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13h
comment Differential equation $y''-4y = e^{-x}$
A similar but more involved differential equation is solved in great detail at Initial Value Problem.
1d
comment The set of infinite sequences with finitely many nonzero values is dense.
(i) is immediate, and (iii) follows from (ii) and the fact that $l^p({\mathbb N})$ contains elements in which infinitely many $x_k$ are nonzero, so let's consider (ii). For (ii), pick $\{x_k\}$ in $l^p({\mathbb N}),$ and note that each of $(x_{1},0,0,0,\ldots),$ $(x_{1},x_{2},0,0,0,\dots),$ $(x_{1},x_{2},x_{3},0,0,\dots),$ etc. belongs to $V$ (why?) and they form a sequence in $V$ that converges to $\{x_k\}$ (why?). (Hint: What do you know about the tails of a convergent series?)
May
19
comment Prerequisites for learning general topology
Erica Flapan's book When Topology Meets Chemistry: A Topological Look at Molecular Chirality might be useful, perhaps as alternate reading for a gentler introduction to some of the topics.
May
18
comment Solving an equation including $e^{-x}$ with the Lambert W function
For what it's worth, in recent years calculus books have been including problems such as this that are intended to be solved approximately with a standard graphing calculator (e.g. the TI-82 and its descendents) without specifically saying that such a device is intended to be used. These problems began showing up in the early 1990s, and back then there would be explicit comments in the text saying that a graphing calculator was intended, but I think with the now omnipresence of graphing calculators, books have been less explicit about when they might be needed.
May
13
revised How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?
Deleted functional analysis tag, which is not an appropriate tag for this question.
May
8
comment Functional equation $f'(x)=cf(x+1)$ has a solution if and only if $c\leq 1/e$
You could try writing to him. See his web page.
May
7
comment Measure-theoretic analogue of a result from elementary calculus
Lebesgue point of a function is what you want to look up.
May
7
comment Crazy Set Theory Analogies
I believe the original symbols used by Cantor for union and intersection was based on this analogy, and they may have been suggested by (or perhaps just indirectly influenced by) Dedekind. José Ferreirós discusses the number-theoretic origins of union and intersection in his book Labyrinth of Thought, but I don't have my copy with me to look at right now.
May
7
comment $x y''+y'+y=0$ How to approach
Typically with power series solutions one doesn't expect to recognize the series, and in your specific case you wouldn't unless you were especially cognizant of Bessel functions.
May
7
comment $x y''+y'+y=0$ How to approach
This looks like it's a problem where you'd seek a power series solution.
May
6
revised Making sense of the expression $\lim_{x \rightarrow k^+}f(x)$ using filters, and a reference request.
Included some information (2nd paragraph of Preface) of Meyer's 1969 book
May
6
answered Making sense of the expression $\lim_{x \rightarrow k^+}f(x)$ using filters, and a reference request.
May
5
comment Why can't differentiability be generalized as nicely as continuity?
Not yet mentioned as far as I can tell are Radon-Nikodym derivatives, stochastic derivatives, and the seemingly endless wilderness of variations associated with people like A. P. Morse, W. W. Bledsoe, C. A. Hayes, H. Kenyon, C. Y. Pauc, M. du Guzmán, and others (e.g. Web Derivatives by Kenyon/Morse, and Derivation and Martingales by Hayes/Pauc). I don't know how any of this might fit into the areas of math your focus seems to be in, however.
May
4
comment Cardinality and Concrete Mathematics
@Asaf Karagila: Some reasonably natural sets with cardinality greater than $c = 2^{{\aleph}_0}$ are: (a) there are $2^c$ many functions from $[0,1]$ to $\mathbb R$ that are Riemann integrable; (b) on $\mathbb R$ there are $2^c$ many complete Borel measures and there are $2^c$ many $\sigma$-finite measures; (c) there are $2^c$ many convex sets in ${\mathbb R}^{2};$ (d) I think there are $2^{2^c}$ many $\sigma$-algebras on ${\mathbb R}.$
Apr
29
comment formula for the $n$th derivative of $e^{-1/x^2}$
For more details (but not all) of what Alamos discussed, see my answer to Examples of applying L'Hôpitals rule ( correctly ) leading back to the same state?.
Apr
28
comment Understanding bounded variation
Regarding your comment about all the partitions having a finite number of points, consider this question: What is the supremum of the number of partition points over all partitions? Each partition has a finite number of points, but the supremum of the number of points is infinity.
Apr
28
comment Real Analysis book with pictures and ideas of proofs
A little more advanced, with lots of intuition (and I think pictures) is Bressoud's A Radical Approach to Real Analysis. Bressoud's book is less advanced than Pugh's book mentioned below (a book I like a lot, by the way) and a little more advanced than Abbott's book mentioned below (another book I like a lot).
Apr
27
comment What are some genuine ways to define the derivative of a fractal?
Maybe something listed at Is there a garden of derivatives? is what you want? See also my answer to Most general $A \subseteq \mathbb R$ to define derivative of $f: A \to \mathbb R$?.
Apr
27
comment Real Analysis book with pictures and ideas of proofs
Look at Victor Bryant's Yet Another Introduction to Analysis. I've written about this book several times since 2000 -- google its title along with my name.
Apr
24
comment Reference for set theoretic topology
Regarding background books besides those in set theory, useful references are General Topology by Stephen Willard (1970/2004) for basic general topology background, General Topology by Ryszard Engelking (1989) for a more advanced and extensive treatment of basic general topology background, and Set-theoretic topology by Gregory L. Naber (1977) for a little-known book that gives a transition treatment between Engelking's book and the set-theoretic topology literature.