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# Blogging some crap…

I had taken a vow not to blog very frequently any more—certainly not any more at least right this month, in April.

But then, I am known to break my own rules.

Still, guess I really am coming to a point where quite a few threads on which I wanted to blog are, somehow, sort of coming to an end, and fresh topics are still too fresh to write anything about.

So, the only things to blog about would be crap. Thus the title of this post.

Anyway, here is an update of my interests, and the reason why it actually is, and also would be, difficult for me to blog very regularly in the near future of months, may be even a year or so. [I am being serious.]

1. About micro-level water resources engineering:

Recently, I blogged a lot about it. Now, I think I have more or less completed my preliminary studies, and pursuing anything further would take a definitely targeted and detailed research—something that only can be pursued once I have a master’s or PhD student to guide. Which will only happen once I have a job. Which will only happen in July (when the next academic term of the University of Mumbai begins).

There is only one idea that I might mention for now.

I have installed QGIS, and worked through the relevant exercises to familiarize myself with it. Ujaval Gandhi’s tutorials are absolutely great in this respect.

The idea I can blog about right away is this. As I mentioned earlier, DEM maps with 5 m resolution are impossible to find. I asked my father to see if he had any detailed map at sub-talukaa level. He gave me an old official map from GSI; it is on a 1:50000 scale, with contours at 20 m. Pretty detailed, but still, since we are looking for check-dams of heights up to 10 m, not so helpful. So, I thought of interpolating contours, and the best way to do it would be through some automatic algorithms. The map anyway has to be digitized first.

That means, scan it at a high enough resolution, and then perform a raster to vector conversion so that DEM heightfields could be viewed in QGIS.

The trouble is, the contour lines are too faint. That means, automatic image processing to extract the existing contours would be of limited help. So, I thought of an idea: why not lay a tracing paper on top, and trace out only the contours using black pen, and then, separately scan it? It was this idea that was already mentioned in an official Marathi document by the irrigation department.

Of course, they didn’t mean to go further and do the raster-to-vector conversion and all.  I would want to adapt/create algorithms that could simulate rainfall run-offs after high intensity sporadic rains, possibly leading also to flooding. I also wanted to build algorithms that would allow estimates of volumes of water in a check dam before and after evaporation and seepage. (Seepage calculations would be done, as a first step, after homogenizing the local geology; the local geology could enter the computations at a more advanced stage of the research.) A PhD student at IIT Bombay has done some work in this direction, and I wanted to independently probe these issues. I could always use raster algorithms, but since the size of the map would be huge, I thought that the vector format would be more efficient for some of these algorithms. Thus, I had to pursue the raster-to-vector conversion.

So I did some search in this respect, and found some papers and even open source software. For instance, Peter Selinger’s POTrace, and the further off-shoots from it.

I then realized that since the contour lines in the scanned image (whether original or traced) wouldn’t be just one-pixel wide, I would have to run some kind of a line thinning algorithm.

Suitable ready made solutions are absent and building one from the scratch would be too time consuming—it can possibly be a good topic for a master’s project in the CS/Mech departments, in the computer graphics field. Here is one idea I saw implemented somewhere. To fix our imagination, launch MS Paint (or GIMP on Ubuntu), and manually draw a curve in a thick brush, or type a letter in a huge font like 48 points or so, and save the BMP file. Our objective is to make a single pixel-thick line drawing out of this thick diagram. The CS folks apparently call it the centerlining algorithm. The idea I saw implemented was something like this: (i) Do edge detection to get single pixel wide boundaries. The “filled” letter in the BMP file would now become “hollow;” it would have only the outlines that are single pixel wide. (ii) Do raster-to-vector conversion, say using POTrace, on this hollow letter. You would thus have a polygon representation for the letter. (iii) Run a meshing software (e.g. Jonathan Schewchuk’s Triangle, or something in the CGAL library) to fill the interior parts of this hollow polygon with a single layer of triangles. (iv) Find the centroids of all these triangles, and connect them together. This will get us the line running through the central portions of each arm of the letter diagram. Keep this line and delete the triangles. What you have now got is a single pixel-wide vector representation of what once was a thick letter—or a contour line in the scanned image.

Sine this algorithm seemed too complicated, I thought whether it won’t be possible to simply apply a suitable diffusion algorithm to simply erode away the thickness of the line. For instance, think that the thick-walled letter is initially made uniformly cold, and then it is placed in uniformly heated surroundings. Since the heat enters from boundaries, the outer portions become hotter than the interior. As the temperature goes on increasing, imagine the thick line to begin to melt. As soon as a pixel melts, check whether there is any solid pixel still left in its neighbourhood or not. If yes, remove the molten pixel from the thick line. In the end, you would get a raster representation one pixel thick. You can easily convert it to the vector representation. This is a simplified version of the algorithm I had implemented for my paper on the melting snowman, with that check for neighbouring solid pixels now being thrown in.

Pursuing either would be too much work for the time being; I could either offload it to a student for his project, or work on it at a later date.

Thus ended my present thinking line on the micro-level water-resources engineering.

2. Quantum mechanics:

You knew that I was fooling you when I had noted in my post dated the first of April this year, that:

“in the course of attempting to build a computer simulation, I have now come to notice a certain set of factors which indicate that there is a scope to formulate a rigorous theorem to the effect that it will always be logically impossible to remove all the mysteries of quantum mechanics.”

Guess people know me too well—none fell for it.

Well, though I haven’t quite built a simulation, I have been toying with certain ideas about simulating quantum phenomena using what seems to be a new fluid dynamical model. (I think I had mentioned about using CFD to do QM, on my blog here a little while ago).

I pursued this idea, and found that it indeed should reproduce all the supposed weirdities of QM. But then I also found that this model looks a bit too contrived for my own liking. It’s just not simple enough. So, I have to think more about it, before allocating any specific or concrete research activities about it.

That is another dead-end, as far as blogging is concerned.

However, in the meanwhile, if you must have something interesting related to QM, check out David Hestenes’ work. Pretty good, if you ask me.

OK. Physicists, go away.

3. Homeopathy:

I had ideas about computational modelling for the homeopathic effect. By homeopathy, I mean: the hypothesis that water is capable of storing an “imprint” or “memory” of a foreign substance via structuring of its dipole molecules.

I have blogged about this topic before. I had ideas of doing some molecular dynamics kind of modelling. However, I now realize that given the current computational power, any MD modelling would be for far too short time periods. I am not sure how useful that would be, if some good scheme (say a variational scheme) for coarse-graining or coupling coarse-grained simulation with the fine-grained MD simulation isn’t available.

Anyway, I didn’t have much time available to look into these aspects. And so, there goes another line of research; I don’t have much to do blogging about it.

4. CFD:

This is one more line of research/work for me. Indeed, as far as my professional (academic research) activities go, this one is probably the most important line.

Here, too, there isn’t much left to blog about, even if I have been pursuing some definite work about it.

I would like to model some rheological flows as they occur in ceramics processing, starting with ceramic injection moulding. A friend of mine at IIT Bombay has been working in this area, and I should have easy access to the available experimental data. The phenomenon, of course, is much too complex; I doubt whether an institute with relatively modest means like an IIT could possibly conduct experimentation to all the required level of accuracy or sophistication. Accurate instrumentation means money. In India, money is always much more limited, as compared to, say, in the USA—the place where neither money nor dumbness is ever in short supply.

But the problem is very interesting to a computational engineer like me. Here goes a brief description, suitably simplified (but hopefully not too dumbed down (even if I do have American readers on this blog)).

Take a little bit of wax in a small pot, melt it, and mix some fine sand into it. The paste should have the consistency of a toothpaste (the limestone version, not the gel version). Just like you pinch on the toothpaste tube and pops out the paste—technically this is called an extrusion process—similarly, you have a cylinder and ram arrangement that holds this (molten wax+sand) paste and injects it into a mould cavity. The mould is metallic; aluminium alloys are often used in research because making a precision die in aluminium is less expensive. The hot molten wax+ceramic paste is pushed into the mould cavity under pressure, and fills it. Since the mould is cold, it takes out the heat from the paste, and so the paste solidifies. You then open the mould, take out the part, and sinter it. During sintering, the wax melts and evaporates, and then the sand (ceramic) gets bound together by various sintering mechanism. Materials engineers focus on the entire process from a processing viewpoint. As a computational engineer, my focus is only up to the point that the paste solidifies. So many interesting things happen up to that point that it already makes my plate too full. Here is an indication.

The paste is a rheological material. Its flow is non-Newtonian. (There sinks in his chair your friendly computational fluid dynamicist—his typical software cannot handle non-Newtonian fluids.) If you want to know, this wax+sand paste shows a shear-thinning behaviour (which is in contrast to the shear-thickening behaviour shown by, say, corn syrup).

Further, the flow of the paste involves moving boundaries, with pronounced surface effects, as well as coalescence or merging of boundaries when streams progressing on different arms of the cavity eventually come together during the filling process. (Imagine the simplest mould cavity in the shape of an O-ring. The paste is introduced from one side, say from the dash placed on the left hand side of the cavity, as shown here: “-O”. First, after entering the cavity, the paste has to diverge into the upper and lower arms, and as the cavity filling progresses, the two arms then come together on the rightmost parts of the “O” cavity.)

Modelling moving boundaries is a challenge. No textbook on CFD would even hint at how to handle it right, because all of them are based on rocket science (i.e. the aerodynamics research that NASA and others did from fifties onwards). It’s a curious fact that aeroplanes always fly in air. They never fly at the boundary of air and vacuum. So, an aeronautical engineer never has to worry about a moving fluid boundary problem. Naval engineers have a completely different approach; they have to model a fluid flow that is only near a surface—they can afford to ignore what happens to the fluid that lies any deeper than a few characteristic lengths of their ships. Handling both moving boundaries and interiors of fluids at the same time with sufficient accuracy, therefore, is a pretty good challenge. Ask any people doing CFD research in casting simulation.

But simulation of the flow of the molten iron in gravity sand-casting is, relatively, a less complex problem. Do dimensional analysis and verify that molten iron has the same fluid dynamical characteristics as that of the plain water. In other words, you can always look at how water flows inside a cavity, and the flow pattern would remain exactly the same also for molten iron, even if the metal is so heavy. Implication, surface tension effects are OK to handle for the flow of molten iron. Also, pressures are negligibly small in gravity casting.

But rheological paste being too thick, and it flowing under pressure, handling the surface tensions effect right should be even bigger a challenge. Especially at those points where multiple streams join together, under pressure.

Then, there is also heat transfer. You can’t get away doing only momentum equations; you have to couple in the energy equations too. And, the heat transfer obviously isn’t steady-state; it’s necessarily transient—the whole process of cavity filling and paste solidification gets over within a few seconds, sometimes within even a fraction of a second.

And then, there is this phase change from the liquid state to the solid state too. Yet another complication for the computational engineer.

Why should he address the problem in the first place?

If the die design isn’t right, the two arms of the fluid paste lose heat and become sluggish, even part solidify at the boundary, before joining together. The whole idea behind doing computational modelling is to help the die designer improve his design, by allowing him to try out many different die designs and their variations on a computer, before throwing money into making an actual die. Trying out die designs on computer takes time and money too, but the expense would be relatively much much smaller as compared to actually making a die and trying it. Precision machining is too expensive, and taking a manufacturing trial takes too much time—it blocks an entire engineering team and a production machine into just trials.

So, the idea is that the computational engineer could help by telling in advance whether, given a die design and process parameters, defects like cold-joins are likely to occur.

The trouble is, the computational modelling techniques happen to be at their weakest exactly at those spots where important defects like cold-joins are most likely. These are the places where all the armies of the devil come together: non-Newtonian fluid with temperature dependent properties, moving and coalescing boundaries, transient heat transfer, phase change, variable surface tension and wall friction, pressure and rapidity (transience would be too mild a word) of the overall process.

So, that’s what the problem to model itself looks like.

Obviously, ready made software aren’t yet sophisticated enough. The best available are those that do some ad-hoc tweaking to the existing software for the plastic injection moulding. But the material and process parameters differ, and it shows in the results. And, that way, validation of these tweaks still is an on-going activity in the research community.

Obviously, more research is needed! [I told you the reason: Economics!]

Given the granular nature of the material, and the rapidity of the process, some people thought that SPH (smoothed particle hydrodynamics) should be suitable. They have tried, but I don’t know the extent of the sophistication thus far.

Some people have also tried finite-differences based approaches, with some success. But FDM has its limitations—fluxes aren’t conserved, and in a complex process like this, it would be next to impossible to tell whether a predicted result is a feature of the physical process or an artefact of the numerical modelling.

FVM should do better because it conserves fluxes better. But the existing FVM software is too complex to try out the required material and process specific variations. Try introducing just one change to a material model in OpenFOAM, and simulating the entire filling process with it. Forget it. First, try just mould filling with coupled heat transfer. Forget it. First, try just mould filling with OpenFOAM. Forget it. First, try just debug-stepping through a steady-state simulation. Forget it. First, try just compiling it from the sources, successfully.

I did!

Hence, the natural thing to do is to first write some simple FVM code, initially only in 2D, and then go on adding the process-specific complications to it.

Now this is something about I have got going, but by its nature, it also is something about you can’t blog a lot. It will be at least a few months or so before even a preliminary version 0.1 code would become available, at which point some blogging could be done about it—and, hopefully, also some bragging.

Thus, in the meanwhile, that line of thought, too comes to an end, as far as blogging is concerned.

Thus, I don’t (and won’t) have much to blog about, even if I remain (and plan to remain) busy (to very busy).

So allow me to blog only sparsely in the coming weeks and months. Guess I could bring in the comments I made at other blogs once in a while to keep this blog somehow going, but that’s about it.

In short, nothing new. And so, it all is (and is going to be) crap.

More of it, later—much later, may be a few weeks later or so. I will blog, but much more infrequently, that’s the takeaway point.

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(Marathi) “madhu maagashee maajhyaa sakhyaa pari…”
Lyrics: B. R. Tambe
Singer: Lata Mangeshkar
Music: Vasant Prabhu

[I just finished writing the first cut; an editing pass or two is still due.]

[E&OE]

# Getting dusty…

I have been getting dusty for some time now.

… No, by “dusty” I don’t mean that dust of the “Heat and Dust” kind, even though it’s been quite the regular kind of an “unusually hot” summer this year, too.

[In case you don’t know, “Heat and Dust” was a neat movie that I vaguely recall I had liked when it had come on the scene some 2–3 decades ago. Guess I was an undergrad student at COEP or so, back then or so. (Google-devataa now informs me that the movie was released in 1983, the same year that I graduated from COEP.)]

Anyway, about the title of this post: By getting dusty, I mean that I have been trying to get some definite and very concrete beginning, on the software development side, on modelling things using “dust.” That is, particles. I mean to say, methods like molecular dynamics (MD), smoothed particle hydrodynamics (SPH), lattice Boltzmann method (LBM), etc. … I kept on postpoing writing a blog post here with the anticipation that I would succeed in tossing a neat toy code for a post.

… However, I soon came face-to-face with the  sobering truth that since becoming a professor, my programming skills have taken a (real) sharp dip.

Can you believe that I had trouble simply getting wxWidgets to work on Ubuntu/Win64? Or to get OpenGL to work on Ubuntu? It took almost two weeks for me to do that! (And, I haven’t yet got OpenGL to work with wxWidgets on Ubuntu!) … So, finally, I (once again) gave up the idea of doing some neat platform-neutral C++ work, and, instead (once again) went back to VC++. Then there was a problem awaiting me regarding VC++ too.

Actually, I had written a blog post against the modern VC++ at iMechanica and all (link to be inserted) but that was quite some time back (may be a couple of years or so). In the meanwhile, I had forgotten how bad VC++ has really become over the years, and I had to rediscover that discouraging fact once again!

So, I then tried installing VC++ 6 on Win7 (64-bit)—and, yet another ugly surprise. VC++ 6 won’t run on Win7. It’s possible to do that using some round-about ways, but it all is now a deprecated technology.

Finally, I resigned myself to using VC++ 10 on Win7. Three weeks of precious vacation time already down!

That‘s what I meant when I said how bad a programmer I have turned out to be, these days.

Anyway, that’s when I finally could begin writing some real though decidedly toy code for some simple MD, just so that I could play around with it a bit.

Though writing MD code seems such a simple, straight-forward game (what’s the Verlet algorithm if not plain and simple FDM?), I soon realized that there are some surprises in it, too.

All of the toy MD code to be found on the Internet (and some of the serious code, too) assumes only a neat rectangular or a brick-like domain. If you try to accommodate an arbitrary shaped domain (even if only with straight-lines for boundaries), you immediately run into the run-time efficiency issues. The matter gets worse if you try to accommodate holes in the domain—e.g., a square hole in a square domain like what my Gravatar icon shows. (It was my initial intention to quickly do an MD simulation for this flow through square cavity having a square hole.)

Next, once you are able to handle arbitrarily shaped domains with arbitrarily shaped holes, small bugs begin to show up. Sometimes, despite no normal flux condition, my particles were able to slip past the domain boundaries, esp. near the points where two bounding edges meet. However, since the occurrence was rare, and hard to debug for (what do you do if it happens only in the 67,238th iteration? hard-code a break after 67,237, recompile, run, go off for a coffee?) I decided to downscale the complexity of the basic algorithm.

So, instead of using the Lennard-Jones potential (or some other potential), I decided to switch off the potential field completely, and have some simple perfectly hard and elastically colliding disks. (I did implement separate shear and normal frictions at the time of collisions, though. (Of course, the fact that frictions don’t let the particles be attracted, ever, is a different story, but, remember, we are trying to be simple here.)) The particles were still managing to escape the domain, at rare times. But at least, modelling with the hard elastic disks allowed me to locate the bugs better. Turned out to be a very, very stupid-some matter: my algorithm had to take care for a finite-sized particle interacting with the boundary edges.

But, yes, I could then get something like more than 1000 particles happily go colliding with each other for more than 10^8 collisions. (No, I haven’t parallelized the code. I merely let it run on my laptop while I was attending in a longish academics-related meeting.)

Another thing: Some of the code on the ‘net, I found, simply won’t work for even a modestly longer simulation run.  For instance, Amit Kumar’s code here [^]. (Sorry Amit, I should have written you first. Will drop you a line as soon as this over-over-overdue post is out the door.) The trouble with such codes is with the time-step, I guess. … I don’t know for sure, yet; I am just loud-thinking about the reason. And, I know for a fact that if you use Amit’s parameter values, a gas explosion is going to occur rather soon, may be right after something like 10^5 collisions or so. Java runs too slowly and so Amit couldn’t have noticed it, but that’s what happens with those parameter values in my C++ code.

I haven’t yet fixed all my bugs, and in fact, haven’t yet implemented the Lennard-Jones model for the arbitrarily shaped domains (with (multiple) holes). I thought I should first factor out the common code well, and then proceed. … And that’s when other matters of research took over.

Well, why did I get into MD, in the first place? Why didn’t I do something useful starting off with the LBM?

Well, the thing is like this. I know from my own experience that this idea of a stationary control volume and a moving control volume is difficult for students to get. I thought it would be easy to implement an MD fluid, and then, once I build in the feature of “selecting” (or highlighting) a small group of molecules close to each other, with these molecules together forming a “continuous” fluid parcel, I could then directly show my students how this parcel evolves with the fluid motion—how the location, shape, momentum and density of the parcel undergoes changes. They could visually see the difference between the Eulerian and Lagrangian descriptions. That, really speaking, was my motivation.

But then, as I told you, I discovered that I have regressed low enough to have become a very bad programmer by now.

Anyway, playing around this way also gave me some new ideas. If you have been following this blog for some time, you would know that I have been writing in favor of homeopathy. While implementing my code, I thought that it might be a good idea to implement not just LJ, but also the dipole nature of water molecules, and see how the virtual water behaves: does it show the hypothesized character of persistence of structure or not. (Yes, you read it here first.) But, what the hell, I have far too many other things lined up for me to pursue this thread right away. But, sure, it’s very interesting to me, and so, I will do something in that direction some time in future.

Once I get my toy MD code together (for both hard-disk and LJ/other potentials models, via some refactorability/extensibility thrown in), then I guess I would move towards a toy SPH code. … Or at least that’s what I guess would be the natural progression in all this toys-building activity. This way, I could reuse many of my existing MD classes. And so, LBM should logically follow after SPH—what do you think? (And, yes, I will have to take the whole thing from the current 2D to 3D, sometime in future, too.)

And, all of that is, of course, assuming that I manage to become a better programmer.

Before closing, just one more note: This vacation, I tried Python too, but found that (i) to accomplish the same given functional specs, my speed of development using C++ is almost the same (if not better) than my speed with Python, and (ii) I keep missing the nice old C/C++ braces for the local scope. Call it the quirky way in which an old programmer’s dusty mind works, but at least for the time being, I am back to C++ from that brief detour to Python.

Ok, more, later.

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A Song I Like:
(Marathi) “waaT sampataa sampenaa, kuNi waaTet bhetenaa…”
Music: Datta Dawajekar
Lyrics: Devakinandan Saaraswat
Singer: Jayawant Kulkarni

[PS: A revision is perhaps due for streamlining the writing. May be I will come back and do that.]

[E&OE]

# A Hypothesis on Homeopathy, Part 4

OK, back to the exercise on the three states of matter. (If joining late, first go through my last post [^].)

First of all, take out your sketches. Then, do a Google search on “states of matter.” Click open the Google link to “Images for states of matter,” and browse further, esp. the Encyclopaedia Britannica entry. Also, see the Wiki entry on States of Matter [^] (version, today’s!!). Compare your sketches with these diagrams.

I can bet a coin of (Marathi) “chaar aaNe” (i.e. in Hindi, “chaar aanaa” or “chawanni”) that for most of you, the sketches you made would look very similar to those reported by the Wiki. Also, by the Britannica Encyclopaedia.

To make the matters somewhat more convenient, I also made a small diagram myself, using MS Paint. It appears below, and shows, from top to bottom, a solid, a liquid, and a gas. This diagram is not my idea of how they ought to be drawn, but, as per my aforementioned bet, it does show all yours’!!:

States of Matter, as Usually Shown

Now, on to the main point I said you all were going to miss (well, almost all of you, anyway!). To illustrate it, let’s take an example.

Take any metal, say, iron. When a piece of solid iron is heated, it melts into liquid iron. Does the above sketch adequately represent this change of state? Think about it.

If you know materials science, you would know that iron exists as a BCC (body-centered cubic) crystal at the room temperature and as an FCC (face-centered cubic) crystal at intermediately high temperatures. Etc. In contrast, my diagram shows iron as a SCC (simple cubic structure) crystal. Point granted. But that’s not the point I have in mind here.

So, my question still is: Do you find anything else wrong with that picture, esp. the sketches for the solid and liquid states?

In case you didn’t, here’s a hint.

What is the density of solid iron at room temperature? (Did at least that question give you a hint, now?) The density is, say, 7800 kg/m^3. OK. Now, what is the density of liquid iron, at a temperature a little beyond its melting point, say, at 1725 C? It is, say, 6900 kg/m^3. Given these data, suppose, you begin with one kg of iron, and melt it, and further heat the liquid iron to 1725 C. What do you think is the overall volume expansion, in percentage terms for such a solid to liquid change of state? (Note, volume and density are inversely related.) Say, roughly, 10 percent. That is, on the volume basis.

How does a volume expansion translate into lineal terms? For a unit cube to have a 10 percent increase in volume, what must be the increase in the length of each of its sides? I will provide some details for the benefit of some of my readers. Does it go like: $\sqrt[3]{0.1} \approx 0.464 \approx 46 \%$? Or does it go like this: $(1.0+e)^3 = 1.1$ where $e$ is the per-unit expansion (in fractional terms), and hence, $1.0 + e = \sqrt[3]{1.1} \approx 1.03228$, implying that $e = 0.03228 \approx 3.2 \%$? 🙂

The second answer is correct. In lineal terms, the expansion accompanying the solid-to-liquid change is just about 3.2 %. Thus, when a cube of solid iron melts, if the molten liquid is to be held in a cube-shaped container, the container will have sides barely 3.2 % longer. That’s all!

For a viewpoint more convenient for us here: If you keep the volume of the liquid the same as that of the solid, how many atoms from the original solid would you expect to find in that same volume? The answer is: In volume terms, about 91% (because, the inverse of 1.1 is 0.909090…). Hence, in lineal terms, 97%. Hence, in areal terms, about 94%.

What we have drawn above is a 2D representation, and hence, areal terms apply. In the sketch for the solid state, there are exactly 100 atoms (10 rows and 10 columns). Hence, in the corresponding sketch for the liquid state, there should be 94 atoms.

Now, go, refer to the above sketch, and count the number of atoms actually drawn for the liquid state. I did. There are 63 of them.

The question is: How do you squeeze the extra 31 atoms—or approximately 50% more atoms—in the same space? Doesn’t it look impossible?

One way to look at it is this. Why not start with the sketch of those end-to-end stacked atoms in the solid state, and simply draw the area they would be permitted to play in, once the state of the matter changes from solid to liquid? That is precisely what I have done in the sketch below. The side of the square in which the solid atoms fit, was 480 pixels. At about 3.2 % increase, it becomes 496 pixels. I have drawn a blue square of 496 pixels to show you the room allowed to the 100 atoms, once they get into their liquid state.

Extra Space Available to a Solid When It Becomes Liquid

Conclusion: Contrary to what (i) your high-school teacher, (ii) Wikipaedia, (iii) Encyclopaedia Britannica, (iv) etc. told you, (and contrary to what their high-school teachers, in turn, had told them) atoms simply don’t find as much room to roam around as the usual sketches show, when they go from the solid state to the liquid state. Indeed, in a great contrast to gases, both liquid and solid states are almost similar in terms of how tightly packed their atoms are.

Now, let’s try to integrate what that conclusion means, assuming the neat hypothetical substance we have sketched above. (Don’t rush to apply this discussion directly to water. We will cover water in some future post in this series. Water shows too much of anomalous behavior. The hypothetical substance we took here is based on the hard-spheres assumption, and as such, for some purposes, it might provide a good abstraction for metals—but not for salts, minerals, silicates, glasses, plastics, liquid crystals, etc., and, above all, in most important respects, not for water either.)

Is our model for the liquid state consistent with the fact that liquids flow readily? What really happens at the atomic level when liquids flow?

If the questions includes: “at the atomic level,” the answer is: “not much.” While you have been taught, perhaps right from your high-school science, that both liquids and gases are fluids, there is a great difference between how the flow processes must be occurring in them at the atomic level. Let me explain.

Here, you must first understand that, in the general sense of the term, a flow cannot be generated by simple direct pulls or pushes, i.e. by applying the normal load. Pressure does not result in transport of matter from one region of space to another. Pressure differences might. But not pressure by itself (whether it is positive or negative). Flow, in general, involves transport of a quantity of matter from one spatial region to another, together with change of shape, too—not just a local change of volume. For flow in this sense to occur, it is shear loading—and not normal loading—which must get applied. Liquids can flow only when the loading type is shear.

Do a physical experiment. Construct a large rectangular frame out of four wooden battons that are hinged at each of its four corners. It should also be possible to easily dis-assemble the sides, if necessary. Place the frame on a flat glass top, so as to make a rectangular tray. The frame should be large enough to carry a great number of (hundreds of) small glass marbles forming a monolayer on the surface of the table-top. Place the marbles in the tray following different systematic ways. For instance, first, let all blue marbles be together and all red marbles be together. Next, create one or two circular clusters of blue marbles in the other red marbles. Etc.

If the lengths of the frame-sides are such that they exceed the dense packing of marbles by a few percentage , the setup will show a liquid state. If the marbles instead fit snugly, it will show a solid state. (For representing the solid state, ideally, the hexagonal packing should be considered, not the square-packing shown above. (Exercise: Find out the lineal expansion factor e when a hexagonal solid becomes liquid.))

Starting with a setup for the liquid state (i.e. with, say, 3% extra length), first, shake the frame horizontally for some time. Observe how the marbles move. In particular, mark a few marbles and observe how they move with respect to their neighbouring marbles. Do the initial neighbours remain neighbours? (To CS guys: The issue here is not primarily the number of connected elements; it is: whether each connection keeps the same neighbouring element all throughout, or not.)

Notice, even if in this experiment, you have not allowed for interatomic bonds between marbles (which is an extremely important point), you would still find something interesting. Unlike in gases, thermal energy (or vibrations) does not serve to break “neighbourhoods.” For most atoms, even over very very long times of shaking, and even when there is no attractive interatomic potential present, the originally directly neighbouring atoms continue to remain direct neighbours. Even in the liquid state, not just the solid one. This is an important point, and though it is so easy to see, it is also very easily possible that you read it here first. Science is like that.

Next, take a fresh setup of liquid-like marbles in a frame. Now, completely remove two opposite sides of the frame, and, using a couple of extra battons, slowly apply shear to the marbles so that the “liquid” deforms in shear in such a way that two large chunks of marbles are made to merely slide against each other. What do you observe?

Obviously, even in such a shear deformation, most atoms remain as closely connected as ever. Only very few atoms, i.e. precisely those which happen to lie on the line of the shear-slip, slide against each other. Only these few atoms acquire new individual neighbours. The rest of the overwhelming majority of neighbourhoods remain intact.

Lesson: When liquids flow, they do not do so via independent motions of individual atoms as happens in gases, but via sliding motions of large blocks of atoms. There simply is not enough room for liquid atoms to behave in a neat conformance to your high school teachers’ imagination. Consequently, the mechanism of each atom moving independent of others is ruled out. And so, the only deformation mode left to explain flow is: movement of large blocks of neighbourhood-preserving atoms, sliding against each other in shear.

Suppose you argue that this is not the only shear deformation possible. Suppose, you advise this course of action: Keep all the four sides of the frame intact and simply apply shear to the overall frame, so as to deform the whole frame from a rectangular shape to a general parallelogram shape. Now, all atoms would move, albeit in a sliding mode. Accordingly, suppose that the battons that are originally alighed to the y-axis become, during shear-deformation, oblique-angled—their angle with the x-axis changes from 90 degrees to, say, 30 degrees. In this case, horizontal rows of atoms slide, and yet, all atoms move. If such a shear movement is carried out to a sufficiently large extent (large, as necessary to explain magnitudes experiences in real flows), then all atoms would have changed all their direct neighbours.

Why shouldn’t a liquid flow in this manner? You may try to buttress your argument by making a reference to the diagram they show while teaching Newton’s law of viscosity: The liquid at the ground surface is shown to remain stationary throughout, whereas planes of liquid are shown sliding at ever incresing velocity as you go away from the surface. Why not apply the suggested idea to the atomic model, you may ask.

To find the answer, just try it. Try it with actual marbles. See whether producing a shear this way is easier or more difficult. You will find that the mode I suggested—two blocks sliding on only one plane—is easier. And, the rows-sliding mode becomes plane-sliding mode for a 3D fluid. Between the block-sliding mode and the rows-sliding mode, the former requires far less energy, and hence, Nature would pursue it.

Thus, real liquids do not flow in the rows-sliding mode, despite the nice abstract illustration they use while teaching viscosity. (The illustration accompanying viscosity assumes a continuum, just in case you missed it!)

In fact, even the description of two blocks shearing away, the way we presented it above, is not an exact description. In reality, the boundary conditions are such that liquids can split into 3D regions—call them globules or cells or whatever. Each of the 3D globules might itself suffer deformation, but the point is, it would do so only as an independent block. Thus, the point is, within each globule/3D region, it’s not just planes of atoms but entire 3D blocks that remain together. Thus, when liquid flow occurs, for most atoms, there are blocks within which neighbouring atoms are never replaced by other atoms.

The idea of blocks makes it necessary to think in terms of scales. At what scale does separation via sliding occurs? How? etc.

It’s perhaps possible that real liquids flow with not just micro- but also meso- and even macro-scale structures.

However, even if you take only nano-scale blocks, precisely because the Avogadro number is so mind-bogglingly large, it still means that literally billions of atoms remain together even in extremely turbulent motion. Not convinced? Do a simple calculation. Take a reasonable 100 pm (1 angstrom or 0.1 nm) as the reasonable effective size for an atom while in the liquid state. For a 50 nm cubic block, it means: $500^3 \approx 125$ million atoms. For a 100 nm cubic block, it means: 1 billion atoms.

Atoms this many (hundreds of millions) move together even in the most turbulent motion. Even if there are nasty mechanisms like generation of gas bubbles and their collapse. (If not convinced, estimate the overall volume fraction of gas bubbles in the given liquid, and the size of the tiniest bubble possible in it.) (BTW, in case you didn’t know, gas-bubbles are more disruptive of a liquid’s fabric than the motions of second phase particles such as undissolved solids, colloidally dispersed particles, etc.)

Now, keep aside the hard-spheres assumption for a while and think what it means for hundreds of millions (or even billions) of atoms to move together i.e. without destroying their neighbourhoods. (Even in the worst turbulent motion.)

It means: their quantum mechanical wavefunction involves sufficiently large number of particles that the configuration space is sufficiently large that it would be capable of supporting an incredibly greater number of at least metastable states brought about by whatever external means. In highly simplified terms, what it in turn means are two things: (i) structures undestroyed by most turbulent flows are possible in liquids in general (let alone in water), (ii) the structures can be of an incredibly great variety. In other words, it is possible that liquids can carry “memory,” at a suitable scale.

Notice, we made an extremely pathetic (i.e. unrealistic) assumption in the development of our argument above: a complete absence of interatomic forces (which allows for great slideability). Real liquids are not like that.

Further, to stay close to what could be happening in homeopathic preparation process, we thought of only nano-scale blocks. Despite this worst-case scenario, we found that we still have millions of atoms staying together, even during the most violent liquid motion, and within a space hardly different from the most rigidly structured solids.

Structure and liquid can only be rather thick friends.

Not surprising. Though liquids don’t keep their shape, they do keep their volume—they don’t fill the container they are poured in. (Philosophers must be thankful for this circumstance; it allows them the debate as to whether the glass is half-full or half-empty.)

The structures formed in liquids may be different in several respects from those in solids. The structures may not be sufficiently periodic. So, it may be impossible to catch them via diffraction techniques (like XRD/TEM). Yet, based even on the simplest of considerations as those touched upon above, a counter-question naturally arises: How can there not be structures—mers/linkages, rings, “fabrics,” clusters, etc. Involving not just hundreds or thousands, but millions and billions of atoms per block of those structures.

Your high-school teachers gave you an incredibly wrong idea when they drew those 60% volume filled representations for the liquid state—which, in turn, augmented in your mind the idea that liquids are structureless.

And, mind you, the “liquids” we considered here were very neat ones. Very hypothetical—without interatomic forces, and with all atoms spherical in shape, i.e., isotropic in the charge distribution. On both these counts, real water differs. A lot. Think how strong then can the evidence that 100% pure water, too, could carry structural imprints.

[PS: Might streamline this a bit or correct any minor mistakes still left, within a few days. If there is any major change, will note it as such.]

Links to my earlier posts on this topic:

A comment on homeopathy [^]
A Hypothesis on Homeopathy, Part 1 [^]
A Hypothesis on Homeopathy, Part 2 [^]
A Hypothesis on Homeopathy, Part 3 [^]

* * * * *    * * * * *    * * * * *

A Song I Like:

(Marathi) “Tap Tap paDati, angaavarati, praajaktaachi phule…”


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# How to interpret a variable that represents a random error between -1 and 1?

In the storage of a stock of $100Kg$ flour bags, a random error $X$ is committed whose density function is of the form $f(x)=k(1-x^2)$, if $-1<x<1$ and $f(x)=0$, otherwise.

• $a)$ Calculate the probability that a sack of flour will pass from $99,5Kg$.
• $b)$ What percentage of sacks will have a weight between $99,8$ and $100,2Kg$?

My question is, the relationship between the error and the $100Kg$. Are we talking about a margin of error of $+1 Kg$ and $-1Kg$ respectively? That is:

In the part $a)$ I must calculate $P (X< -0.5)$ and in the part $b)$ $P (-0.2 <X <0.2)$?

Thank you very much.

• In this case it would be $1-P (X <-0.5)$? – emi Feb 13 '17 at 23:14
The only other possible interpretation I can think of is that $X$ is the percentage error, so that the final weight would be $100 (1+X)$. But, frankly, this would be quite a stretch.

format: plain


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Infinite Series with Numerator Arithmetic and Denominator Geometric

• July 12th 2013, 06:31 PM
Espionage
Infinite Series with Numerator Arithmetic and Denominator Geometric
$(1/1) + (4/4) + (7/16) + (10/64) ...$

This series is supposed to be an infinite series but it's not going up by a set common ratio. How do I find the sum of the infinite series of this series?
• July 12th 2013, 07:01 PM
Soroban
Re: Infinite Series with Numerator Arithmetic and Denominator Geometric
Hello, Espionage!

Quote:

$\text{Evaluate: }\:S \;=\;\frac{1}{1} + \frac{4}{4} + \frac{7}{4^2} + \frac{10}{4^3} + \cdots$

$\begin{array}{ccccccc}\text{We are given:}&S &=& \dfrac{1}{1} + \dfrac{4}{4} + \dfrac{7}{4^2} + \dfrac{10}{4^3} + \cdots \\ \\[-3mm] \text{Multiply by }\frac{1}{4}\!: & \frac{1}{4}S &=& \quad\;\;\; \dfrac{1}{4} + \dfrac{4}{4^2} + \dfrac{7}{4^3} + \cdots \\ \\[-3mm] \text{Subtract:} & \frac{3}{4}S &=& 1 + \underbrace{\frac{3}{4} + \frac{3}{4^2} + \frac{3}{4^3} + \cdots}_{\text{geometric series}} \end{array}$

The geometric series has first term $a = \tfrac{3}{4}$ and common ratio $r = \tfrac{1}{4}$

Its sum is: . $\dfrac{\frac{3}{4}}{1-\frac{1}{4}} \:=\:\dfrac{\frac{3}{4}}{\frac{3}{4}} \:=\:1$

Hence: . $\tfrac{3}{4}S \:=\:1+1 \:=\:2$

Therefore: . $S \:=\:\frac{8}{3}$

**
• July 12th 2013, 07:04 PM
chiro
Re: Infinite Series with Numerator Arithmetic and Denominator Geometric
Edit: Post above answered it way better.
• July 12th 2013, 07:24 PM
MarkFL
Re: Infinite Series with Numerator Arithmetic and Denominator Geometric
If I was going to find the infinite sum, I would begin with the difference equation to obtain the partial sum in closed form:

$S_{n}-S_{n-1}=\frac{3n+1}{4^n}$ where $S_0=1$

The corresponding homogeneous solution is:

$h_n=c_1$

We then look for a particular solution of the form:

$p_n=(An+B)4^{-n}$

We may now determine the coefficients A and B.

Substituting this into the original difference equation, we find:

$(An+B)4^{-n}-(A(n-1)+B)4^{-(n-1)}=(3n+1)4^{-n}$

Multiply through by $4^{n}$:

$(An+B)-4(A(n-1)+B)=3n+1$

$-3An+4A-3B=3n+1$

Equating coefficients, we obtain the system:

$-3A=3\,\therefore\,A=-1$

$4A-3B=1\,\therefore\,B=-\frac{5}{3}$

And thus:

$p_n=-\frac{1}{3}(3n+5)4^{-n}$

By superposition, we have:

$S_n=h_n+p_n=c_1-\frac{1}{3}(3n+5)4^{-n}$

Using the initial value, we find:

$S_0=c_1-\frac{1}{3}(3(0)+5)4^{0}=c_1-\frac{5}{3}=1\,\therefore\,c_1=\frac{8}{3}$

and so we find:

$S_n=\frac{8}{3}-\frac{1}{3}(3n+5)4^{-n}$

Hence, the infinite sum is:

$S=\lim_{n\to\infty}S_n=\frac{8}{3}$
• July 12th 2013, 07:26 PM
Espionage
Re: Infinite Series with Numerator Arithmetic and Denominator Geometric
Thank you very much.


text:
5 added 576 characters in body

It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb spectare (paradigm: specto, -as, -avi, -atum, -are), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.

Edit: after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: spècere (or interchangeably spicere)= "to see", from which comes the root spec- of the latin word spectrum= "something that appears, that manifests itself, vision". Furthermore, spec- = "to see", -trum = "instrument" (like in spec-trum). Also the term "spectrum" in astronomy and optics has the same origin.

In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).

I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from in + spècere...] step by step the deep properties of some cohomological constructions.

Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?

Edit: The following paragraph from the wikipedia article on "primon gas" seems to support my guess:

"The connections between number theory and quantum field theory can be somewhat further extended into connections between topological field theory and K-theory, where, corresponding to the example above, the spectrum of a ring takes the role of the spectrum of energy eigenvalues, the prime ideals take the role of the prime numbers, the group representations take the role of integers, group characters taking the place the Dirichlet characters, and so on"

4 added 10 characters in body

It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb spectare (paradigm: specto, -as, -avi, -atum, -are), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.

Edit: after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: spècere (or interchangeably spicere)= "to see", from which comes the root spec- of the latin word spectrum= "something that appears, that manifests itself, vision". Furthermore, spec- = "to see", -trum = "instrument" (like in spec-trum). Also the term "spectrum" in astronomy and optics has the same origin.

In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).

I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from in + spècere...] step by step the deep properties of some cohomological constructions.

Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?

3 added 3 characters in body

It seems reasonable to me that in operator theory the term "spectrum" comes from the Latin verb spectare (paradigm: specto, -as, -avi, -atum, -are), which means "to observe". After all in quantum mechanics the spectrum of an observable, i.e. the eigenvalues of a self adjoint operator, is what you can actually see (measure) experimentally.

Edit: after having a look to an online etymological dictionary, it seems the relevant Latin verb is another: spèceospècere (or interchangeably spiceospicere), from which comes the root spec- of the latin word spectrum= "something that appears, that manifests itself, vision". Furthermore, spec- = "to see", -trum = "instrument" (like in spec-trum). Also the term "spectrum" in astronomy and optics has the same origin.

In algebraic geometry, I believe the term "spectrum", and the corresponding concept, has been introduced after the development of quantum mechanics became well known. In this context, the concept of spectrum as a space made of ideals is perfectly analogous of that in operator theory (think of Gelfand-Naimark theory, and that the Gelfand spectrum of the abelian C-star algebra generated by one operator is nothing but the spectrum of that operator).

I wouldn't be surprised if the term "spectral sequence" had something to do with "inspecting" [b.t.w. also "to inspect" comes from in + speceospècere...] step by step the deep properties of some cohomological constructions.

Maybe the term "spectrum" in homotopy theory and generalized (co)homology -but I don't know almost anything about these- has to do with "probing", "testing", a space via maps from (or to?) certain standard spaces such as the Eilenberg-MacLane spaces or the spheres. Does it sound reasonable?


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# Common problem: OP deletes question immediately upon receiving answer

I saw a question on this site and solved it and took a very long time to answer it after a lot of typing, and as soon as the asker saw the solution he/she deleted their question immediately.

Is this the way one is grateful to someone else here? This is not really good right?

I literally spent about 15 minutes on posting the solution which I wrote in the most detailed way possible, I couldn't have if I didn't wish to?

l

• Flag the question for moderator attention, too. – amd Apr 22 '20 at 7:23
• How do i flag a deleted question? Or you mean this one? – user732848 Apr 22 '20 at 7:26
• It seems that this particular question has been undeleted by a moderator. – Martin Sleziak Apr 22 '20 at 8:07
• – Martin Sleziak Apr 22 '20 at 8:11
• We've been here before, and there's general agreement that users should not delete questions after getting an answer, and, if they do, those questions should be undeleted. BUT there was a red flag on this question that might have warned you that you might be wasting your time writing out an answer. Namely, the user showed no sign of being willing to put any effort into the question, beyond that required to type it in. We generally frown on that, and ask such a user to show us what progress he had made, and so on. Indeed (continued) – Gerry Myerson Apr 22 '20 at 12:23
• (continued) had the user not deleted the question, there are many users who would have voted to close and delete it on just those grounds, and your answer would have vanished anyway. Also, many of us don't like writing out answers that a user can copy & paste into a homework assignment without thinking about it; we prefer to give hints and to encourage the poster to come to an understanding of the problem on his own. The users who refuse to meet us halfway are the ones most likely to delete the question anyway, I suspect. So maybe think about which questions are worth your full efforts. – Gerry Myerson Apr 22 '20 at 12:28
• Thanks a lot, i will keep in mind from next time. – user732848 Apr 22 '20 at 12:32
• Yah you almost certainly spent 15 minutes helping someone cheat on their homework or at the very least your answer is detailed enough to do that(although the poster did ask for a hint too so maybe they just asked a bad question and not looking to cheat). While deleting shouldn't have been done, it probably should have been closed and you in some sense also acted badly—whether intentional or not. – user29123 Apr 22 '20 at 13:11
• Sorry @PaulPlummer , whenever i see a question lacking the part which they try, i give a downvote and tell in the comments to do the needful or tell them, maybe i made a mistake this time. Will not be repeated. – user732848 Apr 22 '20 at 13:28
• @Shamim: Perhaps you found the problem interesting enough to regard what you learned from studying it sufficient reward for your efforts. I often limit myself to answering such Questions, in which my own learning is an important factor. So you are welcome to tackle "problem statement questions" if you are more interested in what you gain than in the possible downside you experienced in this case. – hardmath Apr 22 '20 at 15:51
• Yes @hardmath i like number theory very much – user732848 Apr 22 '20 at 15:55
• Don't feel too bad, @Shamim. It looks like a lot of seasoned answerers have been doing this asker's work for them for some time. Perhaps your post here today will help curb that. Keep your enthusiasm! – amWhy Apr 22 '20 at 18:13
• Thanks @amWhy very much! :) – user732848 Apr 23 '20 at 1:41
• The question's been deleted and my 40 reputation goes away ;( – user732848 Apr 23 '20 at 4:26
• @Shamim You gained, and then lost, 40 imaginary internet points. Don't sweat it. Your actual Reputation (what people think of you, how people perceive your actions, etc) is far more important than your reputation (a silly number on a website). – Xander Henderson Apr 23 '20 at 14:51

That is a very rude behaviour. What I do in this situation (yes, I've been there too) is to flag the question, telling the moderators that they should warn the OP that this is an inappropriate behaviour. Usually, the questions gets reopend (by a moderator, I guess, but I am not sure about that).

• Yes, i have flagged this question(this one) and i have added to deal with the asker in the appropriate way. Thanks! – user732848 Apr 22 '20 at 8:13

For the record, the question under consideration is How to solve this divisibility problem: when is $$f(m, n) = mn^2 + am^2 + n^2 +8$$ divisible by $$16$$?. No link was provided in the original meta question, and Martin Sleziak's link in the comments is to the revision history.

1. It is considered inappropriate for a user to delete a question immediately after they get an answer. It is rude to the person who put the effort into answering, and is often understood as evidence of an effort to cheat (i.e. they get an answer on MSE, then delete their question to "cover their tracks"; many users don't understand that deleted questions never really go away). When a user does this, it is appropriate to take action:

• Flag the post for moderator attention. Explain what has happened ("This question was deleted immediately after I provided an answer."). It is likely that a moderator will eventually come by to undelete the question.

• Post an answer to the Reopen Request thread on the math meta. This thread is meant to bring attention to questions which users would like to see reopened and/or undeleted.

• Post a comment in the CRUDE chatroom (CRUDE stands for "Close Reopen Undelete Delete Edit"; the goal of this room is to help coordinate janitorial duties on the site of the type described in the acronym). Those that participate in CRUDE are generally pretty quick to undelete questions which have been deleted immediately after receiving an answer.

2. That being said, the question itself is of fairly low quality. In the future, before you answer a question, you might ask yourself if it is going to be worth your time. Some things to consider:

• Has the asker shown a willingness to make an effort? This might mean that the question was well-written from the get-go, or that the asker engaged with users asking questions in the comments. In this case, I don't see any major red flags&mddash;the original title was weak, but the question is correctly formatted otherwise, and the asker responded to the one comment which was left.

• Is the question of broad interest, or is it very narrowly focused? In other words, is the question asking about principle, or is it the kind of thing that might show up on a homework assignment? In the case of the question being discussed here, I think that there is a major red flag: the question is quite specific, and appears to be a homework exercise designed to familiarize a student with a broader principle. These kinds of questions are not a priori off-topic on MSE, but they do require more context.

• Has the asker provided context? The goals of context are (broadly speaking) two-fold: they orient answerers to the general framework in which a question is being answered by providing necessary definitions or references; and they give answerers an idea about the level of the asker so that answers can be written to an appropriate level. This particular question is what is known as a PSQ (a "problem statement question" or, more cynically, a "please solve question"). No context has been provided. We don't know what the asker knows, what they are studying, or what theorems they might bring to bear. These kinds of questions are highly likely to be closed and deleted eventually.

In the future you might consider skipping questions like this one. The question is not broadly appealing (it is likely to help only the original asker and other people assigned the same problem, but is unlikely to be helpful otherwise), and lacks sufficient context. If you are worried about wasting your time on answers that are destined for deletion, be more selective about what you answer.

• "In the future you might consider skipping questions like this one." Or, you might consider trying to engage with the poster in comments, to draw him out, to get him to provide context and such, to get him to edit the question up to standards, to guide him to working out a solution himself, actually add some value to the site. – Gerry Myerson Apr 23 '20 at 0:01
• Thanks a lot! I will surely keep this in mind while answering now. – user732848 Apr 23 '20 at 1:51


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# Where are the limits of this question?

• May 24th 2009, 06:53 AM
thomas49th
Where are the limits of this question?
Question 7
http://i157.photobucket.com/albums/t...nArcLength.jpg

At first I thought it was polar co-ordinates but it was parametric so I used the arc length formula to try and find the length of the curve in ONE of the quadrants. I would later times this by 4 to get the length of the curve all together. So arc length = $\int{\sqrt{\frac{dx}{dt}^{2}+\frac{dy}{dt}^{2}}}dt$

so after lots of cancelling and using everybody's favourite trig identity sin^2(t) + cos^2(t) = 1 I get it down to

$3a\int{sint cos t}dt$

Okay now use a substitution of u = sint

to get $\frac{3a}{2} sin^{2}t$ but where are the limits I need to use to get this done? Can someone show me what the limits I need to use are?

Thanks :)

NOTE using t instead of theta
• May 24th 2009, 07:11 AM
Spec
If you're going to calculate the length of the arc in the first quadrant then simply use $0 \leq \theta \leq \frac{\pi}{2}$
• May 24th 2009, 07:12 AM
HallsofIvy
The problem says " $0\le \theta \le 2\pi$"! But because you will have signs that cancel out, integrate from 0 to $\pi/2$ and multiply by 4.
• May 24th 2009, 07:39 AM
thomas49th
Quote:

Originally Posted by HallsofIvy
The problem says " $0\le \theta \le 2\pi$"! But because you will have signs that cancel out, integrate from 0 to $\pi/2$ and multiply by 4.

Im not quite sure what you mean? Have you simply down 2pi/4 but you mention about signs cancelling? Could you please elaborate

Thanks :)
• May 24th 2009, 09:21 AM
Spec
If you put in $0$ and $2\pi$ as the limits then $\sin^2 x|_0^{2\pi}=0$

A length is obviously not negative, but if you integrate around the entire circle you will have four lengths that cancel each other out because two of them will end up negative.


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# tracial triples

Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the non-realizable set?

This is surely known, but I couldn't find an answer by myself, nor a reference on the www. It's easy to see that the set in question is non-empty (it contains the origin, for instance) and it lives in the 3-dimensional square $(-2,2)\times(-2,2)\times(-2,2)$. I'm guessing that the Fricke polynomial $a^2+b^2+c^2-abc$ should enter the picture as well.

Note that realizability is, well, a real problem but not a complex one.

-
Could you please write down next time the polynomial equations you get in terms of the entries of A and B? I change the notation $$det(A) = ad-bc = 1 , \; det (B) = a'd'-b'c', \; tr (A)=a+d = \alpha, \; tr(B) = a'+d'= \beta, \; tr AB = aa'+dd'+bc'+b'c.$$ Where does the Fricke Polynomial turn up? –  Marc Palm Apr 26 '13 at 6:49
$trAB=aa′+dd′+bc′+b′c$ –  Marc Palm Apr 26 '13 at 6:49
$$trAB=aa'+dd'+bc'+b'c = \delta$$ –  Marc Palm Apr 26 '13 at 6:50
In case you are interested: The seemingly algebraic answer has a geometric interpretation. Namely, these are necessary and sufficient conditions for nonexistence of a spherical triangle with the given side-lengths, provided you use logs of eigenvalues instead of traces. –  Misha Apr 28 '13 at 20:39

Let $x, y, z$ be traces of $A, B, AB$ respectively. Define $$k(x,y,z)= x^2+y^2 + z^2 -xyz -2.$$ Then a triple of real traces $(x, y, z)$ is realizable in $SL(2,R)$, unless it is realizable in $SU(2)$, the latter happens if and only if $x, y, z\in [-2,2]$ and $k(x,y,z)\le 2$. See Goldman's paper "Topological components of spaces of representations", Inventiones, 1988.

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Thank you, Misha, for the answer.

Goldman's proof is part of a bigger picture, and it is hard to rip it out of the context. Besides, it looks a bit too sophisticated for such a simple question. Helped by knowing what should happen, I eventually came up with an elementary and self-contained argument, though not particularly elegant. Feel free to improve.

Proposition (Goldman?) A triple $(a,b,c)$ is not realizable iff $|a|,|b|,|c|<2$, $a^2+b^2+c^2−abc<4$.

(This set looks like a slightly inflated tetrahedron. A decent picture would be nice here. An indecent one even nicer.)

Proof. If $|a|=2$, then $(a,b,c)$ is realizable: \begin{align} \begin{pmatrix} \pm 1 & \pm b-c \newline 0 & \pm 1 \end{pmatrix}\begin{pmatrix} b & 1 \newline -1 & 0 \end{pmatrix}=\begin{pmatrix} c & \pm 1 \newline \mp 1 & 0 \end{pmatrix}. \end{align}

Assume $|a|\neq 2$. Let $tr(A)=a$, $tr(B)=b$, $tr(AB)=c$; in particular, $A\neq \pm I$. Up to conjugating $A$ and $B$ by the same matrix, we may assume that $A$ takes the form \begin{align} A=\begin{pmatrix} a & \pm 1 \newline \mp 1 & 0 \end{pmatrix}.\end{align} Put \begin{align} B=\begin{pmatrix} x & z \newline y & b-x \end{pmatrix} \end{align} so $c=tr(AB)=ax\pm y\mp z$. In other words, $z=\pm ax +y\mp c$. The condition $\det B=1$ becomes \begin{align} x^2+y^2\pm axy -bx \mp cy+1=0 \end{align} We can go backwards as well, so $(a,b,c)$ with $|a|\neq 2$ is realizable iff the above equation is solvable over the reals. The rest is even more basic high-school mathematics. Viewing $x$ as the main quadratic variable, the equation is not solvable iff the discriminant \begin{align} (a^2-4)y^2\pm 2(2c-ab)y+(b^2-4) \end{align} is negative for all real $y$. As $|a|\neq 2$, this happens precisely when $|a|<2$ and the discriminant \begin{align} 16(a^2+b^2+c^2-abc-4) \end{align} is negative. For the symmetry's sake, observe that $|a|<2$ and $a^2+b^2+c^2−abc<4$ entail $|b|<2$ and $|c|<2$.

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The above argument has the graciousness of a geriatric turtle's gait, but here's a redeeming feature: it can be used to show that, over finite fields of characteristic not equal to 2, all triples are realizable. In fact, if I am to believe my doodles, non-realizability over any field of characteristic not equal to 2 could be characterized as Hilbert symbol conditions on a,b,c's. –  BN2 Apr 29 '13 at 9:12


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# 2R/stanreg.R

The main things to deal with here are the coefficients and the linear predictors/fitted values.

The first few conditionals deal with picking up the estimated coefficients. Sometimes object$family$family isn’t sufficient to pick up on this so you might have to use is(object, "class_name") to determine whether the object is of a certain class (in addition to the class “stanreg”).

The linear.predictors should be an N-dimensional vector of predictions that have not been transformed by the link function. The fitted.values are the linear predictors transformed by the link function. (e.g. if object$family$family == "gaussian" then the linear predictor and fitted values will be identical since the link function is the identity function.)

Lastly, at the end of out <- list(...) you should include any other stuff that you might need for the methods (e.g. spatial models need the spatial weight matrix, stan_betareg needs the info associated with the “z” linear predictor if declared, etc.)


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# The appropriate code (Arduino) of Xbee

Can you tell me what is the appropriate code to send two variables (x,y) from an Xbee to another Xbee (series 1). I configured them and I can't write a code (because I don't know how to receive information and send them via Xbee and Arduino). Xbee is connected to the shield to Arduino UNO in 1st side, on the other side Xbee is connected to another shield to Arduino Mega.

I have configured Xbees yet and I tried to send commands from PC (Xbee) to Arduino (Xbee) and it worked.

Would someone help me by providing me with an example of sending variables (x,y) from UNO to MEGA (I use Arduino program which contain C/C++), or explain how to do it.

I'd advise you to start by configuring the XBees in AT "Serial Wire" mode. This turns the XBees into a simple serial connection between the Arduinos. From that point on, you use Serial.print()/Serial.write() and Serial.read() just as if the Arduinos were connected directly to each other by wires.

Let's name the XBees XBee_uno and XBee_mega for the following steps. You have most probably completed the first three already, but I'm writing them in full to help others who may have this question.

• Use X-CTU to set both XBees to AT mode (this may necessitate a new firmware or function set, though Series 1 XBees I think only need the setting of one parameter )

• Set both XBees to use the same PAN ID; set one to be coordinator and the other to be a router/end device. Again, you may need a firmware/function set update for this. It doesn't matter which one is the coordinator, as long as one is the coordinator and the other is a router/end device.

• Set the serial speed of both XBees to a known value, e.g. 9600bps, no parity, 8 data 1 stop bits

Now set them so that they send to each other. This completes the "Serial Wire" configuration.

• From X-CTU read the SH and SL of each (i.e. the serial number high/serial number low)

• For XBee_uno, set DH and DL to the value of the SH and SL respectively from the XBee_mega

• For XBee_mega, set DH and DL to the value of the SH and SL respectively from the XBee_uno

• Save the settings to the XBees.

At this point the XBees are configured to function as a virtual wire-pair between the serial ports of the Arduinos.

Sample code:

//on the sender side
void setup() {
Serial.begin(9600);//this has to match the serial speed you configured for the respective XBee
}
void loop() {
Serial.write(20); //or the value you want to send
}

void setup() {
Serial.begin(9600);//this has to match the serial speed you configured for the respective XBee
}
void loop() {
if (Serial.available()) {


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# 3.17: Earth's Crust

What do these two photos have in common?

Both of these photos are of Earth's crust. The crust on the left is on the continent. It's what you see and walk on every day. The crust on the right is in the ocean. You may have seen pictures, but to see it in person you need special equipment.

## Crust

Crust, mantle, and core differ from each other in chemical composition. It's understandable that scientists know the most about the crust and less about deeper layers. Earth’s crust is a thin, brittle outer shell. The crust is made of rock. This layer is thinner under the oceans and much thicker in mountain ranges.

### Oceanic Crust

There are two kinds of crust. Oceanic crust is made of basalt lavas that flow onto the seafloor (Figure below). It is relatively thin, between 5 to 12 kilometers thick (3 - 8 miles). The rocks of the oceanic crust are denser (3.0 g/cm3) than the rocks that make up the continents. Thick layers of mud cover much of the ocean floor.

Gabbro is a dark dense rock that can be found in oceanic crust.

### Continental Crust

Continental crust is much thicker than oceanic crust. It is 35 kilometers (22 miles) thick on average, but it varies a lot. Continental crust is made up of many different rocks (Figure below). All three major rock types—igneous, metamorphic, and sedimentary—are found in the crust. On average, continental crust is much less dense (2.7 g/cm3) than oceanic crust. Since it is less dense, it rises higher above the mantle than oceanic crust.

The average composition of continental crust is granite.

## Summary

• Oceanic crust is thinner and denser than continental crust.
• Oceanic crust is darker in color and denser (more mafic). Continental crust is lighter in color and density (more felsic).
• Earth's crust is very thin relative to Earth's radius.

## Review

1. What are the properties of crust?
2. Describe the properties of oceanic crust.
3. Describe the properties of continental crust.

## Explore More

Use this resource to answer the questions that follow.

https://www.sciencemag.org/news/2018/09/watch-hurricane-put-dent-earth-s-crust

1. Where did Hurricane Harvey make landfall? How much water did it dump?
2. Where did the water go after the storm?
3. What happened to Earth’s crust?
4. How can scientists determine the weight of the water from Harvey?
5. What is depression of crust called?
6. Where did the water go after Harvey?
7. How long did it take for the water stored on land to dissipate?
8. Why will storms strike with increased frequency and ferocity?
9. What does the narrator’s mom always say?

This page titled 3.17: Earth's Crust is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


text:
## Monday, 30 December 2019

### Question

Consider the three Killing vectors of the two-sphere, (3.188). Show that their commutators satisfy the following algebra:
\begin{align}
\left[R,S\right]=T&\phantom {10000}(1)\nonumber\\
\left[S,T\right]=R&\phantom {10000}(2)\nonumber\\
\left[T,R\right]=S&\phantom {10000}(3)\nonumber
\end{align}

The three Killing vectors at 3.188 were
\begin{align}
R&=\partial_\phi&\phantom {10000}(4)\nonumber\\
S&=\cos{\phi}\partial_\theta-\cot{\theta}\sin{\phi}\partial_\phi&\phantom {10000}(5)\nonumber\\
T&=-\sin{\phi}\partial_\theta-\cot{\theta}\cos{\phi}\partial_\phi&\phantom {10000}(6)\nonumber
\end{align}It is easy to prove the algebra as long as we remember that the $\partial_\theta,\partial_\phi$ are just the basis vectors and Carroll's 2.23 that $\left[X,Y\right]^\mu=X^\lambda\partial_\lambda Y^\mu-Y^\lambda\partial_\lambda X^\mu$ which we proved back in exercise 2.04. So first we could write (4),(5),(6) as $R=\left(0,1\right),\ S=\left(\cos{\phi},-\cot{\theta}\sin{\phi}\right),\ T=\left(-\sin{\phi},-\cot{\theta}\cos{\phi}\right)$.

What is more difficult to understand is what does this neat algebra mean? A common way to show a commutator (which Carroll uses when introducing the Riemann tensor) is thus

The commutator $\left[R,S\right]$ is the difference between doing $R$ then $S$ and $S$ then $R$. So the algebra that the Killing vectors satisfy corresponds to this geometry:

$R$ has always been shown as a unit vector in increasing $\phi$ direction, $T,S$ are more flexible. I'm not sure how this helps.

The proof is given at Ex 3.14 Killing vectors on two-sphere.pdf (2 pages)


text:
# Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.[1] In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.[2] In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).

The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".[3] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

## Formulation

When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε) and implication (⊃, which comes from Peano's reversed 'C'.) Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879.[4] Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.[5]

The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or ${\displaystyle \mathbb {N} .}$ The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S.

The first axiom states that the constant 0 is a natural number:

1. 0 is a natural number.

The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.[5]

1. For every natural number x, x = x. That is, equality is reflexive.
2. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
3. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
4. For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S.

1. For every natural number n, S(n) is a natural number.
2. For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection.
3. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number.[6] This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.

The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.

1. If K is a set such that:
• 0 is in K, and
• for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.

The induction axiom is sometimes stated in the following form:

1. If φ is a unary predicate such that:
• φ(0) is true, and
• for every natural number n, φ(n) being true implies that φ(S(n)) is true,
then φ(n) is true for every natural number n.

In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Models below.


text:
# pymatgen.analysis.elasticity.stress module¶

This module provides the Stress class used to create, manipulate, and calculate relevant properties of the stress tensor.

class Stress[source]

This class extends SquareTensor as a representation of the stress

Create a Stress object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.

Parameters

stress_matrix (3x3 array-like) – the 3x3 array-like representing the stress

property dev_principal_invariants

returns the principal invariants of the deviatoric stress tensor, which is calculated by finding the coefficients of the characteristic polynomial of the stress tensor minus the identity times the mean stress

property deviator_stress

returns the deviatoric component of the stress

property mean_stress

returns the mean stress

piola_kirchoff_1(def_grad)[source]

calculates the first Piola-Kirchoff stress

Parameters

piola_kirchoff_2(def_grad)[source]

calculates the second Piola-Kirchoff stress

Parameters

def_grad (3x3 array-like) – rate of deformation tensor

symbol = 's'
property von_mises

returns the von mises stress


text:
# Trigo-sign

• June 21st 2009, 09:03 AM
dhiab
Trigo-sign
• June 21st 2009, 11:13 AM
HallsofIvy
Quote:

Originally Posted by dhiab

$cos(x)+ \sqrt{3} sin(x)= 0$ when $tan(x)= \frac{sin(x)}{cos(x)}= \frac{1}{\sqrt{3}$ and that happens when $x= \pi/6+ k\pi$. $sin^2(x)- 1/2= 0$ when $sin(x)= \pm\frac{1}{\sqrt{2}}$ and that happens when $x= k\pi/4$. Those points are the only places you expression can change sign.


text:
Properties

 Label 1.9.a Base Field $\F_{3^2}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{3^2}$ Dimension: $1$ Weil polynomial: $1 + 9 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$

This isogeny class is simple.

Newton polygon

This isogeny class is supersingular.

 $p$-rank: $0$ Slopes: $[1/2, 1/2]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 100 730 6400 59050 532900 4782970 43033600 387420490 3486902500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 100 730 6400 59050 532900 4782970 43033600 387420490 3486902500

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.