All Questions
5,908 questions
141
votes
17
answers
38k
views
Why is differentiating mechanics and integration art?
It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/...
- 5,935
12
votes
1
answer
898
views
Converse to Banach’s fixed point theorem for ordered fields?
Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
- 19.7k
11
votes
3
answers
3k
views
Is the supremum of continuous functions integrable?
Let $f_\alpha$ be a family of continuous positive functions $\mathbb R\to \mathbb R$
where the index $\alpha$ runs in a compact metric space
and the map $\alpha\to f_\alpha$ is continuous
with ...
- 29.1k
7
votes
3
answers
3k
views
incompleteness in real analysis
Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories ...
- 19.7k
1
vote
1
answer
1k
views
How to verify the weak convergence?
Given a finite measure on a compact, take $f_n\in L^1$ with norms $\leq 1$ and suppose that $\int f_n g$ tends to a limit for all continuous $g$. Is it true that then $\int f_n g$ converge for any $g\...
- 11
6
votes
3
answers
11k
views
Sums of uncountably many real numbers [closed]
Suppose $S$ is an uncountable set, and $f$ is a function from $S$ to the positive real numbers. Define the sum of $f$ over $S$ to be the supremum of $\sum_{x \in N} f(x)$ as $N$ ranges over all ...
- 15.4k
0
votes
1
answer
604
views
Find a explicit choice function of the "rationally equivalence class"
Define two real numbers to be rationally equivalent provided their difference is a rational number.
from Royden Real Analysis
- 3
20
votes
3
answers
4k
views
Propositions equivalent to the completeness of the real numbers
Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't?
...
- 19.7k
4
votes
0
answers
162
views
Symmetric functions and regularity (II)
My previous question (where $n=2$) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved.
Let $f=\mathbb R^n\rightarrow\mathbb R$ be a ...
- 52.3k
5
votes
1
answer
316
views
Symmetric functions and regularity
Let $f:\mathbb R^2\rightarrow\mathbb R$ be a symmetric function: $f(y,x)=f(x,y)$. It can therefore be written has a function of the elementary symmetric polynomials, here $f(x,y)=F(x+y,xy)$, where $F(\...
- 52.3k
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
0
votes
1
answer
1k
views
A question about regular signed or complex Borel measure under LRN decomposition
Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$, m is the Lebesgue measure on the class of Borel sets $\mathcal B_{\mathbb R^n}$ and the Lebesgue-Radon-Nikodym ...
- 764
11
votes
1
answer
1k
views
Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem
I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$:
$$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
- 4,952
5
votes
0
answers
583
views
Cohomology of Real algebraic Varieities
I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.
...
user2529
7
votes
2
answers
2k
views
Baire Category Theorem Application
In Antoine Henrot Michel Pierre -
Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
- 2,222
1
vote
1
answer
275
views
Shift operator that generates separable orbit
Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$...
- 696
3
votes
0
answers
302
views
functions on intervals with endpoints
Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or
just on $(0,\infty)$?
More generally, is there a standard interpretation of the assertion "$F$ is an ...
- 19.7k
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
1
vote
1
answer
254
views
Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user7807
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
2
votes
1
answer
205
views
Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 657
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
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
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
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
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