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Results tagged with ca.classical-analysis-and-odes
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user 7311
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
34
votes
Do these properties characterize differentiation?
Here is a slightly more general take on this question. First notice that your condition $L(1) = 0$ is redundant. That is because if you take $f = g = 1$ in your third condition, you get $L(1) = L(1) + …
- 15.3k
8
votes
Calculus book in the spirit of the 18th century
Have a look at Sylvanus P. Thompson's "Calculus Made Easy" subtitled, What One Fool Can Do, Another Can.
It was written at the end of the 19h Century but takes a pretty much 18th Century approach (and …
- 15.3k
20
votes
Why is the Laplacian ubiquitous?
In Physics, the essential reason boils down to symmetry. One expects the fundamental laws of physics to be independent of where you are or how you are oriented in space, and if they are described by s …
- 15.3k
21
votes
Atiyah-Singer index theorem
I know it may seem rather "old", but the notes from the IAS "Seminar on the Atiyah-Singer Index Theorem from back in 1965 (published by Princeton Univ. Press) may be just what you are looking for, sin …
- 15.3k
4
votes
Any good books on numerical methods for ordinary differential equations?
If you do not mind a "self-reference" there is "Differential Equations, Mechanics, and Computation",
published by AMS and written by me and my son Bob Palais. See the associated website at:
http://od …
0
votes
Consequence of equidistribution or not?
To say that $n \theta$ is equidistributed means in particular that for any open set $O$ of $(0,1)$ that if $N(n)$ is the the number of $k < n$ such that $k \theta (\mod 1) \in O$, then $N(n) /n $ appr …
- 15.3k
23
votes
Accepted
Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?
Well, I cannot say for certain, but I did know Gleason well (he was my thesis advisor, and we wrote a paper together after that) and I have written an essay about Gleason's work on the Fifth Problem ( …
- 15.3k
2
votes
Analytical solutions of a differential equation (from Archimedes' Spiral)
If you go to the Archimedean Spiral page of the Virtual Math Museum:
http://virtualmathmuseum.org/Curves/archimedean_spiral/archimedean_spiral.html
you will find a lot of material about Archimede's …
- 15.3k
21
votes
1
answer
1k
views
If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a ...
This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, suppos …
- 15.3k
1
vote
Analytic hypoellipticity of linear ordinary differential operators
It is easy to generalize the proof tha a differentiable real ODE has differentiable solutions (using a fxed point argument) from the real to the complex case, and deduce that a holomorphic ODE has loc …
- 15.3k
8
votes
Good differential equations text for undergraduates who want to become pure mathematicians
If you don't mind considering a recommendation from one of the co-authors of an ODE textbook, you sound like just the sort of student that we had in mind when we wrote "Differential Equations, Mechani …