Sam Walters  

@SamuelGWalters

🇨🇦 Math prof at U. Northern B.C. Acting Chair of the Mathematics & Statistics Dept (since March 2016). Presbyterian Christian.

British Columbia, Canada
Vrijeme pridruživanja: ožujak 2015.

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  1. Prikvačeni tweet

    My paper is posted on . (It resulted from my - Symposium talk.)

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  2. Is there a continuous Fibonacci function? The old Fibonacci sequence adds previous two terms. A continuous version might be integrating previous unit interval, so F(x) = ∫ F(t) dt from t = x-1 to t = x. Show that there is a (non-trivial) smooth such function. Is it unique?

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  3. It was asked if there's an anti-Fibonacci sequence where each term is the sum of the two terms that follow it. Thus, x(n) = x(n+1) + x(n+2). Since this can be written x(n+2) = x(n) - x(n+1), we have a normal recursion. E.g. 1, 2, -1, 3, -4, 7, -11, 18, ... etc.

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  4. 2. velj

    Useful theorem: if f_n: [a,b] → R is a sequence of differentiable functions which converge at one point, and their derivatives df_n/dx converge uniformly on [a,b], then f_n converge uniformly to a differentiable function f and f'_n(x) → f'(x).

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  5. 2. velj

    Euler's formula is exp(it) = cos t + i sin t. The hyperbolic version of this is much easier: exp(t) = cosh t + sinh t with all the i's removed!

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  6. 2. velj

    Robert Wald mentions (in his General Relativity) an interesting partial order relation on the set of spacetimes (M,g), with metric g, and citing Zorn's Lemma. It is: (M,g) ≤ (M',g') if M can be isometrically embedded in M' with their Cauchy surfaces preserved.

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  7. 2. velj

    Who are you rooting for this ?

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  8. 2. velj

    Today is , the and the , which I'll be watching. It's also good news from the on .

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  9. 2. velj

    All the homotopy groups π_m of the odd-dimensional spheres (of dimension 2n+1) are finite Abelian groups for m ≠ 2n+1. For all m ≥ 1, the m-th homotopy group π_m of the m-sphere is the integers Z.

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  10. 1. velj

    Abstract is a huge tool in our understanding of topology, geometry, differentiable and complex manifolds, and in the various forms of algebraic structures used to study them -- like homology, cohomology, homotopy groups, K-theory, etc.

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  11. 1. velj

    Let u(x,y) be a harmonic real function on a domain D in R². If u² is also harmonic, then u is constant on D.

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  12. 31. sij

    The integer power usually appearing in De Moivre's Formula can be replaced by any complex number as long as we understand we're working with the "principal power" (as described here).

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  13. 31. sij

    In I like to place my commas inside dollar signs $...$ as opposed to outside (where possible) since when it compiles it looks more like a comma than a period. Thus, would write "$f(x) = x,$" instead of "$f(x) = x$,". (If I'm picky, blame it on my gramma's genes.)

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  14. 29. sij

    The trace of operators on Hilbert space, when defined, is independent of which orthonormal basis one picks (so is an invariant). Here, this is stated for "positive" operators T*T where T* is the analogue of the conjugate-transpose for matrices.

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  15. 28. sij

    Assume there is differentiable F such that F(F(x)) = cos x. Taking derivative of this gives F'(F(x)) F'(x) = -sin x. Evaluating this at the unique number c such that cos(c) = c gives F'(F(c)) F'(c) = -sin c < 0. But F(c) = c, so this says F'(c)^2 < 0, contradiction. qed

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  16. 27. sij

    Eigenvectors are a useful tool in describing the structure of Hermitian (or, more generally, normal) matrices in terms of their similarity with diagonal matrices. The structure of all matrices, however, requires we consider generalized eigenvectors and eigenspaces.

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  17. 26. sij

    The word "computation" has greater illusory powers than the word "calculation."

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  18. 26. sij

    If H is a subgroup of a group G of index 2 (so [G:H] = 2), then H is normal in G, and G/H ≅ Z/2Z.

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  19. 26. sij

    One of my private rules in writing papers is: help the reader, and don't be a smart-arse. You are writing to communicate, not show off. 😁

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  20. 26. sij

    Who said the appendix is useless? I like adding them to the end of my papers to help out. 😃

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  21. 24. sij

    Prove that there is no differentiable function F:R→R such that F(F(x)) = cos x (for all real x). (I.e., the cosine has no differentiable "compositional square-root".) The same result holds if F is only assumed to be continuous.

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