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# Why can't we obtain a Hamiltonian by substituting?

This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't we obtain then a Lagrangian expressed in terms of $t$, $q$ and $p$?

Why do we need to use

$$H(t, q, p) = p\dot{q} - L(t, q, \dot{q})?$$

Or is it that whatever the Lagrangian is the method of finding $\dot{q}=\dot{q}(t,q,p)$. Will give us that equation for $H$?

• Just want to add that if you try to get rid of $\dot{q}$ it's in general a fn of both $q$ and $p$: $\dot{q} = \dot{q}(q,p)$. – nervxxx Jun 25 '13 at 17:14
• More on the Legendre Transformation: physics.stackexchange.com/q/4384/2451 and links therein. – Qmechanic Sep 5 '13 at 14:07

While I appreciate that Wouter's response shows that you can not indeed just perform the aforementioned substitution, it really doesn't answer the question (i.e. why doesn't the substitution work) in my opinion.

The underlying confusion probably comes from the fact that in mathematics if substitution $\dot{q}(p)$ is inserted in $f\left(t,q,\dot{q}\right)$ you will certainly end up with a function $f\left(t,q,\dot{q}\right)=g\left(t,q,p\right)$.

The real answer to the question then, is that Lagrangian and Hamiltonian formalisms are just two different paradigms, and no matter what you put in the Lagrangian instead of $\dot{q}$ you will always end up with another Lagrangian (but for a different problem). In other words the $L$ in front of $\left(t,q,\dot{q}\right)$ has different implications than the $H$, such as stationary action, $n$ second order equations for $n$ coordinates (where $n$ is the number of the degree of freedoms) vs. $2n$ first order equations for $2n$ coordinates, etc. This is unlike the case in mathematics where $f$ carries no other assumption different to $g$.

A fairly basic remark to make is that usually we can plainly identify

$$L = T-U$$

where $T$ is the kinetic energy and $U$ is the potential energy, and

$$H = T+U$$

Expressing these quantities for e.g. a Hooke-like spring (or any system where $U\neq 0$) would give you a problem with the sign of $U$ if you simply substitute the expression you find for $\dot{q}(p)$ into the Lagrangian.$^1$ So the Hamiltonian is definitely not just the Lagrangian with $\dot{q}$ expressed in terms of $p$.

More mathematically expressed, the Hamiltonian is defined as the Legendre transform of the Lagrangian. (for some elaboration on the Legendre transformation - particularly in the context of the Lagrangian and Hamiltonian - see my answer here, as well as the other answers to that question)

$^1$ Indeed, the Lagrangian for such a (1D) system would be

$$L(q,\dot{q}[,t]) = \frac{m\dot{q}^2}{2} - \frac{kq^2}{2}$$

for which the canonically conjugate momentum is

$$p = \frac{\partial L}{\partial \dot{q}} = m\dot{q}$$

and therefore

\begin{align} H(q,p[,t]) &= m\dot{q}\cdot\dot{q} - \frac{m\dot{q}^2}{2} + \frac{kq^2}{2} \\ &= \frac{m\dot{q}^2}{2} + \frac{kq^2}{2} \\ H(q,p[,t]) &= \frac{p^2}{2m} + \frac{kq^2}{2}. \end{align}

Just inserting $p$ into the Lagrangian would yield

$$\frac{p^2}{2m} - \frac{kq^2}{2}.$$

Note the sign difference of the second term.

What you wrote is a variable change in the frame of the same Lagrangian. It remains a Lagrangian.

I would argue that the reason your method fails is because the conversion between Lagrangian and Hamiltonian dynamics is not a simple substitution, but through the mathematical transformation of coordinate bases known as a Legendre transformation, given in terms of the Hamiltonian and Lagrangian functions $$H\left(q,p\right) = \frac{\partial L}{\partial \dot{q}}\dot{q}-L\left(q,\dot{q}\right)$$

As Wouter points out, the transformation you offer does not actually give the correct transformation.

Calculating a Hamiltonian from a Lagrangian leads us to a new quantity, which is a function of coordinates and momenta. If you simply substitute velocity as a function of momentum into the Lagrangian, then the "Hamiltonian" will depend on velocity implicitly. This is not what we want.

Let me show you an example to explain why Hamiltonian (the usual one) does not depend on velocity. Let's say the system is 1-dimensional. So the Lagrangian is $L=L(q,\dot q)$. The Euler-Lagrangian equation is

$\frac{\mathrm{d}}{\mathrm dt}\frac{\partial L}{\partial \dot q}=\frac{\partial L}{\partial q}$.

The canonical momentum is $p=\partial L/\partial \dot q$. If we do Legendre transformation, we have

$H(p,q)=p\dot q-L(q,\dot q(p))$

and consider the variation of $H$:

$\delta H=p\delta \dot q+\dot q\delta p-\frac{\partial L}{\partial q}\delta q-\frac{\partial L}{\partial\dot q}\delta \dot q$.

We can now plug in the definition of $p$:

$\delta H=\dot q\delta p-\frac{\partial L}{\partial q}\delta q$

which shows that $H$ is not an explicit function of $\dot q$, which is what we want.


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The Net Advance of Physics RETRO:Weblog SPIRITUAL DYNAMICS: J. E. Oliver on the Physics of Free Will

2015 March 16

The Cornell faculty in 1882. One of these gentlemen, presumably, is J. E. Oliver.

James Edward Oliver (1829-1895), having trained in almost every branch of science, became professor of mathematics (and eventually chairman of the mathematics department) at Cornell University, serving there for almost a quarter century. His interests were extremely varied. He was an early advocate of (possibly Bayesian, although this is not quite clear) statistical methodology for economics. He studied the "imperfect whiteness" of snow, the soaring of birds, the rotation of the sun ... He also could have been one of the first great American research mathematicians. Unfortunately, as the apostle of non-Euclidean geometry George Bruce Halsted wrote of him in his obituary [Science 1, 544 (1895)], "he seemed to have no ambition to leave an adequate record of his mental life in print. In personal character he resembled Lobachévsky, whom he intensely admired." In an 1896 Biographical Memoir of Oliver for the National Academy of Sciences, G. W. Hill lamented: "All the efforts of his friends to tie him down to a sustained labor in one direction until rounded completion was reached were unavailing."

The following rare specimen of his thought can be read in at least three ways: as an artifact of its age, as an early prefiguring of the debates at the Solvay Conferences forty years after his death, and as a hint of possible future directions in the physics of complexity.

For Oliver, mental events take place in a completely physical world that happens to be "quasi-perpendicular" (i.e. at least formally, and perhaps "actually", orthogonal to the three dimensions of ordinary space). He implies that, rather than a single "fourth dimension", there might be several higher dimensions, and (more importantly) that they are imaginary rather than real, so that the ordinary and quasi-perpendicular worlds together make up what would today be called a Minkowski spacetime manifold (but without time as such).

This idea was not so far from the Nineteenth Century mainstream as one might think: the 1890s were the extended pre-dawn hour of special relativity, when the speculations of Clifford, Zöllner, and others about the fourth dimension were widely discussed but thought to have no contact with experiment. Spiritualism, Theosophy, and Hermeticism were also very much in the air, and attracted several leading figures in mathematical physics. Oliver does not seem to have been an occultist, and his proposal was not necessarily an endorsement of Victorian occultism's vision of unseen worlds, but it is certainly compatible with it.

The founders of Twentieth Century quantum mechanics and their heirs in the second generation despised the old-fashioned table-rapping mysticism of the Kelvin-Tait era æther-theorists, preferring their own austerely modern brand of weirdness. However they, more than their predecessors, had to face the issue of consciousness as they grappled with the paradox of Wigner's Friend and the collapse of the wave-function in the moment of observation. For some (notably the Kaluza-Flint-Fisher school of the 1920s and '30s) the answer to this problem lay in a "hyperdimensional" interpretation of the Dirac equation, an approach quite similar to Oliver's. Some of the "hippies who saved physics" (or at least kept the pre-War philosophical tradition on life support) during the 1970s speculated along much the same lines.

Today, "multiverse" is a scientifically respectable word, neurology has more prestige than physics, and the "hard problem" of consciousness remains unsolved. The philosopher Galen Strawson recently commented [Times Literary Supplement, 2015 February 25] "At the root of the muddle lies an inability to overcome the Very Large Mistake so clearly identified by Eddington and others in the 1920s ... The mistake is to think we know enough about the nature of physical reality to have any good reason to think that consciousness can't be physical. It seems to be stamped so deeply in us, by our everyday experience of matter as lumpen stuff, that not even appreciation of the extraordinary facts of current physics can weaken its hold. To see through it is a truly revolutionary experience." Strawson probably has in mind a certain ineffable, irreducible, apophatic quality of matter that cannot be described by mathematics -- certainly that is what Eddington meant in the 1928 passage to which he is referring. However, it is worth noting that in later life Eddington modified his views, even coming perilously close to identifying the Inner Light of his Quaker religion with photons; although the famously unreadable Fundamental Theory does not (so far as I can tell) locate consciousness in the quasi-perpendicular, Eddington would probably not have dismissed the idea out of hand.

Oliver (also a Quaker, though a lapsed one) was as unembarrassed as Eddington about using the word "spiritual" in his scientific writings; the spiritual, for him, was as much a part of the world, and therefore of the proper subject-matter of science, as the material. Right or wrong, Oliver's proposal seems to me one of the few ways classical mind-body dualism could be reintroduced into biology.

A MATHEMATICAL VIEW OF THE FREE WILL QUESTION
by J. E. Oliver
[ The Philosophical Review 1, 292 (1892)].
Oliver's words are in bold

BEFORE discussing objections to the doctrine of Freedom, let me try to state what, in my own thought, that doctrine really is.

My notion of the matter is more or less analogous to the ordinary theory of Attention. As the current of suggestions, whether determined mechanically or otherwise, passes before the ego, the ego puts forth a free selective power whose result is physically directive. Here [Dugald] Stewart's account of attention may or may not come in : either the ego may merely arrest the current long enough for the mechanism to do the rest, like the "type-wheel" in the telegraph, which prints whenever it stops, or else, possibly, the ego may deflect the current without arresting it.

Does the ego here act apart from motives, feeling no pressure from them, or as one among motives, and often against some of the others? i.e. if by "Volition" we understand, not the observable resultant of the ego's free and original act as combined with other causes, but only that act, pure and simple, then are all volitions alike as to strength, intensity, degree of effort, -- or do they differ in this respect according to the strength of the opposing motives ? Either alternative appears to be tenable ; but, in so far as they really differ, I incline to the second one.

This raises the question of relative strength of will-power and determined motives, and I incline to believe that the will-power, in its possibilities, and perhaps even as commonly exerted, is by no means infinitesimal in comparison.

Thus motives and volitions would belong to a system of what we may call "spiritual forces," because like physical forces they may oppose one another, conspire and perhaps combine into intermediately directed resultants ; but these spiritual forces cannot so conspire with or oppose the actual physical changes or motions of the moment as to "do work," but can only produce deflection or transference of an energy to whose potential they do not contribute.

I regard them as quasi-perpendicular to all physical forces : i.e. their action upon any physical atom is, in the above sense, always as if at right angles to its path, so that of course they cannot increase or diminish its kinetic or potential energy, except indirectly by deflecting it into new positions and conditions, where, however, all transfers of energy will still be made under purely physical laws.

(Here, and in what follows, we use the term "physical force" only in the sense of a push or pull exerted at a material point by attraction, repulsion, or vis inertiæ; and analogously, the term "spiritual force" : so that a group of agencies and phenomena, like electricity or intelligence, would not be called "a force.")

Even if the concept of quasi-perpendicularity, which perhaps has been already suggested by Maxwell and others, should rest upon nothing deeper than a metaphor or convention, yet it is the more presumably natural and helpful here because of its known value in Pure Algebra. Thus, the perfect symmetry between the relations of i (= $\sqrt{\mathrm{- 1}}$) to + 1 and to − 1 ; or of 1 to + i and − i, (i.e. that mutual independence of the two abstract units 1 and i which would seem so analogous to the independence of physical and spiritual forces), has led to the use of the "complex plane," with resulting methods of great power and beauty. On this plane every value of A + Bi is located at a point having longitude A and latitude B, as in the annexed scheme ; so that i is treated as quasi-perpendicular to 1.

In mathematics, this notion of quasi-perpendicularity, though as I think often implied, goes unnamed. It is very flexible : a translation and a rotation would be quasi-perpendicular, however their respective directions were related, and so would be the several parameters of a curve when they were regarded as a new system of variables.

Let us hold the concept in a correspondingly free way. Let it simply accentuate for us the fact that the ego does no physical work, but can only by hypothesis decree something as to the direction of work, while it may or may not connote for us anything that is geometrically more definite.

This quasi-perpendicularity appears to me the natural, and perhaps almost necessary, solution of other difficulties that have nothing to do with freedom, but only with the apparent dualism of mind and matter, and with their apparent interaction ; while it may also sufficiently suggest the deeper unity in which that dualism doubtless rests.

We do somehow receive impressions from the outer material world, and do, whether freely or as mere automata or channels of influence, produce impressions upon it ; while, on the other hand, we are aware of phenomena in ourselves, like intelligence, joy, gratitude, obligation, remorse, which seem to us essentially independent of space and matter, though they often have reference to the external world. Now I know of no other figure by which, so well as by that of quasi-perpendicularity, we can represent to ourselves this apparent independence and this apparent interdependence of the physical and the psychical. Our hypothesis may be held as hardly more than such a figure, and it still suffices for the present purpose : though I incline to think it is not merely a figure of speech.

Admitting in a general way such a solution, many curious questions of detail would remain which we cannot as yet adequately discuss ; nor need we, for the purpose of the present argument.

For instance, shall we think of the quasi-perpendicularity as mutual ; the physical forces exerting a merely directive effect upon the spiritual phenomena, as well as the spiritual forces upon the physical phenomena, and the law of conservation presumably holding in the spiritual as well as in the material system ?

Is there any spiritual force that involves the element of Time, or Inertia, and in virtue of which every system of spiritual forces must be in equilibrio just as a falling stone hangs balanced between the pull of gravity and its own resistance to further acceleration?

Again, must this quasi-perpendicularity be regarded as a mere convenient metaphor, or may it be an actual perpendicularity, the spiritual forces pulling in a space of their own, more or less like our known space ? In the latter case, are the two spaces coincident ? (Coincident perhaps, yet not necessarily coextensive. Our familiar space, if "positively curved," may be said to coincide absolutely with another having just half or double its extent, and yet each space to be complete and perfectly symmetric, two points of one space being at every one point of the other.) Or does the spiritual include the other as that includes a given plane or line ? Or is every line and every plane of one space perpendicular to every line and plane of the other ?

The third alternative seems untenable ; for, admitting it, how could a spiritual force produce even a deflective effect in physical space ? But the other two remain, and the second may fall in with such purely physical speculations as W. W. Rouse Ball's, who seeks to obtain both Newton's law and certain results of spectrum analysis from the virtual hypothesis that our known space, lying like a plane in a larger space, can vibrate in the fourth direction. [Messenger of Mathematics 21, p. 20, June, 1891.(Although this volume of the Messenger has evidently not yet been digitised, a similar, more popular essay by Ball on the subject may be found in his Mathematical Recreations and Essays.)]

It remains now to notice briefly, from the standpoint of mathematics, certain supposed difficulties of the Free Will hypothesis.

We are told that the Determinist theory is forced upon us : ---

1. By the principle of conservation of energy ;
2. By the law of causation ;
3. By statistics and the known effect of environment upon conduct and character ; and even
4. By consciousness.

1. The first difficulty disappears with the admission of quasi-perpendicularity ; and, as I have tried to show, some such admission is almost necessary in order to explain other phenomena than those of our supposed freedom.
2. If this means that all causation is physical, in the sense that excludes independent contributions to the stream, then it simply begs the question. If it means that an infinite mind, knowing perfectly well the present, could with certainty trace thence the stream of events both upwards and downwards, this again is an assumption, made plausible by hereditary ideas as to Fate, and the Divine Decrees and Foreknowledge, and perhaps also by two fallacies as to the fixity of the Past. Doubtless the Past is fixed, if only because it is past ; but since the same fixity would result if the Past were wholly inferrible from the Present, one may illogically deduce the inferribleness from the fixity and assign it as a cause of the fixity.

And again, it is natural to assume a symmetry and reversibleness of relation between Past and Future which does not really exist, and thence to conclude that if one be fixed, the other must be so too. This argument is not that "from Causation," yet one seems to suggest the other. Now, however it be to the philosopher, Past and Future are not alike to the scientist. Even if physical causation be considered as reversible, many of the cycles of actual change are not. These cycles move in fixed directions, and not backward, even when they bring events and successive individuals or systems around to nearly the old conditions. And besides, there are changes going on which, as far as science can yet see, never will be unmade or offset by any cycle : for instance, the progressive residual concentration of mass and diffusion of heat, and the unceasing loss of "motivity." Thus neither from moment to moment nor yet in the long run is the relation of Past to Future a reversible one.

But if the objection merely means this, "Since every effect or phenomenon has a cause, and this cause has a cause, etc., thus giving an infinite number of steps of causation between the effect and its first cause, therefore free volition cannot be a cause, for this infinite number of steps between it and its observed effect must take an infinite time," --- if this be the supposed difficulty, then I think it turns upon the same fallacy as does the story of Achilles and the tortoise. We are told that he could not catch the tortoise, because while he ran 100 rods it crawled one ; and while he ran that one it crawled 1/100 ; and then 1/10,000 and so on -- Thus analyzed, the process did indeed require an infinite number of steps, but then these steps grew so rapidly shorter and shorter that the sum of all their lengths was finite, and might have been microscopic. Very likely it is as true in our problem as in that of Achilles, that one may rightly either introduce the infinite series or not : i.e. we may say "Everything is caused," and so bring in the infinite series, or say "Volition is its own cause " and stop there, replacing the series of steps by its resultant or quasi-sum.

3. Statistics do indeed show that, of a given people under given circumstances, one in so many misdirects a letter, or commits a murder, or can read three languages ; and that the ratios remain very nearly constant from decade to decade unless the environment changes. But this is precisely what would happen if human actions were the resultants of environment and of a factor of spontaneity which, like a throw of honest dice, was absolutely unpredictable. We should get, just as we now do get, agreements where the basis of estimation is large, and discrepancies where it is small ; indeed, if the argument from statistics were worth anything, then we could prove in like manner that the throws of dice are governed by discoverable laws and not by chance, --- which is practically untrue.

So with character: if it be the joint product of environment, heredity, and volition, then in the long run it may seem to result mainly or wholly from environment and heredity, for the reason that volition (unless guided by principles whose development is, itself, largely due to environment, etc.) is like dice-throwing, now this way and now that, so that its net result varies nearly as the square root of the number of cases that have come up, while the result of a steady pull from heredity or environment varies as the number of cases.

4. To examine the facts of consciousness belongs more to the philosopher than to the mathematician. My own consciousness, however, appears on the whole to testify neither to complete determinism nor to complete freedom ; and I suspect that its evidence is really in close accord with the theory sketched out above.

It would seem then that the analogies of physical and mathematical science are not unfriendly to that old faith in freedom to which the conscience and common sense of the race have substantially held ; and if we cannot now hold to it in the old simple way, this is mainly because the world as known to us is so much larger and more complex than as known to our fathers.

We cannot do or become at once just what we would : the ego can act effectively only through, or as part of, an intricate mechanism, and perhaps much that it seems to initiate is mere ideo-motor reaction. Yet I think its directive agency does tell in making results to be somewhat other than they would have been, and thus gradually moulding character and environment, and the heredity of those who shall come after. Moreover, the effect upon the inner character may be more rapid than upon the apparent character and the environment : for if there be a certain element of earnestness, making the inner results fairly constant as to direction, these results should tend to accumulate proportionally to the number of occasions, while the effects of environment upon the inner character would be more conflicting and so would follow more nearly the law of the square root.

Thus, though development has taken the place of sudden creation in our new world, it may still be that we do, little by little, create our own characters, and not merely the character of the race.

J. E. Oliver.
Cornell University.

Cornell University in the early 1900s.

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Encyclopedia > C-News

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C-News

C-News is a newsserver, written by Henry Spencer and Geoff Collyer[?] as a replacement for B-News. C News handles the transport and storage subsystems of the news software. The latest official relase is from 1993, and C News is now mostly obsoleted by INN. It is presumably still used for some small news servers.

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text:
# Different volatility surface ( Local vol, Stochastic vol etc.)

Despite many questions about local and stochastic volatility available on this forum, i still have a few doubts left. Essentially I am seeking validation whether I am interpreting things correctly.

Q1. A very basic question: Why are different volatility surfaces used?

My take (please correct if i am wrong):

Implied volatility surface generated using Black-Scholes model is not able to price exotic options (with barriers) correctly. So we need local volatility and stochastic volatility surfaces. The parameters are calibrated so that these volatility surfaces match the implied volatility surface generated from the Black-Scholes model. For calibration we use vanilla options. Once these local volatility and stochastic volatility surfaces match the implied volatility surface generated using the Black-Scholes model and are able to price vanilla options correctly, it is used for pricing exotic products.

Few follow-up questions:

Q2. Why local volatility and stochastic volatility models try to match the volatility surface generated by the Black-Scholes model? Once these local and stochastic volatility surfaces have been generated, we don't need the volatility surface generated by the Black-Scholes model, the new volatility surface can be used for pricing both vanilla as well as exotic options. Is it the correct inference?

However, we do not calibrate stochastic and local volatily models to the Black-Scholes model but to Black-Scholes implied volatilities. If the set of option prices you observe is nice (say arbitrage free), you can translate the observed option price into implied Black-Scholes volatilities (just numerically solve the equation $$\mathrm{MarketPrice}-\mathrm{BSPrice}(\sigma)=0$$).
Implied volatilities behave $$$$better'' than prices, so people prefer to calibrate models to implied volatilities rather than naked prices. But you still calibrate your stochastic and local volatily models to data observed in the market and you do not assume that the Black-Scholes model is correct in any way. You simply use it to translate market prices into market volatilities. These volatilities are then used to find the parameter of your more advanced models and thus guaranteeing that it matches today's market prices.


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# Infinite Sequences and Series: Alternating Tests

in mathematics •  3 months ago  (edited)

In this video I go over further into the wonderful world of Infinite Sequences and Series and this time look at Alternating Series; which are series that alternate from positive and negative terms. When dealing with alternating positive and negative terms it is important to develop specific theorems and methodologies in order to better analyze whether they converge or diverge. Often times it is best to view the odd and even terms of the series separately since they would have the same signs when grouped together. I go over this and much more throughout the video so make sure to follow along!

The topics and their timestamps in the video are listed below:

1. @ 1:28 - Introduction to Alternating Series
2. @ 6:06 - The Alternating Series Test
• @ 14:00 - Proof of the Alternating Series Test
• @ 27:50 - Example 1
• @ 32:03 - Example 2
• @ 36:28 - Example 3
3. @ 53:43 - Estimating Sums
• @ 55:05 - Alternating Series Estimation Theorem
• @ 59:27 - Example 4
• @ 1:11:26 - Important Note
4. Exercises
• @ 1:11:52 - Exercise 1
• @ 1:48:22 - Exercise 2
• @ 2:08:50 - Proof of the Multiplication Rule for Logarithms
• @2:14:05 - Exercise 3

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# Infinite Series and Sequences: Alternating Tests

## Calculus Book Reference

Note that I mainly follow along the following calculus book:

• Calculus: Early Transcendentals Sixth Edition by James Stewart

## Topics to Cover

1. Introduction to Alternating Series
2. The Alternating Series Test
• Proof of the Alternating Series Test
• Example 1
• Example 2
• Example 3
3. Estimating Sums
• Alternating Series Estimation Theorem
• Example 4
• Important Note
4. Exercises
• Exercise 1
• Exercise 2
• Proof of the Multiplication Rule for Logarithms
• Exercise 3

# Introduction to Alternating Series

The convergence tests that we have looked at so far apply only to series with positive terms.

In this section and the next we learn how to deal with series whose terms are not necessarily positive.

Of particular importance are alternating series, whose terms alternate in sign.

An alternating series is a series whose terms are alternately positive and negative.

Here are two examples:

We see from these examples that that n-th term of an alternating series is of the form:

where bn is a positive number. (In fact, bn = |an|).

The following test says that if the terms of an alternating series decrease toward 0 in absolute value, then the series converges.

# The Alternating Series Test

If the alternating series:

satisfies:

then the series is convergent.

Before giving the proof let's look at the following figure, which gives a picture of the idea behind the proof.

We first plot the partial sum s1 = b1 on a number line.

To find s2 we subtract b2, so s2 is to the left of s1.

Then to find s3 we add b3, so s3 is to the right of s2.

But, since b3 < b2, s3 is to the left of s1.

Continuing in this manner, we see that the partial sums oscillate back and for.

Since bn> → 0, the successive steps are becoming smaller and smaller.

The even partial sums s2, s4, s6, … are increasing and the odd partial sums s1, s3, s5, … are decreasing.

Thus it seems plausible that both are converging to some number s, which is the sum of the series.

Therefore we consider the even and odd partial sums separately in the following proof.

## Proof of the Alternating Series Test

We first consider the even partial sums:

But we can also write:

Every term in brackets is positive, so s2n ≤ b1 for all n.

Therefore the sequence {s2n} of even partial sums is increasing and bounded above.

It is therefore convergent by the Monotonic Sequence Theorem.

Recall the Monotonic Sequence Theorem from my earlier video on Infinite Sequences.

https://steemit.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more

Retrieved: 1 August 2019
Archive: http://archive.fo/i5ZhH

Essentially, if a sequence is always increasing (or decreasing), hence monotonic, and is bounded above (or below) then it will approach a number or limit L ≤ M (or L ≥ m if the sequence is decreasing).

Let's call its limit s, that is:

Now we compute the limit of the odd partial sums:

Since both the even and odd partial sums converge to s, we have limn→∞ sn = s (see Exercise 1a at the end of this video) and so the series is convergent.

## Example 1

The alternating harmonic series:

satisfies:

so the series is convergent by the Alternating Series Test.

The following figure illustrates Example 1 by showing the graphs of the terms an= (-1)n-1/n and the partial sums sn.

Notice how the values of sn zigzag across the limiting value, which appears to be about 0.7.

In fact, it can be proved that the exact sum of the series is ln 2 ≈ 0.693 (See Exercise 2 at the end of this video).

## Example 2

The following series is alternating:

but:

so condition (ii) is not satisfied.

Instead, we look at the limit of the n-th term of the series:

This limit does not exist, so the series diverges by the Test for Divergence from my earlier video on Infinite Series:

https://steemit.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more

Retrieved: 27 July 2019
Retrieved: http://archive.fo/FGAyJ

## Example 3

Test the following series for convergence or divergence:

Solution:

The given series is alternating so we try to verify conditions (i) and (ii) of the Alternating Series Test.

Unlike the situation in Example 1, it is not obvious that the sequence given by bn = n2/(n3 + 1) is decreasing.

However, if we consider the related function f(x) = x2/(x3 + 1), we find that:

Since we are considering only positive x, we see that:

Thus f is decreasing on the interval (21/3, ∞).

This means that:

Calculation Check: 2^(1/3) = 1.2599.

And therefore:

The inequality b2 < b1 can be verified directly but all that really matters is that the sequence {bn} is eventually decreasing.

Alternative Derivation:

Instead of verifying condition (i) of the Alternating Series Test by computing a derivative, we could verify that bn+1 < bn directly by using the technique of cross-multiplying as in Example 12 of my earlier video on Infinite Sequences, and shown below for reference:

Thus the given series is convergent by the Alternating Series Test.

# Estimating Sums

A partial sum sn of any convergent series can be used as an approximation to the total sum s, but this is not of much use unless we can estimate the accuracy of the approximation.

The error involved in using s ≈ sn is the remainder Rn = s - sn.

The next theorem says that for series that satisfy the conditions of the Alternating Series Test, the size of the error is smaller than bn+1, which is the absolute value of the first neglected term.

## Alternating Series Estimation Theorem

If s = ∑ (-1)n-1 bn is the sum of an alternating series that satisfies:

then:

Proof:

You can see geometrically why the Alternating Series Estimation Theorem is true by looking at the earlier figure I covered, and which I have shown below but from a screenshot of the exact figure from my Calculus book.

Notice that s - s4 < b5, |s - s5| < b6, and so on.

Notice also that s lies between any two consecutive partial sums, sn and sn+1.

It follows that:

## Example 4

Find the sum of the following series correct to three decimal places.

Note that by definition: 0! = 1.

Solution:

We first observe that the series is convergent by the Alternating Series Test because:

To get a feel for how many terms we need to use in our approximation, let's write out the first few terms of the series:

Notice that:

Calculation Check: 1 - 1 + 1/2 - 1/6 + 1/24 - 1/120 + 1/720 = 0.368056.

By the Alternating Series Estimation Theorem we know that:

This error of less than 0.0002 does not affect the third decimal place, so we have s ≈ 0.368 correct to three decimal places.

In later videos, I will show that:

So what we have obtained in Example 4 is actually an approximation to the number e-1. #Amazing

## IMPORTANT NOTE

The rule that the error (in using sn to approximate s) is smaller than the first neglected term is, in general, valid only for alternating series that satisfy the conditions of the Alternating Series Estimation Theorem.

The rule does not apply to other types of series.

# Exercises

Here are 3 very in-depth exercises that explore further alternating series as well as sequences and series in general.

## Exercise 1

(a) Show that if limn→∞ a2n = L and limn→∞ a2n+1 = L, then {an} is convergent and limn→∞ an= L.

(b) If a1 = 1 and:

find the first eight terms of the sequence {an}.

Then use part (a) to show that limn→∞ an= 21/2.

This gives the continued fraction expansion:

Solution:

Note that this exercise was originally part of an early section but it is useful to include it in this video for the derivation of the Alternating Series Test.

(a)

Let ε > 0.

Since limn→∞ a2n = L, there exists N1 such that:

Similarly, since limn→∞ a2n+1 = L, there exists N2 such that:

If n is even, then n = 2m where m > N1, so:

If n is odd, then n = 2m + 1, where m > N2, so:

Thus, effectively all we are doing is varying slightly the terms of the definition of a limit which depends on being arbitrarily close to the limit L as the number of terms n is made arbitrarily large.

Calculation Check:

• a1 = 1
• a2 = 1 + 1/(1+1) = 1.5
• 3/2 = 1.5
• a3 = 1 + 1/(1+3/2) = 1.4
• 7/5 = 1.4
• a4 = 1 + 1/(1+7/5) = 1.4167
• 17/12 = 1.4167
• a5 = 1 + 1/(1+17/12) = 1.4138
• 41/29 = 1.4138
• a6 = 1 + 1/(1+41/29) = 1.4143
• 99/70 = 1.4143
• a7 = 1 + 1/(1+99/70) = 1.4142
• 239/169 = 1.414201
• a 8 = 1 + 1/(239/169) = 1.7071
• 577/408 = 1.414216

Notice that:

It appears that the odd terms are increasing and the even terms are decreasing.

Let's use Mathematical Induction to show that:

Recall from my earlier video on Mathematical Induction:

https://youtu.be/WdIr_onvUtE

Retrieved: 2 July 2019
Retrieved: http://archive.fo/4lslc

Suppose that:

We have thus shown, by induction, that the odd terms are increasing and the even terms are decreasing.

Also all the terms lie between 1 and 2 (i.e. we are adding a positive fraction less than 1 to 1), so both {an} (odds) and {bn} (evens) are bounded monotonic sequences are therefore convergent by the Monotonic Sequence Theorem.

## Exercise 2

Use the following steps to show that:

Let hn and sn be partial sums of the harmonic and alternating harmonic series.

(a) Show that s2n = h2n - hn.
(b) From Exercise 3 later in this video, we have:

And therefore:

Use these facts together with part (a) to show that:

Solution:

We will prove this also by Mathematical Induction.

Let P(n) be the proposition that s2n = h2n - hn.

P(1) is the statement s2 = h2 - h1, which is true since:

So suppose that P(n) is true.

We will show that P(n+1) must be true as a consequence:

## Exercise 3

Use the following steps to show that the sequence has a limit:

Note that the value of the limit is denoted by γ and is called Euler's constant.

(a) Draw a picture like the figure below from my earlier video on the Integral Test with f(x) = 1/x and interpret tn as an area (or use the inequality also shown in my earlier video and shown below) to show that tn > 0 for all n.

https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums

Retrieved: 28 July 2019
Retrieved: http://archive.fo/3YMxr

(b) Interpret:

as a difference of areas to show that tn - tn+1 > 0.

Therefore, {tn} is a decreasing sequence.

(c) Use the Monotonic Sequence Theorem to show that {tn} is convergent.

Solution:

Note that this is also an exercise from an earlier section in my calculus book but it is fitting to include it in this video.

The sum of the areas of the n rectangles in the graph below is:

And the integral or area below the curve is less than the area of rectangles because the rectangles extend above the curve y = 1/x, so:

(b) Note that:

The area under f(x) = 1/x between x = n and x = n + 1 is:

Thus clearly the area below the curve is greater than the inscribed rectangle in the figure above (which is 1/(n+1)), so:

Thus tn > tn+1 and so {tn} is a decreasing sequence.

(c) We have shown that {tn} is decreasing and tn > 0 for all n.

Thus, 0 < tn ≤ t1 = 1, so {tn} is a bounded monotonic sequence, and hence converges by the Monotonic Sequence Theorem.


text:
# Re: Is the universe computable?

At 16:37 12/01/04 +0100, Eugen Leitl wrote:

If I'd kill you, you'd have no chance of thinking that thought.
Actually this is pure wishful thinking, unless you mean succeeding
to kill me and my counterparts in some absolute way, but how would
you be able to do a thing like that. I will not insist on this
startling consequence of COMP or QM, giving that you
postulate physicalism at the start. See my thesis for a proof that
physicalism is incompatible with comp. We have discuss the
immortality question a lot in this list.

If I killed
all animals capable of counting, "abstract immaterial numbers" would become
exactly that: immaterial.

OK. But "immaterial" does not mean "not existing". Even a physicalist can
accept that. Only very reductionist forms of physicalism reject that.

> Pebbles can't count themselves, obviously. But it is not because
> pebbles can't count that two pebbles give an even number of pebbles.
> Electron cannot solve schroedinger equation (only a physicist can do that),
> nevertheless electron cannot not follow it (supposing QM).

The universe does what it does, it certainly doesn't solve equations.

So we agree. (but note that anything does what it does, so what is your point).

People
solve equations, when approximating what universe does. As such, QM is a fair
approximation; it has no further reality beyond that.

That is your opinion, which is not really relevant for the question
we are talking about.

H\psi=E\psi in absence of context is just as meaningless as 2+2=4.

I can understand that point and respect that opinion, but
what makes you so sure. Could you give me a context in
which H\psiis not equal to E\psi ? Could you give me a context in
which 2+2 is not equal to 4, and where 2, +, 4, = have their
usual standard meaning?

Perhaps we should put our hypothesis on the table. Mine is
comp by which I mean arithmetical realism, Church thesis, and
the "yes doctor" hypothesis, that is the hypothesis that there is
a level of description of myself such that I don't detect any differences
in case my parts are functionaly substituted by digitalizable  device.
Do you think those postulates are inconsistent?

Bruno


text:
# magni.utils.validation.types module¶

Module providing abstract superclasses for validation.

## Routine listings¶

MatrixBase(object)
Abstract base class of custom matrix classes.
MMSEInputChannel(object)
Abstract base class of a Minimum Mean Squaread Error (MMSE) input channel.
StopCriterion(object)
Abstract base class of a stop criterion.
ThresholdOperator(object)
Abstract base class of a threshold operator.
class magni.utils.validation.types.MatrixBase(dtype, bounds, shape)[source]

Bases: object

Abstract base class of custom matrix classes.

The magni.utils.validation.validate_numeric function accepts built-in numeric types, numpy built-in numeric types, and subclasses of the present class. In order to perform validation checks, the validation function needs to know the data type, the bounds, and the shape of the variable. Thus, subclasses must call the init function of the present class with these arguments.

Parameters: dtype (type) – The data type of the values of the instance. bounds (list or tuple) – The bounds of the values of the instance. shape (list or tuple) – The shape of the instance.
bounds

list or tuple – The bounds of the values of the instance.

dtype

type – The data type of the values of the instance.

shape

list or tuple – The shape of the instance.

Notes

dtype is either a built-in numeric type or a numpy built-in numeric type.

If the matrix has complex values, bounds is a list with two values; The bounds on the real values and the bounds on the imaginary values. If, on the other hand, the matrix has real values, bounds has one value; The bounds on the real values. Each such bounds value is a list with two real, numeric values; The lower bound (that is, the minimum value) and the upper bound (that is, the maximum value).

bounds
dtype
shape
class magni.utils.validation.types.MMSEInputChannel(var)[source]

Bases: object

Abstract base class of a Minimum Mean Squaread Error (MMSE) input channel.

The magni.cs.reconstruction algorithms may make use of input channels to define the prior knowledge on the sought solution. In order for the validation of such an input channel to work, it must be based on this class.

Parameters: var (dict) – The input channel state variables.
compute(var)[source]

Compute the input channel value.

Parameters: var (dict) – The variables used in computing of the input channel value. mean (ndarray) – The computed input channel mean. variance (ndarray) – The computed input channel variance.
class magni.utils.validation.types.MMSEOutputChannel(var)[source]

Bases: object

Abstract base class of a Minimum Mean Squaread Error (MMSE) output channel.

The magni.cs.reconstruction algorithms may make use of output channels to define the observation model. In order for the validation of such an output channel to work, it must be based on this class.

Parameters: var (dict) – The output channel state variables.
compute(var)[source]

Compute the output channel value.

Parameters: var (dict) – The variables used in computing of the output channel value. mean (ndarray) – The computed output channel mean. variance (ndarray) – The computed output channel variance.
class magni.utils.validation.types.StopCriterion(var)[source]

Bases: object

Abstract base class of a stop criterion.

The magni.cs.reconstruction algorithms are typically iterative algorithms that make use of some stop criterion to determine if it has converged. In order for the validation of such a stop criterion to work, it must be based on this class.

Parameters: var (dict) – The stop criterion state variables.
compute(var)[source]

Compute the stop criterion value.

Parameters: var (dict) – The variables used in computing of the stop criterion value. stop (bool) – The indicator of whether or not the stop criterion is satisfied. value (float) – The stop criterion value.
class magni.utils.validation.types.ThresholdOperator(var)[source]

Bases: object

Abstract base class of a threshold operator.

The magni.cs.reconstruction algorithms may make use of threshold operators for “de-noising”. In order for the validation of such a threshold operator to work, it must be based on this class.

Parameters: var (dict) – The threshold operator state variables.
compute_deriv_threshold(var)[source]

Compute the entrywise derivative threshold.

Parameters: var (dict) – The variables used in computing the derivative threshold. eta_deriv (ndarray) – The computed entrywise derivative threshold.
compute_threshold(var)[source]

Compute the entrywise threshold.

Parameters: var (dict) – The variables used in computing the threshold. eta (ndarray) – The computed entrywise threshold.
update_threshold_level(var)[source]

Update the threshold level state.

Parameters: var (dict) – The variables used in computing the threshold level update.


text:
# Dichloromethane solubility in water

I heard dichloromethane is soluble in water 1:50, but I also heard it is not because it can't form hydrogen bonds. I'm confused. Is there a way to extract it? Would putting the water to $40~\mathrm{^\circ C}$ make it evaporate trough water?

• 1:50 is misleading. Wikipedia gives dichloromethane’s solubility in water as $20\,\mathrm{\frac{g}{l}}$ which is approximately 1:50 w/w, but ratios are usually understood v/v. (weight/weight versus volume/volume) – Jan Sep 30 '15 at 9:57
• @Jan That's a good start for an answer... – Martin - マーチン Sep 30 '15 at 11:11
• "it is not because it can't form hydrogen bonds" - Just bear in mind that solubility isn't a black and white thing. Dichloromethane is pretty insoluble in water (as opposed to something like ethanol) because it doesn't H-bond with water, but that doesn't mean it is entirely insoluble. – orthocresol Sep 30 '15 at 12:39

The solubility of dichloromethane in water (as well as the solubility of water in dichloromethane) depends on temperature. At a temperature of $T=20\ \mathrm{^\circ C}$, the solubility amounts to $17.6\ \mathrm g$ dichloromethane per $1\ \mathrm{kg}$ water; i.e. the mass fraction of dichloromethane is $w_{\ce{C2H2Cl2}}=0.0173$ or $w_{\ce{C2H2Cl2}}=1.73\,\%$. [source: “Aqueous Solubility and Henry’s Law Constants of Organic Compounds”, in CRC Handbook of Chemistry and Physics, 90th Edition (CD-ROM Version 2010), David R. Lide, ed., CRC Press/Taylor and Francis, Boca Raton, FL.]
You cannot simply evaporate pure dichloromethane (boiling point: $40\ \mathrm{^\circ C}$) from such mixtures since dichloromethane and water form an azeotrope, which boils at $38.8\ \mathrm{^\circ C}$. The mass fraction of water in this azeotrope is about $w_{\ce{H2O}}=1\,\%$.


text:
# [PATCH v18 00/18] KVM RISC-V Support

Greg KH gregkh at linuxfoundation.org
Fri May 21 11:25:23 PDT 2021

On Fri, May 21, 2021 at 11:08:15AM -0700, Palmer Dabbelt wrote:
> On Fri, 21 May 2021 10:47:51 PDT (-0700), Greg KH wrote:
> > On Fri, May 21, 2021 at 07:21:12PM +0200, Paolo Bonzini wrote:
> > > On 21/05/21 19:13, Palmer Dabbelt wrote:
> > > > >
> > > >
> > > > I don't view this code as being in a state where it can be
> > > > maintained, at least to the standards we generally set within the
> > > > kernel.  The ISA extension in question is still subject to change, it
> > > > says so right at the top of the H extension <https://github.com/riscv/riscv-isa-manual/blob/master/src/hypervisor.tex#L4>
> > > >
> > > >   {\bf Warning! This draft specification may change before being
> > > > accepted as standard by the RISC-V Foundation.}
> > >
> > > To give a complete picture, the last three relevant changes have been in
> > > August 2019, November 2019 and May 2020.  It seems pretty frozen to me.
> > >
> > > In any case, I think it's clear from the experience with Android that
> > > the acceptance policy cannot succeed.  The only thing that such a policy
> > > guarantees, is that vendors will use more out-of-tree code.  Keeping a
> > > fully-developed feature out-of-tree for years is not how Linux is run.
> > >
> > > > I'm not sure where exactly the line for real hardware is, but for
> > > > something like this it would at least involve some chip that is
> > > > widely availiable and needs the H extension to be useful
> > >
> > > Anup said that "quite a few people have already implemented RISC-V
> > > H-extension in hardware as well and KVM RISC-V works on real HW as well".
> > > Those people would benefit from having KVM in the Linus tree.
> >
> > Great, but is this really true?  If so, what hardware has this?  I have
> > a new RISC-V device right here next to me, what would I need to do to
> > see if this is supported in it or not?
>
> You can probe the misa register, it should have the H bit set if it supports
> the H extension.

To let everyone know, based on our private chat we had off-list, no, the
device I have does not support this extension, so unless someone can
point me at real hardware, I don't think this code needs to be
considered for merging anywhere just yet.

thanks,

greg k-h


text:
# Question #d40e3

Apr 17, 2015

Start with the balanced chemical equation for the equilibrium reaction that exists when hydroxylamine reacts with water

$N {H}_{2} O {H}_{\left(a q\right)} + {H}_{2} {O}_{\left(l\right)} r i g h t \le f t h a r p \infty n s N {H}_{3} O {H}_{\left(a q\right)}^{+} + O {H}_{\left(a q\right)}^{-}$

Use the solution's pH to determine the pOHm which will allow you to calculate the concentration of the hydroxide ions

$p {H}_{\text{sol" = 14 - pOH => pOH = 14 - pH_"sol}}$

$p O H = 14 - 10.11 = 3.89$

The concentration of hydroxide ions will be

$\left[O {H}^{-}\right] = {10}^{- p O H} = {10}^{3.89} = 1.29 \cdot {10}^{- 4} \text{M}$

Since you have $1 : 1$ mole ratios between all the species of interest, the initial concentration of hydroxylamine will decrease by the same amount the concentration of hydroxide increased.

At equilibrium, you'll have

$\left[N {H}_{3} O {H}^{+}\right] = \left[O {H}^{-}\right] = 1.29 \cdot {10}^{- 4} \text{M}$

$\left[N {H}_{2} O H\right] = {\left[N {H}_{2} O H\right]}_{0} - \left[O {H}^{-}\right] = 0.15 - 1.29 \cdot {10}^{- 4} = \text{0.14987 M}$

By definition, the base dissociation constant, ${K}_{b}$, will be equal to

${K}_{b} = \frac{\left[O {H}^{-}\right] \cdot \left[N {H}_{3} O {H}^{+}\right]}{\left[N {H}_{2} O H\right]}$

${K}_{b} = \frac{1.29 \cdot {10}^{- 4} \cdot 1.29 \cdot {10}^{- 4}}{0.14987} = \textcolor{g r e e n}{1.1 \cdot {10}^{- 9}}$


text:
# Global maxima of a function subject to a constraint.

I am trying to prove that the global maxima of $f(x_1,x_2,...,x_n)=(x_1x_2···x_n)^2$, subject to $\lVert(x_1,x_2,...,x_n)\rVert_2=r$ is $(r^2/n)^n$

I know I have to find the critical points of the Lagrange function, that is $$F(x_1,x_2,...,x_n,\lambda)=(x_1x_2···x_n)^2+\lambda(\sqrt{\sum_{i=1}^nx_i^2}-r)$$ In order to do that, I've built the system in which all partial derivatives are null, so I have:

$$D_1F(x_1,x_2,...,x_n,\lambda)=2(x_1x_2···x_n)(x_2x_3···x_n)+\frac{\lambda x_1}{\sqrt{\sum_{i=1}^nx_i^2}}=0$$ $$...$$ $$D_jF(x_1,x_2,...,x_n,\lambda)=2(x_1x_2···x_n)(x_1x_2···x_{j-1}x_{j+1}···x_n)+\frac{\lambda x_j}{\sqrt{\sum_{i=1}^nx_i^2}}=0$$ $$...$$ $$D_nF(x_1,x_2,...,x_n,\lambda)=2(x_1x_2···x_n)(x_1x_2···x_{n-1})+\frac{\lambda x_n}{\sqrt{\sum_{i=1}^nx_i^2}}=0$$ $$D_{n+1}F(x_1,x_2,...,x_n,\lambda)=\sqrt{\sum_{i=1}^n}x_i^2-r=0$$

I think, because I have tried to solve reduced forms of this system in a mathematical software, that the solutions are like the following: $$(x_1,x_2,...,x_n)=(\pm\frac{r}{\sqrt{n}},\pm\frac{r}{\sqrt{n}},...\pm\frac{r}{\sqrt{n}})$$ where all the $x_i$ take the positive and negative sign, so there are $2^n$ solutions.

However, I didn't manage to solve the system. I have tried to replace the summation with $r$ in all equations, to sum all the equations, to reduce dividing by $x_1···x_n$... but I didn't get to any interesting end.

Could you help me, please, giving me any hint about how can I solve this system?

Thank you very much.

You can argue that since the function you want to maximize is convex and since all the $x_i$ appear symmetrically in the objective function as well as in the ristriction(s), that there is a solution in which all of them can be equal. So by guessing that $x_1=x_2=\ldots=x_n := x$ you can solve the system with one unknown (the $x$) and then it is straightforward to show that the global maximum is equal to $(r^2/n)^n$. Guessing a property of the solution is common in nonlinear programming.


text:
## LaTeX forum ⇒ Community talk ⇒ Best LaTeX Resources on YouTube?

General discussion about this LaTeX community.
Please no LaTeX topics here! Just community topics!
LaTexLearner
Posts: 138
Joined: Tue Mar 10, 2015 11:06 am

### Best LaTeX Resources on YouTube?

I've really enjoyed Michelle Krummel's LaTeX introduction on YouTube.

https://www.youtube.com/playlist?list=P ... 31D3EBC449

Now that I know that beginner-level stuff, though, I'm ready to move on.

Can anybody suggest a YouTube playlist to follow up with? There are so many videos there I don't know where to go next.

Thanks!

Johannes_B
Site Moderator
Posts: 3582
Joined: Thu Nov 01, 2012 4:08 pm
I picked one of the videos (the first on text formatting) and it started. My first thoughts were, you should never use those commands within the document. I skipped a bit and at minute four was cursing the video because what was told there is simply, excuse me, crap. Stuff like that promotes wrong bits and pieces that never stop floating around.

Nicola Talbot published some nice books for novices and advanced users, all of them reviewed by a large community of experienced LaTeX users. Please have a look at them, they are free. Dickimaw LaTeX books

btw: so, far, i haven't seen a single video on Youtube, that was 100 percent correct.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

LaTexLearner
Posts: 138
Joined: Tue Mar 10, 2015 11:06 am
Johannes_B wrote:I picked one of the videos (the first on text formatting) and it started. My first thoughts were, you should never use those commands within the document. I skipped a bit and at minute four was cursing the video because what was told there is simply, excuse me, crap. Stuff like that promotes wrong bits and pieces that never stop floating around...

btw: so, far, i haven't seen a single video on Youtube, that was 100 percent correct.

What, exactly, was wrong with those videos?

Johannes_B
Site Moderator
Posts: 3582
Joined: Thu Nov 01, 2012 4:08 pm
Commands like textbf and friends shoudl never be used with in the body of a document. Better to use semantic markup and user-defined commands.

Font size switches are simple commands, using them as an environment is a wide spread misunderstanding, distributed over generations and more generations to come. The thing is, that the LaTeX ekrnel defines that stuff, that this still works. But i would never recommend that.

Unfortunately, Nicola introduces that as well. Gotta talk to her about that.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

LaTexLearner
Posts: 138
Joined: Tue Mar 10, 2015 11:06 am
Johannes_B wrote:Commands like textbf and friends shoudl never be used with in the body of a document. Better to use semantic markup and user-defined commands.

Font size switches are simple commands, using them as an environment is a wide spread misunderstanding, distributed over generations and more generations to come. The thing is, that the LaTeX ekrnel defines that stuff, that this still works. But i would never recommend that.

Unfortunately, Nicola introduces that as well. Gotta talk to her about that.

To be honest, I'm completely new to programming and LaTeX and didn't actually understand this message. What is "semantic markup"? I think (I know what user-defined commands are, though.)

Why is it so bad to use environments for font sizing?

Johannes_B
Site Moderator
Posts: 3582
Joined: Thu Nov 01, 2012 4:08 pm
textbf makes something bold, but there is no reason given. You are using semantic markup at different places, you issue the section command instead of changing to a bigger fontsize, stepping the section number by one, typesetting the text, make a note for the toc, switch back to normalfont and normalsize ...

All this is hidden, you can later decide that sections shall be bigger, or use another font and all instances of section will change.

Now, suppose you have latin names of plants species, you are setting them using textbf. Later you decide to change it to italics. You have to redo every instance, which might be ok for 5 occurences, but is highly impractical for 1000 occurences.

Following some source code, please cpy it into your favourite LaTeX editor and view source and pdf side by side. You can also click on »Open in Writelatex« (which is now overleaf) to see everything.

To put it in a nutshell, though placing the commands inuse of an environment with it working pretty well, there are situations where it terribly breaks down. Leaving an inexperienced user confused, looking for odd workarounds. That is why i am discouraging the use of commands as environments.

Please play around with the code above and ask if needed.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.

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# Is this problem in P: Finding a common key for a collection of systems of equations?

Let $B=\{b_1=g_1,\cdots,b_n=g_n\}$ be a set of binary variables $b_i$ and their corresponding values $g_i \in \{0,1\}$. Let $M=\{\sum_{e \in A}e \;:\; A \subset B\}$, i.e., $M$ is the set of all possible linear combinations of the equations in $B$.

Given $S_i \subset B$ for $i=1,\cdots,m$, is that possible to compute, in polynomial time, a $K \subset M$ with minimum size such that $S_i \cup K$ is a full rank system of equations (i.e., the values of all of the variables can be obtained by solving $S_i \cup K$)?

An example: Let $B=\{b_1=1,b_2=0,b_3=1\}$, $S_1=\{b_1=1,b_2=0\}$, and $S_2=\{b_2=0,b_3=1\}$. $K=\{b_1+b_3=0\}$ is the solution because both $S_1\cup K$ and $S_2 \cup K$ can be solved uniquely and $K$ has the minimum size 1.


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# Browse Dissertations and Theses - Mathematics by Contributor "Hieronymi, Philipp"

• (2018-06-29)
This thesis consists of two unrelated research projects. In the first project we study the model theory of the 2-sorted structure (F, C; χ), where F is an algebraic closure of a finite field of characteristic p, C is the ...

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• (2018-07-09)
In this thesis, we study expansions of the real field by multiplicative subgroups of the complex numbers. We first consider expansions by a subgroup generated by an element of the unit circle and a positive real number. ...

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• (2017-12-08)
We study two different problems: generic behavior of a measure preserving transformation and extending partial isometries of a compact metric space. In Chapter $1$, we consider a result of Del Junco--Lema\'nczyk [\ref{DL_B}] ...

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• (2015-01-21)
In this dissertation, we study three problems about Ramsey theory. First, we prove a self-dual Ramsey theorem for parameter systems which is a generalization of the self-dual Ramsey theorem developed by Solecki. Second, ...

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• (2017-07-13)
Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same ...

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• (2017-07-09)
The ordered valued differential field $\mathbb{T}_{\log}$ of logarithmic transseries is conjectured to have good model theoretic properties. This thesis records our progress in this direction and describes a strategy moving ...

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• (2018-01-10)
Being closed under truncation for subsets of generalized series fields is a robust property in the sense that it is preserved under various algebraic and transcendental extension procedures. Nevertheless, in Chapter 4 of ...

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# Viscosity index

(Redirected from Viscosity Index)

Viscosity index (VI) is an arbitrary measure for the change of viscosity with variations in temperature [1]. . The lower the VI, the greater the change of viscosity of the oil with temperature and vice versa. It is used to characterize viscosity changes with relation to temperature in lubricating oil.

The viscosity of liquids decreases as temperature increases. The viscosity of a lubricant is closely related to its ability to reduce friction. Generally, the least viscous lubricant which still forces the two moving surfaces apart is desired. If the lubricant is too viscous, it will require a large amount of energy to move (as in honey); if it is too thin, the surfaces will come in contact and friction will increase.

Many lubricant applications require the lubricant to perform across a wide range of conditions, for example, automotive lubricants are required to reduce friction between engine components when the engine is started from cold (relative to the engine's operating temperatures) up to 200 °C or 392 °F when it is running. The best oils with the highest VI will remain stable and not vary much in viscosity over the temperature range. This allows for consistent engine performance within the normal working conditions.

The VI scale was set up by the Society of Automotive Engineers (SAE). The temperatures chosen arbitrarily for reference are 100 and 210 °F (38 and 99 °C). The original scale only stretched between 0 (lowest VI oil, naphthenic) and 100 (best oil, paraffinnic) but since the conception of the scale better oils have also been produced, leading to VIs greater than 100 (see below).

Viscosity index Classification
Under 35 Low
35 to 80 Medium
80 to 110 High
Above 110 Very high

VI improving additives and higher quality base oils are widely used nowadays which increase the VIs attainable beyond the value of 100. The Viscosity Index of synthetic oils ranges from 80 to over 400.

The viscosity index can be calculated using the following formula:

${\displaystyle \mathrm {VI} =100{\frac {L-U}{L-H}}}$

where U is the oil's kinematic viscosity at 40 °C (104 °F), and L and H are values based on the oil's kinematic viscosity at 100 °C (212 °F). L and H are the values of viscosity at 40°C for oils of VI 0 and 100 respectively, having the same viscosity at 100°C as the oil whose VI we are trying to determine. These L and H values can be found in ASTM D2270.[2]

## References

1. ^ "understanding viscosity index in lubricants | Learn Oil Analysis". learnoilanalysis.com. Retrieved 2017-08-28.
2. ^ Stachowiak, Gwidon W.; Batchelor, Andrew W. (2001). Engineering Tribology (2nd ed.). Boston: Butterworth-Heinemann.


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# How do you align equations parts vertically?

I have read many other solutions but this situation seems unique.

\documentclass{exam}
\usepackage{multicol}
\usepackage{amsmath}
\begin{document}
\begin{questions}
\question Simplify the following:
\begin{multicols}{4}
\begin{parts}
\part $8^\frac{2}{3}$
\part $\dfrac{16^{\frac{2}{3}}}{16^{\frac{1}{6}}}$
\part $9^\frac{1}{3}\cdot 9^{-\frac{5}{6}}$
\part $\left(25^{\frac{1}{3}}\right)^{\frac{3}{2}}$
\end{parts}
\end{multicols}
\end{questions}
\end{document}

-

The heights of the four part letters is different, because the different heights of the formulas. The following example defines \vphantomB with an invisible box of the greatest height of part (b):

\documentclass{exam}
\usepackage{multicol}
\usepackage{amsmath}
\begin{document}
\begin{questions}
\question Simplify the following:
\begin{multicols}{4}
\def\vphantomB{\leavevmode\vphantom{$\dfrac{16^{\frac{2}{3}}}{}$}}
\begin{parts}
\part \vphantomB $8^\frac{2}{3}$
\part $\dfrac{16^{\frac{2}{3}}}{16^{\frac{1}{6}}}$
\part \vphantomB $9^\frac{1}{3}\cdot 9^{-\frac{5}{6}}$
\part \vphantomB $\left(25^{\frac{1}{3}}\right)^{\frac{3}{2}}$
\end{parts}
\end{multicols}
\end{questions}
\end{document}

-

you can use the package enumitem instead:

documentclass{exam}
\usepackage{amsmath}
\usepackage[inline]{enumitem}
\begin{document}
\begin{questions}
\question Simplify the following:

\begin{enumerate*}[label*=(\alph*),itemjoin={\hfill}]
\item $8^\frac{2}{3}$
\item $\dfrac{16^{\frac{2}{3}}}{16^{\frac{1}{6}}}$
\item $9^\frac{1}{3}\cdot 9^{-\frac{5}{6}}$
\item $\left(25^{\frac{1}{3}}\right)^{\frac{3}{2}}$
\end{enumerate*}
\end{questions}

\end{document}

-
In this solution, if you have too many items, there is no vertical alignment betwen the items in different lines. – Bernard Apr 24 '14 at 12:01


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Snow? I know.. I know.. I know.. I don't want to write about snow either. Everywhere's got snow. It's white stuff and it falls on you. And it's cold and Canada gets more of it.... but listen up we had 30 cm of snow.
That's a lot of snow.
For Cheesetown.

Taking one border collie as a useful unit of measurement, this is what six centimetres looks like...
Just past paw level. Pretty huh? Add some tinsel and it'd be festive even...

This is what 30cm of the stuff looks like.

See the difference? This might be funny if you want the dog to suffer for getting you out of bed at 5:30 for the third morning in a row...
But with the snow also over the tops of your own wellies it gets less funny fast.
Trust me on that...

30 centimetres... well 30 centimeters means..
• It means that the snow is six inches higher than your doorstep. Open the door and it falls in on you.
• It means that Cheesetown looks like a village of midgets, as all the gravestones are only a couple of inches above snow level.
• It means that the entire world is silent..because Edinburgh International Airport is closed for days. The whole runway! No airplanes!
• And it means that there's no cars, no buses...there's no transport in or out of Cheesetown. The BRIDGE has closed for the first time in 46 years.

And in a fresh twist to anyone elses Snow Saga, 30 centimetres of snow means that it has fallen on me to help the Cherub with his maths revision.

Yeah with no school it's down to me to explain why the merry hell
$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},$


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# UL HPC MPI Tutorial: Building and Runnning OSU Micro-Benchmarks

 Copyright (c) 2013-2018 UL HPC Team  <hpc-sysadmins@uni.lu>

The objective of this tutorial is to compile and run on of the OSU micro-benchmarks which permit to measure the performance of an MPI implementation. On purpose, we won't check here if an Easybuild recipe is available for this software to conduct a full build by hand and check the difference in build options between the different MPI suits.

You can work in groups for this training, yet individual work is encouraged to ensure you understand and practice the usage of MPI programs on an HPC platform.

In all cases, ensure you are able to connect to the UL HPC clusters.

Advanced users only: rely on screen (see tutorial or the UL HPC tutorial on the frontend prior to running any oarsub or srun/sbatch command to be more resilient to disconnection.

The latest version of this tutorial is available on Github. Finally, advanced MPI users might be interested to take a look at the Intel Math Kernel Library Link Line Advisor.

## Objectives

The OSU micro-benchmarks feature a series of MPI benchmarks that measure the performances of various MPI operations:

• Point-to-Point MPI Benchmarks: Latency, multi-threaded latency, multi-pair latency, multiple bandwidth / message rate test bandwidth, bidirectional bandwidth
• Collective MPI Benchmarks: Collective latency tests for various MPI collective operations such as MPI_Allgather, MPI_Alltoall, MPI_Allreduce, MPI_Barrier, MPI_Bcast, MPI_Gather, MPI_Reduce, MPI_Reduce_Scatter, MPI_Scatter and vector collectives.
• One-sided MPI Benchmarks: one-sided put latency (active/passive), one-sided put bandwidth (active/passive), one-sided put bidirectional bandwidth, one-sided get latency (active/passive), one-sided get bandwidth (active/passive), one-sided accumulate latency (active/passive), compare and swap latency (passive), and fetch and operate (passive) for MVAPICH2 (MPI-2 and MPI-3).
• Since the 4.3 version, the OSU micro-benchmarks also features OpenSHMEM benchmarks, a 1-sided communications library.

In this tutorial, we will build version 5.5 of the OSU micro-benchmarks (the latest at the time of writing), and focus on two of the available tests:

• osu_get_latency - Latency Test
• osu_get_bw - Bandwidth Test

The latency tests are carried out in a ping-pong fashion. The sender sends a message with a certain data size to the receiver and waits for a reply from the receiver. The receiver receives the message from the sender and sends back a reply with the same data size. Many iterations of this ping-pong test are carried out and average one-way latency numbers are obtained. Blocking version of MPI functions (MPI_Send and MPI_Recv) are used in the tests.

The bandwidth tests were carried out by having the sender sending out a fixed number (equal to the window size) of back-to-back messages to the receiver and then waiting for a reply from the receiver. The receiver sends the reply only after receiving all these messages. This process is repeated for several iterations and the bandwidth is calculated based on the elapsed time (from the time sender sends the first message until the time it receives the reply back from the receiver) and the number of bytes sent by the sender. The objective of this bandwidth test is to determine the maximum sustained date rate that can be achieved at the network level. Thus, non-blocking version of MPI functions (MPI_Isend and MPI_Irecv) were used in the test.

The idea is to compare the different MPI implementations available on the UL HPC platform.:

For the sake of time and simplicity, we will focus on the first two suits. Eventually, the benchmarking campain will typically involves for each MPI suit:

• two nodes, belonging to the same enclosure
• two nodes, belonging to different enclosures

## Pre-requisites

If not yet done, clone the ULHPC/tutorials and ULHPC/launcher-scripts repositories once connected to the cluster:

### ONLY if not yet done: setup the tutorials repo
# See http://ulhpc-tutorials.rtfd.io/en/latest/setup/install/
$> mkdir -p ~/git/github.com/ULHPC$> cd ~/git/github.com/ULHPC
# Clone the tutorial repository
$> git clone https://github.com/ULHPC/tutorials.git$> cd tutorials
$> make setup # Initiate git submodules etc... # Clone the launcher-scripts repository$> cd ~/git/github.com/ULHPC
$> git clone https://github.com/ULHPC/launcher-scripts.git  Prepare your working directory $> mkdir -p ~/tutorials/OSU-MicroBenchmarks
$> cd ~/tutorials/OSU-MicroBenchmarks$> ln -s ~/git/github.com/ULHPC/tutorials/parallel/mpi/OSU_MicroBenchmarks/ ref.ulhpc.d   # Keep a symlink to the reference tutorial
$> ln -s ref.ulhpc.d/Makefile . # symlink to the root Makefile$> ln -s ref.ulhpc.d/scripts  .     # symlinkls to launcher/build scripts

Now you can reserve an interactive job on 2 nodes and 1 task per node (for 30 minutes)

############### iris cluster (slurm) ###############
(access-iris)$> si -N 2 --ntasks-per-node=1 -t 0:30:00$> echo $SLURM_NTASKS ############### gaia/chaos clusters (OAR) ############### (access-{gaia|chaos})$> oarsub -I -l nodes=2/core=1,walltime=0:30:00

$> cd ~/tutorials/OSU-MicroBenchmarks$> mkdir src
$> cd src # Download the latest version$> export OSU_VERSION=5.5     # Just to abstract from the version to download
$> wget --no-check-certificate http://mvapich.cse.ohio-state.edu/download/mvapich/osu-micro-benchmarks-${OSU_VERSION}.tar.gz
$> tar xvzf osu-micro-benchmarks-${OSU_VERSION}.tar.gz
$> cd osu-micro-benchmarks-${OSU_VERSION}

## Building the OSU Micro-benchmarks

We will build the OSU micro-benchmarks for each considered MPI suit, thus in a separate directory build.<suit> -- that's a good habbit you're encouraged (as with CMake based projects) In all cases, you should now operate the compilation within an interactive job to be able to use the module command.

# If not yet done
(access)$> si -N 2 --ntasks-per-node=1 # on iris (1 core on 2 nodes) (access)$> oarsub -I -l nodes=2/core=1,walltime=4   # chaos / gaia

### Compilation based on the Intel MPI suit

We are first going to use the Intel Cluster Toolkit Compiler Edition, which provides Intel C/C++ and Fortran compilers, Intel MPI. We will compile the OSU micro-benchmarks in a specific directory (that a good habbit)

$> cd ~/tutorials/OSU-MicroBenchmarks/$> mkdir build.intel    # Prepare the specific building directory
$> cd build.intel # Load the appropriate module$> module spider MPI     # Search for available modules featuring MPI
$> module load toolchain/intel # On iris -- use 'module load toolchain/ictce' otherwise$> module list
1) compiler/GCCcore/6.3.0                   4) compiler/ifort/2017.1.132-GCC-6.3.0-2.27                 7) toolchain/iimpi/2017a
2) tools/binutils/2.27-GCCcore-6.3.0        5) toolchain/iccifort/2017.1.132-GCC-6.3.0-2.27             8) numlib/imkl/2017.1.132-iimpi-2017a
3) compiler/icc/2017.1.132-GCC-6.3.0-2.27   6) mpi/impi/2017.1.132-iccifort-2017.1.132-GCC-6.3.0-2.27   9) toolchain/intel/2017a

# Configure the Intel MPI-based build for installation in the current directory
$> ../src/osu-micro-benchmarks-${OSU_VERSION}/configure CC=mpiicc CXX=mpiicpc CFLAGS=-I$(pwd)/../src/osu-micro-benchmarks-${OSU_VERSION}/util --prefix=$(pwd)$> make && make install

Questions:

• Q1: Why did we request the use of the --prefix at the configure step ?
• Q2: Why did we enforce the environment variables CC and CXX?
• Q3: Why did we enforce the environment variables CFLAGS?

If everything goes fine, you shall have the OSU micro-benchmarks installed in the directory libexec/osu-micro-benchmarks/mpi/.

Once compiled, ensure you are able to run it:

$> cd libexec/osu-micro-benchmarks/mpi/one-sided/ #### On iris$> srun -n $SLURM_NTASKS ./osu_get_latency$> srun -n $SLURM_NTASKS ./osu_get_bw #### On gaia, chaos$> mpirun -hostfile $OAR_NODEFILE -perhost 1 ./osu_get_latency$> mpirun -hostfile $OAR_NODEFILE -perhost 1 ./osu_get_bw  ### Compilation based on the OpenMPI suit Repeat the procedure for the OpenMPI suit: $> cd ~/tutorials/OSU-MicroBenchmarks/
$> mkdir build.openmpi # Prepare the specific building directory$> cd  build.openmpi
# Clean the previously loaded module and load the appropriate OpenMPI one
$> module purge$> module spider OpenMPI
$> module load mpi/OpenMPI$> module list

1) compiler/GCCcore/6.3.0              3) compiler/GCC/6.3.0-2.28              5) system/hwloc/1.11.7-GCC-6.3.0-2.28
2) tools/binutils/2.28-GCCcore-6.3.0   4) tools/numactl/2.0.11-GCCcore-6.3.0   6) mpi/OpenMPI/2.1.1-GCC-6.3.0-2.28

# Configure the OpenMPI-based build for installation in the current directory
$> ../src/osu-micro-benchmarks-${OSU_VERSION}/configure CC=mpicc CFLAGS=-I$(pwd)/../src/osu-micro-benchmarks-${OSU_VERSION}/util --prefix=$(pwd)$> make && make install

Questions:

• Q4: Why do we need to precise the CC variable at the configure step?
• Q5: How to get rid of CFLAGS at the configure step ?

Once compiled, ensure you are able to run it:

$> cd libexec/osu-micro-benchmarks/mpi/one-sided/ #### On iris$> srun -n $SLURM_NTASKS ./osu_get_latency # OR mpirun -npernode 1 --mca btl openib,self,sm ./osu_get_latency$> srun -n $SLURM_NTASKS ./osu_get_bw # OR mpirun -npernode 1 --mca btl openib,self,sm ./osu_get_bw # Or, if you don't want to use PMI2$> mpirun -np $SLURM_NTASKS -perhost 1 --mca btl openib,self,sm ./osu_get_{latency,bw} #### On gaia, chaos$> mpirun -x LD_LIBRARY_PATH -hostfile $OAR_NODEFILE -npernode 1 ./osu_get_latency$> mpirun -x LD_LIBRARY_PATH -hostfile $OAR_NODEFILE -npernode 1 ./osu_get_bw  ## Preparing batch runs We are now going to prepare launcher scripts to permit passive runs (typically in the {default | batch} queue). We will place them in a separate directory (runs/) as it will host the outcomes of the executions on the UL HPC platform . $> cd ~/tutorials/OSU-MicroBenchmarks/
$> mkdir runs # Prepare the specific run directory  ### Slurm launcher (Intel MPI) Copy and adapt the default SLURM launcher you should have a copy in ~/git/ULHPC/launcher-scripts/slurm/launcher.default.sh $> cd ~/tutorials/OSU-MicroBenchmarks/runs
# Prepare a laucnher for intel suit
$> cp ~/git/ULHPC/launcher-scripts/slurm/launcher.default.sh launcher-OSU.intel.sh  Take your favorite editor (vim, nano, etc.) to modify it according to your needs. Here is for instance a suggested difference for intel MPI: --- ~/git/ULHPC/launcher-scripts/slurm/launcher.default.sh 2017-06-11 23:40:34.007152000 +0200 +++ launcher-OSU.intel.sh 2017-06-11 23:41:57.597055000 +0200 @@ -10,8 +10,8 @@ # # Set number of resources # -#SBATCH -N 1 -#SBATCH --ntasks-per-node=28 +#SBATCH -N 2 +#SBATCH --ntasks-per-node=1 ### -c, --cpus-per-task=<ncpus> ### (multithreading) Request that ncpus be allocated per process #SBATCH -c 1 @@ -64,15 +64,15 @@ module load toolchain/intel # Directory holding your built applications -APPDIR="$HOME"
+APPDIR="$HOME/tutorials/OSU-MicroBenchmarks/build.intel/libexec/osu-micro-benchmarks/mpi/one-sided" # The task to be executed i.E. your favorite Java/C/C++/Ruby/Perl/Python/R/whatever program # to be invoked in parallel -TASK="${APPDIR}/app.exe"
+TASK="${APPDIR}/$1"

# The command to run
-CMD="${TASK}" +# CMD="${TASK}"
### General MPI Case:
-# CMD="srun -n $SLURM_NTASKS${TASK}"
+CMD="srun -n $SLURM_NTASKS${TASK}"
### OpenMPI case if you wish to specialize the MCA parameters
#CMD="mpirun -np $SLURM_NTASKS --mca btl openib,self,sm${TASK}"

If you apply the above changes, you can test the script in an interactive job as follows:

$> cd ~/tutorials/OSU-MicroBenchmarks/runs$> si -N 2 --ntasks-per-node=1     # create an interactive job, 1 core on 2 nodes
$> ./launcher-OSU.intel.sh osu_get_bw$> ./launcher-OSU.intel.sh osu_get_latency

And then test it in batch mode:

$> cd ~/tutorials/OSU-MicroBenchmarks/runs$> sbatch ./launcher-OSU.intel.sh osu_get_bw
$> sbatch ./launcher-OSU.intel.sh osu_get_latency  ### Slurm launcher (OpenMPI) $> cd ~/tutorials/OSU-MicroBenchmarks/runs
$> cp launcher-OSU.intel.sh launcher-OSU.openmpi.sh  Take again your favorite editor (vim, nano, etc.) to modify launcher-OSU.openmpi.sh as follows: --- launcher-OSU.intel.sh 2017-06-11 23:41:57.597055000 +0200 +++ launcher-OSU.openmpi.sh 2017-06-11 23:46:04.589924000 +0200 @@ -61,10 +61,10 @@ # Load the {intel | foss} toolchain and whatever module(s) you need module purge -module load toolchain/intel +module load mpi/OpenMPI # Directory holding your built applications -APPDIR="$HOME/tutorials/OSU-MicroBenchmarks/build.intel/libexec/osu-micro-benchmarks/mpi/one-sided"
+APPDIR="$HOME/tutorials/OSU-MicroBenchmarks/build.openmpi/libexec/osu-micro-benchmarks/mpi/one-sided" # The task to be executed i.E. your favorite Java/C/C++/Ruby/Perl/Python/R/whatever program # to be invoked in parallel TASK="${APPDIR}/$1"  And then test it in (passive) batch mode: $> cd ~/tutorials/OSU-MicroBenchmarks/runs
$> sbatch ./launcher-OSU.openmpi.sh osu_get_bw$> sbatch ./launcher-OSU.openmpi.sh osu_get_latency

### Slurm launcher (OpenMPI over Ethernet interface)

By default, the MPI communications are operated over the fast Infiniband interconnect. With OpenMPI, we can force them over the Ethernet network to highlight the performance impact of using such a slower network.

$> cd ~/tutorials/OSU-MicroBenchmarks/runs$> cp launcher-OSU.openmpi.sh launcher-OSU.openmpi-eth.sh

Take again your favorite editor (vim, nano, etc.) to modify launcher-OSU.openmpi-eth.sh as follows:

--- launcher-OSU.openmpi.sh     2017-06-11 23:46:04.589924000 +0200
+++ launcher-OSU.openmpi-eth.sh 2017-06-11 23:55:02.239258000 +0200
@@ -72,9 +72,9 @@
# The command to run
# CMD="${TASK}" ### General MPI Case: -CMD="srun -n$SLURM_NTASKS ${TASK}" +# CMD="srun -n$SLURM_NTASKS ${TASK}" ### OpenMPI case if you wish to specialize the MCA parameters -#CMD="mpirun -np$SLURM_NTASKS --mca btl openib,self,sm ${TASK}" +CMD="mpirun -np$SLURM_NTASKS -npernode 1 --mca btl tcp,self ${TASK}" ### Prepare logfile LOGFILE="${RUNDIR}/$(date +%Y-%m-%d)_$(basename ${TASK})_${SLURM_JOBID}.log"

And then test it in (passive) batch mode:

$> cd ~/tutorials/OSU-MicroBenchmarks/runs$> sbatch ./launcher-OSU.openmpi-eth.sh osu_get_bw
$> sbatch ./launcher-OSU.openmpi-eth.sh osu_get_latency  You can find the obtained results on the iris cluster: ### OAR launcher (Intel MPI) In the case of OAR (i.e. on the gaia and chaos cluster), you can use the MPI generic launcher to run the code: $> ~/tutorials/OSU-MicroBenchmarks/runs
$> ln -s ~/git/ULHPC/launcher-scripts/bash/MPI/mpi_launcher.sh launcher-OSU.intel.sh$> ./launcher-OSU.intel.sh \
--basedir $HOME/tutorials/OSU-MicroBenchmarks/build.intel/libexec/osu-micro-benchmarks/mpi/one-sided \ --npernode 1 --module toolchain/intel --exe osu_get_latency,osu_get_bw  If you want to avoid this long list of arguments, just create a file launcher-OSU.intel.default.conf to contain: # Defaults settings for running the OSU Micro benchmarks compiled with Intel MPI NAME=OSU.intel DATADIR=$HOME/tutorials/OSU-MicroBenchmarks/runs/data

MPI_PROG_BASEDIR=$HOME/tutorials/OSU-MicroBenchmarks/build.intel/libexec/osu-micro-benchmarks/mpi/one-sided/ MPI_PROGstr=osu_get_latency,osu_get_bw MPI_NPERNODE=1  Now you can run the launcher script (interactively, or not): # IF within an interactive job$> ./launcher-OSU.intel.sh
# You might want also to host the output files in the local directory (under the date)
$> ./launcher-OSU.intel.sh --datadir data/$(date +%Y-%m-%d)

# Submitting a passive job
$> oarsub -S ./launcher-OSU.intel.sh  ### OAR launcher (OpenMPI) Again, we will rely on the MPI generic launcher to run the code: $> ~/tutorials/OSU-MicroBenchmarks/runs
$> ln -s ~/git/ULHPC/launcher-scripts/bash/MPI/mpi_launcher.sh launcher-OSU.openmpi.sh$> vim launcher-OSU.openmpi.default.conf
[...] # See below for content
$> cat launcher-OSU.openmpi.default.conf # Defaults settings for running the OSU Micro benchmarks wompiled with OpenMPI NAME=OSU.openmpi MODULE_TO_LOADstr=mpi/OpenMPI MPI_PROG_BASEDIR=$HOME/tutorials/OSU-MicroBenchmarks/build.openmpi/libexec/osu-micro-benchmarks/mpi/one-sided/
DATADIR=$HOME/tutorials/OSU-MicroBenchmarks/runs/data MPI_PROGstr=osu_get_latency,osu_get_bw MPI_NPERNODE=1 # Submit a passive job$> oarsub -S ./launcher-OSU.openmpi.sh

### Benchmarking on two nodes

Operate the benchmarking campain (in the two cases) in the following context:

• 2 nodes belonging to the same enclosure. Use for that:

$> oarsub -l enclosure=1/nodes=2,walltime=8 […]  • 2 nodes belonging to the different enclosures: $> oarsub -l enclosure=2/core=1,walltime=8 […]

## Now for Lazy / frustrated persons

You will find in the UL HPC tutorial repository, under the advanced/OSU_MicroBenchmarks directory, a set of tools / script that facilitate the running and analysis of this tutorial that you can use/adapt to suit your needs.

In particular, once in the advanced/OSU_MicroBenchmarks directory:

• running make fetch will automatically download the archives for the OSU micro-benchmarks in the src/ directory
• The different launcher files in runs/
• Some sample output data in runs/data/
• run make build to build the different versions of the OSU Micro-benchmarks
• run make plot to invoke the Gnuplot script plots/benchmark_OSU.gnuplot and generate various plots from the sample runs.

In particular, you'll probably want to see the comparison figure extracted from the sample run in plots/benchmark_OSU_2H_latency.pdf and plots/benchmark_OSU_2H_bandwidth.pdf

A PNG version of these plots is available on Github: OSU latency -- OSU Bandwidth


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# 14,641

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## Number

$14 \, 641$ (fourteen thousand, six hundred and forty-one) is:

$11^4$

The $4$th power of $11$ after $(1)$, $11$, $121$, $1331$:
$14 \, 641 = 11^4$

The $11$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$, $6561$, $10 \, 000$:
$14 \, 641 = 11 \times 11 \times 11 \times 11$

The $121$st square number:
$14 \, 641 = 121 \times 121$


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# Search results

22 records were found.

## The Gigatracker: An ultra-fast and low-mass silicon pixel detector for the NA62 experiment

The Gigatracker is a hybrid silicon pixel detector developed to track the highly intense NA62 hadron beam with a time resolution of 150 ps (rms). The beam spectrometer of the experiment is composed of three Gigatracker stations installed in vacuum in order to precisely measure momentum, time and direction of every traversing particle. Precise tracking demands a very low mass of the detector assembly (o0:5% X0 per station) in order to limit multiple scattering and beam hadronic interactions. The high rate and especially the high timing precision requirements are very demanding: two R&D options are ongoing and the corresponding prototype read-out chips have been recently designed and produced in 0:13 mm CMOS technology. One solution makes use of a constant fraction discriminator and on-pixel analogue-based time-to-digital-converter (TDC)...

## Measurement of the direct emission and interference terms and search for CP violation in the decay K ±→π ± π 0 γ

We report on the measurement of the direct emission (DE) and interference (INT) terms of the K ±→π ± π 0 γ decay by the NA48/2 experiment at the CERN SPS. From the data collected during 2003 and 2004 about 600k such decay candidates have been selected. The relative amounts of DE and INT with respect to the internal bremsstrahlung (IB) contribution have been measured in the range MeV: where is the kinetic energy of the charged pion in the kaon rest frame. This is the first observation of an interference term in K ±→π ± π 0 γ decays, thus allowing the normalised electric and magnetic amplitudes to be measured, giving X E =(−24±6) GeV−4, and X M =(254±9) GeV−4. In addition, a limit on the CP violating asymmetry in the K + and K − branching ratios for this channel has been determined to be less than 1.5×10−3 at 90% confidence level.

## Empirical parameterization of the K±→π±π0π0 decay Dalitz plot

As first observed by the NA48/2 experiment at the CERN SPS, the π0π0 invariant mass (M00) distribution from K±→π±π0π0 decay shows a cusp-like anomaly at M00=2m+, where m+ is the charged pion mass. An analysis to extract the ππ scattering lengths in the isospin I=0 and I=2 states, a0 and a2, respectively, has been recently reported. In the present work the Dalitz plot of this decay is fitted to a new empirical parameterization suitable for practical purposes, such as Monte Carlo simulations of K±→π±π0π0 decays.

## New high statistics measurement of $K_{e4}$ decay form factors and $\pi\pi$ scattering phase shifts

We report results from a new measurement of the $K_{e4}$ decay $K^{\pm} \pi^{+}\pi^{-}e^{\pm}\nu$ by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670 000 $K_{e4}$ decays in both charged modes collected in 2003. The form factors of the hadronic current (F,G,H) and $\pi\pi$ phase difference ($\delta=\delta_s-\delta_p$) have been measured in ten independent bins of the $\pi\pi$ mass spectrum to investigate their variation. A sizeable acceptance at large $\pi\pi$ mass, a low background and a very good resolution contribute to an improved experimental accuracy, a factor two better than in the previous measurement, when extracting the $\pi\pi$ scattering lengths a00 and a20 . Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in th...

## First Observation and Measurement of the Decay K+- -> pi+- e+ e- gamma

Using the full data set of the NA48/2 experiment, the decay K+- -> pi+- e+ e- gamma is observed for the first time, selecting 120 candidates with 7.3 +- 1.7 estimated background events. With K+- -> pi+- pi0D as normalisation channel, the branching ratio is determined in a model-independent way to be Br(K+- -> pi+- e+ e- gamma, m_eegamma > 260 MeV/c^2) = (1.19 +- 0.12_stat +- 0.04_syst) x 10^-8. This measured value and the spectrum of the e+ e- gamma invariant mass allow a comparison with predictions of Chiral Perturbation Theory.

## Measurement of the Spin-Dependence of the pbar-p Interaction at the AD-Ring

Comment: 51 pages, 23 figures, proposal submitted to the SPS committee of CERN

## Protons in near earth orbit

The proton spectrum in the kinetic energy range 0.1 to 200 GeV was measured by the Alpha Magnetic Spectrometer (AMS) during space shuttle flight STS-91 at an altitude of 380km. Above the geomagnetic cutoff the observed spectrum is parameterized by a power law. Below the geomagnetic cutoff a substantial second spectrum was observed concentrated at equatorial latitudes with a flux ~70m2s1sr1. Most of these second spectrum protons follow a complicated trajectory and originate from a restricted geographic region.

## Search for antihelium in cosmic rays

The Alpha Magnetic Spectrometer (AMS) was flown on the space shuttle Discovery during flight STS-91 in a 51.7 degree orbit at altitudes between 320 and 390 km. A total of 2.86 * 10^6 helium nuclei were observed in the rigidity range 1 to 140 GV. No antihelium nuclei were detected at any rigidity. An upper limit on the flux ratio of antihelium to helium of < 1.1 * 10^-6 is obtained.

## SuperB Progress Reports -- Detector

This report describes the present status of the detector design for SuperB. It is one of four separate progress reports that, taken collectively, describe progress made on the SuperB Project since the publication of the SuperB Conceptual Design Report in 2007 and the Proceedings of SuperB Workshop VI in Valencia in 2008. The other three reports relate to Physics, Accelerator and Computing.