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1 vote
0 answers
174 views

Eigenvalues of a Parametrized Family of Linear Functions

Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number. For each $\alpha$, it is given that $L(\alpha)$ is a ...
21 votes
2 answers
924 views

Codimension of Measurable Sets

I am currently teaching an advanced undergraduate analysis class, and the following question came up. Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the ...
1 vote
1 answer
685 views

This limit converges to the partial derivative?

Let a function $f:X \times \mathbb{R} \rightarrow \mathbb{R}$ continuous, with $X \subset \mathbb{R}$ compact, and supose that $\partial_2 f(x,t)$ exists for all $x \in X$ and is continuous. (here $\...
5 votes
0 answers
369 views

Independent Events Inducing Probability Measures

Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
0 votes
1 answer
937 views

Lebesgue's Majorized Convergence Theorem

Can anyone point me to an explanation and a proof of this theorem? For reference, it is mentioned in Kolmogorov's almost everywhere divergent function in $L$ as given in Zygmund, volume I. In the ...
4 votes
1 answer
1k views

An application of Baire category theorem

Hi, Does somebody know a proof (or a reference) for the following statement: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function. Suppose that for all $x$, $f^n(x)=0$ ...
8 votes
3 answers
813 views

Strange real functions

I know there are a lot of strange functions $f~:~\mathbb R \to \mathbb R$. I'm looking for an "elementary but complete" exposition of a result discovered by W. Sierpi\'nski and A. Zygmund in "Sur une ...
44 votes
3 answers
4k views

Smooth functions for which $f(x)$ is rational if and only if $x$ is rational

A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and ...
5 votes
1 answer
781 views

Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
1 vote
2 answers
641 views

Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$

Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the ...
1 vote
0 answers
615 views

Is there a Real valued function with image of every open interval the whole real line [duplicate]

Possible Duplicate: Function with range equal to whole reals on every open set Hello, My problem is the following "Is there a Real valued function with image of every open interval the whole ...
2 votes
1 answer
465 views

What is the regularity of the argument of a complex function?

Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
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 - ...
3 votes
1 answer
952 views

Geometrical structure of critical points of harmonic functions

For a harmonic function $\Phi$ on a simply connected subset $\Gamma$ of $\mathbb{R}^3$, define a guide curve $\gamma: I \mapsto \Gamma$ of $\Phi$ as a simple regular $C^1$ curve such that all point ...
5 votes
2 answers
917 views

Is the inclusion of Lebesgue spaces compact?

[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-...
2 votes
0 answers
470 views

Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
4 votes
3 answers
794 views

Monotone injection of an ordinal into $[0,1]$

This is related to my recent question and would provide a natural positive answer to Question 2. I am sure this must be known to experts. Question: Is there a monotone injection $(\omega_1,<) \...
4 votes
0 answers
939 views

Proofs of Baire category theorem

I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere). My motivation is the ...
11 votes
3 answers
1k views

Can there be two continuous real-valued functions such that at least one has rational values for all x?

Of course, no continuous real valued non-constant function can attain only rational or irrational values, but can there be a pair of nowhere-constant continuous functions f and g such that for all x, ...
5 votes
2 answers
3k views

Closedness of finite-dimensional subspaces

Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed? I suspect yes, but I can't come up with a proof, and it seems like locally ...
1 vote
2 answers
360 views

Inf of a mutivariate function

Let $f(x_1,\ldots , x_n) = \frac{x_1}{x_2+x_3} + \frac{x_2}{x_3+x_4} + \cdots + \frac{x_n}{x_1+x_2}$, defined for $x_i>0$. Is there $(x_1, \ldots ,x_n)\in {\mathbb{R}^*_+}^n$ such that $f(x_1,\...
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 ...
2 votes
0 answers
299 views

Uniqueness of dimension for topological vector spaces

Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$. Let $B$ be a subset of $V$ satisfying . Linearly Independent: For all functions $f$ in $\mathbb{...
10 votes
4 answers
3k views

Measure 0 sets on the line with Hausdorff dimension 1

I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\...
3 votes
1 answer
362 views

Cartesian product of test function spaces

Mini introduction Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
3 votes
1 answer
367 views

A differential inclusion relating to the slope of a convex function

This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the ...
6 votes
1 answer
369 views

Denominators in the solution to Hilbert's XVII

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...
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 ...
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)?
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 $f(x)=f(2x+1)$, ...
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 ...
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 $w^*(x)=\sup_y \frac{w(x+y)}{w(y)}$. ...
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 ...
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 $v$?
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 ...
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 $h:\mathbb R^n\to\mathbb R$ ...
-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\...
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 ...
-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 ...
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 ...
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 $\mathbb{R}$ ...
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 $V_- \subseteq V$...
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_{...
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 ...
12 votes
4 answers
1k views

Topologizing free abelian groups

For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
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?
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.
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 ...
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.
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 \...