“That is a nit-pick that is also an erroneous nit-pick.

See the discussion of Black-Scholes from the perspective of Brownian motion and Statistical Mechanics Here.

That discussion includes thermodynamics and heat flow.

Equation 10 is the famous Black-Scholes equation. Its solutions are widely adopted for financial analysis by traders, fund managers, economists, and so on. The Equation can be further transformed into standard form of the heat diffusion equation from Physics (Thermodynamics) ”

For the record, placing information in these comments which helps the scientifically untrained folks with scientific distinctions such as I wrote, is not nitpicking; in my view it ups the profile of these discussions by demonstrating that there are professional folks in the sciences posting here. Scordova does not need to take this as a put down. After going to the site of the link he provides I see that an economist equates heat flow with thermodynamics. Not helpful. My Holman thermo text from 1970 makes no mention of Fourier’s law. The decent wikipedia article on thermodynamics makes no mention of heat flow: http://en.wikipedia.org/wiki/T.....d_branches

As so often, the case is not so clear cut. Earlier, he obviously has done some legit work, while his International Committee for Scientific Ethics and Accountability shouts “CRANK”.
He isn’t referred to in the books with which I was introduced to Lie-theory – and that seems to be part of his problem

See:
Hadronic Mathematics I

Hadronic Mathematics II

Many thanks. I figured whatever the outcome of your evaluation you might find his ideas amusing. I’m not familiar with Lie-admissible Algebras at all…

I hear the term Lie-algebra all the time in connection with advanced physics, but I’ve yet to run into it in formal study…

Thank you again, and thanks for responding to my questions here at UD.

Can you tell if R. Santilli is a real mathematician? He claims to have taught at some good schools.

Thanks.

Salvador

However, it appears that condition 3 (absolute integrability of P(x)) will ensure that a normalization integral exists. So that seemed reasonable to me….is this correct?

Even for a finite period, condition 3 (“f(x) must be absolutely integrable over a period”) isn’t sufficient for the existence of a normalization integral, you’d need condition 4 (“f(x) must be bounded”), too.
But, as I stated before, you wouldn’t invoke the Dirichlet conditions in the context of L², as there is a beautiful Fourier transform on this space.
I never used Apostol’s “Modern Analysis” before. If someone is interested in mathematics, I’d usually hint him at books for which more current editions exist.

Condition 3 in which enumeration? In post #159, you give condition 3 as

3. ?(x) and d?(x)/dx must be finite

I don’t see there anything about absolute integrability…

I thought you meant condition 3 from the list of 4 Dirichlet conditions. I mis-interpreted your remarks.

as stated so nicely in the text of Apostol.

Do you have Apostol’s book? Do you like it?

“physically realizable boundary conditions”? That I don’t know neither – I’m a mathematician, not a physicist.

I’ll point you to a simple example in the Britney Spears Guide to Finite-Barrier Quantum Wells. See equation (3)…

One can see the solutions to the schrodinger equation (when they are defined from -infinity to +infinity). In this case the solutions are going to be functions damped by a decaying exponential. In this case, we have two well-defined boundary conditions imposed by 2 physical boundaries. We could in priniciple create n-boundaries, for n-boundary conditions…..

It’s seems it would be impossible (in a universe with finite resources) to construct an infinite number physical boundaries spaced apart in such a way that we have a function Psi(x) that is not absolutely integrable. I believe if we have a finite set of boundaries, with a finite distance from each other, we’ll eventually end up with functions damped with decaying exponentials on the ends that go to +/- infinity. Does that seem correct?

What I presume you meant was the series converges on points except where the the function lies outside the null-set.

What I wanted to say: If a function satisfies the Dirichlet conditions – or Jordan’s test – then it has at most a countably number of discontinuities, i.e., the points of discontinuity form a null-set (regarding λ.) In this points, the series will converge to the mean of the left-hand-side and right-hand side value of the function, as stated so nicely in the text of Apostol.

It is possible in principle that there exists a Psi(x) that is not absolutely integrable over a period of infinity but Psi*(x)Psi(x) is integrable.

yes.

But I don’t know that there exists a set of physically realizable boundary conditions which would imply such a Psi(x) solution to Schrodinger’s equation!

“physically realizable boundary conditions”? That I don’t know neither – I’m a mathematician, not a physicist.

However, it appears that condition 3 (absolute integrability of P(x)) will ensure that a normalization integral exists. So that seemed reasonable to me….is this correct?

Condition 3 in which enumeration? In post #159, you give condition 3 as

3. Ψ(x) and dΨ(x)/dx must be finite

I don’t see there anything about absolute integrability…

Correct me if I’m wrong (heck, you’ll correct even when I’m right…), but isn’t it possible that ? violates condition no. 3, as as ? ? L² per normalization integral?

It is possible in principle that there exists a Psi(x) that is not absolutely integrable over a period of infinity but Psi*(x)Psi(x) is integrable.

But I don’t know that there exists a set of physically realizable boundary conditions which would imply such a Psi(x) solution to Schrodinger’s equation!

However, it appears that condition 3 (absolute integrability of P(x)) will ensure that a normalization integral exists. So that seemed reasonable to me….is this correct?

Using Dirichlet conditions in the context of general Fourier transforms isn’t appropriate. Using them for Fourier series is all fine and dandy: they’ll provide you with the existence of on inverse transform, at least outside of a null-set.

I don’t believe that it is customary to speak of an inverse transform when one is speaking of a fourier SERIES!

What I presume you meant was the series converges on points except where the the function lies outside the null-set.

BUT, does there exist a function x(t) satisfying Dirichlet conditions with transform X(f), where the inverse transform of X(f) yields a function that is not in the same class as x(t) in L1? If not one might complain Dirichlet is too austere for Fourier Transforms.

But then I’ll counter by saying, will Jordan’s Test or Dini’s test for Fourier series lead to convergence (yes according to Apostol), so in that sense Dirichlet is too austere for Fourier Seiries.