Questions tagged [smooth-functions]
For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.
647
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Smooth maximum for functions
Let $f: \mathbb{R}^+_0 \rightarrow \mathbb{R}$ be a scalar function. I am looking for smooth approximations of the function $m: \mathbb{R}^+_0 \rightarrow \mathbb{R}$ where
$$m(t) = \max_{0 \leq t' \...
- 11
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40
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Smooth curve confusion
I am reading a book that somehow gives two different definitions for a smooth curve, so I wonder which one is more precise.
Suppose we're given the curve $L$ defined by the parametric equation
$$(x,y)=...
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2
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1
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Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
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Another definition of a smooth map between manifolds
definition (1) of a smooth map is as follows:
A continuous map $f : X → Y$ is smooth if for any pair of charts $\phi : U →R^m, \psi:V →R^n$ with $U ⊂ X,V ⊂Y$, the map $\phi(U ∩f^{-1}(V)) → R^n$ given ...
- 531
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smoothness of $f(x) = \dfrac{((1-\|x\|_2)_+)^2}{1+\|x\|_2}$ [closed]
Consider $f(x) = \dfrac{((1-\|x\|_2)_+)^2}{1+\|x\|_2}$, where $(x)_+ = \max\{0,x\}$. Is $f$ infinite smooth on $\mathbb{R}^d$?
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Is this double exponential smoothing algorithm valid?
I use this double smoothed prediction algorithm:
https://cs.brown.edu/people/jlaviola/pubs/kfvsexp_final_laviola.pdf
Equations:
$$
First Smoothing Statistic:\;Sy_t' = \alpha y_t + (1-\alpha)...
- 13
2
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0
answers
42
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On the "basis" of the space $C^\infty(a,b)$ of smooth functions
My math methods textbook (Hassani's Mathematical Physics) makes the following (I think dubious) claim:
If we assume that a < 0 < b, then the set of monomials $1,x,x^2,...$ forms a basis for $C^∞...
- 661
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1
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12
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Proof of bound an optimal solution of minimum problem with strong convexity and smoothness
I am studying this set of notes. In 1.2 they proof
the creation of a ball with the optimal solution (x*) inside.
I am having difficulties to undertsand how the betha-strong convexity of f implies the ...
0
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51
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Level sets and continuously differentiable functions
It is not difficult to prove the following theorem.
Theorem 1. Let $f_n\in C^1[a,b]$ such that $f_n\rightrightarrows f$ and $f'_n\rightrightarrows g$. Then $g=f'$ on $[a,b]$. Here by
$\...
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Bump function on the plane
Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $$f(x) =
\begin{cases}
e^{\frac{1}{x^2-1}}, & \text{if }|x|<1 \\
0, & \text{if }|x|\geq 1.
\end{cases}$$ It is not difficult to check that $f\...
- 16k
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0
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42
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Regularity of infimum of translations of a $C_c^{\infty}$ function
I have a question regarding the regularity of a certain infimum. The questions are posed at the end. I first start by providing some context. Thank you for your time spent reading!
Context
I've come ...
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72
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Does $|f(x,y)| \leq g_x(y)$ pointwise imply $\sup_x |f(x,y)| \leq g(y)$ if $f$ is smooth and compactly supported in $x$? [duplicate]
Consider a function $f(x,y)$ smooth in both variables and $f(x,y) = 0$ for $x \in \mathbb R\backslash K$ for some compact set $K$.
I know that for each $x$, there is a constant $M_x$ such that $|f(x,y)...
- 382
1
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1
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Extending an embedding to the boundary
Let $f:int(M^k) \rightarrow N^n$, ($k\leq n$) be an embedding of the interior of a $k$-manifold $M$ into a closed (i.e. compact and without boundary) $n$-manifold $N$.
Can I always extend this map to ...
- 274
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Smoothness of integral of a smooth function
If I have a function $l(w,x)$, defined as $\mathbb R^n \times \mathbb R^m \mapsto \mathbb R$ that is continuously differentiable and $L$-smooth with respect to $w$, $\|\nabla l(w,x) - \nabla l(w',x)\| ...
- 43
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31
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How to check that a typical map is a homeomorphism?
I'm currently doing a geometry course, and I'm struggling a bit on the practical side of how to show a map is a smooth homeomorphism. Can we go through the following example from one of my problem ...
- 387
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20
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Connecting piecewise smooth functions using mollification.
Let's say I am given a function $f: \mathbb{R}\setminus (0,1) \to \mathbb{R}$ defined piecewise as
$$f(x)= \begin{cases}
g(x) \quad x \geq 1 \\
h(x) \quad x \leq 0
\end{cases},$$
where $g$ and $h$ ...
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Boundedness of the norm of gradient by the convex smooth function
$\text { Let }\left\{L_k\right\}_{k=1}^m \text { s be convex and } L \text { smooth } $, where $L$ smooth is defined to be the following:
$$
\left\|\nabla L_k(\boldsymbol{x})-\nabla L_k(\boldsymbol{y})...
- 303
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14
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Identifying K-parameter families with a shift
Working on a lemma related to research - it seems true but I am stuck as to how to prove it in general.
Suppose $\mathcal{F}$ is a family of smooth, real functions that is parametrized uniquely by $\...
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40
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Morse-Bott inequalities in $\mathbb{CP}^n$
I want to prove the following statement (which I heavily believe is true):
Let $g:\mathbb{CP}^n\to \mathbb{R}$ be a non-constant Morse-Bott function and denote by $\text{Icrit}(g)$ the set of ...
- 359
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24
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Mollifier function of $f(x)=|x|$ in Scilab.
Hello. I am trying to graph the different approximations of the function $f(x)=|x|$ by convolutions with the standard Mollifier function. However, my code doesn't output the correct graph. What is ...
- 2,876
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1
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39
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Application of Mollifier function.
According to Wikipedia https://en.wikipedia.org/wiki/Mollifier, one of the uses of Mollifer functions is to smooth a function. How could you smooth with a mollifer function the function $f(x)=|x|$ at ...
- 2,876
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Let $f \in \mathcal C_0$ and $\varepsilon >0$. Is there an explicit construction of $g\in\mathcal C_c^\infty$ such that $\|f-g\|_\infty <\varepsilon$?
Let $\mathcal C_0 (\mathbb R^d)$ be the space of real-valued continuous functions on $\mathbb R^d$ that vanish at infinity. Let $\mathcal C_c^\infty (\mathbb R^d)$ be the space of real-valued smooth ...
- 3,892
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Arbitrary parametrized functions determined by n points
An arbitrary polynomial of degree $n$ is determined by $n+1$ points. In some sense, one can think of the polynomial as having $n+1$ parameters - the $n+1$ coefficients associated to it. Is there a ...
- 21
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1
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129
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Discontinuous function in Smooth infinitesimal analysis
I read that there isn't discontinuous functions in Smooth infinitesimal analysis. But I tried to define discontinuous function ($\varepsilon$ is infinitesimal):
$f(x) =
\begin{cases}
1, & \text{...
- 125
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votes
1
answer
57
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Are Functions Smooth Sections?
In differential geometry we often identify $\Omega^0(M)$ with the smooth functions on
a smooth manifold $M$. But for every $i>0$ we know that:
$$\Omega^i(M)=\Gamma(\Lambda^i(T^*M))$$
I imagine with ...
- 1,413
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0
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38
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On chain rule for composition of local parameterizations and map between open subsets of euclidean spaces
I have been reading Differential Topology by Guillemin and Pollack recently and I came across the definition of derivative for a smooth map between two manifolds.
Let $f:X\rightarrow Y$ is a smooth ...
- 1,756
2
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2
answers
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If two vector fields on a one dimensional manifold commute then they are linearly dependent
Let $X,Y$ be smooth vector fields on the unit circle $M = S^1$ such that $[X,Y] = 0$, i.e. (treating tangent vectors as derivations) $X(x)(Yf) = Y(x)(Xf)$ for all $x\in M$. Assuming that $X(x)\ne 0$ ...
- 309
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2
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51
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Show that $ \| \nabla^2 f(x) \| \le L \implies \| \nabla f(x) - \nabla f(y) \| \le L \| x-y \| $
$f$ is a twice differentiable function from $R^n $ to $R$. I want to show that
$$ \| \nabla^2 f(x) \| \le L \implies \| \nabla f(x) - \nabla f(y) \| \le L \| x-y \| $$ for all $x,y \in R^n $ and $...
- 416
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Show that $F(A)=S$ and that the function is bijective.
Define
$$\tag{1}
\begin{equation}
S:=\{(x, y, z) \in A \times \mathbb{R} \mid z=h(x, y)\},
\end{equation}
$$
where $A = \begin{equation}
\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times\left(-\frac{\...
- 615
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1
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If $\partial_2 f\equiv 0$ on domain, then $f$ is independent of that variable?
Let $z=f(x,y)$ be a function of class $C^1(G;\mathbb{R})$.
a) If $\frac{\partial f}{\partial y}(x,y)\equiv 0$ in $G$, can one
assert that $f$ is independent of $y$ in $G$?
b) Under what condition on ...
- 16k
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0
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38
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If $f \in C_c^\infty (X)$ then $\operatorname{supp} (f^{(n)}) \subset \operatorname{supp} (f)$ for all $n \in \mathbb N$
Let $X:=\mathbb R^n$ and $C_c^\infty (X)$ the space of all smooth functions with compact supports. We adopt the convention that $f^{(0)} := f$. I would like to prove that
Theorem If $f \in C_c^\infty ...
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1
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Proof by density Sobolev Spaces?
When reading about Sobolev Spaces, some results are proven using the density of smooth function.
So they prove the results on smooth function and then conclude by taking limits within the integral.
...
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1
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24
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How the author proves that this function is smooth on the boundary $\partial W$?
I'm reading a proof of Lemma 8.2. from this lecture note.
Lemma 8.3. Let $M$ be an $m$-manifold, $W$ an open set in $M$, and $f: W \to \mathbb{R}$ a smooth function. Suppose that $x \in W$. Then ...
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There is a smooth function $g: M \longrightarrow \mathbb{R}$ which agrees with $f$ on some neighbourhood of $x$ in $W$
I'm trying to prove Lemma 8.2. from this lecture note.
Lemma 8.3. Let $M$ be an $m$-manifold, $W$ an open set in $M$, and $f: W \to \mathbb{R}$ a smooth function. Suppose that $x \in W$. Then there ...
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There is a smooth function $\theta: M \to[0,1]$ such that $\theta =0$ on $M \setminus W$ and that $\theta = 1$ on some neighbourhood of $x$
I'm trying to prove Lemma 8.2. from this lecture note.
Lemma 8.2. Let $M$ be an $m$-manifold and $W$ an open set in $M$. Let $x \in W$. Then there is a smooth function $\theta: M \longrightarrow[0,1]$...
- 3,892
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1
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Is this a typo in the existence proof of a special smooth function?
I'm reading below lemma from this lecture note.
Lemma 8.1. There is a smooth function, $\theta_0: \mathbb{R}^n \longrightarrow[0,1] \subseteq \mathbb{R}$, with $\theta_0(x)=1$ whenever $\|x\| \leq 1$,...
- 3,892
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Let $f(x) := e^{-1/x}$ if $x>0$ and $0$ otherwise. Then $f$ is smooth
I'm trying to prove that below function is smooth.
Theorem Let $f(x) := e^{-1/x}$ if $x>0$ and $0$ otherwise. Then $f$ is smooth.
Could you have a check on my attempt?
Proof Let $x>0$. Then $...
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Issue with proof of theorem 2.1.11 in Audin-Damian
The theorem in question is as follows:
Let $V$ be a closed smooth manifold and $f: V\to \mathbb{R}$ a Morse function. Let $a$ be a critical point of $f$ with index $k$ and $\alpha=f(a)$. Suppose that ...
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votes
1
answer
88
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Is there a simple way to interpolate smoothly between levels of a complex-valued continued fraction?
I have two complex numbers, $a = x_1 + y_1 i$ and $b = x_2 + y_2 i$. These serve as inputs to an infinite continued fraction of the form $f_n = a + \frac{b}{f_{n - 1}}$, with $f_1 = a$. Thus the first ...
- 969
8
votes
2
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372
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Prove / Disprove / Complete the proof That $f$ is Infinitely Differentiable
THE NEW (MUCH SHORTER) POST
My original post perhaps was too long for most people, and understandably not a lot of people tried to go over it. In this new post my question is much shorter please ...
- 267
2
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1
answer
73
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Verify: If every open interval has $x \neq 0, f(x) = 0$, then $f^{(n)} = 0$
Let $f: \mathbb R \to \mathbb R$ such that for any open interval $I$ containing zero, there exists $x \in I, x \neq 0$ with $f(x) = 0$. Show that $f^{(n)}(0)$ (the nth-derivative of $f$ at zero), if ...
- 2,083
3
votes
1
answer
112
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Connection between compactly supported smooth functions on a bounded domain and sobolev spaces
I am interested in finding a direct yes or no answer for the following:
In general, on a bounded domain $\Omega$ in $R^n$ with say $C^1$ boundary, can we that say any function $f$ $\in$ $C_c^{\infty}$(...
- 57
1
vote
1
answer
47
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For which sequence do all associated power series converge? (describe the subset of $\Bbb R^\Bbb N$ "homeomorphic" to the analytic functions)
I am investigating a certain property of real smooth functions (though it is easily extended to complex analytic functions) which requires me to define a topology on $C^\infty(\Bbb R)$ that "...
- 4,440
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votes
1
answer
30
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Smooth functions are uniformly approximated by pieces of circle arcs?
Let $X\subset \mathbb{R}$ be a compact set. Then for a smooth function $f\colon X\to \mathbb{R}$ and a fixed number $\epsilon>0$, can we find numbers $x_1, x_2, \dots, x_n \in X $ and circle arcs $...
- 37
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1
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62
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Are Maps from manifold with global coordinates onto non-empty compact manifold automatically smooth?
Let $M$ be a manifold with a global chart, and let $N$ be a non-empty compact manifold.
Let $f\colon M\to N$ be a function such that $f(M)=N$.
To study the smoothness of $f$, I have to choose a point $...
- 368
1
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0
answers
19
views
Least square optimization problem over the class of smooth functions
I met the following problem when working on non-parametric statistics.
We have $(y_i,t_i)$ a pair of points in $\mathbb{R^{2}}$. We want to find a smooth function that allows the best fit to the $y_i$....
- 888
9
votes
1
answer
369
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Is there a simple way to characterize the smooth functions without using the derivative?
As seen in this question, the derivatives can be easily characterized if we know $C^\infty(\mathbb{R}).$ How can we simply characterize $C^\infty(\mathbb{R})$ if we can't use limits?
- 3,853
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0
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172
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Smooth infinitesimal analysis and law of excluded middle
I read about smooth infinitesimal analysis and I have several questions:
How to prove that in SIA every function on $R$ is continuous? (Every function whose domain is $R$, the real numbers, is ...
- 125
27
votes
1
answer
1k
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Is there any simple set of properties which uniquely characterizes differentiation?
The transformation of differentiation is a linear operator over $C^\infty(\mathbb{R}),$ the vector space of smooth functions over $\mathbb{R}.$ Is there any simple set of properties that uniquely ...
- 3,853
5
votes
1
answer
82
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Does there exist smooth functions $f_i,g_i \in C^{\infty} (\mathbb R)$ such that $\sin (xy) = \sum\limits_{i = 1}^{n}f_i (x) g_i (y)$ for all $x,y\ $?
Does there exist smooth functions $f_i,g_i \in C^{\infty} (\mathbb R)$ such that $\sin (xy) = \sum\limits_{i = 1}^{n} f_i (x) g_i (y)$ for all $x,y \in \mathbb R\ $?
I don't think it's true but ...
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