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2 answers
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A Jordan arc in the unit disk

Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
Jeff's user avatar
  • 95
239 votes
14 answers
76k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
25 votes
9 answers
6k views

Function with range equal to whole reals on every open set

There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$. I want to generalize this in a way to get a function ...
falagar's user avatar
  • 2,821
13 votes
3 answers
2k views

Set of real numbers with positive measure containing no midpoints

Does there exists a subset E of R with positive measure and without containing any midpoints (i.e. x,y distinct in E, (x+y)/2 not in E)?
Lieven's user avatar
  • 133
12 votes
1 answer
5k views

Points of continuity of Baire class one functions

This is an idle question motivated by two comments I made to a previous MO question (which I just searched for, unsuccessfully). That question asked if the characteristic function of the rationals is ...
Pete L. Clark's user avatar
5 votes
1 answer
2k views

Continuous functions remaining constant

I solved a problem in analysis and i was thinking of generalizing this question which i couldn't succeed. If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function which satisfies $f(x)=f(2x+1)$, ...
C.S.'s user avatar
  • 4,795
24 votes
11 answers
8k views

The role of the mean value theorem (MVT) in first-year calculus

Should the mean value theorem be taught in first-year calculus? Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
2 votes
0 answers
1k views

Good sources for linear algebra for convex optimization and graph analysis?

What are some good sources for linear algebra for convex optimization and graph analysis? In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
DoubleJay's user avatar
  • 2,383
4 votes
1 answer
346 views

approximately linear functions -- more

Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that $$f(x)+f(y)=g(x+y)$$ for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
Yiannis's user avatar
  • 123
7 votes
1 answer
2k views

approximately linear functions

i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies $f(x-y)=f(x)-f(y)+const$ then it is necessarily linear. are there any general ...
Yiannis's user avatar
  • 123
3 votes
1 answer
2k views

What is the pure intuition for topological continuity and topology? [closed]

I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity. The ...
Nick's user avatar
  • 191
4 votes
4 answers
385 views

Is anything known about $w^*(x)=\sup_y w(x+y)/w(y)$ for measurable functions w on $R^n$

In my recent studies (fourier multipliers on weighted Lp spaces) I have to deal with this kind of transformation: if w is a measurable function on $R^n$, define $w^*(x)=\sup_y \frac{w(x+y)}{w(y)}$. ...
Nicolò's user avatar
  • 783
1 vote
4 answers
620 views

Do there exist nonconstant functions such that...

Do there exist nonconstant real valued functions $f$ and $g$ such that the expression: $$f(x) -v/g(x)$$ is maximized at $x = v$ for all positive real $v$?
pavel's user avatar
  • 13
23 votes
3 answers
6k views

Density of smooth functions under "Hölder metric"

This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha ...
Vince's user avatar
  • 505
2 votes
1 answer
810 views

On matrices that almost have the same eigenvalues

Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if $$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
Malik Younsi's user avatar
  • 2,154
25 votes
1 answer
8k views

Convergence of Fourier Series of $L^1$ Functions

I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
Jesse Madnick's user avatar
-3 votes
2 answers
260 views

On \ell_3 norm in R^2

Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$, in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\...
user4606's user avatar
1 vote
1 answer
879 views

Countable discrete abelian group amenable

For me the definition of amenability of an at most countable discrete group (with counting measure) is existence of a Folner sequence. Assuming this, why is every countable discrete abelian group ...
Kestutis Cesnavicius's user avatar
7 votes
1 answer
2k views

Hanner's inequalities: the intuition behind them

Hanner's inequalities in the theory of $L^p$ spaces (see http://en.wikipedia.org/wiki/Hanner's_inequalities) look hard to come-up with at the first glance. Their proof (say, the one in Lieb & Loss ...
Kestutis Cesnavicius's user avatar
-4 votes
2 answers
6k views

Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + Cy^...
6 votes
1 answer
778 views

Inverse function theorem for DC-functions

I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it... Definitions: Let $h:\mathbb R^n\to\mathbb R$ ...
Anton Petrunin's user avatar
8 votes
2 answers
3k views

Centralizers in GL(n,p)

There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
Jack Schmidt's user avatar
  • 10.7k
-3 votes
1 answer
590 views

A problem regarding definition of p-norm [closed]

Let ${\bf x}=(x_1,...,x_n)$, the p-norm of x is $(|x_1|^p+...+|x_n|^p)^{1/p}$. If one of the components of x is 0, there will be exponential of the form $0^p$. If p is an irrational, $x^p$ is only ...
zzzhhh's user avatar
  • 764
13 votes
7 answers
35k views

Real analysis has no applications?

I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
17 votes
12 answers
5k views

Looking for an interesting problem/riddle involving triple integrals.

Does anyone know some good problem in real analysis, the solution of which involves triple integrals, and which is suitable for second semester Analysis students? Thanks!
Pandora's user avatar
  • 459
17 votes
4 answers
10k views

Prime/undecomposable matrices

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
Unknown's user avatar
  • 2,855
4 votes
2 answers
734 views

Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
Tom LaGatta's user avatar
  • 8,512
3 votes
2 answers
2k views

Examples of deterministic processes of quadratic variation which are of unbounded variation

In [Föllmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process ...
vonjd's user avatar
  • 5,935
7 votes
2 answers
505 views

The set of non-smooth points of a convex function is (m - 1)-rectifiable

I am looking for a reference to the following result. Let $f:\mathbb R^m\to\mathbb R$ be a convex function. Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-...
Anton Petrunin's user avatar
87 votes
8 answers
16k views

Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
KConrad's user avatar
  • 50.6k
7 votes
4 answers
6k views

The characteristic (indicator) function of a set is not in the Sobolev space H¹

Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$? And if so, what is the best/easiest/...
Spencer's user avatar
  • 1,771
9 votes
3 answers
4k views

Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...
user avatar
7 votes
1 answer
1k views

Is the absolutely continuous image of a nowhere dense set is also nowhere dense?

Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an absolutely continuous map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set? Remarks: The answer is "no" if $f$ is ...
pinaki's user avatar
  • 5,339
39 votes
8 answers
13k views

Can Cantor set be the zero set of a continuous function?

More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth? Some days ago I discovered that in this proof ...
pinaki's user avatar
  • 5,339
9 votes
2 answers
804 views

Partition of R into midpoint convex sets

We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$. My question is: is it possible to partition $\mathbb{R}$ ...
filipm's user avatar
  • 1,359
4 votes
0 answers
352 views

"Cholesky decomposition" X=YY* for p-adic matrices?

Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
Sean Rostami's user avatar
2 votes
0 answers
354 views

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
cheater's user avatar
  • 165
14 votes
6 answers
3k views

What's a natural candidate for an analytic function that interpolates the tower function?

I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
John Jiang's user avatar
  • 4,466
5 votes
0 answers
558 views

continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
gondolier's user avatar
  • 1,839
32 votes
4 answers
18k views

About the Riemann integrability of composite functions

When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions. For the composite function $f \circ g$, He presented three cases: 1) ...
X.M. Du's user avatar
  • 627
4 votes
8 answers
3k views

Theory of cones

Hi all, Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence ...
10 votes
2 answers
3k views

Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
Cristos A. Ruiz's user avatar
0 votes
1 answer
359 views

a unique solution ? iteration involving conditional distributions

consider the following mappings, G and T, $y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$ $z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$ where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in ...
rubin's user avatar
  • 1
7 votes
1 answer
1k views

Can a continuous, nowhere differentiable function have specified "shape" at every point?

I'm a bit embarrassed to admit that: a) This is a rather unmotivated question. b) I can't remember whether or not I've asked this before, but searching doesn't seem to turn anything up so ... ...
Mike Hall's user avatar
  • 793
6 votes
6 answers
4k views

existence of antiderivatives of nasty but elementary functions

In discussing with my honors calculus class the fact that some continuous elementary functions do not have an elementary antiderivative, I realized I was unsure whether every discontinuous elementary ...
James Propp's user avatar
  • 19.7k
2 votes
1 answer
897 views

Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis

Greetings, I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
107 votes
9 answers
36k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
Kevin Buzzard's user avatar
10 votes
4 answers
8k views

Any reference on multilinear algebra [closed]

Do you know any good reference on multilinear algebra?
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