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# Localization of a total quotient ring of a reduced ring

Let $$R$$ be any ring. The total quotient ring of $$R$$, denoted $$K(R)$$, is $$R$$ localized at the set of non zero-divisors.

I am trying to solve Exercise 3.15 in Eisenbud's Commutative Algebra:

a) Let $$R$$ be a Noetherian reduced ring and let $$U$$ be a multiplicative subset. Then $$K(R[U^{-1}])=K(R)[U^{-1}]$$.

b) If $$R$$ is a Noetherian ring, then $$K(R[U^{-1}])=K(K(R)[U^{-1}])$$.

For a), I was able to show that all prime ideals in $$K(R)$$ are maximal. Further, there are finitely many of them. Let these ideals be denoted $$P_1,\ldots ,P_k$$. By the Chinese Remainder Theorem, we have that

$$K(R)=\prod_{i=1}^k K(R/P_i)$$

How should I proceed? Also, can I have a hint to start part b)? Should I use the universal property of the localization somehow?

• I have changed my question to include that $R$ is Noetherian. I assume Eisenbud meant to include it. – Justine Feb 17 '16 at 16:36
• We cannot know what Eisenbud really thought, but the book is called '...with a view towards geometry' and on geometrical practice, one mostly encounters only the noetherian case :) So I think it is pretty satisfying to have a proof in the noetherian case and a counterexample in the nonnoetheriam case, isn't it? – MooS Feb 17 '16 at 18:06

Note that your $P_1, \dotsc, P_k$ are the minimal primes of $R$. Note that they are only finite if $R$ is noetherian, you have to add this assumption as the other answer points out.

For $a)$ you should show the following:

If we localize $\prod_{i=1}^k K(R/P_i)$ at $U$, then the component $K(R/P_i)$ survives if and only if $U \cap P_i = \emptyset$.

Using this, we get:

$K(R)[U^{-1}]$ is the product of all $K(R/P_i)$, where $U \cap P_i=\emptyset$.

Now we start with the reduced ring $R[U^{-1}]$. The minimal primes of this ring are all $P_i[U^{-1}]$ with $P_i \cap U = \emptyset$, i.e. its total quotient ring is the product of all $K(R[U^{-1}]/P_i[U^{-1}])$, where $U \cap P_i=\emptyset$.

Localization commutes with quotients, hence we have

$$K(R[U^{-1}]/P_i[U^{-1}])=K((R/P_i)[U^{-1}])=K(R/P_i),$$ which gives you the result. The latter equality holds, since we have $(R/P_i)[U^{-1}]=R/P_i$ if $U \cap P_i=\emptyset$.

• How does this give the result: $K(R[U^{-1}])=K(R)[U^{-1}]$? – Al Jebr Jun 27 '20 at 6:28
• How do we have $(R/P_i)[U^{-1}]=R/P_i$ if $U \cap P_i=\emptyset$ ? – Al Jebr Jun 27 '20 at 6:51

a) is wrong without the assumption that $$R$$ is noetherian.

Let $$k$$ be a field and let $$R=k[X_1, \dots, X_n, \dots]/(X_iX_j:i\ne j,j>2).$$ $$R$$ is reduced, and its maximal ideal $$\mathfrak m=(x_1,\dots,x_n,\dots)$$ is the set of the zerodivisors. Hence $$K(R)=R_{\mathfrak m}$$. Set $$\mathfrak p=(x_2, \dots, x_n, ...)$$ and $$S=R\setminus\mathfrak p$$. Then $$S^{-1}K(R)=R_{\mathfrak p}$$, while $$K(S^{-1}R)=K(R_{\mathfrak p})$$. (Note that $$R_{\mathfrak p}=k[X_1,X_2]_{(X_2)}$$ is not a field.)

• I have changed my question to include that $R$ is Noetherian. I assume Eisenbud meant to include it. – Justine Feb 17 '16 at 16:36


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# [change of basis] Consider the family β = (1 + x + x 2 , x − x 2 , 2 + x 2 ) of the polynomial space of degree ≤ 2, R2[x].

[change of basis] Consider the family ? = (1 + x + x 2 , x ? x 2 , 2 + x 2 ) of the polynomial space of degree ? 2, R2[x].

- Prove that ? of R2[x] is a basis of R2[x].
- Calculate M?can , the matrix for change of basis from the basis ? to the canonical basis of R2[x].
-Calculate the kernel and range of the matrix M?can
- If we were to change ? for any other basis of R2[x], would the kernel and range of M?can change?

format: plain


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# Gas station without pumps

## 2017 May 18

### Midterm quiz doesn’t tell me much new

Filed under: Circuits course — gasstationwithoutpumps @ 09:53
Tags: , ,

I don’t usually give exams in my courses any more, because I’m more interested in what students can do when they have time and resources than what they can do on toy problems under resource limitations.  But if students don’t do the homework, then they don’t learn the material, so I threaten each class that if too many students don’t turn in the homework, I’ll have to add a quiz (worth as much as one of the lab reports, each of which is equal to all the homework) to the course.

This quarter I had to follow through on that threat, because 12% of the class had turned in half or less of the homework (and by that, I don’t mean answered half the questions—I mean turned in nothing at all for half the assignments).  A quarter of the class had not turned in 25% or more of the assignments.

I gave the quiz yesterday, with 6 easy questions that only tested the very basic material: single-pole RC filters (passive and active) and negative-feedback amplifiers.  I told students ahead of time (and on the exam) that they could use the Bode approximations (the straight-line approximations to the gain of the RC filters) and we even reviewed them in class last week.  There were 60 points possible on the test, and none of the questions were design questions—they were almost all of the form “what is the corner frequency?” or “what is the gain of this circuit?”.

There are a small number of students in the class whose probity I have reason to question, so I took steps to reduce cheating that I would not normally bother with: I made up two versions of the test (same schematics, but different component values) and alternated them in the piles passed along each row.  I also had the students sit in different rows from usual, reversing front and back of the room, with the front row reserved for latecomers. I’ve noticed a high correlation between good homework grades and people being on-time and in the first two rows, so I had those students sit in the back row, where no one would be able to copy from them.

I normally figure that a test is appropriately long if an expert can do it in about a quarter of the time allotted.  So I made up the keys for the test while the students were taking it.  Working through one form with the Bode approximations took about 5 minutes.  Doing exact computation with the formulas for series and parallel impedances and complex numbers using only real-number arithmetic on my calculator extended that by another 15 minutes.  The students had 63 minutes, so the exam was too easy if the students used the Bode approximations (as they were told) but a little too hard if they worked just from the fundamentals of complex impedance and negative-feedback amplifiers.  As a consequence, I decided to give bonus points for exact computations of the gains that didn’t use the Bode approximations, though the class was not informed of this bonus, because I didn’t want them to waste time on the tiny bonus.  (The differences in answers were small, because I had deliberately asked for gains only at points well away from the corner frequency, so that the Bode approximations would be good.)

Even if students really didn’t understand complex impedance or RC filters, 39 of the 60 points could be earned with just DC analysis of the negative-feedback amplifiers and knowing that capacitors don’t conduct DC.   So I was hoping that students would do better on these very easy questions than they did on the harder design questions of the homework.  As a confirmed pessimist, though, I expected that students would show almost exactly the same distribution on the test that they showed on the homework, with the middle of the class being around 20 out of 60 points and showing serious misunderstandings of almost everything, with a long tail out to one or two students who would get almost everything right.  I also expected that the correlation between the homework scores and the quiz scores would be high.

So what happened?  First, I saw no evidence of any cheating (not that I had expected any), so that is one worry removed.  Second, my pessimistic assumption that students really were not learning stuff that they had done many times in homework and in lab was confirmed:

Here is a stem-and-leaf plot of the scores:

OO: 3
05: 6889
10: 011112444444
15: 555667777899
20: 00111112223344
25: 677999
30: 12224
35: 5678
40: 00444
45: 67
50: 01
55:
60: 2

The median is indeed 21 out of 60, as I feared. At least no one got a zero, though the scores at bottom indicated complete failure to apply the basics of the course.

Most students could compute a corner frequency from a resistor and capacitor, but few had any idea what to do with that corner frequency. Many students could compute the DC gain of a non-inverting amplifier, though many could not then apply this knowledge to the DC gain of an active filter (which only requires replacing the capacitors with open circuits). A lot of students forgot the “+1” in the formula of the gain for the non-inverting amplifier.

Inverting amplifiers were even less understood than non-inverting ones, with students forgetting the minus sign or trying to use the formula for non-inverting amplifiers.

A lot of student answers failed simple sanity checks (students were having passive RC filters with gain greater than 1, for example).

Very few students used the Bode approximations correctly, and many tried the exact solution but either couldn’t set up the formulas correctly or couldn’t figure out how to use their calculators, often getting numbers that were way, way off.  Others seem to have ignored the complex numbers and treat $x+jy$ as if it were $x+y$.

One disturbing result was how many students failed to recognize or understand a circuit that they have designed in three different labs: a voltage divider and unity-gain buffer to generate Vref, combined with a non-inverting amplifier. I asked for the output voltage as a function of the input voltage (both clearly labeled on the schematic). This was intended to be almost free points for them, since they had used that circuit so many times, and the formula they needed was one of the few formulas on the study sheet: $\frac{V_{out}-V_{ref}}{V_{in}-V_{ref}} = 1 + Z_{f}/Z_{i}$. The frequent failure to be able to fill in the blanks of this formula for a circuit that they have used several times in design makes me question whether the students are actually learning anything in the course, or if they are simply copying designs from other students without understanding a thing. (Note: the extremely poor performance and group-think duplication of ludicrously wrong answers on pre-lab homework this year has also lead me to the same question.)

Did the quiz tell me anything that the homework had not already told me? Here is the scatter diagram:

Pearson’s r correlation is 0.539 and Kendall’s tau is 0.306, so the homework and quiz scores are highly correlated. There are a few outliers: a diligent student who bombed the quiz and a student who has turned in few of the homeworks who actually understands at least the easy material. The points have a small amount of noise added, so that duplicate points are visible.

The high correlation between the quiz and the homework mostly confirmed my prior belief that the quiz would not tell me much that is new, and that the homework grades were pretty reflective of what students had learned. I will want to talk with a few of the most extreme outliers, to find out what happened (why were students who mostly understood the material blowing off the homework? and why did diligent students who had been doing moderately well on the homework bomb the quiz—is there undiagnosed test anxiety that should be getting accommodations, for example?).

Most of the points that were earned were from students randomly plugging numbers into a memorized formula and (perhaps accidentally) having chosen the right formula and the right numbers. Only a few students showed real understanding of what they were doing, and only one student saw the quiz as the trivial exercise it was intended to be.

It seems that the hands-on active learning that I have been so enthusiastic about is not working any better at getting students to learn the basics than the traditional (and much cheaper) droning lecture that EE uses. I’m not in complete despair about the course, as there is some evidence that students have picked up some lab skills (using oscilloscopes, multimeters, soldering irons, …) and some writing skills (though many are still not writing at a college level). But I’m trying to teach the students to be engineers, not technicians, so I was aiming at them understanding how to design and debug things, not just implementing other people’s designs. Picking up lab skills is not enough for the course.

I need help. How do I reach the lower half of the class? How do I get them to think about simple electronics instead of randomly applying half-remembered formulas? We’ve only got 3 weeks left—I don’t know how much I can salvage for this cohort, but I certainly would like better outcomes next year.

## 2017 February 25

### More writing advice from the electronics course

Filed under: Circuits course — gasstationwithoutpumps @ 20:13
Tags: , , ,

I spent the entire long Presidents’ Day weekend grading and still did not clear my backlog (the cold I’ve had for the past two weeks has really reduced my ability to work long hours).  I did get a homework set graded and a design report set graded during the 3-day weekend, but I was left with a set of 18 redone lab reports that I still hadn’t gotten to.  Wednesday produced another set of redone reports (so I now have about 35), and Friday produced another homework set (which I just finished grading after spending all Saturday—I haven’t even gotten dressed yet and it is almost 8pm). Tomorrow I’ll tackle the first set of redone reports, assuming my cold lets me do anything tomorrow.

In Disappointment with chain stores, I commented on no longer wanting to grade in the Peet’s coffee shop I used to grade in, and commenter “Mike” wanted to know what solution I came up with. This winter, I’ve mainly been grading in my breakfast room at home, using a tiny laptop (an 11″ MacBook Air, which we bought for travel and for lecturing) when needed to look things up (data sheets, my solution sets, the exact wording in the book, …).  This has worked out ok this year, though I do tend to wander into the kitchen for snacks a bit too often.  Still, the snacks at home are healthier than in a coffee shop!

Here are some general comments that I shared with the whole class, based on the most recent design reports:

Content:

Many students reported using a 3.3V power supply without measuring it, resulting in inconsistent information, such as VOH > 3.3V.  PteroDAQ reports the power-supply voltage on the GUI and in the metadata for the data file, so the data was available, even if the students hadn’t thought to measure it while in lab.

Formatting:

• Equations are parts of sentences, not figures and not objects dropped randomly on the page. Treat them grammatically as noun phrases.
• Explicit figure credit is needed any time a figure is copied or adapted. The caption must say something like “figure adapted from …” or “figure copied from …”. Failure to do so is plagiarism, and I’ll have to start academic integrity proceedings if students fail to do proper figure credits in future.
• Don’t bury the lede. Start with the design goal, not with generic background. A lot of students still wanted to give me a bunch of B.S. about what hysteresis was, before telling me that they were designing a capacitive touch sensor using a relaxation oscillator built around a Schmitt trigger.

Grammar:

• The subjunctive mood marked by the auxiliary verb “would” is used for many things in English, but technical writing primarily uses just one: contrary-to-fact statement. “The inverter that we would be using” says that you didn’t use that inverter and are about to say why. A lot of students seem to think that “would be” is some formal form of past tense—they’ve seen it in writing, but never understood what it means.  I fault their middle-school English teachers for not stressing the importance of more advanced grammar than the bare minimum, but the fault could have been corrected in high school or in college composition classes, but still persists.
• Students are still using way too much passive: “It was decided …” should be replaced with “We decided …”.  Part of the problem here is that much of the writing they are exposed to overuses passive also—excessive passive is a common writing error for scientists and engineers, not just students.

Wording:

• “Firstly”, “secondly”, “lastly” ⇒ “first”, “second”, “last”. These are already adverbs and don’t need an “-ly” ending.  Strangely, I never see the corresponding problem with “next”, though it is in the same class of words that are simultaneously adjectives and adverbs.
• There are a lot of words that are compound words as nouns, but separable verb+particle pairs as verbs. For example, “setup” is a noun, but “set up” is a verb. Other examples include layout, turnaround, pickup, putdown, stowaway, flyover, and setback.
• Avoid the unit abbreviation mm2, as it is too hard to tell whether you mean m(m2) or (mm)2. Most often, the (mm)2 interpretation will be made, but a lot of students used it for m(m2).  (Same for cm2.)
• Many students are using “proportional” wrong, for any increasing function. The phrase “f proportional to d” means f=kd for some k, not just that f increases with d. Similarly, “T inversely proportional to d” means T=k/d for some k, not just that C decreases with d.

Punctuation:

• Capitalize at beginning of sentence and proper names only: “Schmitt trigger” not “Schmitt Trigger”, “Figure 3” but “many figures”. Figure names, table names, and equation names are proper nouns, so should be capitalized: “There are three figures on the last page: Figures 4, 5, and 7”.
• Unit names are not capitalized (hertz, volts, amps, …), but symbols for units from people’s names are (Hz, V, A)
• Hyphenate a noun phrase used as a modifier for another noun: “Schmitt-trigger inverter” but “Schmitt trigger”.

## 2017 February 5

### Units matter

Filed under: Circuits course — gasstationwithoutpumps @ 11:37
Tags: , , , , ,

I was a little surprised by how many students had trouble with the following homework question, which was intended to be an easy point for them:

Estimate C2(touching) − C2(not touching), the capacitance of a finger touch on the packing-tape and foil sensor, by estimating the area of your finger that comes in contact with the tape, and assume that the tape is 2mil tape (0.002” thick) made of polypropylene (look up the dielectric constant of polypropylene on line). Warning: an inch is not a meter, and the area of your finger tip touching a plate is not a square meter—watch your units in your calculations!

Remember that capacitance can be computed with the formula $C = \frac{\epsilon_r\epsilon_0 A}{d}~,$
where $\epsilon_r$ is the dielectric constant,  $\epsilon_0=$8.854187817E-12 F/m is the permittivity of free space, A is the area, and d is the distance between the plates.

The problem is part of their preparation for making a capacitance touch sensor in lab—estimating about how much capacitance they are trying to sense.

There is a fairly wide range of different correct answers to this question, depending on how large an area is estimated for a finger touch. I considered any area from 0.5 (cm)2 to 4 (cm)2 reasonable, and might have accepted numbers outside that range with written justification from the students.  Some students have no notion of area, apparently, trying to use something like the length of their finger times the thickness of the tape for A.

People did not have trouble looking up the relative dielectric constant of polypropylene (about 2.2)—it might have helped that I mentioned that plastics were generally around 2.2 when we discussed capacitors a week or so ago.

What people had trouble with was the arithmetic with units, a subject that is supposed to have been covered repeatedly since pre-algebra in 7th grade. Students wanted to give me area in meters or cm (not square meters), or thought that inches, cm, and m could all be mixed in the same formula without any conversions.  Many students didn’t bother writing down the units in their formula, and just used raw numbers—this was a good way to forget to do the conversions into consistent units.  This despite the warning in the question to watch out for units!

A lot of students thought that 1 (cm)2 was 0.01 m2, rather than 1E-4 m2. Others made conversion errors from inches to meters (getting the thickness of the tape wrong by factors of 10 to 1000).

A number of students either left units entirely off their answer (no credit) or had the units way off (some students reported capacitances in the farad range, rather than a few tens of picofarads).

A couple of students forgot what the floating-point notation 8.854187817E-12 meant, even though we had covered that earlier in the quarter, and they could easily have looked up the constant on the web to figure out the meaning if they forgot.  I wish high-school teachers would cover this standard way of writing numbers, as most engineering and science faculty assume students already know how to read floating-point notation.

Many students left their answers in “scientific” notation (numbers like 3.3 10-11 F) instead of using more readable engineering notation (33pF). I didn’t take off anything for that, if the answer was correct, but I think that many students need a lot more practice with metric prefixes, so that they get in the habit of using them.

On the plus side, it seems that about a third of the class did get this question right, so there is some hope that students helping each other will spread the understanding to more students.  (Unfortunately, the collaborations that are naturally forming seem to be good students together and clueless students together, which doesn’t help the bottom half of the class much.)

## 2016 October 5

### Broken bike seat

Filed under: Uncategorized — gasstationwithoutpumps @ 17:25
Tags: , , , , , , ,

Yesterday was not a good day for me.

First, I spent most of the day struggling with the homework for the control-theory class I’m sitting in on. The course is dual listed as an undergrad and grad course, with shared lectures but different homework and projects. The undergrad part of the homework was straight-forward, and I finished it Monday night, but the two additional problems for the grad students were tough.  One of them had a simple “engineering” solution that I got quickly by formal manipulation of the formulæ, but I could not justify some of the steps, since they involved a integral that was not finite.  The other problem was not difficult, but involved a rather tedious amount of algebra to linearize the system—the professor had done the linearization in lecture notes,  and we were just supposed to check it for the homework, but he’d made an error in algebra, so I had to redo the whole thing.

Late in the afternoon, I decided to take a break and replace the sump pump that had failed sometime in the past couple of weeks.  Originally I was going to disassemble the pump and see if the problem was repairable (I think that the switch for the float is not turning on reliably, possibly from corroded contacts), but I decided that I could do that later to have a spare pump, meanwhile getting a working sump pump.  (My house is built over a seep where an aquifer comes to the surface, and the water table is about 3 inches below the surface—during wet years, the water table is sometimes right at the surface.)

I put the old pump in my panniers and headed down to hardware store, when my bike seat suddenly failed.  I tried riding for a block with the failed seat and gave up and returned home.  The failure was right at edge of the block that holds the horizontal crossbar at the front of the seat:

Here is a view from the front showing the tubing displaced vertically from where it belongs.

A closer view shows a very clean break right at the surface of the block that clamps around the tube.

I probably should have had some warning about the imminent failure, as the bike has been creaking a bit more than usual when I pedal for the past several months, but I was never able to track down the creaking. I’m not sure I could have seen the crack that was probably propagating, since it was flush with clamp block.

The seat on my Longbikes Vanguard is not a standard, off-the-shelf component, so I’m probably going to have to custom order a new seat from the manufacturer (who no longer make the Vanguard model, so probably has no spare seats built) and wait weeks or months for one to be built.

I got my old upright bike down from the garage wall, inflated the tires, adjusted one of my panniers to fit the different rack, and headed off to the hardware store, carrying the old pump in the pannier. At the hardware store, I could not find a sump pump with the same outlet size as the old one (they all had bigger outlets). I needed to match, in order to hook the sump pump up to the existing plumbing. Luckily, they did have a reducer that would adjust for the difference.

After buying the pump, I went out to my bike and realized that I couldn’t fit both pumps into one pannier—in fact the new boxed sump pump wouldn’t fit into the pannier even by itself. Normally I carry a bungee cord or two for strapping stuff onto my rear rack, but those were left on the other bike. So I had to go back into the hardware store to buy some new bungee cords—not a big deal, but an irritation.

The bike was a bit wobbly on the way home—I’d forgotten how much difference a high center of gravity makes on an upright bike—and the bike has much twitchier steering than my recumbent anyway—but I got home without incident.

On getting home, I immediately attached the pluming to the new sump pump and lowered it into the sump. Let me correct that—I tried to lower it into the sump, but it wouldn’t fit. The pump was a couple of inches wider than the old pump and though the hole at the top was more than wide enough, it narrowed significantly where the bottom of the foundation for the house spread out, and the remaining hole was simply too small for the new pump. This was particularly frustrating for me, as I was meeting my wife downtown for dinner in less than an hour, and I was going to have to walk rather than bike, so I only had about 10 minutes to come up with a fix.

I then remembered something that should have occurred to me much earlier—I had another one of the small sump pumps in a different sump in the back garden. Quickly pulling it out and attaching the plumbing got the main sump working again (though I still need to recheck the plumbing for leaks). And it turned out that the garden sump was wide enough to accept the new pump—problem solved!

I cleaned up, grabbed a backpack so I could do some shopping after dinner, and walked down to the library to meet my wife. After the stresses of the day, I felt the need for comfort food, so we went to Betty’s Noodles, a hole-in-the-wall Chinese restaurant in the bus station. This restaurant has taken over the niche that Little Shanghai used to fill of providing cheap, tasty Chinese fast food (noodles and rice bowls).  I had ma-po tofu over Chow Fun noodles, which went a long way to de-stress me.  Going to Mission Hill Creamery for a plum sorbet cone afterwards helped also.

On the walk home, a couple blocks before we got home, I realized that I had not done my shopping! I decided not to go back downtown, but to do without my chocolate soymilk for a couple of days, until I can go shopping again.

This morning I finished the homework and submitted it. I’m still a bit bothered about the inverse Laplace transform problem that  can be formally solved but that ends up with a function that doesn’t have a Laplace transform, but I’m pretty sure I did what was expected. After turning in the homework, I realized that there was a possible different interpretation of part of the linearization question than what I did, so I queried the professor about what he really meant.  (The homework isn’t due for a week, so if there is a clarification needed, he can get it to the grad students before the homework is due.)

The TA does not grade my homework, since I’m just auditing, but I’m doing the homework using Python instead of Matlab, so I’m sharing it with the TA and professor anyway, so they can see whether it would be worth switching to free tools.

Currently, the scipy.signal package and matplotlib seem as easy to use at Matlab, but there is no equivalent of SIMULINK, which the professor is relying on for students doing simulations.  I can do the simulations in Python, but setting them up is all text-based, and requires thinking explicitly about the state vector, rather than having a GUI that does all the setup for you.

I bicycled up to campus today on my old upright, after adjusting my other pannier to fit the rack.  I had forgotten how uncomfortable an upright bike is.  This evening my neck and shoulders are sore, and I have chafing on the inside of my thigh.  I really hope I can get the recumbent seat replaced quickly, so that I can go back to riding comfortably!  It might even be worth taking the seat to a local frame-builder and finding out whether they could replace the tube, even if only for a temporary fix. (Although most of the bike is chrome-moly steel, the seat appears to be all aluminum tubing.)

## 2015 April 13

### Too much prelab homework for microphone lab

Filed under: Circuits course — gasstationwithoutpumps @ 22:54
Tags: , ,

In putting together the book for this quarter, I added exercises to some of the chapters, and I assigned a chunk of them due today as the pre-lab exercise for the microphone lab.  I just spent over 3 hours grading the set, just marking questions right or wrong. There were too many questions, and even the best in the class got only 8/11, with the bottom of the class getting 3/11. I think that the class is doing better this year than previous years’ classes, but some of them got discouraged by how much and how difficult the homework was.  The amount was certainly more than I had intended, but next week’s homework should be substantially less.  I’ll have to figure out how to distribute the load more evenly next year.

I spent most of today’s lecture going over two of the questions (in response to student request), and I’ll have to do some of the other ones on Wednesday, in addition to showing them how to model DC behavior of the FET in the electret microphone. I may also ask the group tutor for the class to have an extra help session this week.  The two questions that got asked about were the modeling of the oscilloscope probe and computing the sensitivity of the electret microphone circuit with a different load resistor.

Some of the problems students had were ones that can be easily fixed (like that 1/10000 is 0 in gnuplot, because it looks like integer arithmetic—1/10E3 does the right computation, as does 1./10000).  Other problems were fundamental misunderstandings of complex numbers or complex impedance, which may be harder to address.

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# Code released for “Binary Pursuit” spike sorting

At long last, I’ve finished cleaning, commenting, and packaging up code for binary pursuit spike sorting, introduced in our 2013 paper in PLoS ONE.  You can download the Matlab code here (or on github), and there’s a simple test script to illustrate how to use it on a simulated dataset.

The method relies on a generative model (of the raw electrode data) that explicitly accounts for the superposition of spike waveforms. This allows it to detect synchronous and overlapping spikes in multi-electrode recordings, which clustering-based methods (by design) fail to do.

If you’d like to know more (but don’t feel like reading the paper), I wrote a blog post describing the basic intuition (and the cross-correlation artifacts that inspired us to develop it in the first place) back when the paper came out (link).

# Subunit models for characterizing responses of sensory neurons

On July 28th, I presented the following paper in lab meeting:

This paper proposes a new method for characterizing the multi-dimensional stimulus selectivity of sensory neurons. The main idea is that, instead of thinking of neurons as projecting high-dimensional stimuli into an arbitrary low-dimensional feature space (the view underlying characterization methods like STA, STC, iSTAC, GQM, MID, and all their rosy-cheeked cousins), it might be more useful / parsimonious to think of neurons as performing a projection onto convolutional subunits. That is, rather than characterizing stimulus selectivity in terms of a bank of arbitrary linear filters, it might be better to consider a subspace defined by translated copies of a single linear filter.

Fig 1 illustrates the rationale rather beautifully:

Panel (a) shows a bank of subunit filters — translated copies of a single filter (here scaled by the weighting of each subunit). The output of each subunit filter passes through a point nonlinearity, and the resulting values are summed to form the neural output. Panel (b) gives the corresponding geometric picture: the arrows labeled $\vec k_1, \ldots, \vec k_4$ indicate vectors defined by the subunit filter shifted to positions 1, …, 4, respectively. Note that these vectors lie on a hyper-sphere (since translations do not change the length of a vector). The span of these vectors defines a multi-dimensional space of stimuli that will excite the neuron (assuming their projection is positive, so perhaps we should say half-space). But if we were to attempt to characterize this space using spike-triggered covariance analysis (panel c), we would obtain a basis defined by a collection of orthonormal vectors. (The slightly subtle point being made here is that the number of vectors we can recover grows with the size of the dataset — so in the bottom plot with $10^6$ datapoints, there are four “significant” eigenvalues corresponding to a four-dimensional subspace). The four eigenvectors $\vec e_1, \ldots, \vec e_4$ span the same subspace as $\vec k_1, \ldots \vec k_4$, but they have been orthogonalized, and their relationship to the original subunit filter shape is obscured.

If you think about it, this picture is devastating for STC analysis. The subunit model gives rise to a 4-dimensional feature space with a single filter shifted to 4 different locations (which requires $n + 4$ parameters), whereas STC analysis requires four distinct filters ($4n$ parameters). But the real calamity comes in trying to estimate the nonlinear mapping from feature space to spike rate. For STC analysis, we need to estimate an arbitrary nonlinear function in 4 dimensions. That is, we need to go into the 4-dimensional space spanned by filters $\vec e_1, \ldots, \vec e_4$ and figure out how much the neuron spikes for each location in that 4D hypercube.  In the subunit model, by contrast, the nonlinear behavior is assumed to arise from a (rectifying) scalar nonlinearity applied to the output of each subunit filter. We need only a single 1D nonlinearity and a set of 4 weights.

So clearly, if a neuron’s response is governed by a linear combination of nonlinear subunits, it’s crazy to attempt to go about characterizing it with STC analysis.

This idea is not entirely new, of course. Nicole Rust’s 2005 paper suggested that the derivative-like filters revealed by spike-triggered covariance analysis of V1 responses might be a signature of nonlinear subunits (Rust et al, Neuron 2005). Kanaka Rajan & Bill Bialek made a related observation about STC analysis (Rajan & Bialek 2013), noting that many of the simple computations we might expect a neuron to perform (e.g., motion selectivity) lead to relatively high-dimensional features spaces (as opposed to the low-dimensional subspaces expected by STC). Tatyanya Sharpee’s lab has introduced a method for characterizing translation-invariant receptive fields inspired by maximally-informative-dimensions estimator (Eickenberg et al 2012). And of course, the idea of nonlinear subunits is much older, going back to at least 1960s work in retina (e.g., Barlow & Levick 1965).

The Vintch et al paper describes a particular method for characterizing the subunit model, which they formulate as a linear-nonlinear-linear (LNL) model from stimulus $\vec x$ to response $r$ given by:

$r = \sum_i w_i f_{\theta}(\vec k_{(i)} \cdot \vec x)$ + noise

where $\vec k_{(i)}$ denotes the subunit filter $\vec k$ shifted to the $i$‘th stimulus position, $f_\theta(\cdot)$ is the subunit nonlinearity, and $w_i$ is the weight on the output of the $i$‘th subunit.  I won’t belabor the details of the fitting procedure, but the results are quite beautiful, showing that one can obtain substantially more accurate prediction of V1 responses (with far fewer parameters!) using a model with separate populations of excitatory and inhibitory subunits.

We finished lab meeting with a brief discussion of some new ideas I’ve been working on with Anqi and Memming, aimed at developing fast, scalable methods for fitting high-dimensional subunit models. So watch this space, and hopefully we’ll have something new to report in a few months!

References

1. Efficient and direct estimation of a neural subunit model for sensory coding. Vintch, Zaharia, Movshon, & Simoncelli, NIPS 2012.
2. Spatiotemporal elements of macaque V1 receptive fields.  Rust, Schwartz, Movshon & Simoncelli, Neuron 2005.
3. Maximally Informative “Stimulus Energies” in the Analysis of Neural Responses to Natural Signals.  Rajan & Bialek, PLoS ONE 2013.
4. Characterizing Responses of Translation-Invariant Neurons to Natural Stimuli: Maximally Informative Invariant Dimensions. Eickenberg,Rowekamp, Kouh, & Sharpee.  Neural Comp., 2012.
5. The mechanism of directionally selective units in rabbit’s retina. Barlow & Levick, J Physiol 178(3):1965.

# Quantifying the effect of intertrial dependence on perceptual decisions

Today we discussed a paper on sequential effects in psychophysics by Fründ et al. Although inter-trial dependencies are known to exist, psychophysical responses are typically modeled as independent Bernoulli observations. One way to account for the effect of previous trial outcomes is to use logistic regression with terms that represent previous stimuli or responses (Busse et al). Fründ and colleagues extend this approach by including a lapse rate which captures responses where the subject was not doing the task. They propose a latent variable on each trial, $l_t$ that is in one of three possible states: 0,1,2 (“guesses left”, “guesses right”, “does task”). The probability of a response is dependent on the stimulus on that trial, the outcome of previous trials and the state of the latent variable:

where $x_t$ is the concatenation of stimulus and history terms for that trial, $\omega$ is weights on the stimulus and history terms and $g(x)$ is $\frac{1}{1+e^{-x}}$

The authors use Expectation-Maximization to fit the weights, $\omega$, and the probabilities of the latent states, $p$. They then compare model fits that include history terms to fits without them for four different psychophysical tasks.

The main results are: a small gain in likelihood per trial for the full model with history terms, thresholds are a few percent lower if history is included, and the effect of history is strongest for low signal conditions and disappears with high signal strength. Probably the coolest figure is 4c where they demonstrate how history weights reveal different task strategies for the different psychophysical experiments.

The different colors are the four different tasks, each dot is a different subject.  Red and Blue (both visual 2AFC tasks) seem safely in ‘switch’ or ‘win stay – lose switch’. All subjects maintain a ‘stay’ strategy for Yellow (an Auditory Yes-No task).

Does it matter? The authors point out that all though the bias in estimated parameters introduced by inter-trial dependence is systematic, it is small. They suggest that it is particularly important to account for history effects in the analysis of neurophysiology and it’s correlation with choice.

References:

• Fründ I, Wichmann FA, Macke JH. (2014). Quantifying the effect of intertrial dependence on perceptual decisions. Journal of Vision. 14(7). pii: 9. doi: 10.1167/14.7.9.
• Busse, L., Ayaz, A., Dhruv, N. T., Katzner, S., Saleem, A. B., Scholvinck, M. L., et al. (2011). The Detection of Visual Contrast in the Behaving Mouse. Journal of Neuroscience, 31(31), 11351–11361. doi:10.1523/JNEUROSCI.6689-10.2011

# Partitioning Neural Variability

On July 7, we discussed Partitioning Neural Variability by Gorris et al.  In this paper, the authors seek to isolate the portion of the variability of sensory neurons that comes from non-sensory sources such as arousal or attention.  In order to partition the variability in a principled way, the authors propose a “modulated Poisson framework” for spiking neurons, in which a neuron produces spikes according to a Poisson process whose mean rate is the product of a stimulus-driven component $f(S)$ , and a stimulus-independent ‘gain’ term (G).

Applying the law of total variance, the variance for the number of spikes $N$ over a time period $\Delta t$ can then be written as $var[N | S, \Delta t] = f(S) \Delta t + \sigma^2_G (f(S)\Delta t)^2$.  Moreover, with a gamma prior over $G$, the distribution for $N | S, \Delta t$ has a negative binomial distribution, which can be fit to neural data.

The authors analyzed quite a large data set, from neurons in macaque LGN, V1, V2, and MT stimulated with drifting gratings.  Their decomposition of the variance of $N$ (above) motivates the corresponding decomposition of within-condition sum-of-squares into components arising “from the point process” (i.e. corresponding to the $f(S) \Delta t$ term above) and from the gain signal.  Using this decomposition they can estimate of the fraction of the within-condition variance that comes from the fluctuations of $G$.  They find that the modulated Poisson model fits the model better than the Poisson model, and that the gain share of variance is quite high (47.5%, compared to 47% due to stimulus drive and only 5.5% from the Poisson noise), and that it increases along the visual pathway.

The authors also write down an analogous decomposition for the covariance of two neurons into a “point-process” and gain components.  A strength of this form for the covariance is that it allows for spike-count correlations to vary dramatically with stimulus drive.  This decomposition of pairwise tuning correlation into “point-process” and gain components is then used for an analysis of correlations with respect to stimulus similarity, cortical distance, and temporal structure.

# Fast Marginal Likelihood Maximisation for Sparse Bayesian Models

In this week’s lab meeting, we read “Fast Marginal Likelihood Maximisation for Sparse Bayesian Models, by Tipping & Faul (2003). This paper proposes a highly accelerated algorithm for performing maximum marginal likelihood (or “evidence optimization”) in a sparse linear model (also known as the Relevance vector machine).

The basic setup is:

$\quad y = \phi(x)^T w + \epsilon, \quad$  $\epsilon \sim \mathcal{N}(0,\sigma^2)$

and an independent zero-mean Gaussian prior on each element of $w$:

$w_i \sim \mathcal{N}(0,1/\alpha_i)$,

where $\alpha_i$ denotes the prior precision of each weight.

The basic idea is to set the hyperparameters $\{\alpha_i\}$  by maximizing marginal likelihood $P(Y|X,\{\alpha_i\})$. The resulting estimate for $w$ under the prior $P(w | \{\alpha_i\})$ will be sparse because many of the $\alpha_i$ will go to infinity (meaning prior variance for the corresponding weights is set to zero).

The OLD way to optimize this model (“ARD”) is to update hyperparameters $\alpha_i$ and $\sigma^2$ iteratively, but the draw back is the initial $\alpha$ vector would be a huge, and every matrix operation would be time consuming with large scale data. When an $\alpha_i$ reaches infinity, it’s abandoned and $\alpha$ vector has fewer and fewer elements. Thus it’s a reduction process and any abandoned $\alpha$ would not be re-estimated.

In contrast, THIS paper proposed an adding and modifying process. It initializes with an empty model and sequentially adds basis functions (i.e., sets $\alpha_i$‘s to something finite) to increase the marginal likelihood. Within the same framework, it can increase the objective function by deleting functions which subsequently become redundant.

The key to this approach is to separate each $\alpha_i$ from all other hyperparameters, labelled “$\alpha_{-i}$“. The authors analyze $l(\alpha_i)$ to show that $L(\alpha)$ has a unique maximum with respect to $\alpha_i$. Then they update $\alpha_i$ given:
$\alpha_i=\frac{s_i^2}{q_i^2-s_i},\mbox{ if }q_i^2>s_i$
$\alpha_i=\infty,\mbox{ if }q_i^2\le s_i$
where $s_i$ and $q_i$ are statistics needed for computing $\alpha_i$. Let $\theta_i=q_i^2-s_i$, then
If $\theta_i>0$ and $\alpha_i<\infty$, re-estimate $\alpha_i$;
If $\theta_i>0$ and $\alpha_i=\infty$, add $\phi_i$ to the model with updated $\alpha_i$;
If $\theta_i\le 0$ and $\alpha_i<\infty$, delete $\phi_i$ from the model and set $\alpha_i=\infty$;

After updating $\alpha_i$, it recomputes $\Sigma$ and $\mu$ (covariance and mean of posterior of $w$) and all related statistics then continue until convergence.

The paper shows that this algorithm achieves similar MSE performance and finds an even sparser estimate for $w$ in less time than the old method. It’s quite clever to use the separable property of marginal likelihood to optimize the $\alpha_i$ sequentially. It could also be borrowed to solve similar problems with independent $\alpha$ vector.

# Noise correlation in V1: 1D-dynamics explains differences between anesthetized and awake

We discussed state dependence of noise correlations in macaque primary visual cortex [1] today. Noise correlation quantifies the covariability in spike counts between neurons (it’s called noise correlation because the signal (stimulus) drive component has been subtracted out). In a 2010 science paper [2], noise correlation was shown to be much smaller than previously reported; in the range of 0.01 compared to the usual 0.1-0.2 range and stirred up the field (see [3] for a list of values). In this paper, they argue that this difference in noise correlation magnitude is due to population level covariations during anesthesia (they used sufentanil).

They showed that a factor model with 1-dimensional latent process captures the excess noise correlations observed during anesthesia. They extended GPFA [4] to include a linear stimulus drive component, and fit the trials corresponding to the same stimulus. This model assumes that the square-root transformed spike counts in 100 ms bins $y(t)$ follows,

$y(t) \sim \mathcal{N}(f(s(t)) + cx(t) + d, R)$

$x(t) \sim \mathcal{N}(0, K)$

where $s(t)$ is the stimulus, $f(\cdot)$ is the tuning curve drive, $d$ is the mean, $R$ represents the individual (diagonal) covariance, $c$ is the factor loadings vector, and $x(t)$ is the 1-dimensional population wide latent process with assumed smoothness given by the kernel $K(t_1, t_2) = \exp(-(t_1 - t_2)^2 / 2 / \tau^2 )$. The square-root transformation is for the variance stabilization, making Poisson-like spike count observations more like Gaussian distributed.

They show that the most of the extra noise correlation observed during anesthesia is explained by the 1D latent process, and the remaining noise correlation is in par with awake condition, while noise correlations in the awake data didn’t get explained away by it (figure below).

A remarkable result! (red) anesthesia, (blue) awake (solid) raw noise correlation (dashed) residual noise correlation unexplained by the 1D-latent variable

Note that this nonlinear relation between the mean rate and noise correlation is not a natural consequence of the generative model they assumed (equation above), since the noise is additive while firing rate is modulated by stimulus. That’s why they had to fit different factor loadings vector $c$ for different stimulus conditions. However, they show that a multiplicative LNP model where a 1D latent process multiplicatively interacts with the stimulus drive in the log firing rate does reproduce this quality (Fig. 6). (It’s a shame that they didn’t actually fit this more elegant generative model directly to their data.)

Furthermore, they showed that local field potentials (LFP) in 0.5-2Hz range is correlated with the inferred latent variable. Hence, low-pass filtered LFP averaged over electrodes is a good proxy for L4 population correlation under anesthesia.

1D-latent process, LFP, and LNP results are all consistent with Matteo Carandini’s soloist vs chorister story except his preparations were awake rodent as far as I recall. This story also bears similarity with [5-8]. This is certainly an exciting type of model and there are many similar models being developed (e.g. [9, 10] and references therein). I hope to see more developments in these kinds of latent variable models; they have the advantage of scaling well in terms of number of parameters, but the disadvantage of being non-convex optimization. Also, I would like to see similar analysis for awake but spontaneous (non-task performing) state.

Their data and code are all available online (awesome!).

1. Ecker, A. S., Berens, P., Cotton, R. J., Subramaniyan, M., Denfield, G. H., Cadwell, C. R., Smirnakis, S. M., Bethge, M., and Tolias, A. S. (2014). State dependence of noise correlations in macaque primary visual cortexNeuron, 82(1):235-248.
2. Ecker, A. S., Berens, P., Keliris, G. A., Bethge, M., Logothetis, N. K., and Tolias, A. S. (2010). Decorrelated neuronal firing in cortical microcircuitsScience, 327(5965):584-587.
3. Cohen, M. R. and Kohn, A. (2011). Measuring and interpreting neuronal correlationsNat Neurosci, 14(7):811-819.
4. Yu, B. M., Cunningham, J. P., Santhanam, G., Ryu, S. I., Shenoy, K. V., and Sahani, M. (2009). Gaussian-Process factor analysis for Low-Dimensional Single-Trial analysis of neural population activityJournal of Neurophysiology, 102(1):614-635.
5. Luczak, A., Bartho, P., and Harris, K. D. (2013). Gating of sensory input by spontaneous cortical activityThe Journal of Neuroscience, 33(4):1684-1695.
6. Goris, R. L. T., Movshon, J. A., and Simoncelli, E. P. (2014). Partitioning neuronal variabilityNat Neurosci, 17(6):858-865. [last week’s blog post]
7. Tkačik, G., Marre, O., Mora, T., Amodei, D., Berry, M. J., and Bialek, W. (2013). The simplest maximum entropy model for collective behavior in a neural networkJournal of Statistical Mechanics: Theory and Experiment, 2013(03):P03011+.
8. Okun, M., Yger, P., Marguet, S. L., Gerard-Mercier, F., Benucci, A., Katzner, S., Busse, L., Carandini, M., and Harris, K. D. (2012). Population rate dynamics and multineuron firing patterns in sensory cortexThe Journal of neuroscience : the official journal of the Society for Neuroscience, 32(48):17108-17119.
9. Pfau, D., Pnevmatikakis, E. A., and Paninski, L. (2013). Robust learning of low-dimensional dynamics from large neural ensembles. NIPS 26

# Computational Vision Course 2014 @ CSHL

Yesterday marked the start of the 2014 summer course in COMPUTATIONAL NEUROSCIENCE: VISION at Cold Spring Harbor. The course was founded in 1985 by Tony Movshon and Ellen Hildreth, with the goal of inspiring new generations of students to address problems at the intersection of vision, computation, and the brain. The list of past attendees is impressive.

For the next two weeks, students will have two 3-hour lectures per day from a smattering of leading experimental and theoretical neuroscientists (full list here), punctuated with breaks for conversation and recreation (swimming, soccer, frisbee, tennis, ping pong, squirt gun battles, to name a few) on the lush grounds of the Banbury Center at CSHL. Evenings will be devoted to informal discussions, tutorials, and to course projects inspired by lectures and/or interactions with students, TAs, and lecturers.

The course is a ridiculous amount of fun, despite the number of equations. I first attended as a second-year grad student in the summer of 2000 (part of a cohort that included Alex Huk, Anne Churchland, Hannah Bayer, Stephen David, Odelia Schwartz, and Richard Murray), and I returned in 2002 as TA alongside Anne Churchland. It’s an honor to be back this year as a co-organizer (with Geoff Boynton, Greg Horwitz, and Stefan Treue), and I’m joined by Jake Yates and two other students from UT.

Tony Movshon kicked things off yesterday with characteristic Tony levels of charm and wit. Here’s a short sampling of quotes:

• “Matlab is an abomination. But I want you to learn as much as possible so that when you come to my lab, you will know how to get whatever it is you need to get done, done.”
• “Only a theorist could use the terms ‘actual’ and ‘simulated’ simultaneously when referring to data.”
• “The brain works the way it does because it’s made of meat, and meat is not deterministic.”
• “Did I mention how dumb the motor system is?”


text:
# Projecting points on a Pareto-front onto a (hyper)plane?

I'm trying to implement the Biased Crowding Distance in NSGA-II as described in the paper Integrating User Preferences into Evolutionary Multi-Objective Optimization by Branke and Deb.

Basically, from what I understand, given a set of points on a Pareto-front, we need to project the points onto a hyperplane with a direction vector $\eta$. Doing this will give the new crowding distance $d'$ based on the locations of the individualson the hyperplane with direction $\eta$.

To illustrate, see the figure below.

My question is simple: how do we obtain the hyperplane given a user-specified direction vector $\nu$? Is it simply by getting the perpendicular of the direction vector?

And a follow up question would be: how would I project the points onto the hyperplane? If $p$ is the point and $v$ is the vector representing the hyperplane, is it correct that the new location of $p$, call it $p'$ is $\frac{v.x(p.x) + v.y(p.y)}{||v||^2} (v.x, v.y)$ based on this?

Thanks and any information would be very helpful.

I'm not sure if this should be considered a computer science question or not, it seems more an algebra question than anything else.

A vector is a direction vector of a hyperplane if it is parallel to that hyperplane (source).

For any vector $\vec{v} = (v_1, . . . , v_n)^T$ you can find a plane parallel to it by taking the dot product of $v$ and your variable vector $\vec{x} = (x_1, . . . , x_n)^T$ and setting the result equal to some constant $c$.

Consider $\eta$ as described in the paper, then the hyperplane he is referring to will be $$\eta_1 f_1 + . . . + \eta_n f_n = c$$ for a constant $c$. Because you are only interested in the distance between the points after projection, and not the location of the projected points, you can select $c = 0$, as shifting around the hyperplane won't affect the distances between the projected points.

Since the origin is contained in this hyperplane, the hyperplane is a subspace, call it $H$. We can then represent each $p_i$ as a vector $\vec{p_i}$ and project the vector onto the subspace.

To project a vector onto $H$ we need to find a basis for $H$. This can be done by making $n - 1$ similar vectors to $\eta$, where the difference is a single value is set to $0$ (distinct from previous selections). E.g., the plane $3x_1 + 4x_2 + 5x_3 = 0$ has a basis of $$\Bigg\{\begin{bmatrix}3 \\ 4 \\ 0\end{bmatrix}, \begin{bmatrix}3 \\ 0 \\ 5\end{bmatrix}\Bigg\},\Bigg\{\begin{bmatrix}3 \\ 4 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 4 \\ 5\end{bmatrix}\Bigg\}, \text{ or } \Bigg\{\begin{bmatrix}0 \\ 4 \\ 5\end{bmatrix}, \begin{bmatrix}3 \\ 0 \\ 5\end{bmatrix}\Bigg\}$$

Once you have a basis of $H$ you can use this guide guide to perform the projection. Note, since you have to do this multiple times, coming up with $Q$ as described in the guide and saving it will allow you to perform the process on many points.


text:
# Sketching Domains and Images in Complex Analysis

Sketch the domain $S:=\{x+iy:\;x<0,\;\pi/4<y\leq\pi/2\}$ and its image $T$ under the exponential function.

My question is how do I begin to think about this region? I think we can change $z=re^{i\theta}$, where $\pi/4<\theta\leq\pi/2$, I don't even know where this will take me.

-
The Maple commands $$with(plots):conformal(exp(z), z = -10^9+(1/4)*Pi*I .. 0+(1/2)*Pi*I, scaling = constrained)$$ produce a very good approximation of $T$. – user64494 Aug 26 '13 at 4:35

Consider $z = x + iy \in S$, and $T(z)$:

$$T(z) = e^{x + iy} = e^x e^{iy}$$

Now since $y \in (\pi/4, \pi/2]$, this means that the argument of $T(z)$ lies between $\pi/4$ and $\pi/2$, including the second number. Similarly, the modulus is

$$|T(z)| = e^x < 1$$

So the image is the part of the unit disk whose angle lies in $(\pi/4, \pi/2]$.

-
Okay the math makes sense. Now in layman's terms what is the statement asking? – Mr.Fry Aug 26 '13 at 3:19
It's asking: If you "exponentiate a strip," what region do you get? One decent way to get intuition is to pick some points in the strip, and explicitly compute what $T$ does to them. These maps will generally stretch and bend the region, and watching specific points can help you see exactly how that happens. – user61527 Aug 26 '13 at 3:23


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Factors Affecting Rate of Reaction

Table of Content

Nature of the Reactants

Physical State of Reactants

Physical Size of the Reactants

Chemical Nature of Reactants

Concentration of Reactants

Effect of Temperature

Notable Points

Arrhenius Equation

Catalyst

General Characteristics of Catalyst

Presence of a Positive Catalyst

Presence of Negative Catalyst

Related Resources

Nature of the Reactants

(i) Physical State of Reactants

This has considerable effect over rate of reaction.

Decreasing rate of reaction: Gaseous state> Liquid state>Solid state

In a heterogeneous system collision is not so effective as in homogenous system. Thus, reactions in liquid phase or solution phase will be faster in comparison to heterogeneous conditions when same concentrations of the reactants are taken.

(ii) Physical Size of the Reactants

Among the solids, rate increases with decrease in particle size.

In powdered state rate of reaction is maximum because in powdered state, surface area is maximum.

(iii) Chemical Nature of Reactants

Consider the following two reactions:

2NO(g) + O2(g) →   2NO2(g)              ...... (i)

CH2(g) + 2O2(g) →  CO2(g) + 2H2O    ..... (ii)

The first reaction is faster than the second because in the first reaction only N ≡ O bond is to be broken where as in the second reaction four (C-H) bonds are to be broken.

Similarly consider another example of two similar reactions:

2NO(g) + O2(g) → 2NO2(g)                ...... (iii)

2CO(g) + 2O2(g) → 2CO2(g)             ...... (iv)

NO bond is weaker than CO bond, hence broken easily. Thus reaction (iii) is faster than (iv).

Concentration of Reactants

Let us consider the reaction:

A + B → C + D;    Rate = k[A][B]

Rate of the above reaction decreases with the passage of time because the concentration of reactants A and B will decrease as time pass on.

Let us consider the following reversible reaction: A + B ↔ C + D

Rate of forward reaction = kf[A][B]

Rate of backward reaction = kb[C][D]

Rate of forward reaction decreases and that of backward reaction increases with passage of time.

At equilibrium both the rates become equal.

Refer to the following video for factors affecting rate of reaction

Effect of Temperature

The rate of reaction increases considerably with an increase in temperature.

The rates of many reactions are approximately doubled or tripled for every 10oC rise in temperature.

The temperature coefficient of a chemical reaction is defined as the ratio of the specific reaction rates of a reaction at two temperature differing by 10oC.

μ = Temperature coefficient= k(r+10)/kt

Let temperature coefficient of a reaction be ' μ ' when temperature is raised from T1to T2; then the ratio of rate constants or rate may be calculated as

Its value lies generally between 2 and 3.

In order to explain the effect of temperature on the reaction rate. Arrhenius proposed a theory of reaction rate which states as follows :

A chemical reaction takes place by collision between the reactant molecules, and collision to be effective the colliding molecules must posses some certain minimum energy called threshold energy of the reaction.

Reactant molecules having energy equal or greater than the threshold are called active molecules and those having energy less than the threshold are called passive molecules.

At a given temperature there exists a dynamic equilibrium between active and passive molecules. The process of transformation from passive to active molecules being endothermic, increase of temperature increases the number of active molecules and hence the reaction.

Passive molecules  Active molecules, DH = +ve

Concept of energy of activation (Ea)

The extra amount of energy which the reactant molecules (having energy less than the threshold) must acquire so that their mutual collision may lead to the breaking of bond(s) and hence the reaction, is known as energy of activation of the reaction. It is denoted by the symbol Ea. Thus,

Ea = Threshold energy – Actual average energy

Ea is expressed in kcals mole–1 or kJ mole–1.

The essence of Arrhenius Theory of reaction rate is that there exists an energy barrier in the reaction path between reactant(s) and product(s) and for reaction to occur the reactant molecules must climb over the top of the barrier which they do by collision. The existence of energy barrier and concept of Ea can be understood from the following diagram.

?=  Summation of enthalpies of reactants

?=  Summation of enthalpies of products

H = Enthalpy change during the reaction

Ea1 = Energy of activation of the forward reaction (FR)

Ea2  = Energy of activation of the backward reaction (BR)

Notable Points

dH = Energy activation of FR – Energy of activation of BR

In the diagram mentioned above, FR is exothermic (dH  is –ve) while BR is endothermic (dH is +ve). The minimum activation energy of FR i.e. any exothermic reaction will be zero while minimum activation energy for BR i.e. any endothermic reaction will be equal to DH.

Greater the height of energy barrier, greater will be the energy of activation and more slower will be the reaction at a given temperature.

Rate = Collision frequency  fraction of the total number of collision which is effective.

Rate   = Collision frequency  fraction of the total number of collisions in which K.E. of the colliding molecules equals to Ea or exceeds over it.

Collision frequency is the number of collisions per unit volume per unit time. It is denoted by the symbol Z. Z is directly proportional to ÖT. By 10° rise in temperature. So, it is the fraction of the total number of effective, collisions that increases markedly by 10° rise in temperature resulting into marked increase in the reaction rate.

Arrhenius Equation

When the temperature is increased, heat energy is supplied which increases the kinetic energy of the reacting molecules.

This will increase the number of collisions and ultimately the rate of reaction will be enhanced. Arrhenius suggested an equation which describes k as a function of temperature, i.e.,
k = Ae-Ea/RT
where k → rate constant

A → a constant (frequency factor)

Ea → energy of activation

At two temperature T1 and T2, taking log of Arrhenius equation, we get

loge k1 = log A - Ea/RT1  loge e               .... (i)

and loge k2 = log A - Ea/RT2  loge e        .... (ii)

subtracting Eq. (ii) from Eq. (i) and converting the log to the base 10, we get

log10k1/k2 = -Ea/(2.303 R) [(T2-T1)/(T1 T2)]
or

This equation can be used for calculation of energy of activation.

Logarithmic Arrhenius equation is:

log10 k = log10 A–Ea/(2.303 R) [1/T]

Y = C + M X

It is the equation of straight line with negative slope. On plotting log10 k against [1/T] we get a straight line. The graph gives two kinetic parameters.

The slope gives activation energy and intercept gives frequency factor.

Reactions on the basis of influence of temperature are classified into five types.

Catalyst

A catalyst is a substance, which increases the rate of a reaction without itself being consumed at the end of the reaction, and the phenomenon is called catalysis. There are some catalysts which decrease the rate of reaction and such catalysts are called negative catalyst. Obviously, the catalyst accelerating the rate will be positive catalyst. However, the term positive is seldom used and catalyst itself implies positive catalyst.

Catalyst are generally foreign substances but sometimes one of the product formed may act as a catalyst and such catalyst is called “auto catalyst” and the phenomenon is called auto catalysis.

General Characteristics of Catalyst

A catalyst does not initiate the reaction. It simply fastens it.

Only a small amount of catalyst can catalyse the reaction.

A catalyst does not alter the position of equilibrium i.e. magnitude of equilibrium constant and hence dG0. It simply lowers the time needed to attain equilibrium. This means if a reversible reaction in absence of catalyst completes to go to the extent of 75% till attainment of equilibrium, and this state of equilibrium is attained in 20 minutes then in presence of a catalyst also the reaction will go to 75% of completion before the attainment of equilibrium but the time needed for this will be less than 20 minutes.

A catalyst drives the reaction through a different route for which energy barrier is of shortest height and hence Ea is of lower magnitude. That is, the function of the catalyst is to lower down the activation.

Presence of a Positive Catalyst

The function of a positive is to lower down the activation energy.

The greater the decrease in the activation energy caused by the catalyst, higher will be the reaction rate.

In the presence of a catalyst, the reaction follows a path of lower activation energy. Under this condition, a large number of reacting molecules are able to cross over the energy barrier and thus the rate of reaction increases.

Activation energy is lowered in presence of a catalyst.

Presence of Negative Catalyst

A negative catalyst increases the activation energy of reaction by forming a new intermediate of high energy, i.e., by changing the reaction mechanism.

Due to increased activation energy, some active molecules become inactive, therefore, rate of reaction decreases.

Let 'p' denote presence of catalyst and 'a' denote absence of catalyst.

kp = Ae-Ep/RT                .... (i)

ka = Ae-Ea/RT                 ... (ii)

Dividing (1) by (2), we get

kp/ka = e(Ea - Ep )/RT = e?E/RT

kp/ka = Antilog[?E/(2.303 RT)]

Solved Example

Example :  The rate of a reaction triples when temperature changes from 20oC to 50oC. Calculate energy of activation for the reaction (R = 8.314 JK-1 mol-1).

Solution:

The Arrhenius equation is

log10 k2/k1 = Ea/(R×2.303) [(T2-T1)/(T1 T2 )]

Given:  k2/k1 = 3; R=8.314 JK-1 mol–1; T1 = 20 + 273 = 293 K

and   T2 = 50 + 273 = 323 K

Subtracting the given values in the Arrhenius equation,

log10 3 = Ea/(8.314×2.303) [(323-293)/(323×293)]

Ea = (2.303 × 8.314×323 × 293×0.477)/30

= 28811.8 J mol-1

= 28.8118 kJ mol-1

Example :

In Arrhenius equation for a certain reaction, the value of A and Ea(activation energy) are 5×1013 sec-1 and 98.6 kJ mol-1 respectively. At what temperature, the reaction will have specific rate constant 1.1 × 10-3 sec-1 ?

Solution :

According to Arrhenius equation

k =  Ae-Ea/RT

or    loge k = loge A - Ea/RT loge e

or    2.303 log10 k = 2.303 log10 A - Ea/RT

or    2.303 log (1.1×10-3)

= 2.303 log(4×1013)-(98.6×103)/(8.314×T)

T = (98.6×103)/(8.314×2.303×16.56) K

= 310.96 K

Example :

The rate constant is given by Arrhenius equation

k = Ae-Ea/RT

Calculate the ratio of the catalyst and uncatalysed rate constants at 25oC if the energy of activation of a catalysed reaction is 162 kJ and for the uncatalysed reaction the value is 350 kJ.

Solution :

Let kca and kun be the rate constants for catalysed and uncatalysed reactions.

2.303 log10 kca = 2.303 log10 A -  (162×103)/RT     .... (i)

and   2.303 log10 kun = 2.303 log10 A - (350 × 103)/RT    .... (ii)

Subtracting Eq. (ii) from Eq. (i)

log10 kca/kun = 103/(2.303 RT) (350-162)

= (188× 103)/(2.303×8..314×298) = 32.95

kca/kun = 8.88 × 1032

Question 1:

Which of the following factors does not affect rate of a reaction ?

a. Temperature

b. Catalyst

c. Concentration of reactants

d. Nature of Product

Question 2:

Among the solids, rate increases with decrease in particle size.

a. pressure

b. particle size

c. concentration

d. mass of particle

Question 3:

A negative catalyst increases

a. rate of reaction

b. activation energy

c. amount of product formed

d. rate of backward reaction

Question 4:

The slope of plot  log10 k vs  [1/T] gives activation energy and intercept gives frequency factor

a.frequency factor

b. rate constant

c.rate f reaction

d.temperature coefficient

Q.1
Q.2
Q.3
Q.4

d
b
b
a

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# Shannon Entropy and Euler Characteristic

Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related.

#### Introduction

This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an other direction and compare random variables with simplicial complexes, Shannon entropy with Euler characteristic.
Random variables and simplicial complexes seem to be unrelated as they appear in completely different fields. But they become siblings, if we look at them together. A random variable is defined over a probability space and one can assign interesting quantities to random variables and entropy is one of them. Euler characteristic is an interesting quantity in topology. It is a number one can assign to simplicial complexes.

 The entropy functional is the most important quantity in probability theory. The Euler characteristic functional is the most important quantity in topology.

Despite the fact that these two functionals appear utterly different at first, there are many parallels, both can be identified by some additive and multiplicative conditions as well as a gauge with a fixed element. Alternatively, one can see each of them as a unique fixed point of a renormalizations map, one for random variables, and one for simplicial complexes. The two quantities are then associated with central limit theorems.

#### Some history

Here are four pioneers. Boltzmann and Shannon for entropy and Euler and Poincaré for characteristic.

 Leonhard Euler (1707 – 1783) Ludwig Boltzmann (1844 – 1906) Henry Poincaré (1884 – 1912) Claude Shannon (1916 – 2001)

Seriously guys! You four need to lighten up!

Boltzmann introduced entropy in 1872 in his kinetic theory of gases. The concept has been around earlier, like in 1850 when Rudolf Clausius stumbled upon it. Euler observed Euler characteristic in 1751. Lets first look at Euler characteristic.

There is always the historical question about first appearance. Already Descartes saw experimentally in 1639 that the Euler formula v-e+f=2 holds for a few Platonic solids. This is an experimental verification and is a first step. Proving a theorem is a completely different matter. [This is paralleled for almost every theorem. For example, Pythagoras was first experimentally seen on Clay tablets, but these are special cases. Seeing a special case is not the same than proving a general theorem.]

And then there is the generality. Euler characteristic has for a long time been a murky notion even in higher dimensions. Casual definitions do not state clearly what the k-dimensional faces are. Now for a Platonic solid like a cube we can by inspection (or Euclidean realization as a brick) see what the faces are but this is not clear for general networks, even when restricting to polyhedra. But the notion of Polyhedron has probably the worst track record in the entire realm of mathematics. Unclear definitions often lead to wrong theorems. In the case of Euler characteristic, it was skillfully explained by Lacatos in his dialogs “Proof and refutation”. The story is also well told in “Euler’s gem”. Branko Gruenbaum also outlined the difficulty well at various places. Gruenbaum wisely restricts his book on polytopes to convex ones, where the ambiguities disappear (a finitist not accepting Euclidean space would today maybe reformulate things using discrete CW complexes). The confusion starts much earlier already in one dimension, with the concept of polygon in school mathematics. (The Wikipedia article does a pretty good job with explaining the difficulties in that case, but less so in the case of polyhyedra). The problem with polyhedra and polytopes is that many different notions exist in the literature. The notion of finite abstract simplicial complex elegantly cuts through the historically grown ambiguities and additionally builds the possibly simplest mathematical construct imaginable.

Fortunately, for finite simplicial complexes, one has not only one of the simplest mathematical structures we know (there is only one axiom!) but also one of the most elegant definitions of the Euler characteristic functional: just super count the simplices. We will see why super counting is much better than counting itself: it will lead to a scale invariant notion and so go over to the continuum limit. The elegance of abstract simplicial complexes is similar as the elegance and beauty of set theoretical topology which completely avoids all the historical balast and confusions about the concept of limit for example. The definition of continuity as the property that the preimage of an open set is an open set not only avoids limits (and is so really in a “calculus without limits = quantum calculus) spirit, it also makes perfectly sense for a strict finitist who believes the concept of infinitiy is a doomsday notion. Indeed, one can look at finite topological spaces, for example on simplicial complexes.

Now lets turn to entropy. Probability theory at the time of Boltzmann was in a terrible state of Woodoo. Only Kolmogorov cleared up the mess by placing the theory firmly on measure theory and giving clear definitions of what random variables or stochastic processes are. So, it is not surprising that the notion of entropy which Boltzmann considered was also not very clear. Shannon defined entropy in a discrete setting in 1948. Today, entropy is defined for discrete probability measures (for which no zero probabilities are possible) as well as absolutely continuous random variables for which the function f(x) log(f(x)) is integrable. This is also called differential entropy. For random variables which have a singular continuous law (the generic case), one has to go through a limiting process but this is always a bit unsatisfactory as we need a definition which is independent of the approximation.

Shannon dealt only with discrete distributions and was so mathematically on safe grounds. Many other notions of entropy have been introduced since, in particular when dealing with dynamical systems, the theories of time evolutions. The topological entropy (Adler, Konheim and McAndrew) and the metric entropy (Kolmogorov,Sinai) are examples. The mathematical frame work and language is surprisingly close to the frame work and language seen in statistical mechanics even so these notions are defined on the category of topological or measure theoretical dynamical systems and not just notions of a random variable, as Boltzmann or Shannon did.

#### The objects

 A finite random variable is a random variable taking finitely many values. The data are completely determined by a probability measure p on a finite set V. A finite simplicial complex is a finite set V equipped with a set of non-empty subsets which is invariant under the operation of taking finite non-empty subsets.

Definitions of entropy were first given by Boltzmann, and in a modern context by Shannon. Euler characteristic was first experimentally defined by Descartes. For two dimensional sphere complexes, the formula v-e+f=2 was proven by Euler. The higher dimensional notions were then developed by Schläfli, Poincaré and later to finite abstract simplicial complexes.

 The entropy functional for a random variable X is defined with P[X=x]=px as H(X) = – ∑x px log(px). where the sum is over all points x for which the value of the probability px is not zero. The Euler characteristic functional for a simplicial complex X is defined as χ(G) = ∑x (-1)dim(x) where the sum ranges over simplices x of the complex and where dim(x)=|x|-1 with |x|=cardinality.

Boltzmann was interested in entropy in a statistical mechanics frame work. He defined entropy as the expectation of the logarithm of Wahrscheinlichkeit W=1/P. Shannon was interested in entropy in an information theory frame work. It turns out that we can assign entropy to data. It can be measured by data compression. High entropy data can not be compressed well, while low entropy data can.

The origins of Euler characteristic started with the study of platonic solids, then more general polyhedra, then polytopes, graphs and finally was defined for simplicial complexes. Low dimensional notions are still entrenched widely even among mathematicians. Historically for example, there has been huge interest in one dimensional objects (curves, Riemann surfaces = 1 dimensional complex curves), where the Euler-Poincare relations express Euler characteristic as b0-b1 with two Betti numbers. For a connected curve this is 1-g, where g is the genus. So, for one dimensional connected objects, the genus $1-\chi(G)=g$ plays an important role. It is the number of holes in a Riemann surface. It is no wonder therefore that also in full generality (when not restricting to the one dimensional world), the number $1-\chi(G)$ plays an important role. It appeared for example as diagonal Green function elements in the Super Green story which is still in the works.

#### Question

If one has a functional, it is natural to ask whether it can be singled out from basic principles. This is indeed possible, both for entropy as well as for Euler characteristic. For entropy, this was done already by Shannon. For Euler characteristic, the characterization has been known to combinatorial topologists since about the same time. In both cases we deal with finite structures, finite probability spaces on one side and finite complexes on the other side. To make the comparison more compelling, lets look at the exp entropy ψ(X) = exp(H(X)).

#### The Cartesian product

The defining property deals in both cases with products. In the case of probability spaces we deal with random variable taking finitely many values. A simplicial complex is a finite set V called base equipped with a set of non-empty subsets with the property that it is invariant under the process of taking non-empty subset.

 The Cartesian product of two random variables is obtained by realizing them as independent. This means that we look at the probability space on which the points are the possible pairs of values (x,y) = (X(w),Y(w)) and where p(x,y) = p(x) p(y). The Cartesian product of finite abstract simplicial complexes X,Y is defined as a complex on the Cartesian product of the sets V,W of X,Y. First define the graph on X x Y, where two pairs are connected if one contains the other. Then take the Whitney complex of that graph, the set of cliques in the graph.

 The exponentiated entropy of a product is the product of the entropies of the factors. The characteristic of a product is the product of the characteristics of the factors.

If X is a random variable, then E[f(X)] is the expectation of f(X). In a finite frame work this is $\sum_x f(x) P[X=x]$. For example, E[X] = ∑x x P[X=x] is the expectation of X itself.

If G is a simplicial complex, then v(G) is its f-vector. It is a finite vector with vk(G) counting the number of k-dimensional simplices in G. For example, v0 is the number of vertices, zero dimensional simplices in G, v1 is the number of edges, one-dimensional simplices in G etc. The number of facets, the number of highest dimensional simplices in a complex is called its volume.

 A functional φ on random variables is called additive if it is of the form E[f[X]] for some function f. A functional φ on complexes is called additive if it is of the form F.v(G) for some vector F.

The valuation property is φ(X+Y) = φ(X) + φ(Y) in the sense that φ(X ∪ Y) + φ (X ∩ Y) = φ(X) + φ(Y). Note that the symmetric difference is not a simplicial complex as simplicial complexes can not be added. (They are a subset of the linear space of all chains defined by the complex.)
This parallels that probability measures do not form a linear space but are part of a larger linear space, the space of signed measures.

#### More notation

We say a random variable is non-trivial if it is not constant. We say a simplicial complex is non-trivial if it is not the empty complex.

We say a functional on random variables is non-trivial if it is not constant 0. We say a functional on simplicial complexes is non-trivial if it is not constant 0.

We say, a functional on random variables is normalized if it is 1 on the constant random variables. We say a functional on simplicial complexes is normalized if it is 1 on the one-point simplicial complex.

We say a functional on random variables is multiplicative if it is multiplicative with respect to the Cartesian product of random variables (pairs of IID random variables). We say a functional on simplicial complexes is multiplicative if it is multiplicative with respect to the Cartesian product of simplicial complexes.

#### Characterization of entropy and characteristic.

 The exponential of Shannon entropy is an example of a normalized, non-trivial, additive and multiplicative functional on random variables. The Euler characteristic is an example of a normalized, non-trivial, additive and multiplicative functional on simplicial complexes.

More interesting is that these functionals are unique:

 Shannon Theorem (1948). The exp entropy ψ is the only non-trivial additive and multiplicative normalized functional on random variables. Mayer Theorem (1942). The Euler characteristic χ is the only non-trivial additive and multiplicative normalized functional on simplicial complexes.

The Shannon theorem is essentially Theorem 2 in his 1948 article. The earliest reference for the result on Euler characteristic is W. Mayer, W, “A new homology theory. {I}, {II}, “Ann. of Math. (2),43, 1942. Of course, the notions in the last 70 years have considerably changed.

 Here is a proof sketch of Shannon’s theorem: the product property implies f(t)= k log(t). The normalization implies k=1. Here is a proof sketch of Mayer’s theorem: by the discrete Hadwiger theorem, a valuation is of the form where v(G) is the f-vector of G. The product of G with 1 is the Barycentric refinement. Since v(G_1) = A v(G) for an explicit operator A which has only one non-zero eigenvector (1,-1,1,-1 …), the Euler characteristic is unique.

#### Measures

 A random variable X defines a law μ defined as the derivative μ = F’ of the conditional expectation function F(t) = P[X ≤ t] A finite abstract simplicial complex G has as the law the density of states, the derivative of the integrated density of states of the (Hodge) Laplacian of G.

While probability spaces can look quite complicated and be large, the law is a measure on the real line. The entropy is completely determined by the law. The Euler characteristic is in some sense also determined by the law but only if we look at operators on discrete forms. Euler characteristic is the super trace of 1 and equivalently by McKean Singer the super trace of exp(-t L), where L is the Hodge Laplacian.

#### Renormalization maps

 Given a random variable X, define a new variable X1 = X+X’ normalized so that the variance is 1. Given a simplicial complex X, define a new simplicial complex X1 as the Barycentric refinement of X.

Both renormalizations can be iterated. The entropy increases under the renormalization map on variables and Euler characteristic is invariant under the Barycentric refinement map.

#### Central limit theorems

There is also a parallel on universality:

 When iterating the renormalisation map on random variables, it converges to a universal fixed point, the Gaussian random variable. When iterating the Barycentric renormalization map, the law converges to a universal fixed point, which depends only on the dimension of the complex.

The renormalization story changes if in the target space of the random variable, the non-compact real line is replaced by a compact abelian Lie group. Then the limiting measure (of course) is the Haar measure. The proof is the same as in the central limit theorem case, as the story trivializes after taking the Fourier transform. I have a small chapter in this text. Statisticians are interested in circular data.

For the Barycentric renormaliation story, see this paper (and a earlier announcement).

#### Complexity

Both the concept of entropy as well as the concept of Euler characteristic is related to fundamental enigmas. The first one is the question of “time”. Why is there an arrow of time? The second is related to the NP complexity problem as it is close to an NP complete problem.

 Is entropy preserved under fundamental physical phenomena? Can we compute the Euler characteristic effectively?

The difficulty with entropy is fundamental as long as we do not know what physical laws are approximations and which physical laws are fundamental. The second law of thermodynamics for example is not a fundamental law which can be proven, but a consequence of too large time scales, properties of matter and time preventing Maxwell deamons.

For Euler characteristic the fundamental difficulty is that we know that counting simplices in graphs is difficult. For graphs of small dimension, there are Morse theoretical fast computations of Euler characteristic through Poincare-Hopf theorems but these computations become hard if graphs have high dimension.

#### Confusions

All known basic fundamental physical theories (classical mechanics, quantum mechanics relativity) are reversible and so preserve entropy, there is an obvious clash between the scales. Poincaré recurrence for even very small systems would take too long to be observed in the time scales accessible to us. There is an arrow of time due to the fact that we can not keep track of all information (impossibility of the Laplace daemon). There are limitations to matter preventing a Maxwell daemons. All this is hardly of any mystery any more for mathematicians who know about dynamical systems but physical models having to deal with reasonable time scales and experiments have led often to confusions. The clash is so strong that some researchers like Prigogine have proposed to look at irreversible fundamental laws of nature.

Similarly, there have been confusions about Euler characteristic. The story was best told by Lacatos in the book “proofs and refutations”. The story is amusing since during the development of understanding Euler characteristic, there were theorems which proved to be wrong. Again and again. We know now that similarly as with entropy this came from murky and unprecise definitions. What is nice about finite probability spaces or finite abstract simplicial complexes that everything is crystal clear, pure mathematics and no ideological or cultural ballast needs to be carried around. Already physics has much more trouble with that.

 Is there an “arrow of time”? Is entropy fundamentally preserved? Are there different “complexity classes”? (Church thesis)

The question of entropy occupies also large scale physisists who look at the mathematics of singularities in Pseudo Riemannian geometry. The question of computational complexity is murky too. In principle, quantum computing could smash through complexity barriers and allow us for example to compute Euler characteristic for very large networks.

[Side remark: while at UT Austin, I visited some famous seminars (mostly out of curiosity). One was the seminar of Weinberg,and an other by Prigogine. The Weinberg seminars were regular and went a couple of times just to see what is happening. They took place in regular class rooms. The Prigogine talk was completely different and resembled more like a religious service. The room was fancy, there was tea served and the speaker was lecturing like a pastor before a devoted public. He had just published his book “The end of Certainty”. It was not a book reading event however.]

Anyway, the book illustrates that confusions are great if one does not stay within pure mathematics.

But mathematicians can also leave the realm of mathematics and speculate. It just has to be stated as such. Speculation is very cheap as one could just throw out random conjectures, random models. By probability, one will be right for some.

I personally have opinions about the above “big questions” but it hardly matters what one believes: if needed to guess I would state that entropy is fundamentally preserved by physical processes but that one has to remember that as there are mathematical facts which are un-observable phenomena (winning at the Petersburg casino for example with 100 dollars entrance fee is never observed even so it is a mathematical fact that we will win eventually as the winning expectation ∑n P[win 2n dollars] 2n = ∑n 2-n 2n is infinity). An other example is Poincaré recurrence in a macroscopic box with a few hundred balls. We never, ever will be able to see the recurrence as the probability is just too low. But Poincaré recurrence is one of the simplest mathematical theorems in ergodic theory and only uses the pigeon hole principle: given an automorphism T of a probability space X and a set A of positive area, then it is not possible that all iterates Tn(A) are disjoint. The problem with Prigogine is that as a chemist and taking experiments seriously as one should, he concludes from the fact that we don’t observe any recurrence, conclude that on a fundamental level the physical laws have to be changed. But even if Prigogine would be right, he would be right in the same way than Democritus or Lucretius were right about atomic theory, just by pure luck. There was no reason at the time of Lucretius to believe in atoms. There is no reason to assume on a fundamental level that due to an observed arrow of time, we need to change the fundamental laws. There are many mathematical theories (Martingale theory in probability theory is one, non-equilibrium statistical mechanics, turbulence theory within partial differential equations or then ergodic theory within dynamical systems theory), which allow very well to deal with the arrow of time and explain it within a frame work. Perfectly and by simple means like for example that typically in an infinite dimensional Hamiltonian system, the system will tend to go up in the frequency range. When looking at a Vlasov gas for example which is a perfectly reversible infinite dimensional Hamiltonian system, the density of the gas will tend to converge, a piston within the system will settle down as we observe. here is a 1 page handout for a class from 2005 or this article with Evan Reed, explaining this a bit.

 De Rerum Natura, by Lucretius The end of Certainty, by Prigogine

On the side of possible quantum complexity classes, I would tend to believe that the Church thesis will remain valid and that similarly as in classical mechanics, the limitations of chaos and quantum mechanics will kick in when working with larger number of qubits. Already classically, we not say for example what the one millionth iterate of the Ulam map T(x)=4x(1-x) is when we start with $\sqrt{2}+\pi/10$ is. In order to answer this numerically, we would have to be able to process numbers with $2^{1000000}$ digits. In order to do the computation analytically, we would have to store expressions of this size. Both are impossible because there are not enough elementary particles in the universe to write the objects down. In the case of quantum computing, we have not seen these barriers yet, but experimentally observe entanglement problems which look like rounding error problems in classical computing. Any way, it would be very surprising if P=NP was true (even on a quantum level). When pressed to answer whether we ever will be able to compute Euler characteristic polynomially fast, then my guess is yes, maybe because it is so special. We know already that it can be computed polynomially fast on large parts of Erdoes-Renyi probability spaces of graphs but it could also be that it is fundamentally equivalent to the clique counting problem and NP complete. But we don’t know that yet. The question of whether the computation of Euler characteristic is NP complete is an example of a good scientific question because there is a chance that we can answer it within a fixed computational frame work (like Turing machines or quantum computers). The question about whether on a fundamental level entropy is preserved is also a good question within a fixed theory (like classical mechanics or general relativity).

#### Variational problems

 Maximizing entropy explains some distributions like Gaussian, geometric or uniform distribution. Extremizing Euler characteristic is a combinatorial variational problem on finite simplicial complexes.

It has been known since a long time (Gibbs?) that the Gaussian distribution maximizes the entropy among all distributions on the real line, that geometric distribution maximizes the entropy among all distributions on the half line that the uniform distribution maximizes the entropy among all distributions on an interval. For finite probability spaces, the entropy maximizing problem is boring as the uniform distribution again is maximal. The variational problem becomes there more interesting when adding an energy functional. Maximizing free energy = entropy plus free energy, then leads to the Gibbs distribution.

The problem of extremizing Euler characteristic is largely unstudied. I have looked at it a few times, like here or on even dimensional graphs. The problem is very interesting on geometric 4 dimensional graphs, as we can write the Euler characteristic as a sum of local Euler curvature quantities which are averages over Poincaré-Hopf indices, which are again Euler characteristics of two-dimensional random surfaces and so again a sum of two dimensional curvatures. In some sense therefore, it is an average over all possible sectional curvatures at a point. In the continuum, this is the scalar curvature, when integrated is the Hilbert action producing the Einstein equations in the continuum. So, in some sense we can look at Euler characteristic as a quantized Hilbert action. It is called quantized because it takes integer values only.

 Apollo, son of Zeus and Leto, god of music, Image source Artemis, daughter of Zeus and Leto, goddess of nature Image source.