All Questions
5,657 questions
9
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4
answers
952
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What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?
My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples.
Let me start the discussion with ...
- 3,245
9
votes
2
answers
792
views
Uniformly Lebesgue differentiable functions
Note: Here $\mu$ denotes Lebesgue measure on $\mathbb R$.
We say a function $f: \mathbb R \to \mathbb R$ is uniformly Lebesgue differentiable if there exists some measurable subset $E$ of $\mathbb R$ ...
- 6,215
9
votes
2
answers
244
views
If normal with respect to prime base then normal for all bases
I tried to find it on internet but couldn't so m asking this here. I want to ask if a number is normal with respect to all prime number base then do we know that it is normal with respect to any base. ...
- 335
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
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 ...
user2529
9
votes
3
answers
383
views
convergence of 2nd eigenvalue
Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real.
Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric ...
- 43.2k
9
votes
4
answers
742
views
Distributional derivatives are locally integrable implies the distribution is also locally integrable?
Let $T$ be a distribution on $\mathbb{R}^n$ such that there are functions $f_1,\ldots,f_n \in L^1_\text{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_j} = f_j, \forall j=1,\ldots,n. $
My ...
- 93
9
votes
2
answers
584
views
Does this integral condition characterize $L^\infty$?
Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition ...
- 6,215
9
votes
1
answer
966
views
A question about composition of functions
Recently, I heard this question: are there two functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is strictly increasing on $\mathbb{R}$ and $g\circ f$ is ...
- 343
9
votes
3
answers
657
views
Degree necessary of a polynomial?
Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
- 13.9k
9
votes
1
answer
224
views
Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
- 19.6k
9
votes
1
answer
366
views
Can the canonical Eudoxus-real representatives be defined easily?
(See e.g. here for background on the Eudoxus reals, which motivates this question.)
Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such ...
- 20.9k
9
votes
2
answers
424
views
Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?
This question was firstly asked in mathematics stack exchange. Getting no answer, I copied it to here.
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $...
- 193
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
9
votes
1
answer
764
views
Does the family of fat Cantor sets contain a measurable rectangle?
Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$.
...
- 6,215
9
votes
1
answer
918
views
A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
- 1,245
9
votes
1
answer
458
views
Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
- 43.2k
9
votes
1
answer
378
views
Does “on average” Hölder continuity imply Hölder continuity?
Let $\Omega$ be a smooth, bounded, connected open subset of $\mathbb R^n$.
A function $f: \Omega \to \mathbb R$ is said to be Hölder continuous on average of order $\alpha$, for $0 < \alpha < 1$ ...
- 6,215
9
votes
1
answer
608
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
9
votes
1
answer
805
views
Does every measurable subset of $\mathbb R$ of non zero Lebesgue measure contain arbitrarily long arithmetic progressions?
A subset $E$ of $\mathbb R$ is said to contain arbitrarily long arithmetic progressions, if for every natural $n$, there exists $a, d \in R, d$ nonzero, such that $a + kd$ is in $E$ for all natural $k ...
- 2,069
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
9
votes
2
answers
2k
views
Does the Weierstrass function have a point of increase?
Problem
The Weierstrass function $W(x)$ is given by
$W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$
where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.
A function $f:\mathbb{R}\...
- 491
9
votes
2
answers
313
views
Average as a constant approximation in $L^p$
Let $I=[0,1]$. For $p\in[1,\infty]$ define $C_p$ as the best constant such that for all $f\in L^p(I)$
$$
\left\|f-\int_If\,\right\|_{L^p(I)}\leq C_p\inf_{c\in\mathbb{R}}\left\|f-c\,\right\|_{L^p(I)}.
$...
- 493
9
votes
1
answer
299
views
Can all contours of a function on a disk be made arbitrarily small?
Denote $D=\{x^2+y^2\le1\}\subset\mathbb R^2$ a disk.
Let $f:D\to\mathbb R$ be a continuous function on it. I am interested in restrictions of simple Morse functions on $\mathbb R^2$, but I suspect ...
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
9
votes
1
answer
553
views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
- 541
9
votes
1
answer
301
views
For which Sheaf topoi is Brouwer's fan theorem true?
Brouwer's fan theorem is the standard result that the Cantor space is compact, or equivalently that the Cantor space viewed as a locale is spatial. Since it is a compactness result for a countable ...
- 1,947
9
votes
1
answer
556
views
A non-recursive, explicit formula for the Fabius function
The Fabius function $F\colon\mathbb R\to[-1,1]$ may be defined as the unique solution of the functional integral equation
$F(x)=\int_0^{2x}F(t)\,dt$ for all real $x$ such that $F(1)=1$.
The recent ...
- 128k
9
votes
1
answer
499
views
Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$
It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
- 13.8k
9
votes
1
answer
636
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
9
votes
1
answer
299
views
Sequence of nested sets in $[0, 1]$ with bound on gaps
What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...
user78817
9
votes
3
answers
375
views
Decay of real continuous algebraic functions at infinity
Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\...
- 5,169
9
votes
2
answers
2k
views
Stokes theorem with corners
I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:
Notation:
for $1 \le n \le m$
$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ \...
- 103
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
9
votes
1
answer
1k
views
Why is this generality in Vitali's Lemma useful?
In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?
Vitali's Lemma:
...
- 466
9
votes
2
answers
490
views
Rearrangement, conditional convergence, and "placid" permutations
This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
- 20.9k
9
votes
2
answers
777
views
Can the thief escape (from a smooth, simple closed curve)?
Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite ...
- 2,619
9
votes
2
answers
466
views
Small uncountable cardinals related to $\sigma$-continuity
A function $f:X\to Y$ is defined to be
$\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
- 41.9k
9
votes
1
answer
635
views
De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions
Context: In formulating problems for secondary school mathematics teachers (and students) about absolute value functions, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \...
- 7,831
9
votes
2
answers
2k
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Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
user37929
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
9
votes
1
answer
580
views
Does ODE uniqueness unconditionally implies the flow continuity?
Suppose we have a (say compactly supported) $C^0$-vector field $X:\mathbb R^n\to\mathbb R^n$ such that for every $x\in\mathbb R^n$ there is a unique $C^1$-curve $\gamma:\mathbb R\to\mathbb R^n$ ...
- 938
9
votes
3
answers
362
views
Maximum zero converges to $\sqrt{2}$
In my research I came upon a recursively defined sequence, and I'm pretty sure it converges to $\sqrt{2}$ though I can't prove it easily. I don't think it is a difficult question but I'm not sure.
...
- 342
9
votes
1
answer
734
views
Constructive analysis and synthetic differential geometry
I am curious if (any of) the various inequivalent constructions of the real line in constructive mathematics can be used to build a model of Kock and Lawvere's synthetic differential geometry? In ...
- 6,025
9
votes
1
answer
260
views
Cover of the positive real numbers by intervals
For which real numbers $x$ and $y$ does the following hold?:
$$
\bigcup_{\frac{a}{b} \in \mathbb{Q}^+}
\left[\frac{a}{b},\frac{a}{b}+\frac{1}{a^x b^y}\right]
\ = \ \mathbb{R}^+
$$
- 19.6k
9
votes
1
answer
1k
views
Limit formula for the second derivative
Suppose that $f$ is a real-valued function which is twice differentiable in the interval $(-1,1)$. Does the following hold?:
$$\lim_{h \to 0} \frac{f(h) - 2f(0) + f(-h)}{h^2} = f''(0)$$
If $f''(x)$ ...
- 365
9
votes
1
answer
2k
views
Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 263
9
votes
2
answers
1k
views
Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?
From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
- 2,049
9
votes
1
answer
947
views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
- 3,245
9
votes
2
answers
791
views
Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,283