New answers tagged analysis
1
vote
Ahlfors' complex analysis proof doubt - Heine-Borel theorem
if each [$B(x,\varepsilon_1)$] satisfies Heine-Borel property, the same would be true of $S$
Assume for a later contradiction that $\mathcal C$ is an open covering of $S$ that has no finite subcover. ...
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Ahlfors' complex analysis proof doubt - Heine-Borel theorem
In short, your most recent edit seems to mix up covers.
We start the proof by assuming there's an open cover $\{U_\lambda\}$ of $S$ with no finite subcover.
Then we find $S$ is covered by finitely ...
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votes
Calculate an integration on a sphere
Here is an elementary proof. We have
$$ \text{div}(Bx) = Tr(B)$$
Integrating this over $B_1$ (ball of radius $1$), we have
$$ \int_{S_1} \langle Bx,x\rangle d\sigma = Tr(B) \int_{B_1} dx = Tr(B) |B_1|...
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Ahlfors' complex analysis proof doubt - Heine-Borel theorem
For question 1), it seems to me you do not grasp what a proof by contradiction is. It is simply assumed such a cover exists, and then one seeks a contradiction.
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If $f \in H^{1}$ is it true that $f^{-1} \in H^{1}$ and $\nabla f^{-1} = -\nabla f/f^{2}$?
It is simply not true: We surely have
$$
\exp(-|x|^2) \in H^1(\mathbb{R}^n),
$$
but
$$
\exp(|x|^2) \notin H^1(\mathbb{R}^n).
$$
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Prob. 26, Chap. 5, in Baby Rudin: If $\left| f^\prime(x) \right| \leq A \left| f(x) \right|$ on $[a, b]$, then $f = 0$
Let $g(x)=e^{-Ax}f(x),x\in[a,b]$,
then $g'(x)=e^{-Ax}(f'(x)-Af(x))$.
So when $x\in[a,b]$, we have
\begin{align*}
g(x)g'(x)
&=e^{-2Ax}\big[f(x)f'(x)-Af^2(x)\big]\\
&\leq e^{-2Ax}\big[|f(x)f'(x)|...
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Accepted
Proof, if the left-sided and right-sided limit values are equal, the limit value exists (e-d Definition)
Let $\epsilon >0$. Then there exist $\delta_1 > 0, \delta_2 >0$ such that
$| f(x) - L | < \epsilon$ if $0 < a - x < \delta_1$
$| f(x) - L | < \epsilon$ if $0 < x - a < \...
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8
votes
Accepted
Nonstandard Analysis in ZFC?
Examples of good books that develop nonstandard analysis in ZFC are
Davis, Martin. Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-...
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3
votes
Accepted
Show that $\int_0^{\infty}e^{-yx}\sin(x)dx=\frac{1}{1+y^2}$ for $y>0$ using Feynman's trick
Note that " Feynman's trick " is nearly as old as calculus itself. Now, for this proposed integral to consider there are times to use tricks and there are times to just do the work in a ...
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Is the subspace $\text{End}(\mathbb{R}^n)$ inheriting the product topology normable?
If we let $\mathcal B = \{e_1,\ldots,e_n\} \subset \mathbb R^n$ be the standard basis for $\mathbb R^n$, there is a natural continuous projection map
$$\prod_{v \in \mathbb{R}^n} \mathbb{R}^n \to \...
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2
votes
Accepted
Is the subspace $\text{End}(\mathbb{R}^n)$ inheriting the product topology normable?
Every finite-dimensional Hausdorff TVS is normable. Indeed, it is a standard fact that an $n$-dimensional Hausdorff TVS is linearly homeomorphic to $\mathbb{K}^n$, where the latter is equipped with ...
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vote
Accepted
Confusion regarding expressing an $L^{2}$ function with respect to a basis
The answer is no : here is an example. Let $\Omega = [0,\pi]$ and let
$$f_n(x) = \sin(nx)$$
for $n = 1,2,3,...$.
One can show that $f_n$ is a Hilbert basis of $L^2([0,\pi])$ (this can be shown for ...
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0
votes
A formula for higher order derivatives of inverse function
The Wikipedia article
inverse function rule gives some examples and is easy to digest.
Another article can be found on vixra: Higher order derivatives of the inverse function
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Approximating Euler's number to a certain decimal point
$$S_n=\sum\limits_{k=0}^{n}\frac{1}{k!}=e\,\frac{ \Gamma (n+1,1)}{\Gamma (n+1)}$$ If $n$ is large
$$\Delta_n=e-S_n \sim \frac{e^n\, n^{-n-\frac{3}{2}}}{\sqrt{2
\pi }}$$ and we want that $\Delta_n\...
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4
votes
Accepted
How is $x_n$ bounded above by 3?
If $x_n\leq2$, then
$$
x_{n+1}=\sqrt{x_n+2}\leq\sqrt{2+2}=2.
$$
This allows you to show by induction that $x_n\leq2$ for all $n$.
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4
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Evaluate $\int_{0}^{\infty} e^{-ax} \frac{\sin x}{x} \,dx$.
The function $(\sin x)/x$ is bounded (look at a graph!), so the first part when $\varepsilon > 0$ is a straightforward estimate. The finiteness at $\varepsilon = 0$ is more subtle, since in that ...
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0
votes
Does the series $\sum\limits_{n=1}^\infty \frac{e^{inz}}{\bar zn^3+|z|^3}$ converge uniformly on $[-2,0)\cup(0,2]$ or $[-2)\cup(-2,\infty)$?
For all $n$ where $n^3>|z|^2$, we have:
$$
|a_n|:=\left|\frac{e^{inz}}{\overline zn^3+|z|^3}\right|=\frac{1}{|\,\overline zn^3+|z|^3|}\leq \frac{1}{|\overline z|n^3-|z|^3}=\frac{1}{|z|(n^3-|z|^2)}
$...
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1
vote
Accepted
Bounding a quadratic form using projections
For $u,v\in \mathcal{H}$ we have
$$|\langle u,Tv\rangle|=|\langle v,Tu\rangle|=|\langle T^{1/2}u,T^{1/2}v\rangle|\\ \le \|T^{1/2}u\|\, \|T^{1/2}v\| =\langle u,Tu\rangle^{1/2} \langle v,Tv\rangle^{1/2}...
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1
vote
Accepted
calculating gradient of a function
$
\def\R#1{{\mathbb R}^{#1}}
\def\l{\lambda}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\frob#1{\left\| #1 \right\|_F}
\def\q{\quad} \def\qq{\qquad}
\...
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1
vote
Accepted
1-norm-ball is equal to convex hull of vertices
Assume wlog $x_0=0$, just to simplify the notations. Given $x\in B^{\|\cdot\|_1}_\epsilon(0)$, let $y$ be a boundary point such that $x\in[0,y]$. You already know that $y$ is a convex combination of ...
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3
votes
The continuity of $L^p$ norm of continuous bounded functions
Theorem: For any $p>1$ and $a\in (0,1)$ satisfying $ap>1$, there exists a continuous real-valued function on $[0,1]$ such that
$$\sup_{t>0} t^{-a} ||f(\cdot +t) - f||_p = \infty . $$
Proof: ...
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2
votes
Accepted
The continuity of $L^p$ norm of continuous bounded functions
This is not true in general for any $a>0$ and $1\leq p<\infty$. Consider $L^p[0,1]$, and define a sequence of "continuous spike" functions $(f_{n})\in L^p[0,1]$ defined by, $f_n=0$ if $...
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votes
Least upper bound for $(0,2)\cap \mathbb{Q}$
You can determine this formally by using the following argument.
Suppose $x \in \mathbb Q$ is the greatest upper bound of $(0, 2) \cap \mathbb Q$.
We know that $2$ is an upper bound of $(0, 2)$.
Hence ...
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votes
Let $f_n:[0,1]\rightarrow[-1,1]$ and let $g_n(x)=\int_0^xf_n(t)dt$. There is $g_{n_k}\rightarrow g$ in $C([0,1])$ where $g$ is absolutely continuous
For ease of notation we redenote the subsequence $(g_{n_k})$ of $g_n$ by $\varphi_n(x)=\int_0^x f_n$.
The key is to note that $\varphi_n$ is "uniformly" absolutely continuous:
Given $\...
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3
votes
Accepted
Fubini's Theorem: $\int_{2}^x (\int_2^t \frac{du}{log(u)})\frac{1}{\sqrt{t}}dt = \int_{2}^x \int_u^x \frac{1}{\sqrt{t}\cdot log(u)}du dt=...$
Fubini's Theorem tells you, under proper condition, your integration in area of $A\subset T\times U$ can be done by iterated integral, that is
$$
\iint_{T\times U} f(t)g(u)\,\mathrm{d}(t,u) = \int_{T} ...
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3
votes
Accepted
Show that $ 2^{1-t} \leq \frac{\exp(-t/2)}{1 - \exp(-t/2)}$ for $t > 0$
The desired inequality is written as
$$2\mathrm{e}^{- (t/6) \times 6\ln 2} \le \frac{\mathrm{e}^{-t/2}}{1 - \mathrm{e}^{-t/2}}.$$
Since $6\ln 2 > 4$, it suffices to prove that
$$2\mathrm{e}^{- (t/6)...
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2
votes
Show that $ 2^{1-t} \leq \frac{\exp(-t/2)}{1 - \exp(-t/2)}$ for $t > 0$
The given inequality is equivalent to
$$
2^{t-1} \ge e^{t/2}-1.
$$
Consider the difference
$$f(t) = 2^{t-1} -e^{t/2}+1,$$
for $t\ge 0$. Its derivative
$$f’(t) = (\ln 2) 2^{t-1} -\frac12 e^{t/2},$$
...
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Baby Rudin's proof that non-empty Perfect subsets of $\mathbb{R}^k$ are uncountable
I believe this answers my question:
Need help in Understanding the Proof of Theorem 2.43 on Perfect Sets in Baby Rudin.
But don't know how to resolve my question. In any event, this seems to show the ...
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vote
Accepted
How do I prove the closure of the image of T is uniformly bounded?
Let $y \in X$ and $y_0$ be the constant function $x \mapsto y_0$. For $x \geq x_0$:
$$
\left|y(x) - y_0\right| = \left| \int_{x_0}^x f(t, y(t)) dt \right| \leq
\int_{x_0}^x \left|f(t, y(t))\right| dt \...
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2
votes
Accepted
Why is the inverse of a differentiable function $f: \mathbb{R}^n \to \mathbb{R}^n$ is also differentiable?
The essential point is the interpretation of "invertible". The minimal requirement is that $f$ is bijective so that the inverse function $f^{-1} : \mathbb R^m \to \mathbb R^n$ exists.
...
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Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$
This partial answer piggybacks on Sangchul Lee's earlier suggested transformations to show that, with a few more steps, we can indeed reveal some elliptic integrals. (Applying the fundamental theorem ...
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4
votes
Accepted
On Theorem 2.7 of Montgomery and Vaughan's MNT
Without loss of generality, we can assume that $\delta\in\left(0,\log\left(2\right)\right)$ since, if I remember correctly, in that theorem we want to take $\delta\rightarrow0^{+}$. Then $$\int_{1}^{2}...
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votes
Polynomials with only even terms are dense in the set of polynomials in $[0,1]$ with sup norm.
For $p(x)\in\mathcal{P}$ we have $$p(x)=p_1(x^2)+xp_2(x^2)$$ for some polynomials $p_1$ and $p_2.$ Since $p_1(x^2),p_2(x^2)\in \mathcal{Q},$ it suffices to focus on $x.$ The function $\sqrt{t}$ is ...
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1
vote
Accepted
Linear functionals on Von Neumann algebra tensor products
Yes.
First of all, your extension already applies to all operators on $B(H \otimes K)$, so it’s not just extended to the $\sigma$-WOT closure. Regardless, the $\sigma$-WOT closure and the WOT closure ...
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2
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Geometric mean of the nonzero numbers in the Cantor set $ C $
This may seem odd, but even with a zero entry the limiting geometric mean of the Cantor set is greater than zero. In fact it's greater than or equal to $2/9$.
Define $x_{1,1}=2/3$ as the only nonzero &...
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4
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Polynomials with only even terms are dense in the set of polynomials in $[0,1]$ with sup norm.
Another hack:
Extend $p$ to $[-1,0)$ by defining $p(x) = p(-x)$, then $p$ is even on $[-1,1]$. Let $q_n$ be a sequence of polynomials that converges uniformly on [$-1,1]$ to the extended $p$. Note ...
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Polynomials with only even terms are dense in the set of polynomials in $[0,1]$ with sup norm.
Hope, the following construction is explicit enough.
At first pick any method that suites you to approximate the function $f(x) = \sqrt{x}$ on $[0,1]$ uniformly by polynomials (e.g., partial sums of ...
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votes
Accepted
Is $a_n < \epsilon s_n$ for a strictly increasing sequence $(a_n)$, its partial sums $(s_n)$, and all sufficiently large $n$?
You suspect that if $(a_n)$ is a strictly increasing sequence of real numbers then
$$
\lim_{n \to \infty} \frac{a_n}{a_1 + a_2 + \cdots + a_n} = 0 \, ,
$$
but that is not generally true: A ...
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Solve or find a sharper lower bound for $\int_{1}^t \frac{x^v}{\beta^x} \ \mathrm{d}x$
Using the incomplete gamma function
$$I=\int \frac{x^v}{\beta^x} \,dx=-\log ^{-(v+1)}(\beta )\,\, \Gamma (v+1,x \log (\beta )))$$For the definite integral
$$J=\int_1^t \frac{x^v}{\beta^x} \,dx$$ using ...
- 274k
2
votes
Accepted
Are probability distributions the same as these distributions in mathematical analysis? (simple question)
No, they are not the same. A probability distribution is a measure which assigns $1$ to the full space. A distribution (in the analytic sense) is an element of the dual space of some set of test ...
3
votes
Asymptotic Stability of Origin
A useful observation is that the substitution $v = x-y$ reduces the system to a much simpler one:
$$
v' = y-v^3, \qquad y' = -v-y^3.
$$
From here it is more obvious that $(v,y) = (0,0)$ is the only ...
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Prove that you can rearrange absolutely convergent series in a fully general way?
Every rearrangement in the question (and seemingly every conceivable rearrangement) can be expressed as a sum indexed by an ordinal. A recursive definition of such sums can be found on the Wikipedia ...
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2
votes
Accepted
Intersection of a $C^\infty$ manifold and a sufficiently small ball
This was proved in
Katz, Mikhail G.
Convexity, critical points, and connectivity radius.
Proc. Amer. Math. Soc. 148 (2020), no. 3, 1279–1281
using the ideas around critical points following Grove, ...
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Question about weighted means (V.Zorich Mathematical Analysis 1 Chapter 5.4.6 Ex. 1)
This is clear when each $x_i = 0$, so we may assume some $x_i > 0$.
We are going to apply a useful idea: take advantage of homogeneity in the $x_i$'s to reduce to the special case where the maximum ...
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Question about weighted means (V.Zorich Mathematical Analysis 1 Chapter 5.4.6 Ex. 1)
It suffices that $\alpha_i>0.$ Other assumptions are not essential. Let $x_j=\max x_i$ and $\alpha =\max\alpha_i.$ Then $$\alpha_j^{1/t}x_j\le M_t(x,\alpha)\le (n\alpha)^{1/t}x_j$$ By the squeeze ...
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How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
The accepted answer is a proof by contradiction. Here is a direct proof, inspired by a recent duplicate, Can you find a square rational between two rational numbers? :
Let $x,y,z\in\Bbb N$ be such ...
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0
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Accepted
How to Evaluate $\lim_{t \to 0} \frac{\sin(t) - \cos(t) + e^t}{t^3}$ Without L'Hôpital or Taylor Expansion?
Observe that $\frac{\sin(t) - \cos(t) + e^t}{t^3}$ = $\frac{1}{t^2}\frac{\sin(t) - \cos(t) + e^t}{t}$ = $\frac{1}{t^2}\frac{\sin(t) + [1- \cos(t)] + [e^t - 1]}{t}$ = $\frac{1}{t^2}\left[\frac{\sin(t)}{...
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2
votes
Accepted
$\frac{\Gamma'(\frac{s}{2}+1)}{\Gamma(\frac{s}{2}+1)}$ or $\frac{1}{2}\frac{\Gamma'(\frac{s}{2}+1)}{\Gamma(\frac{s}{2}+1)}$?
First of all, while writing something like "$\log(\xi(s))$" is okay for intuition, I think it is a terrible idea to work with such things in a loose way because there is no such thing as a ...
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How to Evaluate $\lim_{x \to 0} \frac{\cos(x) - \ln(1+x) - \sqrt{1+2x}}{x^2}$ Without L'Hôpital or Taylor Expansion?
You don’t need to think to far, look at the questions more carefully and you’ll see that you don’t need L'Hôpital's Rule or Taylor series expansion to solve this problem
$$\lim_{x \to 0} \frac{\cos(x) ...
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