All Questions
5,632 questions
7
votes
2
answers
2k
views
Baire Category Theorem Application
In Antoine Henrot Michel Pierre -
Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
- 2,222
1
vote
1
answer
275
views
Shift operator that generates separable orbit
Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$...
- 696
3
votes
0
answers
302
views
functions on intervals with endpoints
Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or
just on $(0,\infty)$?
More generally, is there a standard interpretation of the assertion "$F$ is an ...
- 19.7k
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 ...
- 111
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 ...
- 8,493
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 $\...
- 121
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 ...
- 4,952
11
votes
3
answers
2k
views
Hilbert's 17th Problem for smooth functions
Consider an open subset $U \subseteq \mathbb{R}^n$ and a smooth function $f\colon U \longrightarrow \mathbb{R}$ with $f(x) \ge 0$ for all $x \in U$.
It is then known (if I remember correctly: by ...
- 8,098
46
votes
7
answers
10k
views
Are some numbers more irrational than others?
Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
- 1,823
19
votes
4
answers
12k
views
How did Bernoulli prove L'Hôpital's rule?
To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
- 4,254
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 ...
- 505
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 $x$, $f^n(x)=0$ ...
- 678
2
votes
3
answers
5k
views
Smooth approximation of the hinge loss function
I came across a paper but the smooth approximation for the hinge loss function is wrong. Can someone guide me to the proper smooth approximation (using polynomials) of the function $$h(x)=\max(0,1-x)$$...
- 111
49
votes
3
answers
6k
views
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
- 13.1k
10
votes
1
answer
772
views
Nondifferentiability set of an arbitrary real function
A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...
- 367
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 ...
- 2,829
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 ...
- 543
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 8,512
21
votes
1
answer
1k
views
If $f:R^n \to R$ is a smooth real-valued function such that $\nabla f : R^n \to R^n$ is a diffeomorphism, what can one conclude about the behavior of $f(x)$ at infinity?
This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, ...
- 15.3k
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
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 ...
- 8,512
18
votes
4
answers
3k
views
Why is there no Borel function mapping every countable set of reals outside itself?
A choice function maps every set (in its domain) to an element of itself. This question concerns existence of an anti-choice function defined on the family of countable sets of reals. In an answer to ...
- 30.1k
9
votes
1
answer
958
views
Quantitative bounds for multivariate central limit theorem
For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:
https://...
- 93
102
votes
21
answers
15k
views
Proofs of the uncountability of the reals
Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not ...
- 2,855
9
votes
2
answers
791
views
Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,283
11
votes
1
answer
2k
views
Functions whose antiderivative behaves like xf(x)
I'm wondering if a classification of analytic functions, $f\,$ (it may be that $C^1$ is enough, but I'm not taking any chances, if you have a reason why I only need to consider a larger class of ...
- 1,049
1
vote
2
answers
318
views
Finite interpolation by nondecreasing indefinitely differentiable functions in a finite-dimensional space
Some time ago, I asked about inite interpolation by
a nondecreasing polynomial here at Finite interpolation by a nondecreasing polynomial. This turned out to be an already solved problem; it also ...
- 3,595
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 ...
- 201
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
78
votes
5
answers
8k
views
Does pointwise convergence imply uniform convergence on a large subset?
Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.
Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$?
Is there a ...
- 31.5k
9
votes
3
answers
934
views
local behavior of a finite Borel measure
Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
- 1,839
46
votes
2
answers
8k
views
"Closed-form" functions with half-exponential growth
Let's call a function f:N→N half-exponential if there exist constants 1<c<d such that for all sufficiently large n,
cn < f(f(n)) < dn.
Then my question is this: can we prove that no ...
- 9,823
7
votes
2
answers
724
views
Sturm chain analogue for exponential polynomials?
I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...
- 8,688
9
votes
2
answers
616
views
construction of a random measure with a given mean
Let me first pose a trivial question.
Given a Borel probability measure $\mu$ on the real line, is it possible to construct a purely atomic random measure $M$ whose mean is $\mu$?
The answer is ...
- 1,839
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 ...
- 305
7
votes
4
answers
1k
views
The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees
This question is closely related to MO f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. Consider $e^{e^x-1}$, this is the generating function of the Bell ...
user37691
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 29.8k
1
vote
3
answers
845
views
$H^{-1}$ conservative gradient flow and $L^2$ projection
Consider Cahn-Hilliard (see this) equation hich is known as the $H^{-1}$ gradient flow of Cahn-Hilliard energy functional, also it is easy to verify that this equation is mass preserving i.e. measure ...
- 53
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 ...
- 595
20
votes
3
answers
2k
views
Do convex and decreasing functions preserve the semimartingale property?
Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
- 17.1k
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-...
- 39k
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 ...
- 10.1k
22
votes
2
answers
2k
views
Is a real power series that maps rationals to rationals defined by a rational function?
Suppose that the function $p(x)$ is defined on an open subset $U$ of $\mathbb{R}$ by a power series with real coefficients. Suppose, further, that $p$ maps rationals to rationals. Must $p$ be defined ...
- 6,199
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,<) \...
- 25.5k
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 ...
- 315
4
votes
2
answers
767
views
Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
- 7,320
114
votes
34
answers
86k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
4
votes
2
answers
3k
views
Power series with non-integer exponents
Motivation:
For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that ...
- 8,903
10
votes
2
answers
3k
views
Gluing two diffeomorphisms together
A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
$\psi(...
- 8,903