All Questions
5,628 questions
2
votes
0
answers
104
views
Fourier multiplier with a singularity on a convex curve
Let $h$ be a strictly convex function such that $h(0) = h'(0)=0$. Let $\Phi: \mathbb{R}^2 \to \mathbb{R}$ be a $C^{\infty}$-function with compact support (say, $\Phi$ is supported on $[-1,1]\times[-1,...
4
votes
1
answer
280
views
Approximation of an integral over the unit ball of L_1
For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
- 4,705
2
votes
1
answer
208
views
Does a particular iteration produce a weak solution to a non linear pde?
Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...
- 3,245
5
votes
0
answers
428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
2
votes
1
answer
546
views
Limit involving regularized gamma function and its inverse
Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function $\Gamma(...
- 917
7
votes
4
answers
6k
views
Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? ...
- 263
1
vote
1
answer
133
views
Special finite subcover of a compact
Let $(a,b)\in \mathbb R^n$. We consider the following open cover of the compact line segment $[a,b]$: $$[a,b]\subset\underset{x\in [a,b]}{\bigcup}B(x,\rho_x),$$
where for $x\in K,B(x,\rho_x)$ is a ...
0
votes
0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
0
votes
2
answers
720
views
Is there a probability density function satisfying the following conditions?
I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.
...
- 5,470
6
votes
0
answers
223
views
Sum of product maximum
For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?
(...
- 259
12
votes
2
answers
732
views
Is it possible to have the set $f^{-1}(\lbrace x \rbrace)$ perfect for every $x$?
There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that
for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup_n \frac{...
user38090
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
- 16.6k
3
votes
1
answer
403
views
Is there a probability density function providing the least expected value?
Fix constant reals $A>1$ and $D>0$. Let $f:\mathbb{R}\to[0,\infty)$ be a probability density function on $\mathbb{R}$, i.e. $\int_{-\infty}^\infty f(x)\, dx=1$, that is continuous almost ...
- 31
18
votes
2
answers
1k
views
An Entropy Inequality (generalized)
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability measure ...
- 351
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
4
votes
1
answer
461
views
How to get an expression for this integral (Numerically/Analytically)
I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^1 P_l(t)P_{l'}(t) \, dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the $l$-th ...
user37929
8
votes
2
answers
2k
views
Do proper Zariski closed sets of algebraic sets have measure zero
This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...
- 81
1
vote
0
answers
416
views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
- 61
2
votes
0
answers
890
views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
- 692
-1
votes
2
answers
418
views
An inequality involving multi-index [closed]
I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = (\alpha_{...
- 339
0
votes
0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
9
votes
3
answers
4k
views
Is there a reference for compact imbedding theory of Hölder space?
This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
- 1,399
7
votes
1
answer
941
views
Kakeya and Nikodym maximal functions
I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more specifically,...
- 213
37
votes
1
answer
2k
views
Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?
(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...
- 3,017
1
vote
1
answer
183
views
Where find proof of such theorem about uniform convergence of differences
Where to find a proof of theorem which says that:
if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesgue measure or on the set of second category with Baire ...
- 13
4
votes
1
answer
240
views
Ideals in a subalgebra in $C^\infty(M)$
Let $M$ be a (usual, finite dimensional) smooth manifold, and $C^\infty(M)$ the algebra of (real valued) smooth functions on $M$. For each point $a\in M$ and for each subalgebra $A$ in $C^\infty(M)$ (...
- 7,426
2
votes
2
answers
2k
views
convergence of the infima of convex functions
Can one give a reference to a result like this:
If a sequence of convex functions $f_{n}$ on $\mathbb{R}$ converges pointwise to a non-monotonic function $f$, then $\displaystyle\inf_{\mathbb{R}...
- 128k
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
- 24.7k
16
votes
6
answers
3k
views
A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 2,239
4
votes
1
answer
370
views
Norms for complex measures
I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
- 123
5
votes
1
answer
345
views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
- 692
6
votes
1
answer
152
views
Terminology for sequences/functions that approach each other
What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ...
- 2,754
1
vote
0
answers
324
views
Linearization of cones
Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
- 6,962
8
votes
1
answer
838
views
Density of prime pairs whose gap is less than the average gap
By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
- 5,470
1
vote
1
answer
263
views
When can we "displace" an ultrafilter limit with another limit?
Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal A)...
- 565
3
votes
1
answer
133
views
A recurrent sequence related to the Brouwer fixed-point theorem
Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} \sum_{j=0}...
- 7,249
2
votes
1
answer
433
views
bounding the absolute value of a trigonometric polynomial
Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
\...
- 859
0
votes
0
answers
127
views
A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis
I have a question. The question is to prove:
The weight $w \in A_\infty $if and only if
$\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all
...
2
votes
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
- 21
2
votes
0
answers
563
views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.6k
0
votes
0
answers
94
views
Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 1
0
votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
0
votes
1
answer
1k
views
Proper Group action on a metric space
Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {...
- 1
3
votes
2
answers
291
views
on completeness of R_mn, the set of all rational functions of type (m,n)
It is known from finite dimensionality of $P_r$, the space of all polynomials of degree less than or equal to $r$, that $P_r$ is complete with respect to uniform norm.
Considering $R_{m,n}[a,b]=\{p/...
- 31
8
votes
8
answers
6k
views
Is Riemannian integration sufficient in physics?
Are there any applications in physics or engineering which require the Lebesgue integral and cannot be treated by Riemannian integration
2
votes
2
answers
224
views
"Then obviously..." (a bound on strongly differential functions on an interval)
On the fourth page of their 1967 paper Local Behavior of Solutions of Quasilinear Parabolic Equations, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real ...
- 363
4
votes
1
answer
529
views
Find a continuous function with a prescribed continuity set
It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...
- 193
8
votes
1
answer
597
views
complete metric space
Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\...
- 83
7
votes
0
answers
340
views
Polynomials and divided differences
I would greatly appreciate any hint for proving the following.
Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then $...
- 71
1
vote
1
answer
715
views
Importance of Denjoy-Carleman classes as a class.
Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
- 807