All Questions
5,634 questions
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Optimal regularity of polynomial interpolators
Definitions
We define the "complexity" of any polynomial function $p:\mathbb{R}^n\rightarrow \mathbb{R}^m$ as $m\binom{n+\deg(p)}{n}$ (i.e the dimension of $\oplus_{i=1}^m\,\mathbb{R}[X_1,\...
- 5,405
1
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1
answer
169
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How to prove that is a consistent estimator?
Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to ...
- 288
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1
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376
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Equivalence of real numbers in terms of Dedekind cuts and Cauchy nets of rational numbers
We work in weakly predicatively constructive mathematics, in that we accept function sets but do not accept power sets or excluded middle. More specifically, we shall assume a sequential universe ...
- 2,624
1
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1
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72
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Identification of $\lim_{n\to\infty}f_n^{-1}$ with $f_n:\mathbb R_+\to (0,1]$ strictly decreasing and converging pointwise
Let $f_n: \mathbb R_+\to (0,1]$ be continuous and strictly decreasing for every $n\ge 1$. Assume that the pointwise limit of $(f_n)_{n\ge 1}$ exists, denoted by $f$, and is also strictly decreasing. ...
user478492
3
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315
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When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
2
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216
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Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
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1
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165
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Does that exponent of (absolute value of derivative) is constrained implies Lipschitz continuity?
Given $C^1([a, b])$ functions $f_n$ that converge to a continuous real-valued function $f_n \to f$ on a closed interval $[a, b] \subset \mathbb R$, suppose
$$
\int_a^b |f_n'(x)|^{1 + \epsilon} dx < ...
- 43
1
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1
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141
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How to get the estimator?
They introduce a new correlation. For $\pi\in \Pi(\mu,\nu)$ the set of coupling of two probability measures $\mu$ and $\nu$ on a Polish space $(X,d)$. The author introduces a plugin estimator.
...
- 288
17
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2
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"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
Recently, I encountered this problem:
"Given a sequence of positive number $(x_n)$ such that for all $n$,
$$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$
Find the limit $\lim_{n \rightarrow \infty} \...
- 327
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1
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87
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Measuring the quality of real approximation
Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{\big|r-\...
- 48.9k
4
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1
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291
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A “compactness theorem” for measurable functions
Note: Here we consider the Lebesgue measure on $[0, 1]$.
Let $f_n: [0, 1] \to [0, 1]$ be a sequence of measurable functions.
We say a measurable subset $E$ of $[0, 1]$ is a condensation set of the ...
- 6,233
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75
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finite (non-disjoint) open covering a finite set
A collection of sets $O_{\lambda}$ open covers a set $A$ in $\bf R$ that is bounded and measurable. Assume that $A$ can not be open covered by a finite subcollection of $O_{\lambda}$, let $\epsilon>...
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2
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176
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How to compute the unique disintegration w.r.t. the first coordinate?
Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
- 288
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1
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141
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The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$
Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
- 531
1
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1
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134
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Do these two pairs coincide at small time?
Let $\alpha, \beta:\mathbb R_+\to [0,1]$ be continuous and decreasing functions s.t. $\alpha(0)=1=\beta(0)$ and $\alpha, \beta$ are continuously differentiable on $(0,\infty)$ satisfying for some $c&...
- 1,334
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5
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2k
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Axiomatic construction of trigonometric functions
I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties:
$\sin^2 x + \cos^2 x = 1$,
$\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
- 702
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99
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Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
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2
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1k
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Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field
In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
- 2,624
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377
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Intersecting points of increasing convex functions
Can two increasing and differentiable convex functions agree exactly on a countable set of cardinality greater than two?
- 15
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2
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272
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The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$
How to prove the following inequality $$\forall t>0,\quad\int^\infty_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$
for some constants $A>0,c>0$
- 531
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0
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134
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From convergence pointwise to convergence of the supremum for semicontinuous functions
Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
- 449
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1
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323
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An inequality in four variables
Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that ...
- 367
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48
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Notation for dominating (or uniformly bounded) function
While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function.
A situation like this. For some true function $f:\mathbb{R} \to \...
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135
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Uniqueness of a moment style problem
This is a leftover from this question (and I've modified slightly to make the question more natural in the new setting). It's maybe not a very fascinating question by itself, but it seems this is what ...
- 24.2k
3
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535
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If function and derivative extend continuously to boundary of domain, can one extend as $C^1$ function?
Let $D \subseteq \mathbb{R}^n$ be open, connected, bounded. Let $f : D \to \mathbb{R}$ be $C^1$ and assume that $f$ as well as $\partial_i f$ extend as continuous functions to the closure $\overline{D}...
- 403
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Differentiability of functions given as integral of some singular kernel
Let $A: \mathbb R_+\to [0,1]$ be $1/2$-Holder continuous and $k: \{(s,t): 0\le s\le t\}\to\mathbb R$ be continuous. Define $f:\mathbb R_+\to\mathbb R$ by
$$f(t):=\int_0^t\frac{k(s,t)}{\sqrt{t-s}}\big(...
- 1,334
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100
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Existence of a particular positive definite and radially unbounded function
Let $A \subset \mathbb{R}^{n}$ be a closed set. Does there exist an open set $O$ containing $A$, and a smooth function $f : O \to \mathbb{R} $ such that
$f(x) = 0$ for all $x \in A$,
$f(x) > 0$ and ...
- 351
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2
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336
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On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
- 4,153
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213
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Equivalent forms of Fourier restriction conjecture
this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow.
I'm reading Pertti Maattila's book ...
- 196
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1
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207
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Does pointwise convergence yield the convergence under Skorokhod topology?
Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
user478657
1
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0
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142
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Tiling a rectangle with squares
Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle:
The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...
- 48.9k
2
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0
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An inequality in Huisken paper
I am reading a paper is written by Gerhard Huisken 'Flow by mean curvature of convex surfaces into sphere' and I want to show the following statement
Let $p\ge 2$ then for $\eta>0$ and any $0\le \...
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1
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190
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Writing a class of functions in terms of positive semidefinite matrix-valued functions
Consider a continuously differentiable function $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ such that $f(0) = 0$ and $\langle f(x), x \rangle \ge 0$ for all $x \in \mathbb{R}^{n}$. Does there exist a ...
- 351
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What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$
What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$.
I'm reading a paper
Chern-Yamabe flow
which said ...
- 809
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Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
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1
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89
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Determine $\alpha \in (0,1)$ such that $J_{\alpha}(\phi):=\int \psi/\phi^{\alpha}$ exists?
Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$
If $|\phi^{...
- 852
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Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
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1
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165
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Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicative up to an factor?
Related to this question.
For $x_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x_+)$) function with $f := F'$, $f(x) > 0$, ...
- 15
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Harmonic functions as limits of harmonic functions on graphs?
I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...
- 1,075
3
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2
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210
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Bounding integral expression with total variation of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
user139844
1
vote
1
answer
110
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Positive, monotone decreasing function, with limit in 0 equal to ∞ submultiplicative up to an factor?
For $x_+ \in (0,\infty)$ let $f\colon(0,x_+] \to (0,\infty)$ be a continous differentiable function with $f(x) > 0$ and $f'(x) < 0$ for all $x \in (0,x_+]$.
Moreover, we assume that
$$\lim_{x \...
- 15
4
votes
2
answers
348
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Chain rule for $e^f$, where $f$ has bounded variation
Let $f : \mathbb{R} \to \mathbb{R}$ be a function of normalized bounded variation (NBV), meaning that $f$ is of bounded variation, $f$ is right continuous, and $f(x) \to 0$ and $x \to -\infty$. As ...
- 481
16
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1
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888
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Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k
5
votes
1
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415
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Why is it valid to take uncountable infimum of one dimension of a multivariate function of random variables?
let $\xi,\eta: \Omega \to \mathbb R$ be i.i.d. random variables on a measurable space $(\Omega , \mathcal F,\mathbb P)$, and let $f: \mathbb R^2 \to \mathbb R$ be a bivariate measurable function (say ...
- 53
3
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1
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217
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The energy of a semilinear ODE
I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
- 457
3
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1
answer
577
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A constant ratio of integrals? Part II
This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
- 43.2k
4
votes
1
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379
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A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
4
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1
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245
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How to unperiodise a function
We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by
$$
\begin{align}
F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
- 595
7
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2
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786
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Mapping exponentiation onto addition
I was inspired by Does there exist a function which converts exponentiation into addition? to think about mapping exponentiation onto addition.
The question asks whether there exists $f:\mathbb{R}\...
- 333
8
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2
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432
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Can an Osgood curve be almost everywhere differentiable?
It is known that you can “reparametrize” Osgood curves to make them almost-everywhere smooth curves (simply compose one after the Cantor function). However doing this breaks injectivity, stopping them ...
- 700