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The Open Dialogue FoundationNewsLyudmyla Kozlovska speaks at the UK House of Commons on the rule of law in Poland
Lyudmyla Kozlovska speaks at the UK House of Commons on the rule of law in Poland
Polski Русский Українська
On 22 November 2018, our President Lyudmyla Kozlovska had the honour to speak in the UK House of Commons, at a panel debate on the rule of law in Poland and Hungary organised by the Foreign Policy Centre. Below is a transcript of her opening statement.
First of all, thank you Foreign Policy Centre, and Adam in particular, for inviting me to speak in this esteemed panel, in such a prestigious place.
As a quick introduction, the Open Dialog Foundation is a non-political, non-governmental organisation funded in 2009, whose activities – until recently – focused on supporting democracy, human rights and the rule of law in the post-soviet region. Our aim is to protect oppressed activists, journalists, opposition politicians and other victims of political persecution.
Until July 2017, we have never formally spoken out on the internal situation in Poland. Quite the opposite – we used Poland as a role model and inspiration for its Eastern neighbours and other post-Soviet states struggling for a democratic transformation. The Polish people have shown them great solidarity, with my second homeland often being a safe haven for political refugees.
Sadly, it is not the case anymore – instead of being an inspiration and model to follow, since Law and Justice took power, Poland has been adopting post-Soviet standards of state governance and public life.
As you are well aware, and as was mentioned by prof. Matczak, the ruling party, Law & Justice, started dismantling the constitutional order by quickly seizing control of institutions that should always remain independent from the legislative and executive power. But that was just the beginning.
As a direct victim of Poland's rule of law abuse I'd like to focus on how Kaczyński's regime has impacted activists such as myself. Because the escalating assault on the rule of law in Poland is not an abstract issue. It has a very direct, human dimension. According to the latest reports including a periodic summary prepared by Obywatele RP – the Citizens of the Republic of Poland civic movement – there are more than 2,000 ongoing proceedings against peaceful protesters defending the constitution, fighting for women's rights, or standing up to various radical right-wing and fascist groups in Poland.
The current harassment and repressions of activists have no precedent since the fall of communism in 1989. Over the last months there have been cases of surveillance, seizure of documents or mobile phones and computers taken from NGOs' offices. A fellow activist, Justyna Butrymowicz, recently had her hand broken by police officers. Unjustified, excessive use of force advances.
My husband and I were among the thousands of people in Poland protesting in defence of the rule of law in July 2017. My husband's post on possible civil disobedience actions, published at the time on his personal Facebook account, triggered a wave of repression against our organisation, which quickly evolved into hate campaigns from pro-government media and online trolls. What is interesting, up to that point our NGO never had any issues with the authorities, quite the opposite – following an in-depth control by security services we were granted a special permit to export bulletproof vests and helmets to protect protesters at the Revolution of Dignity/Euromaidan in Ukraine. Law and Justice MPs themselves took part in monitoring missions and events organised by our Foundation.
But following my husband's post, critical of the government, key ministers of the Polish government publicly discussed measures that should be taken against us and other civil society organisations critical of the state authorities. Right-wing commentators and even government officials openly suggested deporting me as a possible measure.
We have been accused of intending to provoke a bloody revolution (so-called Maidan) and planning to overthrow the government in order to achieve the interests of foreign states and actors, among them George Soros, the Russians, the Germans or the so-called "Brussels elites". State-owned and other pro-government media outlets have been producing new fake news and conspiracy theories about us on a daily basis, very often simply reproducing disinformation purposefully created by the post-Soviet regimes we oppose.
Two consecutive Ministers of Foreign Affairs attempted to censor us and take over control of the Foundation by appointing their commissioners, but the court rejected their pleas, citing the constitutionally guaranteed freedom of speech. Recently, an analogical ruling was made in the case of a prodemocratic opposition movement – Obywatele RP in their dispute with the Minister of Interior.
Despite the allegations in the media and statements of the government officials, no charges have been brought against me, my husband, nor the foundation itself.
Just a few days ago we were informed that a special fiscal and customs control, initiated on the request of the Minister of Foreign Affairs one year ago and extended multiple times, produced no proof of any particular wrongdoing. The authorities, again, haven't found any illegal activities. They have therefore concluded that our Foundation has to pay a minimum of 700.000 Polish zlotys of unpaid income tax for our numerous human rights and humanitarian aid activities which according to Polish law, as confirmed by our accountants and tax advisors, are tax-exempt. This may set a precedent for a new, highly effective way of persecuting NGOs which the government finds inconvenient.
The final blow against the Foundation – or at least as it was planned – came when I found myself entered into the Schengen Information System with the highest alert level, reserved for terrorism suspects, spies or serious criminals. I was expelled from the EU, with no prior warning and – as leading legal experts have confirmed – in violation of EU law. After 10 years of living in Poland I was separated from my husband, a Polish citizen, and forced to stay for weeks in exile in Kyiv, having to move between random rented rooms and apartments every few days, as I was under constant surveillance and threat from Kazakhstani or Moldovan security services.
Just recently, then-foreign minister Waszczykowski openly admitted, that all the repressions, including my ban, were initiated as a precaution, out of the government's fear that our Foundation might conduct anti-state activities. He basically admitted to violating the possible reasons for an SIS alert, as laid out in EU law.
Officially, I remain a threat to national security in Poland and should be treated as such by all Schengen member states.
Only thanks to the good will of other EU Member States, starting with Germany, then Belgium, France, the UK and others, all of which have disregarded the ban as groundless, I am able now to stay in Brussels and visit you here today.
I would also like to underline at this point the exceptional role that parliaments and cooperation between their members have played here. It is thanks to the invitations of MEPs like Guy Verhofstadt, Rebecca Harms, Julie Ward and Michał Boni – each from a different political group, that I was able to return to Brussels and speak in the EP. Thanks to national MPs like Frank Schwabe and many who joined him later I was able to speak at the Bundestag and in the Council of Europe. I am sad to say the Polish parliament, which used to play a similar role for those persecuted by post-Soviet regimes, has lost this function since Law and Justice came to power. It used to be a beacon of freedom and democracy for the East, but now the roles have reversed, with Kremlin standards and methods being introduced in Poland.
To conclude – Polish president Andrzej Duda, in his recent speech, said that "the EU is an imaginary community which brings little benefits", which, sadly, closely resembles the rhetoric and position of the ruler of the Kremlin. Justice Minister and Prosecutor General in one, Zbigniew Ziobro – who just this Tuesday refused to confront commissioner Frans Timmermans in the European Parliament – has questioned the compatibility of EU law with the Polish constitution, striking fear of a "Polexit". Luckily, no-one in Poland wants to go through the Brexit experience, so this blunder helped convince Poles – 85% of whom are pro-EU – to vote against PiS in the recent local elections.
We believe that the Polish government must not be allowed to disintegrate the EU from within, taking advantage of the SIS to persecute its critics, abusing the trust between member states as well as European laws and values it vowed to uphold.
The populist path towards authoritarianism must be stopped. For now it's the most involved, active citizens and residents who are the first victims, but anyone can be next.
Kozlovska , Poland , Open Dialogue , the rule of law in Poland , #BringHerBack , Foreign Policy Centre , UK House of Commons


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July 2014 podcast: ProPublica reporting on seclusion and restraints results in playbook for reporters interested in pursuing similar story
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https://splc.org/wp-content/uploads/2014/07/187_july14podcasto-2.mp3
Subscribe to SPLC podcasts on iTunes07/17/14 12:00am
ProPublica reporter Heather Vogell talks with Student Press Law Center Executive Director Frank LoMonte about her recent reporting into seclusion and restraint practices at schools across the country.
Frank LoMonte: Welcome to the Student Press Law Center podcast, the monthly rundown of news and developments affecting the rights of student journalists around the country. I'm Frank LoMonte, executive director of the Student Press Law Center and from to time we like to focus the podcast on especially interesting examples of investigative journalism using public records that might be inspiring to students around the country interested in following the example.
We have an especially exemplary one to talk about this month. It's Heather Vogell's work for ProPubica — the nonprofit, investigative news site — that took a deep dive into the world of the restraint of unruly students in America's public schools. Hundreds of thousands of time every year, kids are being confined in locked rooms, pinned down in wrestingly holds, hog tied like calves in a rodeo, or otherwise physically restrained for acting out at school. This is a practice that would be illegal if done in a federally regulated mental institution but not at a public school, as Heather and her reporting team at ProPublica discovered.
Heather Vogell is well known to people in the education journalism field for her groundbreaking work on standardized test cheating while a reporter at the Atlanta Journal-Constitution. That reporting exposed widespread wrongdoing that led to criminal indictments against dozens of people in the Atlanta schools, including the former school superintendent, so we're especially delighted to have Heather Vogell of ProPublica here to talk about her reporting on this subject. Thanks for being with us.
Heather Vogell: Hi. I'm happy to be here.
Frank LoMonte: Well, Heather, I guess if you could just sort of dive in with the origin of this story — how did you become aware that there was a larger problem that needed scrutiny, of students being restrained in public schools and how did you go about taking on that project?
Heather Vogell: I became aware of the issue through a phone call from somebody that I'd been in touch with for awhile, who was an advocate for parents who had children that were special education, special needs students. I got a call from her one afternoon, right as I was transitioning to take the job at ProPublica.org, and she told me about a mom she'd been in touch with who had a kindergartener who was being restrained repeatedly in the classroom. And I think even at one point the mom had seen the kid being restrained and not being allowed to get up and get a drink of water, something like that. And so I hadn't been aware of that happening in classrooms and had also coincidently gotten an email recently about the federal legislation that was being sort of advocated for on the federal level that there had been an investigative report out recently. So, both those things came around the same time and peaked my interested and I started looking into it and found out there was a lot to write about on a national level. There had been a lot of local stories when incidents happened here or there, but nobody had really put it together recently and written a big overall national story or looked at the most recent data.
Frank LoMonte: Well, you mention that the data, and of course that's the backbone of any investigative project like this, so can you talk about how do you went about getting a hold of what had to be a massive number of public records, and maybe some of the obstacles that you encountered on what is, undoubtedly, a sensitive subject, that a lot of schools would probably prefer not to talk about.
Heather Vogell: Yeah sure, and the data was definitely a key part of this. In reading that report to the Senate about the restraints and seclusion issues in public schools, they had mentioned that there was data being collected by the Office of Civil Rights and the U.S. Department of Education, and that that data had shown, I think in the 2009-2010 school year, that there was 66,000 uses of restraints and seclusions. And it struck me because this report had come out in February, and this was 2009-2010 data, and so I just was curious whether there was going to be an update with more recent data coming out soon, so I called the Department of Education and found out that indeed there was some new data coming out in the spring. That was terrific and I was excited about that, but, when it came out, the data had a lot of problems with it. It was very dirty data, meaning that it wasn't something you could take the whole database at face value because there had been errors in inputting it by districts, as well as the fact that it was still clearly incomplete. Do you want me to talk a little bit more about that and how we worked through that?
Frank LoMonte: Definitely, because I think that's something that a lot of people fall into, is that they get a hold of the data and they think they're reporting is over and sometimes it's really just beginning.
Heather Vogell: Yeah. I work with data a lot in my stories and I increasingly see the data not as sort of being an end point, it's really just a beginning point, a tip sheet if you will. You've got to really work hard to validate what you have, one, and two, understand it in context, which can mean that it can be revealing in some ways but data can also hide some nuances in what's happening. In this case, we weren't getting a database that showed, faithfully, every single incidence of restraint and seclusion in the country. What it did show, was that there were a lot more instances of this happening than had ever been previously documented. It turned out that that early sample that I had looked at, that was in the report from February, was just from a sample of districts that were over 3,000 kids, and then were were some states that reported all their districts. The newer dataset, which is the one that I went after for this story, was the first time that the federal government had collected this information from every school district in the country. And despite that, there was acknowledgement that a lot of districts were still not collecting this data, including, potentially, the three largest districts in the country, which all reported zeroes. They may have policies that don't permit restraint as often, or in most cases even, as other districts, but pretty much everywhere that I've seen that has been faithfully reporting, even a state like Connecticut that has a rule that says you can only use restraints in emergencies, they still report thousands of uses because you have fights, you have real things that happen. So, if the data was real from these districts, you'd expect to see at least a handful of situations with a distict that big.
So, first of all, we knew that we were dealing with incomplete data. And second of all, I started to see things in the data that set off alarm bells. Like, a district that reported that it had equals numbers of every category. For instance, it would have 11 restraints, 11 seclusions, 11 restraints for kids with disabilities, 11 restraints for kids without disabilities, it was like every category across had the same number, so clearly the statistical possibility of that happening was so low that we basically went through and excluded those.
Frank LoMonte: I'm sure this is starting to remind you of seeing people cheating on standardized tests. Right? I just bubbled in "B" for everything.
Heather Vogell: Right. It would be sort of the same thing. You sort of know it's probably not valid. It's possible, but unlikely. And so basically what we ended up doing to sort that out as best as we could and be as faithful as we could to the idea of wanting to present a real number, we went back to the people who had collected the data at the department of education and just had a conversation with them about what they thought the limitations were in the data. And so they shared a few tips for sorting through it, to try to catch some of the errors in data input that were in there. So we ended up excluding a good number of incidents that we thought were probably largely over-reports. Although, we probably ended up throwing out some instances that were real in the process but we knew that a lot of them weren't real. So, that process took a few weeks just to get through that and I worked with our data editor here, Jeff Larson, to do that.
Frank LoMonte: So when you got down to actually having to look into individual cases at the institution level, and there's some very powerful ones that are mentioned in your story, I have to imagine that you ran into objections about student privacy and student confidentiality. Can you talk about ways to deal with that?
Heather Vogell: Yeah, sure. You hit the nail right on the head. That is always an issue with reporting on schools. The confidentiality and privacy issue, this sort of formal issue that there's a federal protection under FERPA is the acronym for it, that requires schools to keep students' personal data, academic information, private, and there's also sort of, I think, a moral and ethical component to it for a reporter, in that I don't believe in thrusting a student into the limelight without their being aware of it and in complete agreement of it happening. I mean, when you have a situation with an arrest or something and the name becomes public in another way, that's different, but you have to tread very carefully when dealing with vulnerable populations, kids in this case, and a lot of them had disabilities.
So, what we needed to do was find a situation where everybody was very comfortable with sharing the story. It turned out that original tip that I got, that that family was not comfortable with it. They were pursuing a legal avenue and their attorney wanted them to be patient and not work with the press at this point and so my lead anecdote disappeared, just a couple weeks into the reporting. So I had to figure out, "well, I still think this is really interesting and it seems to be happening a lot, so I've got to find somebody else." The way I did that was to call around to some advocates for kids with disabilities. And it turns out that in every state there is something called a protection and advocacy agency and they're all federally mandated groups that protect the rights, they're there to advocate for disabled people and help them stand up for their rights. And a lot of the complaints about what's happening in schools were funneling to these groups from local attorneys in different states. So, calling around to some of these guys, talking with the National Disability Rights Network, which is sort of the head of all the different protection and advocacy agencies, and getting different names and numbers, sometimes of advocates of attorneys, brought me to a few others families and frankly that happened fairly quickly. I was surprised at how easy it was to find examples that were really heartbreaking and representative of what seemed to be happening out there. And I ended up focusing on two in my story but there were many, many others that were out there, across the country.
Frank LoMonte: So you were able to gather some pretty eye popping, aggregate data, showing that there are literally, we're talking about the magnitude of 200,000 times a year, somewhere across the country a child is being either locked away in a room or put in a restraining hold, or sometimes even literally put in cuffs or something like cuffs, and I guess the question is, were you able to, we know that there are some subset of these where as you mentioned some subsection, like breaking up a fight, where it's necessary to physically grab one of the combatants and immobilize them for a second to keep somebody from getting more badly hurt, so some subset of those probably falls into that legitimate category, were you able to draw any conclusions about whether there is a systematic overuse of restraint when it's not one of these imminent, physical danger situations?
Heather Vogell: I think it's impossible for us with the data that's out there to draw a definite conclusion like that because the data is so spotty. We don't know how much under — we know there's underreporting going on, we don't know how much there is. There's some schools that are reporting very faithfully and they appear to have very high numbers of restraints and seclusions and I think the advocates feel there are too many, that it's a sign that their treatment plans, their work with disabled students, is not, typically with disabled students it's not effective.
But I think that overall there is a consensus that restraints and seclusions — seclusions perhaps never, there's a lot of people who say that there's never a reason for those — restraints, the advocates, and a lot of of the experts say, they have no therapeutic benefit for children. There's a great risk of harm to the children and also to staff members who are trying to implement them. This is a dangerous thing to do. You're putting your hands on a child who's out of control. There's a consensus that they should be used as little as possible and what we're seeing is that there are places that are turning to them very often and so that was sort of the most of a conclusion that we could come up with, that these are things that are supposed to be used rarely, if ever, but there are places that are using them a lot. And we don't have a very clear, there's not a very clear picture of everywhere. We don't have a complete understanding of what's happening out there and so there are sort of injuries and situations that are popping up here and there that maybe we wouldn't have been aware of, that these places have a problem.
I look at Connecticut, they are connecting data perhaps in a more diligent way than even the federal government has been on the use of restraints and seclusions and they have a very, as I said a law that only allows it during emergencies, and what they're doing is they're using — and I think Florida does this too — they're using that data to look for districts that stand out among other districts, as outliers, as having too many. And they're going and saying, "Hey, what's going on here? Why are you using this tool so much, is there something that we can do to help you figure out a better way — perhaps more crisis intervention training, more training in de-escalation techniques for staff or better planning with your special needs students who are at risk for these sorts of outbursts?" There's real concrete things that they can do when they see the problem through the data. The problem is that nationwide, A, there's not one standard nationwide and 2, the data isn't good, so we have sort of a murky picture right now and we're forced to rely on anecdotes more than I think would be really prudent for good policy, if that makes sense. It's a really long answer to a simple question.
Frank LoMonte: No, no, no. And you mention that patchiness of the record keeping and one of the really interesting pieces on the ProPublica.org site — which I recommend that everybody check out — has a graphic looking at how much or how little states regulate the use of seclusion and restraint and you've actually got a map that people can check out so as to compare the state of regulation in their own state and you've got 15 states and the District of Columbia on this map that do not have any state-level regulation whatsoever, so this is really being left to the discretion of districts or of individual people at the building level. So that is one way certainly that a reporter who is interested in this subject might be able to go forward and localize it, by comparing the state of their state to others around it. What other suggestions or ideas might you have for somebody who's interested in doing a project of this kind in their own community?
Heather Vogell: I'll add that, it's a really good point, our map is a really good place to start in terms of getting a sense of where your state is on regulations. The thing that I would look at to compare it to would be the U.S. Department of Education has come up with voluntary guidelines for school districts on policies on restraint and seclusion and we used some of those elements to create this map but if you wanted more details you can look at your state's law and compare it to what the federal government says is best practices. And I think a lot of states, or most states, are still pretty far from that ideal. So, that's a good place to look at the policies. Then you have to add the district policies on top of that. Each of these districts are either writing policies or not having a policy, either of which could be very interesting. That's a good place to start. I always say don't assume that policy is practice. What you're going to need to do is find a way to get at practice, which is the hard thing. And that's when you start talking to parents of special needs students and advocates. Looking for those complaints or concerns, teachers, people who are in the building who can talk about what's happening. A big piece of this that I wrote a little bit about but that I think really deserves more attention, is that districts are supposed to have these fairly sophisticated approaches to handling behavior that's challenging. They're supposed to be using something called Positive Behavior Interventions and Supports, which is sort of a whole system of decisions you can make. That is supposed to reduce the chances of having these sorts of outbursts, basically to help teach students who are having a hard time with their behavior, help them learn how to control their behaviour better. It's a very important foundation for them as they get older and if it's successful it can really help them live independently and get something out of school sometimes in cases where they may not have without that help. And I think I read that only one in five districts nationwide is using it. I'm not 100 percent sure about that statistic, but it was fairly low. So look at how your school district is handling crisis intervention, I guess behavior management, challenging behavior, seeing whether they're doing positive behavior, what kind of training they offer — is it in restraints, is it in these other areas? Those are all good places to look.
Frank LoMonte: Just by way of wrap-up in a minute or two, any sort of closing thoughts or tips. Not about this subject in particular but for a student journalist who is out there and is trying to embark on a really data-heavy project that might involve quite a consumption of time, quite a consumption of manpower. You've been through several of these projects now in the last few years — any advice for somebody that's trying to dive in for the first time?
Heather Vogell: I think that the thing you want to do, again, understand the context of the data that you're getting and analyzing, and that means probably doing your best to get to the source, to talk to the people that, usually government, work closely with the data that are either present or help with the data collection, understand the data collection really well, they know where the holes are in data collection and what it might be missing, what it might be over-reporting, how it might be misleading, and they can also help you figure out how you can use the data to help look for interesting things, anomalies, maybe haven't come out before. That's a good way to start.
Frank LoMonte: I would add to that, as well, that sometimes the absence of the data in itself is the story. If you have something that's sufficiently important, especially something that involves people's physicals safety, and government agencies haven't bothered to count it or keep track of it, that's a very valid question for journalists to be asking. If we don't know the number, if we don't know the frequency, then the question is why don't we know that, and that question in itself can be a story that can lead to real change.
I want to thank Heather Vogell of ProPublica for being with us and sharing some tips about investigative reporting using data. ProPublica.org is the website and that's one of many terrific investigations on this nonprofit newsite that we commend to your attention. So, are there other resources that you could recommend that journalists use, things that ProPublica offers, that might help them get started on an investigative project of this kind?
Heather Vogell: Absolutely. In this issue in particular, what we have done is we have provided a whole bunch of different tools for reporters online to get started, including, what we've been doing is collecting anecdotes from around the country, examples of this, restraint and seclusion, from parents, sometimes educators, and we are matching them up with actual reporters who are interested in learning about their stories and using these anecdotes. We also have the data to share that we've cleaned and we've got, of course, the information about the state by state regulatory information and also a tip sheet of, sort of a reporting recipe, for how to approach reporting on restraints and seclusions. And all these can be found on ProPublica.org.
Frank LoMonte: I'll make sure to put links to some of those resources on the SPLC's Twitter feed, which is just @SPLC. We try to use our social media to share ideas and tips for students interested in doing investigative reporting in their own communities. Heather, thanks for being with us and for everyone who's listening who is involved in a public-records intensive project, Student Press Law Center is there to help. We've got free legal help for you. splc@splc.org is the email, the 703-807-1904 is the phone number and our attorney specializes in helping people get a hold of records from schools and colleges that are difficult to obtain. We'd love to hear from you, help you out, and we encourage you to use all of the free resources available on the SPLC.org website. Thanks for listening and we'll talk to you next month.

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Q: Problem with showing decrypted message in my chat list I would be grateful if someone could advise.
I have a chat with AES encryption. Encryption itself works and everything is OK, but the problem is that I do not understand how to show decrypted message in my Chat List
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private void loadMessages() {
        messagesList.clear();
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MessageModel.java
package com.appeteria.training.cryptcommunicator.chats;

public class MessageModel {
    private  String message;
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    private  String messageId;
    private  long messageTime;
    private  String messageType;

    public MessageModel() {
    }

    public MessageModel(String message, String messageFrom, String messageId, long messageTime, String messageType) {
        this.message = message;
        this.messageFrom = messageFrom;
        this.messageId = messageId;
        this.messageTime = messageTime;
        this.messageType = messageType;
    }

    ...etc...

redpajama_set_name: RedPajamaStackExchange


Serine/Threonine Kinases (STKs), Protein Kinase N (PKN) subfamily, catalytic (c) domain. STKs catalyze the transfer of the gamma-phosphoryl group from ATP to serine/threonine residues on protein substrates. The PKN subfamily is part of a larger superfamily that includes the catalytic domains of other protein STKs, protein tyrosine kinases, RIO kinases, aminoglycoside phosphotransferase, choline kinase, and phosphoinositide 3-kinase. PKN has a C-terminal catalytic domain that is highly homologous to PKCs. Its unique N-terminal regulatory region contains antiparallel coiled-coil (ACC) domains. In mammals, there are three PKN isoforms from different genes (designated PKN-alpha, beta, and gamma), which show different enzymatic properties, tissue distribution, and varied functions. PKN can be activated by the small GTPase Rho, and by fatty acids such as arachidonic and linoleic acids. It is involved in many biological processes including cytokeletal regulation, cell adhesion, vesicle transport, glucose transport, regulation of meiotic maturation and embryonic cell cycles, signaling to the nucleus, and tumorigenesis.


text:
vtk_module(vtkIOFFMPEG
  DEPENDS
   vtkIOMovie
  TEST_DEPENDS
   vtkTestingCore
   vtkImagingSources
  KIT
    vtkIO
  )

redpajama_set_name: RedPajamaGithub


text:
\section{\sc introduction}
In this article, we construct
variants of the Riemann zeta function
with convenient properties
and make
conjectures about their dynamics;
some of the conjectures are based on
an analogy with the
dynamical system of zeta.
More specifically, we study the family
of functions
$V_z: s \mapsto  \zeta(s) \exp (zs)$.
We
observe convergence of
$V_z$ fixed points
along nearly
logarithmic spirals with
initial points at zeta
fixed points and
centered upon Riemann zeros.
We can approximate these
spirals numerically,
so they
might afford a means to
study the geometry
of the relationship
of zeta fixed points
to Riemann zeros.
\newline \newline
When we
examined other zeta variants
we observed behavior
similar to  that of the $V_z$;  we
do not have  a clear idea
of the extent of the phenomenon.
\newline \newline
We formulated  our conjectures
after  computer experiments using
\it Mathematica. \rm Data and
\it Mathematica \rm notebooks
for this project are at
ResearchGate, here \cite{Br3}.
Several other writers
have considered the dynamics of
the Riemann zeta function,
 for example,
Kawahira \cite{Ka} and Woon \cite{W}.
\subsection{Definitions.}
In our experiments we found
evidence that
certain sequences
of complex numbers are interpolated
by nearly logarithmic spirals,
but at first without independent
information about the
underlying spirals.
In order to describe these
observations,  we  use
terminology  that initially avoids
referring to  these
 spirals.
\newline \newline
Given a sequence of complex numbers
$\vec{z}$
$= \{z_0, z_1, ....\}$
with limit $\gamma$, we define a variant
$\theta_{\vec{z}}$
of the argument  concept
 by making
the following choices, which are
always possible and
unique after fixing a branch of the
argument function: \newline \newline
(1) $\theta_{\vec{z}}(z_0) =
\arg(z_0-\gamma)$ \newline
(2) $\theta_{\vec{z}}(z_n)
\equiv \arg(z_n - \gamma)$ mod $2\pi$,
and
\newline
(3) for any non-negative integer $n$,
$\theta_{\vec{z}}(z_{n+1})$ is
the  smallest  real number
greater than
$\theta_{\vec{z}}(z_n)$
compatible with
conditions (1) and (2).
\newline \newline
Let $h$ and $K$ be positive integers.
For each positive integer $n$ let
$r(z_n) := |z_n-\gamma|$ and for
$h = 0, 1, 2, ...$,
let $m, b$ be real numbers
such that the straight line
$y= m x + b$ is the best linear  model
of the data
$(\theta_{\vec{z}}(z_n), \log r(z_n))$,
$n= h, h + 1, h + 2, ..., h+K$.
(For the sake of
definiteness,
we might insist that the model be
chosen by the method of least squares;
in practice, we have relied on
proprietary routines  of \it Mathematica\rm.)
\newline \newline
Now let
$$d_h(n) :=
\frac {|m  \theta_{\vec{z}}(z_n) + b - \log r(z_n)|}
{\log r(z_n)},$$
and let
$d_h := d_h(h+K)$.
Then we  say that the sequence $\vec{z}$
is nearly logarithmic with respect to
$\gamma$ if, for any choice of $K$,
$d_h  \to 0$ with exponential
decay as $h \to \infty$.
\newline \newline
Now we define  a nearly logarithmic spiral
in the complex plane
as a spiral $S$ with center $\gamma$
such that any sequence of points
$\{z_n\}_n$
on $S$
with $|z_n - \gamma|$
decreasing monotonically with $n$
is nearly logarithmic with
respect to $\gamma$.
\newline \newline
Next we say what it means for
$\vec{z}$
 to
be  nearly uniformly distributed
with respect to $\gamma$; namely,
if $\delta_n$ is the quantity
$|\theta_{\vec{z}}(z_n) - \theta_{\vec{z}}(z_{n+1})|$
and
$\Delta_n = |\delta_n - \delta_{n+1}|$,
then $\Delta_n$
decays exponentially with $n$.
\newline \newline
We  also adopt what is more or
less standard usage,
namely, that
an exactly logarithmic spiral with center
$\gamma$ is a curve  consisting
of points $z$ satisfying
$\log r(z) =  m \theta_{\gamma}(z) + b$
for  real constants $m$ and $b$ (with the
above conventions for $\theta_{\gamma}$,
except  that condition (3)
in the definition of
$\theta_{\vec{z}}$ is replaced
with the requirement that $\theta_{\gamma}$ be
continuous and monotonic
increasing as $z \to \gamma$.
An exactly logarithmic spiral
is (according to these definitions)
also a nearly logarithmic spiral.
In \cite{Br2},
we studied  the complex-valued
deviations of
certain nearly logarithmic
spirals  from  exactly logarithmic spirals
to which we had fitted them.  We  displayed
some plots of
these deviations in
Figure 6.1  of  \cite{Br2},
and we  make
a similar analysis in the discussion of
conjecture 3
below.
\newline \newline
\textsection \thinspace
We fix the following notation
for the remainder of the article:
\newline \newline
$\bf{C}^{\it cut}$ := $\{z \in \bf{C}$
s.t. $z$
does not lie on the negative real axis$\}$.
\newline
$\rho$ is a
nontrivial Riemann zero.
\newline
$\Phi_z$ is the set of fixed points of $V_z$.
\newline $\psi$
is an element of  $\Phi_0$
(fixed point of zeta).
\newline $\psi_{\rho}$ is
the  particular
 element of $\Phi_0$
 closest to $\rho$.
\newline $u$ is
a point on the
unit circle.
\newline
$\vec{R}_u$ is the ray
emanating from zero and passing through $u$,
so that $z \in \vec{R}_u$
if and only if $z =xu$ for some
$x \geq 0$.
\newline
$X=(x_0, x_1,..., )$
is an infinite increasing
arithmetic  progression with
$x_0 = 0$.
\newline
$\vec{\phi} = (\phi_0, \phi_1, ...)$
is a member of
$\Phi_{0} \times \Phi_{x_1 u}
\times \Phi_{x_2 u} \times ... := \vec{\Phi}_{X,u}$.
\newline
$V^g_z(s):= g(s)e^{zs}$,
$\Phi^g_z$ is the set of
fixed points of $V^g_z$,
and
$\Phi^g_{0} \times \Phi^g_{x_1 u}
\times \Phi^g_{x_2 u} \times ... := \vec{\Phi^g}_{X,u}$.
(Thus $V_z = V^{\zeta}_z$, etc.)
\subsection{A theorem on limits
of sequences from
\texorpdfstring{$\Phi_{X, u}$}
{TEXT}.}
Before we list the conjectures,
we prove a theorem.
\begin{theorem}
Suppose that
$\vec{\phi}$
in $\Phi^g_{X, u}$
converges to a complex
number $\lambda$,
$g$ is continuous
at $\lambda$,
and that
$\Re(\lambda) \cdot \Re(u)
-\Im(\lambda) \cdot \Im(u) > 0$.
Then $g(\lambda) = 0$.
\end{theorem}
\hskip -.2in
\it Proof  \rm
\thinspace
Let us write $\lambda = L + Mi$
and $u = P + Qi \thinspace (L, M, P, Q$ real.)
If $g(\lambda) \neq 0$, then
we can choose a subsequence
$\vec{\phi^*}$ of $\vec{\phi}$
and a positive number $B$
such that for, some
natural number $n(B)$,
$n > n(B) \Rightarrow
|g(\phi^*_n)| > B$.
Let us write the real
and imaginary parts of
$\phi^*_n$ as $a_n, b_n$, respectively,
and set
$D_n = a_n P - b_n Q, E_n = a_n Q + b_n P$.
From the hypotheses we know that
the $D_n \rightarrow LP - MQ > 0$,
and so there is a
number $C > 0$ and a number
$n(C)$ such that
$n > n(C) \Rightarrow D_n > C > 0
\Rightarrow e^{D_n} > e^C > 1$.
Let us choose a subsequence
$X^* = (x^*_0, x^*_1, ...)$ of
the arithmetic progression $X$
so that the $\phi^*_n$ are fixed points
of the functions $V^g_{x^*_n u}$.
Thus $\phi^*_n = V^g_{x^*_n u} (\phi^*_n)=
g(\phi^*_n) e^{x^*_n u \phi^*_n}$.
Now we have:
$$|e^{x^*_n u \phi^*_n}| =
|e^{x^*_n (P + Qi) (a_n + b_n i)}|
= |e^{x^*_n  D_n + i x^*_n E_n} | = e^{x^*_n D_n}.$$
Therefore,
$$|\phi^*_n| = |g(\phi^*_n)| \cdot
|e^{x^*_n u \phi^*_n}| = |g(\phi^*_n)|e^{x^*_n D_n}.$$
Now let $n > \max (n(B), n(C))$; it follows that
$|\phi^*_n| > B (e^C)^{x^*_n}$.
But $x^*_n \rightarrow \infty$,
hence,
$(e^C)^{x^*_n} \rightarrow \infty$.
Since the $\phi^*_n$ converge to the finite number
$\lambda$, this cannot be true.
$\square$ \newpage
\begin{corollary}
Suppose that
$\vec{\phi}$
in $\Phi_{X, u}$
converges to a complex
number $\lambda \neq 1$
and that \newline
$\Re(\lambda) \cdot \Re(u)
-\Im(\lambda) \cdot \Im(u) > 0$.
Then $\lambda$ is a Riemann zero.
\end{corollary}
\subsection{Conjectures.}
In all of the conjectures to follow,
we assume that $u \neq -1$ and that
(if $u \neq 1$)
$\Im \rho \cdot \Im u < 0$.
Our experiments indicate that
this restriction is necessary.
\begin{conjecture} \rm
\vskip .1in
(1) The  set of imaginary parts
$\{\Im \phi: \phi \in \Phi_{z} \}$
is unbounded
and nonempty.
\newline
(2) A unique $\psi_{\rho}$
exists for each $\rho$.
\end{conjecture}
\begin{conjecture} \rm
(1) Some  $\vec{\phi}$ in
$\vec{\Phi}_{X,u}$
converges
to each $\rho$.
(2) Any such $\vec{\phi}$
is nearly logarithmic
with respect to  $\rho$,
 and nearly uniformly distributed
with respect to $\rho$.
\end{conjecture}
\begin{conjecture} \rm
For each choice
of $\rho$,
there is a continuous function
$f^{cut}: \bf{C}^{\it cut}
\rightarrow \bf{C}$
such that the restriction of
$f^{cut}$ to $\vec{R}_u$
is a function $f$ with the
following properties:
\newline
(1)
$f: \vec{R}_u \rightarrow \bf{C}$
is continuous and one-to-one,
\newline
(2) $f(z) \in \Phi_{z}$
for each $z \in \vec{R}_u$,
\newline
(3)  $f(0) = \psi_{\rho}$,
\newline
(4)  $\lim_{z \to \infty}
f(z) =  \rho$,
\newline
(5) the image of $\vec{R}_u$
under $f$ is a nearly
logarithmic
spiral $S_{u,\rho}$ with center
$\rho$,
\newline
(6)  among the sequences
$\vec{\phi} =
(\phi_0, \phi_1, ...) \in
\vec{\Phi}_{X,u}$
converging to $\rho$,
all the points of one of them
(say, $\vec{\phi}_{u,\rho}$)
are interpolated by $S_{u,\rho}$;
the initial element of
$\vec{\phi}_{u,\rho}$
is
$\psi_{\rho}$.
\newline
(\it n.b. \rm Our notation suppresses the dependence of
$f$ on $u$ and $\rho$, and
the dependence of $f^{cut}$
on $\rho$.)
\end{conjecture}
\hskip -.2in
\textsection  \thinspace
The next conjecture is
motivated by the analogy mentioned in the
introduction and discussed later in the article.
\begin{conjecture} \rm
For each $X$ and $u$,
there is a function $G$ with the following
properties:\newline
(1) $G$ is meromorphic
and independent of $\rho$.
\newline
(2) The sequence  $\vec{\phi}_{u,\rho}$
from conjecture 3
satisfies
$G(\phi_n) =  \phi_{n-1}$
for  $n = 1, 2, ....$
\newline
(3)  Each $\rho$ is a repelling fixed point
of $G$,
\newline
(4) $G$ is many-to-one,
but for each $\rho$  there
is a function
$F_{\rho}$
such that

(i)  $F_{\rho}^{(-1)} = G$,

(ii) consequently (in view of clause 2)
for $n = 0, 1, ...$,
$$F_{\rho}(\phi_n) =  \phi_{n+1},$$

(iii)
 $\rho$ is
an attracting fixed point of
$F_{\rho}$,
and

(iv) $F_{\rho}$
carries $S_{u,\rho}$
into itself.
\newline
(\it n.b. \rm Our notation suppresses the
dependence of $G$,
and the dependence of
$F_{\rho}$ on $X$ and $u$.
\end{conjecture}
\begin{remark}
Obviously, it is
 not surprising
 that the $V_z$ might in some
way pick out
zeta zeros and fixed points
as special.
On the other hand,
our numerical methods make it
feasible
to estimate the
parameters incorporated in
the equations of
simple curves interpolating
$V_z$ fixed points.
This is possible because we are
able to find points on the
curves that are arbitrarily
close to each other, something
we were unable to do for
analogous curves we studied in \cite{Br2},
where to do so would have required
doing something else we do not know
how to do: extending
the iteration of zeta to non-integer heights.
(In the notation we will introduce below,
it would have required evaluating
expressions of the form
$\zeta^{\circ q}(s)$
for non-integer values of $q$.
There is some literature around the
problem of extending the
iteration operator, \it e.g.\rm,
\cite{N2, N1}, but we have not
reduced those
results to code. In the
situation of the present article,
we do not have to
confront this difficulty.)
In the present situation,
conjecturally,
these curves are nearly logarithmic
spirals with
a Riemann zero $\rho$
and the zeta fixed point
$\psi_{\rho}$
as center and initial point,
respectively. At present,
we are able to make
the estimates only for
one $\rho$
at a time. But
more study of these
estimates may be a way
to search for a
dictionary between
zeta zeros and zeta fixed points.
The Riemann hypothesis might
then be rephrased as a
claim about the fixed
points. In another direction,
the convergence properties
of sequences from the
$\Phi_z$
bear upon the
Riemann hypothesis
as well. (See question 1 below.)
\end{remark}
\begin{remark}
Conjecture 3
(in which $\psi = \psi_{\rho}$)
is
supported by some
experimental evidence
(see below). Based on the
analogy with the
situation of \cite{Br2}
promised in the
introduction and discussed in
the next section,
it seems plausible
that conjecture 3
might extend to
arbitrary $\psi$.
To find experimental evidence
for other $\psi$
(corresponding to
evidence for the
analogous claim in \cite{Br2}),
we  would first need to identify
the unknown
zeta-analogue $G$, because
the needed sequences in \cite{Br2}
corresponding to the
$\vec{\phi} \in \vec{\Phi}_{X,u}$
in  conjectures  3 and 4
were obtained by  solving equations
involving zeta.
Conjecture 4 is also
a claim about $\psi_{\rho}$,
since it involves the
curve $S_{u, \rho}$ and
clause (6) of conjecture 3
associates $\psi_{\rho}$
to $S_{u, \rho}$.
Therefore the question arises
of the extension
of conjecture 4 to
arbitrary $\psi$.
Conjecture 4 is
based entirely on the analogy
with the situation of \cite{Br2}.
Thus, the same requirements
for finding experimental
evidence for its extension
to arbitrary $\psi$ apply
to conjecture 4 itself.
\newline \newline
We write down the
contemplated extensions
of
conjectures 3 and 4
(labeled more tentatively there as
``proposals'')
in more detail in the
appendix.
\end{remark}
\vskip .05in
\hskip -.2in
\textsection  \thinspace
When iteration of
a function $f$ is meaningful,
we write
$f^{\circ 0}(s) = s$ and, for
$n$ a positive integer,
$f^{\circ n}(s) = f(f^{\circ (n -1)}(s))$.
\begin{remark}
For particular $\rho$,
by conjecture 1 we
may choose
$\psi_{\rho} \in \Phi_0$.
Then conjecture 4
 implies that
the  forward orbit of $\psi_{\rho}$
under $F_{\rho}$, namely
\newline
$(\psi_{\rho},  F_{\rho}(\psi_{\rho}),
F_{\rho}^{\circ 2}(\psi_{\rho}), ...)$,
is  a sequence
in  $\vec{\Phi}_{X,u}$
converging to  $\rho$.  Therefore,
clause (1) of
conjecture 2 follows from
conjecture 1 and conjecture 4.
\end{remark}
\begin{conjecture} \rm
Here, it will be
convenient to indicate the
dependence of $G$ upon
$X$ and $u$
by writing $G = G_{X, u}$.
Suppose that
$X'$ is a refinement of
$X$ in the sense that
the common difference
between consecutive elements of  $X$
and between  consecutive elements of
$X'$ are  $d, d'$ respectively
and  $d=Kd'$ for some
natural number $K$. Then
$G_{X,u} = G_{X',u}^{\circ K}$.
\end{conjecture}
\subsection{Conjectures 4, 5
and the promised analogy.}
In \cite{Br2}, we studied
the dynamical system of zeta
and several associated
nearly logarithmic
sequences
$\vec{\phi}_{\psi, \rho, \zeta} =
(z_0, z_1, ...)$
such that \newline \newline
(*) $z_0 = \rho$, \newline
(**) $z_{n-1} = \zeta(z_n)$ for all
$n > 0$,
and
\newline
(***)
$\lim \vec{\phi}_{\psi, \rho, \zeta} = \psi$.
\newline \newline
In that article, we
conjectured that such
a sequence existed
for each choice of the
pair $(\psi, \rho)$,
and we remarked that
in view of (for example)
\cite{HY}, Theorem 2.6,
this phenomenon
would follow if
each $\psi$ is a
repelling fixed point
of zeta, hence an attracting
fixed point of a branch of the
inverse $\zeta^{(-1)}$.
(Here \it branch \rm has
its usual meaning in
complex analysis;
it does not denote a branch of a
backward orbit, as defined in
\cite{Br2},
but it is true that
$\vec{\phi}_{\psi, \rho, \zeta}$
constitutes a branch of the
backward orbit of zeta as
defined there.)
The behavior of iterates
of zeta were shown to
be similar in the range of our observations.
So we think it
is plausible to
propose a dynamical explanation
for the attraction of
the sequences in $\vec{\Phi}_X$
to Riemann zeros that draws on an analogy
with the proposals in \cite{Br2}.
In this analogy, in the
context of \cite{Br2},
with
 $X_n$ (say) $= (0, n, 2n, ...)$
$G$ should
correspond to $\zeta^{\circ n}$,
the $F$'s
should
correspond to (complex analysis sense)
branches of $\zeta^{\circ n (-1)}$,
and
the roles of the
zeta fixed points $\psi$ and the
zeta zeros $\rho$ in \cite{Br2}
should be swapped.
This idea results in conjecture 4,
conjecture 5,
and proposal 2 (appendix.)
\newline \newline
We ask
the following question,
because (together with Conjecture 2)
an affirmative answer implies
the truth of the Riemann hypothesis,
and (together with Theorem 1)
a negative answer provides an avenue
to search for counterexamples
to the Riemann hypothesis.
Both possibilities are consistent with
our observations.
\begin{question}
Is the following claim true?
 \it Suppose $0 < \sigma < 1$
 and the real parts
$(\Re \phi_0, \Re \phi_1, ...)$
converge to  $\sigma$. Then
$\sigma = \frac 12$.
\end{question}
\section{\sc methods}
\subsection{Quadrant plots.}
We will make use
of colored plots
(``quadrant plots'').
The  routine that makes
quadrant plots
takes as inputs
a specification
of the image resolution,
the center and dimensions
of a region $R$
in the complex plane,
and a routine to compute some
$\bf{C} \rightarrow \widehat{\bf{C}}$
function
$f$. The
output colors a
small square (the size of
which depends on the
resolution)
around each point $w$
of a regular lattice in $R$;
the color is chosen to
represent
the quadrant of $f(w)$.
Similar methods
that do not use coloring are
put to use in, \it e.g\rm,
\cite{A}.)
\newline \newline
The pixel representing the square
is colored according to the rules in
Table 1.
In the table,
the region $D$ is a disk with center $s = 0$ and
large radius $r$ (chosen as may be convenient.)
We denote the complement of $D$ as $-D$.
\newline \newline
\hskip 1in
\begin{tabular}{|c|c|} \hline
Location of $f(s)$& Color of pixel depicting region containing
$s$\\ \hline \hline
real and imaginary axes & black \\ \hline
$D \hskip .05in\cap $ Quadrant I  &  rich blue\\ \hline
$ - D \hskip .05in\cap$ Quadrant I  &  pale blue\\ \hline
$D \hskip .05in\cap $ Quadrant II  & rich red \\  \hline
$ - D \hskip .05in\cap$ Quadrant II  &  pale red\\ \hline
$D \hskip .05in\cap $ Quadrant III  & rich yellow \\ \hline
$ - D \hskip .05in\cap$ Quadrant III  &  pale yellow\\ \hline
$D \hskip .05in\cap $ Quadrant IV  &  rich green\\ \hline
$ - D \hskip .05in\cap$ Quadrant IV  &  pale green\\ \hline
\end{tabular}
\vskip .1in
\hskip .5in
\sc{table 1: coloring scheme for quadrant plots}
\rm
\vskip .1in
\hskip -.2in
The junction of four rich colors
represents a zero,
the junction of four pale colors represents a pole,
and
the boundary of two appropriately-colored regions is an
$f$ pre-image of an
axis. An example is shown in Figure \ref{leaf2point1.png}.
(We have superimposed a pair of axes on this
quadrant plot.)
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.8]{leaf2point1.png}
\vskip .07in
\caption{Quadrant plot of $s \mapsto
(s - 1)^2 (s - i) (s + 1)^5/(s+i)^3$}
\label{leaf2point1.png}
\end{figure}
\newpage
\subsection{Basins of attraction.}
For $z \in \bf{C} \rm \cup \{\infty\}$,
let $A_z :=\{w \in \bf{C}$ s.t.
$\lim_{n \to\infty}
\zeta^{\circ n}(w) = z\}$
(the ``basin of attraction'' of $z$ under
zeta iteration.)
Then $A_{\infty}$ and its
complement in $\bf{C}$ are fractals \cite{W}.
(As far as we know,
plots of  $A_{\infty}$
were made first by
Woon in \cite{W}.)
Let $\phi \approx -.295905$
be the largest negative
zeta fixed point.
As we noted in \cite{Br2},
plots of
$A_{\phi}$ and the complement in
$\bf{C}$ of $A_{\infty}$
are indistinguishable to the eye.
(But, for example,
the complement in
$\bf{C}$ of $A_{\infty}$
contains the zeta cycles and zeta zeros, and
$A_{\phi}$ does not.)
We reproduce plots of
$A_{\phi}$ and the complement in
$\bf{C}$ of $A_{\infty}$
in Figure \ref{leaf1point1.png}
(appendix.)
\section{\sc Basis of the claims}
\subsection{Conjecture 1.}
\subsubsection{Existence and uniqueness of
\texorpdfstring{$\psi_{\rho}$}{TEXT}.}
The right panel of Figure
\ref{leaf3point2left.png}
(Figure 3.2 of \cite{Br2})
in the appendix is
a quadrant plot
of the function
$s \mapsto \zeta(s) - s$,
the zeros of which are the zeta fixed points.
The quadrant plot is shown superposed on
a plot of $A_{\phi}$.
The major feature
of this escaping set
is that it consists of an infinite family of
irregularly shaped bulbs straddling
the critical strip, each
one of which
(except for the central bulb, referred to
in \cite{Br2} as a
``cardioid'') contains one Riemann
zero (trivial or nontrivial,
but only the cardioid and
the bulbs associated to
nontrivial zeros are visible at the scale
of Figure \ref{leaf3point2left.png}.)
One zero of
$s \mapsto \zeta(s) - s$
evidently lies on
one filament decorating
each of
the visible
non-cardioid bulbs.
These filaments
each consists of smaller bulbs
of the zeta escaping set, and
there is a numerical pattern
(described in \cite{Br2})
dictating the distribution of zeta
fixed points among
these bulbs. The fact that
this distribution is non-random
is evidence (we would argue)
that the association of
zeta fixed points to the Riemann zeros
is one-to-one, and, therefore,
that  there is probably
a unique $\psi_{\rho}$
for each $\rho$.
\subsubsection{The case
\texorpdfstring{$u=1$}
{TEXT}.}
Clause 1 of conjecture 1 is the claim that
the  set of imaginary parts
$\{\Im \phi: \phi \in \Phi_z \}$
is unbounded
and nonempty.
The
set $\Phi_z$ is the set of solutions
of the function
$s \mapsto V_z(s) - s$.
When $z = 0$,
$\Phi_z = \Phi_0$ is the set of fixed points
of the Riemann zeta function.
Plots of these fixed points in \cite{Br2}
provided our evidence for this case of
the conjecture; we will not reproduce
them here.
Using our particular
methods, we can only spot-check
this claim for selected $z$
and selected regions of the complex
plane.
In Figure \ref{seqfig2.png} below, we display a
quadrant plot of the function
$s \mapsto V_1(s) - s$
on a $2000$ by $2000$ square
with center at $s=0$ in the complex plane.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.8]{seqfig2.png}
\vskip .07in
\caption{2000 by 2000 quadrant plot of $s
\mapsto V_1(s) - s$}
\label{seqfig2.png}
\end{figure}
Visible zeros of this function (lying at
the junctions of four colors) lie in the vicinity
of the imaginary axis. There are also zeros (not visible
at this scale) in the vicinity of each nontrivial
Riemann zero in the depicted region,
on the boundary of a triangular region
(barely
visible at this scale) near the center of the plot,
and along the
negative real axis.
The straight
boundaries of the green and blue
regions  do
not (on inspection at smaller scales)
 contain any zeros of this function.
 \newline \newline
 In Figure \ref{seqfig3r1c1.png} below, we  display
quadrant plots of
$s \mapsto V_k(s) - s, k = 2, 4, 8$
and  $16$, each centered at $s = 0$,
in squares with side length
$400,  2400, 1.2 \times 10^5$, and
$2 \times 10^8$ respectively.
The are shown clockwise from upper left in the
order of the size of $k$.
 \begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig3r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig3r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig3r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig3r2c2.png}
\vskip .07in
\caption{Quadrant plots of $s
\mapsto V_k(s) - s, k = 2, 4, 8$ and $16$}
\label{seqfig3r1c1.png}
\end{figure}
\newline \newline
Since the Riemann hypothesis has been
verified well beyond
the range of our experiments,
we are safe in designating the
$n^{th}$-by-height nontrivial zero
in the upper half plane as $\rho_n$.
Figure \ref{seqfig4left.png}
shows (at the four-color junctions)
zeros of
$s \mapsto V_{100}(s) - s$ on two
$1.2$ by $1.2$ squares,
one centered at $\rho_1$
and the other centered at $\rho_{649}$.
The zeros along the the
left sides of the squares
apparently belong (as we will explain below)
to sequences of fixed points  $\phi_n$
of the $V_n$, the real parts of
which converge to zero
(but it is not clear that
the imaginary parts converge at all.)
The zeros very near
the centers of the squares
appear to belong
to such sequences converging to $\rho_1$
and $\rho_{649}$, respectively,
as we will also explain below.
 \begin{figure}[!htbp]
\centering
\includegraphics[scale=.4]{seqfig4left.png}
\includegraphics[scale=.4]{seqfig4right}
\vskip .07in
\caption{Quadrant plots of $s \mapsto V_{100}(s) - s$
near $\rho_1$ and $\rho_{649}$.}
\label{seqfig4left.png}
\end{figure}
These and similar plots, which depict the
fixed points of various $V_z$ as (apparently)
unbounded sets of isolated points
in the complex plane,
are the basis of conjecture 1.
\subsection{Conjecture 2: the case
\texorpdfstring{$u=1$}{}.}
This is the
claim that, for $X$ and $\rho$
as above,
some  $\vec{\phi} \in \vec{\Phi}_{X,1}$
converges to $\rho$,
is nearly logarithmic with respect to
$\rho$, and is nearly uniformly
distributed  with  respect to $\rho$.
We will describe experiments
that tend to support this claim
for
 $X = (0, 1, 2, ...), X' =
 (0, \frac 15, \frac 25, ...)$
 and for
$\rho_n$ with $n$ selected from
the range $1 \leq n \leq 100$.
\subsubsection{Typical plots.}
The  red points
on the pictured spirals
depict elements of the
sequences $\vec{\phi}$ from
conjecture 3, clause 5.
Because the $\vec{\phi}$
converge to $\rho$ so rapidly,
logarithmic scaling was
necessary  to obtain readable plots.
As a result, the spirals are
for the most part everted:
points that appear farther from the center
(which means farthest from the zero)
are, in fact, closer to the center.
(The only exception might be  the
red point  depicting $\psi_{\rho}$,
since  typically
$|\psi_{\rho} - \rho| > 1$.)
Thus,
for $\phi \in \vec{\phi}$,
the corresponding red point
$p$ (say)
in the plots of spirals
below  is situated
on  the  ray  originating
at $\rho$ and passing through
$\phi$, but
$|p -  \rho|  = \log |\phi - \rho|$.
In
these figures, blue chords
connect  representatives (red points)
of consecutive members of
$\vec{\phi}$.
\newline \newline
We  made plots  from which we  formulated
conjecture 2. In one experiment,
we
examined the  zeros
$\rho_n, 1 \leq n \leq 100$.
Figures \ref{seqfig5r1c1.png} and
\ref{seqfig6r1c1.png}  are typical outcomes
for $X = (0, 1, 2, ...)$.
We chose them for display because,
together, they  indicate the dependence
upon $\delta_n$ of the  appearance
of the spirals.
Figure  7  depicts several
spirals centered on other zeros;
we omit the  statistics
for these zeros, which are
consistent with our conjectures.
 \begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig5r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig5r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig5r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig5r2c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig5r3c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig5r3c2.png}
\vskip .07in
\caption{Zero $\rho_1, u = 1, X  = (0, 1, 2, ...)$,
$0 \leq n \leq 300$
\newline row 1, column 1:
$\log |\phi_n - \rho_1|$ vs. $n$
\newline  row 1, column 2:
$\log |\phi_n - \rho_1|$ vs. $\theta_{\vec{\phi}}(\phi_n)$
\newline row 2, column 1: logarithmically scaled plot
of  $\vec{\phi}$
\newline row 2, column 2: $\delta_n/\pi$ vs. $n$
\newline
\newline row 3, column 1: $\log \Delta_n$ vs. $n$
\newline row 3, column 2:  $\log h_n$ vs. $n$ (K =  50)}
\label{seqfig5r1c1.png}
\end{figure}
\newpage
 \begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig6r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig6r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig6r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig6r2c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig6r3c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig6r3c2.png}
\vskip .07in
\caption{Zero $\rho_3, u = 1, X  = (0, 1, 2, ...)$,
$0 \leq n \leq 300$
\newline row 1, column 1:
$\log |\phi_n - \rho_1|$ vs. $n$
\newline  row 1, column 2:
$\log |\phi_n - \rho_1|$ vs. $\theta_{\vec{\phi}}(\phi_n)$
\newline row 2, column 1: logarithmically scaled plot
of  $\vec{\phi}$
\newline row 2, column 2: $\delta_n/\pi$ vs. $n$
\newline
\newline row 3, column 1: $\log \Delta_n$ vs. $n$
\newline row 3, column 2:  $\log h_n$ vs. $n$ (K =  50)}
\label{seqfig6r1c1.png}
\end{figure}
 \begin{figure}[!htbp]
\centering
\includegraphics[scale=.40]{seqfig7zr7left.png}
\hskip .03in
\includegraphics[scale=.40]{seqfig7zr7right.png}
\vskip .03in
\includegraphics[scale=.40]{seqfig7zr65left.png}
\hskip .03in
\includegraphics[scale=.40]{seqfig7zr65right.png}
\vskip .03in
\includegraphics[scale=.40]{seqfig7zr70left.png}
\hskip .03in
\includegraphics[scale=.40]{seqfig7zr70right.png}
\vskip .03in
\includegraphics[scale=.40]{seqfig7zr78left.png}
\hskip .03in
\includegraphics[scale=.40]{seqfig7zr78right.png}
\vskip .03in
\includegraphics[scale=.40]{seqfig7zr82left.png}
\hskip .03in
\includegraphics[scale=.40]{seqfig7zr82right.png}
\vskip .03in
\caption{Row 1, $\rho_7$; row 2, $\rho_{65}$;
row 3,  $\rho_{70}$; row 4, $\rho_{78}$,
row 5, $\rho_{82}$;\newline
column 1, $X = (0, 1, 2, ...)$; column 2,
$X = (0, \frac 15, \frac 25, ...)$}
\label{seqfig7zr7left.png}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig8r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig8r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig8r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig8r2c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig8r3c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig8r3c2.png}
\vskip .07in
\caption{Nontrivial zero $\rho_{70}, X  = (0, 1, 2, ...)$,
$0 \leq n \leq 1500$;\newline
row 1, column 1:
$\log |\phi_n - \rho_1|$ vs. $n$;\newline
row 1, column 2:
$\log |\phi_n - \rho_1|$ vs. $\theta_{\vec{\phi}}(\phi_n)$;\newline
row 2, column 1: logarithmically scaled plot
of  $\vec{\phi}$;\newline
row 2, column 2: $\delta_n/\pi$ vs. $n$;\newline
row 3, column 1: $\log \Delta_n$ vs. $n$;\newline
row 3, column 2:  $\log h_n$ vs. $n$ (K =  50)}
\label{seqfig8r1c1.png}
\end{figure} \hskip -.1in
\subsubsection{An anomaly.}
For the zero
$\rho_{70}$, while the other
statistics we display in these
plots were entirely consistent
with our conjectures, $\delta_n$
was so small in the range
$0 \leq n \leq 300$ that the spiral
structure for $X = (0, 1, 2, ...)$
was not obvious and
we had to extend our observations
to the range $0 \leq n \leq 1500$
to see it.
It appears that,  for
$\rho_{70}$,  $\delta_n$
converges rapidly to a limit $\approx .0016 \pi$,
so that  it requires  $1187$ points of
$\vec{\phi}$ to wind entirely around
the  zero.
We have not encountered
another spiral like this.
(The anomaly does not extend, for example,
to the
case $X = (0, \frac 15, \frac 25,...)$.)
The uniqueness (within the
range of our observations)
of this anomaly suggests that,
for the purpose of searching
for the unknown functions $G$,
one should look for an invariant of
the Riemann zeros that takes
an anomalous value at $\rho_{70}$.
As we said above,
we regard
the Riemann zeta function
as analogous to $G$;
spirals  studied in
\cite{Br2} are centered
upon zeta fixed points $\psi$,
and  $\lim_{n \to \infty} \delta_n$
appears to be determined by
$\arg \frac d{ds}\zeta(s)|_{s = \psi}$.
Therefore
 we might expect
 the  invariant we are looking for
to
coincide with
 $\arg \frac d{ds}G(s)|_{s = \rho_n}$,
which we might hope to identify
without knowing $G$ explicitly.
We  have undertaken a
cursory search for
such an
invariant, so far without success.
The plots for
$\rho_{70}$ are in Figure \ref{seqfig8r1c1.png}.
An interesting feature of the spiral
about $\rho_{70}$ for
$X = (0, 1, 2, ...)$ is that
red points  $p_n$ and $p_{n'}$ on it
are close just if  $|n - n'|$
is small; on most  of the other
spirals we will display
in this article, this is not the case,
as one can  tell  by  keeping track of the
blue connecting chords.
(The spiral about $\rho_3$
depicted in Figure \ref{seqfig6r1c1.png},
for which the $\delta_n$ are
also fairly small, is an
exception.) In the plot
shown in row 3, column 1 of
Figure \ref{seqfig7zr7left.png}
(and reproduced in Figure
\ref{seqfig8r1c1.png}),
consecutive  $p_n$ are
so close that  these chords are
not visible.
\subsection{Conjecture 3.}
For convenience, we reprint the conjecture:
\newline \newline
For each choice
of $\rho$,
there is a continuous function
$f^{cut}: \bf{C}^{\it cut}
\rightarrow \bf{C}$
such that the restriction of
$f^{cut}$ to $\vec{R}_u$
is a function $f$ with the
following properties:
\newline
(1)
$f: \vec{R}_u \rightarrow \bf{C}$
is continuous and one-to-one,
\newline
(2) $f(z) \in \Phi_{z}$
for each $z \in \vec{R}_u$,
\newline
(3)  $f(0) = \psi_{\rho}$,
\newline
(4)  $\lim_{z \to \infty}
f(z) =  \rho$,
\newline
(5) the image of $\vec{R}_u$
under $f$ is a nearly
logarithmic
spiral $S_{u,\rho}$ with center
$\rho$,
\newline
(6)  among the sequences
$\vec{\phi} =
(\phi_0, \phi_1, ...) \in
\vec{\Phi}_{X,u}$
converging to $\rho$,
all the points of one of them
(say, $\vec{\phi}_{u,\rho}$)
are interpolated by $S_{u,\rho}$;
the initial element of
$\vec{\phi}_{u,\rho}$
is
$\psi_{\rho}$.
\newline
(\it n.b. \rm Our notation suppresses the dependence of
$f$ on $u$ and $\rho$, and
the dependence of $f^{cut}$
on $\rho$.)
\newline \newline
\textsection
\thinspace
We are proposing this because
a natural explanation
for our observation that
convergent
sequences $\vec{\phi} \in \Phi_{X, u}$
are apparently always nearly logarithmic
is that the tail of
such a $\vec{\phi}$ lies on
a nearly logarithmic spiral.
The naturalness disappears
(in our opinion) unless,
for a given $u$,
the spiral is independent of
the choice of $X$.
Therefore,
clauses (1) and (4)
both say something stronger,
namely, that
every single point of  $S_{u,\psi,\rho}$
is, in fact, an element of $\Phi_{x,u}$
for some $x \geq 0$.
(For the sake of clarity, we have
used
some redundancy in the
statement of this conjecture.)
If we knew that such an $f$ exists,
at least, independent of the choice
of $X$'s with rational
common differences, it seems
clear that it would extend
by continuity to the
family of $X$'s with
real common differences.
If $X_1, X_2$ with
common differences $d_1, d_2$ both
rational are two instances of $X$
such that neither is a refinement
of the other, they clearly
have a refinement
in common with common difference a
rational number again.
\newline \newline
Thus
(it seems to us)
we can test conjecture 3 by
examining a tower of arithmetic
progressions $X_n$ such that
$m > n \Rightarrow X_m$ is
a refinement of $X_n$.
(This is not the $X_n$ of
section 1.2)
For a given $u$ each such $X$
determines (by way, for example, of the
\it Mathematica
\tt LinearModelFit \rm command)
a linear model of the data
$(\theta, \log r)$ derived from
the points of a sequence
$\vec{\phi} \in \Phi_{X,u}$
converging to a zero $\rho$
as follows: for $z_n \in \vec{\phi},
r = |z_n - \rho|$ and
$\theta = \theta_{\vec{\phi}}(z_n)$
(see section 1.1.)
\newline \newline
In row 1 of Figure \ref{seqfig9r1c1.png}
we have plotted the slopes and intercepts of
such models for $u = 1$ and
$\rho = \rho_1$
against an index on the horizontal
axis that specifies the refinement of a
particular $X = X_1 = (0, \frac 1{1000}, \frac 2{1000}, ..., 100)$
from $x = 50$ to $x = 100$.
(Actually,
for the sake of
efficiency,
we worked in the
opposite direction: from
more highly-refined arithmetic
progressions to coarser ones.)
In row 2, we have plotted
the logarithms of the absolute
values of the
successive differences of the corresponding
values from row 1. The common
differences of the $X$'s in the
tower increase from left to right; thus,
if the point corresponding to an
arithmetic progression $X_1$
lies to the left of the point
corresponding to an arithmetic progression
$X_2$, then $X_1$ is a refinement of $X_2$.
The row 2 values indicate
that the row 1 heights are a Cauchy sequence,
thus a convergent sequence,
and we propose that the limits
of these two Cauchy sequences are the
slope and intercept
of a log-linear model of $S_{u, \psi, \rho}$,
\it i.e., \rm that $S_{u, \psi,\rho}$ is quite
possibly an exactly logarithmic spiral, but
is, at least, very probably a nearly logarithmic
spiral.
\newline \newline
As we have mentioned,
we did not, in fact,
begin with a coarse
$X$ and refine it repeatedly
to create a tower
of $X$'s.
That
procedure, to avoid redundant
operations, would have
required computing the
new fixed points at
each stage and then inserting
them into the sequence
determined from the
previous $X$. This seemed
too baroque.
Instead,
we began with the ``highly-refined''
arithmetic progression $X_1$
and chose from it successively less-refined
subsequences by the following process;
at stage $n$ of the process,
we selected every $n^{th}$ member
of the original
$X_1$ to form a ``coarsening''
$X_n$ of $X_1$.
Consequently, we were working
with subsequences $\vec{\phi}_n$
of the original
$\vec{\phi}_1 = \vec{\phi} \in \Phi_{X_1,u}$,
one for each
of the coarsened arithmetic
progressions $X_n$.
The data we were going to model then
comprised (initially)
a subsequence of the list of
pairs $(\theta, \log r)$ derived from
$\vec{\phi}_1$. But,
because we were working with subsequences,
the values of $\theta$ now violated
the minimality condition in clause (3)
of the definition of
$\theta_{\vec{\phi}_n} =
\theta_{(z_0, z_1, ...)}$ (say);
so we set $\theta_{\vec{\phi}_n}(z_0)=
\arg (z_0 - \rho)$ as per clause (1),
and, for $k > 0$, chose
$\theta_{\vec{\phi}_n}(z_k)$
according to clauses (2), (3) as applied,
not to the first $k$ elements of $\vec{\phi}_1$,
but to the first $k$ elements of $\vec{\phi}_n$.
Figure \ref{seqfig9r1c1.png} shows the plots
resulting from these operations
for $\rho = \rho_1, u = 1$,
and coarsenings $X_n$ of
$X_1 = (0, \frac 1{1000}, \frac 2{1000}, ..., 100)$
from $x = 50$ to $x = 100$.
(In the caption to \ref{seqfig9r1c1.png},
we refer to this $n$ as the ``filter index.'')
We ignore the first $50,000$ members of
$X_1$ because the plots of $\log r$ vs. $\theta$
in this region are visibly non-linear
(even though, because only the tails count in
this matter, the sequences in question do qualify as
nearly logarithmic). We speculate
that the reason
for this observation is that
\it Mathematica\rm's \tt FindRoot \rm command,
when searching for
$\phi_{x,u} \in \Phi_{x,u}$ for relatively small $x$,
has not yet found the special sequence
mentioned in clause (5) of the conjecture--because,
for small $x, \phi_{x,u}$ is not yet close
enough to
$\rho = \lim \vec{\phi}_1$. Nevertheless,
$\phi_{0,u}$, which by definition is a zeta fixed point,
is probably (we conjecture) so close to $\rho$ that it
is the zeta fixed point we have denoted as $\psi_{\rho}$;
this is the reason for clause (2) of conjecture 3.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig9r1c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig9r1c2.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig9r2c1.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig9r2c2.png}
\caption{Linear models of $\log r$ vs.
$\theta$ for $\rho = \rho_1, u = 1$,
and a sequence
$\vec{\phi} \in \Phi_X$ on coarsened, truncated
$X_1 = (50, 50.001, 50.002, ..., 100)$: parameters plotted against
the filter index.
\newline
row 1, column 1: slopes;
\newline
row 1, column 2: intercepts;
\newline
row 2, column 1: logarithms of slope increments;
\newline
row 2, column 2: logarithms of intercept increments
}
\label{seqfig9r1c1.png}
\end{figure}
\vskip .1in
\hskip -.23in
\thinspace
Figure \ref{seqfig10r1c1fi512.png}
shows the spirals (actually: polygons)
induced by the filter indices $512$ (row 1, column 1),
$128$ (row 1, column 2), $64$ (row 2, column 1),
and $16$ (row 2, column 2.) As this parameter decreases, the set of points included in
the plot grows and the polygons appear
to better approximate a spiral,
which we propose is the
$S_{1,\psi_{\rho},\rho}$ conjectured
to exist in conjecture 3. The
fourth row displays two plots of the
arc lengths of the polygons corresponding
to filter index
$\lfloor 2^n \rfloor, 0 \leq n \leq 13$
(column 1) and to filter index
$\lfloor 1.1^n \rfloor, 0 \leq n \leq 100$
(column 2.) The arc lengths appear
to converge to what we propose is the arc length
of the part of $S_{1,\psi_{\rho},\rho}$
with endpoints corresponding to
$x = 50$ and $x = 100$. The convergence
(we wish to argue) is more evidence for
the existence of the spiral $S_{1,\psi_{\rho},\rho}$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{seqfig10r1c1fi512.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig10r1c2fi128.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig10r2c1fi64.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig10r2c2fi16.png}
\vskip .05in
\includegraphics[scale=.45]{seqfig10r3c1fi2to.png}
\hskip .05in
\includegraphics[scale=.45]{seqfig10r3c2fi1_11to.png}
\caption{$\rho = \rho_1, u = 1$,
coarsened, truncated
$X_1 = (50, 50.001, 50.002, ..., 100)$:
rows 1, 2: logarithmically scaled polygons plotted against
filter index 512, 128, 64, and 16.
row 3: polygon lengths for
filter indices $\lfloor 2^n \rfloor$ (left) and
$\lfloor 1.11^n \rfloor$ (right)
}
\label{seqfig10r1c1fi512.png}
\end{figure}
\section{\sc appendices}
\subsection{Extension of conjecture 3 to arbitrary
zeta fixed points.}
\begin{proposal}
For each choice
of $\rho$ and $\psi$
there is a continuous function
$f^{cut}: \bf{C}^{\it cut}
\rightarrow \bf{C}$
such that the restriction of
$f^{cut}$ to $\vec{R}_u$
is a function $f$ with the
following properties:
\newline
(1)
$f: \vec{R}_u \rightarrow \bf{C}$
is continuous and one-to-one,
\newline
(2) $f(z) \in \Phi_{z}$
for each $z \in \vec{R}_u$,
\newline
(3)  $f(0) = \psi$,
\newline
(4)  $\lim_{z \to \infty}
f(z) =  \rho$,
\newline
(5) the image of $\vec{R}_u$
under $f$ is a nearly
logarithmic
spiral $S_{u,\psi,\rho}$ with center
$\rho$,
\newline
(6)  among the sequences
$\vec{\phi} =
(\phi_0, \phi_1, ...) \in
\vec{\Phi}_{X,u}$
converging to $\rho$,
all the points of one of them
(say, $\vec{\phi}_{u,\psi,\rho}$)
are interpolated by $S_{u,\psi,\rho}$;
the initial element of
$\vec{\phi}_{u,\psi,\rho}$
is
$\psi$.
\newline
(\it n.b. \rm For readability,
our notation suppresses the dependence of
$f$ on $u, \psi$, and $\rho$, and
the dependence of $f^{cut}$
on $\psi$ and $\rho$.)
\end{proposal}
\subsection{Extension of conjecture 4 to arbitrary zeta fixed
points.}
\begin{proposal}
For each $X$ and $u$,
there is a function $G$ with the following
properties:\newline
(1) $G$ is meromorphic, independent
of $\psi$,
and independent of $\rho$.
\newline
(2) The sequence  $\vec{\phi}_{u,\psi,\rho}$
from conjecture 3
satisfies
$G(\phi_n) =  \phi_{n-1}$
for  $n = 1, 2, ....$
\newline
(3)  Each $\rho$ is a repelling fixed point
of $G$,
\newline
(4) $G$ is many-to-one,
but for each $\rho$ and  each
$\psi$ there
is a function
$F_{\psi,\rho}$
such that

(i)  $F_{\psi,\rho}^{(-1)} = G$,

(ii) consequently (in view of clause 2)
for $n = 0, 1, ...$,
$$F_{\psi,\rho}(\phi_n) =  \phi_{n+1},$$

(iii)
 $\rho$ is
an attracting fixed point of
$F_{\psi,\rho}$,
and

(iv) $F_{\psi,\rho}$
carries $S_{u,\psi,\rho}$
into itself.
\newline
(\it n.b. \rm Again for readability,
our notation suppresses the
dependence of $G$,
and the dependence of
$F_{\psi, \rho}$ on $X$ and $u$.
\end{proposal}
\newpage
\subsection{Two basins of attraction.}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{leaf1point1.png}
\hskip .05in
\includegraphics[scale=.45]{leaf3point3.png}
\vskip .07in
\caption{left: $A_{\phi}$;
right: $\bf{C} - A_{\infty}$
}
\label{leaf1point1.png}
\end{figure}
\subsection{Figure 3.2 of \texorpdfstring{\cite{Br2}}{TEXT}.}
This figure is reproduced below as
Figure \ref{leaf3point2left.png}.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.45]{leaf3point2left.png}
\hskip .05in
\includegraphics[scale=.45]{leaf3point2right.png}
\vskip .07in
\caption{left: zeros of zeta; right: zeros of
$s \mapsto \zeta(s) - s)$ (\it i.e.\rm, zeta
fixed points);
both superposed on a plot
of $A_{\phi}$
}
\label{leaf3point2left.png}
\end{figure}
\newpage

redpajama_set_name: RedPajamaArXiv


Initially, I wanted to call and make reservation, so I googled and found this number,+65 6337 7337 . But it turned out that the number was for Yoogane Bugis instead.
However, when I managed to call through Yoogane Westgate, they don't take any reservation, only walk-in(s).
Ok, so it's the M.E.A.T makan time again!!! Finally, a time when all of us are free. Decided to try something else this time round other than HaiDiLao!
There wasn't any queue yet, cos we reached before 5.30pm where most people just knocked off from work?
Basically, when I'm out with them... the amount of food they order sure gonna be like a crazy!
Meanwhile I went to do some quick errands.
And came back to see... 4 bottles of beer.
okok, shall let them do the cooking instead.
Free flow of salads at the salad bar.
Strawberry sauce for the cabbage?
hmmm, find it a little weird combi though.
Let Mrs Lim do the rest of the cooking ok?!
This is awesome. A must order dish.
I like eggs, I like cheese. I like tomato sauce.
Lots of topoki, noodles, chicken inside.
Not too spicy, taste not too bad.
The noodles were very QQ. But the noodles kinda clump together though.
I still prefer the one I taste at JP - Baek Doo San.
Quite a variety of ingredients in it.
However, I felt Big Mama's taste better.
We ordered sweet potato topokki too!
Next up is the Marinated Chicken Galbi Cheese Fried rice!
I love cheese with anything.
So I enjoy the fried rice.
After we left the place, there's queue already.
So, its about $50-70 per pax. With the beer and so many other dishes we ordered, I think its still quite affordable.
I would rate a 3/5 for this. As there's quite a variety of choices for the Chicken Galbi, toppings and side dishes to go along. Prices are affordable and cosy place.
That's all for this one!


text:
American Buddhism
http://www.teachersdomain.org/resource/awr09.socst.world.glob.amerbud/
WEB VIDEO & DISCUSSION QUESTIONS This video from Religion & Ethics NewsWeekly examines the differences that have emerged between Asian-American Buddhists and European-American converts since Buddhism first reached America's shores in the 1840s. Asian immigrants to the United States brought their traditions and belief systems with them, giving Buddhism a foothold in the West as new immigrants found comfort in their native faith tradition. At the same time, Americans new to Buddhism discovered value and contentment in Buddhist ideas and ideals. A transcript. background essay and discussion questions support the video.
Buddhism: Religions and Religious Movements edited by Jeff Hay
https://www.amazon.com/Buddhism-Religions-Religious-Movements-Jeff/dp/073772563X
BOOK Buddhists believe in no single deity but ascribe to a guide for living that offers the possibility of transcendent enlightenment. This volume outlines the tenets of Buddhism and traces the spread of the religion from its birth in India to Southeast Asia, China, Japan and ultimately the West. HS-Adult
Buddhist Art and Architecture
http://www.buddhanet.net/gallery.htm
IMAGES & TEXT This site has examples and explanations of art influenced by Buddhist beliefs and traditions.
http://buddhism.about.com/od/basicbuddhistteachings/u/basics.htm
WEB RESOURCE This About.com site is listed as excellent in the Digital Library of Buddhist Studies. It provides an overview of basic ideas and practices, as well as information on Buddhist history and major schools of belief.
http://www.teachersdomain.org/resource/awr09.socst.world.glob.tibetnyr/
WEB VIDEO & DISCUSSION QUESTIONS This video from Religion & Ethics NewsWeekly features Tibetan Buddhists in the U.S. performing the traditional services of the Tibetan New Year. The activities of the Tibetan New Year symbolize purification. They are intended to drive away evil spirits, renew positive spiritual energy and reinforce the resolutions a person has made. Download the video onto your device desktop in order to view it.
http://www.bbc.co.uk/religion/religions/buddhism/
WEB RESOURCE This part of the BBC religions site on Buddhism has information on beliefs, ethics, history, holy days, people and the various sub groups within the faith.
Nen Ordination
http://www.teachersdomain.org/resource/awr09.socst.world.glob.nenordin/
WEB VIDEO and DISCUSSION QUESTIONS This video from Religion & Ethics NewsWeekly shows a family in the U.S. sending off their pre-teenage son to become ordained as a nen, or novice monk, in the tradition of Theravada Buddhism. Theravada, one of the two major schools of Buddhism, is considered the earliest record of the Buddha's teachings. It is the predominant religion of Thailand, Mynamar (Burma), Cambodia, and Sri Lanka, and has also blossomed in the West. To view the video, download it to the disktop of your device.
Japanese Buddhist Art
http://www.asiasocietymuseum.org/region_results.asp?RegionID=6&CountryID=14&ChapterID=38
ART IMAGES The introduction of Buddhism to Japan was one of the most important events in Japanese history and had a lasting effect on the development of the country's thought, art and culture. This Asia Society site has 11 examples.
Internet Sacred Text Archive - Buddhism
http://www.sacred-texts.com/bud/index.htm
SACRED TEXT This Sacred Text Archive has a large selection of modern translations of Buddhist religious texts. For schools, the collection of Jataka fables framed as previous incarnations of the Buddha, many of which either have parallels or derivatives in western folklore and literature, will be especially of value. Although the Jataka is not considered part of the canonical Buddhist scripture, it is very popular. Most tales have a short moral lesson.
Becoming Buddha: The Story of Siddhartha by Whitney Stewart
https://www.barnesandnoble.com/w/becoming-buddha-whitney-stewart/1110898123
CHILDREN'S BOOK This introduction to the story of Buddhism depicts the life of Siddhartha Gautama, a wealthy young prince who gave up his worldly ways to attain supreme enlightenment to become Buddha. Gr. 5-9
Oldest Buddhist Shrine Uncovered In Nepal May Push Back the Buddha's Birth Date
http://news.nationalgeographic.com/news/2013/11/131125-buddha-birth-nepal-archaeology-science-lumbini-religion-history/
WEB NEWS Excavations by a team of British archaeologists uncovered what is thought to be the oldest Buddhist shrine on record at the Lumbini Pilgrimage Center in Nepal, the legendary site of Buddha's birth. Using multiple dating methods, the archaeologists fix the shrine at around 550 BCE which seems to support Buddha's birth at about 100 years before most scholars think.
http://www.pbs.org/thebuddha/birth-and-youth/
WEB VIDEO This three-part PBS film by David Grubin provides an overview of the Buddha's life and teachings. The topics include Early Life, Seeking, Enlightenment, Teachings, Community, Meditation, Compassion, Miracles, and Death & Legacy. Since the web version is in parts, teachers may choose the sections that are useful in the classroom and use other resources to supplement if needed. The lesson materials that support the film will be controversial in many schools so should be used after parental notification.
Ask a Buddhist
http://www.clear-vision.org/Schools/Ask-a-Buddhist.aspx
WEB VIDEO This is a video Q & A featuring clear explanations of Buddhist thought on ethics, enlightenment, basic beliefs, and more. Use the section for teachers with care in that some suggested material crosses the line into practice of religion and is not appropriate for public schools. However, this site has much to offer to learn about Buddhism
Features of a Chinese Buddhist Temples
http://ffh.films.com/PreviewClip.aspx?id=12386
WEB VIDEO This is a short narrated visually rich description of Chinese Buddhist temples are easily recognizable by features such as rolling waves of glazed tiles, architectural symmetry, lots of red paint, lotus flower motif, incense burners, and a centrally located main door. This is a useful site for teachers but the last sentence would create a negative response from most Buddhist scholars.
Buddhism: The "Imported" Tradition
http://afe.easia.columbia.edu/cosmos/ort/buddhism.htm
WEB RESOURCE Buddhism is an interesting form of Chinese religion for many reasons, not least because it was the first major religious tradition in China that was "imported" from abroad.
Buddhism and Its Spread Along the Silk Road
http://www.silk-road.com/artl/buddhism.shtml
WEB RESOURCE Besides silk, paper and other goods, the Silk Road carried Buddhism another commodity which was equally significant in world history. The world's oldest international highway served as the vehicle to spread Buddhism through Central Asia. This site has a brief description of the history of that process.


text:
How to Invest Like a Billionaire and Own a Sports Franchise
Analysts see a potential 25% gain in share price for Madison Square Garden, which owns the NBA's New York Knicks and NHL's New York Rangers.
John Persinos
Why let the super rich have all the fun? You can make money and own a stake in a high-profile professional sports franchise.
It may be a small number of shares, but you'll still be able to brag that you own part of a team like the NBA's New York Knicks.
You can join the ranks of billionaire, big egos like Liberty Media's John Malone (Atlanta Braves), Carnival Cruise's Micky Arison (Miami Heat), and Microsoft's Paul Allen (Seattle Seahawks and Portland Trailblazers) who purchased their teams and are regular sideline presences at home games.
None of these teams are publicly traded. But investors can buy stock in the corporate entities that own other teams.
Consider Madison Square Garden, which owns the Knicks, one of the country's most celebrated sports franchises, the NHL's New York Rangers and the iconic circular arena, Madison Square Garden. MSG represents a good investment opportunity because of its recent maneuvering and the high-profile nature of its teams and other sports and entertainment events.
MSG, which operates through two divisions, MSG Sports and MSG Entertainment, is looking to expand its brand nationwide. Moreover, it meets several of the fundamental criteria that Warren Buffett rigorously applies to any stock before buying it. The company has a highly recognizable brand, and there's clear demand for its products. Analysts see potentially big growth prospects for the stock.
MSG Sports operates the Knicks, the Rangers and the WNBA's New York Liberty. This unit also promotes and produces other live sporting events, including professional boxing, college basketball and tennis.
The MSG Entertainment unit presents and hosts live entertainment events, including concerts, family shows, performing arts and special events in a diverse collection of venues, including The Theater at Madison Square Garden, Radio City Music Hall, the Beacon Theatre, the Forum, The Chicago Theatre and the Wang Theatre.
In addition, MSG Entertainment creates, produces and presents live productions, including the Radio City Christmas Spectacular and New York Spring Spectacular, both featuring the Rockettes.
MSG now represents the sort of untapped market dominance that value investors seek in a stock. Consider the entity's recent history.
On Sept. 30, the spin-off of Madison Square Garden from MSG Networks (MSGN - Get Report) was completed. Under the deal, shareholders in the former Madison Square Garden, backed by the ownership family of James Dolan, received one share in its sports and entertainment business for every three shares they held.
In the spinoff, MSG retained the Knicks and Rangers, among other teams, and Madison Square Garden and Radio City Music Hall.
The new MSG entity is keeping the parent's name and ticker symbol.
MSG Networks is remaining as operator of MSG Network and MSG+, which are regional sports networks that telecast about 700 live sporting events.
MSG's operational results for both the fiscal 2016 and 2015 first quarter were reported as the combined numbers of the sports and entertainment businesses, which had been part of MSG Networks prior to the spin-off.
First quarter revenue for fiscal 2016 was $150.4 million, a year-over-year increase of 26%. The quarter's operating loss of $4.6 million improved by $30.5 million compared to the same period a year ago, which equates to an earnings a share (EPS) loss of 6 cents versus a loss of $1.51 a year ago.
With the costs of the spinoff now in the rear view mirror, MSG can focus on live sports and entertainment with an ambitious growth strategy to extend its operations beyond New York.
MSG is now a more streamlined organization and better positioned to take advantage of its inherently strong assets, not least of which is the fame of its sports and entertainment brands. The company is capitalized with $1.47 billion, giving it enough financial muscle to pursue not only acquisitions but also buyback stock.
MSG closed trading Tuesday at a little over $164.39 a share. But analysts are setting a mean target price of $205, a 25% increase from today's price. If you want to buy a sports and entertainment juggernaut that's poised to beat the market in 2016, MSG is a slam dunk.
As 2016 looms around the corner, which stocks should you buy or sell? We suggest you take your cue from the greatest investor of all time, Warren Buffett. To learn what the Oracle of Omaha is buying and selling right now, download a copy of this free report.
John Persinos is editorial manager and investment analyst at Investing Daily. At the time of publication, the author held no positions in the stocks mentioned.
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