text:
## QIP 2015

The website is up for QIP 2015, which will be held this year in beautiful Sydney, Australia. Here is a timeline of the relevant dates:

• Submission of talks deadline: Sep 12, 2014
• Submission of posters deadline: Oct 25, 2014
• Decision on talks and posters submitted before talk deadline: Oct 20, 2014
• Decision on posters submitted after talk deadline: Nov 15, 2014
• Tutorial Session: Jan 10-11, 2015
• Main Conference: Jan 12-16, 2015

And students, don’t worry, there are plans to include some student support scholarships, so we hope that many of you can attend. We’re looking forward to seeing you all here!

Posted in Announcement, Conferences, QIS Annoucements, Quantum | 3 Comments

## Elsevier again, and collective action

We all know about higher education being squeezed financially. Government support is falling and tuition is going up. We see academic jobs getting scarcer, and more temporary. The pressure for research to focus on the short term is going up. Some of these changes may be fair, since society always has to balance its immediate priorities against its long-term progress. At other times, like when comparing the NSF’s \$7.6 billion FY2014 budget request to the ongoing travesty that is military procurement, it does feel as though we are eating our seed corn for not very wise reasons.

Against this backdrop, the travesty that is scientific publishing may feel like small potatoes. But now we are starting to find out just how many potatoes. Tim Gowers has been doing an impressive job of digging up exactly how much various British universities pay for their Elsevier subscriptions. Here is his current list. Just to pick one random example, the University of Bristol (my former employer), currently pays Elsevier a little over 800,000 pounds (currently \$1.35M) for a year’s access to their journals. Presumably almost all research universities pay comparable amounts.

To put this number in perspective, let’s compare it not to the F-35, but to something that delivers similar value: arxiv.org. Its total budget for 2014 is about 750,000 US dollars (depending on how you count overhead), and of course this includes access for the entire world, not only the University of Bristol. To be fair, ScienceDirect has about 12 times as many articles and the median quality is probably higher. But overall it is clearly vastly more expensive for society to have its researchers communicate in this way.

Another way to view the £800,000 price tag is in terms of the salaries of about 40 lecturers ($\approx$ assistant professors), or some equivalent mix of administrators, lecturers and full professors. The problem is that these are not substitutes. If Bristol hired 40 lecturers, they would not each spend one month per year building nearly-free open-access platforms and convincing the world to use them; they would go about getting grants, recruiting grad students and publishing in the usual venues. There are problems of collective action, of the path dependence that comes with a reputation economy and of the diffuse costs and concentrated benefits of the current system.

I wish I could end with some more positive things to say. I think at least for now it is worth getting across the idea that there is a crisis, and that we should all do what we can to help with it, especially when we can do so without personal cost. In this way, we can hopefully create new social norms. For example, it is happily unconventional now to not post work on arxiv.org, and I hope that it comes to be seen also as unethical. In the past, it was common to debate whether QIP should have published proceedings. Now major CS conferences are cutting themselves loose from parasitic professional societies (see in particular the 3% vote in favor of the status quo) and QIP has begun debating whether to require all submissions be accompanied by arxiv posts (although this is of course not at all clear-cut). If we cannot have a revolution, hopefully we can at least figure out an evolutionary change towards a better scientific publishing system. And then we can try to improve military procurement.

Posted in \$\$\$, Science 2.0, Scientific Publishing | 19 Comments

## Quantum computers can work in principle

Gil Kalai has just posted on his blog a series of videos of his lectures entitled “why quantum computers cannot work.”  For those of us that have followed Gil’s position on this issue over the years, the content of the videos is not surprising. The surprising part is the superior production value relative to your typical videotaped lecture (at least for the first overview video).

I think the high gloss on these videos has the potential to sway low-information bystanders into thinking that there really is a debate about whether quantum computing is possible in principle. So let me be clear.

There is no debate! The expert consensus on the evidence is that large-scale quantum computation is possible in principle.

Quoting “expert consensus” like this is an appeal to authority, and my esteemed colleagues will rebuke me for not presenting the evidence. Aram has done an admirable job of presenting the evidence, but the unfortunate debate format distorts perception of the issue by creating the classic “two sides to a story” illusion. I think it’s best to be unequivocal to avoid misunderstanding.

The program that Gil lays forth is a speculative research agenda, devoid of any concrete microscopic physical predictions, and no physicist has investigated any of it because it is currently neither clear enough nor convincing enough. At the same time, it would be extremely interesting if it one day leads to a concrete conjectured model of physics in which quantum computers do not work. To make the ideas more credible, it would help to have a few-qubit model that is at least internally consistent, and even better, one that doesn’t contradict the dozens of on-going experiments. I genuinely hope that Gil or someone else can realize this thrilling possibility someday.

For now, though, the reality is that quantum computation continues to make exciting progress every year, both on theoretical and experimental levels, and we have every reason to believe that this steady progress will continue. Quantum theory firmly predicts (via the fault-tolerance threshold theorem) that large-scale quantum computation should be achievable if noise rates and correlations are low enough, and we are fast approaching the era where the experimentally achievable noise rates begin to touch the most optimistic threshold estimates. In parallel, the field continues to make contributions to other lines of research in high-energy physics, condensed matter, complexity theory, cryptography, signal processing, and many others. It’s an exciting time to be doing quantum physics.

And most importantly, we are open to being wrong. We all know what happens if you try to update your prior by conditioning on an outcome that had zero support. Gil and other quantum computing skeptics like Alicki play a vital role in helping us sharpen our arguments and remove any blind spots in our reasoning. But for now, the arguments against large-scale quantum computation are simply not convincing enough to draw more than an infinitesimal sliver of expert attention, and it’s likely to remain this way unless experimental progress starts to systematically falter or a concrete and consistent competing model of quantum noise is developed.

## TQC 2014!

While many of us are just recovering from QIP, I want to mention that the submission deadline is looming for the conference TQC, which perhaps should be called TQCCC because its full name is Theory of Quantum Computation, Communication and Cryptography. Perhaps this isn’t done because it would make the conference seem too classical? But TQQQC wouldn’t work so well either. I digress.

The key thing I want to mention is the imminent 15 Feb submission deadline.

I also want to mention that TQC is continuing to stay ahead of the curve with its open-access author-keeps-copyright proceedings, and this year with some limited open reviewing (details here). I recently spoke to a doctor who complained that despite even her Harvard Medical affiliation, she couldn’t access many relevant journals online. While results of taxpayer-funded research on drug efficacy, new treatments and risk factors remain locked up, at least our community is ensuring that anyone wanting to work on the PPT bound entanglement conjecture will be able to catch up to the research frontier without having to pay \$39.95 per article.

One nice feature about these proceedings is that if you later want to publish a longer version of your submission in a journal, then you will not face any barriers from TQC. I also want to explicitly address one concern that some have raised about TQC, which is that the published proceedings will prevent authors from publishing their work elsewhere. For many, the open access proceedings will be a welcome departure from the usual exploitative policies of not only commercial publishers like Elsevier, but also the academic societies like ACM and IEEE. But I know that others will say “I’m happy to sign your petitions, but at the end of the day, I still want to submit my result to PRL” and who am I to argue with this?

So I want to stress that submitting to TQC does not prevent submitting your results elsewhere, e.g. to PRL. If you publish one version in TQC and a substantially different version (i.e. with substantial new material) in PRL, then not only is TQC fine with it, but it is compatible with APS policy which I am quoting here:

Similar exceptions [to the prohibition against double publishing] are generally made for work disclosed earlier in abbreviated or preliminary form in published conference proceedings. In all such cases, however, authors should be certain that the paper submitted to the archival
journal does not contain verbatim text, identical figures or tables, or other copyrighted materials which were part of the earlier publications, without providing a copy of written permission from the copyright holder. [ed: TQC doesn't require copyright transfer, because it's not run by people who want to exploit you, so you're all set here] The paper must also contain a substantial body of new material that was not included in the prior disclosure. Earlier relevant published material should, of course, always be clearly referenced in the new submission.

I cannot help but mention that even this document (the “APS Policy on Prior Disclosure”) is behind a paywall and will cost you \$25 if your library doesn’t subscribe. But if you really want to support this machine and submit to PRL or anywhere else (and enjoy another round of refereeing), TQC will not get in your way.

Part of what makes this easy is TQC’s civilized copyright policy (i.e. you keep it). By contrast, Thomas and Umesh had a more difficult, though eventually resolved, situation when combining STOC/FOCS with Nature.

## Two Cultures in One of the Cultures

A long time ago in a mental universe far far away I gave a talk to a theory seminar about quantum algorithms. An excerpt from the abstract:

Quantum computers can outperform their classical brethren at a variety of algorithmic tasks….[yadda yadda yadaa deleted]… This talk will assume no prior knowledge of quantum theory…

The other day I was looking at recent or forthcoming interesting quantum talks and I stumbled upon one by a living pontiff:

In this talk, I’ll describe connections between the unique games conjecture (or more precisely, the closely relatedly problem of small-set expansion) and the quantum separability problem… [amazing stuff deleted]…The talk will not assume any knowledge of quantum mechanics, or for that matter, of the unique games conjecture or the Lasserre hierarchy….

And another for a talk to kick off a program at the Simons institute on Hamiltonian complexity (looks totally fantastic, wish I could be a fly on the wall at that one!):

The title of this talk is the name of a program being hosted this semester at the Simons Institute for the Theory of Computing….[description of field of Hamiltonian complexity deleted...] No prior knowledge of quantum mechanics or quantum computation will be assumed.

Talks are tricky. Tailoring your talk to your audience is probably one of the trickier sub-trickinesses of giving a talk. But remind me again, why are we apologizing to theoretical computer scientists / mathematicians (which are likely the audiences for the three talks I linked to) for their ignorance of quantum theory? Imagine theoretical computer science talks coming along with a disclaimer, “no prior knowledge of the PCP theorem is assumed”, “no prior knowledge of polynomial-time approximation schemes is assumed”, etc. Why is it still considered necessary, decades after Shor’s algorithm and error correction showed that quantum computing is indeed a fascinating and important idea in computer science, to apologize to an audience for a large gap in their basic knowledge of the universe?

As a counter argument, I’d love to hear from a non-quantum computing person who was swayed to attend a talk because it said that no prior knowledge of quantum theory is assumed. Has that ever worked? (Or similar claims of a cross cultural prereq swaying you to bravely go where none of your kind has gone before.)

## Error correcting aliens

Happy New Year!  To celebrate let’s talk about error correcting codes and….aliens.

The universe, as many have noted, is kind of like a computer.  Or at least our best description of the universe is given by equations that prescribe how various quantities change in time, a lot like a computer program describes how data in a computer changes in time.  Of course, this ignores one of the fundamental differences between our universe and our computers: our computers tend to be able to persist information over long periods of time.  In contrast, the quantities describing our universe tend to have a hard time being recoverable after even a short amount of time: the location (wave function) of an atom, unless carefully controlled, is impacted by an environment that quickly makes it impossible to recover the initial bits (qubits) of the location of the atom.

Computers, then, are special objects in our universe, ones that persist and allow manipulation of long lived bits of information.  A lot like life.  The bits describing me, the structural stuff of my bones, skin, and muscles, the more concretely information theoretic bits of my grumbly personality and memories, the DNA describing how to build a new version of me, are all pieces of information that persist over what we call a lifetime, despite the constant gnaw of second law armed foes that would transform me into unliving goo.  Maintaining my bits in the face of phase transitions, viruses, bowel obstructions, cardiac arrest, car accidents, and bears is what human life is all about, and we increasingly do it well, with life expectancy now approaching 80 years in many parts of the world.

But 80 years is not that long.  Our universe is 13.8ish billion years old, or about 170ish million current lucky human’s life expectancies.  Most of us would, all things equal, like to live even longer.  We’re not big fans of death.  So what obstacles are there toward life extension?  Yadda yadda biology squishy stuff, yes.  Not qualified to go there so I won’t.  But since life is like a computer in regards to maintaining information, we can look toward our understanding of what allows computers to preserve information…and extrapolate!

Enter error correction.  If bits are subject to processes that flip the values of the bits, then you’ll lose information.  If, however, we give up storing information in each individual bit and instead store single bits across multiple individual noisy bits, we can make this spread out bit live much longer.  Instead of saying 0, and watching it decay to unknown value, say 000…00, 0 many times, and ask if the majority of these values remain 0.  Viola you’ve got an error correcting code.  Your smeared out information will be preserved longer, but, and here is the important point, at the cost of using more bits.

Formalizing this a bit, there are a class of beautiful theorems, due originally to von Neumann, classically, and a host of others, quantumly, called the threshold theorems for fault-tolerant computing which tell you, given an model for how errors occur, how fast they occur, and how fast you can compute, whether you can reliably compute. Roughly these theorems all say something like: if your error rate is below some threshold, then if you want to compute while failing at a particular better rate, then you can do this using a complicated larger construction that is larger proportional to a polynomial in the log of inverse of the error rate you wish to achieve. What I’d like to pull out of these theorems for talking about life are two things: 1) there is an overhead to reliably compute, this overhead is both larger, in size, and takes longer, in time, to compute and 2) the scaling of this overhead depends crucially on the error model assumed.

Which leads back to life. If it is a crucial part of life to preserve information, to keep your bits moving down the timeline, then it seems that the threshold theorems will have something, ultimately, to say about extending your lifespan. What are the error models and what are the fundamental error rates of current human life? Is there a threshold theorem for life? I’m not sure we understand biology well enough to pin this down yet, but I do believe we can use the above discussion to extrapolate about our future evolution.

Or, because witnessing evolution of humans out of their present state seemingly requires waiting a really long time, or technology we currently don’t have, let’s apply this to…aliens. 13.8 billion years is a long time. It now looks like there are lots of planets. If intelligent life arose on those planets billions of years ago, then it is likely that it has also had billions of years to evolve past the level of intelligence that infects our current human era. Which is to say it seems like any such hypothetical aliens would have had time to push the boundaries of the threshold theorem for life. They would have manipulated and technologically engineered themselves into beings that live for a long period of time. They would have applied the constructions of the threshold theorem for life to themselves, lengthening their life by apply principles of fault-tolerant computing.

As we’ve witnessed over the last century, intelligent life seems to hit a point in its life where rapid technological change occurs. Supposing that the period of time in which life spends going from reproducing, not intelligent stuff, to megalords of technological magic in which the life can modify itself and apply the constructions of the threshold theorem for life, is fast, then it seems that most life will be found at the two ends of the spectrum, unthinking goo, or creatures who have climbed the threshold theorem for life to extend their lifespans to extremely long lifetimes. Which lets us think about what alien intelligent life would look like: it will be pushing the boundaries of using the threshold theorem for life.

Which lets us make predictions about what this advanced alien life would look life. First, and probably most importantly, it would be slow. We know that our own biology operates at an error rate that ends up being about 80 years. If we want to extend this further, then taking our guidance from the threshold theorems of computation, we will have to use larger constructions and slower constructions in order to extend this lifetime. And, another important point, we have to do this for each new error model which comes to dominate our death rate. That is, today, cardiac arrest kills the highest percentage of people. This is one error model, so to speak. Once you’ve conquered it, you can go down the line, thinking about error models like earthquakes, falls off cliffs, etc. So, likely, if you want to live a long time, you won’t just be slightly slow compared to our current biological life, but instead extremely slow. And large.

And now you can see my resolution to the Fermi paradox. The Fermi paradox is a fancy way of saying “where are the (intelligent) aliens?” Perhaps we have not found intelligent life because the natural fixed point of intelligent evolution is to produce entities for which our 80 year lifespans is not even a fraction of one of their basic clock cycles. Perhaps we don’t see aliens because, unless you catch life in the short transition from unthinking goo to masters of the universe, the aliens are just operating on too slow a timescale. To discover aliens, we must correlate observations over a long timespan, something we have not yet had the tools and time to do. Even more interesting the threshold theorems also have you spread your information out across a large number of individually erring sub-systems. So not only do you have to look at longer time scales, you also need to make correlated observations over larger and larger systems. Individual bits in error correcting codes look as noisy as before, it is only in the aggregate that information is preserved over longer timespans. So not only do we have too look slower, we need to do so over larger chunks of space. We don’t see aliens, dear Fermi, because we are young and impatient.

And about those error models. Our medical technology is valiantly tackling a long list of human concerns. But those advanced aliens, what kind of error models are they most concerned with? Might I suggest that among those error models, on the long list of things that might not have been fixed by their current setup, the things that end up limiting their error rate, might not we be on that list? So quick, up the ladder of threshold theorems for life, before we end up an error model in some more advanced creatures slow intelligent mind!

## Are articles in high-impact journals more like designer handbags, or monarch butterflies?

Monarch butterfly handbag (src)

US biologist Randy Schekman, who shared this year’s physiology and medicine Nobel prize, has made prompt use of his new bully pulpit. In

How journals like Nature, Cell and Science are damaging science: The incentives offered by top journals distort science, just as big bonuses distort banking

he singled out these “luxury” journals as a particularly harmful part of the current milieu in which “the biggest rewards follow the flashiest work, not the best,” and he vowed no longer to publish in them. An accompanying Guardian article includes defensive quotes from representatives of Science and Nature, especially in response to Schekman’s assertions that the journals favor controversial articles over boring but scientifically more important ones like replication studies, and that they deliberately seek to boost their impact factors by restricting the number of articles published, “like fashion designers who create limited-edition handbags or suits.”  Focusing on journals, his main concrete suggestion is to increase the role of open-access online journals like his elife, supported by philanthropic foundations rather than subscriptions. But Schekman acknowledges that blame extends to funding organizations and universities, which use publication in high-impact-factor journals as a flawed proxy for quality, and to scientists who succumb to the perverse incentives to put career advancement ahead of good science.  Similar points were made last year in Serge Haroche’s thoughtful piece on why it’s harder to do good science now than in his youth.   This, and Nature‘s recent story on Brazilian journals’ manipulation of impact factor statistics, illustrate how prestige journals are part of the solution as well as the problem.

Weary of people and institutions competing for the moral high ground in a complex terrain, I sought a less value-laden approach,  in which scientists, universities, and journals would be viewed merely as interacting IGUSes (information gathering and utilizing systems), operating with incomplete information about one another. In such an environment, reliance on proxies is inevitable, and the evolution of false advertising is a phenomenon to be studied rather than disparaged.  A review article on biological mimicry introduced me to some of the refreshingly blunt standard terminology of that field.  Mimicry,  it said,  involves three roles:  a model,  i.e.,  a living or material agent emitting perceptible signals, a mimic that plagiarizes the model, and a dupe whose senses are receptive to the model’s signal and which is thus deceived by the mimic’s similar signals.  As in human affairs, it is not uncommon for a single player to perform several of these roles simultaneously.

## QIP 2014 accepted talks

This Thanksgiving, even if we can’t all be fortunate enough to be presenting a talk at QIP, we can be thankful for being part of a vibrant research community with so many different lines of work going on. The QIP 2014 accepted talks are now posted with 36 out of 222 accepted. While many of the hot topics of yesteryear (hidden subgroup problem, capacity of the depolarizing channel) have fallen by the wayside, there is still good work happening in the old core topics (algorithms, information theory, complexity, coding, Bell inequalities) and in topics that have moved into the mainstream (untrusted devices, topological order, Hamiltonian complexity).

Here is a list of talks, loosely categorized by topic (inspired by Thomas’s list from last year). I’m pretty sad about missing my first QIP since I joined the field, because its unusually late timing overlaps the first week of the semester at MIT. But in advance of the talks, I’ll write a few words (in italics) about what I would be excited about hearing if I were there.

### Quantum Shannon Theory

There are a lot of new entropies! Some of these may be useful – at first for tackling strong converses, but eventually maybe for other applications as well. Others may, well, just contribute to the entropy of the universe. The bounds on entanglement rate of Hamiltonians are exciting, and looking at them, I wonder why it took so long for us to find them.

1a. A new quantum generalization of the Rényi divergence with applications to the strong converse in quantum channel coding
Frédéric Dupuis, Serge Fehr, Martin Müller-Lennert, Oleg Szehr, Marco Tomamichel, Mark Wilde, Andreas Winter and Dong Yang. 1306.3142 1306.1586
merged with
1b. Quantum hypothesis testing and the operational interpretation of the quantum Renyi divergences
Milan Mosonyi and Tomohiro Ogawa. 1309.3228

25. Zero-error source-channel coding with entanglement
Jop Briet, Harry Buhrman, Monique Laurent, Teresa Piovesan and Giannicola Scarpa. 1308.4283

28. Bound entangled states with secret key and their classical counterpart
Maris Ozols, Graeme Smith and John A. Smolin. 1305.0848
It’s funny how bound key is a topic for quant-ph, even though it is something that is in principle purely a classical question. I think this probably is because of Charlie’s influence. (Note that this particular paper is mostly quantum.)

3a. Entanglement rates and area laws
Michaël Mariën, Karel Van Acoleyen and Frank Verstraete. 1304.5931 (This one could also be put in the condensed-matter category.)
merged with
3b. Quantum skew divergence

22. Quantum subdivision capacities and continuous-time quantum coding
Alexander Müller-Hermes, David Reeb and Michael Wolf. 1310.2856

### Quantum Algorithms

This first paper is something I tried (unsuccessfully, needless to say) to disprove for a long time. I still think that this paper contains yet-undigested clues about the difficulties of non-FT simulations.

2a. Exponential improvement in precision for Hamiltonian-evolution simulation
Dominic Berry, Richard Cleve and Rolando Somma. 1308.5424
merged with
2b. Quantum simulation of sparse Hamiltonians and continuous queries with optimal error dependence
Andrew Childs and Robin Kothari.
update: The papers appear now to be merged. The joint (five-author) paper is 1312.1414.

35. Nested quantum walk
Andrew Childs, Stacey Jeffery, Robin Kothari and Frederic Magniez.
(not sure about arxiv # – maybe this is a generalization of 1302.7316?)

### Quantum games: from Bell inequalities to Tsirelson inequalities

It is interesting how the first generation of quantum information results is about showing the power of entanglement, and now we are all trying to limit the power of entanglement. These papers are, in a sense, about toy problems. But I think the math of Tsirelson-type inequalities is going to be important in the future. For example, the monogamy bounds that I’ve recently become obsessed with can be seen as upper bounds on the entangled value of symmetrized games.

4a. Binary constraint system games and locally commutative reductions
Zhengfeng Ji. 1310.3794
merged with
4b. Characterization of binary constraint system games
Richard Cleve and Rajat Mittal. 1209.2729

20. A parallel repetition theorem for entangled projection games
Irit Dinur, David Steurer and Thomas Vidick. 1310.4113

33. Parallel repetition of entangled games with exponential decay via the superposed information cost
André Chailloux and Giannicola Scarpa. 1310.7787

### Untrusted randomness generation

Somehow self-testing has exploded! There is a lot of information theory here, but the convex geometry of conditional probability distributions also is relevant, and it will be interesting to see more connections here in the future.

5a. Self-testing quantum dice certified by an uncertainty principle
Carl Miller and Yaoyun Shi.
merged with
5b. Robust device-independent randomness amplification from any min-entropy source
Kai-Min Chung, Yaoyun Shi and Xiaodi Wu.

19. Infinite randomness expansion and amplification with a constant number of devices
Matthew Coudron and Henry Yuen. 1310.6755

29. Robust device-independent randomness amplification with few devices
Fernando Brandao, Ravishankar Ramanathan, Andrzej Grudka, Karol Horodecki, Michal Horodecki and Pawel Horodecki. 1310.4544

### The fuzzy area between quantum complexity theory, quantum algorithms and classical simulation of quantum systems. (But not Hamiltonian complexity.)

I had a bit of trouble categorizing these, and also in deciding how surprising I should find each of the results. I am also somewhat embarrassed about still not really knowing exactly what a quantum double is.

6. Quantum interactive proofs and the complexity of entanglement detection
Kevin Milner, Gus Gutoski, Patrick Hayden and Mark Wilde. 1308.5788

7. Quantum Fourier transforms and the complexity of link invariants for quantum doubles of finite groups
Hari Krovi and Alexander Russell. 1210.1550

16. Purifications of multipartite states: limitations and constructive methods
Gemma De Las Cuevas, Norbert Schuch, David Pérez-García and J. Ignacio Cirac.

### Hamiltonian complexity started as a branch of quantum complexity theory but by now has mostly devoured its host

A lot of exciting results. The poly-time algorithm for 1D Hamiltonians appears not quite ready for practice yet, but I think it is close. The Cubitt-Montanaro classification theorem brings new focus to transverse-field Ising, and to the weird world of stoquastic Hamiltonians (along which lines I think the strengths of stoquastic adiabatic evolution deserve more attention). The other papers each do more or less what we expect, but introduce a lot of technical tools that will likely see more use in the coming years.

13. A polynomial-time algorithm for the ground state of 1D gapped local Hamiltonians
Zeph Landau, Umesh Vazirani and Thomas Vidick. 1307.5143

15. Classification of the complexity of local Hamiltonian problems
Toby Cubitt and Ashley Montanaro. 1311.3161

30. Undecidability of the spectral gap
Toby Cubitt, David Pérez-García and Michael Wolf.

23. The Bose-Hubbard model is QMA-complete
Andrew M. Childs, David Gosset and Zak Webb. 1311.3297

24. Quantum 3-SAT is QMA1-complete
David Gosset and Daniel Nagaj. 1302.0290

26. Quantum locally testable codes
Dorit Aharonov and Lior Eldar. (also QECC) 1310.5664

### Codes with spatially local generators aka topological order aka quantum Kitaev theory

If a theorist is going to make some awesome contribution to building a quantum computer, it will probably be via this category. Yoshida’s paper is very exciting, although I think the big breakthroughs here were in Haah’s still underappreciated masterpiece. Kim’s work gives operational meaning to the topological entanglement entropy, a quantity I had always viewed with perhaps undeserved skepticism. It too was partially anticipated by an earlier paper, by Osborne.

8. Classical and quantum fractal code
Beni Yoshida. I think this title is a QIP-friendly rebranding of 1302.6248

21. Long-range entanglement is necessary for a topological storage of information
Isaac Kim. 1304.3925

### Bit commitment is still impossible (sorry Horace Yuen) but information-theoretic two-party crypto is alive and well

The math in this area is getting nicer, and the protocols more realistic. The most unrealistic thing about two-party crypto is probably the idea that it would ever be used, when people either don’t care about security or don’t even trust NIST not to be a tool of the NSA.

10. Entanglement sampling and applications
Frédéric Dupuis, Omar Fawzi and Stephanie Wehner. 1305.1316

36. Single-shot security for one-time memories in the isolated qubits model
Yi-Kai Liu. 1304.5007

### Communication complexity

It is interesting how quantum and classical techniques are not so far apart for many of these problems, in part because classical TCS is approaching so many problem using norms, SDPs, Banach spaces, random matrices, etc.

12. Efficient quantum protocols for XOR functions
Shengyu Zhang. 1307.6738

9. Noisy Interactive quantum communication
Gilles Brassard, Ashwin Nayak, Alain Tapp, Dave Touchette and Falk Unger. 1309.2643 (also info th / coding)

### Foundations

I hate to be a Philistine, but I wonder what disaster would befall us if there WERE a psi-epistemic model that worked. Apart from being able to prove false statements. Maybe a commenter can help?

14. No psi-epistemic model can explain the indistinguishability of quantum states
Eric Cavalcanti, Jonathan Barrett, Raymond Lal and Owen Maroney. 1310.8302

### ??? but it’s from THE RAT

update: A source on the PC says that this is an intriguing mixture of foundations and Bell inequalities, again in the “Tsirelson regime” of exploring the boundary between quantum and non-signaling.
17. Almost quantum
Miguel Navascues, Yelena Guryanova, Matty Hoban and Antonio Acín.

I love the part where Daniel promises not to cheat. Even though I am not actively researching in this area, I think the race between surface codes and concatenated codes is pretty exciting.

18. What is the overhead required for fault-tolerant quantum computation?
Daniel Gottesman. 1310.2984

27. Universal fault-tolerant quantum computation with only transversal gates and error correction
Adam Paetznick and Ben Reichardt. 1304.3709

### Once we equilibrate, we will still spend a fraction of our time discussing thermodynamics and quantum Markov chains

I love just how diverse and deep this category is. There are many specific questions that would be great to know about, and the fact that big general questions are still being solved is a sign of how far we still have to go. I enjoyed seeing the Brown-Fawzi paper solve problems that stumped me in my earlier work on the subject, and I was also impressed by the Cubitt et al paper being able to make a new and basic statement about classical Markov chains. The other two made me happy through their “the more entropies the better” approach to the world.

31. An improved Landauer Principle with finite-size corrections and applications to statistical physics
David Reeb and Michael M. Wolf. 1306.4352

32. The second laws of quantum thermodynamics
Fernando Brandao, Michal Horodecki, Jonathan Oppenheim, Nelly Ng and Stephanie Wehner. 1305.5278

34. Decoupling with random quantum circuits
Winton Brown and Omar Fawzi. 1307.0632

11. Stability of local quantum dissipative systems
Toby Cubitt, Angelo Lucia, Spyridon Michalakis and David Pérez-García. 1303.4744

Posted in Conferences, Quantum | 10 Comments

## Portrait of an Academic at a Midlife Crisis

Citations are the currency of academia. But the currency of your heart is another thing altogether. With apologies to my co-authors, here is a plot of my paper citations versus my own subjective rating of the paper. Hover over the circles to see the paper title, citations per year, and score. Click to see the actual paper. (I’ve only included papers that appear on the arxiv.)

If I were an economist I suppose at this point I would fit a sloping line through the data and claim victory. But being a lowly software developer, its more interesting to me to give a more anecdotal treatment of the data.

• The paper that I love the most has, as of today, exactly zero citations! Why do I love that paper? Not because it’s practical (far from it.) Not because it proves things to an absolute T (just ask the beloved referees of that paper.) But I love it because it says there is the possibility that there exists a material that quantum computes in a most peculiar manner. In particular the paper argues that it is possible to have devices where: quantum information starts on one side of device, you turn on a field over the device, and “bam!” the quantum information is now on the other side of the material with a quantum circuit applied to it. How f’in cool is that! I think its just wonderful, and had I stuck around the hallowed halls, I probably would still be yelling about how cool it is, much to the dismay of my friends and colleagues (especially those for which the use of the word adiabatic causes their brain to go spazo!)
• Three papers I was lucky enough to be involved in as a graduate student, wherein we showed how exchange interactions alone could quantum compute, have generated lots of citations. But the citations correlate poorly with my score. Why? Because it’s my score! Haha! In particular the paper I love the most out of this series is not the best written, most deep, practical, or highly cited. It’s the paper where we first showed that exchange alone was universal for quantum computing. The construction in the paper has large warts on it, but it was the paper where I think I first participated in a process where I felt like we knew something about the possibilities of building a quantum computer that others had not quite exactly thought of. And that feeling is wonderful and is why that paper has such a high subjective score.
• It’s hard not to look this decades worth of theory papers and be dismayed about how far they are from real implementation. I think that is why I like Coherence-preserving quantum bit and Adiabatic Quantum Teleportation. Both of these are super simple and always felt like if I could just get an experimentalist drunk enough excited enough they might try implement that damn thing. The first shows a way to make a qubit that should be more robust to errors because its ground state is in an error detecting state. The second shows a way to get quantum information to move between three qubits using a simple adiabatic procedure related to quantum teleportation. I still hope someday to see these executed on a functioning quantum computer, and I wonder how I’d feel about them should that happen.


text:
Investigations off arbitrary tree classifier with other classifiers

Anticipate performance towards the WGBS data and you may get across-program anticipate. Precision–bear in mind contours having get across-platform and you may WGBS prediction. Per accuracy–remember contour signifies the common precision–keep in mind to possess anticipate on kept-aside set for joingy every of your ten regular arbitrary subsamples. WGBS, whole-genome bisulfite sequencing.

We compared brand new prediction show of one’s RF classifier with many different other classifiers which have been commonly used when you look at the associated really works (Desk step 3). Particularly, we opposed all of our prediction is a result of the brand new RF classifier with the individuals out-of good SVM classifier which have a beneficial radial foundation function kernel, a great k-nearby locals classifier (k-NN), logistic regression, and you may an unsuspecting Bayes classifier. We made use of similar function set for everybody classifiers, and the 122 provides useful for prediction out-of methylation standing with the fresh RF classifier. We quantified results having fun with repeated random resampling which have identical training and you will attempt kits across classifiers.

I unearthed that brand new k-NN classifier shown the fresh bad overall performance with this task, having an accuracy off 73.2% and you can a keen AUC off 0.80 (Profile 5B). The brand new naive Bayes classifier presented top accuracy (80.8%) and AUC (0.91). Logistic regression and SVM classifier one another shown a great abilities, with accuracies of 91.1% and you will 91.3% and you will AUCs out of 0.96% and you will 0.96%, correspondingly. We learned that our RF classifier displayed somewhat most readily useful prediction accuracy than logistic regression (t-test; P=3.8?ten ?sixteen ) in addition to SVM (t-test; P=1.3?10 ?thirteen ). I mention together with your computational day needed to train and you will try the brand new RF classifier try considerably below enough time necessary to the SVM, k-NN (try merely), and you can unsuspecting Bayes classifiers. I picked RF classifiers for this task since the, and the growth during the accuracy over SVMs, we were capable assess brand new contribution to prediction of any function, which we identify lower than.

## Region-certain methylation prediction

Degree from DNA methylation features worried about methylation within this promoter countries, restricting predictions to CGIs [40,41,43-46,48]; i although some have demostrated DNA methylation provides some other patterns within the this type of genomic places according to the remainder genome , therefore, the accuracy of them prediction tips outside of this type of nations try not sure. Right here i examined regional DNA methylation anticipate for our genome-greater CpG website forecast means restricted to CpGs inside certain genomic nations (Most file step 1: Dining table S3). Because of it try, forecast is simply for CpG internet sites that have neighboring sites contained in this step 1 kb length of the small size regarding CGIs.

Within CGI regions, we found that predictions of methylation status using our method had an accuracy of 98.3%. We found that methylation level prediction within CGIs had an r=0.94 and a root-mean-square error (RMSE) of 0.09. As in related work on prediction within CGI regions, we believe the improvement in accuracy is due to the limited variability in methylation patterns in these regions; indeed, 90.3% of CpG sites in CGI regions have ?<0.5 (Additional file 1: Table S4). Conversely, prediction of CpG methylation status within CGI shores had an accuracy of 89.8%. This lower accuracy is consistent with observations of robust and drastic change in methylation status across these regions [62,63]. Prediction performance within various gene regions was fairly consistent, with 94.9% accuracy for predictions of CpG sites within promoter regions, 93.4% accuracy within gene body regions (exons and introns), and 93.1% accuracy within intergenic regions. Because of the imbalance of hypomethylated and hypermethylated sites in each region, we evaluated both the precision–recall curves and ROC curves for these predictions (Figure 5C and Additional file 1: Figure S8).

## Predicting genome-greater methylation accounts all over platforms

CpG methylation levels ? in a DNA sample represent the average methylation status across the cells in that sample and will vary continuously between 0 and 1 (Additional file 1: Figure S9). Since the Illumina 450K array measures precise methylation levels at CpG site resolution, we used our RF classifier to predict methylation levels at single-CpG-site resolution. We compared the prediction probability ( $$<\hat>_ \in \left [0,1\right ]$$ ) from our RF classifier (without thresholding) with methylation levels (? we,j ? [0,1]) from the array, and validated this approach using repeated random subsampling to quantify generalization accuracy (see Materials and methods). Including all 122 features used in methylation status prediction, but modifying the neighboring CpG site methylation status ? to be continuous methylation levels ?, we trained our RF classifier on 450K array data and evaluated the Pearson’s correlation coefficient (r) and RMSE between experimental and predicted methylation levels (Table 1; Figure 5D). We found that the experimentally assayed and predicted methylation levels had r=0.90 and RMSE =0.19. The correlation coefficient and the RMSE indicate good recapitulation of experimentally assayed levels using predicted methylation levels across CpG sites.

Sluiten

format: plain


text:
# help in time interval cutout circuit

Discussion in 'The Projects Forum' started by amankarwasra, Aug 1, 2010.

1. ### amankarwasra Thread Starter New Member

Jul 30, 2010
3
0

i need help in makking a circuit which:

run a 1 HP water moter for 2 mins and then stop water moter and run 8 fans for 10 mins................and thus making a loop of------------- water moter 2 min--------------fans 10 mins ---------water motor 2 mins--------- fans 10 mins --------and on and on

but they both can't be on at a given piont of timeonly any one of them

2. ### KMoffett AAC Fanatic!

Dec 19, 2007
2,575
230
I think this should do what you want. The fan and pump motor can never be on at the same time. Since there in interaction between the two time adjustments, set R3 first for 600 sec (10 Min) and then R2 for 120 sec (2 min). The 12V supply can be a 120VAC to 12VDC/500mA "regulated" wall wart or equivalent.

ken

• ###### 555 Astable Alternating Relay.gif
File size:
24.2 KB
Views:
21
amankarwasra likes this.
3. ### amankarwasra Thread Starter New Member

Jul 30, 2010
3
0

hi how do you set R3 and R2 and do we need a 12V battery extra for it or we connect the 12 to the main wires actually am all new to this so please explain in details and yes thanks for your time dear

are there 2 transistors and what is regulated wall wart

Last edited: Aug 1, 2010
4. ### tom66 Senior Member

May 9, 2009
2,613
214
@OP: A regulated wall wart supplies a fairly constant DC voltage. You can pick them up for a few dollars, or use a spare one you have. It must be DC, and it should be regulated. You could also use a car battery, but it would be overkill.

The schematic shows R2 and R3 as potentiometers, i.e. adjustable voltage dividers or a pair of complementary variable resistors. These adjust the on/off time, by turning the pots (potentiometers) you can get your desired settings. You MUST use a CMOS 555 with this (such as the LMC555 as spec'd), you cannot use any other type, because the resistors are too large for another type.

amankarwasra likes this.
5. ### amankarwasra Thread Starter New Member

Jul 30, 2010
3
0
but when i am using it with all AC equipments do i need the car battery etc. and actually i meant by setting potentiometer that do we have to just guess by turning the know or they come with any marking like our fan regulators

6. ### tom66 Senior Member

May 9, 2009
2,613
214
There's a formula for it.

The on time of any 555 astable is, in seconds:

$t_{on} = 0.693 \times (R_1 + R_2) \times C$

The off time is:

$t_{off} = 0.693 \times R_2 \times C$

And from these the frequency in Hz is:

$f = \frac{1}{t_{on}+t_{off}}$

You can measure the resistance with an ohmmeter and capacitance is usually printed on the side of a capacitor, or in this case is specified in the schematic. Remember to normalize all your values, i.e. 470 µF becomes 0.00047 and 1Mohm becomes 1,000,000.

And yes, the circuit posted will need a DC supply, you cannot run it off AC, you will damage the 555 timer. Since it's a CMOS timer, the current consumption is very low, you only need to know about the relay current, which should be no more than about 200mA.

7. ### KMoffett AAC Fanatic!

Dec 19, 2007
2,575
230
R2 in tom's calculations is R2+R3 on my schematic, and R1 is R1+R2 on the schematic.
The nominal values for 2 minutes and 10 minutes are R1+R2=1.5M and R2+R3=390K.
Once you've set those resistances with an Ohmmeter, tweak R3 first to get the 2 minutes and then R2 to get 10 minutes.

This is a handy calculator to get the values and timing: url]http://home.cogeco.ca/~rpaisley4/LM555.html#3[/url]

Ken


text:
# 2017 Grade 9 Math Challenge – Division Finals Team Orals with answer key

Below are the 2017 MTAP Grade 9 Math Challenge Division Finals Team Orals questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.

15 – Second Questions [? ?????? ????]
1.) What value of ? will make $x^2 - 14x + c$ a perfect square trinomial?

2.) Solve for ? in the quadratic equation $x^2 + x - 6 = 0$

3.) What is the sum of the roots of $2x^2 + 6x + 1 = 0$?

4.) Suppose ? varies directly as ?. If ? = 6, then ? = 12. If ? = 2, find ?.

5.) Simplify the expression $\sqrt {49x^7}$

6.) The diagonals of a rhombus are 4 cm and 8 cm. Find the area of the rhombus.

7.) In a parallelogram, two adjacent angles have measures (2? + 2)° and (2? – 2)°. Find x.

8.) If $x^{0.123} = 4$ , what is 1 more than $x^{0.246}$?

9.) If ?: ? = 2: 1 and ?: ? = 2: 1, find the ratio ?: ?

10.) How long is the hypotenuse of a right triangle if its two legs are 1 cm and 2 cm?

11.) Find the vertex of the graph of $y = x^2 - 4x + 5$

30 – Second Questions [? ?????? ????]

1.) Simplify: $4 \sqrt{20} - 2 \sqrt{45}$

2.) If $9x + 19 = (x + 3)^2$

3.) In a triangle, a line parallel to the base divides one side into segments 4 and 14 units long. If the other side has length 27, how long (in units) is each segment formed when the line divides this side?

4.) Find the range of values of $x^2 + kx + 9k = 0$ has no real roots.

5.) Isosceles trapezoid ???? has parallel bases ?? and ??, and the diagonals intersect at ?. If ?? = 8, ?? = 3? – 1, ?? = 5? – 2, how long (in units) is ???

6.) The diagonals of a rhombus are 18 ?? and 24 ??. Find the length of each side.

1 – Minute Questions [? ?????? ????]

1.) Triangle ??? has a right angle at ?. Let ? be the foot of the altitude from ? to AC. If ?? = 10 and ?? = 6, find the length (in units) of ??.

2.) Solve for ? in the equation $\sqrt {x} + 2 = \sqrt{2x - 1}$

3.) Suppose ? varies directly as the cube of ? and inversely as ?. If ? = 6 when ? = . Find ? when ? = 3 and ? = 6.

4.) Fencing material that is 400 ? long is used to enclose 3 sides of a rectangular lot. What are the dimensions of such lot with the largest possible area?

5.) Find the smallest possible values of ? which satisfies $(2x + 1)^2 + 3x \geq 2(x + 1)^2 + 1$

6.) The shortest sides of two similar triangles are 5 ?? and 7 ??. If the sum of their perimeters is 48 ??, find the perimeter of the smaller triangle.

Tiebreaker

1.) Find the smallest positive integer solution of ?2 – 3? – 10 ≥ 0

2.) Rationalize the denominator of $\dfrac {1 + \sqrt{5}}{3- \sqrt{5}}$

3.) If ? and ? are the roots of $2x^2 - 7x + 4 = 0$, find the value of $r^2 + s^2$

Do-or-Die

1.) In a right triangle, the two legs have lengths ? and ? + 7?, while the hypotenuse has length ? + 8?, find the value of $\dfrac {a}{d}$?

15 – Second Questions [? ?????? ????]
1.) 49
2.) x = -3, 2
3.) -3
4.) y = 4
5.) $7x^3\sqrt{x}$
6.) $16 m^2$
7.) 45
8.) 17
9.) 4 : 1
10.) $\sqrt{5} cm$
11.) (2, 1)

30 – Second Questions [? ?????? ????]
1.) $2 \sqrt{2}$
2.) x = -2, 5
3.) 6, 21
4.) 0, 36
5.) 13 units
6.) 15 cm

1 – Minute Questions [? ?????? ????]
1.) $\dfrac {32}{3}$
2.) x = 25
3.)
4.) 100m, 200m
5.) $\dfrac {1}{2}$
6.) 20 cm

Tiebreaker
1.) 5
2.) $2 + \sqrt{5}$
3.) $\dfrac {33}{4}$

Do-or-Die
1.) 5

This entry was posted in Grade 1-2 and tagged , , , , . Bookmark the permalink.


text:
PrivateCore nets $2.25M to build a hardened hypervisor PrivateCore, a startup launched by two VMware(s vmw) veterans, aims to boost the physical security of your virtualized cloud computing workloads and will use$2.25 million in seed funding from Foundation Capital to staff up for that effort.

The need for security in virtualized compute environments will only grow as more workloads flow into public and private clouds. And, for customers wary of putting bare compute technology off premises at third-party cloud or service providers still have nagging concerns about the physical security of their data.

PrivateCore is building a secure hypervisor that should mitigate concerns over the physical attack vector, CEO Oded Horovitz told me in an interview. This hypervisor, now under development, will run VMs without changes on X86 machines.

“Right now, with any other hypervisor running, if someone is physically standing next to the machine, they can look at it, see the memory card and make a copy of all that data,” Oded said.

PrivateCore management has solid security DNA. Oded spent 5 years at VMware, where he developed vShield and VMSafe security products. His co-founder Dr. Stephen Weis has a Ph.D. in cryptography from MIT and designed Google’s(s goog) two-step verification system. Right now they are the extent of the company, with former VMware principal engineer Dr. Carl Waldspurger on as an advisor.

The issue they’re attacking is that computing resources — including those that run off-site in a third-party data center — use components that are now trusted, and that trust may be misplaced, Weis said.

One risk is that someone in the server room can pop out the memory card or boot the machine to a malicious OS that scrapes the memory. But perhaps scarier, is the notion that the very components in these servers and other data center devices are a threat in and of themselves.

“Network cards can be modified, someone can change the firmware to extract their contents and put it out over the network,” Weis said. One nagging issue is that many of these components originate in China so their provenance can be shaky given documented cases of back doors, he said. In addition, nonvolatile RAM will save contents for a long time after powering down…. if someone takes out that RAM, there’s another risk.
From the company’s statement announcing the seed round:
PrivateCore’s software provides a secure hypervisor that enables the private execution of virtual machines. This approach only allows the real owner of the virtual machine to access the data inside – and only with the right credentials (keys). PrivateCore’s technology prevents service provider staff that have physical ownership of the server from bypassing this protection and from gaining access to the owner’s data without having the owner’s keys.

For more on Privatecore, and the problem it’s taking on check out the video:

Image courtesy of Flickr user abardwell


text:
Ponzi Financial Dynamics - DecisionBoundaries

# Ponzi Financial Dynamics

## The Financial Dynamics of Unsustainable Fraudulent Investment Schemes

###### June 3rd, 2019

Abstract

We use a first order differential linear equation to describe the dynamics of an investment scheme that promises more than it can deliver (a Ponzi scheme). The model is based on (i) a promised, unrealistic, rate of return, (ii) the rate at which new investments are gathered, and (iii) the withdrawal rate. We establish the parameters that lead to the scheme’s collapse.

Introduction
The principle of Ponzi schemes is simple: they entice potential investors by promising high returns, which they cannot possibly deliver. The only was to pay the promised returns is by attracting new investors whose money is used to subsidize the returns of those already in the fund.

In this article we posit a simple model that attempts answer the basic questions: How fast must new investments come in? How long can the scheme last? What parameters drive the dynamics of the scheme?

The model is described in Section 2, with detailed results on the behavior of the scheme as a function of seven parameters.

Section 3 highlights the main results and concludes.

The Model
Assumptions

We assume that the fund starts at time $t=0$ with an initial investment $K\geq 0$, followed by a continuous cash inflow $s(t)$. We next assume a promised rate of return, $s_p$ and a nominal rate $r_n$ at which the money is actually invetsed. If $r_n\geq r_p$ then the fund is legitimate and has a profit rate $r_n-r_p$. If $r_n the fund is promising more than it can deliver. We call the promised rate, $r_p$, the “Ponzi rate”.

We need to model the fact that investors withdraw at least part of their money along the way. The simplest way of doing this is to assume a constant withdrawal rate, $r_w$, applied at every time $t$ to the promised accumulated capital. The withdrawal rate at time $t$ by those who invested the initial amount $K$ is $r_wKe^{t(r_p-r_w)}$. If $r_w$ is less that the promised return $r_p$, then these withdrawals increase exponentially; $r_w$ can also be larger than $r_p$, in which case withdrawals decrease exponentially, as these investors are eating into the capital $K$.

In order to calculate the withdrawals at time $t$ from those who added to the fund between times 0 and $t$, we note that those who invested $s(u)$ at time $u$ will want to withdraw a quantity $r_ws(u)e^{(r_p-r_w)(t-u)}$ at time $t>u$. Integrating these withdrawals from 0 to $t$ and adding the previously calculated withdrawals from the initial deposit $K$, results in the following withdrawals at time $t$:

$W(t) = r_w\Big(Ke^{t(r_p-r_w)}+\int_{0}^{t}s(u)e^{e(r_p-r_w)(t-u)}du\Big)$

We note that the nominal interest rate, $r_n$, does not appear in $W(t)$: withdrawals are a function of only the promised rate of return, $r_p$.

The differential Equation

If S(t) is the amount in the fund at time $t$, then $S(t+dt)$ is obtained by adding to $S(t)$, the nominal return $r_nS(t)dt$, the inflow of fresh money $s(t)$, and subtracting the withdrawals $W(t)dt$:

$S(t+dt)=S(t)+dt[r_nS_a(t)+s(t)-W(t)]$

For $dt\rightarrow 0$, the amount $S(t)$ is the solution to the first-order linear differential equation:

$\frac{S(t)}{dt}=r_nS_a(t)+s(t)-W(t)$

We let $C=S(0)$ be the initial condition which may or may not be equal to $K$, the initial investment made by customers. The fund managers can make an initial “in-house” investment $K_0\geq 0$, which will also be invested at the nominal rate $r_n$. In this case, the initial value $C=K_0+K$ is greater than $K$. An initial condition $C formally corresponds to the cae where, for some reason (theft or other), a fraction of the initial investment $K$ is not available. We will see later that the solution to the differential equation with an initial condition $C=S(0)$, other than $K$, will be used when there is, at some subsequent time $t^*$, a sudden change in parameter values (for example, if the cash inflow or withdrawal rate changes at $t^*$).

One often made assumption in the literature on the subject is that the cash inflow $s(t)$ grows at an exponential rate:

$s(t)=s_0e^{r_it}$

were $s_0$ is the initial density of the cash inflows and $r_i$ will be called the investment growth rate. Then, Eq.(1) becomes:

$W(t)=r_we^{t(rp-r_w)}\Big(K+s_0\frac{e^{t(r_w+r_t-r_p)}-1}{r_w+r_i-r_p}\Big)$

where the fraction should equal $t$ when $r_w+r_i-r_p=0$.

The solution, $S(t)$, to the differential equation (3) has a closed-form solution, which will be formulated using the function:

$g(t,a,b,c,d,\alpha) = ae^{bt}+ce^{dt}+\alpha$

Based on this notation, $S(t)$ becomes:

$S(t)=g(t,a,b,c,d,\alpha)e^{r_nt}=ae^{(b+r_n)t}+ce^{(d+r_n)t}+\alpha e^{r_nt}$

where:
$%a = b a = \frac{r_w[s_0-(r_i-r_p+r_w)K]}{(r_p-r_n-r_w)(r_i-r_p+r_w)},$

$b = r_p-r_n-r_w,$

$c = \frac{s_0(r_i-r_p)}{(r_i-r_n)(r_i-r_p+r_w)},$

$d = r_i-r_n,$

$\alpha = C-\frac{s_0(r_n-r_p)+Kr_w(r_i-r_n)}{(r_i-r_n)(r_n-r_p+r_w)}.$

The solution $S(t)$ to Eq.(7) is a linear combination of three exponentials which we are not able to tackle directly by elementary methods. Instead, the zeros of $s(t)$ and of its derivative can be calculated numerically but providing no insights into the function’s behavior.

On the other hand, if we know the number of positive zeros of $S(t)$ we can shed light on the conditions under which the fund is solvent ($S(t)$ remains positive). We will see below that, depending on parameter values, $S(t)$ of Eq. (7) has 0, 1, or 2 positive zeros. When there is no positive zero then $S(t)$ remains positive and the fuind is solvent. One positive zero means that $S(t)$ becomes negative and the scheme has collapsed. Two positive zeros mean that $S(t)$ becomes negative, reaches a negative minimum, then becomes positive again. The fund has collapsed but could recover with a bailout equal to the absolute value of the negative minimum. To simplify, we will say that, in this case, the scheme has collapsed and then recovered.

Analytical results on the number of positive zeros will result from noting that the zeros of $S(t)$ are also those of $g(t,a,b,c,d,\alpha)$ of Eq. (6). This function is a linear combination of only two exponentials plus a constant but the zeros still cannot be found in closed form by elementary method. However, the derivative of $g(t,a,b,c,d,\alpha)$ is a linear combination of two exponentials with no constant, which can be analytically studied. The following proposition provides results on the number of positive zeros of $g(t,a,b,c,d,\alpha)$.\\[12pt]

Proposition 1. We consider the function $g(t,a,b,c,d,\alpha)$ of Eq. (6) in the non-trivial case where $a,b,c,d\neq 0$ and $b\neq d$. We also assume that $g(0,ab,c,d,\alpha)\geq 0$. We further consider the following set of two conditions:

$U = \frac{cd}{ab},\;\; V = \frac{1+cd/ab}{b-d}\leq 0$

The function $g(t,a,b,c,d,\alpha)$ then has an extremum:

$m = a\Big(\frac{-cd}{ab}\Big)^{\frac{b}{b-d}}+c\Big(\frac{-cd}{ab}\Big)^{\frac{d}{b-d}}+\alpha$

at the positive value:

$t_c = \frac {ln\Big(\frac{-cd}{ab}\Big)}{b-d}$

if and only if Condition (13) is satisfied.

Depending on the number of positive zeros of $g(t,a,b,c,d,\alpha)$, we get results for four different cases $A_1,A_2,A_3,A_4$:

$A_1$: Condition (13) is satisfied and $ab+cd<0$. If $m>0$ then $g(t,a,b,c,d,\alpha)$ has no positive zero (and, therefore, remains positive). If $m<0$ and $b,d,\alpha<0$, then the function $g(t,a,b,c,d,\alpha)$ has exactly one positive zero at a value smaller than $t_c$. In all other cases, with $m<0$, the function has one positive zero on each side of $t_c$.

$A_2$: Condition (13) is satisfied and $ab+cd>0$. If $b,d<0,\,\alpha>0$, then the function has no positive zero. In all other cases there is one positive zero.

$A_3$: Condition (13) is not satisfied and $ab+cd<0$. If $b,d<0,\,\alpha>0$, the function has no positive zero. In all other cases there is one positive zero.

$A_4$: Condition (13) is not satisfied and $ab+cd>0$. There is no positive zero.

Proof. The proof is elementary and hinges on the following facts:

1. The derivative $g^{'}(t,a,b,c,d,\alpha)$ equals 0 at $t_c$ and $m$ is the value of $g(t,a,b,c,d,\alpha)$ at $t_c$; $t_c$ is positive if and only if Condition (13) is satisfied. The derivative $g^{'}(0,a,b,c,d,\alpha)$ at 0 is equal to $ab+cd$.
2. If $b$ and $d$ are negative, the function $g(t,a,b,c,d,\alpha)$ tends to $\alpha$ for $t\rightarrow \infty$. If $b$ or $d$ is positive, the function tends to $\pm \infty$ (depending on the signs of $a,c$).
3. If Condition (13) is not satisfied, then either $g(t,a,b,c,d,\alpha)$ has an extremum for a negative value of $t$ or no extremum at all. In both cases, the function $g(t,a,b,c,d,\alpha)$ for $t>0$ is monotone increasing if $ab+cd>0$ and monotone decreasing otherwise.

Main Result

In order to apply Proposition 1 to the parameters $a,b,c,d,\alpha$ of Eqs. (8)-(12), we first define:

$%a = b \rho = r_i-r_p,\; \sigma_K = \frac{Kr_w}{s_0}-1.$

We will need the function:

$C_1(K) = \frac{s_0(r_n-r_p)+Kr_w(r_i-r_n)}{(r_i-r_n)(r_n-r_p+r_w)}$

which is the critical value of $C$ above which $\alpha$ of Eq. (12) is positive.

We then define the function:

$Z(K) = \frac{\frac{K}{s_0}(r_w+r_i-r_p)-1}{(r_i-r_n)(r_n-r_p+r_w)},$

and note that $Z(K)=1$ if and only if $k=s_0/r_w$ (i.e., $\sigma_K=0$).

The extremum $m$ of Eq. (14) and the corresponding $t_c$ of Eq. (15) are:

$m=s_0\frac{(r_p-r_i)Z(K)^{\frac{r_n-r_i}{r_p-r_i-r_w}}}{(r_i-r_n)(r_n-r_p+r_w)}+C-C_1(K),$

$t_c=\frac{ln(Z(K))}{r_w+r_i-r_p}.$

We also define the function $C_2(K)$ as:

$C_2(K) = C_1(K)+s_0\frac{(r_p-r_i)Z(K)^{\frac{r_n-r_i}{r_p-R_i-r_w}}}{(r_i-r_n)(r_n-r_p+r_w)}\;if\;K\geq s_0/r_w,$

$C_2(K) = 0\; if\;K

The quantity $C_2(K)$ of Eq. (21) is the critical,value of $C$ above which the extremum $m$ of Eq. (19) is positive.

With these notations, we have the following result on the number of positive zeros of $S(t)$ of Eq. (7).

Theorem 1. We consider the solution $S(t)$ to Eq. (7) defined by the non-negative parameters $K,C,s_0,r_i,r_w,r_p$ and $r_n$. The number of positive zeros of $S(t)$ is given as a function of the sign of $\rho$:

Case $B_1$: $\rho>0,\,(r_i>r_p)$.

Sub-Case $B_{1,1}$: $\sigma_K<0\,(K. $S(t)$ has no positive zero.

Sub-Case $B_{1,2}$: $\sigma_K>0\,(K>s_0/r_w)$. We first consider the case $r_n>r_i$.
If $C>C_2(K)$ (which includes the case $C=K$), then $S(t)$ has no positive zero for $t>0$ and, therefore, remains positive for all $t>0$. For $C_1(K) the function $S(t)$ has one positive zero on each side of $t_c$. For $C the function $S(t)$ has one positive zero. When $r_p the function $S(t)$ has one positive zero for $C (which includes the case $C=K$) and none if $C>C_2(K)$. When $r_n the function $S(t)$ has one positive zero for $C (which includes the case $C+K$ if $K$ is larger than the fixed point $K^*=C_2(K^*)$ of $C_2(K)$) and none if $C>C_2(K)$ (which includes the case $C=K$ if $K$ is smaller than the fixed point $K^*$).

Case $B_2$: $\rho<0,\,(r_i

Sub-Case $B_{2,1}$: $r_w or $r_n. The function $S(t)$ has one positive zero.

Sub-Case $B_{2,2}$: $r _w>r_p-r_n$ and $r_n>r_i$. For $C>C_1(K)$ (which includes the case $C=K$ if $r_n>r_p$ the function $S(t)$ has no positive zero. For $C (which includes the case $C=K$ if $r_n), then $S(t)$ has one positive zero.

Proof. The application of Proposition 1 hinges on the following observations:

1. The parameter $\alpha$ of Eq. (12) is positive if and only if $C>C_1(K)$.
2. The extremum $m$ of Eq. (14) is positive if and only if $C>C_2(K)$.
3. The difference $C_2(K)-C_1(K)$ has the same sign as $(r_p-r_i)/[(r_i-r_n)(r_n-r_p+r_w)]$.
4. For $\rho>0$ the function $C_2(K)$ is a non-decreasing function of $K>0$ that has no positive fixed point if $r_n>r_p$ and one positive fixed point $K^*=C_2(K^*)$ if $r_n.
5. The parameter $\sigma_K$ and the derivative

$g^{'}(0,a,b,c,d,\alpha)=ab+cd=s_0-Kr_w=-s_0\sigma_K$

of $g(t,a,b,c,d,\alpha)$ at 0 have opposite signs.

With $a,b,c,d,\alpha$ of Eqs. (8)-(12) the quantities $U$ and $V$ of (13) are:

$U=\frac{\rho}{-r_w\sigma_K-(\sigma_K+1)\rho},\;\;V=\frac{\sigma_K}{-r_w\sigma_K-(\sigma_K+1)\rho},$

and are both negative if and only if $\rho$ and $\sigma_K$ have the same sign (because $\sigma_K+1>0$).

In the case $B_1$ ($\rho>0$) we considered two sub-cases:

Sub-Case $B_{1,1}$: $\sigma_K<0$. Condition (13) is not satisfied and the derivative of $g(t,a,b,c,d,\alpha)$ at 0 is positive. This result follows from $A_4$ of Proposition 1.

Sub-Case $B_{1,2}$: $\sigma_K>0$. Condition (13) is satisfied and the derivative of $g(t,a,b,c,d,\alpha)$ at 0 is positive. The sub-cases $B_{2,1}$ and$B_{2,2}$ correspond to $b=r_p-r_n-r_w$ or $d=r_i-r_n$ positive and to $b$ and $d$ negative, respectively. The results follow from $A_2$ of proposition 1. When $\sigma_K>0$ Condition (13) is not satisfied and the derivative of $g(t,a,b,c,d,\alpha)$ at 0 is negative. These results follow from $A_3$ of Proposition 1.

Interpretation of Results

Theorem 1 breaks down the results depending on whether the rate of growth $r_i$ of new investments is larger or smaller than the promised return $r_p$.

We first consider the case when $r_i$ is larger than $r_p$ (legitimate fund). In the case $B_{1,1}\;(K the fund is solvent ($Z=0$) regardless of the initial condition $C$. In the sub-case $B_{1,2}\;(K>s_0/r_w)$ the fund remains solvent when $C$ remains above $C_2(K)$ (which includes the case $K=C$). For $r_n$ between $r_p$ and $r_i$ the scheme collapses ($Z=1$) as soon as $c$ drops below $C_2(K)$. For $R_n$ larger than $r_i$ the fund collapses but recovers ($Z=2$) if $C$ does not fall too much below $C_2(K)\;(C_1(K). If $C$ is too small($C) then the scheme collapses ($Z=1$).

Case $B_1$ illustrates what happens to a Ponzi scheme $r_n) even if the rate of growth of new investments $r_i$ is larger than $r_p$. The fund will remain solvent for $C=K$ only if $K$ is not too large ($K$ less than the fixed point $K^*$). If $C=K$ and is larger than the fixed point $K^*$, then the combined withdrawals by the initial and subsequent investors eventually cause the scheme to collapse.

For Case $B_2$ (the rate of growth $r_i$ of new investments is smaller than the promised return $r_p$) we first consider sub-case $B_{2,2}$ with $r_n>r_p$ (legitimate fund). The fund remains solvent for $C>C_1(K)$, which included the case $C=K$. In the Ponzi sub-case $B_{2,2}$, with $r_i and $r_w>rp_{r,n}$ the fund does not grow too fast and is solvent if $C$ is larger than $C_1(K)$, which is itself larger than $K$. This means that, despite an $r_i$ and an $r_n$ smaller than $r_p$, the Ponzi scheme is solvent if the manager can add to $K$ an “in-house” investment $K_0$ at least equal to $C_1(K)-K$. This type of scheme is unprofitable of the manager and is often called a “philanthropic Ponzi scheme”. This scenario relies on $r_i$ remaining smaller than the nominal return $r_n$. If the manager does not invest enough ($C) the scheme collapses.

The Ponzi sub-case $B_{2,1}$ consists of the values $r_n and of the values $(r_n,r_w)$ for which $r_w and $r_n$ is between $r_i$ and $r_p$. In this sub-case $B_{2,1}$ the fund grows too fast and collapses ($Z=1$).

This analysis shows that the role of $r_w$ is ambiguous when $r_n$ is between $r_i$ and $r_p$. Although a small $r_w\;(r_wmay seem desirable, the fund will grow more in the long run and eventually collapses. A large $r_w\;(r_w>r_p-r_n,\,B_{2,2})$ may seem dangerous but depletes the fund and means smaller withdrawals in the long run. The fund is ultimately solvent if $c$ is large enough to absorb the large early withdrawals (“philanthropic Ponzi scheme”).

Actual and Promised Amounts in the Fund

In order to describe the dynamics of a fund that includes a suden parameter change at some time $t^*$, we need to make explicit the role of the parameters by denoting $S(t,K,C,s_0,r_i,r_w,r_p,r_n)$ the solution of differential equation (7).

We introduce the actual and promised amounts $S_a(t)$ and $S_p(t)$. The actual amounts $S_a(t)$ in the fund (resulting from the nominal rate of return $r_n$ and the initial condition $C$), is the one given in Eq. (7) and rewritten explicitly as:

$S_a(t)=S(t,K,C,s_0,r_i,r_w,r_p,r_n)=\frac{r_w[s_0-(r_i-r_p+r_w)]K}{(r_p-r_n-r_w)(r_i-r_p+r_w)}e^{(r_p-r_w)t}$

$+\frac{s_0(r_i-r_p)}{(r_i-r_n)(r_i-r_p+r_w)}e^{r_it}$

$+C-\frac{s_0(r_n-r_p)+Kr_w(r_i-r_n)}{(r_i-r_n)(r_n-r_p+r_w)}e^{e^{r_nt}}$

The value $S_p(t)$ is the amount that is promised to and belongs to investors; $S_p(t)$ is obtained by setting in Eq. (25) the parameter $r_n$ equal to $r_p$ and the initial condition $C$ equal to $K$. Under these conditions, the third term in Eq. (25) becomes zero and:

$S_p(t)=S(t, K, K,s_0,r_i

Contrary to the actual amount $S_a(t)$ in the fund, the promised amount $S_p(t)$ is positive regardless of the parameter values.

Change in Parameter Values

The assumption of an exponentially increasing density of new investments is not realistic in the long run and we may wish to examine what happens in the particular case where the irate of growth of new investments $r_i$ suddenly drops to 0. This means that the flow of new investments becomes a constant. More generally, it would be useful to describe the future dynamics of the fund if at a point $t^*$ the parameters$(s_0,r_i,r_w,r_p,r_n)$ experience a sudden (discontinuous) change of values and become $(s_0^\prime,r_i^\prime,r_w^\prime,r_p^\prime,r_n^\prime)$.

We call $C^\prime$ and $K^\prime$ the actual and promised amounts in the fund at time $t^\ast$, respectively:

$C^\prime=S_a(t^\ast)=S(t^\ast,K,C,s_0,r_i,r_w,r_p,r_n)$

and

$K^\prime=S_p(t^\ast)=S(t^\ast,K,K,s_0,r_i,r_w,r_p,r_p).$

These values will be the initial condition and initial investment starting at time $t^\ast$. The actual and promised amounts at any time $t$ then become:

$S_a(t)=$

$S(t,K,C,s_0,r_i,r_w.r_p,r_n)$ if $t\leq t^*$
$S(t-t^\ast,K^\prime,C^\prime,s_0^\prime,r_i^\prime,r_w^\prime,r_p^\prime,r_n^\prime)$ if $t>t^\ast.$

and

$S_p(t)=$

$S(t,K,K,s_0,r_i,r_w.r_p,r_p)$ if $t \leq t^\ast$

$S(t-t^\ast,K^\prime,K^\prime,s_0^\prime,r_i^\prime,r_w^\prime,r_p^\prime,r_p^\prime)$ if $t>t^\ast$

Several discontinuous parameter changes at different times can be dealt with in this fashion.

Conclusion

As we expected, the fund is always solvent with $C=K$ in the case of a legitimate fund characterized by $r_n>r_p$. In a Ponzi scheme ($r_n), the fund can remain solvent depending on the values of the rate of growth of new investments $r_i$ and the withdrawal rate $r_w$. Our model sheds light on the ambiguous role played by these two parameters. If $r_i$ is too large or $r_w$ too small, the fund grows fast and can be in jeopardy as withdrawals increase. If $r_i$ is too small or $r_w$ too large, the fund may not keep up with withdrawals.

Our model yields a variety of increasing trajectories that may look alike initially, but are fundamentally different in their long-run behavior. Some will continue to increase as long as new investments come in — others will increase possibly for a long time before they collapse. This happens when parameter values in the phase spaces of solutions are close to border regions between different qualitative behaviors (for example, between no zero and one zero for the function $S(t)$). In some cases $S(t)$ initially decreases, reaches a positive or negative minimum, and then recovers.


text:
First Measurement of Charm Production in its Fixed-Target Configuration at the LHC

09 April 2019

Abstract: The first measurement of heavy-flavor production by the LHCb experiment in its fixed-target mode is presented. The production of $J/\psi$ and ${D}^{0}$ mesons is studied with beams of protons of different energies colliding with gaseous targets of helium and argon with nucleon-nucleon center-of-mass energies of $\sqrt{{s}_{\mathrm{NN}}}=86.6$ and 110.4 GeV, respectively. The $J/\psi$ and ${D}^{0}$ production cross sections in $p\mathrm{He}$ collisions in the rapidity range [2, 4.6] are found to be ${\sigma }_{J/\psi }=652±33\left(\text{stat}\right)±42\left(\text{syst}\right)\text{}\text{}\text{}\mathrm{nb}/\text{nucleon}$ and ${\sigma }_{{D}^{0}}=80.8±2.4\left(\text{stat}\right)±6.3\left(\text{syst}\right)\text{}\text{}\mu b/\text{nucleon}$, where the first uncertainty is statistical and the second is systematic. No evidence for a substantial intrinsic charm content of the nucleon is observed in the large Bjorken-$x$ region.

Published in: Physical Review Letters 122 (2019)


text:
# Sum of Arithmetic Progression

Sum of arithmetic progression is very interesting because of its application in real life from basic arithmetic to Olympiad math levels. When dealing with the sum of arithmetic series the most important number is the middle term. I have read a story about a little boy how he easily sum up the sum of the first 100 numbers when he was 10 years old. That kid is no other than Carl Friedrich Gauss.

Formula:

Let,

$S_n$– Sum of series

$a_m$– middle term

$d$– common difference

$n$– number of terms

$a_1$– first term

$a_n$– last term

Formula Derivation:

Given the middle term:

$S_n=a_m\cdot n$  – 1st formula

Middle term can be calculated by the formula

$a_m=\displaystyle\frac{a_1+a_n}{2}$ $S_n=a_m\cdot n$

$S_n=\displaystyle\frac{n(a_1+a_n)}{2}$ –  2nd formula

We also learned from the previous post that $a_n=a_1+(n-1)d$

From 2nd formula,

$S_n=\displaystyle\frac{n(a_1+a_n)}{2}$ $S_n=\displaystyle\frac{n}{2}[a_1+a_1+(n-1)d]$

$S_n=\displaystyle\frac{n}{2}[2a_1+(n-1)d]$3rd formula

Sample Problem 1:

The first term of an arithmetic sequence is 4, the last term is 60. The second term is obtained by adding 4. Find the sum of the series.

Solution:

We have three formula available, we need to find the best formula according to the given. Better to list down the given.

$a_1=4$, $a_n=60$, $d=4$

The three formulas need the number of terms and that is what we need to find out.

Using the formula for nth term,

$a_n=a_1+(n-1)d$ $60=4+(n-1)4$ $n=15$

Solving for sum of the series, we can solve one of the three formulas.

$S_n=\displaystyle\frac{n(a_1+a_n)}{2}$ $S_n=\displaystyle\frac{15(4+60)}{2}$ $S_n=480$

Sample Problem 2:

The nth term of an arithmetic series is given by the formula $a_n=5+2n$. Find the sum of the first 100 terms.

Solution:

We’re doomed right? Where is our a1,an,d,n?

With the previous post there is the same problem and we know that the coefficient of n is the common difference. Thus, d=2.

Solving for $a_1$ :

Let n=1

$a_n=5+2n$ $a_1=5+2(1)$ $a_1=7$

Solving for $a_{100}$

Let n=100

$a_n=5+2n$ $a_{100}=5+2(100)$ $a_{100}=205$

Solving for the sum of the first 100th terms:

$S_n=\displaystyle\frac{n(a_1+a_n)}{2}$ $S_{100}=\displaystyle\frac{100(7+205)}{2}$ $S_{100}=10,600$

Worked Problem 3:

Find the sum of the first 50 positive multiples of 3.

Solution:

The first multiple of 3 is obviously 3, then 6, 9, 12,…

The last term then is 3(50)=150.

$S_n=\displaystyle\frac{n(a_1+a_n)}{2}$ $S_{50}=\displaystyle\frac{50(3+150)}{2}$ $S_{50}=3825$

Worked Problem 4:

The logs of lumber are piled in a shape of a triangle where the tip of the pile has one lumber. The next row has two, three and so on. If the bottom part of the pile have 15 lumbers. How many logs of lumber are there?

Solution:

The solution is like finding the sum of first 15 counting numbers. There is a special formula for that which is ${n+1\choose 2}$

$S_n={n+1\choose 2}=\displaystyle\frac{(n+1)n}{2}$ $S_n=\displaystyle\frac{(15+1)15}{2}$ $S_n=120$

Worked Problem 5: Adapted from Engr. Joselito Torculas of Adamson University Team

Find the value of $a_2+a_4+a_6+\ldots + a_{96}+a_{98}$

If $a_1,a_2,a_3,\ldots$is an arithmetic progression with common difference 1 and

$a_1+a_2+a_3+\dots+a_{97}+a_{98}=137$

Solution:

$a_1+a_2+a_3+\dots+a_{97}+a_{98}=137$

We know that $a_2=a_1+1 \to a_1=a_2-1$, generally $a_{n-1}=a_n-1$

$(a_2-1)+a_2+(a_4-1)+a_4+\dots+(a_{98}-1)+a_{98}=137$ $2(a_2+a_4+a_6+\dots+a_{96}+a_{98})-49=137$ $2(a_2+a_4+a_6+\dots+a_{96}+a_{98})=186$ $a_2+a_4+a_6+\dots+a_{96}+a_{98}=93$


text:
### Predator-Prey Models

Next week’s post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here.

Solution for X (blue) and Y (red) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.

The Lotka-Volterra Model

Many ecological process can be modelled by simple systems of equations. An early example of this is the predator-prey model, developed independently by American mathematician Alfred Lotka (1880–1949) and Italian Vito Volterra (1860-1940).

The interaction between two species of animals, one preying on the other, can be simulated by two first-order nonlinear ordinary differential equations. As an example, we consider rabbits as prey and foxes as predators. Letting ${x}$ be the number of rabbits and ${y}$ the number of foxes, the rates of change of the two populations are given by

$\displaystyle \frac{dx}{dt} = (\alpha - \beta y) x \qquad\qquad \frac{dy}{dt} = -(\gamma - \delta x) y$

These are the Lotka-Volterra equations. Of their nature, ${x}$ and ${y}$ are positive quantities. We also assume that all the parameters are positive. Parameters ${\alpha}$ and ${\gamma}$ are the reproduction rates. If only rabbits are present ($x>0, y=0$) their exponential growth is determined by ${\alpha}$. If there are only foxes (${x=0,y>0}$), they die out at a rate determined by ${\gamma}$, as there is no food available to them. When both species are present, ${\beta}$ and ${\delta}$ determine the interaction between them.

The equation for the prey ${x}$ says that the change in population is determined by the growth rate ${\alpha}$ minus the predation rate ${\beta y}$. For the predators, the second equation says that their population is determined by the food supply ${\delta x}$ minus their natural death rate ${\gamma}$.

Scaling and Solution

We can re-scale ${x}$, ${y}$ and the time ${t}$ so that the system depends on a single parameter. Let ${X = \delta x/\gamma}$, ${Y = \beta y/\alpha}$ and ${T = \alpha t}$. Then we get the canonical form

$\displaystyle \frac{dX}{dT} = (1 - Y) X \qquad\qquad \frac{dY}{dT} = - k(1 - X) Y$

where ${k=\gamma/\alpha}$. The right hand sides give the velocity in phase space:

$\displaystyle \mathbf{V} \equiv (X^\prime, Y^\prime) = ( X - XY , kXY -kX )$

In Fig. 1 we plot the velocity in the ${(X,Y)}$ phase-plane for the case ${k=1}$.

Fig. 1. Velocity field for the Lotka-Volterra equations with k=1. Prey on X-axis and predators on Y-axis.

Although the equations are nonlinear, they can be solved. Taking the ratio of the two equations, we get a separable equation which can be integrated immediately to yield a constant quantity:

$\displaystyle Z(k) \equiv k\log X + \log Y - (kX + Y) = \mathrm{Const.}$

Fig. 2 shows the contours of ${Z(k)}$ for three values of the parameter ${k}$. It is clear that both ${X}$ and ${Y}$ vary periodically, but out of phase. The phase-point traces out a contour in an anti-clockwise direction, with ${Y}$ lagging about a quarter-cycle behind ${X}$. First, the rabbits increase in number. The extra food allows fox numbers to grow. But as predation increases, rabbit numbers fall leading to reduced food for the foxes, which decline in numbers.

Fig. 2. Contours of Z=Const. Left: k=0.1, Middle: k=1.0. Right: k=10.

The cycle repeats indefinitely: Fig. 3 shows the solution for ${X(T)}$ and ${Y(T)}$ for 30 time units with ${X(0)=0.5, Y(0)=0.2}$ and ${k=0.5}$.

Fig. 3. Solution for X(Blue) and Y(Red) for 30 time units.


text:
# Category: SageMath

## Seminar Talk: Analysis of Growth Processes for Plane Trees (Purdue Univ.)

##### By behackl, 7th July 2017

Today I gave a talk (actually, the first of two parts) in Mark Ward‘s informal seminar on Analytic Combinatorics at Purdue University. The abstract and additional resources can be found below.

## The Register Function and Reductions of Binary Trees and Lattice Paths — AofA’16

##### By behackl, 9th July 2016

On July 8 I gave a contributed talk on “The Register Function and Reductions of Binary Trees and Lattice Paths” at this years 27th edition of the International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA’16), which took place in Kraków, Poland.

My Slides can be found here.

## Jupyterhub with SageMath kernel

##### By behackl, 16th January 2016

Recently I made some efforts to get a Jupyterhub instance with a SageMath kernel up and running on one of my servers — with success! 🙂 Actually, the installation is really not that difficult. Nevertheless, I want to write a few lines describing the troubles I ran into and my solutions.

### Installing Dependencies

The dependencies of Jupyterhub are well-explained on their github page, and their installation should not result in big trouble. However, as my server runs CentOS 7.2, I had to compile a recent Python version by myself:

Then, in order to find the library, you have to add the line /usr/local/lib to /etc/ld.so.conf.d/local.conf and reload the config by calling ldconfig. These steps (usually) have to be completed as root. Afterwards, Python 3.5.1 should be installed (you can check with python3.5 -V or pip3.5 -V).

The goal of these steps was to obtain a useable installation of pip.

The installation of the second dependency, the node package manager “npm” was straightforward:

### Installation of Jupyterhub and linking the SageMath kernel

The next part is really easy: Jupyterhub (together with the notebook) can be installed with the help of pip just by calling

The available kernels then can be listed with jupyter-kernelspec:

In order to add a SageMath-kernel to your installation, you first need a working Sage install. For example, I’ve installed sage-6.10 in the /opt-directory, so that I can run Sage by calling

Then, all that needs to be done is linking the kernel into a directory, where Jupyter is aware of them:

After linking the kernel, jupyter-kernelspec list should already list the new kernel like this:

### Configuring the Webserver

In my case, I configured Jupyterhub to run behind a reverse proxy, powered by an Apache web server, such that the service can be reached from a subdomain like https://jupyter.example.com.

First, note that Jupyterhub can be started just by using the command jupyterhub. With jupyterhub --port 98765, you can run the serivce at localhost:98765. The complicated part with the Apache configuration is that you also have to take care of the websockets — otherwise the kernels are not able to connect. The following snippet comes from my Apache configuration:

And finally, in order to have Jupyterhub in a systemd-setting, I’ve put the following into /usr/lib/systemd/system/jupyterhub.service:

… and that’s it! The service can then be started by reloading the configuration via sudo systemctl daemon-reload and sudo systemctl start jupyterhub.service.


text:
# How can I systematically detect unknown barcode/adapter sequences within a set of samples?

I have often downloaded datasets from the SRA where the authors failed to mention which adapters were trimmed during the processing.

Local alignments tend to overcome this obstacle, but it feels a bit barbaric.

fastQC works occasionally to pick them up, but sometimes fails to find the actual adapter sequences.

Usually, I ended up looking up the kits they used and trying to grep for all the possible barcodes.

Is there a more robust/efficient way to do this?

• This doesn't answer your question, but I hope there is a possibility to report such problems to SRA so that they ask the authors to publish the missing information. – bli May 31 '17 at 10:24
• Why do you feel that local alignment is a bit barbaric? It should be the default method in this day and age, unless you’re working with smallRNA sequencing. I tend to trim adapters to be on the safe side but I did lots of work without bothering, and just relying on local alignment. – Konrad Rudolph May 31 '17 at 11:01

The minion utility from the kraken/reaper toolkit may be helpful for this: http://wwwdev.ebi.ac.uk/enright-dev/kraken/reaper/src/reaper-latest/doc/minion.html

• This looks like exactly the right type of tool. Although too bad it was designed mainly for the 3' end adapter. I wonder if you could just flip all your reads and apply it to the 5' end. – story Jun 2 '17 at 14:43

You mention that FastQC "fails to find the actual adapter sequences" - I guess you mean in the Adapter Sequence Contamination plot. However, the kmer and Sequence Content Plots are often useful even when the former fails. I've used these in the past - you can sometimes just read off the adapter sequence from the start of the Sequence Content Plot (or at least see how many bases to trim).

I'm not aware of any existing methods to do this, but here are a couple of ideas about how it might be done:

Canu has a method of adapter trimming which involves looking for the absence of overlap for reads. If there are no other reads which share sequence across a particular region, then the read is broken up at the point of low coverage, and small pieces are discarded. It would be possible to use a method like this to hunt for possible adapter/barcode sequences by preserving the short reads.

Another option is to do a kmer search at the start of reads, and see if any of the high-abundance kmers can be assembled together and/or matched to existing known adapters or barcodes.

If you happen to know a sequence that should be highly abundant in the library, you can grep its beginning or end (with pattern match highlighting) and see if the same sequence systematically comes just before or just after respectively. This kind of visual inspection can help you finding the adaptor.

For instance, in a previous lab, we were working on D. melanogaster small RNA sequencing data and my colleague knew from previous experience with this kind of data that the following small RNA was likely to be abundant: http://flybase.org/reports/FBgn0065042.html

We just had to grep it in the fastq file to see many lines with this sequence, next to another sequence that happened to be always the same: the unknown adapter.

• May I know the reason of the downvote? I've seen this method applied in a case of small RNA-seq, where one highly abundant sequence was expected. Visually inspecting the output of grep of this sequence (with pattern highlight) gave a very good hint of what the adaptor was (the non-highlighted part). – bli Jun 1 '17 at 12:29
• The question is asking about how to detect unknown adapter sequences, so the OP will not know about abundant sequences in advance. That's kind of the point of the question... – ewels Jun 2 '17 at 7:49
• @tallphil I don't see the link between not knowing the adapter and not knowing of an abundant sequence expected to be present in the data. If I remember well, in the example I mention in my comment, my colleague knew from previous experience with this kind of data that the following small RNA was likely to be abundant: flybase.org/reports/FBgn0065042.html We just had to grep it in the fastq file to see many lines with this sequence, next to another sequence that happened to be always the same: the unknown adapter. – bli Jun 2 '17 at 8:04
• Actually I just re-read your post and now I see what you meant. This is a reasonable idea. However, I think you explained it poorly in the sense that a reader might be confused and think you meant that searching for the most abundant sequence might come up with the barcode. You should have specified that the "abundant sequence" in this case was a known nucleic acid sequence that would be expected to have adapters ligated to one or both ends. – story Jun 2 '17 at 14:36
• I tried to clarify my explanations. – bli Jun 3 '17 at 9:10


text:
# Show how to prove a function is convex.

## Question:

Show how to prove a function is convex.

## Concavity of Function:

For proving a convex function we need to understand both concave and convex functions:

Concave up/ Concave: {eq}y = f(x) {/eq} is concave up on an interval {eq}I {/eq} when the second derivative is greater than zero that is: {eq}\displaystyle f''(x) > 0 {/eq} on the interval {eq}I {/eq}.

Concave down / Convex: {eq}y = f(x) {/eq} is concave down on an interval {eq}I {/eq} when the second derivative is greater than zero that is: {eq}\displaystyle f''(x) < 0 {/eq} on the interval {eq}I {/eq}.

We can understand the convex function using an example. Prove that the given function {eq}\displaystyle f(x) = - 3 x^2 + 18 x + 4 {/eq} is a convex function for all real values of {eq}x {/eq}.

Take the derivative of {eq}f(x) {/eq} with respect to x

\begin{align*} \displaystyle f'(x) &= \frac{d}{dx} [- 3 x^2 + 18 x + 4 ] \\ &= - 3 \times 2 x + 18 \times 1 + 0 &\text{(Differentiating using the power formula } \frac{d}{dx}[x^n] = n x^{n - 1} \text{)}\\ &= - 6 x + 18 \\ f''(x) &= \frac{d}{dx}[ - 6 x + 18] &\text{(Taking the derivative of } f'(x) \text{ with respect to x)}\\ &= - 6 \times 1 + 0 &\text{(Differentiating using the power formula } \frac{d}{dx}[x^n] = n x^{n - 1} \text{)}\\ &= - 6 \\ \end{align*}

Here {eq}f''(x) < 0 {/eq} for all real values of x, so {eq}f(x) {/eq} is a convex function on for all real values of x.


text:
# What is the appropriate statistical test to perform a difference of means within a difference of means?

Let's say I am interested in measuring the effect of three different types of interventions on test scores. Let's say the intervention is book quality, of which there are three kinds of book quality (poor quality, good quality, highest quality). There's also a number of demographic characteristics that my population can further be subdivided into (whites vs blacks, poor vs non poor, male vs female).

I want to see if mean effect of book quality on test scores is different for different races, economic status, and sex.

Let's say I'm interested first in the race question: My idea was to first construct a t-test for the difference in means for Whites for two categories of textbook type (poor and good quality), then construct an additional t-test for the difference in means for Blacks. I then subtract those differences and divide by the square root of the sum of their standard errors squared.

This was my strategy in R (forgive the bad code):

whites <- subset(data, data$race == "white" & data$textbook != "highest quality" )
whiteTable<- describeBy(whites$testScore, whites$textbook, mat=TRUE)
numeratorWhites <- whiteTable[, "mean"][1] - whiteTable[, "mean"][2]
standardErrorWhite <- (whiteTable[, "se"][1]^2 + whiteTable[, "se"][2]^2)

So I'm manually constructing a t-test. numeratorWhites contains the difference in mean Test Score for Whites for two possible textbook categories (poor and good). I do the same for blacks:

black <- subset(data, data$race == "black" & data$textbook != "highest quality" )
blackTable<- describeBy(black$testScore, black$textbook, mat=TRUE)
numeratorblack <- blackTable[, "mean"][1] - blackTable[, "mean"][2]
standardErrorWhite <- (blackTable[, "se"][1]^2 + blackTable[, "se"][2]^2)

I then calculate another t-value for the difference between whites and blacks like this:

(numeratorWhites - numeratorBlacks) / sqrt(standardErrorWhite + standardErrorBlack)

The issue with this strategy is that it's extremely time consuming and I still do not know if there's a racial difference in the effect between different permutations of textbook quality such as (good and highest) and (highest and poor). Is there a more straightforward statistical way to do this in R?