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6 votes
7 answers
5k views

Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
mshafrir's user avatar
  • 163
7 votes
4 answers
3k views

completeness axiom for the real numbers

Do any treatises on real analysis take the following as the basic completeness axiom for the reals? "Let $A$ and $B$ be set of real numbers such that (a) every real number is either in $A$ or in $B$; ...
James Propp's user avatar
  • 19.7k
13 votes
1 answer
2k views

Hausdorff Dimension and Hölder Continuity

Suppose we have a curve γ : [0,1] -> ℝn. It is well known that if this curve is Hölder continuous for some exponent α then the Hausdorff dimension of γ[0,1] is bounded above ...
Brent Werness's user avatar
13 votes
1 answer
1k views

Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them. Background The Wiener integral is an analytic tool to define certain "...
Theo Johnson-Freyd's user avatar
4 votes
2 answers
340 views

Embeddings of Weighted Banach Spaces

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
Leandro's user avatar
  • 2,044
3 votes
1 answer
363 views

"exchange" of real analyticity and integration

Sorry for the impreciseness of the title. It is merely meant for an analogy. Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For ...
gondolier's user avatar
  • 1,839
2 votes
0 answers
517 views

When deRham curve is bijection?

Motivation: Suppose we have deRham curve. From wikipedia: Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M: $d_0:\ M \...
kakaz's user avatar
  • 1,626
10 votes
0 answers
439 views

Evaluating Shintani cone zeta functions

Hi everyone I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ ...
user3628's user avatar
  • 265
6 votes
3 answers
1k views

Dependence of error on mesh for Riemann sums

Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$, and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest $\delta > 0$ such that every Riemann sum arising from a ...
James Propp's user avatar
  • 19.7k
7 votes
1 answer
727 views

Reference for Tate vector spaces

... aka locally linear compact vector spaces. The one reference I know is http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov3-10(CentExt).pdf. Does anyone know another good reference?
Dinakar Muthiah's user avatar
6 votes
1 answer
802 views

Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set

For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
Elgrimm's user avatar
  • 143
2 votes
2 answers
3k views

Statement of Lagrange's theorem on determinants(elementary question).

Apologies for this elementary question; but I was unable to find a reference otherwise. Let $A, B, C$ be square matrices of the same dimension. Then, $$\begin{vmatrix} A & C \\\ 0 & B \end{...
Anweshi's user avatar
  • 7,442
7 votes
4 answers
639 views

Explicit bounds for the asymptotics of oscillatory integrals

Recall the following theorem (c.f. LC Evans, M Zworski, "Lectures on semiclassical analysis", Theorem 3.15, depending on the version): Theorem: Let $\varphi: \mathbb R^n \to \mathbb R$ be smooth and ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
1k views

Why is this generality in Vitali's Lemma useful?

In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets? Vitali's Lemma: ...
S. Donovan's user avatar
28 votes
7 answers
5k views

Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be. "Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
2 votes
1 answer
1k views

How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?

I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
S. Donovan's user avatar
2 votes
2 answers
317 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.
Learner's user avatar
  • 143
12 votes
2 answers
812 views

Inequality in Gaussian space -- possibly provable by rearrangement?

The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem. Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
Ryan O'Donnell's user avatar
5 votes
5 answers
2k views

Cardinality of Equivalence Classes of Cauchy Sequences

What's the cardinality of a single equivalence class of Cauchy sequences in ℚ? To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
SLaks's user avatar
  • 153
2 votes
3 answers
946 views

How can I measure the Morse index in infinite dimensions?

Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
Theo Johnson-Freyd's user avatar
2 votes
4 answers
3k views

Splitting a space into positive and negative parts

Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
Theo Johnson-Freyd's user avatar
4 votes
2 answers
730 views

Decomposition of Hölder continuous functions

Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is ...
user17240's user avatar
  • 852
23 votes
4 answers
2k views

Which is the correct ring of functions for a topological space?

There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one. ...
Theo Johnson-Freyd's user avatar
51 votes
5 answers
18k views

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in ...
Mark Meckes's user avatar
  • 11.4k
3 votes
1 answer
263 views

Asymptotically multiplicative functions and matrices

Hi, Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
M.G.'s user avatar
  • 7,127
8 votes
1 answer
638 views

Composite residues with determinant denominators

I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
Jared Kaplan's user avatar
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
Gil Kalai's user avatar
  • 24.7k
38 votes
26 answers
57k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
26 votes
2 answers
9k views

Maximal ideals in the ring of continuous real-valued functions on ℝ

For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
Alon Amit's user avatar
  • 6,734
72 votes
9 answers
16k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
Yoo's user avatar
  • 1,093
3 votes
1 answer
2k views

Minkowski inequality

In the Wikipedia proof of the Minkowski inequality (http://en.wikipedia.org/wiki/Minkowski_inequality), the following inequality is used: $$|f+g|^p\leq2^{p-1}(|f|^p+|g|^p).$$ I was just wondering if ...
mornington's user avatar
18 votes
3 answers
2k views

Elementary $\mathrm{Ext}^1$ intuition

$\DeclareMathOperator{\Hom}{\operatorname{Hom}}\DeclareMathOperator{\Ext}{\operatorname{Ext}}$I am wondering what sort of basic basic intuitive meaning $\Ext^1(M,N)$ has. As a base case: if $M$ and $N$...
alekzander's user avatar
40 votes
5 answers
10k views

Is there a natural measures on the space of measurable functions?

Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first ...
Kenny Easwaran's user avatar
7 votes
2 answers
477 views

Characterizing the Radon transforms of log-concave functions

$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex). Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$, $$ ...
Darsh Ranjan's user avatar
  • 5,992

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