All Questions
5,672 questions
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 ...
CommunityBot
- 1
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 ...
- 29.8k
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 ...
CommunityBot
- 1
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
- 73.2k
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 ...
CommunityBot
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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(\...
- 19.6k
9
votes
2
answers
1k
views
Question on eigenvalue square root subadditivity
ORIGINAL QUESTION
Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a
$2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller
eigenvalue of a $2\times2$ matrix. Is it true ...
- 9,486
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 ...
- 981
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 ...
- 237
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.
...
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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
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}{...
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- 1
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 ...
- 8,298
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 $\...
- 1,883
8
votes
1
answer
1k
views
Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 28.6k
2
votes
2
answers
308
views
Analogue of an orthogonal subspace in a noneuclidian normed space
This question is related to https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question.
...
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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
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 ...
- 13k
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$ ...
- 1,468
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 ...
- 236k
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 ...
CommunityBot
- 1
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 ...
- 5,743
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 ...
CommunityBot
- 1
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 ...
CommunityBot
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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 ...
- 27.5k
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 - ...
- 60.5k
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 ...
- 9,389
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-...
- 60.5k
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
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,<) \...
CommunityBot
- 1
6
votes
3
answers
1k
views
Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 9,389
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 ...
- 29k
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, ...
- 19.6k
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,\...
- 479
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 ...
- 16.2k
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 $\...
- 2,135
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 \...
- 22.3k
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 ...
- 2,210
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 ...
- 39.9k
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 ...
- 833
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)?
- 22.3k
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 $f(x)=f(2x+1)$, ...
- 39k
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 ...
- 22.3k
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
$w^*(x)=\sup_y \frac{w(x+y)}{w(y)}$.
...
CommunityBot
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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 ...
- 191
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 $v$?
- 3,631
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 ...
- 11.4k