All Questions
5,871 questions
3
votes
2
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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.
...
CommunityBot
- 1
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}{...
CommunityBot
- 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
-1
votes
1
answer
185
views
eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 56.6k
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
2
votes
1
answer
3k
views
Is it possible to decompose a symmetric, positive definite matrix in this way?
Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist ...
- 96.4k
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
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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
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
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2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
CommunityBot
- 1
1
vote
1
answer
399
views
Which linear transformations between f.d. Hilbert spaces contract the inner product?
Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$
...
- 10.3k
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
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
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
- 1
3
votes
3
answers
1k
views
Structure theorem for finitely generated Z[G] modules
For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?
- 57.6k
5
votes
4
answers
2k
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Diagonalization of Infinite Hermitian matrices
We know that $n\times n$ square Hermitian matrices can be diagonalized and have real eigenvalues.
Suppose I have a countable sized Hermitian matrix $A=(a_{ij})$ where the indices $i$ and $j$ run ...
- 60.5k
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
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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
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4
votes
0
answers
939
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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
7
votes
3
answers
2k
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Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
- 39.9k
11
votes
3
answers
1k
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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
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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
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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
7
votes
3
answers
744
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Looking for applications of a nice result in linear algebra
Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M_n(k)$ which commutes with $...
- 14.5k
10
votes
4
answers
3k
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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
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1
answer
362
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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
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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
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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
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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
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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
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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
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1
answer
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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
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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
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1
answer
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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
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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
3
votes
2
answers
376
views
Invariant subspaces of subalgebras of $M_n(C)$
Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, ...
- 18.7k
7
votes
1
answer
2k
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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
6
votes
1
answer
778
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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 $h:\mathbb R^n\to\mathbb R$ ...
- 45k
-3
votes
2
answers
260
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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\...
- 12.6k