text:
American Institute of Mathematical Sciences

September  2018, 17(5): 2139-2147. doi: 10.3934/cpaa.2018102

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

 Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

* Corresponding author

Received  February 2017 Revised  June 2017 Published  April 2018

We consider the bifurcation problem
 $-u''\left(t \right) = \lambda \left(u\left(t \right)+g\left(u\left(t \right) \right) \right),\;u\left(t \right)>0,\;\;t\in I: = \left(-1, 1 \right),\;\; u\left(\pm 1 \right) = 0,$
where
 $g(u) ∈ C^1(\mathbb{R})$
is a periodic function with period 2π and
 $λ > 0$
is a bifurcation parameter. It is known that, under the appropriate conditions on $g$, $λ$ is parameterized by the maximum norm
 $α = \Vert u_λ\Vert_∞$
of the solution
 $u_λ$
associated with
 $λ$
and is written as
 $λ = λ(α)$
. If
 $g(u)$
is periodic, then it is natural to expect that
 $λ(α)$
is also oscillatory for
 $α \gg 1$
. We give a simple condition of
 $g(u)$
, by which we can easily check whether
 $λ(α)$
is oscillatory and intersects the line
 $λ = π^2/4$
infinitely many times for
 $\alpha \gg 1$
or not.
Citation: Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102
References:
 [1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. Google Scholar [2] S. Cano-Casanova and J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations, 244 (2008), 3180-3203. Google Scholar [3] S. Cano-Casanova and J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl., 352 (2009), 166-174. Google Scholar [4] S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production, Appl. Anal., 93 (2014), 1115-1134. Google Scholar [5] Y. J. Cheng, On an open problem of Ambrosetti, Brezis and Cerami, Differential Integral Equations, 15 (2002), 1025-1044. Google Scholar [6] R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Anal., 13 (1989), 871-878. Google Scholar [7] J. M. Fraile, J. López-Gómez and J. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, J. Differential Equations, 123 (1995), 180-212. Google Scholar [8] A. Galstian, P. Korman and Y. Li, On the oscillations of the solution curve for a class of semilinear equations, J. Math. Anal. Appl., 321 (2006), 576-588. Google Scholar [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Translated from the Russian. Translation edited and with a preface by Daniel Zwillinger and Victor Moll. Eighth edition. Elsevier/Academic Press, Amsterdam, 2015.Google Scholar [10] P. Korman and Y. Li, Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities, J. Differential Equations, 257 (2014), 3730-3737. Google Scholar [11] P. Korman and Y. Li, Computing the location and the direction of bifurcation for sign changing solutions, Differ. Equ. Appl., 2 (2010), 1-13. Google Scholar [12] P. Korman and Y. Li, Infinitely many solutions at a resonance, Electron. J. Differ. Equ. Conf. 05, 105–111.Google Scholar [13] P. Korman, An oscillatory bifurcation from infinity, and from zero, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 335-345. Google Scholar [14] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.Google Scholar [15] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13. Google Scholar [16] T. Shibata, New method for computing the local behavior of $L_q$-bifurcation curve for logistic equations, Int. J. Math. Anal., 7 (2013), 29-1541. Google Scholar [17] T. Shibata, S-shaped bifurcation curves for nonlinear two-parameter problems, Nonlinear Anal., 95 (2014), 796-808. Google Scholar [18] T. Shibata, Asymptotic length of bifurcation curves related to inverse bifurcation problems, J. Math. Anal. Appl., 438 (2016), 629-642. Google Scholar [19] T. Shibata, Oscillatory bifurcation for semilinear ordinary differential equations, Electron. J. Qual. Theory Differ. Equ., 44 (2016), 1-13. Google Scholar

show all references

References:
 [1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. Google Scholar [2] S. Cano-Casanova and J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations, 244 (2008), 3180-3203. Google Scholar [3] S. Cano-Casanova and J. López-Gómez, Blow-up rates of radially symmetric large solutions, J. Math. Anal. Appl., 352 (2009), 166-174. Google Scholar [4] S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production, Appl. Anal., 93 (2014), 1115-1134. Google Scholar [5] Y. J. Cheng, On an open problem of Ambrosetti, Brezis and Cerami, Differential Integral Equations, 15 (2002), 1025-1044. Google Scholar [6] R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term, Nonlinear Anal., 13 (1989), 871-878. Google Scholar [7] J. M. Fraile, J. López-Gómez and J. Sabina de Lis, On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems, J. Differential Equations, 123 (1995), 180-212. Google Scholar [8] A. Galstian, P. Korman and Y. Li, On the oscillations of the solution curve for a class of semilinear equations, J. Math. Anal. Appl., 321 (2006), 576-588. Google Scholar [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Translated from the Russian. Translation edited and with a preface by Daniel Zwillinger and Victor Moll. Eighth edition. Elsevier/Academic Press, Amsterdam, 2015.Google Scholar [10] P. Korman and Y. Li, Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities, J. Differential Equations, 257 (2014), 3730-3737. Google Scholar [11] P. Korman and Y. Li, Computing the location and the direction of bifurcation for sign changing solutions, Differ. Equ. Appl., 2 (2010), 1-13. Google Scholar [12] P. Korman and Y. Li, Infinitely many solutions at a resonance, Electron. J. Differ. Equ. Conf. 05, 105–111.Google Scholar [13] P. Korman, An oscillatory bifurcation from infinity, and from zero, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 335-345. Google Scholar [14] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.Google Scholar [15] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970/1971), 1-13. Google Scholar [16] T. Shibata, New method for computing the local behavior of $L_q$-bifurcation curve for logistic equations, Int. J. Math. Anal., 7 (2013), 29-1541. Google Scholar [17] T. Shibata, S-shaped bifurcation curves for nonlinear two-parameter problems, Nonlinear Anal., 95 (2014), 796-808. Google Scholar [18] T. Shibata, Asymptotic length of bifurcation curves related to inverse bifurcation problems, J. Math. Anal. Appl., 438 (2016), 629-642. Google Scholar [19] T. Shibata, Oscillatory bifurcation for semilinear ordinary differential equations, Electron. J. Qual. Theory Differ. Equ., 44 (2016), 1-13. Google Scholar
λ(g; α) with (OP)
graph of sinx + $\epsilon\psi$(x)
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2018 Impact Factor: 0.925


text:
# B Proton decay probability (Bayesian or frequentist?)

1. Sep 6, 2016

### cube137

If an event has probability occurring say 1/100000000000000000000000000000000 times.. if you do the experiment 100000000000000000000000000000000 times.. you are supposed to get the hit at least once? This is the proton decay experiment.. it's more than the above probability figure.. but if you use bayesian analysis instead of frequentist, could you really get that probability and is it really a frequentist thing?

2. Sep 6, 2016

No, and no.

3. Sep 6, 2016

### cube137

No.. what I meant was if the experiment has 100000000000000000000000000000000 protons. At least one must decay if GUT is right.. but does this really happen..

4. Sep 6, 2016

No.

5. Sep 6, 2016

### cube137

The following is typical proton decay experiment i read "Much rests on the existence of proton decay, and yet we’ve never seen a proton die. The reason may simply be that protons rarely decay, a hypothesis borne out by both experiment and theory. Experiments say the proton lifetime has to be greater than about 10^34 years: That’s a 1 followed by 34 zeroes."....... "Because of quantum physics, the time any given proton decays is random, so a tiny fraction will decay long before that 10^34-year lifetime. So, “what you need to do is to get a whole bunch of protons together,” he says. Increasing the number of protons increases the chance that one of them will decay while you’re watching."...

But is the technique sound.. by having so many protons at experiment.. could one of them really decay now? What if one of them decays thousands of years from now.. and not really now in spite of having so many protons in the experiment sample..

6. Sep 6, 2016

### Staff: Mentor

When you flip a coin, it has probability 1/2 of coming up "heads." Does that mean that if you flip the coin 2 times, you will get "heads" at least once?

When you flip two coins, must at least one of them come up "heads"?

7. Sep 7, 2016

### dlgoff

Where's the Like Like button?

8. Sep 7, 2016

### cube137

This may work for small numbers.. but for something on the level of 10^34 years and proton decay. Maybe probability fails and that's why we don't see any decays. Hm.

9. Sep 7, 2016

Staff Emeritus
If anyone needs an example on why PF shouldn't allow personal theories, this is it. Do you really want to argue that math is wrong?

10. Sep 7, 2016

### Staff: Mentor

We can actually make the probabilities come out the same for proton decay as for coin-tossing if we state the question the right way.

You probably know about the "half life" of a particle, right? That's the time for which the probability of decay is 1/2. After one half-life, half of a collection of particles has decayed, on the average.

When physicists talk about the "lifetime" of a particle, they usually mean the "mean lifetime" ($\tau$) of an exponential-decay distribution, which is related to the half-life ($t_{1/2}$) by $t_{1/2} = \tau \ln 2 \approx 0.69 \tau$.

After one mean lifetime, the decay probability is about 36.8%. Or: after one mean lifetime, about 36.8% of a collection of particles has decayed.

So if by "lifetime" you mean "half-life", the probabilities are exactly the same as with coin-flipping. If you have two particles, after one half-life the probablility is 25% that neither of them has decayed; 50% that exactly one of them has decayed; and 25% that both of them have decayed. After one "mean lifetime" the probabilties are 13.5%, 46.6%, and 39.9% (if I did the arithmetic correctly).

11. Sep 7, 2016

### anorlunda

Many threads would be more interesting if we did this. "Let me help you state your question the right way." Thank you @jtbell

12. Sep 7, 2016

### Staff: Mentor

The conversion to a half-life in years is a purely mathematical thing - obviously we cannot run the experiment for 1034 years. It does not matter, because both theory and experiment make statements about a different quantity: the probability that a proton decays within a year, or similar relevant timescales. If we observe 4*1034 protons for a year and see no decay, we know that the probability for a proton to decay within a year is smaller than 10-34 - otherwise we should have seen a decay (or probably more than one). This can be compared to GUT predictions.

13. Sep 7, 2016

### Jehannum

That sounded good in the voice of Master Po from Kung Fu.

Sorry for being flippant.

14. Sep 7, 2016

### Avodyne

Here is how to do the calculation.

Suppose the probability of a single proton decaying in one year is x, where x is a very small number. Suppose we have N protons, where N is a very large number. We watch the N protons for one year. The probability that none of them decay is (1-x)N. For large N and small x, this is well-approximated by exp(-Nx).

15. Sep 8, 2016

### cube137

Can anyone please share papers on the details of the experiment or the best reference or paper that shows beyond the shadows of a doubt
(with no possibility of error that protons don't decay). I need to make important decisions on donations and investments on GUT and Superstrings.

16. Sep 9, 2016

### davenn

you have to be kidding !!
now one here is likely to get involved in giving you info that may or may not help you with how to spend your money
when it fails for you are you going to come back here and rant about the bad info ???

Dave

17. Sep 9, 2016

Indeed.

format: plain


text:
sparsity of QR decomposition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:55:44Z http://mathoverflow.net/feeds/question/94198 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94198/sparsity-of-qr-decomposition sparsity of QR decomposition Mike 2012-04-16T10:40:40Z 2012-04-17T07:18:26Z
Hi, everyone!

I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and triangular matrix $R$. My problem is that : can we use some methods to control the sparsity of $Q$ satisfying $nnz(Q) \le nnz(A)$.
 http://mathoverflow.net/questions/94198/sparsity-of-qr-decomposition/94212#94212 Answer by Brian Borchers for sparsity of QR decomposition Brian Borchers 2012-04-16T13:33:12Z 2012-04-17T03:16:11Z
Unfortunately you can't. With any orthogonal factorization (e.g. QR, LQ, or SVD) you have the problem that because some of the columns of the orthogonal matrix have to span a particular subspace, and because the remaining columns have to form an orthogonal basis for the complement to this subspace, and because these spaces can be completely arbitrary, the orthogonal matrix won't be sparse unless you happen to be very lucky.

There are "sparse QR" methods that effectively represent Q as a product of Givens rotations rather than storing Q explicitly. The Givens rotations are transformations that are extremely sparse/structured matrices.
 http://mathoverflow.net/questions/94198/sparsity-of-qr-decomposition/94277#94277 Answer by Felix Goldberg for sparsity of QR decomposition Felix Goldberg 2012-04-17T07:18:26Z 2012-04-17T07:18:26Z
Also, if $A$ is invertible and $R$ is required to have positive diagonal, the $QR$ decomposition is unique, so you can't really control it. If you are looking for approximate decompositions, though, it's a whole different game.


text:
## Archive for the ‘public health’ Category

December 10, 2022

### Another paper on hepatitis C: absence of backward bifurcations

June 13, 2022

In a previous post I wrote about a paper by Alexis Nangue, myself and others on an in-host model for hepatitis C. In that context we were able to prove various things about the solutions of that model but there were many issues we were not able to investigate at that time. Recently Alexis visited Mainz for a month, funded by an IMU-Simons Foundation Africa Fellowship. In fact he had obtained the fellowship a long time ago but his visit was delayed repeatedly due to the pandemic. Now at last he was able to come. This visit gave us the opportunity to investigate the model from the first paper further and we have now written a second paper on the subject. In the first paper we showed that when the parameters satisfy a certain inequality every solution converges to a steady state as $t\to\infty$. It was left open, whether this is true for all choices of parameters. In the second paper we show that it is not: there are parameters for which periodic solutions exist. This is proved by demonstrating the presence of Hopf bifurcations. These are obtained by a perturbation argument starting from a simpler model. Unfortunately we could not decide analytically whether the periodic solutions are stable or unstable. Simulations indicate that they are stable at least in some cases.

Another question concerns the number of positive steady states. In the first paper we showed under a restriction on the parameters that there are at most three steady states. This has now been extended to all positive parameters. We also show that the number of steady states is even or odd according to the sign of $R_0-1$, where $R_0$ is a basic reproductive ratio. It was left open, whether the number of steady states is ever greater than the minimum compatible with this parity condition. If there existed backward bifurcations (see here for the definition) it might be expected that there are cases with $R_0<1$ and two positive solutions. We proved that in fact this model does not admit backward bifurcations. It is known that a related model for HIV with therapy (Nonlin. Anal. RWA 17, 147) does admit backward bifurcations and it would be interesting to have an intuitive explanation for this difference.

In the first paper we made certain assumptions about the parameters in order to be able to make progress with proving things. In the second paper we drop these extra restrictions. It turns out that many of the statements proved in the first paper remain true. However there are also new phenomena. There is a new type of steady state on the boundary of the positive orthant and it is asymptotically stable. What might it mean biologically? In that case there are no uninfected cells and the state is maintained by infected cells dividing to produce new infected cells. This might represent an approximate description of a biological situation where almost all hepatocytes are infected.

### Advances in the treatment of lung cancer

May 1, 2022

Now let me come back to the lectures. The first important message is nothing new: most cases of lung cancer are caused by smoking. Incidentally, the secretary I mentioned above smoked a lot when she was young but gave up smoking very many years ago. The message is: if you smoke then from the point of view of lung cancer it is good to stop. However it may not be enough. In the first lecture it was emphasized that the first step these days when treating lung cancer is to do a genetic analysis to look for particular mutations since this can help to decide what treatments have a chance of success. In the case of the secretary the doctors did look for mutations but unfortunately she belonged to the majority where there were no mutations which would have been favourable for her prognosis under a suitable treatment. In the most favourable cases there are possibilities available such as targeted therapies (e.g. kinase inhibitors) and immunotherapies. These lectures are intended to be kept understandable for a general audience and accordingly the speaker did not provide many details. This means that since I have spent time on these things in the past I did not learn very much from that lecture. The contents of the second lecture, on surgical techniques, were quite unfamiliar to me. The main theme was minimally invasive surgery which is used in about 30% of operations for lung cancer in Germany. It is rather restricted to specialized centres due to the special expertise and sophisticated technical equipment required. It was explained how a small potential tumour in the lung can be examined and removed. In general the tumour will be found by imaging techniques and the big problem in a operation is to find it physically. We saw a film where an anaesthetised patient is lying on an operating table while the huge arms of a mobile imaging device do a kind of dance around them. The whole thing looks very futuristic. After this dance the device knows where the tumour is. It then computes the path to be taken by a needle to reach the tumour from outside. A laser projects a red point on the skin where the needle is to be inserted. The surgeon puts the point of the needle there and then rotates it until another red point coincides with the other end. This fixes the correct direction and he can then insert the needle. At the end of the needle there is a microsurgical device which can be steered from a computer. Of course there is also a camera which provides a picture of the situation on the computer screen. The movements of the surgeon’s hands are translated into movements of the device at the end of the needle. These are scaled but also subject to noise filtering. In other words, if the surgeon’s hands shake the computer will filter it out. There is also a further refinement of this where a robot arm connected to the imagining device automatically inserts the needle in the right way. The result of all this technology is that, for instance, a single small metastasis in the lung can be removed very effectively. One of the most interesting things the surgeon said concerned the effects of the pandemic. One effect has been that people have been more reluctant to go to the doctor and that it has taken longer than it otherwise would have for lung cancer patients to go into hospital. The concrete effect of this on the work of the surgeon is that he sees that the tumours he has to treat are on average in a more advanced state than they were than before the pandemic. Putting this together with other facts leads to the following stark conclusion which it is worth to state clearly, even if it is sufficiently well known to anyone who is wiling to listen. The reluctance of people to get vaccinated against COVID-19 has led to a considerable increase in the number of people dying of cancer.

### My COVID-19 vaccination, part 3

December 31, 2021

### Uğur Şahin, Özlem Türeci and The Vaccine

October 4, 2021

I now come to the story itself. Şahin and Türeci are both Germans whose parents came to Germany from Turkey. They studied medicine and they met during the practical part of their studies. They were both affected by seeing patients dying of cancer while medicine was helpless to prevent it. They decided they wanted to change the situation and have pursued that goal with remarkable consistency since then. They later came to the University of Mainz. They founded a biotechnology company called Ganymed producing monoclonal antibodies which was eventually sold for several hundred million Euros. They then went on to found BioNTech with the aim of using mRNA technology for cancer immunotherapy. An important role was played by money provided by the Strüngmann brothers. They had become billionaires through their company Hexal which sold generic drugs. They were relatively independent of the usual mechanisms of the financial markets and this was a big advantage for BioNTech. (A side remark: I learned from the book that the capital NT in the middle of the company name stands for ‘new technology’.) In early January 2020 Şahin foresaw the importance of COVID-19 and immediately began a project to apply the mRNA technology of BioNTech to develop a vaccine. The book is the story of many of the obstacles which he and Türeci had to overcome to attain this goal. In the US the vaccine is associated with the name Pfizer and it is important to mention at this point what the role of Pfizer was, namely to provide money and logistics. The main ideas came from Şahin and Türeci. Of course no important scientific development is due to one or two people alone and there are many contributions. In this case a central contribution came from Katalin Karikó.

How does the BioNTech vaccine work? The central idea of an mRNA vaccine is as follows. The aim is to introduce certain proteins into the body which are similar to ones found in the virus. The immune response to these proteins will then also act against the virus. What is actually injected is mRNA and that is then translated into the desired proteins by the cellular machinery. To start with the sequences of relevant proteins must be identified and corresponding mRNA molecules produced in vitro based on a DNA template. The RNA does not only contain the code for the protein but also extra elements which influence the way in which it behaves or is treated within a cell. In addition it is coated with some lipids which protect it from degradation by certain enzymes and help it to enter cells. Karikó played a central role in the development of this lipid technology. After the RNA has been injected it has to get into cells. A good target cell type are the dendritic cells which take up material from their surroundings by macropinocytosis. They then produce proteins based on the RNA template, cut them up into small peptides and display these on their surface. They also move to the lymph nodes. There they can present the antigens to T cells, which get activated. For T cells to get activated a second signal is also necessary and it is fortunate that mRNA can provide such a signal – in the language of vaccines it shows a natural adjuvant activity. In many more popular accounts of the role of the immune system in the vaccination against COVID-19 antibodies are the central subject. In fact according to the book many vaccine developers are somewhat fixated on antibodies and underestimate the role of T cells. There Şahin had to do a lot of convincing. It is nevertheless the case that antibodies are very important in this story and there is one point which I do not understand. Antibodies are produced by B cells and in order to do so they must be activated by the antigen. For this to happen the antigen must be visible outside the cells. So how do proteins produced in dendritic cells get exported so that B cells can see them?

I admire Şahin and Türeci very much. This has two aspects. The first is their amazing achievement in producing the vaccine against COVID-19 in record time. However there is also another aspect which I find very important. It is related to what I have learned about these two people from the book and from other sources. It has to do with a human quality which I find very important and which I believe is not appreciated as it should be in our society. This is humility. In their work Şahin and Türeci have been extremely ambitious but it seems to me that in their private life they have remained humble and this makes them an example to be followed.

### My COVID-19 vaccination, part 2

August 5, 2021

Since my last post on this subject a few things have changed. In Germany 53% of people are fully vaccinated against COVID-19, which is good news. We are now in a situation where in this country any adult who wants the vaccination can get it. Of course this percentage is still a lot lower than what is desirable and the number of people being vaccinated per day has dropped to less than half what it was in mid June. I find it sad, and at the same time difficult to understand, that there are so many people who are not motivated enough to go out and get the vaccination.

Yesterday my wife and I got our second vaccination. In the meantime the relevant authority (STIKO) has recommended that those vaccinated once with the product of AstraZeneca should get an mRNA vaccine the second time. The fact that we waited the rather long time suggested to get our second injection meant that the new recommendation had already come out and we were able to get the vaccine of Biontech the second time around. There have not been many studies of the combination vaccination but as far as I have seen those that there are gave very positive results. So we are happy that it turned out this way. This time the arm where I got the injection was sensitive to pressure during the night but this effect was almost gone by this morning. The only other side effect I noticed was an increased production of endorphins. In other words, I was very happy to have reached this point although I know that it takes a couple of weeks before the maximal protection is there.

Every second year there is an event in Mainz devoted to the popularization of science called the Wissenschaftsmarkt. It has been taking place for the last twenty years. Normally it is in the centre of town but due to the pandemic it will be largely digital this year. This year it is on 11th and 12th September and has the title ‘Mensch und Gesundheit’ [rough translation: human beings and their health]. I will contribute a video with the title ‘Gegen COVID-19 mit Mathematik’ [against COVID-19 with mathematics]. The aim of this video is to explain to non-scientists the importance of mathematics in fighting infectious diseases. I talk about what mathematical models can contribute in this domain but also, which is just as important, about what they cannot do. If the public is to trust statements by scientists it is important to take measures against creating false expectations. I do not go into too much detail about COVID-19 itself since at the moment there is too little information available and too much public controversy. Instead I concentrate on an example from long ago where it is easier to see clearly. It also happens to be the example where the basic reproductive number was discovered. This is the work of Ronald Ross on the control of malaria. Ross was the one who demonstrated that malaria is transmitted by mosquito bites and he was rewarded for that discovery with a Nobel Prize in 1902. After that he studied ways of controlling the disease. This was for instance important in the context of the construction of the Panama Canal. There the first attempt failed because so many workers died of infectious diseases, mainly malaria and yellow fever, both transmitted by mosquitos. The question came up, whether killing a certain percentage of mosquitos could lead to a long-term elimination of malaria or whether the disease would simply come back. Ross, a man of many talents, set up a simple mathematical model and used it to show that elimination is possible and was even able to estimate the percentage necessary. This provided him with a powerful argument which he could use against the many people who were sceptical about the idea.

### The basic reproduction number for infectious diseases, part 2

July 9, 2020

Here I continue the discussion of the previous post. There I mentioned that when defining a basic reproduction number it is necessary, in addition to defining a set of ODE, to make a choice of infected and uninfected compartments. This means partitioning the unknowns into two subsets. In addition it is necessary to distinguish processes regarded as new infections from others. This means splitting the right hand side of the equations into a sum of two terms. There are also some other biologically motivated assumptions on the right hand side. As indicated in the previous post, a reproduction number is a feature of a disease-free steady state of the model (i.e. a steady state where the unknowns belonging to the infected compartment are zero). In fact it is a feature of the linearization of the model about that point. Intuitively, it has to do with a situation where a population contains a very small number of infected individuals. The matrix defining the right hand side of the linearization is a sum of two terms, each of which is partitioned in a certain way. We now consider a situation where the disease-free steady state is perturbed by introducing a small number of individuals into each of the infected compartments while supposing that they cannot cause a significant number of secondary infections. The numbers of individuals in the infected compartments then satisfy a linear system. Under the given assumptions all eigenvalues of the matrix defining this system have negative real parts. Thus the solution tends to zero exponentially. We are interested in the average number of individuals in these compartments over all positive times. To understand this consider first the simpler example of a single compartment with an exponential behaviour $x(t)=x_0e^{-at}$. The average value of $x$ is $\int_0^\infty x_0e^{-at}dt=\left[-a^{-1}x_0e^{-at}\right]_0^\infty=x_0a^{-1}$. In fact what we are interested in is the number of infections expected from the individuals in the infectious compartment. We have the equation $\psi'(t)=-V\psi (t)$, where all eigenvalues of $V$ have positive real parts and we want to calculate $F\int_0^\infty e^{-Vt}dt=FV^{-1}\psi (0)$, where $F$ describes the process of new infections. The matrix $FV^{-1}$ is the next generation matrix mentioned in the previous post. It is now a matter of some rather involved linear algebra to show that the modulus of the largest eigenvalue of this matrix is greater than one if and only if the greatest real part of an eigenvalue of the linearization of the system of ODE is greater than zero.

In the current reporting of epidemiological data the quantity $R$ is time-dependent. I am far away from being able to link this with the above discussion. I looked at the web page of the Robert Koch Institute to see what they say about how they produce their curves. This led me to a paper of Cintron-Arias et al. (Math. Biosci. Eng. 6, 261), which might be able to provide a gateway into this but it is clear that it involves many things which I do not understand at present. Perhaps I should first look at the classic book ‘Infectious Diseases of Humans’ by Anderson and May. Robert May died in April 2020 so that he did not have the opportunity to observe the present pandemic.

### The basic reproduction number for infectious diseases

June 12, 2020

These days reproduction numbers for epidemiology are prominent in the popular media. Many people are familiar with the idea that stopping a resurgence of COVID-19 infections in a region has to do with making and keeping something called $R$ less than one. They may also be familiar with the informal definition of $R$ that it is the number of new infections caused by an infected individual. But how is $R$ (or $R_0$ as it is more commonly called by scientists) defined? A mathematician might not expect to find an answer in the media but it might be reasonable to expect one in the scientific literature on epidemiology. In the past I have been frustrated by the extent to which this fails to be the case. What is typically given is a description in words which I never found possible to convert into a precise mathematical account, despite considerable effort. Now, in the context of a project on hepatis C which I have been working on with colleagues from Cameroon, my attention was drawn to a paper of van den Driessche and Watmough (Math. Biosci. 180, 29) which contains some answers of the type I was looking for. I was vaguely aware of this paper before but I had never seriously tried to read it because I did not realise its nature.

The context in which I would have liked to find answers is that of models given by systems of ordinary differential equations where the unknowns are the numbers of individuals in different categories (susceptible, infected, recovered etc.) as functions of time. How the numbers reported in the media are calculated (on the basis of discrete data) is something I have not yet tried to find out. At the moment I would like an answer in the context which bothered me in the past and this is the context treated in the paper mentioned above. A typical situation is that found in the basic model of virus dynamics, a system of three ODE describing the dynamics of a virus within a host, with the unknowns being uninfected cells, infected cells and virions. There is a quantity $R_0$ which can be expressed in terms of the coefficients of the system. If $R_0\le 1$ then the only non-negative steady state is virus-free. This is the uninfected state and it is globally asymptotically stable. If $R_0>1$ there is an uninfected state which is unstable and an infected state which is positive and globally asymptotically stable. This kind of situation is not unique to this example and similar things are seen in many models of infection. There is a reproductive number (or perhaps more than one) which defines a threshold between different types of late-time behaviour.

It is not obvious that the analysis of van den Driessche and Watmough applies to models of in-host dynamics of a pathogen since it is necessary to make a choice of infected and uninfected compartments which is related to the biological interpretation of the variables and not just to the mathematical structure of the model. Their analysis does apply to the basic model of virus dynamics if the infected compartments are chosen to be the infected cells and virions and the reaction fluxes are partitioned in a suitable way. The simple picture of the significance of the reproductive number given above does not always hold. There is also another scenario which can occur and does so in many practical examples and involves the notion of a backward bifurcation. It goes as follows. For $R_0$ sufficiently small the disease-free steady state is globally asymptotically stable but as $R_0$ is increased this property breaks down before $R_0=1$ is reached. A fold bifurcation occurs which creates a stable and an unstable positive steady state. The unstable steady state moves so as to meet the disease-free state when $R_0=1$. For $R_0>1$ there are exactly two steady states and the positive one is globally asymptotically stable. There is bifurcation for $R_0=1$ but it has a different structure from that in the classical scenario (which is a transcritical bifurcation). It bears some resemblance to a sub-critical Hopf bifurcation.

The most useful insights I got from reading and thinking about the paper of van den Driessche and Watmough are as follows. The primary significance of $R_0$ concerns the disease-free steady state and its stability. The fact that it can sometimes characterise the stability of an infected steady state is a kind of bonus which does not apply to all models. What it can do is to provide information about the stability of a positive steady state in a regime where it is close to the bifurcation point where it separates from the disease-free steady state. This circumstance is analysed in the paper using centre manifold theory. The significance for the stability of a steady state which is far away is a weak one. Continuity arguments can be used to propagate information about stability through parameter space but only as long as no bifurcations happen. When this is the case depends on the details of the particular example being considered. What is the definition of $R_0$ given in the paper? It is the largest modulus of an eigenvalue of a certain matrix (the next generation matrix) constructed from the linearization of the system about the disease-free steady state, whereby the construction of this matrix incorporates information about the biological meaning of the variables. Consider the example of the basic model of virus dynamics with the choices of infected and uninfected compartments as above. There is more than way of partitioning the reaction fluxes. I first tried to put both the production of infected cells and the production of virions into the category of fluxes called $\cal F$ in the paper. Applying the definition of the reproduction number given there leads to $\sqrt{R_0}$, where $R_0$ is the reproduction number usually quoted for this model. If instead only the production of infected cells is put into the category $\cal F$ then the general definition gives the conventional answer $R_0$. The two quantities defining the threshold are different but the definition of being above or below the threshold are the same. ($\sqrt{x}<1\Leftrightarrow x<1$). That this kind of phenomenon can occur is shown by example in the paper.

### Herd immunity

February 14, 2019

I have a long term interest in examples where mathematics has contributed to medicine. Last week I heard a talk at a meeting of the Mainzer Medizinische Gesellschaft about vaccination. The speaker was Markus Knuf, director of the pediatric section of the Helios Clinic in Wiesbaden. In the course of his talk he mentioned the concept of ‘herd immunity’ several times. I was familiar with this concept and I have even mentioned it briefly in some of my lectures and seminars. It never occurred to me that in fact this is an excellent example of a case where medical understanding has benefited from mathematical considerations. Suppose we have a population of individuals who are susceptible to a particular disease. Suppose further that there is an endemic state, i.e. that the disease persists in the population at a constant non-zero level. It is immediately plausible that if we vaccinate a certain proportion $\alpha$ of the population against the disease then the proportion of the population suffering from the disease will be lower than it would have been without vaccination. What is less obvious is that if $\alpha$ exceeds a certain threshold $\alpha_*$ then the constant level will be zero. This is the phenomenon of herd immunity. The value of $\alpha_*$ depends on how infectious the disease is. A well-known example with a relatively high value is measles, where $\alpha$ is about $0.95$. In other words, if you want to get rid of measles from a population then it is necessary to vaccinate at least 95% of the population. It occurs to me that this idea is very schematic since measles does not occur as a constant rate. Instead it occurs in large waves. This idea is nevertheless one which is useful when making public health decisions. Perhaps a better way of looking at it is to think of the endemic state as a steady state of a dynamical model. The important thing is that this state is asymptotically stable in the dynamic set-up so that it recovers from any perturbation (infected individuals coming in from somewhere else). It just so happens that in the usual mathematical models for this type of phenomenon whenever a positive steady state (i.e. one where all populations are positive) exists it is asymptotically stable. Thus the distinction between the steady state and dynamical pictures is not so important. After I started writing this post I came across another post on the same subject by Mark Chu-Carroll. I am not sad that he beat me to it. The two posts give different approaches to the same subject and I think it is good if this topic gets as much publicity as possible.

Coming back to the talk I mentioned, a valuable aspect of it was that the speaker could report on things from his everyday experience in the clinic. This makes things much more immediate than if someone is talking about the subject on an abstract level. Let me give an example. He showed a video of a small boy with an extremely persistent cough. (He had permission from the child’s parents to use this video for the talk.) The birth was a bit premature but the boy left the hospital two weeks later in good health. A few weeks after that he returned with the cough. It turned out that he had whooping cough which he had caught from an adult (non-vaccinated) relative. The man had had a bad cough but the cause was not realised and it was attributed to side effects of a drug he was taking for a heart problem. The doctors did everything to save the boy’s life but the infection soon proved fatal. It is important to realize that this is not an absolutely exceptional case but a scenario which happens regularly. It brings home what getting vaccinated (or failing to do so) really means. Of course an example like this has no statistical significance but it can nevertheless help to make people think.

Let me say some more about the mathematics of this situation. A common type of model is the SIR model. The dependent variables are $S$, the number of individuals who are susceptible to infection by the disease, $I$, the number of individuals who are infected (or infectious, this model ignores the incubation time) and $R$, the number of individuals who have recovered from the disease and are therefore immune. These three quantities depend on time and satisfy a system of ODE containing a number of parameters. There is a certain combination of these parameters, usually called the basic reproductive rate (or ratio) and denoted by $R_0$ whose value determines the outcome of the dynamics. If $R_0\le 1$ the infection dies out – the solution converges to a steady state on the boundary of the state space where $I=0$. If, on the other hand, $R_0>1$ there exists a positive steady state, an endemic equilibrium. The stability this of this steady state can be examined by linearizing about it. In fact it is always stable. Interestingly, more is true. When the endemic steady state exists it is globally asymptotically stable. In other words all solutions with positive initial data converge to that steady state at late time. For a proof of this see a paper by Korobeinikov and Wake (Appl. Math. Lett. 15, 955). They use a Lyapunov function to do so. At this point it is appropriate to mention that my understanding of these things has been improved by the hard work of Irena Vogel, who recently wrote her MSc thesis on the subject of Lyapunov functions in population models under my supervision.

### ECMTB 2018 in Lisbon

July 24, 2018

I am attending a meeting of the ESMTB in Lisbon. I am happy that the temperatures are very moderate, much more moderate than those at home in Mainz. The highest temperatures I encountered so far were due to body heat, for the following reason. The first two sessions on reaction networks were in rooms much too small for the number of participants with people sitting on extra chairs brought in from outside, windowsills and the floor, as available. This caused a rise in temperature despite the open windows. In any case, the size of those audiences is a good sign for the field.


text:
# Problem with gobble listing option inside a tcblisting

Please, compare next two listings inside a beamer frame. The upper one is made with listings and the second with tcblisting (form tcolorbox). Both use same source and listing options.

Following code shows that although latex code is indented by 2 tabs, once tabs are converted to spaces (tab=2), they are partially suppressed with gobble (gobble=2), but only in listings output. tcblisitng fails and all tabs are shown into the result. Is this a tcolorbox known behavior or a bug? Do you know some way of suppressing indenting spaces with tcblisting?

\documentclass{beamer}
\usepackage{lmodern}
\usepackage[listings]{tcolorbox}

\lstdefinestyle{mystyle}{language=[LaTeX]TeX,
basicstyle=\ttfamily,
texcsstyle=*\color{blue},
breaklines=true,
keywordstyle=\color{green!30!black},
commentstyle=\color{red},
morekeywords={},
otherkeywords={$,\{,\},[,],&}, backgroundcolor=\color{gray!30}, escapeinside=<>, moretexcs={maketitle}, showtabs=true, showspaces=true, tabsize=2, gobble=2 } \begin{document} \begin{frame}[fragile] \frametitle{Listing} \begin{lstlisting}[style=mystyle] \documentclass{} preamble \begin{document} document text \end{document} \end{lstlisting} \begin{tcblisting}{listing options={style=mystyle}, listing only} \documentclass{} preamble \begin{document} document text \end{document} \end{tcblisting} \end{frame} \end{document}  • If I add gobble=x to the listings options, I get no error, which makes me think the option is completely ignored. – egreg May 29 '15 at 10:43 ## 2 Answers Unfortunately, gooble really is ignored. This key has no effect on \lstinline or \lstinputlisting (taken from the listings manual) and tcblisting uses \lstinputlisting internally. The following links consider some tricks to circumvent the problem: autogobble for lstinputlistings How to extend the \lstinputlisting command For the current MWE, this could be applied as the following: \documentclass{beamer} \usepackage{lmodern} \usepackage[listings]{tcolorbox} \lstdefinestyle{mystyle}{language=[LaTeX]TeX, basicstyle=\ttfamily, texcsstyle=*\color{blue}, breaklines=true, keywordstyle=\color{green!30!black}, commentstyle=\color{red}, morekeywords={}, otherkeywords={$,\{,\},[,],&},
backgroundcolor=\color{gray!30},
escapeinside=<>,
moretexcs={maketitle},
showtabs=true,
showspaces=true,
tabsize=2,
gobble=2
}

\begin{document}
\begin{frame}[fragile]
\frametitle{Listing}
\begin{lstlisting}[style=mystyle]
\documentclass{}
preamble
\begin{document}
document text
\end{document}
\end{lstlisting}

\begin{tcblisting}{
listing options={style=mystyle,framexleftmargin=-14pt,numbersep=-7pt,xleftmargin=-14pt},
listing only,
}
\documentclass{}
preamble
\begin{document}
document text
\end{document}
\end{tcblisting}
\end{frame}

\end{document}

Without showspaces this would look OK. Another option for tcblisting would be to play with the left key to shift everything to the left.

Hopefully, \lstinputlisting will support gobble some day.

Another simple example to artificially gobble leading spaces (or tabs) in a \begin{tcblisting} block. That is, based on @ThomasFSturm's tcolorbox package and his above example.

\documentclass[12pt,a4paper]{article}

\usepackage[listings]{tcolorbox}
% Uncomment to reverse the default  order.
%\tcbset{text and listing}

% tabsize: increasing this increases the xleftmargin required.
% xleftmargin: only acts on the text, rather than the listing.
\lstdefinestyle{mystyle}{
tabsize=1,
xleftmargin=-14pt
}

\begin{document}

\begin{itemize}
\item Lorem ipsum dolor sit amet.

\begin{tcblisting}{
listing options={style=mystyle},
title=Nibh nunc massa mauris enim dolor}
Quis libero Ut habitant Phas \LaTeX.
\end{tcblisting}
\end{itemize}

\end{document}

Without styling the undesirable result is:

In my IDE I've got intents set to tabs (rather than spaces) and each tab is set to equal 4 spaces. Nevertheless in the code abovetabsize is set to 1 in order to keep the required multiple of xleftmargin low. It's strange behaviour but it produces the required result in my environment.

Thomas, your package is very impressive.


text:
A rental car agency purchases fleet vehicles in two sizes: a : GMAT Data Sufficiency (DS)
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# A rental car agency purchases fleet vehicles in two sizes: a

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A rental car agency purchases fleet vehicles in two sizes: a [#permalink]

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A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs$9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars. [Reveal] Spoiler: OA Math Expert Joined: 02 Sep 2009 Posts: 37113 Followers: 7255 Kudos [?]: 96618 [11] , given: 10770 Re: Car Dealer - Data Sufficiency [#permalink] ### Show Tags 02 Dec 2010, 16:03 11 This post received KUDOS Expert's post 5 This post was BOOKMARKED Yalephd wrote: A rental car agency purchases fleet vehicles in two sizes: a full-size car costs$10,000, and a compact costs $9,000. How many compact cars does the agency own? (1) The agency owns 7 total cars. (2) The agency paid$66,000 for its cars.

Don't have the OA. I think this is an easy question but I am afraid that it might be deceptively simple. I just want to double check.

This is classic C-trap question. C-trap questions are the questions which are obviously sufficient if we take statements together. When we see such questions we should become very suspicious.

Let # of full-size car be F and # of compact cars be C. Question: C=?

(1) The agency owns 7 total cars --> F+C=7. Clearly insufficient to get C.

(2) The agency paid $66,000 for its cars --> 10,000F+9,000C=66,000 --> 10F+9C=66. Here comes the trap: generally such kind of linear equations (ax+by=c) have infinitely many solutions for x and y, and we cannot get single numerical values for the variables. But since F and C represent # of cars then they must be non-negative integers and in this case 10F+9C=66 is no longer simple linear equation it's Diophantine equation (equations whose solutions must be integers only) and for such kind on equations there might be only one combination of x and y (F and C in out case) possible to satisfy it. When you encounter such kind of problems you must always check by trial and error whether it's the case. Now, it's quite easy to check whether 10F+9C=66 has one or more solutions. 9C=66-10F so 66 minus multiple of 10 must be multiple of 9: 66 is not a multiple of 9; 56 is not; 46 is not; 36 IS A MULTIPLE OF 9 (F=3 and C=4); 26 is not; 16 is not and 6 is not. So only one combination of F and C satisfies equation 10F+9C=66, namely F=3 and C=4. Sufficient. Answer: B. For more on this type of questions check: eunice-sold-several-cakes-if-each-cake-sold-for-either-109602.html martha-bought-several-pencils-if-each-pencil-was-either-a-100204.html a-rental-car-agency-purchases-fleet-vehicles-in-two-sizes-a-105682.html joe-bought-only-twenty-cent-stamps-and-thirty-cent-stamps-106212.html a-certain-fruit-stand-sold-apples-for-0-70-each-and-bananas-101966.html joanna-bought-only-0-15-stamps-and-0-29-stamps-how-many-101743.html at-an-amusement-park-tom-bought-a-number-of-red-tokens-and-126814.html collections-confused-need-a-help-81062.html Hope it helps. _________________ Manager Joined: 03 Aug 2010 Posts: 106 GMAT Date: 08-08-2011 Followers: 1 Kudos [?]: 69 [0], given: 63 Re: Car Dealer - Data Sufficiency [#permalink] ### Show Tags 02 Dec 2010, 16:46 Thank you, Bunuel. I thought it was C. Your explanation was clear as always. Bunuel wrote: Yalephd wrote: A rental car agency purchases fleet vehicles in two sizes: a full-size car costs$10,000, and a compact costs $9,000. How many compact cars does the agency own? (1) The agency owns 7 total cars. (2) The agency paid$66,000 for its cars.

Don't have the OA. I think this is an easy question but I am afraid that it might be deceptively simple. I just want to double check.

This is classic C-trap question. C-trap questions are the questions which are obviously sufficient if we take statements together. When we see such questions we should become very suspicious.

Let # of full-size car be F and # of compact cars be C. Question: C=?

(1) The agency owns 7 total cars --> F+C=7. Clearly insufficient to get C.

(2) The agency paid $66,000 for its cars --> 10,000F+9,000C=66,000 --> 10F+9C=66. Here comes the trap: generally such kind of linear equations (ax+by=c) have infinitely many solutions for x and y, and we can not get single numerical values for the variables. But since F and C represent # of cars then they must be non-negative integers and in this case 10F+9C=66 is no longer simple linear equation it's Diophantine equation (equations whose solutions must be integers only) and for such kind on equations there might be only one combination of x and y (F and C in out case) possible to satisfy it. When you encounter such kind of problems you must always check by trial and error whether it's the case. Now, it's quite easy to check whether 10F+9C=66 has one or more solutions. 9C=66-10F so 66 minus multiple of 10 must be multiple of 9: 66 is not multiple of 9; 56 is not; 46 is not; 36 IS MULTIPLE OF 9 (F=3 and C=4); 26 is not; 16 is not and 6 is not. So only one combination of F and C satisfies equation 10F+9C=66, namely F=3 and C=4. Sufficient. Answer: B. Similar problems: gmat-prep2-92785.html?hilit=linear%20type Hope it helps. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7187 Location: Pune, India Followers: 2170 Kudos [?]: 14036 [1] , given: 222 Re: Car Dealer - Data Sufficiency [#permalink] ### Show Tags 02 Dec 2010, 18:46 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED In my blog post given below, I have discussed in detail how to find integer solutions to linear equations in 2 variables (i.e. equations such as this one) See if it helps. http://gmatquant.blogspot.com/2010/11/integral-solutions-of-ax-by-c.html _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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31 May 2013, 21:16
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Yalephd wrote:
A rental car agency purchases fleet vehicles in two sizes: a full-size car costs $10,000, and a compact costs$9,000. How many compact cars does the agency own?

(1) The agency owns 7 total cars.

(2) The agency paid $66,000 for its cars. F.S 1 clearly insufficient. F.S 2 states that 10*f+9*c = 66 [f is the # full size cars;c of compact car] The only way in which we could get a 6 in the units digit for the RHS is through the factor 9*c --> 9*4 = 36. Thus, the no of compact cars = 4.Sufficient. B. _________________ Manager Joined: 27 Apr 2012 Posts: 62 Location: United States GMAT Date: 06-11-2013 GPA: 3.5 WE: Marketing (Consumer Products) Followers: 1 Kudos [?]: 53 [1] , given: 21 Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] ### Show Tags 02 Jun 2013, 13:13 1 This post received KUDOS Stmt 1 : f + c = 7 …insufficient since we can not determine a specific number Stmt 2 : 10000f + 9000c = 66000 Dividing both sides by 1000 10f + 9c = 66 Since cars can only be represented by integers, we can set the following equation : 9c = 66 – 10f C = (66 – 10f)/9 We are looking for a number that is divisible by 9.We need to plug in numbers and arrive at a multiple of 9. C = (66 – 10*3)/9 = 36/9 = 4 Hence the number of compact cars owned is 4. Answer : B Intern Joined: 20 Jun 2011 Posts: 46 Followers: 1 Kudos [?]: 96 [0], given: 1 Re: A rental car agency purchases fleet vehicles in two sizes: a [#permalink] ### Show Tags 18 Nov 2013, 10:32 Alt.: st. 2) $10.000 \$9000 Total # of cars 7 0 70.000 6 1 69.000 5 2 68.000 4 3 67.000 3 4 66.000 ---> Bingo it matches
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# Homework Help: Isotropic rank 3 pseudotensor help

1. Mar 2, 2009

### latentcorpse

Can anybody show me how any isotropic rank 3 pseudotensor can be written as

$a_{ijk}=\lambda \epsilon_{ijk}$

for the isotropic rank 2 tensor case [i.e. $a_{ij}=\lamda \delta_{ij}$ ], my notes prove it by considering an example i.e. a rotation by $\frac{\pi}{2}$ radians about the z axis.

2. Mar 3, 2009

Re: Tensors

anyone?

3. Mar 4, 2009

### Phrak

Re: Tensors

No help yet? It's not clear to me how you're using indices. Are you writing vectors with lower indices?

Secondly, are you defining pseudotensors as tensors that change sign under a reflection of the coordinates such as (x,y,z)-->(-x,y,z)? There are two definitions of a pseudotensor.

Edit: Actually, it should be obvious from your question that you are using the first definition.

So the claim is that the only pseudotensors, whos elements remain unchanged under a general rotation, are equal to the totally antisymmetric tensor multiplied by some scaling factor.

Last edited: Mar 5, 2009
4. Mar 5, 2009

### latentcorpse

Re: Tensors

yes.
i guess then i meant that such a pseudotensor is proprotional to the levi civita rank 3 pseudotensor.

how does one actually show that though?

5. Mar 5, 2009

### Phrak

Re: Tensors

They're not giving you much to go on, are they?

The easy problem (1) is to show that the elements of $\lambda \epsilon_{ijk}$ are unchanged under othonormal transformations. The hard part is (2) showing that the elements of all rank 3 pseudotensors that remain unchanged are of the form $\lambda \epsilon_{ijk}$.

The only difference between tensors and pseudovectors in this proof is that you have to eventually account for an optional reflection of coordinates.

I would tackle the first problem first:

Tensors transform as the product of vectors.

An orthornormal transformation in three dimensions can be obtained as the product of the transformations that rotate vectors about the X, Y and Z axis.

Last edited: Mar 5, 2009


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Zbl 1181.16024
Kelarev, A.V.; Yearwood, J.L.; Vamplew, P.W.
A polynomial ring construction for the classification of data.
(English)
[J] Bull. Aust. Math. Soc. 79, No. 2, 213-225 (2009). ISSN 0004-9727

Let $\bbfF[X]/I_P$ be the factor algebra of the polynomial algebra in $m$ variables over the finite field $\bbfF$ of characteristic $p$ modulo the ideal $I_P$ generated by the differences of monomials $p-q$, $(p,q)\in P\subset [X]\times [X]$. Assuming that the ideal $I_P$ is of codimension $N$, the authors identify $\bbfF[X]/I_P$ with $\bbfF^N$. Let $C=C(U)$ be the ideal of $\bbfF^N$ generated by the set $U\subset\bbfF^N$. The authors call $U$ a visible set of generators of $C$ if the minimal Hamming weight of $C$ is equal to that of $U$.\par In 1988 the reviewer and Lakatos introduced a class of ideals of the factor algebra $\bbfF[X]/(x_i^p-1$, $i=1,\dots,m)$ with visible sets of generators. The class includes several important error correcting codes realized as ideals of modular group algebras.\par In the paper under review the authors extend essentially the class of the reviewer and Lakatos to ideals of the algebra $\bbfF[X]/I_P$ when the commutative semigroup $[X]/(p=q$, $(p,q)\in P)$ is a subsemigroup of a direct product of a semilattice, an elementary Abelian 2-group and an elementary Abelian $p$-group. The authors give examples which show that their results cannot be extended to larger classes of ring constructions and cannot be simplified or generalized. The main results are considered from the point of view of applications to design multiple classifiers and to use them to correct errors of the individual classifiers which constitute the multiple ones.
[Vesselin Drensky (Sofia)]
MSC 2000:
*16S36 Ordinary and skew polynomial rings and semigroup rings
20M25 Semigroup rings, multiplicative semigroups of rings
16S34 Group rings (assoc. rings)
16D25 2-sided ideals (assoc. rings and algebras)
94B60 Other types of codes
68Q45 Formal languages
68T10 Pattern recognition

Keywords: ring constructions; group rings; classification of data; error-correcting capability; combined multiple classifiers; visible sets of generators

Highlights
Master Server


text:
Arithmetic Golfing

A recent PCG golfing question When do I get my sandwich? asked to find a mapping between seven input strings (sandwich names) and the seven days of the week (indexed by number).

The first answer was made by a user named i cri everytim and utilized a string of characters which uniquely appear at the same position in all seven input strings, enklact, to perform the mapping in Python 2 requiring 29 bytes. After their answer, a lot of answers appeared using the same magic string in different languages to reduce the number of bytes needed. Yet nobody reduced the byte count in Python.

Trying to solve the problem on my own, my first attempt was using only the input strings’ last decimal digit to perform the mapping, though this approach did not save on bytes (read my PCG answer for more on this 30 byte solution).

After a few more hours of working on this problem, however, I achieved to bring down the byte count by one entire byte.

I did so by using a simple brute-force algorithm to check for Python expressions which can be used to perform the sought after mapping. To do so, I use Python’s apostrophes (...) to turn the found expression into a string — str(...) is three whole bytes longer — and index that string with the input strings’ lengths. It sure is not very readable, but only takes 28 bytes — and that is all that matters.

lambda S:6793**164[len(S)]

After finding the 28 byte function which uses a 9 byte expression (6793**164), I attempted to find an even shorter expression. And even though I did not yet find one, I did write a more general brute-force Python program (source code shown below; can also be downloaded) than the one I linked to in my PCG answer.

Brute-forcing takes exponentially more time the more digits you have to check, so my brute-forcer still requires the user to decide for themselves which expressions should be tried.
There are three parameters that define the search; a regex pattern that should be contained in the expression’s string, an offset that pattern should ideally have and a target length. If an expression is found that takes as many bytes as or less bytes than the target length, an exclamation point is printed.
Though this program did not prove useful in this case, there may come another challenge where an arithmetic expression golfer could come in handy.

My program may not have found shorter expressions, but definitely some impressive ones (the +... at the end refers to an additional offset from the string index which — unsurprisingly — take additional bytes):

• 2**2**24+800415
• 2**2**27+5226528
• 2**7**9+11719750
• 7954<<850

I also considered using division to generate long strings of digits which may match; the only problem is that Python floating-point numbers only have a certain precision which does not produce long enough strings. Again, using exponentiation (**) and bitshifting (<<) I could not come up with a working expression that takes less bytes.

# Python 2.7 code; 7th, 8th of September 2017

A285494

The On-Line Encyclopedia of Integer Sequences gets regularly updated with new integer sequences. One of the recent updates was contributed by me, A285494.

A285494 is the list of all numbers $k$ so that its digit sum equals its number of distinct prime factors.
A number’s digit sum is the sum of all of its decimal digits. The number $62831853$, for example, has a digit sum of $6+2+8+3+1+8+5+3 = 36$.
A number’s number of distinct prime factors is the number of different prime numbers that multiply together to result in the original number. As an example, $62831853 = 3^2 \cdot 7 \cdot 127 \cdot 7853$, so it has five prime factors of which four are distinct.
Thereby one can conclude that $62831853$ is not an entry in this sequence, as $36 \neq 4$.

The sequence is certainly infinite, as the number $k = 2 \cdot 10^n$ with $n \in \mathbb{N}^*$ has a digit sum of $2 + (0 \cdot n) = 2$ and — because $k = 2^{n+1} \cdot 5^n$ — exactly two distinct prime factors.

In the encyclopedia entry, I provided a Mathematica one-liner to compute the first few entries of this sequence. Since then, I have also written a Python two-liner to achieve the same goal.

(* Mathematica *)
Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&]
Out = {20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010}

# Python 2.7
>>> def p(n):exec"i,f=2,set()\nwhile n>1:\n\tif n%i==0:f.add(i);n/=i;i=1\n\ti+=1";return len(f)
>>> print filter(lambda n:p(n)==sum(map(int,str(n))),range(2,10001))
[20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010]

Multibrot Set

The Mandelbrot Set is typically defined as the set of all numbers $c \in \mathbb{C}$ for which — with $z_0 = 0$, $z_{n+1} = f_c(z_n)$ and $f_c(z) = z^2 + c$ — the limit $\lim\limits_{n \to \infty} z_n$ converges. Visualizations of this standard Mandelbrot Set can be seen in three of my posts (Mandelbrot Set, Mandelbrot Set Miscalculations and Mandelbrot Set II).

However, one can extend the fractal’s definition beyond only having the exponent $2$ in the function to be $f_c(z)=z^\text{exp}+c$ with $\text{exp} \in \mathbb{R}$. The third post I mentioned actually has some generalization as it allows for $\text{exp} \in \{2,3,4,5\}$, although the approach used cannot be extended to real or even rational numbers.

The method I used in the aforementioned post consists of manually expanding $(a+b\cdot i)^n$ for each $n$. The polynomial $(a+b\cdot i)^3$, for example, would be expanded to $(a^3 - 3 \cdot a \cdot b^2) + (3 \cdot a^2 \cdot b - b^3) \cdot i$.
This method is not only tedious, error-prone and has to be done for every exponent (of which there are many), it also only works for whole-number exponents. To visualize real Multibrots, I had to come up with an algorithm for complex number exponentiation.

Luckily enough, there are two main ways to represent a complex number, Cartesian form $z = a+b\cdot i$ and polar form $z = k\cdot e^{\alpha\cdot i}$. Converting from Cartesian to polar form is simply done by finding the number’s vector’s magnitude $k = \sqrt{a^2+b^2}$ and its angle to the x-axis $\alpha = \mbox{atan2}(\frac{a}{b})$. (The function $\mbox{atan2}$ is used in favor of $\arctan$ to avoid having to divide by zero. View this Wikipedia article for more on the function and its definition.)
Once having converted the number to polar form, exponentiation becomes easy as $z^\text{exp} = (k \cdot e^{\alpha\cdot i})^\text{exp} = k^\text{exp} \cdot e^{\alpha \cdot \text{exp} \cdot i}$. With the exponentiated $z^\text{exp}$ in polar form, it can be converted back in Cartesian form with $z^\text{exp} = k^\text{exp} \cdot (\cos{(\alpha \cdot \text{exp})} + \sin{(\alpha \cdot \text{exp})} \cdot i \big)$.

Using this method, converting the complex number to perform exponentiation, I wrote a Java program which visualizes the Multibrot for a given range of exponents and a number of frames.
Additionally, I added a new strategy for coloring the Multibrot Set, which consists of choosing a few anchor colors and then linearly interpolating the red, green and blue values. The resulting images have a reproducible (in contrast to randomly choosing colors) and more interesting (in contrast to only varying brightness) look.

The family of Multibrot Sets can also be visualized as an animation, showing the fractal with an increasing exponent. The animated gif shown below was created using ImageMagick’s convert -delay <ms> *.png multibrot.gif command to stitch together the various .png files the Java application creates. To speed up the rendering, a separate thread is created for each frame, often resulting in 100% CPU-usage. (Be aware of this should you render your own Multibrot Sets!)

To use the program on your own, either copy the source code listed below or download the .java file. The sections to change parameters or the color palette are clearly highlighted using block comments (simply search for ‘/*’).
To compile and execute the Java application, run (on Linux or MacOS) the command javac multibrot.java; java -Xmx4096m multibrot in the source code’s directory (-Xmx4096m tag optional, though for many frames at high quality it may be necessary as it allows Java to use more memory).
If you are a sole Windows user, I recommend installing the Windows 10 Bash Shell.

// Java 1.8 Code
// Jonathan Frech, 11th of September 2016
//          edited 17th of April     2017
//          edited 18th of April     2017
//          edited 20th of April     2017
//          edited 21st of April     2017
//          edited 22nd of April     2017


text:
My academical trajectory in a glimpse:

1.    Ph.D thesis "On symplectic linearization of singular Lagrangian foliations" at the University of Barcelona in September 2003.

2.    October 2004-May 31, 2007, Marie Curie EIF postdoc at Laboratoire Émile Picard, Université Paul Sabatier.

3.   June 1st, 2007 – August 31, 2009, Juan de la Cierva Fellow at the Departament de Matemàtiques at Universitat Autònoma de Barcelona.

4.   September 2009-August 2013- Professor Lector (Lecturer) at the Department of Applied Mathematics I, UPC.

5.   September 2013-September 2015- Professor Agregat Interí/Interim Associate Professor at Universitat Politècnica de Catalunya.

6. Currently (since October 2015), Associate Professor (Professor Agregat) at UPC.

7. Since January 2016, I am principal investigator of the MTM Research project MTM2015-69135-P, Geometría y Topología de variedades, álgebra y aplicaciones.

8. Since December 2016, I am the head of the Laboratory of Geometry and Dynamical Systems at EPSEB.

9. Since January 2017 and for 5 years I am an ICREA Academia Professor.

10. From September 2017 until February 2018, I have been a Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris and my work has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0098).

11. I have been selected under the Programa Càtedres-UPC 2017 for promotion to Full Professor.

12. I am the principal investigator of  the AGAUR SGR research project 2017SGR932 (21 members).

13. Since May 2018 I am the group leader of the UPC research group GEOMVAP (Geometry of varieties and applications).

14. Since November 2018, Full Professor (Professor Catedrático) at UPC.

Affiliations and memberships:

1.      Full Professor at UPC, Chercheur Affilié at Observatoire de Paris, Honorary Doctor Vinculado at ICMAT and member of Barcelona Graduate School of Mathematics BGSMath.

2.      I am a member of the European Mathematical Society and a member of Societat Catalana de Matemàtiques.

3.      Since 2011 I am the corresponding member for the SCM and EMS.

Accreditations:

• I am qualified as Full Professor in the area of Geometry and Topology by AQU since October 2016.

• Since February 2010, I am qualified as Professeur des Universités en Section 25 (Mathématiques) by the French CNU (Conseil des Universités). I was qualified as MCF in February 2008.

•  I am qualified as Professor agregat  by AQU since 2010.

•   I am qualified as Profesor Titular de Universidad since April 2015.

Services for the mathematical community:

1-     Since January 2010, I am an editor of the Journal Newsletter of the European Mathematical Society. Since January 2016 I am responsible for the section of Research Centres.

2-     I have been a guest editor for Journal of Geometry and Physics in 2014 and I am currently an editor for another special issue at Journal of Geometry and Physics.

3-     In the past I have been guest editor for Geometriae Dedicata (You can find this issue here) and for Acta Mathematica Vietnamica ( You can find this issue here.)

4-     I am an editor (together with Vladimir Matveev and Enric Ventura) of a volume of the collection Advanced Courses at CRM, Birkhauser on Integrable Systems (publication expected in 2016).

5-     Since October 2011, I am the corresponding member for the EMS-SCM societies.

6-     I have organized a total of 23 workshops and 2 Research programmes at CRM-Barcelona. You can have a look at the section links.

7-     In the past Jaume Amorós and I organized a seminar on geometry for Graduate Students called Teen Seminar.

8-     I am organizing a regular one day intensive seminar called HIS. (HIS stands here for Hamiltonian Intensive Seminar).

9-     In the past I organized a special Seminar on Symplectic and Poisson Geometry in Barcelona.

10-I have served as referee for a list of journals including: Acta Mathematica, Nonlinearity, Annales de l’Institut Fourier, Journal of the London Mathematical Society, Journal of Geometry and Physics, Journal of Mathematical Physics, Journal of Symplectic Geometry, Quaterly Journal of Mathematics, Journal of Regular and Chaotic Dynamics, Revista Matemática Complutense,  Differential Geometry and its applications,  Acta Mathematica Applicandae, AIP Publications, Birkhauser Advanced CRM courses.

11-I've served as research expert for several (Inter)National Research agencies.

Distinctions:

Curriculum Vitae

My work in a glimpse...some slides and lectures online

1. My minicourse Geometry and Dynamics of singular symplectic manifolds, Course in the framework of my chaire d'Excellence FSMP at IHP-Paris is available online here and in vimeo.

2. My minicourse for Poisson 2014 is on youtube.

3. A video of my talk  at the

4. Slides for my talk at the EMS-RSME meeting in Bilbao, September 2011.

5. Slides for my talk at the Oberwolfach meeting: Geometric Quantization in the non-compact setting, February 2011.

6.  A video of my talk in the Poisson 2010, Rio de Janeiro, July 26, 2010.

7.     Slides for my talk "Poisson Geometry of b-manifolds" at the Poisson 2010 confererence, July 26, 2010.   A in the Conference Conservative Dynamics and Symplectic Geometry, Rio de Janeiro, August 6th 2009.

8. Una presentació del meu curriculum i projectes en Català. (Juny 2009).

9.   My presentation for the Workshop Higher Symmetries in Physics, November 2008.

10.   A of my work in French, May, 2008. en français.

11.   A prepared for CSIC, minicourse in December 2007.

12.  A presentation for the "Colloque: Intégrabilité" Luminy, December 2006.

12.Slides for my talk at the Proceedings of the International Workshop on Geometry and Physics, September 2006.

13.  at PennState University, April 2006.

14.  Thesis Presentation, September 2003.

Publications:

·        My thesis:

Eva Miranda, ,  June, 2003, edicions de la UB, http://www.tdx.cbuc.es/  (the material of this thesis is copyrighted.)

·         My research papers (the versions posted here might differ slightly from the final published versions):

Publications (as author):

1. Universality of Euler flows and flexibility of Reeb embeddings (with R. Cardona, D. Peralta-Salas and F. Presas), preprint, 2019, https://arxiv.org/abs/1911.01963.

2. Geometric Quantization of almost toric manifolds (with F. Presas and R. Solha), arXiv:1705.06572, Journal of Symplectic Geometry, to appear, J. Symplectic Geom (2019).

3. Euler flows and singular geometric structures (with R. Cardona and D. Peralta-Salas), https://arxiv.org/abs/1902.00039, 2019, DOI 10.1098/rsta.2019.0034.

4. On the volume elements of a manifold with transverse zeroes (with R. Cardona), https://arxiv.org/abs/1812.03800, accepted for publication in Regular and Chaotic Dynamics, scheduled for Volume 24, Issue 2 of 2019.

5. Cotangent models for group actions on b-Poisson manifolds (with R. Braddell and A. Kiesenhofer), https://arxiv.org/abs/1811.11894

6. Contact structures with singularities (with C. Oms), https://arxiv.org/abs/1806.05638 (a new version containing a study of the singular Reeb dynamics which I explained at MSRI available on arxiv).

7. Open Problems, Questions, and Challenges in Finite-Dimensional integrable systems, (with A. Bolsinov, V. Matveev and S. Tabachnikov), arXiv:1804.03737,

8. On geometric quantization of b-symplectic manifolds (with V. Guillemin and J. Weitsman), arXiv:1608.08667, Adv. Math. 331 (2018), 941–951.

9. Volume 23, Issue 4 of 2018.

10. Integrable systems and closed one forms (with R. Cardona), arXiv:1712.08156,  J. Geom. Phys. 131 (2018), 204–209.

11. (with D. Bouloc and N.T Zung), arXiv:1803.08332,

12. The geometry of E-manifolds, (with G. Scott), arXiv:1802.02959, preprint 2018, submitted.

13. On geometric quantization of $b^m$-symplectic manifolds, (with V. Guillemin and J. Weitsman), arXiv:1801.03762 preprint, 2018, submitted.

14. (with A. Planas), C. R. Math. Acad. Sci. Paris 356 (2018), no. 1, 92–96.

15. Zeroth Poisson homology, foliated cohomology and perfect Poisson manifolds (with D. Martínez-Torres),  Regul. Chaotic Dyn. 23 (2018), no. 1, 47–53.

16. An invitation to Singular Symplectic Geometry (with R. Braddell, A. Delshams, C. Oms and A. Planas), arXiv:1705.03846,  Int. J. Geom. Methods Mod. Phys. 16 (2019), suppl. 1, 1940008, 16 pp.

17. (with V. Guillemin and J. Weitsman), arXiv:1512.05303, IMRN, International Mathematics Research Notices

rnx126, https://doi.org/10.1093/imrn/rnx126,  Int. Math. Res. Not. IMRN 2019, no. 10, 2981–2998.
18. Comm. Math. Phys. 350 (2017), no. 3, 1123–1145.

19. (with D. Martínez-Torres),http://arxiv.org/abs/1602.03542, J. Geom. Phys. 115 (2017), 131–138.

20. Examples (with A. Delshams and A. Kiesenhofer), http://arxiv.org/abs/1512.08293 , J. Geom. Phys. 115 (2017), 89–97.
21. Non-commutative integrable systems on b-symplectic manifolds (with A. Kiesenhofer), https://arxiv.org/abs/1606.02605, Regul. Chaotic Dyn. 21 (2016), no. 6, 643–659.

22. A note on symplectic topology of b-symplectic manifolds (with P. Frejlich and D. Martinez Torres), arXiv:1312.7329, J. Symplectic Geom. 15 (2017), no. 3, 719–739.
23. Action-angle variables and a KAM theorem for b-Poisson manifolds (with A. Kiesenhofer and G. Scott) , Journal des Mathématiques Pures et Appliquées, J. Math. Pures Appl. (9) 105 (2016), no. 1, 66-85.
24. Geometric Quantization of real polarizations via sheaves (with F. Presas)
25. Toric actions on b-manifolds (with V. Guillemin, A.Pires and G. Scott), Int Math Res Notices (2015) 2015 (14): 5818-5848
26. A Poincaré lemma in Geometric Quantisation (with R. Solha), Journal of Geometric Mechanics,  473 - 491, Volume 5, Issue 4, 2013.
27. On a Poincaré lemma for foliations (with R. Solha), Foliations 2012: pp. 115-137 .
28.  Integrable systems and group actions,  Cent. Eur. J. Math. 12 (2014), no. 2, 240–270.
29.   (with C. Laurent-Gengoux) Acta Math. Vietnam. 38 (2013), no. 1, 21–32.,
30. Rigidity for Hamiltonian actions on Poisson manifolds( with P. Monnier and N. T. Zung) ,    Adv. Math. 229 (2012), no. 2, 1136–1179.
31. Codimension one symplectic foliations and regular Poisson manifolds (with V. Guillemin and A. Pires)Bull Braz Math Soc, New Series 42(4), 603-623, 2011.
32.  From action-angle coordinates to geometric quantization: a round trip,
33. with C. Laurent-Gengoux and  P.Vanhaecke, (for previous version see math.SG arxiv.0805.1679) (31 pages),  Int. Math. Res. Not. IMRN 2011, no. 8, 1839–1869.
34. , , Annales de l'Institut Fourier, vol. 60 (1) p. 51-85, 2010. (36 pages)
35.  Symmetries and singularities of Hamiltonian systems, Conf. Ser. 175 012011, 2009.
36.  Symplectic linearization of semisimple actions, preprint 2008.
37.  Rigidity for Poisson group actions, Oberwolfach report for the meeting on Poisson Geometry and its applications, Oberwolfach Report, Report number, 25, 2007, pages 33—36 (ISSN: 1660-8933), April 2007.
38.  XV International Workshop on Geometry and Physics,, Publ. R. Soc. Mat. Esp., 11,  177-183, R. Soc. Mat. Esp., Madrid, 2007.
39.  A note on Equivariant normal forms of Poisson structures, (with N. T. Zung)   Mathematical Research Letters, vol 13-6, pages 1001-1012, 2006.
40.  A normal form theorem for integrable systems on contact manifolds,  Proceedings of the XIII Fall Workshop on Geometry and Physics, Publ. R. Soc. Mat. Esp., 9, R. Soc. Mat. Esp.,  (2005), 240--246.
41.  A singular Poincaré Lemma (with Vu Ngoc) , IMRN International Mathematics Research Notices, n 1, 2005.
42.  Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems  Ann. Sci. Ecole Norm. Sup.,37 (2004), no. 6, 819--839 2004.
43.  Symplectic linearization of singular Lagrangian foliations (with C. Curras-Bosch), Differential Geometry and its applications,  18, (2), 195-205, 2003.
44.
45. , Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltrú, 2000), 239--244, Publ. R. Soc. Mat. Esp., 3, R. Soc. Mat. Esp.,2001.

Book (New!):

I am currently working on a book project entitled "Geometry and Dynamics of singular symplectic manifolds" (by invitation of the Societé Mathématique de France jointly with Fondation Sciences Mathématiques de Paris) for the collection Cours Spécialisées of the SMF.  By special agreement with FSMP the publication will be made available in open access.

This book aims to be an enhanced and (hopefully) comprehensive version of the course I delivered in October-November-December 2017  at Institut Henri Poincaré during my Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris.  The webpages of the course are available here: https://www.sciencesmaths-paris.fr/fr/le-cours-deva-miranda-933.htm https://mat-web.upc.edu/people/eva.miranda/coursIHP.htm

Stay tuned!

Divulgatio:

1. Miranda, Eva; Muñoz, Miguel Carlos Maryam Mirzakhani, a light that will never go out. (Catalan) SCM Not. No. 42 (2017), 53–55.
2. Columnes de la EMS al Noticies de la SCM 2011-2017.
3. Maryam Mirzakhani, una luz que nunca se apagará (con M. Muñoz Lecanda), Boletín de la RSME. 544, 2017.
4. Congresos y charlas plenarias (Elisa Lorenzo García, Eva Miranda Galcerán, Teresa E. Pérez Fernández y Elena Vázquez Cendón), Boletin de la RSME, 571, 2018.
5. Faces of Women in mathematics, Boletin de la RSME, 573, 2018.
6. El efecto Matilda Columna Mujeres y Matemáticas, Boletin de la RSME, 576, 2018.
7. Entrevista a Sílvia Casacuberta (Boletin de la RSME), Mayo 2018, Boletin 577.
8. En recuerdo a Maryam Mirzakhani, la exploradora de Superfícies, El Pais, July 14, 2018.
9.  La matemática ucraniana que podría haber ganado la medalla fields, El País 14 de Agosto de 2018,
10. La matemática de los fenómenos que se repiten, El País, 9 de Noviembre de 2018.

Media:

Interview at ABC, March 2017

A video promoting the bid ICM2022 for Paris

Publications (as editor):

1. Newsletter of the European Mathematical Society. (editor since 2010).
2. Guest editor of Special Issue at Journal of Geometry and Physics (deadline to submit: December 2015).
3. Guest editor of a Topical Issue on Integrable Systems at Journal of Geometry and Physics.
Journal of Geometry and Physics Volume 87, Pages 1-496 (January 2015)
Finite dimensional integrable systems: on the crossroad of algebra, geometry and physics
Edited by Vladimir Matveev, Eva Miranda, Vladimir Roubtsov, Sergei Tabashnikov and San Vu Ngoc
4. Guest Editor of new Special Issue at Journal of Geometry and Physics. More information available at: http://fdis2015.wfa.uz.zgora.pl/
5. Editor (together with Vladimir Matveev) of a book in the collection CRM-Advanced Courses (Springer) entitled Geometry and Dynamics of Integrable systems http://www.springer.com/gp/book/9783319335025  ISBN 978-3-319-33503-2.
6. Guest editor of a Special issue of Geometriae Dedicata for the conference GESTA 2011. This issue was published in 2013 (volume 1, issue 265, August).
7. Guest editor of a Special issue of Acta Mathematica Vietnamica for the conference GEDYTO. This issue has been published in January 2013 (issue
38).

Book reports:

Carlos Curras-Bosch i Eva Miranda,  Report sobre el llibre “Symplectic geometry of integrable Hamiltonian systems ,Notícies de la Societat Catalana de Matemàtiques, 19, 43-44, 2003.

Students

1.    During the period 2006-2007 during my postdoc at Université de Toulouse I was informally co-supervising (together with Nguyen Tien Zung) Iman Alamiddine who defended her thesis in July 2009 on Singularities of the Gelfand-Ceitlin System.

2.      Romero Barbieri Solha (Start-Up Brazil, Erasmus Mundus fellow) defended his Ph.D thesis in October 2013 on Geometric Quantization with singularities with the qualification of Excellent cum Laude. He is currently a postdoctoral fellow at Universidade de Minas Gerais.

3.      Since Fall 2012, I am supervising  the Ph.D thesis of Anna Kiesenhofer (FI grant). She works on Geometry and Dynamics of b-Poisson manifolds. Her defense is scheduled for the academical year 2015-2016.

4.       2014-2015: Alex Thiele did his undergraduate thesis on Transversality old and new” under my supervision (co-supervised with Marc Chaperon at Université de Paris 7) with the qualification of Excellent/10

5.      2014-2015: Arnau Planas did his Master thesis on Symplectic surfaces with singularities” under my supervision with the qualification of Excel.lent/Matrícula d’Honor.

6. Arnau Planas is doing the thesis under my supervision on Symmetries and singularities on Poisson manifolds.

7. Cédric Oms has stated the thesis under my supervision on Global Hamiltonian Dynamics of singular symplectic manifolds. He is financed via a Luxembourguish AFR-FNR project.

8. Roisin Braddell has started the thesis under my co-supervision (with Amadeu Delshams) on “New geometrical and dynamical techniques for problems in Celestial Mechanics”. She has a contract funded by my ICREA Academia Prize.

9. Robert Cardona has started the thesis under my supervision. He has an FPI grant starting in August 2018. The title of his thesis is "Singularities, integrable systems and foliations in Symplectic and Poisson Geometry".

10. Anastasia Matveeva will start her thesis under my supervision in September with an INPHINIT PhD grant. Her thesis topic is to study Interactions between symmetries and singularities in Geometry and Physics.

11. We plan to add Sándor Hajdú as PhD student to our team in 2018 in cosupervision with Amadeu Delshams.

Postdocs

I have acted as supervisor for the following postdocs:

1-    Anne Pichereau (2008-2009)

2-    Marco Zambon (2008-2009)

3-    Mark Hamilton (2008-2009)

4-    Pablo Suárez-Serrato (i-math postdoc) (2010-2011)

5-      Ainhoa Aparicio (3 months doctoral stay in 2009)

6-      Chiara Esposito (2013).

7- Alfonso Tortorella (2018, Juan de la Cierva, Scheduled).

"Qu'as tu fait de ta vie, pitance de roi ?
- J'ai vu l'homme.
Je n'ai pas vu l'homme comme la mouette, vague au ventre, qui file rapide sur la mer indéfinie.
J'ai vu l'homme à la torche faible, ployé et qui cherchait. Il avait le sérieux de la puce qui saute, mais son saut était rare et réglementé."

excerpt from Ecce Homo by Henri Michaux