Category Archives: Math

How to think about two tricky theorems and other matters

I’m continuing to plow through classic differential geometry en route to studying manifolds en route to understanding relativity. The following thoughts might help someone just starting out.

Derivatives of one variable functions are fairly easy to understand. Plot y = f(x) and measure the slope of the curve. That’s the derivative.

So why do you need a matrix to find the derivative for more than one variable? Imagine standing on the side of a mountain. The slope depends on the direction you look. So something giving you the slope(s) of a mountain just has to be more complicated. It must be something that operates on the direction you’re looking (e.g. a vector).

Another point to remember about derivatives is that they basically take something that looks bumpy (like a mountain), look very closely at it under a magnifying glass and flatten it out (e.g. linearize it). Anything linear comes under the rubric of linear algebra — about which I wrote a lot, because it underlies quantum mechanics — for details see the 9 articles I wrote about it in — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.

Any linear transformation of a vector (of which the direction of your gaze on the side of a mountain is but one) can be represented by a matrix of numbers, which is why to find a slope in the direction of a vector it must be multiplied by a matrix (the Jacobian if you want to get technical).

Now on to the two tricky theorems — the Inverse Function Theorem and the Implicit Function theorem. I’ve been plowing through a variety of books on differential geometry (Banchoff & Lovett, McInenery, DoCarmo, Kreyszig, Thorpe) and they all refer you for proofs of both to an analysis book. They are absolutely crucial to differential geometry, so it’s surprising that none of these books prove them. They all involve linear transformations (because derivatives are linear) from an arbitrary real vector space R^n — elements are ordered n-tuples of real numbers to to another real vector space R^m. So they must inherently involve matrices, which quickly gets rather technical.

To keep your eye on the ball let’s go back to y = f(x). Y and x are real numbers. They have the lovely property that between any two real numbers there lies another, and between those two another and another. So there is no smallest real number greater than 0. If there is a point x at which the derivative isn’t zero but some positive number a to keep it simple (but a negative number would work as well), then y is increasing at x. If the derivative is continuous at a (which it usually is) then there is a delta greater than zero such that the derivative is greater than zero in the open interval (x – delta, x + delta). This means that y = f(x) is always increasing over that interval. This means that there is a one to one function y = g(x) defined over the same interval. This is called an inverse function.

Now you’re ready for the inverse function theorem — but the conditions are the same — the derivative at a point should be greater than zero and continuous at that point — and an inverse function exists. The trickiness and the mountains of notation come from the fact that the function is from R^n to R^m where n and m are any positive integers.

It’s important to know that, although the inverse and implicit functions are shown logically to exist, almost never can they be written down explicitly. The implicit function theorem follows from the inverse function theorem with even more notation involved, but this is the basic idea behind them.

A few other points on differential geometry. Much of it involves surfaces, and they are defined 3 ways. The easiest way to understand two of them takes you back to the side of a mountain. Now you’re standing on it half way up and wondering which would be the best way to get to the top. So you whip out your topographic map which has lines of constant elevation on it. This brings to the first way to define a surface. Assume the mountain is given by the function z = f (x, y) — every point on the earth has a height above it where the land stops and the sky beings (z) — so the function is a parameterization of the surface. Another way to define a surface in space is by level sets: put z equal to some height — call it z’ and define the surface as the set of two dimensional points (x, y) such that f (x, y ) = z’. These are the lines of constant elevation (e.g. the contour lines) – on the mountain. Differential geometry takes a broad view of surfaces — yes a curve on f (x, y) is considered a surface, just as a surface of constant temperature around the sun is a level set on f(x,y,z). The third way to define a surface is by f (x1, x2, …, xn) = 0. This is where the implicit function theorem comes in if some variables are in fact functions of others.

Well, I hope this helps when you plunge into the actual details.

For the record — the best derivation of these theorems are in Apostol Mathematical Analysis 1957 third printing pp. 138 – 148. The development is leisurely and quite clear. I bought the book in 1960 for $10.50. The second edition came out in ’74 — you can now buy it for 76.00 from Amazon — proving you should never throw out your old math books.

An old year’s resolution

One of the things I thought I was going to do in 2012 was learn about relativity.   For why see http://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.  Also my cousin’s new husband wrote a paper on a new way of looking at it.  I’ve been putting him off as I thought I should know the old way first.

I knew that general relativity involved lots of math such as manifolds and the curvature of space-time.  So rather than read verbal explanations, I thought I’d learn the math first.  I started reading John M. Lee’s two books on manifolds.  The first involves topological manifolds, the second involves manifolds with extra structure (smoothness) permitting calculus to be done on them.  Distance is not a topological concept, but is absolutely required for calculus — that’s what the smoothness is about.

I started with “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee.  I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them. Eventually I got through a third of its 380 pages of text.  I thought that might be enough to help me read his second book “Introduction to Smooth Manifolds” but I only got through 100 of its 600 pages before I could see that I really needed to go back and completely go through the first book.

This seemed endless, and would probably take 2 more years.  This shouldn’t be taken as a criticism of Lee — his writing is clear as a bell.  One of the few criticisms of his books is that they are so clear, you think you understand what you are reading when you don’t.

So what to do?  A prof at one of the local colleges, James J. Callahan, wrote a book called “The Geometry of Spacetime” which concerns special and general relativity.  I asked if I could audit the course on it he’d been teaching there for decades.  Unfortunately he said “been there, done that” and had no plans ever to teach the course again.

Well, for the last month or so, I’ve been going through his book.  It’s excellent, with lots of diagrams and pictures, and wide margins for taking notes.  A symbol table would have been helpful, as would answers to the excellent (and fairly difficult) problems.

This also explains why there have been no posts in the past month.

The good news is that the only math you need for special relativity is calculus and linear algebra.  Really nothing more.  No manifolds.  At the end of the first third of the book (about 145 pages) you will have a clear understanding of

l. time dilation — why time slows down for moving objects

2. length contraction — why moving objects shrink

3. why two observers looking at the same event can see it happening at different times.

4. the Michelson Morley experiment — but the explanation of it in the Feynman lectures on physics 15-3, 15-4 is much better

5. The Kludge Lorentz used to make Maxwell’s equations obey the Galilean principle of relativity (e.g. Newton’s first law)

6. How Einstein derived Lorentz’s kludge purely by assuming the velocity of light was constant for all observers, never mind how they were moving relative to each other.  Reading how he did it, is like watching a master sculptor at work.

Well, I’ll never get through the rest of Callahan by the end of 2012, but I can see doing it in a few more months.  You could conceivably learn linear algebra by reading his book, but it would be tough.  I’ve written some fairly simplistic background linear algebra for quantum mechanics posts — you might find them useful. https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/

One of the nicest things was seeing clearly what it means for different matrices to represent the same transformation, and why you should care.  I’d seen this many times in linear algebra, but seeing how simple reflection through an arbitrary line through the origin can be when you (1) rotate the line to the x axis by tan(y/x) radians (2) change the y coordinate to – y  – by an incredibly simple matrix  (3) rotate it back to the original angle .

That’s why any two n x n matrices X and Y represent the same linear transformation if they are related by the invertible matrix Z in the following way  X = Z^-1 * Y * Z

Merry Christmas and Happy New Year (none of that Happy Holidays crap for me)

The New Clayden pp. 1029 – 1068

p. 1034 — “Small amounts of radicals are formed in many reactions in which the products are actually formed by simple ionic processes.”  Interesting — how ‘small’ is small?  

p. 1036 — A very improbable mechanism (but true) given in the last reaction involving breaking benzene aromaticity and forming a cyclopropene ring to boot.  

p. 1043 — Americans should note that gradient (as in Hammett’s rho constant) means slope (or derivative if the plot of substituents vs. sigma for a particular reaction isn’t a straight line).  However we are talking log vs. log plots, and you can fit an elephant onto a log log plot.  It’s worth remembering why logarithms are necessary iin the first place.  Much of interest to chemists (equilibrium constants, reaction rates) are exponential in free energy (of products vs. reactants in the first case, of transition state vs. reactions in the second).

p. 1044 — Optimally I shouldn’t have to remember that a positve rho (for reaction value) means electrons flow toward the aromatic ring in the rate determining step), but should gut it out from the electron withdrawing or pushing effects on the transition state, and how this affects sigma, by remembering what equilibrium constant is over what for sigma, and rho), but this implies a very high working memory capacity (which I don’t have unfortunately).  I think mathematicians do, which is why I’m so slow at it.  They have to keep all sorts of definitions in working memory at once to come up with proofs (and I do to follow them).  

If you don’t know what working memory is, here’s a link – http://en.wikipedia.org/wiki/Working_memory.  

Here are a few literature references 

        [ Proc. Natl. Acad. Sci. vol. 106 pp. 21017 - 21018 '09 ] This one is particularly interesting to me as it states that differences among people in working memory capacity are thought to reflect a core cognitive ability, because they strongly predict performance in fluid inteliigenece, reading, attentional control etc. etc.  This may explain why you have to have a certain sort of smarts to be a mathematician (the sort that helps you on IQ tests).  

       [ Science vol. 323 pp. 800 - 802 '09 ] Intensive training on working memory tasks can improve working memory capacity, and reduce cognitively related clinical symptoms.  The improvements have been associated with an increase in brain activity in parietal and frontal regions. 

I think there are some websites which will train working memory (and claim to improve it).  I may give them a shot. 

Unrelated to this chapter, but Science vol. 337 pp. 1648 – 1651 ’12, but worth bringing to the attention of the cognoscenti reading this –as there is some fascinating looking organometallic chemistry in it.  This is a totally new development since the early 60′s and I look forward to reading the next chapter on Organometallic chemistry.   Hopefully orbitals and stereochemistry will be involved there, as they are in this paper.  Fig 1 C has A uranium atom bound to 3 oxygens and 3 nitrogens, and also by dotted bonds to H and C.

p. 1050 — The unspoken assumption about the kinetic isotope effect is that the C-D and C-H bonds have the same strength (since the curve of potential energy vs. atomic separation is the same for both — this is probably true — but why?    Also, there is no explanation of why the maximum kinetic isotope effect is 7.1.  So I thought I’d look and see what the current Bible of physical organic chemistry had to say about it. 

Anslyn and Dougherty (p. 422 –> ) leave the calculation of the maximum isotope effect (at 298 Kelvin) as an exercise.  They also assume that the force constant is the same.  Exercise 1 (p. 482) says one equation used to calculate kinetic isotope effects is given below — you are asked to derive it 

kH/kD = exp [ hc (vbarH – vbarD)/2KT }, and then in problem #2 plug in a stretching frequency for C-H of 3000 cm^-1 to calculate the isotope effect at 298 Kelvin coming up with 6.5

Far from satisfying.  I doubt that the average organic chemist reading Anslyn and Dougherty could solve it.  Perhaps I could have  done it back in ’61 when I had the incredible experience of auditing E. B. Wilson’s course on Statistical Mechanics while waiting to go to med school (yes he’s the Wilson of Pauling and Wilson).   More about him when I start reading Molecular Driving Forces. 

On another level, it’s rather surprising that mass should have such an effect on reaction rates.  Bonds are about the distribution of charge, and the force between charged particles is 10^36 times stronger than that between particles of the same mass. 

p. 1052 — Entropy is a subtle concept (particularly in bulk thermodynamics), but easy enough to measure there.    Organic chemists have a very intuitive concept of it as shown here.

p. 1054 — Very slick explanation of the inverse isotope effect.  

Again out of context — but more chemistry seems to be appearing in Nature and Science these days.   A carbon coordinated to 6 iron atoms ( yes six ! ! ! ) exists in an enzyme essential for life itself — the plant enzyme nitrogenase which reduces N2 to usable ammonia equivalents for formation of amino acids, nucleotides.   No mention seems to be made about just how unusual this is.  See Science vol. 337 pp. 1672 – 1675 ’12. 

p. 1061 — The trapping of the benzyne intermediate by a Diels Alder is clever and exactly what I was trying to do years ago in a failed PhD project — see https://luysii.wordpress.com/2012/10/04/carbenes-and-a-defense-of-pre-meds-and-docs/

p. 1064 — In the mechanism of attack on epichlorohydrin, the reason for the preference of attack on the epoxide isn’t given — it’s probably both steric and kinetic, steric because attack on the ring is less hindered — the H’s are splayed out, and kinetic, because anything opening up a strained ring should have a lower energy transition state. 

The New Clayden pp. 931 – 969

p. 935 — I don’t understand why neighboring group participation is less common using 4 membered rings than it is using  3 and 5 membered rings.  It may be entropy and the enthalpy of strain balancing out.  I think they’ve said this elsewhere (or in the previous edition).   Actually — looking at the side bar, they did say exactly that in Ch. 31.  

As we used to say, when scooped in the literature — at least we were thinking well.

p. 935 — “During the 1950′s and 1960′s, this sort of question provoked a prolonged and acrimonious debate”  – you better believe it.  Schleyer worked on norbornane, but I don’t think he got into the dust up.  Sol Winstein (who Schleyer called solvolysis Sol) was one of the participants along with H. C. Brown (HydroBoration Brown).

p. 936 — The elegance of Cram’s work.  Reading math has changed the way I’m reading organic chemistry.  What you want in math is an understanding of what is being said, and subsequently an ability to reconstruct a given proof.  You don’t have to have the proof at the tip of your tongue ready to spew out, but you should be able to reconstruct it given a bit of time.   The hard thing is remembering the definitions of the elements of a proof precisely, because precise they are and quite arbitrary in order to make things work properly.  It’s why I always leave a blank page next to my notes on a proof — to contain the definitions I’ve usually forgotten (or not remembered precisely).

I also find it much easier to remember mathematical definitions if I write them out (as opposed to reading them as sentences) as logical statements.  This means using ==> for implies | for such that, upside down A for ‘for all’, backwards E for ‘there exists, etc. etc. There’s too much linguistic fog in my mind when I read them as English sentences.

       So just knowing some general principles will be enough to reconstruct Cram’s elegant work described here.  There’s no point in trying to remember it exactly (although there used to be for me).   It think this is where beginning students get trapped — at first it seems that you can remember it all.  But then the inundation starts.  What should save them, is understanding and applying the principles, which are relatively few.  Again, this is similar to what happens in medicine — and why passing organic chemistry sets up the premed for this style of thinking. 

p. 938 – In the example of the Payne rearrangement, why doesn’t OH attack the epoxide rather than deprotonating the primary alcohol (which is much less acidic than OH itself).

p. 955 – Although the orbitals in the explanation of why stereochemistry is retained in 1,2 migrations are called  molecular orbitals (e.g. HOMO, LUMO) they look awfully like atomic orbitals just forming localized bonds between two atoms to me.  In fact the whole notion of molecular orbital has disappeared in most of the explanations (except linguistically).  The notions of 50 years ago retain their explanatory power.  

p. 956 — How did Eschenmoser ever think of the reaction bearing his name?  Did he stumble into it by accident? 

p. 956 — The starting material for the synthesis of juvenile hormone looks nothing like it.  I suppose you could say its the disconnection approach writ large, but the authors don’t take the opportunity.   The use of fragmentation to control double bond stereochemistry is extremely clever.   This is really the first stuff in the book that I think I’d have had trouble coming up with.  The fragmentation syntheses at the end of the chapter are elegant and delicious.

On a more philosophical note, the use of stereochemistry and orbitals to make molecules is exactly what I mean by explanatory power.  Anti-syn periplanar is a very general concept, which I doubt was brought into being to explain the stereochemistry of fragmentation reactions (yet it does).  It appears over and over throughout the book in various guises.

Urysohn’s Lemma

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”.  . . .  It is the crucial tool used in proving a number of important theorems. . . .  Why do we call the Urysohn lemma a ‘deep’ theorem?  Because its proof involves a really original idea, which the previous proofs did not.  Perhaps we can explain what we mean this way:  By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.  (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.)  But the Uyrsohn lemma is on a different level.  It would take considerably more originality than most of us possess to prove this lemma.”

The above quote is  from  one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on  page 207 of 514 pages of text.  Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405).  For why I’m reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26.   Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 - 281 '12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem.  It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology).  Actually what topology is about, is space, and what it means for objects to be near each other in space.  Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively.  In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality.   As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that).  Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway.  Urysohn talks about normal spaces.  Just think of a closed set as a Russian Doll with a bright shiny surface.  Remove the surface, and you have a rather beat up Russian doll — this is an open set.  When you open a Russian doll, there’s another one inside (smaller but still a Russian doll).  What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted.  Where did they get a weird idea like this?  From the number system of course.  Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can  always insert a new one halfway between.  This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough.  The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers.  So there are an infinite number of non-rational numbers between any two rational numbers.  In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space?  It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish.  This is the calculus of Newton and the relativity of Einstein.  It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level.  You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?

The New Clayden pp. 877 – 908

p. 878 — “The transition state has 6 delocalized pi electrons and thus is aromatic in character”.  Numerically yes, but the transition state isn’t planar, and there is all sorts of work showing how important planarity is to aromaticity. 

p. 881 — It seems to me that the arrow is wrong in the equation at the bottom. Entropy should increase when a Diels Alder product is broken apart, and since deltaG = deltaH  - T * deltaS heating the product should break it apart not cause it to form.  I guess the heat shown is required to increase molecular velocity so that collisions result in reaction.   Enough kinetic energy will blow anything apart (see Higgs particle).

p. 890 — “It is not cheating to use the regioselectivity of chemical reactions to tell us about the coefficients of the orbitals involved.”    I do think that this sort of thing is  cheating when you use the regioselectivity of chemical reactions as an explanation.  They are adding nothing new.  A real explanation predicts new phenomena, the way the anomeric effect does, for example.  You should contemplate the point at which a description of something becomes an explanation (e.g. epistemology).   It’s not the case here, but it was the case for Newton’s laws of gravitation.  Famously he said Hypotheses non dingo (“I frame no hypotheses”).  It appears in the following

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses.

Yet his laws of gravity were used to predict all sorts of events never before seen, so they are explanatory in some sense.  

This sort of thing is just what a neurologist experiences learning functional neuroanatomy (e.g.  which part of the nervous system has which function).  Initially almost all of it was developed by studying neurologic deficits due to various localized lesions of the brain and spinal cord.  There’s a huge caveat involved — pulling the plug on a radio will stop the sound, but that isn’t how the sound is produced.  People with lesions of the occipital lobe lose the ability to see in certain directions (parts of their visual fields).  Understanding HOW the occipital lobe processes sensory input from the eyes has taken 50 years and is far from over.  

p. 892 — Unfortunately the rationale behind  the Woodward Hoffmann rules isn’t covered, so it appears incredibly convoluted and arbitrary.  Read the book — “The Control of Orbital Symmetry” which they wrote.  Also, unfortunately, the description of the rules uses the term ‘component’ in two ways.  At step two butadiene and the dienophile are each considered a component, as they are in steps 3, 4, and 5, then the two are mushed together into a single component in step 6. 

p. 894 — I haven’t been looking at the animations for a while, but those of the Diels Alder type reactions are incredible, and almost sexual.  You can rotate the two molecules in space and watch them come together and react.

p. 894 –”Remember, the numbers in brackets, [ 4 + 2 ] etc., refer to the numbers of atoms.  The numbers (4q +2)s and (4r)s in The Woodward Hoffman (should be Hoffmann) refer to the numbers of electrons.”  This is so very like math, where nearly identical characters are used to refer to quite different things.  Bold capital X might mean one thing, italic x another, script X still another.  They all sound the same when you mentally read them to yourself.  It makes life confusing. 

p. 894 — The Alder ene reactions — quite unusual.  The worst thing is that I remember nothing about them from years ago.  They must have been around as they were discovered by Alder himself (who died in 1958).  They produce some rather remarkable transformations, the synthesis of menthol from citronellal being one.  I wonder if they are presently used much in synthetic organic chemistry. 

p. 900 — How do you make OCN – SO2Cl, and why is it available commercially?

p. 904 — The synthesis of the sulfur containing 5 membered ring of biotin is a thing of beauty.  It’s extremely non-obvious beginning with a 7 membered ring with no sulfur at all. 

Why math is hard (for me) and organic chemistry is easy

I’ve been reading a lot of hard core math lately (I’ll explain why at the end), along with Clayden et al’s new edition of their fabulous Organic Chemistry text.  The level of sophistication takes a quantum jump about 2/3 of the way through (around pp. 796) and is probably near to the graduate level.  The exercise is great fun, but math and orgo require quite distinct ways of thinking.  Intermixing both on a daily basis brought home just how very different they are.

First off, the concepts in organic chemistry are fuzzy.  On p. 796 the graph of the Karplus relationship between J splitting in NMR and the dihedral angle of the hydrogens being split is shown.  It’s a continuous curve as the splitting is maximal at 180, zero at 90 and somewhat less than maximal at 0 degrees.

There is nothing like this in math.  Terms are defined exactly and the logic is that of true, false and the excluded middle (e.g. things are either true or false).   Remember the way that the square root of 2 was proved not to be the ratio of two whole numbers.  It was assumed that it could be done, and than it was shown no matter how you sliced it, a contradiction was reached.   The contradiction then implied that the opposite was true — if the negative of a proposition leads to a contradiction (it’s false) than the proposition must be true.  Math is full of proofs like this.Or if you are trying to prove A implies B, proving the contrapositive ( not B implies not A) will do just as well.  You never see stuff like this in orgo.

There just aren’t that many concepts in organic chemistry, even though the details of each and every reaction are beyond the strongest memory.  The crucial points are to have the orbitals of the various atoms firmly in mind and where they are in space.  This tells you how molecules will or won’t react, or how certain conformations will be stable (see anomeric effect).  Entropy in physics is a very subtle concept, but pretty obvious as used by organic chemists.  Two molecules are better than one etc. etc.  Also you see these concepts over and over.  Everything you study (just about) has carbon in it.  Chair and boat, cis and trans, exo and endo become part of you, without thinking much about them.

Contrast this with math.  I’m currently reading “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee.  I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them.  Each is quite precise.  A topological embedding is (1) a continuous function (2) a surjective function (3) a homeomorphism.  No more no less.  Remove any one of the 3 (examples are given) and you no longer have an embedding.  The definitions are abstract for the most part, and far from intuitive.  That’s because  the correct definitions were far from obvious even to the mathematicians formulating them.  Hubbard’s excellent book on Vector Calculus says that it took some 200 years for the correct definition of continuity to be thrashed out.  People were arguing about what a function actually was 100 years ago.

As you read you are expected to remember exactly (or look up) the  170 or so defined concepts and use them in a proof.  So when you read a bit of Lee’s book, I’m always stopping and asking myself  ‘did I really understand what I just read’?  Clayden isn’t at all like that — Oh that’s just an intramolecular Sn2, helped because of the Thorpe Ingold effect, which is so obvious it shouldn’t be given a name.

Contrast this with:

After defining topological space, open set, closed set, compact, Hausdorff, continuous, closed map, you are asked to show that a continuous map from a compact topological space to a Hausdorff topological space is a closed map, and that such a map, if surjective as well is an embedding.   To get even close to the proof you must be able to hold all this in your head at once.  You should also remember that you proved that in a Hausdorff space compact sets are closed.

No matter how complicated the organic problem, you can always look at the molecule, and use the fabulous spatial processing capacity your brain gives you.   The interpretation of NMR spectra in terms of conformation certainly requires logical thinking — it’s sort of organic Sudoku.

I imagine a mathematician would have problems with the fuzzy concepts of organic chemistry.  Anslyn and Dougherty take great pains to show you why some reactions fall between Sn1 and Sn2, or E1cb.

So why am I doing this?  Of course there’s the why climb Everest explanation — because it’s there, big and hard, and maybe you can’t do it.  That’s part of it, but not all.  For reasons unknown, I’ve always like math, even though not terribly good at it.  Then there’s the surge for the ego should I be able to go through it all proving that I don’t have Alzheimer’s at 74.5 (at least not yet).  Then there is the solace (yes solace) that math provides.  Topology is far from new to me in 2011.  I started reading Hocking and Young back in ’70 when I was a neurology resident, seeing terrible disease, being unable to help most of those I saw, and ruminating about the unfairness of it all.  Thinking about math took me miles away (and still does), at least temporarily.  When I get that far away look, my wife asks me if I’m thinking about math again.  She’s particularly adamant about not doing this when I’m driving with her (or by myself).

The final reason, is that I went to college with a genius.  I met him at our 50th reunion after reading his bio in our 50th reunion book.  I knew several self-proclaimed geniuses back then, and a lot a physics majors, but he wasn’t one of them.  At any rate, he’s still pumping out papers on relativity with Stephen Hawking, and his entries in the index of the recent biography by Kitty Ferguson take up almost as many entries as Hawking himself.  He’s a very nice guy and agreed to answer questions from time to time.  But to understand the physics you need to really understand the math, and not just mouth it.

In particular, to understand gravity, a la relativity, you have to know how mass bends 4 dimensional space-time. This means you must understand curvature on manifolds, which means you must understand smooth manifolds, which means that you must understand topological manifolds which is why I’m reading Lee’s book.
So perhaps when the smoke clears, I might have something intelligent to say to my classmate.

The New Clayden pp. 757 — 800

 

Chapter 30: There’s a lot more use of the disconnection approach in the discussions of the synthesis of heterocylic aromatic compounds than there was in the previous edition.  The analysis of the Viagra synthesis pp. 768 –> is particularly fascinating.

The sophistication of the chapter is much higher than what went on previously.  It’s great !  The writer assumes that you have all the previous reactions well under your belt, as well as disconnection and moves rapidly on from there.  

In a sense it’s like the switch from undergraduate math books where proofs are laid out in detail, to the graduate lectures, where proofs are sketched and you are expected to fill in the dots.  I wonder how a neophyte hitting this chapter for the first time would take it. 

One can take the analogy a bit further.  The target molecule can  be considered the theorem and and the synthesis the proof.  This is exactly why math is harder than organic chemistry.  The target molecule is almost telling you (thanks to the disconnection approach) how to make it.  The examples in this chapter are fairly simple.  Yet most accounts of syntheses focus on one or two most difficult steps and the target is far more complex — for an example see ttp://heterocyclist.com/2012/08/24/synthesis-of-kopsia-lapidilecta-alkaloids-the-rcm-approach-takes-a-hit-retraction/.

In medical school, the importance of taking an accurate history was stressed — “The patient is telling you the diagnosis” was said over and over, just as the structure of a synthetic target is telling you how to make it.  Certainly, with each passing year, the MD finds the history more and more valuable, and the physical exam less.  Medicine has one further wrinkle that math and synthetic organic chemistry do not.  The manner in which  the patient gives the history and answers your questions is incredibly important.  It’s not just the words, it’s the tune.  Is the patient depressed, angry, confused, hyped-up etc. etc.  That’s why I always took the history myself, and never had the patient fill out some checklist, it throws away information you can get in no other way

I don’t know enough math to know if proofs break down this way.  But there is another huge difference between math and orgo.  In math the definitions are incredibly precise.  A collection of subsets of a given set either satisfies 3 extremely specific criteria to make them open sets and the containing set into a topology, or they don’t.  Chemical reactions aren’t like that — Anslyn and Dougherty take you through Sn1 and Sn2 and their variants, and then show you how there are reactions that fall between them, containing aspects of both.   The idea of a Diels Alder reaction, is independent of any particular exemplar — so the concepts in chemistry are inherently fuzzy.  If you’re good at reasoning by analogy, then chemistry is your oyster.  Don’t try this in a mathematical proof.  So the zillion mathematical definitions (first countable, compactness, path connected in its varieties) must be memorized exactly as they are, and used in proofs that way, and that way only.   Medical concepts are even fuzzier.  It takes a very different type of mind to do math well, one which, unfortunately, I don’t posses, even though I love the stuff.

 Back to chemistry

p. 772 The example of the tautomer of the thioamide interacting with an alpha haloketone is a great example of hard/hard nucleophile/electrophile and soft/soft nucleophile/electrophile interactions occuring specifically in the same pair of molecules, while quite near to each other.  It should probably be pointed to in the next edition when  hard/soft nucleophiles and electrophiles are first discussed. 

p. 775 — Interesting that they didn’t call the reaction of an alkyne and an azide ‘click chemistry‘ which is what Sharpless calls it.  It has proved extremely useful in linking together molecules of biologic interest — e.g. seeing where a protein is binding to other proteins or to DNA.  The uses are endless and still being discovered. 

Here are a few examples:

       [ Proc. Natl. Acad. Sci. vol. 98 pp. 4740 - 4745 '01 ] Propargyl choline is a choline derivative which can be used to label choline containing phospholipids using Click chemistry  (forming cycloaddition products with a fluorophore containing an azide.  Total lipid analysis of labeled cells shows strong incorporation of propargyl choline into all classes of choline phospholipids — and the fatty acid composition of these lipids is quite normal. 

        [ Proc. Natl. Acad. Sci. vol. 105 pp. 2415 - 2420 '08 ] It was used to quickly label DNA using 5 ethynyl 2′ deoxy uridine — which can be detected using fluorescence. 

       [ Science vol. 320 pp. 868 - 869 '08 ] It is a modification of the Huisgen reaction — the trick was using Copper Iodide as a catalyst.  Polymer scientists love it.

        Another type of click reaction adds a thiol across an olefin using light. 

 

       [ Proc. Natl. Acad. Sci. vol. 107 pp. 15329 - 15334 '10 ] Oligonucleotides can be produced by automated solid phase phosphoramidite synthesis — chains over 100 (deoxy) nucleotides can be formed.  It’s harder with RNA because of the reactivity of the 2′ OH group which requires selective protection.  So the limit here is 50 nucleotides.  This work describes click ligation as a way to put them together. 

p. 793 — A very useful explanation of the nomenclature of heterocyclic ring compounds (which is actually or logical than it first appears). 

p. 794 — Aziridine is less basic than pyroldine and piperidine, because the hybridization of the nitrogen has more s character. But no mention is made of why this should mean less basicity — it’s because the s orbital experiences the positive charge of the nucleus more intensely than a p orbital (which has a node at the nucleus), lowering its energy and making it less likely to share (like a spoiled child). 

p. 796 – While coupling NMR is through bonds rather than throught space (e.g. more coupling between H’s trans to each other on a double bond, than cis — they never explained why this is so, nor do they here. 

p. 796 — It don’t see why the dihedral angle in the bicyclic compound shown is any different from 60 degrees, the axial equatorial bond separation, unless the ring configuration by compressing the C – C – C angle, expands the H – C – H angle.

p. 797 — Why is the shift of the  hydrogen on the carbon containing the OH groups so different between axial (3.5) and equatorial (4.0)? 

p. 799 — Neurologists are excellent at reading MRI scans of patients (or they should be), these vary in appearance depending on whether they use T1 or T2 relaxation.  But the whole issue of relaxation from a higher energy state to a lower one is rarely discussed.  

The text says “So far we have assumed that the drop back down (to a lower energy state) is spontaneous, just like a rock falling off a cliff.  In fact it isn’t — something needs to ‘help’ the protons to drop back again — a process called relaxation”.  Why is this the case? Is it similar to laser action, where something needs to stimulate the drop down to a lower energy state with the emission of laser light.  Perhaps one of the cognoscenti reading this can explain why help is needed for a transition to a lower energy state.  I don’t understand it.

Being able to admit you don’t know something and publicly asking for help is one of the joys of being a non-academic.  I doubt that I’d be able to do this if I were a chemistry department chair, as at least 3 – 4 of my fellow Harvard graduate students 52 years ago became (one of them is still at it and going strong — also to be noted is that he came out of a State University). 

The New Clayden pp. 222 – 268

p. 228 A picture of a Dean Stark head would be useful. 

p. 230 — Second paragraph from the bottom, 4th line ‘hemiacetal’ should be ‘hemiaminal’

p. 231 — Hard (for me) to see why stereoisomers of imines should interconvert and those of oximes should not.  Perhaps the oxygen is also hybridized sp2?  On p. 232 this appears to be the case, in explaining the stability of oximes, hydrazones and semicarbazones. 

p. 238 — The 4 membered ring of the Wittig intermediate isn’t as strained as it might look because the P – O bond is 1.76 Angstroms, and the P-C bond is 1.87 giving the ring a bit more room, but the spiro bond shown can’t change much due to the cyclohexane ring.  If anyone has actually seen the intermediate and measured its shape it would be interesting to know exactly what it looks like. 

p. 246 — A way to think of enthalpy change, is to remember the first law of thermodynamics — energy is neither created or destroyed. Crudely, enthalpy is just a measure of internal energy.   Also rather crudely, if the products of a reaction are of lower energy than the starting material, some form of energy must be given off (usually heat), and the products have lower enthalpy (a measure of internal energy).  Why ‘crudely’?  Because the discussion ignores entropy. 

p. 247 — Amusing, how chemists possess a very intuitive understand of entropy and enthalpy giving them essentially all the thermodynamics they need.  Think of the hard intellectual work involved in the Clausius’ definition of entropy  – the reversible heat supplied divided by the temperature at which it is supplied.  Chemical thinking is far closer to Boltzmann –  S = k log W. 

p. 248 — It’s worth thinking why the enthalpy of a molecule doesn’t change much with temperature.  It’s basically the energy of the bonds it contains, which is pretty much the same until the bonds are broken (and the molecule changes). 

It’s also worthwhile pausing and thinking what we mean by a ‘strong bond’ — it’s one that requires a large input of energy to break.  So even though we describe a strong bond as high energy, it’s really much lower in enthalpy than separating the atoms that make it up. 

p. 248 — Le Chatelier’s principle may be the basis of a treatment to dissolve the senile plaques of Alzheimer’s disease (and help the condition if they are what’s causing the problem — something rather contentious as of 5/12).  For details see – http://luysii.wordpress.com/2012/03/04/could-le-chateliers-principle-be-the-answer-to-alzheimers-disease/

p. 251 — The concept of a transition state is valid (because we use it all the time and it appears to work).  But, by definition, a transition state can’t be isolated (unlike reaction intermediates), so is it as scientifically valid as the number of Angels which can fit on the head of pain or is it ‘real’ in a truly scientific sense?  

The transition state concept assumes that the states of a molecule are ‘complete’ in the following mathematical sense.  By complete, I mean the following.  Consider the rational numbers (ratios of whole numbers).  We can get as close as we wish to the square root of 2, but the fact that sort(2)  cannot be a rational number is said to have driven the Pythagoreans to murder one of their own who threatened to divulge this to the laity.  So while we today regard sqrt(2) as a number, it exists essentially by the assumption of any number of equivalent postulates. 

Personally, I find equivalence classes of Cauchy sequences the most intuitive definition of real number (of which rational numbers are a part).  The completeness property of the real numbers allows us to prove that any continuous function on a closed interval of real numbers reaches a maximum (the transition state) and a minimum.

Since energy levels are quantized, why not reaction states?  Then there would be no such thing as a continuous transition between reactant and product, and no transition state.

p. 258 — Proton transfers are fast.  Well how fast?

Going to the mat with representation, characters and group theory

Willock’s book (see https://luysii.wordpress.com/category/willock-molecular-symmetry/) convinced me of the importance of the above in understanding vibrational spectroscopy.  I put it aside because the results were presented, rather than derived.  From the very beginning (the 60′s for me) we were told that group theory was important for quantum mechanics, but somehow even a 1 semester graduate course back then didn’t get into it.  Ditto for the QM course I audited a few years ago.

I’ve been nibbling about the edges of this stuff for half a century, and it’s time to go for it.  Chemical posts will be few and far between as I do this, unless I run into something really interesting (see https://luysii.wordpress.com/2012/02/18/a-new-way-to-study-reaction-mechanisms/).  Not to worry –plenty of interesting molecular biology, and even neuroscience will appear in the meantime, including a post about article showing that just about everything we thought about hallucinogens is wrong.

So, here’s my long and checkered history with groups etc.  Back in the day we were told to look at “The Theory of Groups and Quantum Mechanics” by Hermann Weyl.  I dutifully bought the Dover paperback, and still have it (never throw out a math book, always throw out medical books if more than 5 years old).  What do you think the price was — $1.95 — about two hours work at minimum wage then.  I never read it.

The next brush with the subject was a purchase of Wigner’s book “Group Theory and its Application to the Quantum Mechanics of Atomic Spectra”  – also never read but kept.  A later book (see Sternberg later in the post)  noted that the group theoretical approach to relativity by Wigner produced the physical characteristics of mass and spin as parameters in the description of irreducible representations.  The price of this one was $6.80.

Then as a neurology resident I picked up “Group Theory” by Morton Hammermesh (Copyright 1962).  It was my first attempt to actually study the subject.  I was quickly turned off by the exposition.  As groups got larger (and more complicated) more (apparently) ad hoc apparatus was brought in to explain them — first cosets, then  subgroups, then normal subgroups, then conjugate classes.

That was about it, until retirement 11 years ago.  I picked up a big juicy (and cheap) Dover paperback “Modern Algebra” by Seth Warner — a nice easy introduction to the subject.

Having gone through over half of Warner, I had the temerity to ask to audit an Abstract Algebra course at one of the local colleges.  I forget the text, but I didn’t like it (neither did the instructor).  We did some group theory, but never got into representations.

A year or so later, I audited a graduate math abstract algebra course given by the same man.  I had to quit about 3/4 through it because of an illness in the family.  We never got into representation theory.

Then, about 3 years ago, while at a summer music camp, I got through about 100 pages of “Representations and Characters of Groups” by James and Liebeck.  The last chapter in the book (starting on p. 366) was on an application to molecular vibration.  The book was hard to use because they seemed to use mathematical terms differently than I’d learned — module for example.   I was used to.  100 pages was as far as I got.

Then I had the pleasure of going through Cox’s book on Galois theory, seeing where a lot of group theory originated  (along with a lot of abstract algebra) — but there was nothing about representations there either.

Then after giving up on Willock, a reader suggested  “Elements of Molecular Symmetry” by Ohrn.  This went well until p. 28, when his nomenclature for the convolution product threw me.

So I bought yet another book on the subject which had just come out “Representation Theory of Finite Groups” by Steinberg.  No problems going through the first 50 pages which explains what representations, characters and irreducibles are.  Tomorrow I tackle p. 52 where he defines the convolution product.  Hopefully I’ll be able to understand it an go back to Ohrn — which is essentially chemically oriented.

The math of it all is a beautiful thing, but the immediate reason for learning it is to understand chemistry better.  I might mention that I own yet another book “Group Theory and Physics” by Sternberg, which appears quite advanced.  I’ve looked into it from time to time and quickly given up.

Anyway, it’s do or die time with representation theory.  Wish me luck

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