\eta \in (J)$, we have where $x \neq 0$, that $(J) \subset (I)$
and where $x \neq const.$, $V \subset \{ y = const \}^{c}$. $(J)$ is algebraic, if $(I)$ is algebraic.
When $y$ is linear, $V$ is linear.
Assume $x,\eta(x)$ does not have presence of essential singularities in the $\infty$ and consider
$x(1,\eta) \rightarrow (x,y)$, that is given $\eta$ analytic we have existence $y=y(x)$,
such that $\eta(x)=y(x)/x$. Given $x$ reduced, we can represent
a global base.
\newtheorem*{res9}{Intermediate ideals}
\begin{res9}
Assume $d v$ a measure on the boundary $\Gamma$ that is joint for $(I_{1})$ and $(I_{2})$.
Define $d \mu_{1}=\rho_{1} d v$ and a corresponding functional $B_{1}(f)=\int f d \mu_{1}$
and in the same manner for $ d \mu_{2}$ and $B_{2}$, with the condition that $\rho_{2}/\rho_{1} \rightarrow 0$
when we approach $\Gamma$. If we can find $\rho_{0}$, such that $\rho_{2}/\rho_{0} \rightarrow 0$
and $\rho_{0}/\rho_{1} \rightarrow 0$ as we approach $\Gamma$, we have existence of a corresponding
functional $B_{0}$, such that $B_{2} \leq B_{0} \leq B_{1}$ and an ideal between $(I_{1})$ and $(I_{2})$.
\end{res9}
Note that $(I_{Hyp}) \subset (I_{Phe})^{\bot} \subset (I_{He})^{\bot}$. Further if we have existence of
$\alpha$ of bounded variation such that $\int_{\Omega} g d \alpha=0$ and $d \alpha$ reduced,
we have that $\Omega$ is a strict carrier to $\alpha$ (\cite{Martineau}), that is $\alpha$
can be represented with compact support.
Assume $d \beta=e^{\phi} d \alpha$, where $\phi \in L^{1}$, here we associate $\phi$ to a hyperbolic
movement.
When $\phi$ is linear, we can define the continuation $g \rightarrow \tilde{g}$ using the Fourier-Borel duality.
Note that if $e^{\phi} \rightarrow 0$ on a radius $L$, we have that $e^{\phi} \rightarrow 0$
on a disc. In this case $\beta$ can be chosen as summable.
Given existence of $(I_{0})$ such that $(I_{Hyp}) \subset (I_{0}) \subset (I_{He})^{\bot}$ we can chose $(I_{0})=(I_{Phe})^{\bot}$.
Given a point and a normal in the point, if we consider a neighbourhood of the point with regular boundary,
then the boundary can be oriented. This corresponds to the concept of a pseudo vector (\cite{Janc89}).
The regularity conditions for dynamical systems and the corresponding conditions for first surfaces,
for instance the condition (N) (\cite{Nishino68}), can be used instead with advantage. Thus we can replace
the concept of pseudo vectors for first surfaces and transversals, given that we assume the first
surfaces oriented. Note that this condition is necessary for the transversals to be algebraic.
Note the following problem: Determine a dynamical system corresponding to a hyperbolic system
such that $\gamma$ in this system has normal $n_{1}$ in $p_{0}$ and with $p_{0}$ a zero to the system.
We assume further $n_{1}$ locally algebraic. In this case the regularity conditions indicate a pseudo normal orientation
(\cite{Janc89}). In the same manner for $n_{2}$ and a dynamical system corresponding to a partially hypoelliptic system.
Thus $\Delta \rightarrow n_{j}$ approximates singular points that are given by the right hand side.
When $n_{j}$ are locally algebraic, we have that $\gamma$ is defined as locally analytic.
Assume $d_{j}$ the respective distance functions to the joint boundary (first surfaces) with $d_{2}$
reduced, such that $d_{1}/d_{2}$ a distance.
Using that $d_{1}/d_{2} < \epsilon$, we can use the type (A) condition (\cite{Nishino75}),
that is assume $S_{1},S_{2}$ first surfaces closely situated,
we then have that $S_{1}$ intersects the normal in the same manner as $S_{2}$.
In the same manner if $\lambda \bot \eta_{2}$ and $\frac{d \lambda}{d t}=(X_{2},Y_{2})$.
We consider $(X_{1},Y_{1}) \rightarrow_{\mu} (X_{2},Y_{2})$, as a continuation,
that is $\mu_{T} \rightarrow \frac{d \lambda}{d T} \mid_{T=0}$
when $T \rightarrow 0$ and $\mu_{T} \rightarrow (X_{2},Y_{2})$.
Assume now $\gamma_{1} \in (I_{1})$ satisfies a dynamical system $(X_{1},Y_{1})$ and in the same
manner that $\gamma_{2} \in (I_{2})$ satisfies $(X_{2},Y_{2})$. We argue that there exists a dynamical system $(X,Y)$
associated to $(I)$. More precisely $(I_{1}) \subset (I_{j}) \rightarrow (I_{2})$ such that $F_{j}$ are
Hamiltonians corresponding to the approximating ideals $(I_{j})$,
that is $(-\frac{d F_{j}}{d y},\frac{d F_{j}}{d x})$ give the dynamical system we look for in limes.
We assume for this reason $\eta \bot \frac{d F_{j}}{d x}$ and $\eta \bot \frac{d F_{j}}{d y}$,
why assuming the conditions in Weyl's lemma, $\eta \in C^{1}$ and $\mbox{ supp } \eta \downarrow \{ 0 \}$.
In the same manner we can consider $(I) \subset (I_{j}) \rightarrow (I_{2})$ where again $\mbox{ supp }\eta \downarrow \{ 0 \}$.
In a discussion of desingularization, we note that if $\Omega_{1}$ is associated to $(I_{1})$
and $\Omega_{2}$ to $(I_{2})$ we have that we only assume $\gamma_{1} \rightarrow \gamma_{2}$ continuous,
why we can not establish $\Omega_{1} \rightarrow \Omega_{2}$ as a continuous mapping. Note that a desingularization
is not established for non-hyperbolic operators.
Concerning $MV(x,y)=\frac{x-y}{\log x - \log y}$ (\cite{Vuorinen07}), if $r_{T}'$ is locally algebraic and
$r_{T}' y(x)=y(r_{T}' x)$, we have $M(r_{T}' x,r_{T}'y)=r_{T}' M(x,y)$. Let
$MV(x,y)=\frac{x (1 - \frac{d y}{d x})}{-x \frac{d \eta}{d x} \log \eta(x)}$, where
$\eta(x)=\frac{y(x)}{x}$ and $=\frac{-y (1 - \frac{d y}{d x})}{x \frac{d \eta}{d x}}$,
without degenerate points in the denominator. Given that $g=\frac{d}{d x} e^{\phi}$, we then have
$\frac{G(g)}{g} \leq \frac{M(g)}{g} \sim \frac{d x}{d \phi}$, that is we have a lower bound
for the inverse mapping to the phase. If we assume $f_{j}=e^{\phi_{j}}$ with $\phi_{j} \in L^{1}$ for $j=1,2$ and
$\frac{\phi_{1}}{\phi_{2}} \rightarrow 0$, we have $\phi=\phi_{1} + \phi_{2}$ and $\phi_{1} \bot \phi_{2}$,
which implies $e^{\phi}=f_{1}f_{2}$.
In (\cite{Riesz15}) we note Theorem 17, that is if a Dirichlet series
is summable through arithmetic mean, it is summable through logarithmic mean.
The proof is based on a discussion on the function $(\frac{\log w - \log t}{w-t})^{k-1}$
$=(\frac{1}{L(w,t)})^{k-1}$ that is increasing steadily from $t=1$ to $t=w$ and the limit
as $t \rightarrow \infty$ is $w^{1-k}$. Riesz gives a proof for $k>0$. For the converse result
(in general false), we refer to theorems 19,20 (\cite{Riesz15}).
Consider now $\gamma \rightarrow \eta=y/x$, then we have that reflection through $\gamma \rightarrow -i \gamma$
can be written $\eta \rightarrow 1/\eta$ and in the same manner reflection through the real axes, $\eta \rightarrow - \eta$.
The condition on vanishing flux is then $\int_{\beta} d F(\frac{1}{\eta})=0$. Assume $\eta_{2}$ denotes
continuation of $\eta_{1}$ to $(I_{1})^{\bot}$, such that $\eta_{1}^{*} \sim 1/\eta_{2}$. Assume
$F(\eta_{1}^{*})=F^{*}(\eta_{1})$, we then have if $F(\eta_{1})-F^{*}(\eta_{1}) \sim F(\eta_{1}) - F(1/\eta_{2}) \sim 0$,
that $F$ is symmetric over the continuation and we have a transmission property in this case.
Assume now $G(f)=e^{M(\phi)}$ for $f=e^{\phi}$ and $g=\frac{d}{d x} f=e^{\psi} =\frac{d \phi}{d x} M(g)$
that is $\frac{M(g)}{g} \sim \frac{d x}{d \phi}$. Assume $\phi,\psi$ sub-harmonic. Given a lower bound for
$\frac{d x}{d \phi}$, we can conclude that if $\phi \rightarrow \phi_{0}$, we see that $x \rightarrow x_{0}$.
Given $1 \leq \frac{d x}{d \phi}$, we have that $\phi \leq \phi \frac{d x}{d \phi}=z(\frac{1}{\phi})$,
given $x$ is analytic considered as a function of $\phi$ in $0 < \mid \phi \mid < \infty$. Given
$z (\frac{1}{\phi}) \rightarrow 0$, we have that $\phi \rightarrow 0$, why we have existence of $w$,
such that $\frac{1}{w}(\frac{1}{\phi}) \rightarrow 0$ implies $\phi \rightarrow 0$, that is $\frac{1}{w}(\phi) \rightarrow 0$
implies $\phi \rightarrow \infty$, why $w$ is downward bounded. Thus we have where $x$ is analytic,
a mapping $x \rightarrow w$, such that $w$ is locally 1-1, that is if $w \rightarrow w_{0}$ implies
$\phi \rightarrow \phi_{0}$, it is sufficient to consider the phase. Note that
if $\frac{d \phi}{d x} \rightarrow 0$ implies $x \rightarrow 0$, that is
$f'/f \rightarrow 0$ implies $x \rightarrow 0$, then we have that $f \in (I_{He})^{\bot}$ for
all $x \neq 0$.
\section{The transmission property}
\subsection{The transmission property for summable distributions}
Immediately, in the case when
$F$ can be represented by a linear functional, we can define $\bot$ using for instance annihilators.
If $I(f)=\delta_{0}*f$ we have that $I$ reflects the support. If $f$ has symmetric support we have that
$I(f) \sim f$. If further $f$ is proper, then $I(f)$ has the same property.
Assume in particular $U_{j}$ closed in some set $Z$, assuming Schwartz type topology,
we have that $(x,\varphi) \in V \times U_{j}$ is closed iff $x \rightarrow \varphi(x)$ continuous.
If singularities are approximated through transversals $L$, there are points in $Z \cap L$
that do not belong to $U_{j}$. More precisely, if $u_{0} \in L \cap U_{j}$ and
$u_{j} \rightarrow u_{0}$, we have that $u_{j} \in Z \backslash U_{j}$.
Define $F(I x)=F(-x)$ and ${}^{t} I F(x)=-F(x)$, this can be seen as a weak definition of odd
operators. Thus if $F I = I F$ we have that $F$ is odd. If $F I = F$ we have that $F$ is even.
Note that $\mbox{ supp } \big[ I,E \big]=- \mbox{ supp }E$. If $E,F$ are both one-sided,
we may have that $\big[ I,E \big] \big[ I,F \big]$ is one-sided
but $\big[ I,E \big] F \equiv 0$.
In the terminology of oscillating integrals, $F(x)=\int a(x,\theta)e^{i s(x,\theta)} d \theta$
where $a(x,\theta)$ are poly homogeneous, for $x \in X$ and $\theta \in \mathbf{R}^{N}$.
The phase function $s(x,\theta)$ is assumed real, 1-homogeneous in $\theta$ with
$s_{m}(x,\theta) \neq 0$, as $s \neq 0$. Assume $S=\{ s_{\theta}(x,\theta)=0 \}$ closed and conical
and $WF(u)=\{ (x,\theta) \quad \theta=s_{x} \quad s_{\theta}=0 \}$. Using a bijection
$\varphi$, $WF(u (\varphi(x)))=\{ (\varphi(x),{}^{t} \varphi'(x) \theta) \in WF(u) \}$. Assume
$x^{*}=\xi$, $\eta^{*} \sim \varphi(x)$, why $\frac{d}{d x} s(\varphi(x),\xi)=\frac{d \varphi}{d x} s_{x}=\varphi'(x) \xi$,
then we have $\eta = \frac{d \varphi}{d x} \xi$ and ${}^{t} \big[ \frac{d \varphi}{d x} \big]^{-1} \eta=\xi$.
In particular $\int a(x,\xi) e^{i < x,\xi> - H(\xi)} d \xi \sim \widehat{(A/H)}$ (\cite{Garding87}).
Note for the dual $a'$ to a polynomial, if $a'-a \equiv 0$ on the radius in a disc,
we have that $a'-a \equiv 0$ on the disc.
For a pseudo differential symbol $a$ we must except the imaginary axes.
Assume that $a$ has a symmetric kernel with respect to the imaginary axes, for instance
$a'-a = \Sigma_{j} \delta^{(j)}$, where $\delta$ is assumed to have support on
the imaginary axes.
Given a very regular boundary and $\eta_{2}$ reduced
we can consider $1/\eta_{2} \in \dot{B}$, why $(I_{1})$ can be seen in $\mathcal{D}_{L^{1}}^{\bot}$
that is $P(D) \tilde{f}_{0}$
is associated to $1/\eta_{2}$, which can be seen as a topological transmissions property.
\subsection{The transmission property for two bases}
The transmission property, when it is derived from involutive reflections $U$, implies
presence of a normal model. In the last section, we see that a global model does not imply a normal
model. If we can map bijectively a broken ray
onto a transversal, we have one-sidedness for one segment with respect to the first ($x-$) axes and one-sidedness
for another segment with respect to the second ($y-$) axes.
Now consider one-sidedness with respect to two planes, assume $E(x,y)$ the kernel corresponding to
the first segment and $F(y,z)$ the kernel to the last segment. Assume that we have existence of
a segment $ y \rightarrow y'$
(transversal) according $\frac{d y'}{d y}=1$. We assume again the segment normalised using distances.
We write $G(y,y')$ for the kernel corresponding to the middle segment.
In order to compare with one-sidedness, we assume $G(y,y') \rightarrow 1$
as $y \rightarrow y'$. Note that $F$ is assumed symmetric after reverting the orientation
for the last segment.
If the composition is algebraic, we do not have that the factors are algebraic.
Assume we use distances to represent the convergence, let $d_{1}$ be the distance
to $\xi_{2}=0$ and $d_{2}$ the distance to $\xi_{1}=0$. When the distances are reduced we can assume the inverse
a distance, otherwise a pseudo distance. Assume $A \rightarrow A' \rightarrow B' \rightarrow B$, where $A,B$ on $\mid \xi \mid=1$ and
$A',B'$ on axes. Given symmetry for the kernel, we can assume $d_{2} \rightarrow 0$ has a
reflection in $d_{1}^{-1} \rightarrow 0$.
If $d_{1}(\xi) \rightarrow 0$ as $\xi \rightarrow 0$, then $\frac{1}{d_{1}}(\frac{1}{\xi}) \rightarrow 0$,
when $\xi \rightarrow 0$
that is $\overline{\xi}=1/\xi$.
Let $i(d_{j})=1/d_{j}$ that maps distances onto pseudo distances.
If $A \sim B$
(congruence) we can assume $d_{1}d_{2} \sim 1$. In particular $d_{1}=0$ for $z=A$,
$\frac{1}{d_{1}}=d_{0}=0$ for $z=A'$, $\frac{1}{d_{0}}=d_{2}=0$ for $z=B'$ and $\frac{1}{d_{2}}=0$,
for $z=B$.
In this case we have that $d_{1}d_{2}=0$ implies $z =A$ or $z=B'$.
As $\frac{d_{1}}{d_{2}}=0$ we have $z=A$ or $z=B$. Under the symmetry condition for $f_{\lambda}=f-\lambda$ we have
$\{ d_{1} \mid f_{\lambda} \mid < c \}$ $\sim \{ \frac{1}{d_{2}} \mid f_{\lambda} \mid < c \}$, that is we can assume
$f_{\lambda}$ has algebraic growth at the boundary.
If we instead assume $A$ singular in the sense that $ d_{1} \frac{1}{\mid f_{\lambda} \mid} < c$ at the boundary,
we have that $f$ is downward bounded at the boundary and $\{ \mid f_{\lambda} \mid < c \}$ can be seen
as bounded. Simultaneously if $B$ is regular, we have that the corresponding set is unbounded at
the boundary.
Consider $f$ with algebraic continuation in $\{ d_{1} \frac{1}{d_{2}} \mid f \mid < c \}$. We are assuming
$\tilde{f}(z,w)$, where $z$ is the reflection of $w$.
Assume now $z,w$ a continuation of path $A \rightarrow A' \rightarrow B' \rightarrow B$.
Define instead $d_{1} \rightarrow 0$ as $z \rightarrow A$ and $d_{1} \rightarrow 1$ as $z \rightarrow A'$
and $\frac{1}{d_{2}} \rightarrow 0$ as $z \rightarrow B$ and $d_{2} \rightarrow 1$ as $z \rightarrow B'$.
We assume $d_{0}(A')=d_{0}(B')=1$, we have that $d_{0} \frac{d_{1}}{d_{2}} \mid f \mid(z,w) \rightarrow $
$0$ as $z \rightarrow A$, $1$ as $z \rightarrow A'$, $1$ as $w \rightarrow B'$ and $0$ as $w \rightarrow B$.
Starting from a point at the boundary $A$, assume that we have existence of a path $\tau$ between
$A$ and $B$, that does not contain any other singularities than $A$, in the sense that $A$ can
be identified with a regular point $B$, we could represent $A$ and $B$ in the same leaf.
The points can be seen as isolated on the path. If not all points on the boundary
$\Gamma$ are singular, we can always find a path to $B$, if all singularities are situated on the
boundary, we have that the path is regular.
\subsection{General remarks}
The condition $\log f \in L^{1}$ can be seen as $\log \mid f \mid \in L^{1}$, that is
$\mid f \mid = \mid f^{*} \mid$ is a symmetric condition. For selfadjoint operators, it is sufficient
to conclude hypoellipticity, to consider translation invariant sets.
Polynomial operators have solutions with one-sided support, that is we can assume $F(\gamma)(\zeta)=I(\zeta)$,
where $F=\widehat{E}$ and $E$ has one-sided support, when
$\gamma$ is polynomial outside the kernel to $E$. Thus we have for the continuation
$\gamma_{1} \rightarrow \gamma_{\delta}$ algebraic
that the corresponding $F_{\delta}$ can be selected with one-sided support.
Starting with $F(e^{-v} \gamma_{1})=F(\gamma_{2})$, with $v \in L^{1}$, such that $v \equiv 0$ over the
set $\Delta$ where $\gamma_{1}=\gamma_{2}$. Assume $V \gamma_{2} \sim \eta$ and $\eta \rightarrow \eta^{*}$
reflection through the axes $\eta=\eta^{*}$, $\eta = y/x$. Assume $v(\eta,\eta^{*}) \in L^{1}$ and note
that $v=0$ implies $\mid \eta \mid \leq 1$ ($\geq 1$). Let $H(\alpha)=\int H(\eta,\eta^{*}) \alpha(\eta) d \eta$,
then for $\alpha \in \dot{B}$, we can assume $H \in \mathcal{D}_{L^{1}}'$ a Schwartz kernel. The condition
$v \in L^{1}$ means isolated singularities and we can assume $v \sim v_{1}$, where $v_{1}$ is analytic.
When $H \sim_{m} H_{1}$ locally algebraic, we do not have a normal model for $H_{1}$, even when $\eta \rightarrow \eta^{*}$
is involutive. However we have a global model according to the moment problem.
\newtheorem*{res12}{A global representation}
\begin{res12}
The representation $F(\gamma_{1} \rightarrow \gamma_{2})(\zeta) = f(\zeta)$ and $F(\gamma_{1})=F(e^{v} \gamma_{2})$,
with the condition that $v \in L^{1}(\eta,\eta^{*})$, gives a global model. It is not necessarily a
normal model.
\end{res12}
Note that $v$ is defined by the movement. By considering $w=v + v^{\bot}$, where $v^{\bot}$ is defined
by the orthogonal movement and where both $v,v^{\bot}$ are assumed continuous, if we assume the
compatibility condition $w=0$ on invariant points and on the
intersection of the supports to $v,v^{\bot}$, we can obviously give a global representation for $w$.
The conclusion is that if the singularities are reached using $w$, we have a global base for the model.
Assume $A=A^{*}$ a hypoelliptic ps.d.o, then $A$ has very regular parametrices, with trivial kernel.
Thus the parametrices to $A$ have a transmission property modulo $C^{\infty}$, in the sense that the regularity
is symmetric with respect to diagonal. For a partially
hypoelliptic operator, we do not have a trivial kernel, but the Schwartz kernel has hypoelliptic
representation over kernel (zero space) and outside
the zero space (\cite{Dahn13}). Obviously we do not have that the sum of kernel has the symmetric regularity property
Note that if we define $\bot$ using $g(\frac{1}{x})=0$ on $V \ni 0$ (a bounded set)
we must separate between the case where $g \sim \frac{\mbox{ Im} f}{\mbox{ Re }f}$ analytic
and the case where $g \equiv 0$ in a disc-neighbourhood of $0$. When $\bot$ is defined by
a rational function $g$ we have that $\bot$ is of order 0, Further, when $g$ is analytic we have that
$\bot$ is Cartier.
\subsection{Final remarks}
Starting with the observation, that if $\varphi$ is a closed form in the plane and $\varphi^{*}$ (harmonic conjugate)
with $\varphi^{*}=-i \varphi$, we have that $\varphi$ analytic. A mapping that maps
$(x,y) \rightarrow (y,-x)$ is pure, that is preserves analyticity.
Assume $L$ the line $x=y$ (the light cone), then reflection through $L$ can be seen as parabolic
movements. Assume $L_{0}$ the positive
imaginary axes, $L_{1}$ line $x=y$, $x>0$, $L_{2}$ the positive real axes,
$L_{3}$ the line $x=-y$ and so on.
Assume $c_{0}(x,y)=(-x,y)$ that is reflection through $L_{0}$ and so on. We then have that
$c_{j+1}c_{j}(x,y)=(y,-x)$ that is pure mappings.
Further, $c_{j}c_{j+1}(x,y)=-(y,-x)$ and so on.
Consider $\frac{f_{2}}{f_{1}}=\frac{1}{\varphi}$. Note that given $W$ planar in $\mathcal{O}_{AD}$
(\cite{AhlforsSario60}), that is we have existence of $u$ analytic with finite Dirichlet integral,
such that $u$ is constant on $W$.
We then have for every single valued function $u$, that $u$ is linear. We can in this context consider
$\frac{1}{\varphi}$ as a function of $f$. For instance $\frac{1}{\varphi}(f+g) \rightarrow 0$
in $\infty$, when $f_{1} \bot f_{2}$ and $g_{1} \bot g_{2}$.
When a domain is defined by a distance $d$, the domain is said to be $d-$ pseudo convex if
$- \log d$ is plurisubharmonic.
Note that $d_{j}(x) - d_{j}(y)=0$ iff $e^{d_{j}(x)}-e^{d_{j}(y)}=0$, $\mid d(x) - d(y) \mid \leq d(x-y)$.
In particular, on the set where $d^{*}=d$ and $d^{2} \sim 1$ we have if $d (f)=0$ that
$\mid f \mid =1$ and $\frac{d}{d x} d(f)=0$ implies $f=1$. Assume $d>0$ a distance on the phase space
to symbols, then we have
$d(\phi)=d(\psi)$ implies $\tilde{d}(e^{\phi})=\tilde{d}(e^{\psi})$, where $\tilde{d}(e^{\phi})=e^{d(\phi)}$.
When we have equality in the tangent space, we have $e^{\phi}=C e^{\psi}$. Further if $\tilde{d}$
is locally 1-1,
we have that $\tilde{d}>0$ where $d>0$. When $d<0$ we can consider
$\tilde{d}(f) \tilde{d}(\frac{1}{f})=e^{d(f) - d(f)}=1$, that is $\tilde{d}(\frac{1}{f}) \sim 1/\tilde{d}(f)$
and $\tilde{d}$ can be seen as ''algebraic``. Note that this means that $d$ corresponds to $(f_{1},f_{2})$, $\overline{d}$ to $(f_{1},e_{2})$,
$-d$ to $(e_{1},e_{2})$,
$-\overline{d}$ to $(e_{1},f_{2})$, $i d$ to $(e_{2},f_{1})$, $i \overline{d}$ to
$(f_{2},f_{1})$, $-i d$ to $(f_{2},e_{1})$ and $- i \overline{d}$ to $(e_{2},e_{1})$.
We have in this case a two-sided solution $f_{1}e_{1}=e_{1}f_{1}=1$.
Assume $A \rightarrow B$ through reflection points $s_{1},s_{2}$. Assume
$\frac{1}{2} d \overline{d}$ $=\frac{1}{2} (d_{1}^{2} + d_{2}^{2})$ defines the distance between
$A$ and $B$. Given that the domain for $d \overline{d}$
is connected and $d \overline{d}$ monotonous on the path, there is a $\mu$ between $s_{1}$ and
$s_{2}$, using the theorem of intermediate values. For instance if $d_{1}=1$ in $s_{1}$ and
$d_{2}=1$ in $s_{2}$, we have that $\frac{1}{2} d \overline{d}=1$ on the path between $s_{1}$ and
$s_{2}$. If $d_{1}^{2}$,$d_{2}^{2}$
are polynomial, we can consider a semi algebraic set and the path $\mu$ between the reflection
points as a part of the boundary.
\section{The boundary}
\subsection{First surfaces}
First surfaces are naturally invariant for all movements and existence of a regular approximation
outside the first surface implies existence of a $V$ such that $\frac{d V}{d T} \neq 0$ over
this set.
More precisely, assume the first surfaces are defined by movements with the compatibility condition
$U^{\bot} \gamma \subset S_{0}$ or $U^{\bot} \gamma \cap S_{0} = \emptyset$, where $S_{0} = S \backslash \{ (x_{0},y_{0}) \}$, and $(x_{0},y_{0})$ is a point
that is invariant for both $U$ and $U^{\bot}$. When $V$ is translation we have that $V^{\bot}$
can be seen as a transversal,
when $V$ is an elliptic movement, then $V^{\bot}$ is a line through $0$, when $V$ is scaling
then $V^{\bot}$ is scaling.
Note that the invariance principle does not include the movements $\bot$ lorentz movements.
However we can define $U^{\bot} \rightarrow V^{\bot}$ using Radon-Nikodym's theorem.
Note that $L(U \gamma, \xi)=0$ iff $L(\gamma,{}^{t} U \xi)=0$. If $\xi=U^{\bot} \gamma$
we see that ${}^{t} U=(U^{\bot})^{-1}$ implies $\gamma \in K$ (the light cone).
Starting with a normal tube, where $L$ is an analytic line ($\Delta$), that is assumed to intersect
the first surfaces transversally, we apply a movement $U \rightarrow V$, with the property that $\gamma \in \Sigma$
implies $V \gamma \subset \Sigma$, where $\Sigma=\{ F(\gamma)=const \}$. We consider a representation
$F \sim G$, that is regular and with equivalent first surfaces. We have that (\cite{Nishino75})
the bases $\gamma$ can be selected as algebraic for $G$, given that $G$ has first surfaces of type (A).
Thus, we can regard $F(V^{\bot} \gamma)$ as an associated function to $F(\gamma)$ (and $F(U \gamma)$).
Assume, $V$ an euclidean movement, that preserves first surfaces to $f$ and $e^{v}F$ the corresponding associated function,
then we have that $e^{v}F$ has type (A), if $f$ has type (A).
More precisely given $L=\{ \gamma_{1}=\gamma_{2} \}$ (formed on $\Delta$) analytic, we can define
the boundary $\Gamma$, as first surfaces that intersect $L$ transversally. We can continue $\Gamma$
with first surfaces $\Sigma \ni \gamma \rightarrow V \gamma \in \Sigma$. Given the compatibility
conditions, we can further continue the boundary by replacing $L$ with $V^{\bot}$ and repeat the
argument.
A simple very regular boundary is, $F_{T}$ holomorphic in $T$ and $\frac{d}{d T} F_{T}$ holomorphic in $T$,
where $T \notin \Sigma$ and $\Sigma$ are isolated points close to the boundary. Regular approximations
are formed as $V_{1} \cup V_{2}$, where $V_{1}=\{ F_{T} \quad \mbox{not const} \}$ and
$V_{2}=\{ \frac{d F_{T}}{d T} \quad \mbox{ not const} \}$.
Note that given $f$ holomorphic with first surfaces $\{ S_{j} \}$ and $S_{0}=\{ f - \alpha_{0} \}$
and let $E = \lim_{j} S_{j}$, with $E \cap S_{0} \neq \emptyset$, this implies $S_{0} \subset E$.
The type for the class of first surfaces depends on the representation of the symbol. For instance
$F(\gamma)=\int A d \gamma$ is not dependent of $V \gamma$, where $V$ is translation. When $F(\gamma)=G(\frac{y}{x})$,
where $G$ is assumed regular, the representation is not dependent of $V \gamma$, when $V$ denotes scaling. Finally
$F(\gamma)=F(\mid \gamma \mid)$ is not dependent of $V \gamma$, where $V$ is rotation and when the
singularities are given by the condition $\log \mid f \mid \in L^{1}$
they are not dependent of rotation of the symbol.
\newtheorem{res13}{Boundary condition}
\begin{res13}
We assume all singularities are situated at first surfaces for $f(\zeta)$. We assume $\Delta$ corresponds
to a set $\Sigma$, where $\gamma_{1}=\gamma_{2}$. The condition $\log \mid f \mid \in L^{1}$ implies algebraic singularities.
When the boundary is represented in $\mathcal{D}_{L^{1}}'$, through $B(\gamma)=\int \gamma d \mu$, where
$d \mu$ is assumed very regular, if we let $F(\gamma_{1})=B_{1}(\gamma_{1})$, we assume as involution
condition $d \mu_{1}/d \mu_{2} = e^{v}$, where $v \in L^{1}$ and $v=0$ over $\Sigma$. Further, we assume
$d \mu_{1}=0$ iff $d \mu_{2}=0$, which when $d \mu_{j}=X_{j} d x_{j} + Y_{j} d y_{j}$, $j=1,2$, corresponds to
Lie's point continuation.
\end{res13}
\subsection{Radon-Nikodym's theorem}
Starting with the lineality we can define the boundary as follows. Assume $\Delta^{\bot}$ are
orthogonals (tangents to $\Gamma$) to $\Delta$, we can describe $\Gamma$ using
$e^{\varphi} \gamma$ regular relative $\Delta^{\bot}$.
The Radon-Nikodym's model is, given that $I(\frac{y}{x}) \rightarrow 0$ implies $A_{1}(\frac{y}{x}) \rightarrow 0$,
we have $A_{1}(\frac{y}{x})=I(\alpha \frac{y}{x})$ for some $\alpha \in L^{1}$.
Note that as $A(0,0)=(0,0)$, we must consider $A_{1}(\frac{0}{0})=\frac{0}{0}$, (compare L'Hopital).
Assume $F_{1}(\eta(x))$ where $\eta(x)=y(x)/x$, we then have that
if the limits are on the form $F_{0/0}$, we must calculate $F_{1}(\frac{dy}{dx})$,
that is we assume that we have non degenerate points and $x \frac{d \eta}{d x} \neq 0$.
Consider $(I_{1}) \subset (I_{0}) \subset (I_{2})^{\bot}$ and $\gamma_{1} \rightarrow \widetilde{\gamma_{1}} \rightarrow \gamma_{2}^{\bot}$
that is $< \gamma_{2},\gamma_{2}^{\bot} > \sim 0$ (or $\sim 1$), if $\gamma_{2}$ polynomial,
we do not necessarily have $\gamma_{2}^{\bot}$ polynomial, but it can always be represented as
an annihilator (or $(1-\gamma_{2}^{\bot})$). The moment problem means,
given $\int g d \mu=0$ with $g \in (I_{1})$ analytic, that the solution if it exists
is in $(I_{2})$, for instance $d \mu=P(D) d v$, with $d v$ reduced implies $d \mu \in \mathcal{E}'$.
If $W$ is open in $V$ we have that $H(V)$ is dense in $H(W)$, then $W$ has Runge's property in
$V$.
Given Runge's property we can select
$(I_{0})$ in $C_{0}$, that is $d \mu \in C_{0}'$, a Radon measure. Oka's property, in this context
existence of a global base,
is dependent on the domain.
Consider the problem of localisation: assume that $V$ a geometric set and $f(\zeta)$
a symbol over $\Omega$ and $V$ not a subset of $\Omega$. Further that we have $F(\gamma)(\zeta)=f(\zeta)$
over $\Omega$. We can construct $\gamma_{1}$ over $\Omega$,
such that there exist $\gamma \rightarrow \gamma_{1}$ continuous (continuation), that is
$F(\gamma) \rightarrow \gamma \rightarrow \gamma_{1} \rightarrow V$ continuous. Assume for instance
that $\gamma_{1}^{\bot} \rightarrow V$ continuous (closed) and locally 1-1,
such that $\gamma^{\bot}=r_{T}' \gamma_{1}^{\bot}$ surjective, existence of $\gamma_{1}^{\bot}$
defines $\gamma_{1}$ (as an annihilator), why $\gamma_{1} \in (I_{He})^{\bot}$ implies
$\gamma_{1} \in C_{0}$, using the moment problem.
The boundary is given by first surfaces $\{ \gamma \quad F(\gamma)(\zeta)=const \}$,
dependent on the involution condition and by $\{ \zeta \quad F(\gamma)(\zeta)=const \}$.
When $\gamma$ is algebraic, we can assume a continuous mapping between these sets.
Let $\big[ F , E_{\delta} \big](\gamma_{2})=const$, as $\delta \rightarrow 0$
and $\log E_{\delta} \in L^{1}$. Then $F$ corresponds to a solution to a (partially) hypoelliptic
operator and we see that $\mbox{ ker }E_{\delta}$ is the kernel to $J \gamma_{1}$.
We thus assume ${}^{t} J_{\delta} F(\gamma_{1})=
F(J_{\delta} \gamma_{1})=\big[ F,E_{\delta} \big](\gamma_{2})$. Assume $F$ such that
$\mbox{ ker }F=\{ 0 \}$ which implies $\mbox{ ker } F \subset \mbox{ ker } {}^{t} J_{\delta} F$.
Further, $E_{\delta} \rightarrow 1$, as $\delta \rightarrow 0$.
Assume first surfaces to $F(\gamma_{1})$
corresponds to a desingularization, then we have $\tilde{S}_{j}=\{ \zeta \quad \big[ F,E_{\delta} \big](\gamma_{2})=const \}$
assuming ${}^{t} J_{\delta} : const \rightarrow const.$
The condition for the normal $n$, $d t=\sigma d n$ can be compared with Radon-Nikodym's theorem.
We assume in this case $\sigma \in L^{1}$ over an unbounded set. When $\sigma$ is independent
of some variable, that is we have a planar obstacle, the condition is not satisfied.
When $\sigma$ have bounded sublevel-surfaces, it can be given by a regular function.
\subsection{Distance functions}
Assume $d_{T}(\gamma)=0$ implies $r_{T}' \gamma \in \Delta$ and that $\Delta'$ defined through the symmetry
condition $F(r_{T}' x,y)=F(x,r_{T}' y)$, that is we assume $\Delta \subset \Delta'$. We define
$d_{T}'(\gamma)=0$ implies $r_{T}' \gamma \in \Delta'$. Define $\Delta''$ through $F(r_{T}'x,y)=F(-y,r_{T}'x)$,
that is invariance for harmonic conjugation. We define the corresponding pseudo distance $d''$.
Finally, we define $d^{0}$ as a pseudo distance to $K=\{ x=y \}$. Thus $(r_{T}' x,y) \rightarrow (x,r_{T}' y)$,
that is reflection through $K$ can be seen as $d^{0}=const$. Further, $(r_{T}'x,y) \rightarrow (-y,r_{T}' x)$
can be seen as $d''=const$, $(-y,r_{T}'x) \rightarrow (-r_{T}'x,y)$ through $d^{0}=const$. The composition,
that is $d'' d^{0}=const$ contains $(r_{T}' x,y) \rightarrow (-r_{T}' x,y)$.
Note if $d_{1}d_{2}=const$ and $d_{1},d_{2}$ distances, we have that $d_{1},d_{2}>0$ and
$d_{1}d_{2}=0$ implies that $d_{1}=0$ or $d_{2}=0$, thus $A \sim B$ (congruence), that is
$d_{1}$ denotes the distance to $A$ and $d_{2}$ denotes the distance to $B$ we have that
$z=A$ or $z=B$. In particular if $A=0$ and $B=\infty$, we have that
$\frac{d_{1}}{d_{2}}=0$, for $\frac{1}{d_{2}}(z)=d_{2}(\frac{1}{z})$.
The example $d_{1}d_{2}=const$ with $d_{1} \rightarrow 0$ as $z \rightarrow \Gamma$ under
the condition $d_{2}(\frac{1}{z})=1/d_{2}(z)$ implies $1/d_{2}(z) \rightarrow 0$ and if
$d_{2}$ reduced, we have $\frac{1}{z} \rightarrow \infty$, that is $z \rightarrow 0$.
We are assuming no essential singularities in the $\infty$, why if $1/d_{\Gamma} \rightarrow 0$,
as $z \rightarrow \infty$, it is a boundary point but not a singularity.
Note that the condition $\log f \in L^{1}$, means that for the ideals defined by $d_{j}^{2}$
polynomials, such that $d_{j} \rightarrow d_{j+1}$ compact, the singularities in the zero space
to $\lim d_{j}^{2}$, can be algebraic.
Assume $V=\{ \frac{1}{v}=const \}$, we then have that $V$ is bounded if
$\mid \xi \mid^{\sigma} \leq C \frac{1}{v}$ when
$\mid \xi \mid \rightarrow \infty$, that is if $v$ algebraic in $\infty$ we have that
$V$ is regular and bounded in $\infty$. If $V' = \{ v=const \}$ and $v$
downward bounded in $\infty$ and in the same way if $\frac{1}{v}$ is algebraic in
$\infty$ we have that $V'$ is regular and bounded.
Assume $F(\gamma) \rightarrow \gamma_{1} \rightarrow \gamma_{2} \rightarrow \zeta$,
where $\gamma_{1}$ is hyperbolic and $\gamma_{2}$ is partially hypoelliptic. The problem is to
determine if $\gamma_{1}$ and $\gamma_{2}$ can be constructed starting from the same lineality $\Delta$.
An important difference between hyperbolic and partially hypoelliptic symbols, is that $\gamma^{N}_{1} \in (I_{1})$
iff $\gamma_{1} \in (I_{1})(\Delta)$ and $\gamma^{N}_{2} \in (I_{2})$ does not imply that $\gamma_{2} \in (I_{2})$.
Analogously, we have that $d \gamma_{1} \in (I)$ implies $\gamma_{1} \in (I)(\Delta)$ and $d \gamma_{2} \in (I)$
is implied by $\gamma_{2} \in (I)$, but not conversely, that is for partially hypoelliptic operators,
we have $\Delta_{(j)} \downarrow \{ 0 \}$ and for hypoelliptic operators, we have that $\Delta_{(j)}=\Delta$.
The proof of this is based on the condition that $\Delta \subset$, a domain of holomophy.
Assume $(I_{1}) \subset (I_{He})^{\bot}$ is defined by distances, $d_{2}^{\bot}/d_{1} \rightarrow 0$,
for $\zeta \in \Sigma$ and $d_{1}=\inf_{\zeta} \mid f-c \mid$, that is distance to the first surface
$f(\zeta)=F(\gamma)(\zeta)$. Then if $f$ is an entire function, we have that $f \rightarrow c$, as $d_{1} \rightarrow 0$.
Note that as $\gamma$ is a pseudo differential operator, we can have that $\{ \gamma < \lambda \}$
is not relative compact. Further, if $d_{2} /d_{2}^{\bot} \rightarrow 0$, then $d_{2} / d_{1} \rightarrow 0$
(one-sided orthogonality).
Note that $H(U \gamma) \equiv 0$ defines a regular domain, if $H$ has an analytic kernel, that is
$H (U \gamma) = \int U \gamma d \mu =0$, for some $d \mu$ defined by the boundary. Isolated singularities
implies that $U$ is monotropical with an analytic $H$.
Assume that all singularities can be approximated by
$U^{\bot}$, we then have that given a fixed $U$, we can assume $U \gamma_{1},V \gamma_{2}$ algebraic.
If for $F \in \mathcal{D}_{L^{1}}'$, $F(\gamma_{1} \rightarrow \gamma_{2})$ is analytic over $(I_{1}) \rightarrow (I_{2})$, we assume
merely continuity for $F$ over $U^{\bot} \gamma_{1}$. Given $\gamma_{1},\gamma_{2}$ a polynomial,
we can chose $F$ analytic over $\gamma_{1} \rightarrow \gamma_{2}$.
\subsection{Localization}
If the boundary is given locally by one single function, we can represent this function as the
solution to a differential operator (boundary operator).
Assume $f$ analytic on a domain $\Omega$, assume $\gamma_{1} \in (I_{Hyp})(\Omega)$ and
that we have existence of $F_{1} \in H'(\Omega)$ such that $F_{1}(\gamma_{1})=\gamma_{2} \in (I_{Phe})$.
Given that we have existence of $F_{2}$ analytic over $\gamma_{2}$,
such that $\big[ F_{2},F_{1} \big]$ analytic over $\gamma_{1}$ we have a lifting principle
over $\gamma_{1} \rightarrow \gamma_{2}$. The base is reversible, if $\big[ F_{1},F_{2} \big]$
analytic over $\gamma_{2}$.
For hyperbolic spaces we have that every pair of points can be linked through a chain of
analytic discs (the transversal analytic). Assume $S_{j}$ a first surface to a hyperbolic
base and $\tilde{S}_{j}$ the continuation to the partially hypoelliptic base
as simply connected domains. If we assume the continuation algebraic, with
$\log \tilde{f} \in L^{1}$, we still have singularities of finite order,
but this does not imply that the transversal is analytic (or locally algebraic).
Assume that the continuation is given by $d_{1}d_{2}$, such that $S_{1} \rightarrow S_{2}$
through reflection with respect to a reflection axes in $\pi$. Transversals can be seen as
$\bot \pi$. On the hyperboloid, we can always represent proper movements as reflection
$\sim$ a normal model. Movement can be continued to euclidean metric, also the reflection axes,
but orthogonals (corresponding to the transversal) are not necessarily analytic.
Note that in the problem of localisation, the condition of surjectivity is not necessary.
It is sufficient with existence of $\gamma$, such that $f(\zeta) \rightarrow \gamma \rightarrow V$ continuous.
Assume $\forall \gamma \in (J)$ we have existence of $R$ hypoelliptic with $R \gamma_{2}=\gamma$.
Assume $P^{*}=R$ in $\mathcal{D}_{L^{2}}'$.
When $E$ is very regular, we have that $E \gamma=\gamma_{2}$ hypoelliptic and $E \sim \delta_{0} - \eta$
implies $\gamma=\gamma_{2} + \eta * \gamma$ gives an approximative solution.
In this article, the mixed problem is dealt with as $F(\gamma_{1} \rightarrow \gamma_{2})(\zeta)
=f(\zeta)$ and in the converse direction over $\gamma_{2} \rightarrow \gamma_{1}$, the mixed
problem is already extensively dealt with (for instance \cite{Ikawa85},\cite{Ikawa88}).
Assume $f_{1} \sim f + \delta$. This equivalence can be used in connection with Cousin's model
of monotropy. For isolated singularities we have $f_{1}(\xi)=f(\xi) + \delta=f(\xi + \epsilon)$.
We do no have in this case a normal covering in the sense of Weyl (\cite{Weyl09}), but
using a reduced complement, if the singular points are $\notin \mbox{ supp }\mu$,
we can assume $f \bot \mu$ iff $f-\delta \bot \mu$. In connection with double transform,
$P(D) \delta \rightarrow P(\xi) \rightarrow P(\xi^{*})$,
when $f(\zeta) + \delta(\zeta)=F(r_{T}' \gamma)(\zeta)$, where $F$ is linear in $\gamma$.
Given $\gamma$ analytic,
we can locally write $F \sim 1/Q$ for a polynomial $Q$, in this case $F(\gamma)$ is constant.
Given $\int g d \mu=0$ with $g$ algebraic, we have that the intersection is of measure zero.
When $g$ is not algebraic, we are discussing one-sidedness. $F(g_{+})=G(g_{-})$
where the regularity is preserved.
\subsection{Continuation of the boundary}
Assume the boundary $\Gamma$ extended with mirrors (congruent to tangents) to
$\tilde{\Gamma}$. Starting from a boundary point in $\Gamma$ selected as origo,
the problem is to reach the others through a chain of broken rays. If also the chain
is regarded as $\tilde{\Gamma}$ then we reach in this manner a subset of the boundary of
positive measure (we assume the symbol $f=0$ over the chain). Assume that $d \mu$ a boundary measure
(\cite{Garding64}) and denote with $\tilde{d \mu}$ the measure corresponding to the extended
boundary. Thus $\tilde{\Gamma} \rightarrow \{ \mid z \mid=1 \}$
Consider now $A(x,y)=A(x(1,\frac{y}{x}))$. If $(1,\frac{y}{x}) \in \Omega$, where $\Omega$
is the domain for analyticity for $A$, we have that $A$ is analytic on $1 + \mid \frac{y}{x} \mid^{2} \leq 1$,
In particular, if $f$ is analytic on $(1,\frac{y}{x})$ with $\mid \frac{y}{x} \mid < 1$ and
on $(x,0)$ where $\mid x \mid < 1$, we have that $f$ is analytic on $\mid x \mid^{2} + \mid y \mid^{2} \leq 1$.
The symmetry condition $F(r_{T}' x,y)=F(x,r_{T}' y)$ means that the zero set is symmetric in
$(x,y)$. If $F(x,y)=0$ we have that $F=0$ in $r_{T}' x$, for $y$ fixed, that is on a cone
relative $r_{T}'$ (relative homogeneity).
The condition $y_T / x_{T}=y/x$ can be written $F(r_{T}' x,r_{T}' y)=F(x,y)$.
Assume $P$ hyperbolic and $Q$ partially hypoelliptic with $\Delta(P)=\Delta(Q)$,
consider for a parmetrix $E_{1}$ to $P$, $\tau P v =P$, that is $vE_{1} = E_{1}$ over $\Delta$.
If $E_{2}$ is parametrix to $Q$ and if we consider the parametrices as Fredholm operators,
with the difference that $\mbox{ ker }E_{2}$ can be reduced to a trivial space through iteration.
Assume $\Delta^{\bot}$ is given by $S_{j}$, continued to $\tilde{S}_{j}$ through
$M(e^{\psi} d \gamma) \rightarrow e^{\phi} M(d \gamma)$ continuous,
where $\phi \in L^{1}$. If we have $\phi>0$, we have trivial first surfaces,
corresponding a reduced operator.
We are assuming $\tilde{S}_{j}$ can be continuously deformed to $S_{j}$ using $\phi$.
If we assume the symmetry condition for $\phi$, we can
assume the deformation independent of parabolic movements.
If $F(\gamma)$ is of type (A), which is the case where $\gamma=\gamma_{1}$, we can chose
$F(\gamma)=P(D) \tilde{f}_{0}$, where $\tilde{f}_{0}$ is very regular and the compatibility condition
according to Kiselman (\cite{Kiselman65}) means that we have existence of $Q_{j} \bot P$. Assume $\phi_{j}$ corresponds to
$\gamma_{2}^{j}$, such that in this case $\phi_{j} \rightarrow 0$, when $j \rightarrow \infty$.
Note the difference between the zero space and constant surface, $\phi_{1}+\phi_{2}=const$.
If we describe the first surfaces as $V_{1} \cup V_{2}$ where $V_{j}$ is connected,
this corresponds to a multiply connected boundary, where every $\phi_{j}$
is assumed such that the transversal is locally represented as a polynomial
(at least analytic).
Assume $\mu$ a reduced measure of bounded variation and $\mu_{2}=P_{1} \ldots P_{N} \mu$,
we have in this case not approximations on the form of orbits. In this case, even if we
for monotropic functions do not have have presence of a normal covering in the sense of Weyl,
we do not have problems with orbits, given that we start with a reduced measure in the
representation of the boundary.
Note the moment problem, if we assume $\alpha$ reduced, we then have $\alpha(g)=0$
implies $g=0$ on the domain $\Omega$, that is given $\alpha \in \mathcal{D}_{L^{1}}'$ is
$\Omega$ a strict carrier to $\alpha$. Since $g \equiv 0$ on $\Omega$
or $\alpha(\Omega)=0$, if $g$ is a polynomial and if $\alpha$ reduced, we must $\Omega$ is trivial.
Finally, if the condition is $\phi \in L^{1}$ and we consider $\psi \in H(V)$ such that
$\psi \sim_{m} \phi$, we do not longer have a normal covering,
but according to the moment problem, we can uniformly approximate $C \cap L^{1}$ with
$H(V)$, over a strict carrier to the measure $\alpha$. If we only assume the measure
$\alpha$ of bounded variation, but instead we chose $g$ as polynomial, we can assume '
$\Omega$ has $\alpha-$ measure zero.
Given a measure of bounded variation and positive definite $d \mu$, such that
$\tilde{d \mu}=d \mu + d \mu_{0}$, where $d \mu_{0}$ is assumed with point support and
$\tilde{d \mu}$ holomorphic, we have a continuation according to Cousin (\cite{Dahn13}, compare also \cite{Garding64}).
When $\tilde{ d \mu}$ locally reduced, we have $\int_{\gamma_{T} - \gamma_{0}} \tilde{d \mu}=0$
implies $\gamma_{T}=\gamma_{0}$.
When $\tilde{d \mu}$ is not reduced, there is the centre case among the possible approximations,
that is orbits around a singular point.
Given $g$ a regular approximation and
$\tilde{ d \mu}$ with support on first surface (the boundary), assume $\int g \tilde{d \mu}=0$
under the approximations. Given $\tilde{d \mu}$ reduced, we can assume $g=0$ outside a compact set.
It is clear that $L$ (transversals) are included in the support for $g$, in this manner we can regard $L$ as a strict carrier for limits.
The proposition is that given the transversal as a strict carrier for limits, the transversal can
be represented through polynomial locally.
The boundary is assumed very regular in the sense Parreau (\cite{Parreau51}). We assume all
singularities on first surfaces and that all first surfaces can be reached.
In the context of the moment problem, we can allow monotropical functions,
that is $g$ can be continued to continuous functions.
Assume $M=\tilde{d \mu}/d z$ and $\frac{d G}{d z}=g$, we then have if $g$ regular that $dG$
is a closed form.
Assume $\Sigma=\{ f(\zeta)=const \}$ and $J U = V J$ according to the invariance principle
and $S=\{ F(\gamma_{1})=const \}$ and $\tilde{S}=\{ F(\gamma_{1} \rightarrow \gamma_{2})=const \}$.
If $\varphi$ is a continuous function in a neighbourhood of $(x^{0},y^{0})$ and $\sigma$
a characteristic surface through this point, we can then write $\sigma \subset \{ \varphi(x,y) >0 \} \backslash (0,0)$
(pseudo convexity) and $\sigma=\{ x_{n}=P(x_{1},\ldots,x_{n-1}) \}$ (strict pseudo convexity (\cite{Oka60})).
We define $U \gamma_{1}=V \gamma_{2}$ in a neighbourhood of $\Delta=\{ U=V=I \}$ and assume
$F(U \gamma_{1}) \rightarrow const$. For all singularities that can be reached in this manner,
given $U$ is a fixed movement, we can chose $\gamma_{1} \rightarrow \gamma_{2}$ as analytic
(polynomial) .
Note that, if $\Delta$ is a plane $x_{j}=0$, we have that $e^{v}$ is constant in some direction,
why the function is not in $L^{1}$ on an unbounded domain.
Given $0 = \int_{V} g d \mu$ we have that $g \equiv 0$ on $V$ or $\mu(V)=0$ (measure zero).
Given $g$ polynomial (exponential of a polynomial) we have from a result by Hurwitz (\cite{Dahn13})
that the measure for all singularities is zero. In the transposed case $0 = \int M d g$
implies $M=0$ or $g(V)=0$. If for instance $M$ is locally 1-1, we have that $x=x_{0}$.
\section{The moment problem}
Note the moment problem, assume $E_{0}$ a set in $C$, the problem can be solved if we existence of $\alpha(x)$ of bounded variation, such that $\int g(x) d \alpha(x)=0$
for $g \in E_{0}$, we have $\int g(x) d \alpha(x)=0$ for $g \in C$ (\cite{Riesz56}). Note that
when the orthogonal (transversal) is algebraic, this corresponds to a measure zero complement .
Concerning mixed problems, assume $(Au,v)=(u,A^{*}v)$, such that $A^{*}v=0$, that is
$v \bot Au=f$. Given that $Bu=v$, we have thus $A \bot B$ or $A^{*}B \bot I$.
Note $R(B)=D(A)^{\bot}$ and $R(A)=D(B)^{\bot}$. If
$R(A)^{\bot}=R(B)$ we have that $B$ has closed range, even when $A$ does not.
If $B$ is algebraic in $\mathbf{R}^{n}$, we have that given $v$, we have existence of $u$ such that
$Bu=v$. The domain for $\bot$ is then $u$, such that $u \in D(A) \cap D(B)$ that is closed
when $D(A)$ is closed. Note if $A$ is a polynomial, there are complex polynomials $B^{*}$,
such that the symbol to $B^{*}A \equiv 0$ (\cite{Kiselman65}). If we consider $B^{*}$ as annihilator, $(B^{*}A u,u)=0$
for all $u \in D(A)$. When $A$ is considered in $X'$, we have $B^{*}A = 0$, that is $B^{*}$ is an
annihilator for $A$.
Compare with the moment problem, where a sufficient condition, given finite order singularities, for existence of $B^{*}$ is an analytic
representation of $A$ (Parseval).
Consider now $A$ hyperbolic and $B^{N}$ hypoelliptic, it is sufficient to consider $u$ with $A^{N}u=f_{N}$ and $B^{N} v=0$ for $v \bot f$.
Note that if $E_{N}$ is a parametrix to $B^{N}$, it has (modulo $C^{\infty}$) a trivial kernel, that is
if $v'=E_{N}v$, $f_{N} \bot v$, we have $v'=0$.
\newtheorem*{res14}{The mixed model}
\begin{res14}
Assume $\gamma \rightarrow \gamma_{\delta}$ is a continuation of $\gamma$ and $F_{\delta}$ is constructed,
so that $F_{\delta}(\gamma_{\delta})=f$ and respects $f \bot v$, where $v$ is continuous. Assume ${}^t F_{\delta}$ has a trivial
kernel, then the continuation gives a global mixed model.
\end{res14}
Note that $\delta_{0} - C^{\infty}$ maps $\mathcal{D}' \rightarrow \mathcal{D}^{' F}$,
that is if the boundary is defined modulo regularizing action instead of modulo $H$,
we do not longer have that $r_{T}$ is injective in $H$, however it is injective in $L^{1}$.
We will use the moment problem (and when possible the transmission problem (\cite{Schechter60})) to solve the mixed problem. In the two mirror model
we assume that a hyperbolic operator $A$, is reflected through the boundary into an operator $A^{*}$
(geometrical optics) and in the same manner for $B$ partially hypoelliptic. Existence of a continuous
mapping between the respective boundary points, implies an abstract transmission property.
Note that only one of the systems has to be invertible.
Assume the condition $B \gamma_{1}=0$ on $\Sigma_{i}$ corresponds to $\Delta(f)$ (lineality), then there is a $B'$ such that
$B \gamma_{1}=0$ iff $B' \gamma_{2}=0$ on $\Sigma_{i}$. Thus, if $\gamma_{2}$ is seen as a
continuation of $\gamma_{1}$ according to $F(\gamma_{1} \rightarrow \gamma_{2})$
with $\gamma_{1} \bot \gamma_{2}$, we have a ``global model''. We have $< F(\gamma_{1}), v>=0$ and
${}^{t} F(v) \bot \gamma_{1}$ and ${}^{t} F(v)=0$ implies $v \bot F(\gamma_{1})$,
where $F$ is a parametrix (localization) to $\gamma_{1} \rightarrow \gamma_{2}$. Further, $< F(\gamma_{2}),v>=0$
and $v \in \mbox{ ker } {}^{t} F=\{ 0 \}$, which implies $v=0$. Assume $J_{\delta} : \gamma_{1} \rightarrow \gamma_{2}$,
when $\delta \rightarrow 0$ and $J_{\delta} \gamma_{1} \bot \gamma_{1}$, further that $({}^{t} J_{\delta} F) \rightarrow E$
with $\mbox{ ker }E=\{ 0 \}$. Define $F_{\delta} \sim ({}^{t} J_{\delta} F)$ with
$< \gamma_{1}, {}^{t} F_{\delta} v > \rightarrow 0$, as $\delta \rightarrow 0$. According to the above,
this implies $v=0$, which implies $F_{\delta}=id$. Thus, if $F_{\delta}(\gamma_{1})=f$ with $F_{\delta}=id$,
then this implies that $f$ hyperbolic and when $F_{\delta}(\gamma_{2})=f$ with $F_{\delta}=id$, then this
implies that $f$ hypoelliptic.
Generally, for a radical geometric ideal, it is sufficient to give the boundary condition on derivatives to the symbol.
Consider otherwise the problem for $M(f)$ (arithmetic mean).
The boundary condition, when it is radical, can be given for the derivative, but does not necessarily define
the domain. The radical boundary condition, does not imply a radical ideal.
Consider $J_{\delta} : \gamma_{1} \rightarrow \gamma_{2}$, such that $\lim_{\delta \rightarrow 0} J_{\delta} \gamma_{1} \rightarrow $
a hypoelliptic symbol. Write $J_{\delta} \gamma_{1}=\gamma_{\delta}$, we then have $F(J_{\delta} \gamma_{1})=
\big[ F , E_{\delta} \big](\gamma_{2})$, where $E_{\delta} \sim e^{v_{\delta}}$, where $v_{\delta}=0$
on invariant sets. On the other hand, $F \gamma_{2} = 1$, as $v_{\delta}=0$ implies $\mbox{ ker }F = \{ 0 \}$.
Assume $F(\gamma) \rightarrow \gamma \rightarrow \zeta$. When $U \gamma$ is analytic, we have $\tilde{U} \zeta$
continuous. Note that $\{ \zeta \quad U \gamma_{1} = \gamma_{1} \}= \{ \zeta \quad U \gamma_{1}^{2}= \gamma_{1}^{2} \}$
and $\{ \zeta \quad V \gamma_{2}^{2} = \gamma_{2}^{2} \} \subset \{ \zeta \quad V \gamma_{2} = \gamma_{2} \}$. As $J_{\delta} U \gamma_{1}^{2} = V J_{\delta} \gamma_{1}^{2} = V \eta_{2}$
where $\eta_{2}=J_{\delta} \gamma_{1}^{2}$ and $\eta_{2} \in (I_{Phe})$.
Consider the continuation $S_{j} \rightarrow \tilde{S}_{j}$ simply connected.
Given Oka's condition for $\tilde{S}_{j}$ and the continuation of the symbol, the transversal can be defined
as analytic (algebraic) even for $\tilde{S}_{j}$. Concerning the set $\{ f < \lambda \}=V$,
if we have $S_{j} \subset \subset V$ and $S_{j}$ is a bounded set, we do not necessarily have the same
for $\tilde{S}_{j}$. The proposition for the continuation implies a downward bounded symbol.
According to Weyl (\cite{Weyl09}), we have for a normal covering, that closed curves corresponds to
closed curves. According to (\cite{Poincare87}) (Ch. XXVII), we have modulo monotropy, that closed curves
correspond to open curves with points in common with its closed correspondent.
We can use the moment problem, given $\int_{(\Gamma)} g d \alpha=0$ for a reduced
measure of bounded variation,
for $0 \neq g \in E_{0}$ implies the same relation for $g \in C$. Thus, in the plane we can use a
reduced measure (modulo removable sets) to solve the problem.
Assume $(I)=(\mbox{ ker }h)$ and $d h(f)=g(z)d z$, where $g \sim_{m} g_{1}$ analytic. Assume
$g-g_{1}$ algebraic and $g_{1}$ regular over closed contours. We can assume one-sided regularity for
$g$. For instance, $f(1/x)g(x)$
where $f$ is bounded close to the $\infty$ and $g$ is bounded close to $0$.
Assume a two-sided limit and $F_{+}(g_{+}) - F_{-}(g_{-}) \rightarrow 0$ at the boundary. Assume
$g_{-} \sim g_{+}^{*}$ and $F_{-}(g_{-}) \sim F_{+}^{*}(g_{+})$, we then have $F_{+} - F_{+}^{*}(g_{+}) \rightarrow 0$
and through Radon-Nikodym's theorem, we have $F_{+}(g_{+})=F_{-}(e^{v} g_{-})$, where $v \in L^{1}$.
The Schr\"odinger operators (\cite{Riesz46}) give a global model, that is not normal.
If we select $d \mu$ very regular, that is hypoelliptic in $L^{2}$,
the representation of the symbol is locally 1-1, why we have two-sided limits.
Compare with $\int g (d \mu - d \mu_{0})=0$, where $d \mu_{0}$ has
point support. Then $d \mu = v d z$ and $< g,d \mu>=<g,v>$ and $v=v_{1} + v_{0}$, we can assume
$v_{0} \sim \delta / g$. Note that $V^{\bot} \gamma \rightarrow x_{0}$ does not imply $V \gamma \rightarrow x_{0}$.
Consider $V^{\bot} \rightarrow x^{*}$ continuous and $V \rightarrow x$, further that $x_{0}$ is a
joint point. For instance, if $\widehat{f}(x^{*}) \rightarrow 1$, we have simultaneously
$f(x) \rightarrow \delta$. We can then have $x^{*} \rightarrow x_{0}$ without simultaneously $x \rightarrow x_{0}$
Concerning pseudo vectors, consider a neighbourhood of a point on $H_{m}$. An infinitesimal displacement
can be performed as parabolic, elliptic or hyperbolic. The normal is considered relative an axes
for invariance, that is we consider the normal as independent of the neighbourhood. Assume for $x,y$ real,
$\eta(x)=y(x)/x = e^{\phi(x)}$. We have then three possibilities, $\phi=0$ parabolic, $\phi<0$ hyperbolic and $\phi>0$ elliptic.
The three possibilities induce three possible orientations for the normal. In particular in the parabolic case,
if the normal is dependent on scaling parameter $h$, as $h \rightarrow 0$ or $h \rightarrow \infty$, we have
a `` scaling orientation'' for the normal. In the case where the transversal is a strict carrier or a carrier,
the limit is not dependent of choice of neighbourhood.
Form the normal in a point $\zeta_{0}$ at the boundary $\Gamma=\{ \zeta \quad F(\gamma)(\zeta) = const. \}$
and consider $\gamma(\zeta_{j} + h) - \gamma(\zeta_{j} - h)$, where $h$ is a scalar. Obviously, if
$\gamma$ is symmetric with respect to $\zeta_{0}$ over the segment, we have that $\gamma$ is absolute
continuous, as $h \rightarrow 0$. In this case the definition of the normal, does not depend on $h$.
Consider $G(h)=\int N(\widetilde{\zeta}) \gamma'(\widetilde{\zeta}) d \widetilde{\zeta}$,
where for instance $\widetilde{\zeta}=(\zeta_{1},\ldots,\zeta_{j}+h,\ldots,\zeta_{n})$. When $N$ is polynomial
over the segment $I_{h}$, we have when $\gamma$ is absolute continuous and $G(h) \equiv 0$, that $\gamma=const.$
over $I_{h}$. Further, $N \bot d \gamma$, independent of $h \rightarrow 0$. Consider now $N \in \mathcal{D}_{L^{1}}'$,
summable with respect to $d F$, of bounded variation, we then have that $G$ is absolute continuous with respect to $dF$,
that is $F(I_{h}) \rightarrow 0$, as $h \rightarrow 0$, implies $G(h) \rightarrow 0$. When $G \equiv 0$,
we have that $N \bot d F$. Thus, when $N \in L^{1}(dF)$, we have if $G$ absolute continuous with respect to $dF$,
that $N \bot dF$, as $h \rightarrow 0$. When $N$ is dependent on $h$, we may have $G \nrightarrow 0$, as $h \rightarrow 0$.
Further, when $N$ has a strict carrier, then $N$ is independent of $h$ for large $h$. This ``orientation''
of $N$ is represented using regularity conditions.
\bibliographystyle{amsplain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Die Bibanca S.p.A. (ehemals Banca di Sassari) ist eine italienische Universalbank, deren Tätigkeitsgebiet hauptsächlich in der Region Sardinien liegt. Das in Sassari ansässige Unternehmen war ab 1993 eine Tochtergesellschaft des Banco di Sardegna und ist damit seit dem Jahr 2000 Teil der BPER Banca Gruppe.
Geschichte
Das Bankinstitut wurde 1888 von 58 Unternehmern aus Sassari als Banca Cooperativa fra Commercianti Società Anonima gegründet. Zweck der genossenschaftlich organisierten Bank war es, seinen Mitgliedern Kredite zur Verfügung zu stellen und diese über das Spargeschäft zu finanzieren.
1925 schloss sich das Institut dem italienischen Volksbanken-Verband an und änderte seinen Namen in Banca Popolare Cooperativa Anonima di Sassari. Ab 1938 wurden die ersten Filialen eröffnet, zunächst in der Provinz Sassari und ab 1947, mit der Eröffnung einer Filiale in Cagliari, auch in anderen Teilen der Insel. 1948 wurde der Bankname auf Banca Popolare di Sassari gekürzt.
Nachdem das Bankinstitut in den Nachkriegsjahren zu einer über die Provinz Sassari aus etablierten Bank herangewachsen war, erlebte es in den 1980er Jahren mit einem stark steigenden Geschäftsvolumen und dem Ausbau des Filialnetzes auf 34 Standorte einen kräftigen Wachstumsschub. Die Expansion stellte sich jedoch als zu ambitiös heraus und brachte die Bank in der Wirtschaftskrise der frühen 1990er Jahre in eine finanzielle Schieflage. Das Bankinstitut wurde daraufhin ab Oktober 1991 rund 18 Monate lang kommissarisch verwaltet.
Im Anschluss an diese Phase wurden die Geschäftsaktivitäten im Juni 1993 unter der in Form einer Aktiengesellschaft neu gegründeten Banca di Sassari S.p.A. unter dem Dach des Banco di Sardegna weitergeführt. Unter der Leitung eines neuen Managements wurde die Banca di Sassari wieder auf eine stabile und rentable Basis gebracht.
Weblinks
Website der Bibanca S.p.A.
Einzelnachweise
Kreditinstitut (Italien)
Unternehmen (Sardinien)
Sassari
BPER Banca
Gegründet 1888
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,818
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Q: Mergext, Mergbanner features and setup I was all geared up to sign up for an app monetisation account with inner-active for livecode but have since found out that livecode don't use them anymore.
They simply sent me a link for MergExt and said "use that".
Can anyone tell me how it works? How do I register an account to get payment from the ads displayed? Does this all work automatically through my apple developer account or do I have to set up a specific account? Suddenly I'm a bit lost and I've been given no information which is frustrating.
I'd really appreciate some guidance from anyone who has already set theirs up. My app is for iOS so I'm open to any suggestions on inserting ads using livecode.
Thanks
Dave :)
A: The mergExt external mergBanner implements iAd banner and interstitial views so everything you need to know caan be gleaned from either the documentation and demo that comes with mergBanner or Apple's iAd documentation
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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