All Questions
5,700 questions
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Extension of super harmonic functions
The following is stated as an exercise in the "Classical Potential Theory" of Armitage and Gardiner (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open set ...
- 411
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58
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The solutions of a system of differential equations
Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y $, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$
Consider this system ...
- 31
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95
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A property of the Hilbert transform involving the cotangent function
A lemma of a paper by T. Elgindi and I.-J. Jeong (Arch. Rational Mech. Anal. 235 (2020) 1763–1817, Lemma 2.2) states the following:
Let $g(z)=\operatorname{sgn}(z)k(|z|^\alpha)$ with $k$ smooth and $k(...
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0
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80
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On the weak convergence of sub-probability measures
Denote by $M(\mathbb R)$ the collection of sub-probability measures. Let $(\mu_n)_{n\ge 1}\subset M(\mathbb R)$ and $\mu\in M(\mathbb R)$. Do we have the equivalence of the following claims :
$\mu_n$ ...
user128095
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58
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How many branches does the reverse Hilbert curve mapping have at most?
Consider the unit square $[0,1]\times[0,1]$, the unit interval $[0,1]$
and the Hilbert curve mapping of this interval onto the unit square.
There also exists a reverse mapping from the unit square to ...
- 1
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106
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Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
- 617
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981
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Weak $H^1$ convergence implies strong $L^p_{\text{loc}}$ convergence
On page 210 of Analysis by by Elliott H. Lieb, Michael Loss (GSM 14), there is a theorem stating that if $f_j \to f$ weakly in $H^1(\mathbb R^n)$ then $\mathbf 1_A f_j \to \mathbf 1_A f$ strongly in $...
- 1,755
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247
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Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
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67
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Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
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239
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Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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53
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Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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0
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81
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What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
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150
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Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
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46
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Superharmonic extension 3
This question is related to the MO post
Superharmonic extension 2. Let $u$ be a superharmonic function on $\mathbb{R}^m$ ($m>2$) such that for some $\alpha\in\mathbb{R}$ and $\beta$, $R>0$,
$$u(...
- 411
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353
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Inverse of the Riesz potential of a measure
Let $0<\alpha<d$ and let $I_{\alpha}(f)$ be the Riesz potential of a function $f$ on $\mathbb{R}^{d}$,
$$
I_{\alpha}(f)(x)=\int_{\mathbb{R}^{d}}\frac{f(y)}{|x-y|^{d-\alpha}}dy.
$$
Assuming $f$ ...
- 4,034
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161
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Superharmonic extension
We know the following classic result. If $K\subset\mathbb{R}^m$ ($m>1$) is a compact set and $u$ is superharmonic on a neighborhood of $K$, then we can extend $u$ to a superharmonic function $\...
- 411
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299
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Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
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84
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Can convex functions on product space be approximated by product of convex functions?
I am working on a problem where I need the following property that I guess should be true but I am not able to prove it.
I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\...
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92
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Can $\ell_1(E)$ be embedd into the dual of continuous function space?
Let $E$ be a compact seperable space, for example $E=[0,1]\subset\mathbb{R}$. Denote by
$$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\...
- 131
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votes
1
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114
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Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
- 161
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168
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Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
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votes
1
answer
141
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Arbitrarily bad rates of convergence in Wasserstein metric
Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...
- 13
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152
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Predual of $BMO(\mathbb{T}^d) $
In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$.
Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
- 11
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1
answer
897
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Error function of multivariate Gaussian
I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$
$$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$
where $r>0$, $k>2$ and
Edit: I have forgot to ...
- 93
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48
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definition of functions that "weakly vanishes as $y\to\infty$" and find a proof of Theorem 9 in the article
I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions:
in Theorem 7 the author use the state "weakly vanishes as $...
- 339
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1
answer
603
views
A concave function as supremum of upper semi continuous is upper semi continuous
We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
- 1,043
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0
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32
views
Minimization of a palindromic-like sequence and asymptotics
Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from
the real interval $[0,1]$.
I am interested in minimizing the infinity norm of the vector
$$ v = \left( \frac{x_{1}}{x_2},...
- 313
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votes
1
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88
views
Recurrence results for an "on average" measure preserving transformation
I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$
$$
\lim_{n\rightarrow \...
- 1,087
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1
answer
84
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One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?
In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...
- 288
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0
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40
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To show a analytic map is zero from a property regarding logarithmic integral
Let $F$ be analytic on $\mathbb{H}=\{z\in\mathbb{C}:Im(z)>0\},$ continuous upto $\overline{\mathbb{H}}$ and bounded on each of the half plane $\{Im(z)\geq h>0\}.$ How to show that if $F$ ...
- 173
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62
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Fractional laplacian on $H^s(\mathbb{R}^n)$ and symmetry
Let $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n),$ i define the fractional laplacian of $u$ in the following way:
$$(-\Delta)^su(x)=C(n,s)P.V.\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad\...
- 339
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0
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207
views
Functions that satisfy a reverse triangle inequality: do they have a name?
Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy
$$f(a) - f(b) \le C f(a - b)$$
$\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...
- 337
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0
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95
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When does a potential function with given partial derivatives exist
I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...
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0
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82
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Integral equality involving fractional laplacian
Let $s\in(0,1)$, let $u\in H^s(\mathbb{R}^n)$. For all $\psi\in\mathcal{S}(\mathbb{R}^n)$, let:
$$ (-\Delta)^s\psi(x)=c(n,s)\lim_{\epsilon\to0^+}
\int_{\mathbb{R}^n\setminus B_\epsilon(0)}\frac{\psi(...
- 339
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0
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82
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A question about Fourier transform of a function defined by an integral
I have the function:
$$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$
for all $x\in\mathbb{R}^n$ and $k>0$....
- 339
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0
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49
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Non-square multiplication operator matrix
Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...
- 617
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1
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152
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A simple clarification on Riesz decomposition theorem
Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions&...
- 411
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0
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53
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Are the densities of a continuous stochastic process locally positive in time?
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...
- 463
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0
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146
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Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
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votes
1
answer
84
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Vanishing sequence and subsequence with particular decay [closed]
Assume I have a sequence $\{a_m\}$ that is vanishing and strictly positive:
$$
0<a_{m+1}\leq a_m\leq\ldots\leq a_1<\infty, \quad \lim_{m\to \infty}a_m = 0
$$
Is it true or false that this has a ...
- 379
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votes
0
answers
76
views
Constructing a small Radon-Nikodym derivative
Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that:
$0<h(x)$.
$\int_{x \in \mathbb{R}^n} |h(x)|<\infty$,
$\sup_{x \in ...
- 5,405
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0
answers
211
views
Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$
Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
- 11
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0
answers
61
views
Weak topology of Gaussian measures
Let us consider a space of Dirac measures $\delta_{x}$ on a Tychonoff space $X$. I know that this space is homeomorphic to $X$. A space of Gaussian measures (weak topology) on some loсally convex ...
- 11
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1
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169
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How to obtain envelope equation for oscillating functon? [closed]
Some oscillating function is given. How can I obtain its envelope? For example, for $ \sin x$ I should get $\pm 1$.
Particularly, I am interested in envelope for $$\begin{equation}\frac{(1-x \cot (2 ...
- 145
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votes
2
answers
156
views
Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$
I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
user157103
0
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0
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143
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A Riemannian manifold with a non-degenerate metric and an inner product $u_{\beta} u^{\beta}=1$
The question is: given a Riemannian manifold with a non-degenerate metric g and an inner product $u_{\beta}u^{\beta}=1$, is $\nabla_{\mu} (u_{\alpha}u_{\beta})=0$ without demanding the trivial ...
- 9
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0
answers
263
views
Existence of the inverse Fourier transform, Carr Madan
I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
- 109
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votes
0
answers
151
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Making area/volume calculations that use SIA rigorous
There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples:
A proof that $\sin'(0) = 1$.
A proof that the surface area of a cone is ...
- 4,943
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0
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73
views
Sufficient and necessary condition for the continuity of an improper integral
Let $f(\cdot) \in \mathscr{C}\left( \mathcal{D}; \mathbb{R} \right)$ where $\mathcal{D} \subseteq \mathbb{R}$ is open with $0 \in \mathcal{D}$ and
$$ f(0) = 0, \quad \forall x \in \mathcal{D}\...
- 81
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votes
0
answers
77
views
Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:
Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is ...
- 375