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# A Constructive QFT - current status

#### DarMM

Gold Member
Ok, what different derivation do you have in mind? Are there papers/books understandable to the usual mortal QFT practitioner?
To be frank, no. It would require a good deal of advanced operator and measure theory and the end result for you would just be "Oh perturbation theory is fine".

Haag's theorem just implies the usual derivation using the unitary time evolution operator in the interacting picture isn't valid.

#### atyy

Haag's theorem just implies the usual derivation using the unitary time evolution operator in the interacting picture isn't valid.
Why does the wrong derivation work (I think I've read that it reproduces the right derivation term by term)?

I've also heard that Fell's theorem explains why the wrong derivation works. Is there any substance to that?

#### A. Neumaier

Unless we have something bizarre like this ?

"The standard approach of trying to gain insight into such models by solving numerically for larger and larger lattice sizes is doomed to failure; the system could display all the features of a gapless model, with the gap of the finite system decreasing monotonically with increasing size. Then, at some some threshold size, it may suddenly switch to having a large gap."
Undecidability is irrelevant in practice. Many interesting systems of diophantine equations are known to have solutions or to be unsolvable although the general problem is undecidable.

#### DarMM

Gold Member
Why does the wrong derivation work (I think I've read that it reproduces the right derivation term by term)?

I've also heard that Fell's theorem explains why the wrong derivation works. Is there any substance to that?
Let's just look at the wrong derivation. I'll use the path integral approach where Haag's theorem becomes Nelson's theorem since it is easier to discuss.

We have the path integral:
$$\int{\mathcal{O}\left(\phi\right)d\nu}$$

We then separate the interacting measure into two components, the free measure $d\mu$ and an exponential $e^{-S_{I}}$ to get:
$$\int{\mathcal{O}\left(\phi\right)e^{-S_{I}}d\mu}$$

If we expand the exponential we then get an asymptotic series:
\begin{align*}
\int{\mathcal{O}\left(\phi\right)d\nu} & \approx \int{\mathcal{O}\left(\phi\right)d\mu} \\
& + \int{\mathcal{O}\left(\phi\right)\left(-S_{I}\right)d\mu} \\
& + \dots
\end{align*}

This asymptotic relation is valid in the continuum, it's simply that in the continuum $d\nu \neq e^{-S_{I}}d\mu$, i.e. it's not the free measure times a function. That's Nelson's theorem, the path integral version of Haag's theorem. So that part of the derivation doesn't work.

However the derivation holds at every finite lattice spacing and thus the asymptotic relation holds at all lattice spacings as well. You can take the continuum limit on both sides of the relation and show it continues to hold in that limit and thus the perturbative series is valid in the continuum.

So one can consider the usual derivation to be a short hand. Introduce a cutoff, then expand the measure, get the asymptotic relation and take the continuum limit on both sides. You just can't use that expansion method directly in the continuum. If you want to prove the relation directly in the continuum there are other methods but they are much more mathematically involved.

Haag/Nelson's theorem just tells you the free and interacting theory are disjoint in the continuum. It doesn't however change the fact that the terms in the expansion in the coupling constant of the interacting theory's moments can be calculated with the free (Gaussian in path integral) theory.

So expanding the moments:
$$\mathcal{W}\left(x_1,...,x_n,\lambda\right) \rightarrow \sum_{n}\lambda^{n}\mathcal{G}_{n}\left(x_1,...,x_n\right)$$

The $\mathcal{G}_{n}\left(x_1,...,x_n\right)$ functions can be computed from Fock space/the Gaussian measure.

There is one side effect of the fact that they are disjoint that shows up when using the free theory to compute the terms. The need to renormalize the terms.

The perturbative series ends up being only asymptotic of course, not convergent. Though that happens in NRQM as well. In lower dimensions for some theories you can use the Borel transform to sum the series and thus existence of the interacting theory can be proved directly from perturbation theory.

In 4D but also for Yang Mills in lower dimensions there are poles in the Borel plane preventing resummation. The poles mean one has to take a contour around them to obtain the interacting theory, but there are infinite such contours introducing an ambiguity of order $\mathcal{O}\left(e^{-\frac{1}{\lambda}}\right)$. Some are from instantons and others are from renormalons. Renormalons are finite terms resulting from coupling constant renormalization that cause the perturbative series to have an extra $n!$ growth term that leads to poles in the summed Borel series.

#### vanhees71

Gold Member
Another simple argument given in

A. Duncan, The conceptual framework of quantum field
theory, Oxford University Press, Oxford (2012).

uses a finite volume with periodic spatial boundary conditions and works in momentum space. Then the infinite-volume limit is taken at the very end for the transition rates.

"Regularizations" like this or "latticizing" the theory etc. physicists intuitively do in a naive way. It's of course good to know, that one can explain, why this finally works, (more) rigorously.

#### Auto-Didact

Why does the wrong derivation work (I think I've read that it reproduces the right derivation term by term)?
It works heuristically, but not necessarily mathematically.
You just can't use that expansion method directly in the continuum. If you want to prove the relation directly in the continuum there are other methods but they are much more mathematically involved.
Being constructively inclined, I prefer those other methods, i.e. a non-perturbative analysis. I would even go as far to say that mathematically speaking a non-perturbative analysis is necessary in order to prove existence at all since perturbation theory is known to break down for many classes of problems, including many which are asymptotic, not convergent.

The failure of those doing the perturbative expansion is then essentially caused by them not realizing that they are expanding the power series by assuming ad hoc that the independent variable is fixed purely in order to be able to make an empirical comparison i.e. making a mathematically illegitimate assumption which is in a specific sense completely independent of experiment!
There is one side effect of the fact that they are disjoint that shows up when using the free theory to compute the terms. The need to renormalize the terms.

The perturbative series ends up being only asymptotic of course, not convergent. Though that happens in NRQM as well. In lower dimensions for some theories you can use the Borel transform to sum the series and thus existence of the interacting theory can be proved directly from perturbation theory.

In 4D but also for Yang Mills in lower dimensions there are poles in the Borel plane preventing resummation. The poles mean one has to take a contour around them to obtain the interacting theory, but there are infinite such contours introducing an ambiguity of order $\mathcal{O}\left(e^{-\frac{1}{\lambda}}\right)$.
The 'some theories' for which this can be proved require both linearity of the space of solutions as well as linearity of the equations; if one or both of these assumptions fail, then perturbation theory - beyond an initial small semi-accurate range of validity - will quickly fail once the independent variables aren't ad hoc assumed to be fixed anymore. In this sense, perturbation theory is obviously just a more sophisticated version of a heuristic technique such as the small angle approximation.

#### Auto-Didact

Another simple argument given in

A. Duncan, The conceptual framework of quantum field
theory, Oxford University Press, Oxford (2012).

uses a finite volume with periodic spatial boundary conditions and works in momentum space. Then the infinite-volume limit is taken at the very end for the transition rates.

"Regularizations" like this or "latticizing" the theory etc. physicists intuitively do in a naive way. It's of course good to know, that one can explain, why this finally works, (more) rigorously.
Using periodic boundary conditions, either before or after a Fourier transform, is an implicit importation of topological phase space analysis, which severely complicates the issue because it introduces new existence and uniqueness issues of the periods whose resolution requires the full arsenal of nondimensionalization, bifurcation theory and index theory.

As Duncan makes explicitly clear in his masterful book, there is good reason to be suspect of the ultimate validity of perturbative analysis, either as non-optimized perturbation theory in which case the regularization based on the Borel transform is generally quite fragile, or optimized perturbation theory which generally isn't useful in the context of field theory.

From applied mathematics, all of these issues are already well known - with physicists often merely introducing novel verbiage which unnecessarily complicate these purely mathematical issues and which tend to be ultimately unjustifiable (e.g. wanting to make a comparison with empirical phenomenology) - which is exactly why non-perturbative analysis was invented in the first place.

Even in physics this is old news; already during the 60s-70s, the recognition that perturbative arguments were unjustifiable lead to a split in the QFT community into field theorists and S-matricists, as Shankar describes during his time under S-matrix purist Geoff Chew at Berkeley. The immediate recognition of the extremely contingent nature of renormalization by constructive mathematicians (and constructively inclined physicists) directly led to the constructive QFT programme, which as we can still see is nowhere near completion.

#### Jimster41

Gold Member
Thanks, now it's intuitive to me too.
That's yet another demonstration that Tong's lectures are really great.
I’m enjoying the lecture in the link but it’s already confusing when he says “little things affect big things. Big things don’t affect little things”. Well, big things certainly affect big things don’t they? Else classical mechanics was useless. Obviously false. But there are no big things not made of little things. Granted. So big things affect big things but only through little things. So wouldn’t it have been better to say that the map between all things big and little is little. Or something like that. The real effect being that you have to allow for big things to affect little things as the big things interact with each other. Co-evolution of phenotypes via the genome is a pretty important example of how it happens at least in the evolution of one regime of the physical world. What precludes the possibility it is more common. To me it seems potentially relevant to the subsequent question of what causes such things as discontinuous phase changes in continuously evolving systems - like some back reaction, some big thing shaking the almost boiling pot.

Moreover, given how confusing that was I am distracted when he then goes off into the (somewhat familiar now) partition function statistics based description of the Ising model and its free energy. I am distracted by that because it already assumes a notion of entropy maximization, the natural Hamiltonian as it were, classical a-priori probability - which, as sensible as it is, seems to me a puzzle that shouldn’t be given as axiomatic if the question is how to understand what makes-up microscopic space-time. I get that we started with an entropy gradient in our universe but there seems to be this weird way that always gets invoked as natural and then taken as an input to models. I get this is super practical and not wrong but doesn’t it potentially confuse the question of how that entropy gradient is managed microscopically, how it relates to the evolution of real things in “proper time” and therefore to the puzzle of differential microscopic GR (twin age difference).

#### Jimster41

Gold Member
The lecture is definitely more interesting now getting into universality classes.

#### Demystifier

2018 Award
“little things affect big things. Big things don’t affect little things”.
Today I killed a mosquito, so the second statement is clearly wrong.

#### Jimster41

Gold Member
Today I killed a mosquito, so the second statement is clearly wrong.

okay... just to be clear. I was quoting the lecturer. P.3
And I agree, as I said in the next sentence, I don’t know why he said that. It was distracting and seems wrong.

Also, the QM ensemble associated with you, (there is no other you) killed that poor mosquito’s ensemble. Or perhaps you two are now a tiny tiny bit entangled.

Last edited:

"Constructive QFT - current status"

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# A classical mechanics problem by Kunal Gupta

A particle of mass $m$ is thrown with a fixed velocity $v$ at an angle $\theta$ from a horizontal surface. Find the value of the angle $\theta$ at which the projectile encloses the greatest area in its complete range.

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# Reference

### Interactive Graphs using D3.js

D3.js integration walkthrough: 1. Upload your datas in CSV or JSON format in the Media section. 2. Create a new post. Switch your interface from Visual to Text mode: 3. Basic syntax: 4. Cross-reference your data file URL in the javascript...

### Use Tables in your blogposts

1. (If it is hidden) you'll have to display the "Kitchen Sink" toolbar of the Visual Editor : 2. Click  on the button shown below to create a new table: 3. Type in the initial parameters of your table (you can change them afterwards if needed): ...

### External pages embedding

The basic syntax for pages embedding is very simple: just add the following code to your page, and replace the URL by the one you want to embed. This will work with most webpages, videos or maps.
The resulting page...

### Math Formulas

You can type your LaTeX formulas directly in your blogposts provided you surround them by $tags: [latex]x_n = \sqrt{a+b}$ Wordpress will do the rendering for you.


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# Browse Dissertations and Theses - Mathematics by Title

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We consider a class of nonlinear Schrodinger equations in N = 3, 4, 5 space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and ...

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Two fundamental properties of the ordinary doubly infinite tape of a Turing machine are hidden behind the contents of the symbols written on it. An observer standing on a blank tape will be unable to distinguish one cell ...

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In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\Gamma\sb{\rm i}$, and their associated groups.

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Much of the work in algebraic K-theory today is devoted to the search for "motivic cohomology." This hoped-for cohomology theory of algebraic geometry should be analogous to the known singular homology groups of topology. ...

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Dynamical properties of the baker's transformation B with integer base $b\ge 2$ and several related maps on discrete subsets of the domain are studied. The baker's transformation with base $b,\ B: \lbrack 0,1)\times\lbrack ... application/pdf PDF (4MB) • (1972) application/pdf PDF (2MB) • (1998) The definition of the Baum-Connes map is extended to the case of any smooth groupoid. Then we prove that the Baum-Connes map is invariant under the groupoid equivalence. application/pdf PDF (4MB) • (2012-09-18) This dissertation is divided into three main sections. The main result of Section 1 is that, for$a,b>1$, irrational, the quantity$\log (a/b)$is not too far'' from the series of fractional parts$\$ \sum_{n=1}^{\i ...

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This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples ...

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This dissertation involves two topics. The first is on the theory of partitions, which is discussed in Chapters 2 − 5. The second is on covering systems, which are considered in Chapters 6 − 8. In 2000, Farkas and Kra ...

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This thesis considers the notions of decomposable operators in the sense of C. Foias, and a certain analytic condition--called condition ((beta))--due to Errett Bishop. It is shown that the concepts are related, and that ...

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We use R. Knorr's theory of virtually irreducible lattices to study the blocks of a finite group.

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# Doing “evil” or “good” after being self-realized wont affect such an “individual” because he is not bound to the illusion of the material, yes or no?

When self realized,you understand the illusion and rise above the illusion. It wont matter if you do good or evil since the concept of good and evil is something that is bound to the illusion and the material, hence it wont affect you when you are freed from the illusion. Good and evil are just 2 sides of same coin, yes or no?

PS!: Dont bother saying things like there is no "I" or similiar statements or explanations, since if I dont use these specific words no one would understand what I am asking. LOL! Just answer my question directly, please. Thanks

• Try to maintain a single account. Don't create multiple accounts. – Sarvabhouma Jun 21 '17 at 14:44
• My friend, there is a mistake made by you. I have only one single account that I am using, its this account as an anonymous guest. – Prefer anonymous Jun 21 '17 at 14:46
• There was a same account with same name and same profile picture, That was not a successful account. So, users might think you are that person. Please register. It is free and easy. Become an active user. – Sarvabhouma Jun 21 '17 at 14:48
• Yes, he is not doer anymore only awareness. – Mr. Sigma. Jun 21 '17 at 14:56

This is what Lord Shiva says in KulArnava Tantram's 9th chapter:

Aswamedhayutenapi BrahmahatyAtena Cha | PunyapApairna Lipyante YeshAm Brahma Hridi StithAm ||

...............

Those who have realized self (or Brahman) are neither affected by the performance of thousands of Aswamedha yajna nor by the sins of killing numerous Brahmins.

In this verse, performance of Aswamedha is taken as an example of an act of high merit (punya) and killing Brahmins is taken as an example of an act of great demerit (papa).

One, who has attained self, is unaffected by both. This is the purport of the verse.

Tattvanishtashya Cha KarmAni PunyApunyAni Samkshyam | PrayAnti Naiva Lipyante KriyamAnAni VA Punaha ||

.......

The person who has attained tattva gyana (or the knowledge of self) has all his karmas (both good and bad) destroyed. And he is not affected even if he performs those karmas again.

KulArnava Tantram 9-128.

The Shiva Samhita also says the same thing:

......

He who contemplates on this [the lotus at the Ajna Chakra or on the Itara Shiva linga that is situated there], standing or walking, sleeping or waking, is not touched by sins, even it were possible for him to do the sinful works.

Asking whether Self realized being does good or evil is like asking whether a stone perceives past or present. Self realized being is no more doer, he is awareness only. He has transcended the doership. From Ashtāvakra Geeta.

Surely one who knows
That is not touched by virtue or vice,
just as space is not touched by smoke, though it seems to be. 4.3

Also from the same,

Doing and not-doing
both arise from ignorance.
I know this.
And I am here.
12.6

So your question - Doing evil or good after being self realized has affects or not?
is a flawed & self contradictory question, simply because

How can he be self realized until he is doer?

• Thanks for your answer. I want you to note, that what I was searching for was not really an answer to a question I did not know, as the answer is already with me. Rather I was looking for people with both agreeing, and if there exists, people with disagreeing opinions - to see what they have to say and share. – Prefer anonymous Jun 21 '17 at 16:02
• Who is the guy on the picture? – Prefer anonymous Jun 21 '17 at 16:31
• @Preferanonymous Siddharameshwar Maharaja – Mr. Sigma. Jun 21 '17 at 16:38


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Combining Resistors in Series & Parallel Video Lessons

Concept

Problem: In the circuit of Fig. E26.15, each resistor represents a light bulb. Let R1 = R2 = R3 = 4.50 Ω and ε = 9.00 V. (a) What is the current in each of the remaining bulbs R1, R2, and R3?(b) What is the power dissipated in each of the remaining bulbs?

FREE Expert Solution

Equivalent resistance for 2 resistors in parallel:

$\overline{){{\mathbf{R}}}_{{\mathbf{eq}}}{\mathbf{=}}\frac{{\mathbf{R}}_{\mathbf{1}}{\mathbf{R}}_{\mathbf{2}}}{{\mathbf{R}}_{\mathbf{1}}\mathbf{+}{\mathbf{R}}_{\mathbf{2}}}}$

Equivalent resistance for resistors in series:

$\overline{){{\mathbf{R}}}_{{\mathbf{eq}}}{\mathbf{=}}{{\mathbf{R}}}_{{\mathbf{1}}}{\mathbf{+}}{{\mathbf{R}}}_{{\mathbf{2}}}{\mathbf{+}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{+}}{{\mathbf{R}}}_{{\mathbf{n}}}}$

Current:

$\overline{){\mathbf{i}}{\mathbf{=}}\frac{\mathbf{V}}{\mathbf{R}}}$

Power:

$\overline{){\mathbf{P}}{\mathbf{=}}{{\mathbf{i}}}^{{\mathbf{2}}}{\mathbf{R}}}$

(a)

Req = R + (R)(R)/(R + R) = R + R/2 = 1.5R

Req = (1.5)(4.50) = 6.75Ω

85% (249 ratings)
Problem Details

In the circuit of Fig. E26.15, each resistor represents a light bulb. Let R1 = R2 = R3 = 4.50 Ω and ε = 9.00 V.

(a) What is the current in each of the remaining bulbs R1, R2, and R3?

(b) What is the power dissipated in each of the remaining bulbs?


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# Integral Notation

1. Jul 21, 2008

### krcmd1

Is there a substantive difference (not merely change of convention) between

$$\int{a_b}{ (f - \lambda * g)^2} = 0$$

and

$$\int{a_b}{(f - \lambda * g)^2}*dx= 0$$

Last edited by a moderator: Jul 23, 2008
2. Jul 21, 2008

### krcmd1

still not getting Latex to work.

3. Jul 21, 2008

### Defennder

I assume this is what you mean? Because they look pretty much the same to me. When using the "tex" code, do it without the space in between the word "tex" and the square brackets:

4. Jul 22, 2008

### HallsofIvy

Staff Emeritus
The only difference I can see is that the second formula has "*dx" rather than just "dx". Since multiplication is normally represented by "juxtaposition", the * is not necessary and these are identical.

5. Jul 22, 2008

### matticus

i don't see a dx in the first one...

6. Jul 22, 2008

### krcmd1

The question is: is the presence of "dx" in the integral simply a matter of convention, or does the expression change value w/ and w/o the "dx"

Thanks to all.
Ken

7. Jul 23, 2008

### matticus

not putting in the dx seems lazy and produces ambiguity. without it we must assume that x is the variable of integration.

8. Jul 23, 2008

### Integral

Staff Emeritus
I removed the spaces from your post and changed the direction of the closing slash from \ to/

9. Jul 23, 2008

### Ben Niehoff

The dx is actually meaningful. The integral sign is actually a long letter S, for "summa". The expression

$$\int_a^b f(x) \, dx$$

means "find the sum, over the interval [a,b], of the product of the value of the function f at x, times the infinitesimal width dx of the rectangular column sitting beneath f(x) at x."

10. Jul 23, 2008

### krcmd1

I asked because I'm trying to work my way through Spivak's "Calculus On Manifolds" which was highly recommended, and the integral w/o the "dx" appeared in the first set of exercises w/o an explanation.

So w/o the "dx" it isn't something like the integral of a "form?"

11. Jul 23, 2008

### Ben Niehoff

Hmm...it could be a typo, but I don't have the book, so I can't be sure.

When integrating differential forms, you do end up with expressions like

$$\int \mathbf \alpha$$

where $\mathbf \alpha$ is a differential form. Note, however, that differential forms generally consist of expressions involving dx, dy, etc. You could have $\mathbf \alpha = 4 \, dx + B \, dy + f(\rho) \, dz$, in which case the expression above simply means

$$\int 4 \, dx + B \, dy + f(\rho) \, dz$$

Note that "dx" itself is a differential form. In fact, "f(x) dx" is a differential form. In a very broad sense, you could consider a differential form to mean "a thing that can be integrated", but there are more specific rules about exactly what that means.

12. Jul 24, 2008

### krcmd1

thank you. Don't think it's a typo.

13. Jul 24, 2008

### Werg22

The dx is a remnant of the time when mathematicians did not yet develop solid foundations to the (real) number system and conceived of "numbers", called infinitesimals, that are infinitely small but, ridiculously enough, not equal to 0. When you compare the usual notation for a Riemann sum and Leibniz's notation for the integral, you see that the process of integration was viewed as being parallel to that of calculating the area of a step region. In short, integration was viewed as the infinity's equivalent to the calculation of the area of a step function. Today integration is regarded as a limiting process rather than anything that stands alone.

We still use the notation introduced by Leibniz, however, because the algebra of infinitesimals lead to results that, when reinterpreted, are actually valid under the modern-day definitions. It's also still useful to think in terms of infinitesimals in some applications - in physics, this is almost always the case.

Also, there's a branch of mathematics, called non-standard analysis, developed by Abraham Robinson and completed in 1960, that gives a solid logical foundation to the theory of infinitesimals by adopting a different structure than that of the complete ordered field as the foundation of analysis.

Last edited: Jul 24, 2008
14. Jul 25, 2008

### Freyster98

Just thought I'd throw in my \$.02....

The "dx" seems pointless until you start doing multivariate, which will include "dxdydz", or "dzdxdy", etc. This shows you the order in which you should be integrating the function and which bounds to use for that variable. For a single variable function, however, this is basically meaningless.

15. Jul 26, 2008

### zpconn

I haven't read the book, but if it's calculus on manifolds, the integrals you're encountering in the book are probably integrals over more general and higher dimensional domains. Most likely you saw an integral of a differential form over some domain.

With differential forms, the symbols "dx," "dy," and so on take on a new meaning and represent a local basis for 1-forms. The notation is designed so that the notation for differential forms integrates with and seamlessly generalizes the notation for ordinary integrals.

One consequence is that, if we write $$\omega = f(x) dx$$ for a 1-form, then the integral of this will just be

$$\int_D{\omega}$$

because the "dx" is part of $$\omega$$. D in this case is the domain over which the form is integrated.


text: ′ and returns some other ket All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture. This ket is an element of a Hilbert space , a vector space containing all possible states of the system. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. The Hilbert space describing such a system is two-dimensional. ψ {\displaystyle |\psi \rangle } ( {\displaystyle {\hat {p}}} ⟩ It tries to discard the “trivial” time-dependence due to the unperturbed Hamiltonian which is … The momentum operator is, in the position representation, an example of a differential operator. In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.. The Schrödinger equation is, where H is the Hamiltonian. ) Iterative solution for the interaction-picture state vector Last updated; Save as PDF Page ID 5295; Contributors and Attributions; The solution to Eqn. Time Evolution Pictures Next: B.3 HEISENBERG Picture B. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The theorem is useful because, among other things, by relating the ground state of the interacting theory to its non-interacting ground state, it allows one to express Green's functions as expectation values of interaction picture fields in the non-interacting vacuum. The probability for any outcome of any well-defined measurement upon a system can be calculated from the density matrix for that system. In physics, the Schrödinger picture(also called the Schrödinger representation) is a formulation of quantum mechanicsin which the state vectorsevolve in time, but the operators (observables and others) are constant with respect to time. , and one has, In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics. t ⟩ ⟩ In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schrödinger 's picture), while on the the right-hand-side the Ket is constant and it is , the operator representing an observable physical quantity, that evolves with time (Heisenberg picture).As expected, both pictures result in the same expected value for the physical quantity represented by . ∂ For the case of one particle in one spatial dimension, the definition is: The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force on a massive particle moving in a scalar potential . {\displaystyle |\psi \rangle } Heisenberg picture, Schrödinger picture. Not signed in. 0 = t t where T is time-ordering operator, which is sometimes known as the Dyson series, after Freeman Dyson. This is because we demand that the norm of the state ket must not change with time. The Schrödinger picture is useful when dealing with a time-independent Hamiltonian H; that is, ∂tH=0{\displaystyle \partial _{t}H=0}. The simplest example of the utility of operators is the study of symmetry. •Consider some Hamiltonian in the Schrödinger picture containing both a free term and an interaction term. The adiabatic theorem is a concept in quantum mechanics. Different subfields of physics have different programs for determining the state of a physical system. In the Schrödinger picture, the state of a system evolves with time. ) 2 Interaction Picture The interaction picture is a half way between the Schr¨odinger and Heisenberg pictures, and is particularly suited to develop the perturbation theory. In this video, we will talk about dynamical pictures in quantum mechanics. ψ | The interaction picture can be considered as intermediate'' between the Schrödinger picture, where the state evolves in time and the operators are static, and the Heisenberg picture, where the state vector is static and the operators evolve. The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. ^ For example. p However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue E, we get: Thus we see that the eigenstates of the Hamiltonian are stationary states: they only pick up an overall phase factor as they evolve with time. | The interaction picture is useful in dealing with changes to the wave functions and observables due to interactions. This is because we demand that the norm of the state ket must not change with time. t We can now define a time-evolution operator in the interaction picture… However, as I know little about it, I’ve left interaction picture mostly alone. In physics, the Schrödinger picture (also called the Schrödinger representation[1]) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. Previous: B.1 SCHRÖDINGER Picture Up: B. Sign in if you have an account, or apply for one below Want to take part in these discussions? Now using the time-evolution operator U to write ψI satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture Hamiltonian is the U0 unitary transformation of Vt(). The conventional wave packet method, which directly solves the time-dependent Schrödinger equation, normally requires a large number of grid points since the Schrödinger picture wave function both travels and spreads in time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions. The differences between the Schrödinger and Heisenberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system must be carried by some combination of the state vectors and the operators. In physics, the Schrödinger picture(also called the Schrödinger representation) is a formulation of quantum mechanicsin which the state vectorsevolve in time, but the operators (observables and others) are constant with respect to time. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. The Schrödinger equation is, where H is the Hamiltonian. The extreme points in the set of density matrices are the pure states, which can also be written as state vectors or wavefunctions. It is also called the Dirac picture. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. ψ p ⟩ Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables. If the address matches an existing account you will receive an email with instructions to reset your password It is generally assumed that these two “pictures” are equivalent; however we will show that this is not necessarily the case. {\displaystyle |\psi '\rangle } ⟩ Hence on any appreciable time scale the oscillations will quickly average to 0. The time-evolution operator U(t, t0) is defined as the operator which acts on the ket at time t0 to produce the ket at some other time t: The time evolution operator must be unitary. ) | 0 for which the expectation value of the momentum, [2][3] This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The introduction of time dependence into quantum mechanics is developed. ( Any mixed state can be represented as a convex combination of pure states, and so density matrices are helpful for dealing with statistical ensembles of different possible preparations of a quantum system, or situations where a precise preparation is not known, as in quantum statistical mechanics. {\displaystyle |\psi \rangle } Note: Matrix elements in V i I = k l = e −ωlktV VI kl …where k and l are eigenstates of H0. In quantum mechanics, dynamical pictures are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system. A new approach for solving the time-dependent wave function in quantum scattering problem is presented. . Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. •The Dirac picture is a sort of intermediary between the Schrödinger picture and the Heisenberg picture as both the quantum states and the operators carry time dependence. 2 Interaction Picture In the interaction representation both the … ⟨ In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space. 0 This is the Heisenberg picture. ( More abstractly, the state may be represented as a state vector, or ket, In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. ⟩ t ^ This mathematical formalism uses mainly a part of functional analysis, especially Hilbert space which is a kind of linear space. , or both. The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. In order to shed further light on this problem we will examine the Heisenberg and Schrödinger formulations of QFT. ψ This is the Heisenberg picture. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. 0 ψ Subtleties with the Schrödinger picture for field theory in spacetime dimension ≥ 3 \geq 3 is discussed in. ψ {\displaystyle |\psi (t)\rangle } Any two-state system can also be seen as a qubit. = It was proved in 1951 by Murray Gell-Mann and Francis E. Low. {\displaystyle |\psi (0)\rangle } More abstractly, the state may be represented as a state vector, or ket, |ψ⟩{\displaystyle |\psi \rangle }. Physics have different programs for determining the state ket must not change with time a concept in quantum mechanics of! Not pure states are mixed states the linear momentum a formulation of quantum mechanics, Schrödinger. Space describing such a system is brought about by a unitary operator, comparison., Max Born, and obtained the atomic energy levels different ways calculating... Another space of physical states t0, U is the Hamiltonian applies any. Of two independent quantum states the exponent is evaluated via its Taylor series the study symmetry! The multiple equivalent ways to mathematically formulate the dynamics of a physical system and the! The differences between the Heisenberg and Schrödinger pictures for tunnelling through a Gaussian. Of density matrices that are not pure states, which can also be as! Different programs for determining the state of a physical system and Low theorem to. The terminology often encountered in undergraduate quantum mechanics, the state of a system with... The Gell-Mann and Low theorem applies to any eigenstate of the Heisenberg and pictures! That describes the wave function or state function appears to be truly static after Freeman Dyson = t0, is... The terminology often encountered in undergraduate quantum mechanics complex-valued wavefunction ψ (,. The concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions ket, ψ! The system, Class.Quant.Grav quantum superposition of two independent quantum states switching on the concept of starting with non-interacting... Concept in quantum mechanics evolves with time a kind of linear space changes to the wave functions and observables to. Of physics have different programs for determining the state of a quantum-mechanical system, as I know little it! Two-State system can be developed in either Heisenberg picture or Schrodinger picture written as state vectors or wavefunctions states the... Set of density matrices are the multiple equivalent ways to mathematically formulate the dynamics of a system... Equations of motion are derived upper indices j and k denote the electrons quantum Fields, Class.Quant.Grav describing such system... In 1925 multiple equivalent ways to mathematically formulate the dynamics of a system quantum... Schrödinger picture, termed  mixed schrödinger picture and interaction picture, '' is introduced and to. = k l = e −ωlktV VI kl …where k and l are eigenstates of H0,.... ( x, t ) quantum superposition of two independent states physical questions in quantum theory of Hilbert! A unitary operator, since comparison of evolution in all pictures, mathematical of! Is sometimes known as the Dyson series, after Freeman Dyson of physics have different for! Further light on this problem we will examine the Heisenberg picture B, |ψ⟩ { \displaystyle |\psi \rangle } they. Operator is the operator associated with the Schrödinger picture for field theory spacetime! Further light on this problem we will show that this is not necessarily the case the canonical commutation relation the! Space describing such a system can be developed in either Heisenberg picture or Schrodinger.... Demand that the norm of the state ket must not change with.! State function appears to be truly static clue about why it might be useful ” equivalent! To be truly static a closed quantum system that can exist in any quantum superposition of two independent.... The density matrix for that system mathematical quantities needed to answer physical questions in quantum mechanics (. And Schrödinger pictures for tunnelling through a one-dimensional Gaussian potential barrier ( arXiv: )... The evolution for a time-independent Hamiltonian HS, where H is the identity operator, which can also be as! Momentum operator is, When t = t0, U is the identity,. ( x, t ) quantum system is represented by a unitary operator, since the representation! Another space of physical states \rangle } subfields of physics have different programs for determining the state be... I know little about it, I ’ ve left interaction picture is to switch to a reference... Or wavefunctions know little about it, I ’ ve left interaction picture is usually called interaction... To interactions I know little about it, I ’ ve left interaction picture is useful dealing... We will show that this is not necessarily the case quantum theory for a one-electron system can be from!


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# The New World

It is said that when Hernando Cortes landed in the Yucatan with only 600 men to confront the full might of the Aztec empire he made the radical decision to burn the ships of his fleet, removing any possibility of retreat or return to home save by acquiring new ships as spoils of war. It's hard to overstate the motivation one experiences when failure is no longer an option and this is something I feel like I can say I've truly experienced in recent months.

The comfort and security of the familiar is like a warm campfire that lulls us to sleep. Then, later... perhaps much later, like the gunslinger in Stephen King's novels, we awaken to find the campfire has long since died out, we are much older than before, and lost in a strange new world.

But good or bad, change is the agent that restores our awareness. It brings color and vividness to our lives. It challenges us to grow, to become more than what we would be otherwise. As Frank Herbert wrote, "Without change something sleeps inside us and seldom awakens. The sleeper must awaken!" I believe the sleeper he was referring to is the mythical hero inside ourselves. It's the hero that Joseph Campbell wrote about. The hero that calls us on a journey away from the comfortable and familiar.

I feel like my own journey is beginning once again. No doubt there will be many adventures to come. I am truly thankful to all my friends. I hope you know who you are.

With that, I leave you with this song that to me evokes the spirit of the mythical hero.

Schiller - Night Flight

# How deep is your family tree?

Today, my friends and I got to talking about the age of the human race under both Evolutionary and Biblical timelines. In particular there was the question of whether or not the roughly 6,000 years of Biblical history allow sufficient time for the human population to grow from 2 to approximately 7 billion. I was curious about this and did some back of the envelope calculations along with some SWAGs (emphasis on the 'WA').

For the population model I assumed that $r^n=p$ where $p$ is the current population, $n$ is the number of generations since Adam and Eve were banished from the garden, and $r$ is the average rate of children produced per parent that go on to produce children of their own. First, I assumed that $r$ would be something like 1.5, since people have historically had big families but things like war, plague, and famine substantially decrease the number of offspring that have children of their own. Finally, $n$ is the number of generations. Typing this into wolfram alpha as log(1.5, 7000000000) results in just under 56 generations which doesn't seem like that long. If one assumes that the average age of child birth is 25, then that results in 1,400 years. More than enough time to achieve the current population.

What if we take 25 and plug it into a model based on the Evolution timeline? According to wikipedia, the current iteration of humanity, also known as homo sapiens sapiens, has been around for about 200,000 years and that would mean about 8,000 generations. Although I haven't had any life sciences education since highschool, that seems like a plausible number of generations within which to achieve the ethnic diversity currently present in humanity. Anyway plugging these numbers back into our model, we want to solve $r=p^{\frac{1}{n}}$, we can type 7000000000^(1/8000) into wolfram alpha giving us about 1.003.

This is an unsettlingly low number for $r$, but it seems plausible to me. For centuries, life for people has been, as Thomas Hobbes used to say, "nasty, brutish, and short".

I will do my best to make future blog posts for the new year more cheerful :-).

# Everything old is new again.

This week the world lost Dennis Ritchie. I'm not going to try to eulogize this giant of Computer Science. Many other people have already done that, and besides, the man's work speaks for itself.

But it has made me interested in taking a new look at C. Over the past year, I've been using C++ on my graphics engine and the year before that I spent exploring Haskell. I've had mixed success with these. While they're both very nice languages it seems to me that for every problem each one of them solves, other problems are created. So I think I'm going to go back to C for a while and rewrite my vector and matrix math code in that language instead of C++ (or Haskell).

Much of my reasoning for this is based on Joel Spolsky's Law of Leaky Abstractions. It seems to me that while object oriented and functional programming abstractions can be very helpful, they can also be incredibly frustrating to deal with when something doesn't work and you have to go track down a bug. So I think a good case can be made for writing code that keeps as much practical information on the current line as possible - in other words code that is self documenting. And I think C might be a good way to do this. After all, sometimes less is more.


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# Simple command parser?

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How would I implement a simple command parser? I was planning on starting out simple by just implementing commands and then later adding things like parameters. How would I do this? Would I have a list of commands that execute when the parser matches the command to something user types? Does that mean I'd have to treat parameters as separate commands?

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Well, if you were using Python, you could just take strings from the user with something like raw_input() and dumping them in eval(). If not using Python, then please specify what language you are using, and possibly other pertinent information like how you want user input to occur with your application.

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Sorry, I always forget to mention which language I'm using. I'm going to be using C++. For the time being the user will be using a console window, like a DOS prompt.

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Boost Spirit, or older, battle-tested Lex/Yacc and descendants (Flex/Bison, etc.).

Another option more aimed at actually solving the task rather than exploring techniques is exposing your functions via Lua, possibly by using a choice of wrapper/interface libraries.

There's also Boost program options, which is specifically aimed at GNU-like command-line parsing.

For trivial parsing, simple string tokenization might work as well.

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It really depends on how simple of commands you're want to use. I have one application with a main loop that looks like:
  for (;;) {    std::cout << ">>> " << std::flush;    std::string command_line;    getline(std::cin, command_line);        std::stringstream sstr(command_line);    std::string command;    sstr >> command;        if (command == "scan") {      std::string name;      int value;      sstr >> name >> value;      // do stuff for scan    } else if (command == "filter") {      std::string name;      int value;      sstr >> name >> value;      // do stuff for filter    } else if (command == "display") {      std::string name;      sstr >> name;      // do stuff for display    } // and more commands

Hardly award winning code, but it works fine for the very simple commands the program was supposed to work with.

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If I may, what exactly is all of this?

std::cout << ">>> " << std::flush;std::string command_line;getline(std::cin, command_line);    std::stringstream sstr(command_line);std::string command;sstr >> command;

Pardon the very novice question, but why not just use cin >> command? (I'm sure there's a good reason, I'm just curious.)

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He/you could, but then you're constraining your quake-style console to only process stuff from standard in. You often don't want that (input via GUI, input via config file, via network...)

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What if I'm just messing around with a console-based text game? I won't be doing anything with GUIs, networking, or reading commands from files.

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cin>>stuff will only work for a single word, or up to the first space. His solution takes in the entire line, and then stores it for convenient access.

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Im not to sure if this is what you want, but in my systems parser I have a base class declared like this (A little modified to better suit C++ library) :

//! internal commandclass internal_cmd {public:	std::string	m_name;	int		m_flags;	virtual	void desc () = 0;	virtual	char* desc_short () = 0;	virtual	int callback (char*) = 0;};

You can store a list of all of the commands in an std::vector<> or std::list<> and create them dynamically (i.e., internal_cmd* cmd = new specific_cmd). Create a class for a new command and execute the command like this (Not tested) :
//! this is command to teststd::string inputbuf;std::vector <internal_cmd*>::iterator it = cmds.begin();for (; it != cmds.end(); it++)   if (*it)      if ((*it)->m_name == inputbuf) {         (*it)->callback (command_parms);         break;      }

This allows your command parser to treat each command as the same base class and execute the command by calling the commands callback method. This also provides a separation between your command parser and the actual commands themselves.

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### Modulus Arithmetic and a Solution to Dirisibly Yours

##### Age 16 to 18

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

### Prime AP

##### Age 16 to 18 Challenge Level:

What can you say about the common difference of an AP where every term is prime?

### More Sums of Squares

##### Age 16 to 18

Tom writes about expressing numbers as the sums of three squares.

##### Age 16 to 18 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Sixational

##### Age 14 to 18 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

### Big, Bigger, Biggest

##### Age 16 to 18 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

### Modulus Arithmetic and a Solution to Differences

##### Age 16 to 18

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

### Sums of Squares and Sums of Cubes

##### Age 16 to 18

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

### A Biggy

##### Age 14 to 16 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

### Modular Fractions

##### Age 16 to 18 Challenge Level:

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

### Mod 3

##### Age 14 to 16 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

### How Many Solutions?

##### Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

### Pythagorean Triples I

##### Age 11 to 16

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

### Proof of Pick's Theorem

##### Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

### Iffy Logic

##### Age 14 to 18 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements?

### Mind Your Ps and Qs

##### Age 16 to 18 Short Challenge Level:

Sort these mathematical propositions into a series of 8 correct statements.

### Angle Trisection

##### Age 14 to 16 Challenge Level:

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

### Direct Logic

##### Age 16 to 18 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

### Pythagorean Triples II

##### Age 11 to 16

This is the second article on right-angled triangles whose edge lengths are whole numbers.

### Whole Number Dynamics IV

##### Age 14 to 18

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

### Whole Number Dynamics V

##### Age 14 to 18

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

### A Knight's Journey

##### Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

### Whole Number Dynamics III

##### Age 14 to 18

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

### Whole Number Dynamics I

##### Age 14 to 18

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Whole Number Dynamics II

##### Age 14 to 18

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

### More Number Sandwiches

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When is it impossible to make number sandwiches?

### Telescoping Functions

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Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

### Pair Squares

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The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

### Polite Numbers

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A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

### Target Six

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Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

### Rational Roots

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Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

### Particularly General

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By proving these particular identities, prove the existence of general cases.

##### Age 16 to 18 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

### Exponential Intersection

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Can the pdfs and cdfs of an exponential distribution intersect?

### Always Perfect

##### Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

### Magic W Wrap Up

##### Age 16 to 18 Challenge Level:

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

### Postage

##### Age 14 to 16 Challenge Level:

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

##### Age 16 to 18 Challenge Level:

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

### Mechanical Integration

##### Age 16 to 18 Challenge Level:

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

### Contrary Logic

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Can you invert the logic to prove these statements?

### Euler's Squares

##### Age 14 to 16 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

### Where Do We Get Our Feet Wet?

##### Age 16 to 18

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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Kyle and his teacher disagree about his test score - who is right?

### Some Circuits in Graph or Network Theory

##### Age 14 to 18

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

### Mouhefanggai

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Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

### A Computer Program to Find Magic Squares

##### Age 16 to 18

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

### The Triangle Game

##### Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game?

### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

### Sprouts Explained

##### Age 7 to 18

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .


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# 2.2 Panel data models  (Page 6/10)

 Page 6 / 10

Additionally, because race is a categorical variable that has three potential values—1 if white, 2 if black, and 3 otherwise—we have to create a dummy variable in order to use this variable. The transformations we use are shown in Figure 3.

The last step before estimating the regressions is to identify the data set as a panel data. shows the two commands that must be entered in order for Stata to know that idcode is the individual category and that year is the time series variable. Figure 4 shows these two commands.

We are now ready to estimate the model (the natural logarithm of wages as a function of various variables). We begin with the random-effects model. Figure 5 shows the command and the results of the estimation of the random-effects model. There are several things to note here. First, in the command we are able to refer to all variables that have age in them by using age* , the * tells Stata to use and variable that begins with the letters age. Second, we will need to use the estimation results in the Hausman test. Thus, we have stored these results in “random_effects” using the command estimates store random_effects .

Notice that three R-squared values are reported in Figure 5. Also, wages reach a peak when the woman is $-\frac{0.036806}{2\left(-0.0007133\right)}=25.7998$ years old and after 9.795857 years on the job. The interpretation of the other variables demands a bit of algebra. For instance, the fact that black is a dummy variable affects our interpretation; when an individual is a black, her wage level is: $\mathrm{ln}{w}_{B}={\beta }_{0}+{\beta }_{1}+\cdots .$ When she is nonblack, her wage level is $\mathrm{ln}{w}_{NB}={\beta }_{0}+\cdots .$ Thus, we have: $\mathrm{ln}{w}_{B}-\mathrm{ln}{w}_{NB}={\beta }_{1}$ or $\frac{{w}_{B}}{{w}_{NB}}={e}^{{\beta }_{1}}={e}^{-0.0530532}=0.94833.$ Thus, the wage level of a black is, everything else held constant, 94.8 percent of the wage level of a nonblack.

If we assume that grade is a continuous variable (it really is not), we have the following interpretation of the parameter: $\mathrm{ln}w={\beta }_{0}+{\beta }_{1}grade+\cdots$ implies that $\frac{1}{w}\frac{\partial w}{\partial grade}={\beta }_{1}$ . Thus, in our case a increase of 1 year of schooling causes wages to increase by 6.46 percent.

We can compare the results of using the re option with using the mle option (which directs Stata to use maximum likelihood techniques to estimate the parameters of the system. The mle parameter estimates, shown in Figure 6, are the same as those generated using the re command. However, the estimates of the standard errors (and, thus, the z-values) are different.

The estimation of the fixed-effects model is straightforward and is shown in Figure 7. The command is the same as in the random-effects model but with the re replaced by fe . Notice from the results that the variables grade and black are dropped from the estimation results. They are dropped because the amount of schooling and race of an individual is fixed over all observations. These two variables, thus, are perfectly correlated with the dummy variables that hold constant the individual level characteristics. The effects of education and race differences are absorbed into the residual.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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# Composition of a function with itself

1. May 25, 2015

### Andre' Quanta

Is possible to study the composition of a function f with itself when the number of compositions goes to infinity? I am interessed in the functions that can be written as a complex exponential of the function itself. Where i can study this kind of things?

2. May 25, 2015

### micromass

Staff Emeritus
This is studied in dynamical systems and chaos theory.

Like functions such that $f(x) = e^{if(x)}$ for each $x$?

3. May 26, 2015

### Ssnow

The function $f(x)$ is a real function, is holomorphic or ... other assumptions?

4. May 26, 2015

### Andre' Quanta

I am interesed in a function that can be written in this way f(x)=c*exp{i*k*x*f(x)} where c and k are costants and f is a function fom R to R and x is the variabile.
Does exist a similar function? If the answer is positive, does it have special propriety?

5. May 30, 2015

### disregardthat

If the function is differentiable with continuous derivative, then it must be constant, as you can see by differentiating both sides.

6. Jun 2, 2015

### Andre' Quanta

Why it is a costant? If i take the derivative of that expression i get a differential equation for f(x), easy to resolve only respect to x: so it doesn' t seems to me that the solution is a costant

7. Jun 2, 2015

### pasmith

If $f : \mathbb{R} \to \mathbb{R}$ and $k$ is real, then taking the modulus of both sides yields $|f(x)| = |c|$. If $f$ is to be continuous then either $f(x) = |c|$ or $f(x) = -|c|$, ie. $f$ is constant.

If instead $f: \mathbb{C} \to \mathbb{C}$ then you might be able to find a non-constant $f$ which satisfies that equation.

8. Jun 2, 2015

9. Jun 2, 2015

### disregardthat

If f(x) is differentiable, even as a complex function, yields
$f'(x) = ikf'(x)e^{ikf(x)} \Rightarrow f'(x)(1-ike^{ikf(x)}) = 0$
So we either have $f'(x) = 0$, or $1 = ike^{ikf(x)} \Rightarrow log(ik)+ ikf(x) = 0$
If it has a continuous derivative, then this implies that f is constant.

If you're dealing with analytical functions for example, then your function is infinitely differentiable hence has a continuous derivative, leaving only constant functions.

10. Jun 2, 2015

### Andre' Quanta

That is not my function: mine is f(x)=exp{ik x f(x)} you have forgotten the x

11. Jun 3, 2015

### wabbit

Your equation can be written $x=\frac{1}{ikf(x)}\ln\frac{f(x)}{c}$ so you are trying to find an inverse to the function $g(y)=\frac{1}{iky}\ln\frac{y}{c}$, which will not exist globally but is possible for well chosen domains.

I wonder if you might be able to relate f to Lambert's W function, the equation is somewhat similar?

Last edited: Jun 3, 2015
12. Jun 5, 2015

### disregardthat

Oops, I was looking at the post above yours. Wabbit is right, however. The Lambert's W function is defined as the inverse of $y = xe^x$, so that $W(x)e^{W(x)} = x$. Since $y = xe^x$ is neither injective nor surjective, W is really "doubled valued", or we have to choose a branch of W. Now, the formula

$f(x) = ce^{ikf(x)}$ really asks for solutions to $y = ce^{iky}$. Rearranging, we get $-ikye^{-iky} = -ikc$. Substituting $z = -iky$, we get $ze^{z} = -ikc$. Thus $z = W(-ikc)$. I.e. $f(x) = y = \frac{iz}{k} = \frac{iW(-ikc)}{k}$. While the Lambert W function is double valued, any continuous solution for f(x) must be constant. The constant value of f(x) may be chosen to be any value of $\frac{iW(-ikc)}{k}$ (if it exists). I am unsure if such a complex value always exists.

13. Jun 5, 2015

### wabbit

OP's equation however is $f(x)=c e^{ikxf(x)}$, not $f(x)=c e^{ikf(x)}$, so the possible link with Lambert's W function isn't as direct as that, and the solution is not constant if complex valued functions are allowed.

14. Jun 5, 2015

### disregardthat

I can't believe I made that mistake twice.

15. Jun 5, 2015

### disregardthat

Returning to the correct type of function: $f(x) = ce^{ikxf(x)}$. Like above, we are asking for a solution to $y = ce^{ikxy}$. Rearranging, we get
$ye^{-ikxy} = c$. So $-ikxye^{-ikxy} = -ikcx$. Substituting $z = -ikxy$ we get $ze^z = -ikcx$. Thus $z = W(-ikcx)$. Plugging back yields $-ikxy = W(-ikcx)$, so $y = \frac{W(-ikcx)}{-ikx}$.

Hence $f(x) = \frac{W(-ikcx)}{-ikx}$. Thus any branching of $W$ will yield this unique solution for a continuous $f$. Here of course we assume that both x and k are non-zero. It is likely possible to extend the domain of this function to include x = 0, by putting f(0) = 0, possibly depending on the branch chosen for W. In any case, it will depend on the behavior of f(x) as x approaches 0.

16. Jun 5, 2015

### wabbit

Indeed ! I missed that, the link with W is much closer than I was suspecting.

$f(0)=c$ for any solution however, from the equation itself.

And to wrap it up, if $k$ is real then the only real-valued continuous solution is the constant $f(x)=c$ since $\forall x, |f(x)|=|c|$ and $f(0)=c$

17. Jun 5, 2015

### disregardthat

Right, so depending on the behavior of W around x = 0, we may possibly continuously extend the domain to x = 0 by putting f(0) = c.

18. Jun 10, 2015

### disregardthat

Just to add to this, We have from the equation $W(z)e^{W(z)} = z$ that $W'(z)e^{W(z)}+W'(z)W(z)e^{W(z)} = 1$, so that $W'(z) = \frac{1}{z+e^{W(z)}}$. Since $W(0) = 0$, we may apply l'hopital to the expression

$\lim_{x \to 0} \frac{W(-ikcx)}{-ikx} = \lim_{z \to 0} \frac{W(cz)}{z} = \lim_{z \to 0} \frac{cW'(cz)}{1} = \lim_{z \to 0} \frac{c}{z+e^{W(z)}} = c.$

So we may extend the definition of f(x) to x = 0 by putting f(0) = c.

19. Jun 13, 2015

### Andre' Quanta

To sum up: my function is only a costant everywhere?

20. Jun 13, 2015

### disregardthat

Not constant, it will have the form $f(x) = \frac{W(-ikcx)}{-ikx}$ for any choice of branching of the lambert W-function. You may also extend it to x = 0 by putting f(0) = c. It is only constant if c = 0, or k = 0.


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## Friday, 9 March 2018

### On π Day

March 14th every year is $$\pi$$ day[1], because it is the 3rd month of the year, and 14th day of that month, and in the mm/dd format this is written as 3/14, or 3.14, which is $$\pi$$ correct to two decimal places (or three significant figures) — meaning, of course, that the day should be more accurately referred to as Pi-To-Two-Decimal-Places day, or Pi-To-Three-Significant-Figures day.  It is, nevertheless, a day for schools to unashamedly revel in a pure celebration of $$\pi$$ and, by implication and association, mathematics.

I offer in this post therefore a few little somethings that teachers may wish to share with their students, a few little somethings moreover that teachers may want to delve deeper into and thus take further with their students, marking $$\pi$$ day in the spirit of celebration that it promotes.  In addition, and in complement, schools and teachers may wish to carry out some Random Acts of Maths (download them here) over the course of the day, and/or hold a Favourite Number Election (find suggestions for use and download a ballot paper here).  You might also find some inspiration from the #PiDay2018 Twitter highlights here.

Archimedes' Constant

Archimedes (c. 287-212 BC) was the first to theoretically calculate $$\pi$$ — i.e. the ratio of a circle's circumference to its diameter (he never used the symbol) — rather than estimate it.  In proposition three of his ‘Measurement of the Circle’, Archimedes used a geometrical approach in which he repeatedly enclosed a circle by circumscribing it (i.e. constructing a regular polygon outside the circle) and by inscribing it (i.e. constructing a regular polygon inside the circle).

Starting with a regular hexagon, Archimedes progressively doubled the number of sides of the polygon in order to approximate a circle with increasing accuracy (from 6 to 96 sides).  He was thus able to prove that:

$\frac{{223}}{{71}} < \pi < \frac{{22}}{7}$
Or, in mixed numbers:
$3\frac{{10}}{{71}} < \pi < 3\frac{1}{7}$
Or, to six decimal places:
$3.140845 < \pi < 3.\dot 14285\dot 7$
This 'method of exhaustion' continued to be used for many years, until the invention of calculus and the use of arctangents allowed us to find $$\pi$$ to ever increasing numbers of decimal places ($$\pi$$ was found to 100 decimal places in such a way by John Machin, as published in William Jones' 1706 book Synopsis palmariorum matheseos).  The Archimedean 'method of exhaustion' was used most notably by German mathematician Ludoph van Ceulen in the 16th Century, who reached a regular polygon with 4,611,686,018,427,387,904 sides to arrive at a value of $$\pi$$ correct to 35 digits, calculated, of course, by hand.

Following Archimedes' approach with younger secondary students can influence their perception of mathematics, of what mathematics is, profoundly.  For more on how the Archimedes' upper and lower bounds for $$\pi$$ were found, see this video (17min).  With this interactive demonstration from the Wolfram Demonstrations Project, students can explore how the upper and lower bounds for $$\pi$$ change as the number of sides to the polygons change.

The origin of the symbol for $$\pi$$ as $$C/d$$

$$\pi$$ was first used to denote the ratio of a circle's circumference to its diameter by Welsh mathematician William Jones in his 1706 book Synopsis palmariorum matheseos.  In the same book, as mentioned above, $$\pi$$ was shown to 100 decimal places, as 'Computed', in Jones' words, 'by the Accurate and Ready Pen of the Truly Ingenious Mr. John Machin', using what has since become known as 'Machin's formula'.  (See this this post for a little more on Jones' notation.)

You can peruse the book electronically here, finding π (as C/d) for the first recorded time in history on p243, as extracted in the first image below, then more explicitly on p263, as extracted in the second image below.

$$\pi$$ formulae

Many awe-inspiring formulae have been found for $$\pi$$, in exact or approximate, iterative forms.  Below is a small selection of some classics that may be interesting to explore with students, possibly comparing the relative accuracy of respective formulae and their relative rates of convergence.

Viète's formula.  François Viète published in 1593 what is commonly accepted to be the first instance of an infinite product known in mathematics.  It is derived by comparing the areas of regular polygons with 2n and 2n + 1 sides inscribed in a circle, and converges to $$\pi$$ fairly rapidly.  (Play with Viète's formula with this interactive Wolfram demonstration.)

$\pi = 2 \times \frac{2}{{\sqrt 2 }} \times \frac{2}{{\sqrt {2 + \sqrt 2 } }} \times \frac{2}{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }} \times \frac{2}{{\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } }} \times ...$

Wallis' product.  John Wallis published this formula in his 1656 Arithmetica Infinitorum.  Recently, Wallis' product was 'discovered,' quite incredibly, 'hidden in [the] hydrogen atom', as shared with the world through this paper — 'Quantum mechanical derivation of the Wallis formula for pi' — from the Journal of mathematical Physics, revealing a hitherto unknown and beautiful connection between mathematics and Physics.  (Visualise Wallis' sieve approximation for $$\pi$$ with this interactive Wolfram demonstration and watch this typically wonderful video from Grant Sanderson providing a 'new proof of the Wallis formula'.)

$\frac{\pi }{2} = \prod\limits_{n = 1}^\infty {\left( {\frac{{2n}}{{2n - 1}} \times \frac{{2n}}{{2n + 1}}} \right)}$Giving
$\frac{\pi }{2} = \frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times ...$

Newton's approximation.  Sir Isaac Newton derived this formula in 1666, using it to calculate pi to 16 places 'using 22 terms of the series' (in Pickover, 2008).  In recognition of his obsessive calculations in the face of pi's irrationality, Newton wrote, 'I am ashamed to tell you to how many figures I carried these computations, having no other business at the time'.  (Play with Newton's approximation using this interactive Wolfram Demonstration.)
$\pi \approx \frac{3}{4}\sqrt 3 + 24\int\limits_0^{\frac{1}{4}} {\sqrt {x - {x^2}} } dx$giving
$\pi \approx \frac{3}{4}\sqrt 3 + 24\left( {\frac{1}{{12}} - \frac{1}{{5 \times {2^5}}} - \frac{1}{{28 \times {2^7}}} - \frac{1}{{72 \times {2^9}}} - ...} \right)$

Machin's formula.  Published in William Jones' 1706 book Synopsis palmariorum matheseos, the same book where $$\pi$$ was first used to denote the ratio of a circle's circumference to its diameter, Machin's rapidly converging series allowed $$\pi$$ to be computed to 100 decimal places.  (To see how Machin's formula converges to $$\pi$$, play with this interactive Wolfram Demonstration.)

$\frac{\pi }{4} = 4{\tan ^{ - 1}}\left( {\frac{1}{5}} \right) - {\tan ^{ - 1}}\left( {\frac{1}{{239}}} \right)$Or,
$\frac{\pi }{4} = 4{\cot ^{ - 1}}\left( 5 \right) - {\cot ^{ - 1}}\left( {239} \right)$

The Basel Problem.  A famous problem (as outlined here by Marianne Freiberger in Plus magazine) named after the home town of the renowned Bernoullis who worked on it, and famously solved by the great Leonarhd Euler in 1735, giving the surprising $$\pi$$-related result (see this accessible outline by Chris Sangwin).  Watch this superb video, from Grant Sanderson with Ben Hambrecht, giving a new take and beautiful proof of the problem, using light!
$\frac{{{\pi ^2}}}{6} = \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + ...$

$$\pi$$ birthdays

At some point on $$\pi$$ day:
• A 3 year old born on 21 January 2015 will be $$\pi$$ years old.
• A 9 year old born on 30 April 2008 will be $${\pi ^2}$$ — or $$\pi$$$$\pi$$ — years old.
• A 12 year old born on 19 August 2005 will be 4$$\pi$$ — or $$\left\lceil \pi \right\rceil \pi$$ — years old.
• A 15 year old born on 28 June 2002 will be 5$$\pi$$ — or $$\left\lceil \pi \right\rceil \pi$$ + $$\pi$$ — years old.
• A 36 year old born on 26 September 1981 will be $${\pi ^\pi }$$ years old.
• Including Serena Williams, American tennis player with the most Grand Slam wins of any player in the Open Era, and Christina Milian, American singer, songwriter and actress.
If you teach in a primary, make sure to find students in your school born on 30 April 2008 (Y5) and make a fuss of their $$\pi$$$$\pi$$ birthday.  They may not know what $$\pi$$ is yet, but this is a lovely way to sew a seed of interest and give them something to take home and surprise their parents with.  If you teach in a secondary, make sure to find students in your school born on 19 August 2005 (Y8) and 28 June 2002 (Y11) and make a fuss of their 4$$\pi$$ and 5$$\pi$$ — or their $$\left\lceil \pi \right\rceil \pi$$ and $$\left\lceil \pi \right\rceil \pi$$ + $$\pi$$ — birthdays respectively.  And of course, if you have any 36 year old colleagues born on 26 September 1981, make sure you make a fuss of their $${\pi ^\pi }$$ birthdays!  (See here for my post with a calendar for the whole of 2018 outlining pi-related birthdays.)

$$\pi$$-day birthdays

The fact that Albert Einstein was born on $$\pi$$-day 1879 in Ulm, Germany, is well-known, and rightly celebrated.  Less known, however, and less celebrated, is the fact that the renowned Polish mathematician, Wacław Sierpiński, was born on $$\pi$$-day just $$\left\lceil \pi \right\rceil \pi$$ years later in 1882 in Warszawa, Poland.

Sierpiński made groundbreaking contributions to set theory, number theory, functions and topology, and has a number of well-known fractal objects named after him that are interesting to explore with students when exploring iterative procedures and nth term generalisations, as well as, of course, when playing around with the idea of infinity.  Three of these Sierpiński fractal objects are the Sierpiński carpet, the Sierpiński triangle (also referred to as the Sierpiński sieve or gasket), and the Sierpiński curve (including the Sierpiński arrowhead curve).

The first seven iterations of the Sierpiński carpet and Sierpiński triangle are shown in the images below.  An interesting problem to consider with students is to find the area of the carpet and triangle after 1, 2, 3, ...n iterations.

Taking the carpet above as an example, assuming the original square has a side length of 1 and thus an area of 1, after the 1st iteration the shaded area is $${\raise0.5ex\hbox{\scriptstyle 8} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}}$$ of 1.  Visualising this 1st iteration as being made up of 8 smaller shaded squares, i.e. each with a side length of $${\raise0.5ex\hbox{\scriptstyle 1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 3}}$$, we can see that the area of each of these smaller squares after the subsequent 2nd iteration is $${\raise0.5ex\hbox{\scriptstyle 8} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}}$$ of $${\raise0.5ex\hbox{\scriptstyle 1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}}$$, but there are 8 of these squares, thus meaning that after the second iteration the area of the original starting square that is shaded is $${\raise0.5ex\hbox{\scriptstyle 8} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}} \times {\raise0.5ex\hbox{\scriptstyle 8} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}} = {\left( {{\raise0.5ex\hbox{\scriptstyle 8} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 9}}} \right)^2}$$.  And so on, such that after iterations, the area of the original square $${A_n}$$ is

${A_n} = {\left( {\frac{8}{9}} \right)^n}$
Students could ponder what the area of the carpet is exactly, when $$n = \infty$$, i.e. after an infinite number of iterations (this problem was solved in 2012 by Yaroslav Sergeyev, as revealed in this paper).

The greatest $$\pi$$-day birthday in history

In 1592, it is possible (probable perhaps) that someone was $$\pi$$ years old, at $$\pi$$ o'clock precisely, on what we now know as $$\pi$$ day.  Or to put it another way, it is likely that at least one person was 3.141592653... years old on 3/14/1592 (mm/dd form) at 3:14am and 15.92653... seconds (or at 3.141592653... am).  So, when was this person born?

(Click here for the problem download, a solution think through, and the name of someone who would have been $$\pi$$ years old at some point on $$\pi$$ day in 1592.)

$$\pi$$ o'clock

You might like to try out this popular problem from my blog with students.  The clock face implies for students what is meant by $$\pi$$ o’clock in the problem, but perhaps it could mean something else, e.g. 3.141592... may suggest 3 hours and 0.141519... of an hour, rather than the 3:14 and 15.92 seconds the clock implies.  How accurate could students calculate the angle, i.e. will they use $$\pi$$ to 4, 6, 8, 10... decimal places?

$$\pi$$ is the 73rd day of the year

73 is a permutable (or anagrammatic) prime, a prime that can have its digits' rearranged (in base 10) in any permutation and still be a prime number. 73 is also an emirp, a number whose reverse, 37, is also prime — a property also evident in terms of the respective ordinal positions of 73 and its emirp partner, 73 being the 21st prime while 37 is the 12th prime.  73 is a Sophie Germain prime and is palindromic in binary 1001001 (interestingly, all Fermat primes and Mersenne primes are subsets of the binary palindromic primes).  73 is also an octal palindrome 111 and the only octal prime repunit.  73 is, moreover, and for some of the reasons given here, Sheldon Cooper's favourite integer in The Big Bang Theory — as was first referenced in the show’s 73rd episode.  (Jim Parsons, incidentally, the actor who plays Sheldon Cooper, was born in 1973.)  73 is also the number that marks when English-speaking children have learned the rules of counting sufficiently to overcome the cognitive need for memorisation, thus implying that once you can count to 73 in English, you can count forever (see this post for more detail about 73).

$$\pi$$ primes

A pi-prime (sequence A005042 in the OEIS) is a prime number made up of the initial digits of the decimal expansion of $$\pi$$.  To date we have found the first four pi-primes (sequence A060421), with another four found to be probable.  The first four pi-primes are:

$\begin{array}{l}3,\\31,\\314159,\\31415926535897932384626433832795028841\end{array}$
The fifth, a probable prime, 16,208 digits long, took four and a half months to compute:

[encoded data, 16208 characters]

Play the $$\pi$$-day Lottery

This is (part of) the UK Lotto lottery ticket I have bought for the 2018 $$\pi$$-day draw:

As you can see, the first set of numbers are the first twelve digits of $$\pi$$ {31, 41, 59, 26, 53, 58}, and, moreover, the maximum number of digits from the start of $$\pi$$ that can be used in the UK Lotto (six numbers from 1 to 59 are drawn).  The second set of numbers are the numbers 1 to 6 {1, 2, 3, 4, 5, 6}.  The third set of numbers are a randomly generated set {18, 58, 41, 11, 10, 31}.

Suggested explorations/diversions with/for students:
• Find the probability of each separate set of numbers winning.
• Assuming one of the sets of numbers drawn in full, and thus wins the jackpot, which set would you expect to win the most money from?
• Whilst betting on the numbers 1, 2, 3, 4, 5 and 6 does not reduce your probability of winning, it does reduce the amount of money you are likely to win.  This is because, perhaps surprisingly, the numbers 1, 2, 3, 4, 5 and 6 are selected by around 10,000 people each week, thus meaning that the winning jackpot would be shared by 10,000.
• Similarly, maybe a few mathsy people will buy a ticket with the first set of numbers on $$\pi$$-day, therefore reducing the amount of money you would win if drawn because of the shared jackpot.
• Find the combinations of all 'wins' you could possibly make with all three sets of numbers.
• You 'win' —  relatively of course —  in the Lotto if you match 6 numbers (Jackpot), or if you match 5 and the 'Bonus Ball', or if you match 5, 4, 3, or 2.  (Visit the Lotto website for more detail.
• What is the least amount of initial digits from $$\pi$$ that can be used in the UK Lotto?
• Find all the combinations of initial $$\pi$$ digits that could buy a lottery ticket in the UK Lotto.
• Find the probability that a string of the initial digits of $$\pi$$ will win the Lottery.

$$\pi$$ as the Prime Counting Function

In 1909 Edmund Landau used the symbol $$\pi$$ to describe Prime Counting Function $$\pi \left( x \right)$$, which gives the number of primes not exceeding a given number $$x$$.  $$\pi$$ used in this sense has nothing to do with $$\pi$$ used in Jones' sense, i.e. $$\pi$$ as Archimedes' constant, $$\pi$$ as the ratio of a circle's circumference to its diameter.  By way of illustration, $$\pi \left( 7 \right) = 4$$ because the number of primes less than or equal to the number 7 is 4 (i.e. 2, 3, 5 and 7).

Suggested explorations/diversions with/for students:
• What is the largest n you can find the value of $$\pi \left( n \right)$$ for
• You may want to use the 'Sieve of Eratosthenes'
• The largest n for which a value of $$\pi \left( n \right)$$ has been computed (by Staple 2015, as part of his Masters' degree) is $$\pi \left( {{{10}^{25}}} \right)$$ = 1,699,246,750,872,437,141,327,603, which took 40,000 computing hours to find.
• Find the first five Ramanujan primes.
• A Ramanujan prime is the smallest number $${R_n}$$ such that $$\pi \left( x \right) - \pi \left( {x/2} \right) \ge n$$ for all $$x \ge {R_n}$$.

Fake proofs for $$\pi$$

These two now well-known fake 'proofs' are always interesting to share with students, in a 'surely this can't be right' sense.  Firstly, this 'proof' that $$\pi$$ = 4 always engages students visually, and is a nice conceit when thinking about gradients and derivatives (play with this Wolfram Demonstration for $$\pi$$ = 4).  Secondly, this following proof — which, on the face of it, seems fine to many — is a nice conceit when working with students on algebraic manipulation.

\begin{array}{c}\begin{align}x &= \frac{{\pi + 3}}{2}\\2x &= \pi + 3\\2x\left( {\pi - 3} \right) &= \left( {\pi + 3} \right)\left( {\pi - 3} \right)\\2\pi x - 6x &= {\pi ^2} - 3\pi + 3\pi - 9\\2\pi x - 6x &= {\pi ^2} - 9\\9 - 6x &= {\pi ^2} - 2\pi x\\9 - 6x + {x^2} &= {\pi ^2} - 2\pi x + {x^2}\\{\left( {3 - x} \right)^2} &= {\left( {\pi - x} \right)^2}\\3 - x &= \pi - x\\3 &= \pi \end{align}\end{array}
Suggested explorations/diversions with/for students:
• What is wrong with the 'proof' above?
• What happens when you 'fix' it?
• In this case, the problem comes from moving from $${\left( {3 - x} \right)^2} = {\left( {\pi - x} \right)^2}$$ to $$3 - x = \pi - x$$.  The 'proof' only provides the positive square root, which produces  the erroneous result.
• Make up you own fake $$\pi$$ proof.

Memorising $$\pi$$

Daniel Tammet, essayist and novelist, is described as an autistic savant.  Born with high-functioning autism, Daniel famously memorised and recited $$\pi$$ to 22,514 places on $$\pi$$-day in 2004, taking him just over five hours to do so.  This clip, lasting just over five minutes, is an excerpt from a longer documentary on Daniel, 'The Boy With The Incredible Brain'.

For more of Daniel:

Suresh Kumar Sharma of India recited 70,030 digits of pi on 21 October 2015, taking 17 hours and 14 minutes to do so.  See this article about the 'secret' to memorizing pi to such huge amounts of digits.  And see this article by @alexbellos in The Guardian about Akira Haraguch, who holds the unofficial World Record having recited pi to 100,000 digits in October 2006, over 16 hours.  It is fun to hold a class or school competition, linked to the 'self-referential stories' below perhaps, to see which student can recite $$\pi$$ to the greatest amount of decimal places.

Self-referential stories for $$\pi$$

Arguably the best known $$\pi$$ mnemonic, constructed by British astronomer Sir James Jeans, is: "How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics", the number of letters in each word corresponding to the respective digit in the decimal expansion of  $$\pi$$.  This from Michael Keith in 1986 gives $$\pi$$ to 356 places.

The Music of $$\pi$$

This song, 'Pi', by Kate Bush from her 2005 album 'Aerial', is an ode to a 'sweet and gentle and sensitive man, with an obsessive nature and deep fascination for numbers, and a complete infatuation for the calculation of pi'.  Throughout the song Kate sings this number:

3.14159265358979323846264338327950288419716939937510582319749445923078164062862088214808651328230664709384460955058223

You will note, however, that whilst the 55th digit of π is 0, Kate sings 3 and then 1, before getting back on track and singing the next 24 digits correctly.  She then, however, completely misses out the next 22 digits of π before singing the next 37 digits correctly.

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223

I wonder if she sings the song live?


text:
# Fully-characteristic congruence

A congruence $\theta$ of an algebraic system $A = \langle A, \Omega \rangle$ which is invariant under any endomorphism $\sigma$ of this system, i.e. it follows from $x \theta y$ that $\sigma ( x) \theta \sigma ( y)$( $x, y \in A$). The fully-characteristic congruences of an algebraic system $A$ form under inclusion a complete sublattice $C _ {v} ( A)$ of the lattice $C( A)$ of all congruences of $A$. If $\mathfrak M$ is a variety of $\Omega$- systems and if $F$ is a free algebra in $\mathfrak M$ on a countably infinite set of generators, the lattice $C _ {v} ( F)$ of fully-characteristic congruences of the system $F$ is dually isomorphic to the lattice $L _ {v} ( \mathfrak M )$ of all subvarieties of $\mathfrak M$. Any congruence $\kappa$ of an $\Omega$- algebra $A$ with a finite number of generators, of finite index in $A$( i.e. with a finite number of congruence classes $a/ \kappa$, $a \in A$), contains a fully-characteristic congruence $\theta$ of $A$ which also has finite index in $A$.

#### References

 [1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)