All Questions
5,657 questions
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Is there a dense planar rational point set within which the distance of any two points is an irrational number?
i.e. could we find a subset $X\subset \mathbb{Q}^2$ such that $\overline{X}=\mathbb{R}^2$ and that for any $x,y\in X$ the distance $|x-y|$ is an irrational number?
I'm considering the following ...
user178596
7
votes
1
answer
2k
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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 ...
- 123
7
votes
1
answer
561
views
How are real numbers defined in elementary recursive arithmetic?
I am currently reading about elementary function arithmetic and Harvey Friedman's grand conjecture.
In Number theory and elementary arithmetic, Jeremy Avigad expressed Fermat's last theorem, ...
user507115
7
votes
2
answers
607
views
If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?
Let $\{a_k\}_{k\in \mathbb{Z}} \subset \mathbb{R}$ a real sequence and $a\in \mathbb{R}$ such that $$ \lim_{n\to +\infty} \frac{1}{n} \sum_{k=1}^n a_k = a = \lim_{n\to +\infty} \frac{1}{n+1} \sum_{k=0}...
- 81
7
votes
1
answer
1k
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Extending continuous functions from $\mathbb Q$ to $\mathbb R$
Definitions:
Let $E$ be a subset of $X$. By an extension of a function $f: E \to \mathbb R$, I mean a function $\bar f: X \to \mathbb R$ such that $f = \bar f$ on $E$.
Question: For every continuous ...
- 6,215
7
votes
1
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547
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Is this operator bounded?
Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space.
We then study
$$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$
I would assume that this norm is ...
- 192
7
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1
answer
371
views
An elementary inequality for three complex numbers
The following problem arose in asymptotic analysis of difference equations.
Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have
$$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
- 128k
7
votes
2
answers
999
views
If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.?
If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that
...
- 73
7
votes
2
answers
470
views
Continuous functions and infinity
Suppose $f(x)$ is continuous on $\mathbb{R}$, for $\forall \delta>0, \forall x\in\mathbb{R}, \lim_{n\rightarrow\infty}f(x+n\delta)=+\infty$. Is it correct that $\lim_{x\rightarrow+\infty}f(x)=+\...
- 183
7
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1
answer
192
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On the zero set of a $C^2$ function on $[0,1]^2$
Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I_x\subset [0,1]$ such that $f(x,y)=0$ for all $...
- 73
7
votes
2
answers
437
views
Radial limit does not exist almost everywhere
Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
- 171
7
votes
2
answers
518
views
Morse lemma with least amount of regularity.
I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
- 1,211
7
votes
2
answers
997
views
$L^p$ bounds on tails of bounded $L^q$ sequences
Note: This is a generalisation of an earlier problem as suggested by user Jochen Glueck in the comments.
Let $1 \leq p < q \leq \infty$, and $f_n: [0, 1] \to \mathbb R$ be a sequence of functions ...
- 6,215
7
votes
1
answer
856
views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
- 395
7
votes
3
answers
841
views
Distance function to $\Omega\subset\mathbb{R}^n$ differentiable at $y\notin\Omega$ implies $\exists$ unique closest point
I am trying to show the following two statements are true:
(1) For any nonempty set $\Omega\subset\mathbb{R}^n$, the set $B$ consisting of points $y\notin\Omega$ where there is not a unique closest ...
- 323
7
votes
2
answers
1k
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Intermediate value for a vector-valued function
Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f_1(x), ..., f_n(x)\}$ with $x\in[0,1]^n$, where the $f_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the ...
- 41
7
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2
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508
views
Why is $\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ harmonic if $u$ is harmonic?
I found myself trying to prove the following, but I had to compute everything explicitly.
It is well known that if $u:\mathbb{R}^n\to\mathbb{R}$ is an harmonic function on $\mathbb{R}^n$, then the so-...
- 213
7
votes
3
answers
547
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,215
7
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1
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348
views
Descartes' rule of signs for infinite series
Consider the function given by
$$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$
where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
- 43.2k
7
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2
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1k
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Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction
This is cross-posted in MSE.
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\...
user14319
7
votes
2
answers
787
views
Riemannian distance functions on the real line
A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties:
$d$ is a length metric (...
- 13.5k
7
votes
3
answers
709
views
On the inequality $\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\right)^2$
I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\...
- 191
7
votes
4
answers
639
views
Explicit bounds for the asymptotics of oscillatory integrals
Recall the following theorem (c.f. LC Evans, M Zworski, "Lectures on semiclassical analysis", Theorem 3.15, depending on the version):
Theorem: Let $\varphi: \mathbb R^n \to \mathbb R$ be smooth and ...
- 54.6k
7
votes
2
answers
477
views
Characterizing the Radon transforms of log-concave functions
$f:\mathbf{R}^d\to \mathbf{R}_{\ge 0}$ is log-concave if $\log(f)$ is concave (and the domain of $\log(f)$ is convex).
Theorem: For all $\sigma$ on the sphere $\Bbb S^{d-1}$ and $r\in \mathbf{R}$,
$$
...
- 5,992
7
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1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 128k
7
votes
1
answer
580
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
7
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1
answer
834
views
Representing $\Gamma(a-x)$ in terms of $\Gamma(kx)$ and $\Gamma(a)$ and elementary functions
I asked this question on MSE here.
I wonder if it is possible to represent $\Gamma(a-x)$ in terms of powers of $\Gamma(a)$, powers of $\Gamma(kx)$, and elementary functions. I am not looking for any ...
- 541
7
votes
2
answers
567
views
Intuition for Agmon-Douglis-Nirenberg ellipticity
First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...
- 1,429
7
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1
answer
250
views
A trapping set with finite measure
Does there exist a measurable subset $T$ of $[0, \infty)$ with finite measure and some $\epsilon > 0$ such that for every $r$ with $0 < r < \epsilon$, $nr$ is in $T$ for infinitely many ...
- 2,069
7
votes
2
answers
588
views
$\sum_{k =1, k \neq j}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)\csc^2\left(\pi \frac{j-k}{N} \right)=?$
It is well-known that one can evaluate the sum
$$\sum_{k =1}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)=\frac{N^2-1}{3}.$$
The answer to this problem can be found here
click here.
I am now ...
- 192
7
votes
2
answers
2k
views
The norm of tensor product operator on Lp spaces
Let $X, Y$ be two $\sigma$-finite measure spaces and $p,q\in [1,\infty]$. Let $T_1, T_2:L^p(X)\rightarrow L^q(Y)$ be two bounded linear operators. Then one can define a linear operator $$T_1\otimes ...
- 71
7
votes
1
answer
355
views
Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?
Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$?
It is known that for $n = 2$, the function $\displaystyle ...
- 211
7
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1
answer
450
views
Convergence of Lagrange interpolation polynomials to entire functions
Consider an entire function $\ f:\mathbb C\rightarrow\mathbb C.\ $ Let $\ (a_n\in\mathbb C:n=0\ 1\ \ldots)\ $ be an infinite sequence, where $\ a_k\ne a_n\ $ whenever $\ k\ne n.\ $ Let $\ L_n\ $ be ...
- 7,500
7
votes
1
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313
views
Surprisingly simple minimum of a rational function on $\mathbb R_+^n$
Motivation:
The following problem has occurred in a study of energy dissipation in a chain of coupled, damped oscillators.
The problem:
Let me define specific rational functions $f$, $g$, and $...
- 686
7
votes
1
answer
2k
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If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?
Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
- 483
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
7
votes
2
answers
178
views
Separating domains in $\mathbb{R}^{2n}$ by a real algebraic variety
Suppose $\Omega_1$ and $\Omega_2$ are two disjoint unbounded domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be ...
- 171
7
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1
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409
views
A property of $C^2$ functions
Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?
- 73
7
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1
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204
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Are $\log(\sigma(A(z))$ subharmonic functions?
Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$.
Is it ...
- 124
7
votes
2
answers
455
views
On a monotonicity property of Fourier coefficients of truncated power functions
Is it true that
$$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$
is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$?
This question is related to this previous one.
Twice integrating by parts, one ...
- 128k
7
votes
2
answers
269
views
Box dimension of the graph of an increasing function
This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
- 439
7
votes
2
answers
665
views
Non-separable metric probability space
Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...
- 6,541
7
votes
2
answers
913
views
Optimal Talmudic Zigzag
I have a finite sequence of positive real numbers $p_1,\dots, p_n$ and I am looking for a monotonically ascending sequence of indices $z_1,\dots, z_k$ that starts with $z_1 = 1$ and ends with $z_k = n$...
7
votes
3
answers
602
views
Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$
During research involving the Born–Jordan quantization I came across the expression
$$
\frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1
$$
for $k\in\mathbb N_0$. It is not too ...
7
votes
3
answers
986
views
Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
- 128k
7
votes
1
answer
1k
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Is there a bound for Lipschitz constant in terms of second differences?
It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where ...
- 16.6k
7
votes
2
answers
5k
views
Relationship between the derivative of a matrix and its eigenvalues
Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing?
...
- 71
7
votes
2
answers
724
views
Sturm chain analogue for exponential polynomials?
I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...
- 8,688
7
votes
3
answers
524
views
Rigorous estimates on roots of function
We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...
7
votes
1
answer
598
views
A lecture by Rudin
Is it available a written version of this lecture by Rudin on the relation between Fourier analysis and the birth of set theory?
https://youtu.be/hBcWRZMP6xs
If not Rudin himself, maybe someone else ...
- 4,459