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# Why don't we define potential due to a magnetic field?

We define electric potential and gravitational potential and use them quite often to solve problems and explain stuff. But I have never encountered magnetic potential, neither during my study (I am a high-schooler), nor during any discussion on physics.

So, does magnetic potential even exist? According to me, it should because a magnetic field is a conservative one and so we can associate a potential with it? Also if it's defined, then why don't we encounter it as often as we do the others (electric potential, gravitational potential, etc.)?

I have encountered magnetic potential energy only in the cases where a dipole is subjected to a magnetic field. Is magnetic potential limited only to this scenario, or is there a general expression for the magnetic potential?

I should have said this earlier, but don't restrain your answers due to my scope. You can use vector calculus as I am quite familiar with it. Also, this question is meant for everyone, so even the answers which are out of my scope are appreciated.

• FWIW, there is a magnetic vector potential. – Qmechanic Dec 26 '19 at 15:28
• Why do you think that magnetic field is a conservative one? It is not. That's why you don't see magnetic scalar potentials (except in more advanced books) – FGSUZ Dec 26 '19 at 15:46
• @FakeMod you don't need to take the curl of the force, but rather the field (as you started with, but switched midstream). Just consider an infinite current, and it's pretty easy to take the curl around a loop concentrric with the current. Keep it up :) – DeltaG Dec 27 '19 at 0:30
• @FakeMod yes, that's right, which is why you don't learn about one in HS or use it with cons. of E. There's some more info about issues with trying to define a scalar mag. potential and with what you really need to do (vector potential) in the answers below. It's a different beast from the electric field, which has a well-defined scalar potential, due to its irrotational nature. – DeltaG Dec 27 '19 at 5:42
• You can define magnetic scalar potentials. but usually such a thing is useless. – Shing Dec 28 '19 at 11:31

Actually, we do!

It's just that it's not the same "kind" of potential - and the reason for this is that magnetic forces work differently than electric forces.

Magnetic fields, if you know, do not directly exert a force on charged particles, simply for being charged. (They would exert such on hypothetical "magnetically charged particles", but we've never found any to exist.) Rather, the force they exert does one thing only: to change the direction of motion of moving charged particles.

The usual thing we call "potential" is what you can think of as a kind of "specific potential energy": it is the potential energy that a unit quantity of charge has from sitting at a certain place in an electric field, and if the particle moves between two areas of different potential, it gains or loses energy as a result of the shifting - but ever-present - pull of the electric force upon it.

Magnetic fields, though, cause no changes in energy - changing something's direction of motion takes no energy, only speeding or slowing its motion does. Think about how that a bullet fired from a gun doesn't hurt more or less depending on the direction from which it comes, only from how powerful the gun is. In theory, to deflect the bullet from one to the other direction in flight, likewise, would not take any energy (though you would need a rather strong source of deflecting force).

But nonetheless, that doesn't mean you can't still describe them using something like a potential, but it doesn't have quite the same meaning any more. As I just see you mentioned you've tried some vector calculus, I'll give this a shot. You see, there's a kind of "duality", if one will, between two operations one can do with at least three-dimensional vectors: the dot product and the cross product, which gives rise to related differential notions of the divergence and gradient, vs. curl, respectively.

The usual ideal of a "potential", that is, for a field like the electric field (and also the Newtonian gravitational field), is based on the following result. "Under certain reasonable conditions", the following implication holds true. If $$\mathbf{F}$$ is some sort of force field, and

$$\oint_C \mathbf{F} \cdot d\mathbf{l} = 0$$

for all closed paths $$C$$, then there exists a scalar function $$V$$ (i.e. with 3 real spatial coordinate arguments and outputting one real number) such that

$$\mathbf{F} = -\nabla V$$

Intuitively, the first equation is a kind of "conservation of energy": the left hand integral is, in effect, a work integral if $$\mathbf{F}$$ is serving as a force field, describing the amount of energy gained or lost (positive is gain, negative is loss) by a particle moving in a closed circuit through that field as it is pushed and pulled by the exerted force. The above implication, then, says that "if the force field conserves energy, we can describe it by a potential energy". This is how you get the usual electric potential, which is the "specific" potential energy: energy per unit of charge, which in SI units comes to be joules per coulomb, which we call as "volts". Moreover, the first equation, "under certain reasonable conditions", corresponds to the one

$$\mathbf{\nabla} \times \mathbf{F} = \mathbf{0}$$

where the left-hand side is a differential operation called "curl", and intuitively represents the amount by which a vector field, thought of as force, locally *fails* to conserve energy.

Now, it turns out that, however, there is another, analogous form but involving this integral: if

$$\mathop{\vcenter{\huge\unicode{x222F}}}_S \mathbf{F} \cdot d\mathbf{S} = 0,$$

a surface integral over a closed surface $$S$$, then it follows that another, vector field $$\mathbf{A}$$ exists such that

$$\nabla \times \mathbf{A} = \mathbf{F}$$

which is very much like the relationship to potential and, indeed, we call this $$\mathbf{A}$$ a vector potential.

Again, we should think about the intuitive meaning of the first integral: this integral now is a flux integral - in effect, if you imagine the field as representing the stream lines of some fluid, i.e. if the vectors returned by $$\mathbf{F}$$ are mass-flow, i.e. mass per time, with direction of flow, vectors, the flux integral would represent the net amount of fluid flowing into or out of that space - and to set it to zero says that, in effect the field "conserves fluid": no new fluid is destroyed or created at any point. Analogously, this corresponds to a similar "local" statement per the divergence:

$$\nabla \cdot \mathbf{F} = 0$$

which, you may recognize, is exactly the equation satisfied by the magnetic field, $$\mathbf{B}$$:

$$\nabla \cdot \mathbf{B} = 0$$

and says there are "no magnetic sources", i.e. no magnetic charges. In a sense, magnetic "flux", which can be thought of as a kind of "fluid", swirls around magnetic objects but none is created or destroyed, and this conservation of flux gives rise to a magnetic vector potential, also typically denoted $$\mathbf{A}$$. This "potential" is a vector, not scalar, quantity - and this is the answer to your question. It doesn't represent energy, but more like "specific flux", I'd suppose, though it's hard to answer and, moreover, interestingly, is much less unique.

• "Magnetic fields, though, cause no changes in energy" Wouldn't magnets cause ferromagnetic objects to have "magnetic potential energy" the same way that gravity wells cause objects with mass to have gravitational potential energy, due to the force attracting them to the magnet, though? – nick012000 Dec 28 '19 at 5:50
• @nick012000 : Good question. This is essentially the common question of "given that magnetic forces do no work, how then can you attract metal with a magnet, which clearly is doing work?" And the answer is that actually, it is electric forces that do the work in those cases. It would be complicated to describe exactly how in the comments - I believe though that there are answers given to this question elsewhere on this site, though, if you search for something like "magnetic forces do no work" etc. – The_Sympathizer Dec 28 '19 at 14:22
• @FakeMod that was a bad idea (screenshot is of Firefox@Android). – Ruslan Dec 29 '19 at 10:20
• @Ruslan Funnily enough, I was inspired from your comment on this post :). – user243267 Dec 29 '19 at 10:30
• @FakeMod yeah, I guess so. In turn, my test case was taken from the comment following my suggestion :) – Ruslan Dec 29 '19 at 10:31

The potential is a kind of primitive function of a vector field, primitive in the sense of being the reverse of a differentiation, ie., an integral with a variable upper limit. The derivatives of the potential in all directions represent the vector field; not all vector fields can be represented that way but some do, for example the electrostatic field. Some other vector fields do not have such representation but one may define another type of differentiation with a corresponding primitive function but it is not a scalar potential and the magnetic field is such an entity. The field usually denoted by B has a primitive function called the vector potential. The mathematical difference between the electric and magnetic fields is that the electric field is an "along a line thing" while the magnetic fields is a "surface thing".

In the electrostatic case the total work done in any loop is zero from which it follows the existence of a potential function. In the case of the B field the total flux passing through any closed surface being zero is the reason for the existence of a vector potential such that the flux through any simple surface spanned by an arbitrary loop is the same as the loop integral of the primitive function, here vector potential integrated around the loop.

• And how do we use this potential? Like, we often use electric and gravitational potential along with energy conservation to solve problems. Similarly, how do we use magnetic potential? Because the way it is defined I don't think it would be any useful while conserving energy. Please add the answers to these questions in your answer rather than posting a comment. It would help us avoid having a long discussion in commentsa and also enrich your answer. – user243267 Dec 26 '19 at 14:07
• Maxwell called it electrokinetic momentum and gave the letter A to show its primacy over other quantities... The vector potential A (not the magnetic field B) shows up directly in the Hamiltonian and is fundamental in QM and being a line thing not a surface thing analysis/calculation is easier with it. – hyportnex Dec 26 '19 at 14:14

There is a "magnetic potential" that appears in more advanced books and is defined as $${\bf B}= \nabla \phi$$, just as $${\bf E}= -\nabla V$$. (please excuse my equations if you are not familar with the $$\nabla$$ symbol) It's useful in regions where there is no current or no time-changing electric field, but it is "multivalued." Because of Ampere's law, if you encircle a current carrying wire, the potential decreases all the way round so when you get back to where you are started you have a different potential -- rather like the Escher prints of water-driven perpetual motion machine in https://en.wikipedia.org/wiki/Waterfall_(M._C._Escher). In many cases we can live with this multivaluedness because $$\phi$$ is not a potential energy (as is $$V$$), but is just a mathematical convenience.

This scalar $$\phi$$ is a simpler thing than the vector potential $${\bf A}$$ (defined so that $${\bf B}=\nabla \times {\bf A}$$) mentioned by hyportnex, but it is less useful.

A charged particle in a magnetic field maintains its kinetic energy. You can see this from the fact that $$\vec{F}_B \propto \vec{v} \times \vec{B}$$ which means that $$\vec{v}\cdot\vec{F}_B=0$$ and therefore the force by the magnetic field cannot preform any work on the particle $$W_B = \int \vec{F}_B \cdot \vec{dx} = \int dt \vec{F}_B\cdot\vec{v} = 0$$ This means that there cannot be a corresponding conserving potential from which it will be derived.

• But what about two magnets kept at a small distance without touching each other? They would either repel or attract, and in either of these cases, there would be an increase in the kinetic energy. Where does that kinetic energy come from? I suppose it is the energy store in the magnetic field which gets converted to kinetic energy. So , there is energy stored in magnetic fields and thus we even define magnetic field energy density, then why not define magnetic potential? – user243267 Dec 26 '19 at 13:57
• That is true, but a bit more complicated than that. In order to fully define this problem one has to consider both electric and magnetic fields, and the corresponding potential that describes both of them is called a vector potential. See the excellent answer of @hyportnex below – yu-v Dec 26 '19 at 14:03
• I see! You are talking about something like "Lorentz force potential", right? – user243267 Dec 26 '19 at 14:08

There is the vector potential $$\bf A$$ for which $${\bf \nabla} \times {\bf A} = {\bf B}$$. So $$B_z = dA_x/dy - dA_y/dx$$ and similar for other components.

The classical field theory of electromagnetism is based on the four potential $$A^\mu = (\phi,{\bf A})$$ . $$\phi$$ is the Coulomb potential .


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# Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$

I have the following problem:

Be the equation:

$$u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw$$

Show that $u\rightarrow 0$ as $x\rightarrow \infty$, even when $e^{-iwx}$ does not falter if $x\rightarrow \infty$.

The problem gives the hint to use integration by parts. I was hoping you explain this problem or help me solve it.

I asked this same question in Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$, however, I found no answer. But I remembered that this equation is a solution of many physical systems, so I decided to put it here hoping for a little more luck.

• what are the hypotheses on $c(w)$? – V. Moretti May 11 '14 at 19:07
• This seems to me like a pure math question devoid of a physical context, which our help center specifies is off topic... if so I'll just send it back to Mathematics. Thoughts from anyone else? – David Z May 11 '14 at 20:55
• Do it, I agree. I hope our answers will be moved there as well. – V. Moretti May 12 '14 at 6:30
• proofwiki.org/wiki/Riemann-Lebesgue_Lemma – Kenshin May 12 '14 at 13:29

If $\mathbb R \ni w\mapsto c(w)e^{-kw^2t}$ is $L^1(\mathbb R)$, the result you want is a trivial consequence of Riemann-Lebesgue theorem (also known as Riemann-Lebesgue lemma).

However some direct proof can easily be produced under fair hypotheses on $c$. For instance, if $\mathbb R \ni w \mapsto c(w)$ is $C^1$ with support in, say $[a,b]$, where $a, b$ are finite, you can write $$x u(x,t)=\int_{-\infty}^{+\infty}c(w)x e^{-iwx}e^{-kw^2t}dw = \int_{-\infty}^{\infty}c(w)x \left(i\partial_w e^{-iwx}\right) e^{-kw^2t}dw = -i\int_{a}^{b} e^{-iwx}\partial_w \left(c(w)e^{-kw^2t}\right) dw + \mbox{vanishing boundary terms}.$$ Finally, since $c$ and its derivative are supported in $[a,b]$ you find: $$|x u(x,t)|=\left|\int_{a}^{b} e^{-iwx}\partial_w \left(c(w)e^{-kw^2t}\right) dw\right| \leq \int_a^b (M + |kt| N) dw = K <+\infty \quad (1)$$ where $M,N,K\geq 0$ are finite constants depending on $a,b, k,t$. Since $K$ in the right-hand side of (1) does not depend on $x$, as $|x| \to + \infty$ in the left-hand side, you should also have $|u(x,t)|\to 0$, for every fixed $t$ and $k$. More precisely, for $|x|\neq 0$, $$|u(x,t)| \leq \frac{K}{|x|}$$

This to me looks as if $c(w)$ is the Fourier transformation of some initial condition of a diffusion equation. Then intuitively the claim holds: When the diffused distribution "reaches" infinity, it will be infinitely dilute.

First of all, to substantiate my claim of diffusion, consider the diffusion (heat) equation $$\frac{\partial u}{\partial t} = c\frac{\partial^2 u}{\partial x^2}$$ Take the Fourier transformation of this and solve the resulting first order diff. equation: $$\hat{u}(k, t) = \hat{u}_0(k) \exp(-c k^2 t)$$ Transforming to real space, $$u(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat{u}_0(k) \exp(-c k^2 t) \exp(i k x)dk$$

Write the initial condition back in real space: $$u(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty u_0(y)\exp(-iky)dy \exp(-c k^2 t) \exp(i k x)dk$$

Switch the order of integration $$u(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty u_0(y) \int_{-\infty}^\infty \exp(-iky -c k^2 t + i k x) dk dy$$

Fill the square, $$u(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty u_0(y) \int_{-\infty}^\infty \exp\left(-\left(i\sqrt{ct}k - \frac{y-x}{2\sqrt{ct}}\right)^2 - \frac{(y-x)^2}{4ct}\right) dk dy$$

i.e.

$$u(x,t) = \frac{1}{2\pi}\int_{-\infty}^\infty u_0(y) \exp\left( - \frac{(y-x)^2}{4ct}\right) \int_{-\infty}^\infty \exp\left(-\left(i\sqrt{ct}k - \frac{y-x}{2\sqrt{ct}}\right)^2\right) dk dy$$

I think we can now recognize that the Gaussian should come out independent of $x$ and $y$ as $\sqrt{\pi/ct}$, and thus we have $$u(x,t) = A \int_{-\infty}^\infty u_0(y) \exp\left(- \frac{(y-x)^2}{4ct}\right) dy$$

i.e. convolution of the initial condition with a Gaussian distribution (that it should take this form is not surprising, for the propagator of diffusion is Gaussian). It is then obvious that if $u_0(y)$ has a finite support, that at infinity $u$ tends to zero.

So what I basically did was to Fourier transform $c(w)$ and change the order of some integrals.

This is not mathematically very rigorous even if we suppose I didn't make mistakes (although I guess it can be made so), but this is the physics stackexchange, not the maths one, after all. V. Moretti's answer was very impressive indeed, but I thought I'd write one up with maybe some more physical intuition behind it.

• I agree with your final comment. My answer was purely mathematical but here a more physical approach should be desirable. – V. Moretti May 12 '14 at 10:02

I started trying to find a first order differential equation for $$u(x,t),$$ such that for instance..

$$\frac{\partial u}{\partial x} = \int c(w)( -iw )e^{-iwx}e^{-kw^2t}dw = i\int c(w)(- w) e^{-iwx}e^{-kw^2t}dw = i\int c(w) e^{-iwx} (1/2kt) \frac{\partial}{\partial w}(e^{-kw^2t})dw = -i(1/2kt)\int \frac{\partial}{\partial w}(c(w)e^{-iwx}) e^{-kw^2t}dw = -(\frac{x}{2kt})u(x,t) - \frac{i}{2kt}\int \frac{\partial}{\partial w}(c(w)) e^{-iwx} e^{-kw^2t}dw =$$

if c is constant that means $$u(x,t) = u(x=0,t)e^{-x^2/4kt}$$. Do not know if this can help you..

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# Is mathematics empirical?

1. Dec 2, 2004

### Aquamarine

There is no absolute truth in mathematics and logics. There are many consistent mathematical systems, consistent in the sense that they are not breaking the basic rules. But there is no absolute way of knowing that the basic rules are true. And there are limitless basic rules that can be chosen. Different basic rules gives different systems, like the intuitionist logic,predicate logic or fuzzy logic.

So why then is mathematics/logic interesting and why are some systems studied instead of others? I would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.

2. Dec 2, 2004

### matt grime

I wouldn't use the word true, and I would never use a comparitive form for it in this sense. It [the set of rules/things we use] is not "more true", but it is more reasonable, and better yet more *useful* than anything else we have. That is why we use the things we do in the models we make: it works. If there were a better one, we'd use that.

3. Dec 2, 2004

### Aquamarine

I tend to agree, but then I see little difference between "truth" and "useful".

Since there is no known ultimate cause and therefore "truth", I think the best we can get is more "useful". In the sense that it allows us to better manipulate the world.

Regarding the current prefered system, predicate logic. Suppose physicists start making better predictions using some theories based on intuitionist logic. Would not this require that most of current mathematical proofs be examined again, and accepted or discarded depending on how they fare in this new logical system that seems better suited to the real world?

Last edited: Dec 2, 2004
4. Dec 4, 2004

### CrankFan

How do you derive from physics, the truth that $$\pi$$ is transcendental?

Can you provide an example of what you have in mind? I seem to recall that intuitionistic theories are weaker than classical theories, any theorem of the former is a theorem of the later, but I may be wrong. An example of what you have in mind might make things clearer.

5. Dec 5, 2004

### Aquamarine

Pi is transcendental according to the basic rules chosen. Physics (and in extension the real world) do not affect the theorems that can be proven when the basic rules have been chosen. The real world affects which basic rules to chose.

We can look at an example higher up in hierarchy. Euclidean geometry was long considered the only possible system and that it also reflected the real world, restricting mathematical research to within this area. Then is was discovered that non-euclidean geometry could be constructed in logically consistent way. And also that physicists can make better predictions assuming non-Euclidean geometry. Since then, much research has been done in non-Euclidean geometry, probably much more than in Euclidean geometry.

Regarding intuitionist logic, let us assume that it is found that the most basic fundamental particles seems to follow the basic rules of intuitionist logic better than predicate logic. That when constructing theories and when using
Reductio_ad_absurdum one gets poor predictions. But if instead if using intuitionist logic one can construct simple theories that give good predictions. I think this would lead to a paradigm shift in mathematics, away from predicate to intuitionist logic.

6. Dec 5, 2004

### matt grime

OK, next challenge - can we have one physical theory constructed using reductio ad absurdum?

Surely, since physics is attempting to explain what is there, as opposed to what one may show may exist, irrespective of its apparent use, or lack of, physics is reasonably intuitionist.

And, as someone has pointed out, anything that can be proved true in intuitionistic logic is true in predicate logic.

Fuzzy logic already is used in some of the applied science, as are other logic systems.

Last edited: Dec 5, 2004
7. Dec 6, 2004

### HallsofIvy

Why do you choose physics specifically? If calculus is used in economics would you say that it works because of physics?? As far as your "basic rules" are concerned, the whole point of mathematics is that all statements in mathematics are of the form "if A then B". One doesn't have to know IF A is true or not, only follow the consequences IF it were true.

8. Dec 9, 2004

### Aquamarine

I say physics since this since some people here question whether economics or psychology is science and I do not want to discuss that in this thread. However, with physics I mean empirical science. My point is that mathematics is also an empirical science, but one step further removed from the real world than for example physics. Another step away would be logic. And logic is not automatically true, many different kinds of logics can be constructed. For example:
http://en.wikipedia.org/wiki/Sequent_calculus

Yes. But if you follow the the sequence backward you will find axioms that have been picked among many possible other axioms. There is no ultimate principle from which everything can be derived. And again, my point is that is the real world that determines which axioms to pick. Those that help makes better predictions are chosen. Chosen in the sense that mathematicians concentrate most of their efforts on those mathematics.

One could maybe argue that predicate logic have already failed in physics. That wave-particle duality would be an example.

I do not know enough physics or logic to know if QM is consistent with predicate logic. But if not, another logic would seems to be better choice since QM seems to make the best current predictions in its field of explanation.

Another interesting idea. What if the it turns out that that real world is discrete, that there is no infinity or continuity in nature. Would this mean that the current concept of limit and differentiation would have to be reworked?

Last edited: Dec 9, 2004
9. Dec 10, 2004

### matt grime

What has wave particle duality to do with predicate logic? Apart from your opinion that anything that *ought* to be dichotomic is the same as predicate calculus. The answer by the way is nothing.

As was pointed out in another thread, discrete models of space currently don't work, and for the umpteenth time limits and derivatives are mathematics, they are not physical objects! The nature of the real world would not invalidate it, in fact it is quite obviously independent of the nature of the physical world in some sense.

QM in case you didnt' notice has not stopped Newton's Laws of Motion from still being taught and used.

10. Dec 10, 2004

### HallsofIvy

Then I think the problem is that you have only studied mathematics that was designed for applications (and you seem to be using the word "physics" in a way I find peculiar). I know many forms of mathematics in which the axioms (what you call the "basic rules") have nothing to do with "the real world", for example the various finite geometries. Of course, those are not then used in applications.

I think you are looking through the "wrong end of the telescope". Mathematics, in its truest sense, does not derive axioms from "the real world". Of course, when you want to apply mathematics to "the real world", you choose the particular types of mathematics whose axioms do, in fact, appear to correspond to the real world.
Sometimes, of course, it happens that newly discovered properties of the "real world" turn out to correspond to axioms of forms of mathematics that had been developed previously without reference to the real world. I'm thinking in particular of the application to General Relativity of Riemannian spaces which were develope before the "basic rules" of General Relativity were known.

11. Dec 10, 2004

### Aquamarine

http://en.wikipedia.org/wiki/Is_logic_empirical?
http://en.wikipedia.org/wiki/Quantum_logic

I am not saying that the universe is discrete or continuous, in my limited understanding that is one point where QM and GR disagree.

It is true that the current limits and derivatives would not be logically falsified if the universe proved discrete. They are consistent according to the basic rules chosen. My point was that if physicists could make better predictions with limits and derivatives defined somewhat differently, they and mathematicians would abandon the current limits and derivatives. Hypothetically, a discrete universe could simulate a continuous universe at the macro level while being discrete at the micro level. The interesting predictions would be in the borderland, where the discrete and continuous mix. If the new limits and derivatives would make better predictions here than the old which ignore the discrete, they would be better suited for the real world.

Again, I am not saying this will be. Just an example of how mathematics could change which axioms to study, similar to the shift from Euclidean to non-Euclidean before.

12. Dec 10, 2004

### Aquamarine

Yes, there are areas of mathematics that seem to have no connection with the real world today But most mathematics are used and studied because it is useful in the real world. If it was not useful, why should society spend resources on it? Then society could as well spend the resources on teaching dead languages. So if mathematicians want to keep their status and pay, they have to choose to study systems built from axioms suited to the real world.

Regarding the second paragraph: Sometimes theory is before experiments. Sometimes experiments are before theory. Still, in the end both experiments and theory must fit together.

13. Dec 10, 2004

### matt grime

Well, that's an interesting opinion, but factually flawed from my experience of being employed by mathematics departments (I would estimate in my current department less than 1/2 do maths that is applied to the real world, and it's predominantly an applied maths department too; this is what engineering departments are for). There is nothing to suggest all maths must eventually translate to the physical world (or it'll have it's funding cut). Intellectual rigour need not have a practical direct use.

Sometimes theory never meets experiment, thank goodness.

Interestingly I appear to be an impossible object in your opinion - a mathematician who's paid, by the state, to do mathematics that isn't practically applicable. At least it's nice when people tell me what I "have" to be.

Good to see you're so against all the higher aspects of humanity at least - utility should be one of the last things we think about in lots of cases. As the old saying goes, it'll be a great day when the schools are fully funded by the state and the military has to hold a bring and buy say to pay for its aircraft carriers.

14. Dec 10, 2004

Staff Emeritus
matt, I want to bring up another sense of empirical. Take a mathematician who sets out to create something new - that is what math research is. The guy has in his mind a conspectus of prior mathematics in his area and he considers the relationships he knows about. Sometimes ther is a little "mix-and-match" novelty he can do, sometimes it's "monkey see monkey do" analogy with other math, and sometimes by building out a train of thought with these humble techniques he can come to a point where insight supervenes and he makes a discovery. (I want to apologize here to all the excellent women mathematicians for my choice of pronouns).

Now isn't a creation path like this "empirical" in its relationship to the world of existing mathematics, which to a mathematician is as real and present as the sensual world?

15. Dec 10, 2004

### matt grime

I missed the post replying to me quoting back my post.

Lots of quantum things in general (such as quantum groups, cohomology etc) use quantum in a certain mathematical sense.

I still stand by my assertion that wave particle duality and predicate logic are not necessarily linked, but that is a philosophical position about the nature of logic (and mathematics) not being physical entitites. I would take it to say that the ideas of QM have led to a new way of thinking about things by introducing measurements of failure to commute (deformations), rather than it being the other way round, but again that is a personal opinion about philosophy.

However it is all hypothetical isn't it?

Mathematicians are obtuse people, they will study what they want to. Some will make connections to the real world some won't.

The development of mathematics and physics has to varying degrees been linked throughout its course. It diverged strongly in the 40's, but they are starting to come together now, it appears.

I don't think calculus wll die (something I'm a little sad about...)

It may interest you to know that there is a non-analytic concept of derivative; we algebraists have been using it for years. It uses something called the space of Dual Numbers.

Self Adjoint: I haven't actually used the word empirical directly, or addressed what may or may not be meant by it.

16. Dec 10, 2004

### CrankFan

What does it mean for an arbitrary physical theory to be "consistent" with an arbitrary mathematical theory?

No.

17. Dec 10, 2004

### Aquamarine

You may not have noticed but we live in a world of limited resources. These should be allocated where they are most useful. One must always think of the alternative use, for example saving lives in health-care with more money and intelligent people.

That many must work so that a few can spend their time in useless games seems absurd. And if theoretical mathematics is of no use in the real world, then it is no better than spending time with a complex computer game. Interesting maybe for the individual, but not something others should be forced to support. Regarding the state, is is noted for its inefficiency in all areas due to lack of competition.

But of course theoretical mathematics with no apparent utility today is much better than a computer game. Exactly because it may be of use in the future. So that is one of the better ways the state spends its money, compared to many others which often even have a negative effect. But note that since it is possible to construct and study countless useless mathematics which could consume all resources in the world, some calculation regarding future use should be made when giving money to today useless mathematics.

Regarding the use of mathematics, it is of course used today in most of society. And most of those using mathematics are certainly not using axioms with no connection to the real world.

18. Dec 10, 2004

### Aquamarine

Two arbitrarily chosen theories are mostly useless without some connection to the real world, for example data from experiments.

If we have data, then we can construct and test a physical theory using a mathematical theory. Note that the mathematical theory will restrict which physical theories are possible to make or affect how good the predictions are or how complex the physical theory must be to fit the data.

It would be quite difficult to construct a good physical theory that fits the data if one were forced to work only within the mathematical theory of Euclidean geometry.

19. Dec 10, 2004

### CrankFan

What makes you think THE theory of predicate calculus is related to QM in some obvious way? As far as I can tell, substituting "theory of recursive functions" or "theory of formal languages" in place of "theory of predicate calculus" is an arbitrary choice.

We all know that mathematics can be used as a tool to build physical theories. What I'm wondering is; what exactly is the process you used to come to the determination that predicate calculus is false based on the empirical evidence that suggests that light has a dual nature?

20. Dec 11, 2004

### matt grime

Can we get back to the original post? It seems that this isn't a discussion about mathematics or anything in particular of that nature. It is purely a debate about what people mean by "true". And in particular comparitive ideas.

21. Dec 11, 2004

### Aquamarine

That question has been asked and answered earlier. And I did not claim that QM has falsified predicate logic.

22. Dec 11, 2004

### Aquamarine

You are misquoting my post.

23. Dec 11, 2004

### Nereid

Staff Emeritus
Reading this thread for the first time, in one sitting, it seems surreal ... an awful lot of people talking past each other ...

Aquamarine has, IMHO, posed some very interesting questions, and certainly in the right place in PF.

First though, some personal clarifications - economics is just as much a 'science' as physics is (the scientific method is applied just as rigourously). Whatever 'reality' the latest and best physics or economics theories suggest, the maths used to describe those theories has its own, independent 'reality' (and if a successful theory can be described using several different mathematical frameworks - insert your favourite examples here - isn't this a practical demonstration of what I just said?). Throughout the history of maths and science (go back as far as you like ... even pre-Greece), there has been a fertile interchange between the two; for a great many of those who have contributed to our current body of theory - directly or indirectly - it would be hard to make a clear distinction; for others, their working lives have had only the most tenuous connection with the other domain (tho a detailed study of how mathematicians have worked may show a greater influence of 'external' ideas than is popularly believed).

So, to what I see as the core of Aquamarine's post: of the truly vast potential 'space' mathematicians could explore (workers in the 'truth mines' - anyone read the SF novel "Diaspora"?), the regions most heavily explored have tended to be those with an apparent connection to 'models of the physical universe'. This is NOT to say that all (or nearly all) mathematicians work in areas that are seen to be (potentially) 'useful', nor that there aren't wonderful results far from the beaten track (e.g. Cantor's work?).

Perhaps we could use the scientific method to examine this idea? I read somewhere that there are ~250,000 new 'theorems' published every year. Perhaps we could take the collective works of mathematicians over the last 50 (500?) years and analyse them in some way?

Finally, there is Aquamarine's 'utility' question (the economics of mathematics? not to be confused with 'usefulness' - utility is a term in economics with a specific and precise meaning): the allocation of scarce resources (salaries for people to do math). I suspect that this is somewhat beyond the farthest shores of economics today ... for example, innovation has only recently begun to be decently addressed (economically, so to speak), and I suspect 'multi-generational' returns are a grey area. This matters not one bit at the individual level (there will always be those who are independently wealthy - free to pursue whatever interests them, and free to fund whomever they like, for example) - but it may be of great interest in the broad.

24. Dec 11, 2004


text:
Following is mwe of what I'm doing. I want to get a couple of things.

1. Instead of bold 1, 2, I need it to be replaced by Chapter 1 and Chapter 2 respectively, followed by the name.

Thank you for any suggestions.

\documentclass[11 pt]{article}

\usepackage{amsmath, mathrsfs}
\usepackage{amssymb}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{blindtext}

\title{This is the title}
\author{Author A and Author B}
\date{}
\begin{document}
\let\conjugatet\overline
\maketitle

\tableofcontents

\section{This is section 1}
\subsection{This is a subsection}

\section{This is section 2}
\subsection{This is a subsection}

\bibliographystyle{abbrv}
\bibliography{References}{}

\end{document}

marked as duplicate by Werner cross-referencing StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 13 '18 at 17:38

• Pardon me, but why do you load blindtext if you don't use it? – Bernard Mar 13 '18 at 16:09
• @Bernard, I am USING it. I just got rid of all the packages for the mwe. – Eureka Mar 13 '18 at 16:12
• Another point: in the article class there's no chapter. If you really want chapters, you should use report or book. – Bernard Mar 13 '18 at 16:29
• @Bernard, my bad. I changed it to report. Still I don't get "Chapter 1" (say), just a bold one, with the chapter name. – Eureka Mar 13 '18 at 16:34
• Use, for instance, titletoc. – Denis Mar 13 '18 at 16:51

tocloft provides the means to add some prefix to chapter number, say, Chapter, with \renewcommand{\cftchappresnum}{...}. However, this would overprint the chapter number, so the \cftchapnumwidth has to be increased accordingly.

For adding the Bibliography to the ToC use \usepackage{tocbibind}

\documentclass[11pt]{report}

\usepackage{amsmath, mathrsfs}
\usepackage{amssymb}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{blindtext}

\usepackage{tocloft}
\usepackage[nottoc,notlot]{tocbibind}

\renewcommand{\cftchappresnum}{\textbf{\chaptername}}

\AtBeginDocument{%
\renewcommand{\bibname}{References}
}

\title{This is the title}
\author{Author A and Author B}
\date{}
\begin{document}
\maketitle

\cite{Lam94}

\tableofcontents

\chapter{Foo}

\section{This is section 1}
\subsection{This is a subsection}

\section{This is section 2}
\subsection{This is a subsection}

\chapter{Foobar}

\clearpage
\bibliographystyle{abbrv}
\bibliography{biblio}

\end{document}

• Thank you. Is there a way to change the heading "Bibliography" to say, "References" ? – Eureka Mar 13 '18 at 20:08
• @Fib1123: \renewcommand{\bibname}{References} should work – user31729 Mar 13 '18 at 20:53
• @Fib1123: You have to use the change after \begin{document} or wrap it in \AtBeginDocument, I have edited the code. – user31729 Mar 14 '18 at 19:15


text:
# Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux:

Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and open book decompositions of $M$ up to positive stabilization.

Given such a contact structure, the existence of an open book supporting the contact structure is proven, for example, in Giroux's Géométrie de contact: de la dimension trois vers les dimensions supérieures.

I am asking for a complete proof of the uniqueness part of the result. At this point I would be surprised if somebody provided me with a link to a peer-reviewed paper containing a proof of the result (which is funny because this is widely acknowledged as a "theorem" inside and outside the field of contact geometry). Usual citations include the above paper (which does not contain a proof of the statement), some book that has been "in preparation" for years or even "transparencies from a seminar"!

So, links to detailed lecture notes or a proof itself will be appreciated. I know for example the existence of these Lectures on open book decompositions and contact structures by J. Etnyre, but they are somehow sketchy to my taste. I am not an expert in the field and I can't complete all the exercises left to the reader or fill in all the gaps in the "sketches of a proof".

• According to John Etnyre, someone claimed that he has a written proof. But no-one saw it. – Anubhav Mukherjee Aug 23 '18 at 22:05

As far as I know, there is no publicly available written proof of uniqueness. Goodman's thesis pointed out by Chris proves neither uniqueness nor existence. What he did was to provide some of the first steps towards understanding the link between open books and tightness. Before that, he does sketch a proof of the open book theorem but, if I remember correctly, this sketch contains less information than what Giroux wrote in the ICM proceedings. In particular it entirely fails to cite Siebenmann's paper that Giroux cites twice in his uniqueness sketch and is the crucial starting point. This paper has been very hard to find for 30 years, but eventually got published as
Les bissections expliquent le théorème de Reidemeister-Singer: Un retour aux sources
Annales de la Faculté des sciences de Toulouse: Mathématiques, Série 6: Volume 24 (2015) no. 5

I'm almost certainly the mysterious person that Anubhav Mukherjee mentions in his comment, but writing a proof of this theorem is way beyond the scope of a mathoverflow answer, I'm sorry. I could probably answer more specific questions though.

• Thank you for your answer. I have not yet look through Goodman’s thesis but, if what you say is true (which I would not be surprised if it were) my question is, why is it considered a Theorem? And who would get the credit after a complete written proof was carried out? – Paul Sep 1 '18 at 19:32
• Is there a chance that, for example, there is a gap that nobody has dealt with before? – Paul Sep 1 '18 at 19:48
• It is considered a theorem because mathematical research is a complex sociological process involving large amounts of intuition and trust. Again this is a huge topic we cannot properly discuss here. And the publication status is not so relevant here. There are also published papers that are not trusted, although sometimes nobody has a specific error to point out. You can google "Coq mathcomp", "lean mathlib", "isabelle archive of formal proof" to get a sense of what is really verified in maths. But even there you'll need to convince yourself that definitions are correctly formalized. – Patrick Massot Sep 2 '18 at 9:11
• About gaps in this particular proof, it's hard to tell, because the available sketch is so sketchy that it can't really be wrong. I never managed to surprise Giroux when discussing subtle points of the proof with him. There are real mistakes in attempted detailed exposition that have been mentioned here, but it's a bit unfair to insist on this point. So we have direct evidence that it's not easy for "any expert" to work out the details. – Patrick Massot Sep 2 '18 at 9:17
• Here is an exercise illustrating the importance of carefully reading what Giroux wrote. On page 409 you read "On épaissit ensuite le 1-squelette L de ∆ en une surface compacte F (presque) tangente à ξ le long de L". (I just copied this sentence into DeepL translator, and the translation is perfect!). Try to find this "presque" in other accounts of the story. Then prove, using only the definition of a contact form, that it shouldn't have been removed. – Patrick Massot Sep 2 '18 at 9:22

This might suffice for you, it is not published and only slightly longer than Etnyre's sketch, but without exercises. This has been shown in the PhD thesis of Noah Daniel Goodman (a student of Etnyre), specifically Theorem 3.4.4:

Contact Structures and Open Books

• Thank you! Since my question is quite open and I did not know this source, I think it is fair that I up-vote you and check the answer (at least while I take the time to read the source). – Paul Aug 30 '18 at 17:33


text:
# A PN 200 Fundamentals of Nursing IT Case Study Bronchial Asthma Michael y dinosed with bronchial...

###### Question:

A PN 200 Fundamentals of Nursing IT Case Study Bronchial Asthma Michael y dinosed with bronchial asthma. His mother her have to the past year, Michael has had three asthma attacks that ne trend th e and autol Preventi inhaler. At the office visit today. pod prethod for a week and the order is written as follows: 7 tablet by mouth four times a day Day? Day 2 Day 3 Day - Table by mouth three times a day tablet by mouth two times a day 1 tablet by mouth in the morning 6 tablet by mouth in the moming 1. Explain the use of prednisone during an asthma attack. Explain why the dosage is decreased tapered) over a period of 5 days. Day 3 2. Can Cromolyn sodium (Nasal Crom) be substituted for prednisone during an asthma attack? Explain your answer. 3. Michael is prescribed albuterol (Proventil, Ventolin). What effect does albuterol have on controlling asthma? 4. For each drug dose, Michael is to take two puffs of albuterol administered by an inhaler What instructions should be given to the patient concerning the use of an inhaler? minimize the frequency of Michael's asthma attacks, the health care provider prescribes phylline (Elixophyllin) 200 mg. twice a day. The albuterol inhaler is to be taken as need Eng interventions include patient history of asthma attacks and physical assessment. When taking the patient's history, what should the nurse include concerning asthm attacks? What type of drug is Elixophyllin? Why should the nurse ask if Michael smokes

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text:
EE235 Course Information

## EE235: Continuous Time Linear Systems

### Course Information, Fall 2006

Course Information/Syllabus: PS version; PDF version. This is electronic version of the form that was passed out the first day of class. Most course information is on the syllabus, including the problem set assignments, due dates, and class schedule.

ANNOUNCEMENTS-CLASS BLOG:

• 12/6/2006: Solutions to our midterm are located here along with the other exams.
• 12/6/2006: All homework solutions are now posted in the solutions page.
• 11/28/2006: For the laplace transform, you will need to look at the following handout.
• 11/27/2006: Several people asked me for the code and scripts of today's matlab demo. I've placed them all in the directory here which includes a documented list of commands and the other scripts that I called.
• 11/27/2006: HW5 and HW6 solutions are now posted in the solutions area. HW7 will be posted later this week. Also, a remimnder to everyone that there are no lates accepted for homework 9 (see our syllabus for details).
• 11/5/2006: midterm histogram is now availalbe. Class mean (median) is 26.4 (28) which corresponds to a final grade of roughly a 3.2.
• 10/31/2006: A student asked for the worked solutions to the problems on your notes (pages 55, 56, and 89), and they are now here
• 10/27/2006: Solutions for the latest hws 3-4 are now posted in the solutions area.
• 10/26/2006: Sample midtern exams are here. Note, some of them have the solutions written right with the exam, so if you don't want to see the solutions, you might want to look at them incrementally. These are exams that I gave myself (the ones whose name start with 'ee235...') and also other example exams. There are also a few finals there as well, but you don't need to worry about them until your final :-)
• 10/23/2006: Question on problem 3.31: The time constant is defined on page 42 of the text. If we have a complex exponential $Ce^\left\{at\right\}$ where $a$ is the variable, then we set the following $Ce^\left\{at\right\} = Ce^\left\{-t/\tau\right\}$, where $\tau$ is known as the time constant in this case we have $\tau = -1/a$. Also, you'll need to turn in short printouts of matlab code for this problem.
• 10/19/2006: HW1 and HW2 solutions are now posted in the solutions area.
• 10/18/2006: Homework solutions and many examples of previous midterms (all with solutions) will be posted here in pdf form very soon!!
• 10/13/2006: As you probably noticed, we have a grader. His name is Dan Luong and his email is dluong@ee.washington.edu. If you have questions about grading on your homework, the first person to ask is Dan. If you still have questions, next ask our TAs ( Evan and/or Eric). And if you then still have questions, please ask me.
• 10/11/2006: Eta Kappa Nu Study Center opens Monday, October 16. Hours: Monday-Friday 11:00am - 1:00 pm, Location: EE Building Upper Basement. Offering high-quality tutoring in a low-pressure environment, focusing on EE215, 233, and 235.
• 10/10/2006: Matlab tutorial Friday the 13th, look here
• 10/2/2006: TA office hours will be in the SIEG tutorial rooms, which are Sieg 126/128.
• 9/28/06: It turns out that the course notes will be at the bookstore after all (this is a new thing they're starting this quarter). I'm told that you can get your course notes starting tomorrow morning at the bookstore located here. I don't know the price (but if you find out, let me know and I'll post it here).
• 9/27/06: A new matlab primer for EE students is now online (other matlab links are below)
• 9/27/06: The class course notes have been now sent to the communications (CMU) copy center, and should be available to pick up starting later tomorrow afternoon. These notes will not be made available electronically unfortunately.
• 9/27/06: No discussion sections this week (first section will be Monday, Oct 2nd).
• 9/27/06: HW0 is now ready in PS and PDF formats.
• 9/27/06: Everyone enrolled in the class must provide a photograph/picture of yourself so we can get to know you. The photo must be recent, and must look like the way you'll typically look in class. You have two options to get us this photo:
1. Email your full name, your SID, and a digital photo of yourself to ee235picture@ssli.ee.washington.edu (please remember to include your SID, student ID as some people have the same name).
2. We will be bringing cameras to class and can take your picture after lecture.
Photos are an assignment, for which you will receive credit, and are due no later than Wednesday, October 4th, 2005.
• 9/27/06: web page up and running

Homework assignments:
All homework assignments are listed in the course information/syllabus above.
Homework FAQ page (general questions about homework that are not in the syllabus).

Laboratory assignments:
All of the five laboratory assignments are avaialble on the connections web site. They are to be finished during discussion in the week as indicated in the syllabus.

You are strongly encouraged to fully read and understand the laboratories before the discussion section in which you are to do them. Failure to read beforehand might mean that you can't complete the lab in discussion. Labs will be graded on a pass/no-pass basis, so again you must prepare before your discussion.

Other handouts

• none yet.

### Other Important Information

Information on EE Computer Access

Send TAs anonymous feedback (this will send anonymous email to both TAS, Evan Hanusa and Eric Rombokas, but to no one else).

Riskin's Interactive Demos
Warning: The class notes mostly parallel these lessons, but there will be some differences, particularly for Lessons 21-24.
(Send email about errors on these web pages to riskin@ee.)

Page last updated $Id: index.html,v 1.24 2006/12/06 23:36:50 bilmes Exp bilmes$


text:
# Math Help - Isomorphic Groups

1. ## Isomorphic Groups

Another question I dont get - I need to show that D & E are isomorphic

2. how about showing that their dimensions are equal..

3. How do i do that?

4. What is a possible basis for $D$?

Let $M = \left[{\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}}\right]$ where $a,b,...,i \in \mathbb{Z}$

compute for $TM$. That $TM$ is in $E$.

based on $TM$, what is a possible basis for $E$.

if number of basis elements of D = number of basis elements of E, then you are done..


text:
# Why are the two dot product definitions equal?

I have an intuitive understanding of why $a\dot{}b=|a||b|\cos{\theta}$ geometrically. The projection of one vector onto another makes sense to me when explaining the origin of this geometric definition.

What I don't understand is why $a\dot{}b=a_xb_x + a_yb_y = |a||b|\cos{\theta}$. How does the algebraic version of the dot product connect to the geometric version? Can you derive the algebraic definition from the geometric? I read the answers to this question, but the proofs seem to depend on the actual algebraic definition to arrive at it.

My main question is, why are the two definitions really equal?

• Can you prove it in two dimensions? Also, some places this is taken as the definition of angles in higher dimensions. So it's not a matter of proving that this is the case, but rather that $\theta$ is defined so that it works. – Arthur Aug 2 '17 at 16:19
• You can find a proof of $a\cdot b=\|a\|\|b\|\cos\theta$, if that's helpful for you, here: proofwiki.org/wiki/Cosine_Formula_for_Dot_Product – Edu Aug 2 '17 at 16:26
• @Edu So the geometric version is derived from the algebraic, not the other way around? – name Aug 2 '17 at 16:27
• Geometrically, consider the triangle formed by the vectors $a$, $b$, and $a-b$ (where we think of $a$ and $b$ having a common origin, and the origin of $a-b$ placed at the end of $a$). This triangle has sides of length $\|a\| = a\cdot a$, $\|b\| = b\cdot b$, and $\|a-b\| = (a-b)\cdot (a-b) = a\cdot a - 2a\cdot b + b\cdot b$. Play with the Law of Cosines, and you should get what you want. – Xander Henderson Aug 2 '17 at 16:30
• Absolutely. The dot product is a particular case of an inner product. For the Euclidean dot product you have this formula, which is slightly different for complex vectors, for instance. – Edu Aug 2 '17 at 16:32

Note that with $a$ and $b$ two sides of a triangle and $\theta$ the angle between them, the third side is $b-a$ and (cosine rule) $$|b-a|^2=|a|^2+|b|^2-2|a||b|\cos \theta$$ so that $$2|a||b|\cos\theta=\Sigma a_i^2+\Sigma b_i^2-\Sigma (b_i-a_i)^2=2\Sigma a_ib_i$$ so that $$|a||b|\cos\theta=\Sigma a_ib_i$$ and the two definitions coincide. You can work the calculations backwards if necessary.

• You are missing the case in which $a$ and $b$ are scalar multiples. – Edu Aug 2 '17 at 16:43
• @Edu then $\cos \theta=1$ and everything works? The cosine rule applies to the degenerate triangle you get. – Mark Bennet Aug 2 '17 at 17:00
• Of course, is a trivial case, no problems with your proof, not even close to think about downvoting or something. I just think is worth it mention what happens if $a=cb$, for some scalar $c$. – Edu Aug 2 '17 at 17:03
• Btw, $\cos\theta=\pm 1$ in that case, depending on the sign of $c$. – Edu Aug 2 '17 at 17:07

First of all you can prove from the geometrical definition that distributive property holds for scalar product: $(\vec a+\vec b)\cdot\vec c=\vec a\cdot\vec c+\vec b\cdot\vec c$ (see diagram below for a sketch of the proof).

Then, you just have to decompose two vectors along an orthonormal coordinate system: $\vec a= a_x\vec i+a_y\vec j+a_z\vec k$,$\quad$ $\vec b= b_x\vec i+b_y\vec j+b_z\vec k$, and apply twice the distributive property, taking into account that $\vec i\cdot\vec i=\vec j\cdot\vec j=\vec k\cdot\vec k=1$ and $\vec i\cdot\vec j=\vec j\cdot\vec k=\vec k\cdot\vec i=0$: $$\vec a\cdot\vec b= (a_x\vec i+a_y\vec j+a_z\vec k)\cdot(b_x\vec i+b_y\vec j+b_z\vec k)= a_xb_x+a_yb_y+a_zb_z.$$

Assuming $a\cdot b = |a||b|\cos( \theta)$

and that $|a| = \sqrt{(x_1)^2 + (y_1)^2}$

and that $|b| = \sqrt{(x_2)^2 + (y_2)^2}$

Let $\gamma -\theta = \alpha$

For $\theta$ is the angle between the two vectors, $a$ and $b$, and not necessarily the angle between the 'outermost' line and say the x-axis.

Thus $\gamma$ is the angle between the outermost vector and the x-axis, whilst $\alpha$ is the angle between the other line and the x-axis.

Thus $\theta=\gamma-\alpha$ and more importantly by the trig identity: $\cos(a+b) = \cos(a)\cos(b)-\sin(a)\sin(b)$

\begin{aligned} a\cdot b &= |a||b|\cos(\theta)\\ &=|a||b|\cos(\gamma - \alpha)\\ &=|a||b|\biggl(\cos(\gamma)\cos(\alpha) - \sin(\gamma)\sin(\alpha)\biggl)\\ &=|a||b|\cos(\gamma)\cos(-\alpha) - |a||b|\sin(\gamma)\sin(-\alpha)\\ \end{aligned}\tag{1}

Then if $|a|$ was the outermost vector you realize that

$|a|\cos(\gamma) = (x_1)$, and that

$|a|\sin(\gamma) = (y_1)$

Making $|b|$ the innermost line, and that

$|b|\cos(-\alpha) = |b|\cos(\alpha) = (x_2)$, and that

$|b|\sin(-\alpha) = -|b|\sin(\alpha) = (-y_1)$

Because $\cos(-a) = \cos(a)$ and $\sin(-a) = -\sin(a)$

Plugging all that into 1 you get that $a\cdot b = (x_1)(x_2) - (y_1)(-y_2)$

And you finally get $a\cdot b = (x_1)(x_2) + (y_1)(y_2)$

Define A, B, C is vector, and a, b and c scale value (distance) of A,B,C
define C=A-B, so A, B and C become a triangle

based on Law of cosines, c² = a² + b² - 2abcos(θ)

because c.c=C²
C.C=A.A + B.B - 2abcos(θ)
because C = A - B
C·C => (A -B)·(A - B) => (A·A - 2A·B + B·B)
so A·A - 2A·B + B·B=A.A + B.B - 2abcos(θ)
because A.A=A.A and B.B=B.B
=> - 2A·B B=- 2abcos(θ)
=> A.B=abcos(θ)

• Try to use math formatting for better readability. See notation help. – ja72 Aug 2 '17 at 17:36
• Thanks I just joined this community, I will try to use the math formatting next time – Bin Yu Aug 2 '17 at 20:27


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## 7.1 Everything is (secretly) unitary

At the fundamental level — and this should be your quantum mantra119 — there is only unitary evolution, and if there is any other evolution then it has to be derived from a unitary evolution. From this perspective, any non-unitary evolution of an open system is induced by a unitary evolution of a larger system — all evolutions become unitary when you make your system large enough! But how? The short answer is: by adding (via tensoring) and removing (via partial trace) physical systems. A typical combination of these operations is shown in the following diagram:

Let’s explain what this diagram is really saying.

First, as always, we prepare our system of interest in an input state \rho.

Next we dilate the system by “adding” (or “taking into account”) an auxiliary system120 which is large enough to include everything our system will interact with, and also large enough to be in a pure state |a\rangle. Mathematically, we do this by tensoring the input state \rho with |a\rangle\langle a| to obtain |a\rangle\langle a|\otimes\rho (here we place the auxiliary system first and our system of interest second). Importantly, we assume that we have “added in” a large enough auxiliary system that the resulting dilated system is closed, and thus undergoes unitary evolution, described by some U, resulting in the state U(|a\rangle\langle a|\otimes\rho)U^\dagger.

Finally, after all the (unitary) interactions have taken place, we trace out the auxiliary system, turning the joint state U(|a\rangle\langle a|\otimes\rho)U^\dagger of the dilated system into the final state of our system of interest: the output state \rho'.

We shall later show that the net effect of these three operations (adding, unitary evolution, and tracing out) can be written, as long as the initial state |a\rangle of the auxiliary system is not correlated with the input state \rho, in a nice compact way: \rho\longmapsto\rho' = \sum_i E_i\rho E_i^\dagger where the E_i are some operators that satisfy \sum_i E_i^\dagger E_i=\mathbf{1}. Such a linear map is called a completely positive trace-preserving map, or, in the parlance of quantum information science, a quantum channel.

We will elaborate on the mathematics behind quantum channels shortly, but for now let us only check the essential properties, i.e. that this map preserves both trace and positivity (as its name suggests).

• Trace preserving. Since the trace is linear, invariant under cyclic permutations of operators, and we ask that \sum_i E_i^\dagger E_i=\mathbf{1}, we see that \operatorname{tr}\left(\sum_k E_k\rho E_k^\dagger\right) = \operatorname{tr}\left(\sum_k E^\dagger_k E_k \rho\right) = \operatorname{tr}\rho.
• Positivity preserving. Since \rho is a positive121 (semi-definite) operator, so too is \sqrt{\rho}, and we thus see that \sum_k E_k\rho E_k^\dagger = \sum_k (E_k\sqrt{\rho})(\sqrt{\rho} E_k^\dagger).

These conditions are certainly necessary if we want to map density operators into legal density operators, but we shall see in a moment that they are not sufficient: quantum channels are not just positive maps, but instead completely positive maps.

We will discuss the special properties of completely positive trace preserving maps, describe the most common examples, and, last but not least, specify when the action of quantum channels can be reversed, or corrected, so that we can recover the original input state. This will set the stage for our subsequent discussion of quantum error correction.

1. …There is only unitary evolution. There is only unitary evolution. There is only unitary evolution… …and everything else is cheating.↩

2. Depending on the context, the auxiliary system is either called the ancilla (usually when we can control it) or the environment (usually when we cannot control it).↩

3. Recall that an operator is positive if and only if it can be written in the form XX^\dagger for some X (here X=E_k\sqrt{\rho}). Also, the sum of positive operators is again a positive operator.↩


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# Which mean value should I use

I am to test the transformational shifts in three different directions for 3 different imaging techniques in a group of 30 patients. I am trying to prove one is not statistically different to the gold standard. Should I get the mean shift values in each direction for the 270 images OR should I get the mean shift per patient, then the mean of those 30 patients?VERY confused.

If I use the 270 figures I get very good normal distribution on histograms and pp plot, however when finding the average per patient and getting the average of that I am getting the mean of 30 mean figures, and the distribution is no longer normal!

• The question is not clear. It will be helpful if you can show a sample of your data in table format. Will it be like this: PtID, x_modality1, y_mod1, z_mod1, x_mod2, y_mod2, z_mod2, x_mod3, y_mod3, z_mod3. And you want to check if modality1 and modality2 are not inferior to modality3, which is the gold standard? – rnso Nov 25 '14 at 12:57
• yes exactly! do you think i should compare each image in each modality and get the mean of that or get the mean for each patient using each modality and then the mean of the patients mean?! – Zoe Campbell Nov 25 '14 at 13:07
• x,y and z are the three different directions (ant post, left right and sup inf) the value of x in the new modality is hopefully similar to that of the gold standard. the first modality is doing a simple match on a cone beam ct, the second one is adding an additional step to this match and the third is having a doctor match from the simple match, this being the gold standard – Zoe Campbell Nov 25 '14 at 13:59
• Sorry, but your question and your replies in comments are difficult to follow because of lack of detail and lack of attention to grammar, spelling and punctuation. (You are not texting to friends who can decode easily.) – Nick Cox Nov 25 '14 at 16:41
• See advice at meta.stats.stackexchange.com/questions/1479/… – Nick Cox Nov 25 '14 at 16:48

In R, following can be done:

For one direction (x):

dd = data.frame(ptid=1:len, mod=sample(1:2, len, replace=T), x=sample(10:20, len, replace=T))
ptid mod  x
1    1   1 20
2    2   2 12
3    3   2 19
4    4   1 13
5    5   1 14
6    6   1 19

Code for plotting boxplot:

library(ggplot2)
ggplot(dd, aes(factor(mod), x))+geom_boxplot()

Image:

Paired t.test etc can be applied after rearranging data as follows:

head(dd2)
ptid x_mod1 x_mod2
1    1    315    311
2    2    288    225
3    3    341    459
4    4    322    272
5    5    223    461
6    6    445    178

t.test(dd2[,2], dd2[,3], paired=T)

Paired t-test

data:  dd2[, 2] and dd2[, 3]
t = 0.7188, df = 49, p-value = 0.4757
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-27.976  59.136
sample estimates:
mean of the differences
15.58

wilcox.test(dd2[,2], dd2[,3], paired=T)

Wilcoxon signed rank test with continuity correction

data:  dd2[, 2] and dd2[, 3]
V = 712.5, p-value = 0.472
alternative hypothesis: true location shift is not equal to 0

A paired line plot can also be plotted to see how x values of 2 modalities compare with each other:


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# Category Archives: The Wilmott Magazine

My published (or soon to be published) column pieces in The Wilmott Magazine

# Functional Programming

Functional programming is the programming methodology that puts great emphasis on statelessness and religiously avoids side effects of one function in the evaluation any other function. Functions in this methodology are like mathematical functions. The conventional programming style, on the other hand, is considered “imperative” and uses states and their changes for accomplishing computing tasks.

Adapting this notion of functional programming may sound like regressing back to the pre-object-oriented age, and sacrificing all the advantages thereof. But there are practitioners, both in academia and in the industry, who strongly believe that functional languages are the only approach that ensures stability and robustness in financial and number crunching applications.

Functional languages, by definition, are stateless. They do everything through functions, which return results that are, well, functions of their arguments. This statelessness immediately makes the functions behave like their mathematical counterparts. Similarly, in a functional language, variable behave like mathematical variables rather than labels for memory locations. And a statement like x = x + 1 would make no sense. After all, it makes no sense in real life either.

This strong mathematical underpinning makes functional programming the darling of mathematicians. A piece of code written in a functional programming language is a set of declarations quite unlike a standard computer language such as C or C++, where the code represents a series of instructions for the computer. In other words, a functional language is declarative — its statements are mathematical declarations of facts and relationships, which is another reason why a statement like x = x + 1 would be illegal.

The declarative nature of the language makes it “lazy,” meaning that it computes a result only when we ask for it. (At least, that is the principle. In real life, full computational laziness may be difficult to achieve.) Computational laziness makes a functional programming language capable of handling many situations that would be impossible or exceedingly difficult for procedural languages. Users of Mathematica, which is a functional language for symbolic manipulation of mathematical equations, would immediately appreciate the advantages of computational laziness and other functional features such as its declarative nature. In Mathematica, we can carry out an operation like solving an equation for instance. Once that is done, we can add a few more constraints at the bottom of our notebook, scroll up to the command to solve the original equation and re-execute it, fully expecting the later constraints to be respected. They will be, because a statement appearing at a later part in the program listing is not some instruction to be carried out at a later point in a sequence. It is merely a mathematical declaration of truism, no matter where it appears.

This affinity of functional languages toward mathematics may appeal to quants as well, who are, after all, mathematicians of the applied kind. To see where the appeal stems from, let us consider a simple example of computing the factorial of an integer. In C or C++, we can write a factorial function either using a loop or making use of recursion. In a functional language, on the other hand, we merely restate the mathematical definition, using the syntax of the language we are working with. In mathematics, we define factorial as:

$n! = left{begin{array}{ll}1 & n=1 \n times (n-1)! & textrm{Otherwise}end{array}right.$

And in Haskell (a well known functional programming language), we can write:

```bang 1 = 1
bang n = n * bang (n-1)```

And expect to make the call bang 12 to get the factorial of 12.

This example may look artificially simple. But we can port even more complicated problems from mathematics directly to a functional language. For an example closer to home, let us consider a binomial pricing model, illustrating that the ease and elegance with which Haskell handles factorial do indeed extend to real-life quantitative finance problems as well.

# Magic of Object Oriented Languages

Nowhere is the dominance of paradigms more obvious than in object oriented languages. Just take a look at the words that we use to describe some their features: polymorphism, inheritance, virtual, abstract, overloading — all of them normal (or near-normal) everyday words, but signifying notions and concepts quite far from their literal meaning. Yet, and here is the rub, their meaning in the computing context seems exquisitely appropriate. Is it a sign that we have taken these paradigms too far? Perhaps. After all, the “object” in object oriented programming is already an abstract paradigm, having nothing to do with “That Obscure Object of Desire,” for instance.

We do see the abstraction process running a bit wild in design patterns. When a pattern calls itself a visitor or a factory, it takes a geekily forgiving heart to grant the poetic license silently usurped. Design patterns, despite the liberties they take with our sensitivities, add enormous power to object oriented programming, which is already very powerful, with all the built in features like polymorphism, inheritance, overloading etc.

To someone with an exclusive background in sequential programming, all these features of object oriented languages may seem like pure magic. But most of the features are really extensions or variations on their sequential programming equivalents. A class is merely a structure, and can even be declared as such in C++. When you add a method in a class, you can imagine that the compiler is secretly adding a global function with an extra argument (the reference to the object) and a unique identifier (say, a hash value of the class name). Polymorphic functions also can be implemented by adding a hash value of the function signature to the function names, and putting them in the global scope.

The real value of the object oriented methodology is that it encourages good design. But good programming discipline goes beyond mere adaptation of an object oriented language, which is why my first C++ teacher said, “You can write bad Fortran in C++ if you really want. Just that you have to work a little harder to do it.”

For all their magical powers, the object oriented programming languages all suffer from some common weaknesses. One of their major disadvantages is, in fact, one of the basic design features of object oriented programming. Objects are memory locations containing data as laid down by the programmer (and the computer). Memory locations remember the state of the object — by design. What state an object is in determines what it does when a method is invoked. So object oriented approach is inherently stateful, if we can agree on what “state” means in the object oriented context.

But in a user interface, where we do not have much control over the sequence in which various steps are executed, we might get erroneous results in stateful programming depending on what step gets executed at a what point in time. Such considerations are especially important when we work with parallel computers in complex situations. One desirable property in such cases is that the functions return a number solely based on their arguments. This property, termed “purity,” is the basic design goal of most functional languages, although their architects will concede that most of them are not strictly “pure.”

### Sections

Paradigms permeate almost all aspects of computing. Some of these paradigms are natural. For instance, it is natural to talk about an image or a song when we actually mean a JPEG or an MP3 file. File is already an abstraction evolved in the file-folder paradigm popularized in Windows systems. The underlying objects or streams are again abstractions for patterns of ones and zeros, which represent voltage levels in transistors, or spin states on a magnetic disk. There is an endless hierarchy of paradigms. Like the proverbial turtles that confounded Bertrand Russell (or was it Samuel Johnson?), it is paradigms all the way down.

Some paradigms have faded into the background although the terminology evolved from them lingers. The original paradigm for computer networks (and of the Internet) was a mesh of interconnections residing in the sky above. This view is more or less replaced by the World Wide Web residing on the ground at our level. But we still use the original paradigm whenever we say “download” or “upload.” The World Wide Web, by the way, is represented by the acronym WWW that figures in the name of all web sites. It is an acronym with the dubious distinction of being about the only one that takes us longer to say than what it stands for. But, getting back to our topic, paradigms are powerful and useful means to guide our interactions with unfamiliar systems and environments, especially in computers, which are strange and complicated beasts to begin with.

A basic computer processor is deceptively simple. It is a string of gates. A gate is a switch (more or less) made up of a small group of transistors. A 32 bit processor has 32 switches in an array. Each switch can be either off representing a zero, or on (one). And a processor can do only one function — add the contents of another array of gates (called a register) to itself. In other words, it can only “accumulate.”

In writing this last sentence, I have already started a process of abstraction. I wrote “contents,” thinking of the register as a container holding numbers. It is the power of multiple levels of abstraction, each of which is simple and obvious, but building on whatever comes before it, that makes a computer enormously powerful.

We can see abstractions, followed by the modularization of the abstracted concept, in every aspect of computing, both hardware and software. Groups of transistors become arrays of gates, and then processors, registers, cache or memory. Accumulations (additions) become all arithmetic operations, string manipulations, user interfaces, image and video editing and so on.

Another feature of computing that aids in the seemingly endless march of the Moore’s Law (which states that computers will double in their power every 18 months) is that each advance seems to fuel further advances, generating an explosive growth. The first compiler, for instance, was written in the primitive assembler level language. The second one was written using the first one and so on. Even in hardware development, one generation of computers become the tools in designing the next generation, stoking a seemingly inexorable cycle of development.

While this positive feedback in hardware and software is a good thing, the explosive nature of growth may take us in wrong directions, much like the strong grown in the credit market led to the banking collapses of 2008. Many computing experts now wonder whether the object oriented technology has been overplayed.

# Zeros and Ones

Computers are notorious for their infuriatingly literal obedience. I am sure anyone who has ever worked with a computer has come across the lack of empathy on its part — it follows our instructions to the dot, yet ends up accomplishing something altogether different from what we intend. We have all been bitten in the rear end by this literal adherence to logic at the expense of commonsense. We can attribute at least some of the blame to our lack of understanding (yes, literal and complete understanding) of the paradigms used in computing.

Rich in paradigms, the field of computing has a strong influence in the way we think and view the world. If you don’t believe me, just look at the way we learn things these days. Do we learn anything now, or do we merely learn how to access information through browsing and searching? Even our arithmetic abilities have eroded along with the advent of calculators and spreadsheets. I remember the legends of great minds like Enrico Fermi, who estimated the power output of the first nuclear blast by floating a few pieces of scrap paper, and like Richard Feynman, who beat an abacus expert by doing binomial expansion. I wonder if the Fermis and Feynmans of our age would be able to pull those stunts without pulling out their pocket calculators.

Procedural programming, through its unwarranted reuse of mathematical symbols and patterns, has shaped the way we interact with our computers. The paradigm that has evolved is distinctly unmathematical. Functional programming represents a counter attack, a campaign to win our minds back from the damaging influences of the mathematical monstrosities of procedural languages. The success of this battle may depend more on might and momentum rather than truth and beauty. In our neck of the woods, this statement translates to a simple question: Can we find enough developers who can do functional programming? Or is it cheaper and more efficient to stick to procedural and object oriented methodologies?

# House of Cards

We are in dire straits — no doubt about it. Our banks and financial edifices are collapsing. Those left standing also look shaky. Financial industry as a whole is battling to survive. And, as its front line warriors, we will bear the brunt of the bloodbath sure to ensue any minute now.

Ominous as it looks now, this dark hour will pass, as all the ones before it. How can we avoid such dark crises in the future? We can start by examining the root causes, the structural and systemic reasons, behind the current debacle. What are they? In my series of posts this month, I went through what I thought were the lessons to learn from the financial crisis. Here is what I think will happen.

The notion of risk management is sure to change in the coming years. Risk managers will have to be compensated enough so that top talent doesn’t always drift away from it into risk taking roles. Credit risk paradigms will be reviewed. Are credit limits and ratings the right tools? Will Off Balance Sheet instruments stay off the balance sheet? How will we account for leveraging?

Regulatory frameworks will change. They will become more intrusive, but hopefully more transparent and honest as well.

Upper management compensation schemes may change, but probably not much. Despite what the techies at the bottom think, those who reach the top are smart. They will think of some innovative ways of keeping their perks. Don’t worry; there will always be something to look forward to, as you climb the corporate ladder.

Nietzsche may be right, what doesn’t kill us, may eventually make us stronger. Hoping that this unprecedented financial crisis doesn’t kill us, let’s try to learn as much from it as possible.

# Free Market Hypocrisy

Markets are not free, despite what the text books tell us. In mathematics, we verify the validity of equations by considering asymptotic or limiting cases. Let’s try the same trick on the statement about the markets being free.

If commodity markets were free, we would have no tariff restrictions, agricultural subsidies and other market skewing mechanisms at play. Heck, cocaine and heroine would be freely available. After all, there are willing buyers and sellers for those drugs. Indeed, drug lords would be respectable citizens belonging in country clubs rather than gun-totting cartels.

If labor markets were free, nobody would need a visa to go and work anywhere in the world. And, “equal pay for equal work” would be a true ideal across the globe, and nobody would whine about jobs being exported to third world countries.

Capital markets, at the receiving end of all the market turmoil of late, are highly regulated with capital adequacy and other Basel II requirements.

Derivatives markets, our neck of the woods, are a strange beast. It steps in and out of the capital markets as convenient and muddles up everything so that they will need us quants to explain it to them. We will get back to it in future columns.

So what exactly is free about the free market economy? It is free — as long as you deal in authorized commodities and products, operate within prescribed geographies, set aside as much capital as directed, and do not employ those you are not supposed to. By such creative redefinitions of terms like “free,” we can call even a high security prison free!

Don’t get me wrong. I wouldn’t advocate making all markets totally free. After all, opening the flood gates to the formidable Indian and Chinese talent can only adversely affect my salary levels. Nor am I suggesting that we deregulate everything and hope for the best. Far from it. All I am saying is that we need to be honest about what we mean by “free” in free markets, and understand and implement its meaning in a transparent way. I don’t know if it will help avoid a future financial meltdown, but it certainly can’t hurt.

# Quant Culprits

Much has been said about the sins of the quants in their inability to model and price credit derivatives, especially Collateralized Debt Obligations (CDOs) and Mortgage Backed Securities (MBSs). In my opinion, it is not so much of a quant failure. After all, if you have the market data (especially default correlations) credit derivatives are not all that hard to price.

The failure was really in understanding how much credit and market risks were inter-related, given that they were independently managed using totally different paradigms. I think an overhauling is called for here, not merely in modeling and pricing credit risks, also in the paradigms and practices used in managing them.

Ultimately, we have to understand how the whole lifecycle of a trade is managed, and how various business units in a financial institution interact with each other bearing one common goal in mind. It is this fascination of mine with the “big picture” that inspired me to write The Principles of Quantitative Development, to be published by Wiley Finance in 2010.

# Where Credit is Due

While the market risk managers are getting grilled for the financial debacle we are in, the credit controllers are walking around with that smug look that says, “Told you so!” But systemic reasons for the financial turmoil hide in our credit risk management practices as well.

We manage credit risk in two ways — by demanding collateral or by credit limit allocation. In the consumer credit market, they correspond to secure lending (home mortgages, for instance) and unsecured loans (say, credit lines). The latter clearly involves more credit risk, which is why you pay obscene interests on outstanding balances.

In dealing with financial counterparties, we use the same two paradigms. Collateral credit management is generally safe because the collateral involved cannot be used for multiple credit exposures. But when we assign each counterparty a credit limit based on their credit ratings, we have a problem. While the credit rating of a bank or a financial institution may be accurate, it is almost impossible to know how much credit is loaded against that entity (because options and derivatives are “off balance sheet” instruments). This situation is akin to a bank’s inability to check how much you have drawn against your other credit lines, when it offers you an overdraft facility.

The end result is that even in good times, the leverage against the credit rating can be dangerously high without counterparties realizing it. The ensuing painful deleveraging takes place when a credit event (such as lowering of the credit rating) occurs.

# Hedging Dilemma

Ever wonder why those airfares are quick to climb, but slow to land? Well, you can blame the risk managers.

When the oil price hit \$147 a barrel in July ’08, with all the pundits predicting sustained \$200 levels, what would you have done if you were risk managing an airline’s exposure to fuel? You would have ran and paid an arm and a leg to hedge it. Hedging would essentially fix the price for your company around \$150 level, no matter how the market moved. Now you sit back and relax, happy in the knowledge that you saved your firm potentially millions of dollars.

Then, to your horror, the oil price nosedives, and your firm is paying \$100 more than it should for each barrel of oil. (Of course, airlines don’t buy WTI, but you know what I mean.) So, thanks to the risk managers’ honest work, airlines (and even countries) are now handing over huge sums of money to energy traders. Would you rather be a trader or a risk manager?

And, yes, the airfares will come down, but not before the risk managers take their due share of flak.

### Sections

Just as 9/11 was more of an intelligence failure rather than a security lapse, the subprime debacle is a risk management breakdown, not merely a regulatory shortcoming. To do anything useful with this rather obvious insight, we need to understand why risk management failed, and how to correct it.

Risk management should be our first line of defense — it is a preventive mechanism, while regulatory framework (which also needs beefing up) is a curative, reactive second line.

The first reason for the inadequacy of risk management is the lack of glamour the risk controllers in a financial institution suffer from, when compared to their risk taking counterparts. (Glamour is a euphemism for salary.) If a risk taker does his job well, he makes money. He is a profit centre. On the other hand, if a risk controller does his job well, he ensures that the losses are not disproportionate. But in order to limit the downside, the risk controller has to limit the upside as well.

In a culture based on performance incentives, and where performance is measured in terms of profit, we can see why the risk controller’s job is sadly under-appreciated and under-compensated.

This imbalance has grave implications. It is the conflict between the risk takers and risk managers that enforces the corporate risk appetite. If the gamblers are being encouraged directly or indirectly, it is an indication of where the risk appetite lies. The question then is, was the risk appetite a little too strong?

The consequences of the lack of equilibrium between the risk manager and the risk taker are also equally troubling. The smarter ones among the risk management group slowly migrate to “profit generating” (read trading or Front Office) roles, thereby exacerbating the imbalance.

The talent migration and the consequent lack of control are not confined merely within the walls of a financial institution. Even regulatory bodies could not compete with the likes of Lehman brothers when hunting for top talent. The net result was that when the inevitable meltdown finally began, we were left with inadequate risk management and regulatory defenses.