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A game with Pi

November 8, 2014 2 comments

Here’s an image of something I wrote down, took a photo of, and posted here for you. It’s a little game you can play with any irrational number. I took \pi as an example.

You just learned about an important math concept/process called continued fraction expansions.

With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish.

As an example, if you truncate the above last expansion where the 292 appears (so you omit the “1 over 292” part) you get the rational number 335/113 which approximates \pi to 6 decimal places. (Better than 22/7.)

You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers.

Exercise: for the square root of 2, show that the sequence you get is
1, 2, 2, 2, 2, …
(all 2’s after the 1). For the square root of 3 the continued fraction sequence is
1, 1, 2, 1, 2, 1, 2, 1, 2, …
(so it starts with 1 and then the pair “1, 2” repeat periodically forever).

Matter and Antimatter don’t always annihilate

October 11, 2014 2 comments

It is often said that when matter and antimatter come into contact they’ll annihilate each other, usually with the release of powerful energy (photons).

Though in essence true, the statement is not exactly correct (and so can be misleading).

For example, if a proton comes into contact with a positron they will not annihilate. (If you recall, the positron is the antiparticle of the electron.) But if a positron comes into contact with an electron then, yes, they will annihilate (yielding a photon). (Maybe they will not instantaneously annihilate, since they could for the minutest moment combine to form positronium, a particle they form as they dance together around their center of mass – and then they annihilate into a photon.)

The annihilation would occur between particles that are conjugate to each other — that is, they have to be of the same type but “opposite.” So you could have a whole bunch of protons come into contact with antimatter particles of other non-protons and there will not be mutual annihilation between the proton and these other antiparticles.

Another example. The meson particles are represented in the quark model by a quark-antiquark pair. Like this: p\bar q . Here p and q could be any of the 6 known quarks u, d, c, b, t, s and the \bar q  stands for the antiquark of q . If we go by the loose logic that “matter and antimatter annihilate” then no mesons can exist since p  and \bar q  will instantly destroy one another.

For example, the pion particle \pi^+ has quark content u\bar d  consisting of an up-quark u and the anti-particle of the down quark. They don’t annihilate even though they’re together (in some mysterious fashion!) for a short while before it decays into other particles. For example, it is possible to have the decay reaction

\pi^+ \to \mu^+ + \nu_\mu

(which is not the same as annihilation into photons) of the pion into a muon and a neutrino.

Now if we consider quarkonium, i.e. a quark and its antiquark together, such as for instance \pi^0 = u\bar u or \eta = d\bar d , so that you have a quark and its own antiquark, then they do annihilate. But, before they do so they’re together in a bound system giving life to the \pi^0, \eta particles for a very very short while (typically around 10^{-23} seconds). They have a small chance to form a particle before they annihilate. It is indeed amazing to think how such Lilliputian time reactions are part of how the world is structured. Simply awesome! 😉

PS. The word “annihilate” usually has to do when photon energy particles are the result of the interaction, not simply as a result of when a particle decays into other particles.

Sources:

(1) Bruce Schumm, Deep Down Things. See Chapter 5, “Patterns in Nature,” of this wonderful book. 🙂

(2) David Griffiths, Introduction to Elementary Particles. See Chapter 2. This is an excellent textbook but much more advanced with lots of Mathematics!

Comparing huge numbers

Comparing huge numbers is often times not easy since you practically cannot write them out to compare their digits. (By ‘compare’ here we mean telling which number is greater (or smaller).) So it can sometimes be a challenge to determine.

Notation: recall that N! stands for  “N factorial,” which is defined to be the product of all positive whole numbers (starting with 1) up to and including N. (E.g., 5! = 120.) And as usual, Mn stands for M raised to the power of n (sometimes also written as M^n).

Here are a couple examples of huge numbers (which we won’t bother writing out!) that aren’t so easy to compare but one of which is larger, just not clear which. I don’t have a technique except maybe in an ad hoc manner.

In each case, which of the following pairs of numbers is larger?

(1) (58!)2 and 100!

(2) (281!)2 and 500!

(3) (555!)2 and 1000!

(4) 500!  and 101134

(5) 399! + 400! and 401!

(6) 8200 and  9189

(The last two of these are probably easiest.)

Have fun!

 

Escher Math

http://upload.wikimedia.org/wikipedia/en/a/a3/Escher%27s_Relativity.jpg

Escher Relativity

You’ve all seen these Escher drawings that seem to make sense locally but from a global, larger scale, do not – or ones that are just downright strange. We’ll it’s still creative art and it’s fun looking at them. They make you think in ways you probably didn’t. That’s Art!

Now I’ve been thinking if you can have similar things in math (or even physics). How about Escher math or Escher algebra?

Here’s a simple one I came up with, and see if you can ‘figure’ it out! 😉

(5 + {4 – 7)2 + 5}3.

LOL! 🙂

How about Escher Logic!? Wonder what that would be like. Is it associative / commutative? Escher proof?

Okay, so now … what’s your Escher?

Have a great day!

 

 

Richard Feynman on Erwin Schrödinger

I thought it is interesting to see what the great Nobel Laureate physicist Richard Feynman said about Erwin Schrödinger’s attempts to discover the famous Schrödinger equation in quantum mechanics (see quote below). It has been my experience in reading physics that this sort of “heuristic” reasoning is part of doing physics. It is a very creative (sometimes not logical!) art with mathematics in attempting to understand the physical world. Dirac did it too when he obtained his Dirac equation for the electron by pretending he could take the square root of the Klein-Gordon operator (which is second order in time). Creativity is a very big part of physics.

 

“When Schrödinger first wrote it down [his equation],
he gave a kind of derivation based on some heuristic
arguments and some brilliant intuitive guesses. Some
of the arguments he used were even false, but that does
not matter; the only important thing is that the ultimate
equation gives a correct description of nature.”
                                    — Richard P. Feynman
(Source: The Feynman Lectures on Physics, Vol. III, Chapter 16, 1965.)
 
 
 

Principles of quantum theory

December 22, 2013 Leave a comment

One beautiful summer morning I spent a couple hours in a park reflecting on what I know about quantum mechanics and thought to sketch it out from memory. (A good brain exercise to recapture things you learned and admire.) This note is an edited summary of my handwritten draft (without too much math).

Being a big subject, I will stick to some basic ideas (or principles) of quantum theory that may be worth noting.

Two key concepts are that of a `state’ and that of an ‘observable’.

The former describes the state of the system under study. The observable is a thing we measure. So for example, an electron can be in the ground state of an atom – which means that it is in `orbital’ of lowest energy. Then we have other states that it can be in at higher energies.

The observable is a quantity distinct from a state and one that we measure. Such as for example measuring a particle’s energy, its mass, position, momentum, velocity, charge, spin, angular momentum, etc.

QM gives us principles / interpretations by how states and observables can be mathematically described and how they relate to one another. So here is the first principle.

Principle 1. The state of a system is described by a function (or vector) ψ. The probability density associated with it is given by |ψ|².

This vector is usually a mathematical function of space, time (sometime momentum) variables.

For example, f(x) = exp(-x^2) is one such example. You can also have wave examples such as g(x) = exp(-x^2) sin(x) which looks like a localized wave (a packet) that captures both being a particle (localized) and a wave (due to the wave nature of sin(x)). This wave nature of the function allows it to interfere constructively or destructively with other similar functions — so you can have interference! In actual QM these wavefunctions involve more variables that one x variable, but I used one variable to illustrate.

Principle 2. Each measurable quantity (called an ‘observable’) in an experiment is represented by a matrix A. (A Hermitian matrix or operator.)

For example, energy is represented by the Hamiltonian matrix H, which gives the energy of a system under study. The system could be the hydrogen atom. In many or most situations, the Hamiltonian is the sum of the kinetic energy plus the potential energy (H = K.E. + V).

For simplicity, I will treat a measurable quantity and its associated matrix on equal footing.

From matrix algebra, a matrix is a rectangular array of numbers – like, say, a square array of 3 by 3 numbers, like this one I grabbed from the net:

https://i0.wp.com/www.mathsisfun.com/algebra/images/matrix-3x3-det-a.gif

Turns out you can multiply such things and do some algebra with them.

Two basic facts about these matrices is:

(1) they generally do not have the commutative property (so AB and BA aren’t always equal), unlike real or complex numbers,

(2) each matrix A comes with `magic’ numbers associated to it called eigenvalues of A.

For example the matrix

http://upload.wikimedia.org/math/4/4/c/44c1569404535221ffac85abef04e4cd.png

(called diagonal because it has all zeros above and below the diagonal) has eigenvalues 1, 4, -3. (When a matrix is not diagonal we have a procedure for finding them. Study Matrix or Linear Algebra!)

Principle 3. The possible measurements of a quantity will be its eigenvalues.

For example, the possible energy levels of an electron in the hydrogen atom are eigenvalues of the associated Hamiltonian matrix!

Principle 4. When you measure a quantity A when the system is in the state ψ the system `collapses’ into an eigenstate f of the matrix A.

Therefore the system makes a transition from state ψ to state f (when A is measured).

So mathematically we write

Af = af

which means that f is an eignstate (or eigenvector) of A with eigenvalue a.

So if A represents energy then `a’ would be energy measurement when the system is in state f.

For a general state ψ we cannot say that Aψ = aψ. This eigenvalue equation is only true for eigenstates, not general states.

Principle 5. Each state ψ of the system can be expressed as a superposition sum of the eigenstates of the measurable quantity (or matrix) A.

So if f, g, h, … are the eigenstates of A, then any other state ψ of the system can be expressed as a superposition (or linear combination) of them:

ψ = bf + cg + dh + …

where b, c, d, … are (complex) numbers. Further, |c|^2 = probability ψ will `collapse’ into the eigenstate g when measurement of A is performed.

These principles illustrate the indeterministic nature of quantum theory, because when measurement of A is made, the system can collapse into any one of its many eigenstates (of the matrix A) with various probabilities. So even if you had the ‘exact same’ setup initially there is no guarantee that you would see your system state change into the same state each time. That’s non-causality! (Quite unlike Newtonian mechanics.)

Principle 6. (Follow-up to Principles 4 and 5.) When measurement of A in the state ψ is performed, the probability that the system will collapse into the eigenstate vector φ is the dot product of Aψ and φ.

The latter dot product is usually written using the Dirac notation as <φ|A|ψ>.  In the notation above, this would be same as |c|^2.

Next to the basic eigenvalues of A, there’s also it’s `average’ value or expectation value in a given state. That’s like taking the weighted average of tests in a class – with weights assigned to each eigenstate based on the superposition (as in the weights b, c, d, … in the above superposition for ψ). So we have:

Principle 7. The expected or average value of quantity A in the state described by ψ is <ψ|A|ψ>.

In our notation above where ψ = bf + cg + dh + …, this expected value is

<ψ|A|ψ> = |b|^2 times (eigenvalue of f)  + |c|^2 times (eigenvalue of g) + …

which you can see it being the weighed average of the possible outcomes of A, namely from the eigenvalues, each being weighted according to its corresponding probabilities |b|^2, |c|^2, … .

In other words if you carry out measurements of A many many times and calculate the average of the values you get, you get this value.

Principle 8. There are some key complementary measurement observables. (Classic example: Heisenberg relation QP – PQ = ih.)

This means that if you have two quantities P and Q that you could measure, if you measure P first and then Q, you will not get the same result as when you do Q first and then P. (In Newton’s mechanics, you could at least in theory measure P and Q simultaneously to any precision in any order.)

Example, position and momentum are complementary in this respect — which is what leads to the Heisenberg Uncertainty Principle, that you cannot measure both the position and momentum of a subatomic particle with arbitrary precision. I.e., there will be an inherent uncertainty in the measurements. Trying to be very accurate with one means you lose accuracy with they other all the more.

From Principle 8 you can show that if you have an eigenstate of the position observable Q, it will not be an eigenstate for P but will be a superposition for P.

So `collapsed’ states could still be superpositions! (Specifically, a collapsed state for Q will be a superposition, uncollapsed, state for P.)

That’s enough for now. There are of course other principles (and some of the above are interlinked), like the Schrodinger equation or the Dirac equation, which tell us what law the state ψ must obey, but I shall leave them out. The above should give an idea of the fundamental principles on which the theory is based.

Have fun,
Samuel Prime

The 21 cm line of hydrogen in Radio Astronomy

June 30, 2013 2 comments

 

This has been a wonderful discovery back in the 1950s that gave Radio Astronomy a good push forward. It also helped in mapping out our Milky Way galaxy (which we really can’t see very well!).File:Galaxy NGC 1232.jpg

It arose from a feature of quantum field theory, specifically from the hyperfine structure of hydrogen. (I’ll try to explain.)

You know that the hydrogen atom consists of a single proton at its central nucleus and a single electron moving around it somehow in certain specific quantized orbits. It cannot just circle around in any orbit.

That was one of Niels Bohr’s major contributions to our understanding of the atom. In fact this year we’re celebrating the 100th anniversary of his model of the atom (his major papers written in 1913). Some articles in the June issue of Nature magazine are in honor of Bohr’s work.

Normally the electron circles in the lowest orbit associated with the lowest energy state – usually called the ground state (the one with n = 1).

It is known that protons and electrons are particles that have “spin”. (That’s why they are sometimes also called ‘fermions’.) It’s as if they behave like spinning tops. (The Earth and Milky Way are spinning too!)

The spin can be in one direction (say ‘up’) or in the other direction (we label as ‘down’). (These labels of where ‘up’ and ‘down’ are depends on the coordinates we choose, but let’s now worry about that.)

When scientists looked at the spectrum of hydrogen more closely they saw that even while the electron can be in the same ground state – and with definite smallest energy – it can have slightly different energies that are very very close to one another. That’s what is meant by “hyperfine structure” — meaning that the usual energy levels of hydrogen are basically correct except that there are ever so slight deviations from the normal energy levels.

It was discovered by means of quantum field theory that this difference in ground state energies arise when the electron and proton switch between spinning in the same direction to spinning in opposite directions (or vice versa).

When they spin in the same direction the hydrogen atom has slightly more energy than when they are spinning in opposite direction.

And the difference between them?

The difference in these energies corresponds to an electromagnetic wave corresponding to about 21 cm wavelength. And that falls in the radio band of the electromagnetic spectrum.

So when the hydrogen atom shows an emission or absorption spectrum in that wavelength level it means that the electron and proton have switched between having parallel spins to having opposite spins. When the switch happens you see an electromagnetic ray either emitted or absorbed.

It does not happen too often, but when you have a huge number of hydrogen atoms — as you would in hydrogen clouds in our galaxy — it will invariably happen and can be measured.

Now it’s a really nice thing that our galaxy contains several hydrogen clouds.  So by measuring the Doppler shift in the spectrum of hydrogen — at the 21 cm line! — you can measure the velocities of these clouds in relation to our location near the sun.

These velocity distributions are used together with other techniques to map out the hydrogen clouds in order to map out and locate the spiral arms they fall into.

That work (lots of hard work!) showed astronomers that our Milky Way does indeed have arms, just as we would see in some other galaxies, such as in the picture shown here of NGC 1232.

The one UNKNOWN about the structure of our Milky Way is that we don’t know whether it has 2 or 4 arms.

References:
[1] University Astronomy, by Pasachoff and Kutner. 
[2] Astronomy (The Evolving Universe), by Michael Zeilik.

(These are excellent sources, by the way.)

August Kekule’s Benzene Vision

June 30, 2013 6 comments

The first time I heard of August Kekule’s dream/vision was from my dear mother! (My mom is a geologist who obviously had to know a lot of chemistry.)  I am referring to Kekule’s vision while gazing at a fireplace which somehow prompted him onto the idea for the structure of the benzene molecule C6H6. And then I heard that the story is suspect maybe even a myth cooked up by unscientific minds. Now I have learned that Kekule himself recounted that story which was translated into English and published in the Journal of Chemical Education (Volume 35, No. 1, Jan. 1958, pp 21-23, translator: Theodor Benfey). Here is an excerpt from that paper relevant to the story where Kekule talks about his discovery.

I was sitting writing at my textbook but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background.    My mental eye, rendered more acute by repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes.    As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis. Let us learn to dream, gentlemen, then perhaps we shall find the truth.

And to those who don’t think The truth will be given. They’ll have it without effort.

But let us beware of publishing our dreams till they have been tested by the making understanding.

Countless spores of the inner life fill the universe, but only in a few rare beings do they find the soil for their development; in them the idea, whose origin is known to no men, comes to life in creative action.  (J. Von Liebig)

I believe it is unnecessary to rule out or ridicule dreams, trances, visions in the pursuit of scientific truth. Because, after all, they still have to be tested and examined in our sober existence (as Kekule already alluded). I see them as extensions of thinking and contemplation, and surely there is nothing wrong with these.

Einstein on theory, logic, reality

 

https://i0.wp.com/www.soundswitheinstein.com/wp-content/uploads/2012/12/einstein-guitar.jpg

Long ago (late 1980s) I attended a lecture by Einstein biographer I. Bernard Cohen. (Cohen actually interviewed Einstein and published it in the Scientific American in the 1955 issue.)

 
In his lecture, Cohen described Einstein’s view of scientific discovery as a sort of ‘leap’ from experiences to theory. That theory is not logically deduced from experiences but that theory is “jumped at” — or “swooped” is the word Cohen used, I think — thru the imagination or intuition based on our experiences (which of course would/could include experiments). This reminds one of the known Einstein quote that “imagination is more important that knowledge.”

In his book Ideas and Opinions, Albert Einstein said:

“Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and end in it. Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo saw this, and particularly because he drummed it into the scientific world, he is the father of modern physics—indeed, of modern science altogether.”

(See Part V of his “Ideas and Opinions” in the section entitled “On the Method of Theoretical Physics.”)

In a related passage from the same section, Einstein noted:

“If, then, it is true that the axiomatic foundation of theoretical physics cannot be extracted from experience but must be freely invented, may we ever hope to find the right way? Furthermore, does this right way exist anywhere other than in our illusions? May we hope to be guided safely by experience at all, if there exist theories (such as classical mechanics) which to a large extent do justice to experience, without comprehending the matter in a deep way?

To these questions, I answer with complete confidence, that, in my opinion, the right way exists, and that we are capable of finding it. Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.”

Note his reference to theory as being ‘freely invented’ (and even ‘illusion’) which echo the role of intuition and imagination in the scientific development of theory (but which are probably not completely divorced from experience either!).

The last two quotes above incidentally can be found online in Standford’s Encyclopedia of Philosophy: Einstein’s Philosophy of Science

 

The sphere in dimensions 4, 5, …

June 23, 2013 2 comments

The volume of a sphere of radius R in 2 dimensions is just the area of a circle which is π R2. (The symbol π is Pi which is 3.1415….)

The volume of a sphere of radius R in 3 dimensions is (4/3) π R3.

The volume of a sphere of radius R in 4 dimensions is (1/2) π2 R4.

Hold everything! How’d you get that? With a little calculus!

Ok, you see a sort of pattern here. The volume of a sphere in n dimensions has a nice form: it is some constant C(n) (involving π and some fractions) times the radius R raised to the dimension n:

V(n,R) = C(n) Rn —————– (1)

where I wrote V(n,R) for the volume of a sphere in n dimensions of radius R (as a function of these two variables).

Now, how do you get the constants C(n)?

With a bit of calculus and some ‘telescoping’ as we say in Math.

(In the cases n = 2 and n = 3, we can see that C(2) = π, and C(3) = 4π/3.)

First, let’s do the calculus. The volume of a sphere in n dimensions can be obtained by integrating the volume of a sphere in n-1 dimensions like this using the form (1):

V(n,R) = ∫ V(n-1,r) dz = ∫ C(n-1) rn-1 dz

where here r2 + z2 = R2 and your integral goes from -R to R (or you can take 2 times the integral from 0 to R). You solve the latter for r, plug it into the last integral, and compute it using the trig substitution z = R sin θ. When you do, you get

V(n,R) = 2C(n-1) Rn ∫ cosnθ dθ.

The cosine integral here goes from 0 to π/2, and it can be expressed in terms of the Gamma function Γ, so it becomes

V(n,R) = 2C(n-1) Rn π1/2 Γ((n+1)/2) / Γ((n+2)/2).

Now compare this with the form for V in equation (1), you see that the R’s cancel and you have C(n) expressed in terms of C(n-1). After telescoping and simplifying you eventually get the volume of a sphere in n dimensions to be:

V(n,R) = πn/2 Rn / Γ((n+2)/2).

Now plug in n equals 4 dimensions and you have what we said above. (Note: Γ(3) = 2 — in fact, for positive integers N the Gamma function has simple values given by Γ(N) = (N-1)!, using the factorial notation.)

How about the volume of a sphere in 5 dimensions?  It works out to

V(5,R) = (8/15) π2 R5.

One last thing that’s neat: if you take the derivative of the Volume with respect to the radius R, you get its Surface Area!  How crazy is that?!!

Can a product of 4 consecutive odds be a perfect square?

June 17, 2013 2 comments

Someone on twitter asked if a product of four consecutive odd positive numbers can be a perfect square.

My answer: No.

For example, 3 x 5 x 7 x 9 = 945 which is not a perfect square. (30^2 = 900 and 31^2 = 961.) Similarly, 7 x 9 x 11 x 13 = 9009, again is not a perfect square.

Here is my proof. Write the four consecutive positive odd numbers as:

2x+1, 2x+3, 2x+5, 2x+7

where x is a positive integer.

The middle two numbers 2x+3, 2x+5 cannot both be perfect squares since their difference is 2 — the difference between two positive consecutive perfect squares is at least 3. Let’s suppose it is 2x+3 that is not a perfect square. (The argument can still be adapted if it was 2x+5.) So 2x+3 is divisible by an odd prime p that has an odd power in its prime factorization. This prime p cannot divide the other three factors 2x+1, 2x+5, 2x+7, or else it would divide their differences from 2x+3, which are 2 or 4 (and p is odd). Therefore the prime p appears in the prime factorization of the product

(2x+1)(2x+3)(2x+5)(2x+7)

with only an odd power, hence this product cannot be a perfect square. QED

Progress on the Twin Prime Conjecture

June 11, 2013 1 comment

The Chinese mathematician Yitang Zhang made great strides in one of the big problems in number theory called the Twin Prime Conjecture. He did not solve it but proved a related result that is considered quite remarkable — and he got his paper accepted for publication in the very prestigious math journal Annals of Mathematics.

If you wish you can read the sources below regarding the news on this, but I’d like to say things my way.

First, let’s explain some things. The ‘numbers‘ we’re talking about here are mainly positive whole numbers 1, 2, 3, 4, …, (which we call positive integers, or natural numbers). (So we’re not talking fractions here.)

A number is prime when it is not divisible by any other number beside 1 and itself. So for example, 5 is a prime, while 6 is not prime (we call it composite, since it is divisible by 2 and 3, beside 1 and 6).

So the sequence of prime numbers starts off and proceeds like this:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, …

and they go on and on forever. We know that there are infinitely many prime numbers (and the proof is not hard).

Now if you look closely at this list of primes, you notice something interesting. That there are ‘consecutive’ pairs of primes. Ones whose difference is 2.  Like the pair 5, 7, and the pair 11, 13, and the pair 29, 31, etc etc. These are called twin primes.

The Twin Prime Conjecture says that there are infinitely many of these twin pairs of primes.

But unlike Euclid’s proof that there are infinitely many primes (which is easy), the proof of the Twin Prime conjecture is extremely hard or just unknown. (Or maybe it’s not true!) No one has been able (so far) to settle the question or find a proof of it.

Since that is too hard, mathematicians try to ask and maybe settle simpler, but related, questions that they may have a chance at answering.

For example, instead of looking at pairs of primes whose difference is 2, why not look at ones whose difference is some number L?

As a related conjecture, one can ask:

Q: Is there a number L such that there are infinitely many pairs of primes whose difference is no more than L?

When L = 2, this question reduces to the Twin Prime Conjecture. For L = 10, let’s say, we can have pairs like 7, 11, whose difference is 4 which is no more than 10. Another pair would be 17, 23, whose difference is 6 (again no more than 10). Etc.

Now what Professor Yitang Zhang proved is that the answer to question Q is YES! There is such a fixed difference L. He proved that if you take L to be 70,000,000 (70 million) then there are infinitely many pairs of primes whose difference is no more than 70 million.

(So in particular we don’t know if there infinitely many pairs of primes whose difference is not more than 100 (or 1000, or even a million). These would still be interesting but apparently quite hard to answer.)

Once again, we have very simple sounding questions (a babe can ask them!) that are very hard to prove, and indeed we do not even have proofs for them.

REFERENCES:

These two reference are short reports on the result.

Nature MagazineNew Scientist

The following reference is slightly longer and more detailed:

Simon’s Foundation

Understanding Israel

December 9, 2012 Leave a comment

Israel’s decision to make plans on the E-1 region (not yet building, however, just planning at this stage) after the United Nations General Assembly vote on the Palestinian nonmember observer state has been much criticized.  In this post, I express my opinion on the Israeli response by contextualizing the many growing dangers in the region to Israel so as to understand where it is coming from.

My theory is that, yes, Israel has made its E-1 response to the UN vote but that this was idea was at the back of their minds only to be used as the last nail on the coffin amidst a serious growing regional threat. (I am aware of the US position, from back in the 1980s I think, that Israel should not build in E-1.)

Let us enumerate all these dangers that Israel faces to better appreciate its position and response.

1) There is the Iran nuclear threat to wipe Israel off the map. This is probably Israel’s biggest security concern right now – which only serves to justify Israel’s hardening position.

2) There are the regular Hamas rockets and missiles fired at Israel. These have increased in range and danger — e.g., the Fajr-5 rockets supplied by Iran to Hamas in Gaza.

3) There is the Hizbullah threat in Lebanon with 1000s of rockets and missiles aimed at Israel – and which are more deadly and with longer reach. (They can now reach Tel Aviv, Jerusalem, and other Israeli population centers.) Notice: the UN has done nothing about these.

4) There is the Islamist threat in the region, and especially in Egypt. Egypt is lead by the Muslim Brotherhood whose primary foreign policy is to establish an Islamist Caliphate empire with Jerusalem as its capital. There’s a trend to scrap the Peace Treaty. So that Treaty is only hanging by a hair.

5) Syria is quite unstable with its 2-year civil war. The danger exists, and is more probable, for Islamists to take it over the way they have in Egypt, Tunisia, Morocco, etc.

6) Turkey is an Islamist state though somewhat more ‘moderate’ but still quite hostile to Israel — and pretty thin-skinned about the flotilla fiasco which was their fault for not restraining the IHH. They have been confrontational with Israel and sympathetic to Hamas’ firing rockets at Israel. Being a NATO state, that does not bode well for Turkey and is certainly a danger to Israel.

7) Jordan looks to be quiet so far, but it can be affected by these regional trends – 40% of its population are Palestinian. Jordan could be in the news at any moment.

8) Sudan is a major arms depot/route for Iran to transfer and supply weapons to Hamas, Islamic Jihad, and other Islamists who want to infiltrate Israel from both Gaza and the Sinai. (Iran also supplies weapons to Hizbullah.)

You have all these dangers facing Israel and in addition you now have the Palestinians threatening Israel with the diplomatic vote at the United Nations.  All these threats put together from all around do a lot to undermine a two-state solution for a Palestinian state and a Jewish state living side by side in peace. These can only harden Israel’s position.

As a matter of fact, Palestinian collusion with all these regional players who threaten Israel have contributed to making the two-state solution much less viable. (It may in fact be dead.)  What is Israel going to do? Support a hostile Palestinian state right on its border in addition to the already existing threatening Islamist states? Accede to a Palestinian state that looks to become an Islamist Palestinian state?  Not in your life. Look at Gaza! Is that the kind of Palestinian state they’re hoping for? (One that is inspired by Egypt’s Muslim Brotherhood?)  In any case, we have two Palestinian visions and there’s no way for Israel to make peace with both of them if those two Palestinian camps can’t live together in peace.

Therefore, the way to fight the UN Palestinian vote, and to fight these many regional dangers, is to threaten the contiguous geography for the Palestinians. So in sum: no Palestinian state so long as they collude with those who want to wipe Israel off the map, fire rockets at Israel, and threaten Israel diplomatically. Actions not aimed at contributing to peace will not lead to a Palestinian state. Israel will build where it wishes. That, in my view, is a much better response than bloodshed.

The UN can make all the laws it wishes, but so long as it makes these laws without regard to Israel’s security given the all-round dangers, these laws will be one-sided, naive, futile, and they will be opposed and fought. Israel’s laws will take precedence over laws cooked up by an antisemitic body (comprising many nations some of whom are already ruled by thugs).  It will also mean that the UN has yet again failed to be the institution that it was designed to for. After all, the UN already failed several times before; e.g., in Congo, in Darfur, Sudan, in Syria.

How I see prayer

August 28, 2012 2 comments

This is a brief outline of my many approaches / perspectives on prayer. I do not simply see it as making demands of a higher power and expecting a response according to your timetable.

I may pray out of a desire or wish for something, but I do not look nor expect it to occur – and in the way that I may expect. If things work out as I wished, that’s great, and I’m happy; and if they don’t, that’s ok too – I’m modest enough to take a No!

1. prayer has the effect of cleaning one’s heart and soul. It’s like when your very confused about a very troubling personal issue, but then you find a solution that settles it – no confusion. It’s like your house is dirty and now it’s clear and clean.


2. prayer as a way for organizing your life. Similar to meditation.

3. prayer reflects and affirms to yourself how you think about something (or maybe that you should rethink it!). Nurturing a healthy attitude by means of prayer (or meditation) could have a positive impact on how you relate to your circumstances. It could make a difference in your life.

4. when someone is told that I’m praying for them, it is a kindly act and affects how they feel in their hearts and how you feel for them. It is a human way of caring for others.

5. you know the expression “what goes around, comes around.” Maybe by praying and showing goodwill to others, those good things may come back to you. Just as when you care for others, others will show care for you.

6. prayer is also a form of love. When you pray sometimes you express love and affection for the people you think of when you pray for them. What you build in your heart and in your attitude toward people reflects in the way you become a person and how others perceive you and how you perceive them.

7. prayer is a time for ‘divine’ guidance. Or, a time for personal devotion to your thoughts about something and on which you are seeking wisdom and guidance. A way for your thinking to rethink itself, possibly by the help of a Higher source. (That’s what I think the Bible means by “not leaning on your own understanding” – a phrase that might seem confusing.)

8. prayer is also a time for seeking inner (or divine) strength and peace. Life is full of challenges. If prayer can be a tool or weapon for dealing with these challenges, even when we are down and weak, then it can’t be such a bad thing. Many people have been invigorated thru prayer when they were at their lowest moments.

9. why do some people pray when they’re facing a crisis or a stressful situation? Because it can be a life stabilizer; a ‘tool’ to get your mind around personal issues in depth, seeking foresight, and maybe seeking the proper perspective on it.

They say “perspective is everything”. If you’re facing a daunting perspective, you might find another perspective.

10. prayer can shape a person. You can shape yourself from the inside out. How you are on the inside reflects on how you are on the outside. (Jesus said something like this.)

11. you don’t have to be religious or even spiritual to pray. Prayer is a very human activity. It’s a mode of self-communication. A reflection of an evolved mind that knows to seek venues in its mind or outside Source from which it can glean solutions.

I’m sure there are many other ways to express prayer, but these are probably enough to give a flavor of my philosophy on prayer.

Children of older fathers

August 23, 2012 Leave a comment

Sometimes when you put separate studies together you could get what seems a confusing picture (or maybe no picture at all?) — particularly seeing how they word their conclusions. Here is an example on studies related to offspring of older fathers.

Study 1. Older fathers have longer telomeres in their chromosomes as they age, so the offspring of older fathers inherit these longer telomeres, enhancing the life expectancy of the offspring.  This was reported by the BBC and is based on a study published in Proceedings of the National Academy of Sciences.

Study 2. Children born to older fathers have a greater likelihood of developing autism or schizophrenia.  Reported by Nature Magazine (here’s Nature’s summary of the actual Nature study) and also quite recently by the BBC.

Study 3. The life expectancy of people with schizophrenia is lower than average, and can be lower by as much as 10 to 15 years. Reported by the BBC and based on a study done by the British Biomedical Research Centre for mental health, and published in the online journal PLoS ONE.

 
These conclusions aren’t necessarily in contradiction to one another (even if verbally they seem to be). It depends on the various rates (such as the smaller fraction of schizophrenics, for example, compared with the many who don’t get mental illness). So if you’re normal and descended from an older father ‘the chances’ are better that your longevity will get a boost (statistically!) if you don’t get a mental illness like schizophrenia. (Barring a Study 4 and Study 5, about which I know nothing and which could wreck my post!)

I’ve often wondered if in some societies around the world these effects and results could be varied depending on the people being studied. There are still numerous factors beyond our control. (Anyway, whatever your condition, I hope this makes you feel better!) 🙂

Earth’s Magnetic Field

August 18, 2012 Leave a comment

 

I know just a little about the Earth’s magnetic field – also called the geomagnetic field. The following are from some notes I wrote a few years ago and came across lately (thought maybe worthwhile sharing in my own words). My notes were based on: Ency. Britannica; Wiki article on geomagnetic field; and ‘chapter 3′ of a physicist’s notes (whose name I’m missing).

1. Magnetic north and Earth’s true north aren’t the same!

2. In fact, it is the magnetic south pole that’s closer to the Earth’s north, by something like 11 degrees. (That is called ‘magnetic declination’, the angle difference from true north.) For precise navigation this 11 degrees could be taken into account.

3. The magnetic field lines (usually written as B in physics) start from magnetic north and end at magnetic south. (At least, that is the convention.) Magnetic fields affect only charged particles (like electrons and protons). These particles move along the field lines by spiraling around them (like a coiled wire).

(They go back and forth. The reason they spiral in doing so is explained by the magnetic force being F = q v x B, where q is the charge on the particle, v is its velocity, and B is the magnetic field. The force is always perpendicular to B and v, which is why they spiral.)

4. The Chinese appear to have been the first to discover the geomagnetic field in their effort to perfect their navigation technology. (About 1100′s AD or so.) Later Sir Edmond Halley (of Halley’s comet) mapped the magnetic field.

5. It was believed 100s of years ago that the geomagnetic field had extra-terrestrial origin. It was the brilliant mathematician Carl Friedrich Gauss (mid 1800s) who showed that the field actually had its origin from the Earth itself, and he gave a mathematical expression for it (using spherical harmonics).

6. The magnetic field of the earth can experience reversal — where magnetic north and south poles are interchanged. But this happens irregularly, from some 700,000 or so years to a few million years. (I think it’s still a mystery as to why this happened in the past.) These reversals are recorded in rocks that register the field’s direction in the past. And in turn, this has been valuable in determining the history and motion of plate tectonics – and discovery of the mid ocean ridges which affect continental motion.

7. The motion of molten iron in the core of the Earth is generally credited for the creation of the geomagnetic field — this is called the geodynamo theory of Sir Bullard (about 1940s-50s). (The Earth’s crust has its contributions too, but they are fairly smaller.) Scientists study the inside of the Earth in part from how the geomagnetic field behaves and changes.

8. There are still some mysteries about geomagnetism. For example, why do the magnetic poles move about 10 km each year?

Why I’m supporting Mitt Romney

August 10, 2012 Leave a comment

Here are my short & quick reasons for supporting Mitt Romney for US president.

1. Move to repeal Obamacare. Some businesses are already raising commodity prices because of it (e.g., one pizza chain) — and food prices are already on the rise outside of that.

2. Give the US economy a boost — a change in leadership helps give the economy a better turn. I’m with former mayor of Carmel, California, Mr Clint Eastwood who said we need a change in leadership and is endorsing Romney. I think he’s right. (And I won’t argue with the man with no name.)

3. Get the unemployment rate down, as it was under Bush. It went up when Obama became president.

4. Stronger and assertive foreign policy — not one that apologizes and bows to other leaders in shame. Many in Europe and Mideast are now taking a lesser view of Obama than in 2008 — his weakness loses the respect a president ought to have. Stand up to China and Russia more. The reset button gotten rusty and ain’t working. (Remember the reset button Hillary gave Putin?) Stand up to Iran more, and treat our friend Israel as a friend.

 

IAEA recent report: Iran enriched U-235 to 27%

The IAEA has just released its report on Iran’s nuclear program. (PDF file.)

On page 6 of this report it says:

“The results of analysis of environmental samples taken at FFEP on 15 February 2012 showed the presence of particles with enrichment levels of up to 27% U-235, which are higher than the level stated in the DIQ.”

(FFEP = Fordow Fuel Enrichment Plant; DIQ = Design Information Questionnaire.)

The report also says that Iran installed 100s more centrifuges (in Production Hall A).

This has been happening even while the Ayatollah said it was a sin to develop a nuclear weapon and while they denied working for that goal.  If this does not stop, I expect a war soon.

Iran enriching uranium to over 20%

The IAEA has just recently reported that samples from their visit to Iran in February 2012 show that Iran has enriched uranium to more than 20% and may even be as high as 27%.  BBC reportJPost report.

Update: the second day meeting of the P5+1 and Iran failed to reach agreement on key issues, as expected. They’re planning a second meeting in Moscow in mid June. More play for time.

EU-Iran nuclear meeting goes for 2nd day

The description of the P5+1 meeting with Iran (Wed May 23 2012) on its nuclear program as “atmosphere was businesslike” is actually the diplomatic jargon used for a rather tense scene — seeing that discussions failed to yield agreement (contrary to Mr Amano’s premature optimism the day before that agreement was expected “quite soon”).

That’s why they’ve made an unscheduled 2nd day meeting on Thursday. Of course, in the meantime, between the last time they met, over a month ago, and now, Iran gained lots more time to install and spin its centrifuges and continue to enrich uranium further. The West now looks like the idiots who would be fooled twice, thrice, as many times as Iran dictates. Of course, Israel is the only nation that knows Iran better than the other (bleeding heart) wimps. Israel’s suspicions are in my view quite valid. This farce has been going on for 9 years and the West is still falling for it. Outrageous.