New answers tagged real-analysis
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Compact-open Topology for Partial Maps?
There is a way to promote function space topologies so that convergence of nets behaves the way that one would expect it to behave. There are probably other ways of getting a topology for spaces of ...
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Uniqueness of solutions of Young differential equations
We will follow the Lemma 8.10. (Rough Gronwall) and Proposition 8.12 from "a course in rough paths" but modify them for this particular setting of studying
$$Y_{t}=Y_{0}+\int^{t}_{0} Y dX,$$
...
- 3,833
3
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Functions that are Khinchin integrable but not Henstock-Kurzweil integrable
Let $F$ be the function from example 6.20 c) in [1]. That is, fix a perfect nowhere dense subset $E$ of $[0, 1]$ such that $0, 1\in E$ such that $0 < |E| < 1$, for example the fat Cantor set. ...
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Can a nowhere locally Hölder function be differentiable almost everywhere?
Define $$\psi(x)=\begin{cases} 1/|\log x| &\text{if } x\in (0,1/2] \\ 0 &\text{if }x\leq 0\\
1/\log 2& \text{ if } x>1/2.\end{cases}$$
Note that $\psi$ is increasing and bounded (and ...
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Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$
$\newcommand\R{\mathbb R}\newcommand\S{\mathcal S}\newcommand\ep{\varepsilon}\newcommand\th{\theta}\newcommand\de{\delta}\newcommand\De{\Delta}$Take any $\tau\in\S(\R^n\times\R^n)$. Then for any $\phi\...
- 110k
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Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$
This a comment (really warning) but too long. Firstly, the good news: if we consider the two dimensional distribution $\delta(x-y)$, this is perfectly well-defined on the plane as is $\delta (x_0-y)$...
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How to control Wasserstein distance in terms of characteristic function
You might want to look for the concept called "Fourier-based metrics/distances", also known as Toscani distances, developed by Giuseppe Toscani and his colleagues.
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6
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High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?
The answer is still (surprisingly to me!) yes. As you've already established, there must exist $L = \lim \frac{||x_n||}{n}$ (I will never write a vector symbol over the variable). If $L = 0$ then ...
- 3,891
4
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Is this constraint convex?
Rewrite the constraint as
$$x_n \le f_n(\rho_1,\dots,\rho_n):=\ln\Big(B\log_2\Big(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\rho_i} g_i^2+\sigma^2}\Big)\Big).$$
The problem is then to prove the ...
- 110k
3
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Can a solution to this parameterized ODE converge to zero?
The answer is yes and the proof splits in some steps. In what follows $\sigma$ is just positive, decreasing and in $L^1(0, \infty)$.
If $y$ is a bounded solution of the ODE (forgetting the initial ...
- 4,765
3
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Continuous selectors of a continuous multifunctin on a compact metric space
Let $X = Y = \mathbb{S}^1$.
Take $f(\theta) = \{ \theta / 2, \theta/2 + \pi\}$.
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7
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If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?
Since $f$ is equal a.e. to $g$, $f$ is Lebesgue integrable. Now for $a < x < y < b$,
$$F(y) - F(x) = \int_x^y f(t)\; dt = \int_x^y g(t)\; dt$$
(see e.g. Rudin, Real and Complex Analysis, ...
- 53.2k
8
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Are “most” bounded derivatives not Riemann integrable?
In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...
- 3,097
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Average distance between points of lower dimensional simplices in $\mathbb R^n$
It is highly unlikely that an explicit expression exists.
Even the calculation of the volume of a polytope is a nontrivial problem, solved by Lawrence for simple polytopes. One can possibly use ...
- 110k
5
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Macroscopic sets - a notion of largeness for Lebesgue null sets
By Frostman's lemma, if $E$ is a compact set of positive $\alpha$-Hausdorff content, then there exists a probability Borel measure $\mu$ supported in $E$ such that $\mu(I) \leq c |I|^\alpha $ for ...
4
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Can we approximate a Hölder pdf by higher-order Hölder pdf's?
No, take any $p$ such that $p(x)=|x|^\alpha$ in some neighbourhood of the origin.
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5
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On the Riemannian integrability of the bounded derivative
The answer to the question in the body of your post is no, for the reason that if $f' = g$ almost every where and $g$ is Riemann integrable and such that (*) holds, then $f'$ must be Riemann ...
- 35.4k
11
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How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$
Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...
- 95k
24
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Writing a function on $\mathbb{R}$ as a sum of two injections
The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...
- 219k
1
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On the additive property of the subdifferential of lower semicontinuous functions
In part (P3) of Definition 2.1 in the paper you linked, it is also required that $g$ be $\partial$-differentiable at $x$ (meaning that both $\partial g(x)$ and $\partial(-g)(x)$ are nonempty), which ...
- 110k
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Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Yes, that's a null set.
Pick a sequence of continuous functions $f_n: S^1 \times \mathbb R \to \mathbb{R}$ such that $\|f_n-f\|_1\leq 4^{-n}.$ Let $g_n(t)=\int_{S^1}|f(x,t)-f_n(x,t)|dx.$ Let $Z=\...
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"Simple" integral equation
Alternative simple proof - integration by parts:
$$
\int_0^{1-a}\frac{H(z)}{1-z}dz=\int_0^{1-a}\frac1{(1-z)^2}\int_z^1\frac{2\zeta}{1+\zeta}H(\zeta^2)d\zeta=
$$
$$
\frac1{1-z}\int_z^1\frac{2\zeta}{1+...
- 283
13
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Writing a function on $\mathbb{R}$ as a sum of two injections
It works at least for (locally) absolutely continuous functions.
Such a function is the integral of a locally $L^1$ function.
This weak derivative can be written as a sum of a positive and negative ...
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3
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Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
No. Let $Y$ be uniform on $[0,1]$ and $X_n$ have density $f_n=1+\sin (2\pi nx)$. Then $X_n\to Y$ in distribution. You can represent them on the same probability space $(0,1)$ (with Lebesgue measure) ...
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13
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Accepted
"Simple" integral equation
That is a rather tough puzzle (took me two full days) with a rather short solution.
The first step is the differential equation Fred already mentioned:
$$
(1-z^2)H'(z)-(1+z)H(z)+2zH(z^2)=0\,.
$$
Now ...
- 58.2k
3
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"Simple" integral equation
This is an incomplete answer, the last step is missing (yet).
We can differentiate the OPs equation to get
\begin{align}\tag{1}\label{eq:1}
(1-z^2) H'(z)-(z+1) H(z)+2 z H(z^2)=0.
\end{align}
The ...
- 1,952
2
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Accepted
Is the Boltzmann entropy continuous in the supremum norm?
The answer is no. For instance, let $d=1$, $\rho(x):=e^{-x}\,1(x>0)$, $$\rho_n(x):=c_n\big(e^{-x}\,1(0<x\le n)+p_n\,1(n<x\le2n)\big),$$
where $c_n:=1/(1-e^{-n}+np_n)$, $p_n\in(0,\infty)$ for ...
- 110k
0
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Comparison of solutions of Hamilton-Jacobi equations with different initial conditions
Consider the general case
$u_t+H(x,t,u_x)=0$ in $\mathbb R^n\times [0,T],$
where $H(x,t,p)$ is uniformly continuous for bounded $p$. The desired result holds when we have
$|H(x,t,p)-H(y,t,p)|\leq m(|x-...
4
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Accepted
If all mixed partials of a $C^1$ function exist and are continuous, is the function $C^2$?
No. Let $f(x,y) = x|x|$.
It is easy to check $df = |x| ~dx$ is continuous.
The mixed partial $\partial^2_{xy} f = 0$ exists and is continuous. But the function is not $C^2$.
- 35.4k
3
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Accepted
Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$?
For each $n$ we set $q_n = \frac{\text{smallest number more than $nr$ coprime to $n$}}n$. Note that the next prime following $nr$ or the next prime following it is coprime to $n$ for large enough $n$ (...
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5
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A fractional weighted Poincaré inequality
It is not true. Start with a function $u$ which vanishes for $x<0$ and is equal to $1$ for $0 \leq x \leq \frac 12$ and then smooth from $x \geq \frac 12$. The Fourier coefficients behave like $1/n$...
- 4,765
2
votes
Accepted
Matrices and vectors of intervals
$\newcommand\R{\mathbb R}$Any operation you can define on intervals on the real line, you can define (entry-wise) on any arrays of such intervals.
For any function $f\colon\R^n\to\R$, you can define ...
- 110k
2
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If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point?
The answer is negative. Take a bounded nowhere differentiable continuous map $f : \mathbb{R} \to \mathbb{R}$ and consider the map $g(x) = f(x) \cdot x^2$, which is differentiable at $x = 0$, but not ...
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Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Take $a=1/4$.
Maple says
$$
{}_1F_2\left(1;\frac14,\frac34;-x^2\right) =
1-2\,\sqrt {x}\sqrt {\pi}\sin \left( 2\,x \right)\, {C}\!
\left( 2\,{\frac {\sqrt {x}}{\sqrt {\pi}}} \right) +2\,\sqrt {x}
\...
- 40.1k
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Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Might be useful - an expression in terms of the incomplete Gamma function. The series
$$
{}_1F_2(1;a,a+\frac12;-x^2)=1-\frac{4x^2}{2a(2a+1)}+\frac{16x^4}{2a(2a+1)(2a+2)(2a+3)}-\frac{64x^6}{2a(2a+1)(2a+...
- 17.2k
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$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
This should follow from the ACL (absolute continuity on lines) characterisation of Sobolev spaces, see for instance, Theorem 4.1.10 here.
Indeed, since $f \in W^{1, 2} (B_1 \setminus \{f = 0\})$, it ...
- 3,802
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