Monthly Archives: May 2014

Fracture Growth Dynamics

Fracture growth dynamics — how cracks grow as some suspension (eg paint) dries. I mentioned that topic in a post last April 13th, and have now taken the time to ready the authors’ paper carefully. It is a beautiful paper and an excellent model for careful scientific work.

The paper is by E. R. Dufresne, and six colleagues, all at Harvard. They thank several other members of the “Cracking Club” at Harvard. I think there must be quite a group studying cracking of materials. It is an important topic.

Reference: E. R. Dufresne, et al, “Dynamics of Fracture in Drying Suspensions”, in “Langmuir” (journal title), vol 22, 2006, pp 7144-7147.

The thing that initially attracted me to this paper is that it mentions the Lambert W function. I’ve come to appreciate the other merits of the paper too — its careful blending of experimental and theoretical work.

But specifically to the Lambert W situation. The authors obtain a considerable amount of data, regarding the growth of a 1-dimensional crack as a function of time. Cracking proceeds by jumps. Stress builds up as the suspension dries (evaporation), then a threshold is reached, and the crack jumps forward, within perhaps a thousandth of a second, until it reaches a stopping condition when the stress has been reduced to a level that cannot sustain additional cracking. Evaporation continues, another jump, etc. The length of the crack is thus a staircase graph. As time passes, the average jump lengths L(t) grow longer, but jumps are considerably less frequent. It is the function L(t) that has a Lambert W solution. Of course one can integrate or transform to get other properties as Lambert W solutions. What is important is that Lambert W seems to be the appropriate function to model the dynamics of crack growth.

The jump length L(t) as a function of time, if length and time are normalized suitably, satisfies the differential equation L(t)=(1/2)*((1/L(t))-1). That equation can be solved exactly, using the Lambert W function. I’ve now confirmed that, in a slightly more general case, with 1/2 replaced by a constant B. There is of course a family of solutions, based upon a constant C that comes in during the integration, and C is determined by the boundary conditions to be satisfied by the solution.

It really is a very nice result. Not only the specific physics/chemistry/materials problem, but the differential equation which is quite general in its structure. That equation, and the fracture growth dynamics illustration, makes a good example for introducing people to applications of the Lambert W function. The solution, incidentally, is L(t)=1+W(C*exp(-Bt)) where B=1/2 in this instance and C is a constant to fit the boundary conditions.

Best wishes,
Ken Roberts
31-May-2014

Delay Differential Equation

Differential equations are pretty useful for describing nature: the rate of change of something (first derivative), or the rate of acceleration (second derivative) is related by an equation to the quantity of that something, or to some other things, or to a potential or a force field (think of gravity), and so on.

Sometimes, the rate of change of a quantity is not related to the quantity here and now, but to the quantity over-there or back-then. That leads to the concept of what are called differential-difference equations, or (if time is the independent variable), delay differential equations. Fortunately, either term abbreviates to DDE, so I will just call such equations DDEs. Ordinary differential equations (without time-shifting) are called ODEs.

One can argue that, in general, delay or difference relationships are a closer fit with nature. Light or force does not travel instantaneously. Fibonacci’s rabbits (or their continuous analogue) do not become fertile at birth. It is advisable to develop a methodology for working with DDEs, just as ODEs have been studied for a couple of centuries with great advantage to our understanding of nature.

That work, developing a methodology for DDEs, has been done, or rather systematized, extended and placed in book form, by Richard Bellman and Kenneth Cooke, in “Differential-Difference Equations” published in 1963. Bellman and Cooke’s book is just the beginning of the field, of course. There has been lots of subsequent work. Some of the early work, prior to Bellman and Cooke’s book, was done by E. M. Wright, who is better known to me as the co-author of G. H. Hardy and E. M. Wright’s wonderful book “An Introduction to the Theory of Numbers”. I’ve had some enjoyment from browsing a couple of Wright’s earlier papers (not about DDEs), and he is worth one’s attention. If he thought DDEs were worth study, then I’m certainly going to give him my attention for a while.

How was my attention first drawn to DDEs? A mention in the 1996 paper of Rob Corless, et al, about the Lambert W function. Much cited, it should be possible to find online — or at least a free-access version of the technical report which preceded its publication as a paper. These authors note that the DDE y'(t)=A*y(t-1), where A is a constant, and t is greater than 1, has as one of its solutions y(t)=exp(t*Wk(A)) where Wk() is any branch of the Lambert W function. That is, where Wk(A) represents a number, say w, such that w*exp(w)=A.

A linear combination of such exp(t*Wk(A)) terms is also a solution of y'(t)=A*y(t-1). So, for instance, if gk are a set of coefficients, k running from -infinity to +infinity, such that y(t)=sum(gk*exp(t*Wk(A))) converges nicely (or is summable nicely, just like Fourier series are sometimes summable even if not ordinarily convergent), then that y(t) will solve the DDE. The various exp(t(Wk(A)) functions are linearly independent. The coefficients gk can be determined by the initial condition: whatever values y(t) for t in (0,1) are assumed to start the process.

It’s fun to generalize a bit: instead of a time delay of 1, assume a time delay of symbol c. That is of course just a scale change, but it is worth looking at the functional form of the LamW solution representation. One can think about other Fourier-series-like manipulations of a representation of the solution via LamW exponentials. One can wonder about the robustness or stability of a solution obtained by numerical estimation of the gk coefficients. There is plenty of other interesting work. A search for references to the Bellman and Cooke book should be a good lead-in to the modern literature.

Best wishes,
Ken Roberts
30-May-2014

Acceptable Methods

What is an Acceptable Method of solution for a problem? Let’s say a physics problem, such as determining the bound state energy levels in a 1-dimensional finite square well potential. I wrote blog posts about that particular task a couple of months ago (March 28th and 31st). The quantum mechanics textbooks, ones from 1951 to 2006 (the latest date I’ve checked), state (correctly) that the FSW problem is a transcendental equation and (incorrectly) that it can only be solved using graphical or numerical methods. That is, there is no analytic solution.

The truth is, that there IS an analytic solution, known in effect since a paper of Burniston and Siewert, and known even more explicitly in a paper published by Siewert in 1978, where he states the fact of the solution in the title of his paper, and addresses the problem directly. Two later papers present variant methods of solution, one by Paul and Nkemzi in 2000, and one by Blumel in 2006. So there are three methods of exact solution of the FSW problem, all rather related but with variations.

The common feature of those three solution methods is that they use contour integration (usually taught at first year of graduate school in the math methods in physics course), and consider the exact FSW solution as a Riemann-Hilbert problem (2nd year grad school level, if it gets covered at all — not very likely). Whereas introductory quantum mechanics finite square well is a 2nd year undergraduate course topic. The students in the QM course are not expected to understand contour integration, etc. So the exact solution cannot reasonably be given time within the beginning QM course. It is much more important to get on with the FSW as an example of how to think about QM and how to do some practical calculation.

But, it is NOT RIGHT to outright mislead the students, by telling them “the FSW problem CANNOT be solved exactly”. Rather, what should be said is that “the FSW problem can be solved exactly by an advanced math method known as contour integration. We will however use simpler graphical and computational techniques here…”

So, one property of an Acceptable Method seems to be that it will suit the mathematical maturity of the audience.

As mentioned in the prior blog posts, the geometric-analytic technique devised by my colleague and myself is quite simple to describe, definitely accessible at the 2nd year undergraduate level. But is it an Acceptable Method? We had an interesting discussion of this question after a talk which I gave yesterday, to some grad students, detailing the solution technique. The solution is a method, in effect a sort of geometric construction (though beyond what Euclid would have used) using conformal maps of straight lines and circles. I’ll not go into the details here — see the earlier blog posts and our paper (Arxiv 1403.6685) if you want details.

Think about Newton. Newton had published his optics book (Optiks), written in English. He must have gotten some feedback, adverse, about having written in English instead of Latin, the acceptable language of science at the time. So when it came to his Principia (published about 327 years ago), he wrote in Latin. Also, despite having devised (as did Liebnitz) the differential and integral calculus and using that calculus for obtaining his deductions, Newton presented his demonstrations of those deductions in the Acceptable mathematical manner, using geometry.

Newton wanted his book Principia to be accepted, so he wrote it in Latin and using Geometry.

Nowadays, one has to write in English and use Algebra/Calculus. That is the Acceptable Method of our times.

What will be the acceptable method some 327 years in the future?

I read an anecdote (don’t recall the details) of a paper in particle physics, which presented some good and novel ideas, having been rejected by the reviewer, because the paper had not used Feynman diagrams. “I cannot evaluate this paper until the calculations are presented via Feynman diagrams.”

Incidentally, if you are looking for an enjoyment, check out “Feynman’s Lost Lecture”. It consists of an audio recording of a talk which Richard Feynman gave, in which he explained the details of one of Isaac Newton’s geometric demonstrations. Photographs had been made of the chalkboard, and so the lecture was reconstructed. The whole thing is a pleasurable read, and if you wish, you can hear Feynman speak while you read along with the text and following the diagrams.

I guess the term “Feynman diagram” has multiple meanings.

Best wishes,
Ken Roberts
29-May-2014

Wobbly Legged Design

How to design a compost sifter? Years ago, I had a sifting tray for compost. It consisted of a screen mesh (2.5-cm mesh works well), on the bottom of a tray frame made of wood. That was fine, except it had to be held up and that was tiring. Suited to small amounts of sifting, not an hour’s activity.

After moving to a place with a bigger garden, I wanted to figure out how to make a sifter that would have legs, and not have to be held up in the air while sifting. So I came up with a sketch of some legs, good solid legs, and then a lever-operated mechanism that would allow the tray to be shuttled back and forth to do the sifting. Complicated!

But here is a better design, with wobbly legs. It is the one I actually built, and have rebuilt when it eventually deteriorates. (The wooden legs will slowly decompose if they are left standing in the compost pile.)

Because the legs are wobbly, the tray on top can be shuttled back and forth easily. No need for lots of effort.

The design point: use the properties of your material, and the “defects” (aka properties) of your design, to your advantage. If you want something that will move around randomly, design it with wobbly legs. Make the wobble your friend.

Here is a photo of the compost sifter in place. It is standing in a portion of my compost pile that contains ready-to-use sifted compost. The material to the right is additions to the compost pile in the present year. Each year I have a compost-sifting session, shoveling ready-to-sift compost into the tray, sifting it, and throwing the residue (not yet digested) back into the digesting portion of the compost pile. During the gardening season, requirements for compost are met from the already-sifted bin.

Best wishes,
Ken Roberts
28-May-2014

Photocopier Memories

Today, while operating the photocopier, I realized that my activities have come full circle. My first job was operating a photocopier. At that time (about 1965) photocopiers (Xerox machines, we called them) were very expensive, and the Faculty of Social Science at Univ of Chicago had only one machine. A student was assigned to operate the photocopier, in order to keep the work flowing through at maximum speed. There was an endless stream of requests to make copies — articles from bound journals, and other documents.

The copier was rather slow, so there was time to try to read the prior page while the current page was being scanned. Of course half the time the prior page had come out in the feed tray “upside down”, because the journal would be flipped around for half the pages. I could not read very fast in that orientation. Thus my brief and fragmentary introduction to many topics of social science.

Did I meet some famous people there? Yes, a few. I also was the Saturday morning mail clerk, keeping the mail room open so faculty members could come in and pick up their mail. Some people were open and friendly, and others rather aloof. Saul Bellow had been awarded some prize (Nobel for literature, perhaps) about that time, and I had read his recent novel “Herzog”. I mentioned that I had read it, and there was an error. His protagonist had exited the freeway onto a particular street in Chicago, but there was actually no freeway exit at that street. He grunted at me. Likely thinking “Oh, preserve me from literal minded science students!”.

Best wishes,
Ken Roberts
27-May-2014

Latex Slides — Beamer Package

I have the need to prepare slides for a talk, and want an alternative to Powerpoint or Impress, which are tedious to edit with math formulas, sometimes have weird side-effects etc. The Beamer package of Latex is very useful. I can prepare the slides in a text file, and typeset them to a pdf file, which is then displayed full screen. One pages back and forth with the arrow keys. I don’t have to learn a new math-formula typesetting language! My thanks to the author, Til Tantau (2004-ish work it appears), and the many others who have improved the package over the past ten years.

There are many web pages and documents which describe how to use Beamer, which you will find on a search. The “official” package, with many templates and examples, is available for download as a zip file from this URL:
http://ctan.org/pkg/beamer

I tried out the template which Tantau gives for a talk, and decided to start with the theme “Warsaw” which he mentions. That was very fortuitous, as the colour scheme is quite close to my university’s “official colour”. Clearly this pathway is meant to be taken! Here is an example slide from the pdf file.
$latex-beamer-example$

And here is a Latex file which would produce that slide.
$latex-beamer-example-input$

As you can see from the above slide, my talk is going to be about the Lambert W function and the finite square well problem. I’ve already discussed that topic here in prior posts. The significant discovery for me right now is Beamer! It is going to be a tremendous time-saver.

One technicality which may save you some trouble. I am using MikTex — TexWorks on a Windows system. The default font of Beamer is Computer Modern (sans serif). However, that failed (error message) when I tried to typeset a math formula. The solution: use Helvetica font. If you zoom in on the example of the Latex file above, you will see two lines
\usepackage{mathptmx}
and
\usepackage{helvet}
which are the method for switching to Helvetica font. It may not be the optimal technique, but it certainly has been a sufficient workaround for the math-typesetting difficulty with the default Computer Modern font. I can focus on the content of the slides, and not worry about the underlying slide preparation software.

Best wishes,
Ken Roberts
25-May-2014

Atomic Collisions and Spectroscopy Lectures

Prof. P. C. Deshmukh has mentioned his lectures in atomic collisions and spectroscopy which are available online at NPTEL (link below). He discusses Levinson’s theorem in lectures 10-11-12 of the series. I’ve had a preliminary look at the slides and believe that they will be very helpful with my study of Levinson’s theorem.

I had the pleasure of sitting in on PCD’s classical mechanics lectures three years ago, which are also available via NPTEL, and found them very enjoyable, and stimulating of ideas and opportunities for further investigation. The greatest benefit I obtained from that course was the start of learning how to really “do physics work”.

There is a blending of ideas in physics work. Real physics does not fit in tidy little packages, courses called classical mechanics, quantum mechanics, statistical mechanics, atomic physics, whatever. Rather these are all perspectives of the physics view of nature. One cannot learn subject A without also learning at least a grounding in subjects B, C, D, etc. Each time one revisits a topic, new insight is obtained into material that was likely previously familiar. It is like listening to good poetry or music, or looking at art. Aside: the best art is that which will still be of interest after the 150-th time one has looked at it. Physics is like that. It is an approach to looking at nature, which is still interesting after many more than only 150 re-considerations.

Here are links to the three NPTEL courses by Prof. P. C. Deshmukh.

The classical mechanics course mentioned above. Introductory:
http://nptel.ac.in/courses/115106068/

A course in atomic physics. Logically precedes collisions course:
http://nptel.ac.in/courses/115106057/

The atomic collisions and spectroscopy course. Advanced:
http://nptel.ac.in/courses/115106085/

Perhaps you will find something useful for your own interests, in these courses or in the many other excellent courses which NPTEL is making available.

There is a great deal that I do not know, which is a tremendous opportunity to learn interesting stuff. Perhaps that could be the idea behind a toast: “May your glass always be half-empty!”

Best wishes,
Ken Roberts
24-May-2014

Pi is Irrational

There’s a very nice proof of the irrationality of Pi at the Vigorous Handwaving blog. Link below. The proof is set to the lyrics of the Major General’s song from the Gilbert and Sullivan comic opera, The Pirates of Penzance. Those lyrics are from Kevin Wald, math at U of Chicago, and are at his website (link in the blog post).

I found the proof very interesting, and very accessible. As the blog post author Matthew Calvin says, it could be assigned as a (guided, sequential) exercise in first year calculus — for advanced students, with a flair for math reasoning and a serious interest in math. I think it might be better as a topic for a math club, accessible at 1st year level so suitable for all, as a sort of 20-min seminar and providing a good discussion topic.

I got interested in the math of the proof. It’s fine. There are a couple of minor discrepancies in the final recursion formula — I get an extra factor of n in one term, and a minus between the two terms, but the overall proof still goes thru successfully. Also, I turned the logic around. Instead of supposing at the start an “on the contrary, suppose that Pi is rational and equals m/n”, I just let n be any positive constant, and derived the recursion formula. It is valid for any positive n, and is a useful thing to know. Then, at the very end of the revised proof, one supposes the contrary, Pi=m/n, and easily shows that is contradictory because with that n all the recursion coefficients would be positive integers, and not converge to 0.

There is more to this method than just this proof. One can replace the nt(pi-t) by a positive function g(t) which is symmetric about pi/2. The ak coefficients remind me of Fourier sine coefficients, though I haven’t sat down to make the exact correspondence yet. The corresponding cosine coefficients are zero because g is symmetric about pi/2 and cos(t) is anti-symmetric about pi/2. The sum function f(t)=exp(g(t)) does not have to be exp() actually — all that one needs for the proof is that the sum function is bounded on (0,pi). One can consider extending into the complex plane. Then think about fiddling with contours and other complex analysis techniques. Alternatively, for proving other numbers are irrational, think about crafting a similar line of reasoning which used integration of a suitable g(t) (or its sum function f) over another interval, let’s say (0,Q) where Q is the number in question. One will get a recursion formula. Under what conditions will the recursion formula allow one to obtain a contradiction, eg the ak all positive integers thus contradicting their convergence to zero, if one supposes that Q is rational?

Further, just as a potentially really enjoyable possibility if it works out, wonder about the feasibility of casting the proof into geometric language. Could Archimedes have come up with a proof that Pi is irrational? He was very comfortable with limiting processes, integrations and the like. And let’s not forget my favourite curve, the Quadratix of Hippias! There is plenty of opportunity to do interesting work with this proof technique. (See Hardy and Wright, Intro to the Theory of Numbers, for geometric proof techniques for sqrt(N), and for a variety of proof techniques used for e and pi.)

I really enjoyed the Vigorous Handwaving blog, and recommend it. Many thanks to Matthew Calvin and to the several authors!

Here is a link to their blog:
http://vigoroushandwaving.wordpress.com/

And a link to the specific post about the irrationality of Pi:

The Pi Proof of Penzance

Best wishes,
Ken Roberts
23-May-2014

Chandrasekhar Book — Kamesh Wali

I finished reading the Chandrasekhar book by Kameshwar C. Wali — Chandra: A Biography of S. Chandrasekhar. It’s a very enjoyable read. Wali is the best of biographers. He became friends with Chandrasekhar, for many years, and taped numerous conversations with him. Wali knows his science — physics and astronomy — so was able to write about Chandrasekhar’s scientific activities, and his interactions with other personalities of science, without getting caught into the “gee whiz, oh wow, wonders of nature, awe of genius” pitfalls of pop writing about science. Science in this book is a variety of matter of fact background frames for a long-span story about a person and his interactions with other people. One gets a genuine picture of Chandrasekhar’s personality and life.

I think for me the most favourite part of the book is the tape transcripts at the end of the book. Chandra in his own words.

A tidbit — There was no good photograph of Ramanujan. Chandra visited R’s mother, and obtained R’s passport photo. That is the photo upon which most of the portrayals of R have been based.

Read the book if you are interested in how great science is done. Especially if you are 20-ish and starting out.

Best wishes,
Ken Roberts
22-May-2014

Blending Functions

Blending functions are functions like g(x)=1/exp(-1/x) for positive x, and zero for negative or zero x. They are smooth at all x, including at the origin x=0. They are connected with the idea of mollifier families, and described in my two previous posts regarding mollifiers.

Here is a small writeup about blending functions. I’m not sure it’s entirely correct; I may have reached too far in section 4, and will have to pull back for a re-assessment of the conditions for h(y) to be a co-function. But it shows the direction of my present thinking on the topic of blending functions. Scientific comment emails of course welcome.

This method of attaching a pdf file to a WordPress post is an experiment for me. I had not realized previously that a “media file” could be a pdf. Very nice capability. I may use it more often for attaching technical details to a post.

Best wishes,
Ken Roberts
21-May-2014

Mollifier Calculation 2

Carrying on with topic of a mollifier family. Actually, I want to talk about blending functions. They are, for me, the more interesting aspect of mollifiers.

A blending function is a function like g(x)=exp(-1/x) for x positive, and g(x)=0 for x zero or negative. The aspect of a blending function that is useful in defining a mollifier family is that the blending function is smooth, ie has derivatives of all orders, at its “blend” point of x=0. By defining a mollifier function f_A(z)=g(A-abs(z)) or similar structure, times a suitable normalizing coefficient, one obtains a “blip” function around the point Z which is nonzero only on a radius-A ball around the point z. This blip function f_A can be used as a smooth test function to average (by convolution) the values of an observation function (provided by data or by theoretical model) around z. Because f_A is nonzero only within a ball of radius A (it has bounded support, is the terminology) one can make various manipulations that are most convenient with integrals over finite regions. Applications include electrodynamics (Maxwell mentions blending functions), and statistical mechanics or particle physics — see the writings of C.N. Yang and T.D. Lee, or Wigner and Eisenbud — where it is convenient to make a hypothesis, for the development of the model, that one is working with functions which are bounded in their effect. To some extent the assumption of a blending function is a given. However, I’ve become interested in the blending functions themselves. So I’m going to focus on families of blending functions, not applications such as the definition of mollifier families.

I had an unfortunate mishap with WordPress’s post editor. I had written quite a bit of info, and making reference to taking of limits, wrote the character sequence dash-dash-greaterthan. What I had forgotten is that sequence of three characters is used to denote a comment in HTML coding. When I saved my post as a draft, the editor discarded the comments! Or rather, what it thought were comments. Very disappointing. Anyway, I’m going to put my thoughts on blending functions into a Latex/PDF document and post it to a website. Will link from here in another post.

I will mention here very briefly the core idea. Given a polynomial Q(y) which has degree at least 1, and positive top coefficient (ie, coefficient of highest nonzero term). Consider the set H of functions P(y)/exp(Q(y)) where P(y) is any polynomial. That set is closed under differentiation (try it out for yourself), and of course it is closed under multiplication by another polynomial in y. All those functions tend to zero as y tends to infinity, because the polynomial in the numerator is dominated by the exponential in the denominator. The prototype is h(y)=1/exp(y), which is the set H defined by Q(y)=y, and the particular numerator P(y)=1, but there are of course a whole family of functions h() in the set H.

Given a nonzero member h() of the set H, one can define the function g(x)=h(1/x) for positive x, and g(x)=0 for x zero or negative. Following the same line of reasoning as in the John Loftin writeup (linked via prior post), one proves that the functions g(x) are smooth, notably at x=0. So they are blending functions. Some care is needed to be sure that g(x) based upon arbitrary nonzero h() cannot be always zero after some number of differentiations (but that is true), and to verify that g(x) is eventually of one sign in a small enough neighborhood of x=0 (but that also is true), and to word the definition of blending function carefully enough to make the reasoning flow well (but that can be done). It is longer than I wish to risk trying to post via the WordPress editor! But you can work it out yourself in the meantime.

Then, one might wish to consider other functions in the denominator. For example, what about the set H of functions of the form P(y)/(exp(y)+const) which is getting into something like Fermi-Dirac integral behavior, seen for instance in the free electron gas model of metals.

Enough for now. I’m going to see if I can get the website set up. Will let you know a URL when have it ready.

Best wishes,
Ken Roberts
19-May-2014

Mollifier Calculation

Mollifiers are a concept that I missed in my prior math education; it is pleasant to encounter the concept now. Sometimes one has idea-knowledge about something, but there is not an attached vocabulary. The mental constructs, the objects and the ways they can be manipulated, do not have names. When someone else describes the objects and manipulations precisely and gives them names, suddenly one’s ideas become much clearer. Vocabulary facilitates thought. Thus it has been for me with the concept of a mollifier.

Briefly, a mollifier is a family of smoothing functions, consisting of localized blips. I will describe only in one dimension, real axis, but the concept extends to multiple dimensions. Suppose each blip function f_A to be non-zero only within distance A of the origin, non-negative, smooth, and have integral 1. You will recognize a family of approximations to the Dirac delta function. As A becomes smaller, approaching zero, f_A becomes more concentrated about the origin, but still maintains the area 1.

An example of a basis for a mollifier family is the blending function g() defined by g(x)=exp(-1/x) for x positive, zero otherwise. This is a smooth gradual rise, starting at x=0 and going to 1. Smoothness of g() at x=0 needs some proof, and I will give a reference below to a nice introductory article. Here is a graph of g(x) for x between 0 and 1.

The blending function g() starts very gradually at x=0; in fact smoothly, ie an infinitely number of derivatives exist at x=0 (and hence at all x values). As x goes to infinity, g(x) goes to 1, always monotonic. The important property of g() is that it blends smoothly from being always-zero for x negative, to becoming close to 1 as x gets large positive.

We wish to use the blending g() to define, given some positive parameter A, a smooth function f_A(z) which is suitable as one member of a mollifier family. That is, f_A(z) will be zero outside distance A from the origin, and have integral 1. The definition is f_A(z)=g(A-abs(z))/C_A where C_A is a constant chosen to make the integral of f_A equal 1. The A-abs(z) part will be negative for z outside the radius A from the origin. Taking g() of that provides a smooth transition on the boundary abs(z)=A. And dividing by C_A equal to the integral of g(A-abs(z)) normalizes f_A() to have integral 1.

You can see how to use the blending function g() to define other mollifier families, for instance one based upon g(A^2-abs(z)^2). In more than one dimension, for instance z=(z1,z2…zn), you can define abs(z) as the distance of the point z from the origin, hence abs(z)^2=z1^2+z2^2+…+zn^2. The important part, the core of the construct, is the availability of a smooth blending function g().

[OOPS — Hit Publish too soon. Will edit and close this off quickly. Return to topic in subsequent post.]

References: Article by John Loftin, Rutgers, about Mollifiers. Good introduction.

Click to access mollifier.pdf

Wikipedia article about Mollifiers (pretty abstract).
http://en.wikipedia.org/wiki/Mollifier

Best wishes,
Ken Roberts
18-May-2014

Chandrasekhar

A friend gave me an article from Physics Today, about Chandrasekhar. I’ve always been curious about Chandra (as he is known informally, I have learnt) and, when an undergraduate student, would sometimes see him walking across the campus. That was 1967 or 1968, Chandrasekhar had recently won the Nobel prize, and we youngsters were in awe of his intellect. I did not approach him!

The article is “Chandra: A biographical portrait”, by Kamesh Wali, in the December 2010 issue of Physics Today. I’ve obtained Wali’s book from the library and am reading it with great interest and enjoyment. That book is “Chandra: A Biography of S. Chandrasekhar”, by Kameshwar C. Wali, published by Univ of Chicago Press, 1991. It has much more detail than the article, of course.

Can I say anything novel about Chandrasekhar the person, or about his scientific work? Nope. But I can provide a snapshot of the building where Chandrasekhar worked when I would see him walking on campus. This photo was taken in July-2012. The building is still called the Laboratory for Astrophysics and Space Research.

Best wishes,
Ken Roberts
17-May-2014

Yellow Trilliums

4 Replies

You have likely seen white trilliums — they are the provincial flower of Ontario. And perhaps you have seen red trilliums. But have you ever seen yellow trilliums? Here is a photo of one.

A yellow trillium is essentially an albino red trillium. Here is a red. You can see it has the same petal structure.

White trilliums, in contrast, have a different petal structure, much more ruffled around their perimeter. Here you can see that the whites grow in company with the reds. Actually, whites tend to blossom several days after the reds and yellows, and prefer a more open sunny warmer lightly wooded area, compared to the reds and yellows which are more in-the-woods plants. The whites are still in-the-woods plants, just not as deep woods as reds.

There is a virus infection that the whites can get, and it produces a stripe along the central lines of the petals. Those are called green trilliums. Here are a couple of photos of green trilliums, with narrow or broad green stripes. You can see that the petal shape is that of the whites.

There are no green trilliums in the woods nearby, so I cannot say if the reds and yellows are susceptible to the virus that produces green trilliums.

Reds are abundant in the nearby woods (tens of thousands), and whites are plentiful (thousands). There are only four yellows. The yellows were first noticed two years ago, and the following two years have been observed to reappear. That is, the same plants coming out with yellow blossoms each year. The yellow must be a distinct variety of the red trillium, and its flower colour must be genetically based, not simply something produced by a variation in soil pH or other environmental factors. Yellows grow adjacent to, among, the reds, with the same soil, light, temperature and other environment.

Trilliums take a long time to spread. I’ve not noticed any yellow offspring. The process, according the booklet that I have read, is that trillium seeds have a waxy coating that ants like. The ants gather the seeds, take them down to their burrows, eat the waxy coating and eventually clean house and distribute the trillium seeds around. So one gets a patch of trilliums where before there were only a few.

I have also noticed a trillium seed pod (from a red) having been eaten by a chipmunk or squirrel. So ants may not be the only way to spread trillium seeds around.

One must simply wait for the ants, or chipmunks, to do their thing and hopefully propagate the yellows. Don’t mess with trilliums, is the message. They are best left to themselves, and they will be happy, plentiful, and beautiful.

Just to provide some clarity about terminology — there is another variety of trillium, commonly called the yellow, which is quite different from what I have here called a yellow trillium. And there is a different name for the reds. It’s advisable to look at the plants, not their names!

Timing … trilliums are found in Carolinean woods, and bloom for about three weeks, from when the ground thaws until when the tree leaves fluff out. Once the tree leaves are out, there is too much shade for trillium blooms. The plants are still there, and you can recognize their leaves and sometimes their seed pods. The best time for a trillium walk is the last week of April and the first two weeks of May, at least hereabouts in southwestern Ontario. There are many wooded areas which have public access. You may want to wear mud-durable shoes or boots.

If you are planning a trillium walk, it will have to be next year! But trilliums are worth the trouble to see.

Best wishes,
Ken Roberts
16-May-2014

Galactic Group Angular Momentum

Galaxies have preferred orientations — their spiral axes of rotation. Solar systems have prefered orientations; the planets tend to go in the same direction. These two phenomena reflect that the respective systems — galaxies or solar systems — condensed from clouds, and as they condensed, their angular momentum became concentrated in their extremities.

So, do galactic groups tend to have preferred orientations? I asked an expert and he said, no, there is no (or little?) evidence of preferred orientations in groups of galaxies. That is curious. It suggests that the process of condensation in a large cloud, which became a group of galaxies, was somehow different from the process of galaxy formation or solar system formation from clouds. Yet, groups of galaxies are still at the scale at which the universe exhibits local density fluctuations. One has to go to very large distances (larger than groups of galaxies) to get a uniform density of matter.

Something to think about.

Best wishes,
Ken Roberts
15-May-2014

Norman Levinson

A recent discussion has led me to the study of Levinson’s theorem, which describes particle scattering. During the course of that investigation, I’ve been reading several papers by Norman Levinson, and have come to greatly admire him as a person. There are not many people whose conclusions I would accept on trust, without feeling the need to go through all their supporting work. But I can imagine making a list “statements by Norman Levinson” that I might rely upon unhesitatingly. Of course it is such a pleasure to read his papers that I can review the details anyway. And Levinson is refreshing in that he does not overburden his papers with references; one can take them as starting points, and delve into the few references for background as needed.

So … can I improve upon the resources already available about Levinson, the person? Nope; I have no prior info. The Wikipedia page is useful. The few biographical notes at the front of “Selected Papers of Norman Levinson” are important. One bit from there, is a suggestion that Levinson did most of the writing for Paley and Wiener’s book “Fourier Analysis in the Complex Domain”. That is quite conceivable. It is interesting to read the pair of books, Paley and Wiener, and Levinson’s book “Gap and Density Theorems”.

I told a friend recently that I think Levinson’s theorem is pure geometry. So probably I should validate that claim — for myself, not necessarily for this blog. It will make an interesting topic for study.

By the way, if you’re looking for the paper which contains the original “Levinson’s theorem”, it is hard to get online — or rather, it is not accessible by ordinary methods. See volume 1 of Levinson’s selected papers, pages 164-191.

The Wikipedia article about Levinson has mentions of and links to some other resources.

http://en.wikipedia.org/wiki/Norman_Levinson

Best wishes,
Ken Roberts
14-May-2014

A Toast

A Toast

I look to the poems of Edward Lear
for inspiration! But take no fear,
I shall never write a poem
about Prophets, Sand or Foam.
I do, though, like foam on my Beer.

Best wishes,
Ken Roberts
09-May-2014

Woodpecker Holes

When a woodpecker drills a hole, looking for a bug, and there is no bug … move a bit aside and try again! Here is a photo of a line of holes in the trunk of a pine tree (trunk diameter about 25 cm).

It reminds me of my experiences, sometimes, when trying to locate a stud behind the drywall to put up some shelf or other heavy object that must be anchored firmly. I tap, listen, feel the springiness of the wall, then drill a tiny hole. No resistance? Move a bit left or right and try again. Behind some of the shelves in my house there is a line of little holes.

Best wishes,
Ken Roberts
05-May-2014

Where is Down?

Where is down from here? That seems a simple question, but has three different answers. Geometric down, static down, and dynamic down.

We live on the surface of a large rotating ball. The ball is slightly flattened, because of rotation, but the three meanings of down that I’m talking about do not depend upon any oddity of the shape of the earth. Let’s suppose that we live on the surface of a rotating ball which is made of sufficiently strong and stiff material that its deformations are negligible. Also that the ball has uniform density — no lumps of denser material within to produce gravitational anomalies. (I’m interested in gravitational anomalies, and the Earth’s geoid, etc, but those are subjects for another time.)

If we use geometry, we can identify the line from where we are standing, to the center of the earth, as being down. Let’s call that geometric down.

Now imagine hanging a plumb bob, a lead weight, at the end of a long almost weightless string (nylon cord). If we are not standing at one of the poles, there will be a centifugal force (or effect) which will cause the bob to hang out from the geometric down-line. That is static down. Or maybe static force down.

Suppose instead we drop another lead weight from the position above where we are standing. It will follow the line of static down, but will curve as it falls, accelerating along the static-down vector at each point in its path of falling, that acceleration adjusting the existing velocity vector of the weight. The result is that the weight will not pass thru the position of the lead weight at the end of the plumb bob. The name given to this effect is the Coriolis effect, or Coriolis force. That is dynamic down, or dynamic force down.

So imagine, you’re on stilts. Balanced on very long, straight stilts, maybe 100 meters tall. You’re standing “vertically”, with your stilts pointing along the static down-line. You’re using your hand to point towards the center of the earth, a direction which is a bit off to one side of your stilts — that is the geometric down-line. And it’s tricky, you’re sweating, and drops of sweat fall from your brow. Those drops follow a path towards the ground, starting along the static down-line parallel to your stilts, but then they deviate, curve away, and land elsewhere than the feet of your stilts.

What looks so simple a concept, “down”, fits within one word. But the world is not as simple as the language.

A popular book, some 45 years ago, was “Been Down So Long It Looks Like Up to Me”, by Richard Farina. It’s an enjoyable story, and the title reminds me of the ambiguity of “down” in our vocabulary.