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Activity for Snoopy‭

Type On... Excerpt Status Date
Answer A: Fourier transform of an $L^1$ function is uniformly continuous
Following the definition of uniform continuity, one basically needs to estimate: $$ \begin{align} |\hat{f}(x)-\hat{f}(y)| &= |\int{\Rbb^n}f(t)(e^{-2\pi i x\cdot t}-e^{-2\pi i y\cdot t})\ dt|\\\\ &\le \intw |f(t)||e^{-2\pi ih\cdot t}-1|\ dt, \end{align} $$ where $h=x-y$. If one can show t...
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2 months ago
Question Fourier transform of an $L^1$ function is uniformly continuous
$\def\Rbb{\mathbf{R}}$$\def\Cbb{\mathbf{C}}$$\def\intw{\int{\Rbb^n}}$If $f\in L^1(\Rbb^n)$, denote the Fourier transform of $f$ as $$ \hat{f}(x) = \int{\Rbb^n}f(t)e^{-2\pi x\cdot t}\ dt $$ > Problem. If $f\in L^1(\Rbb^n)$, show that $\hat{f}:\Rbb^n\to\Cbb$ is uniformly continuous. Note. Thi...
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2 months ago
Answer A: For any real number $m$, $ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $
Without loss of generality, one can assume that $m>0$. For each positive integer $n$, since the function $x\mapsto\frac{m}{x^2+m^2}$ is decreasing, one has $$ \frac{m}{n^2+m^2} = \int{n-1}^{n}\frac{m}{n^2+m^2}\ dx <\int{n-1}^{n}\frac{m}{x^2+m^2}\ dx $$ It follows that $$ \begin{align} \sum{n=...
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2 months ago
Question For any real number $m$, $ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $
> Problem. Prove that for any real number $m$, $$ \left|\sum{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $$ Notes. This is an exercise in calculus. There are different ways to approach this problem. One may find the sum on the left explicitly with hyperbolic functions. One can also s...
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2 months ago
Answer A: The derivatives of a function at a boundary point
Your question is better phrased and answered in Whitney's paper: Analytic extensions of differentiable functions defined in closed sets. Here is an excerpt from the introduction section of the paper, which can be seen from the given AMS link: >Imagealttext
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2 months ago
Answer A: Using convexity in the proof of Hölder’s inequality
The first question can be rephrased in a clear way as follows: Let $A$ and $B$ be two positive real numbers. Show that the function $F(t):=A^{p(1-t)}B^{qt}$ for $t\in[0,1]$ is convex using the convexity of the exponential function. If one rewrites: $$ A^{p(1-t)}B^{qt} = e^{p(1-t)\log(A)+(qt)...
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2 months ago
Question Using convexity in the proof of Hölder’s inequality
A key fact for the algebra properties of $L^p$ spaces is Hölder’s inequality: > Let $f \in L^p$ and $g \in L^q$ for some $0 Our task is now to show that $$\intX |fg|\ d\mu \leq 1. \tag{1}$$ >Here, we use the convexity of the exponential function $t \mapsto e^t$ on ${}[0,+\infty)$, which implies ...
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2 months ago
Answer A: If both $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim_{x\to\infty}f'(x)=0$.
Idea. If $\displaystyle \lim{x\to\infty}f'(x)\ne 0$, then for large enough $x$, $f'(x)$ is bounded away from $0$, which implies by the mean value theorem that $|f(x+1)-f(x)|$ is bounded away from $0$. But $|f(x+1)-f(x)|$ must also be small for large $x$ since $\displaystyle \lim{x\to\infty}f(x)$ ex...
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2 months ago
Question If both $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim_{x\to\infty}f'(x)=0$.
Question. Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function. If both the limits $\displaystyle \lim{x\to\infty}f(x)$ and $\displaystyle \lim{x\to\infty}f'(x)$ exist, how does one show that $\displaystyle \lim{x\to\infty}f'(x)=0$? Note. This statement is intuitively obvious: if $f$ has a...
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2 months ago
Answer A: What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?
We say that a measurable function $f: [0,1] \to {\mathbf C}$ is essentially bounded if there exists an M such that ${|f(x)| \leq M}$ for almost every $x\in[0,1]$, and define $\\\|f\\\|{L^\infty}$ to be the least $M$ that serves as such a bound. Let $\mathscr{L}^\infty([0,1])$ denote the set of me...
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3 months ago
Question What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?
>Question. What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$? Notes. This is an example of questions that are quite "obvious" to experienced readers in analysis but may be very confusing to beginners who take the statement "literally": an element in $C([0,1])$ is a comp...
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3 months ago
Answer A: $\liminf (a_n+b_n) = \liminf(a_n)+\liminf(b_n)$ provided that $\lim a_n$ exists
We will use the following facts in the proof: - (1) If the limit of $(xn)$ exists, so does that of $(-xn)$ - (2) If the limit of $xn$ as $n\to\infty$ exists, then $$\liminfnxn=\limsupnxn=\limnxn$$ - (3) One obvious direction of the inequality is proved by definition: $$ \liminf{n\to\infty}(an...
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3 months ago
Question $\liminf (a_n+b_n) = \liminf(a_n)+\liminf(b_n)$ provided that $\lim a_n$ exists
> Question. Suppose $(an)$ and $(bn)$ are two sequences of real numbers such that $\displaystyle \lim{n\to\infty}an=a.$ Show that $$ \liminf{n\to\infty}(an+bn)=\liminf{n\to\infty}(an)+\liminf{n\to\infty}(bn). $$ > To avoid tedious discussion with infinity, assume in addition that both $an$ an...
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3 months ago
Answer A: $\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbers
Another way to write the proof is to use an alternative equivalent definition of supremum: > For a bounded set of real numbers $A$, $L=\sup A$ if >- $L$ is an upper bound for $A$, i.e., for all $a\in A$: $a\le L$; >- any number smaller than $L$ is not an upper bound of $A$: for every $\epsil...
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3 months ago
Answer A: $\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbers
Let $L:=(\sup A)(\sup B)$. By definition of "supremum", one needs to show the following two things, - $L$ is an upper bound for $A\cdot B$; - $L$ is the smallest upper bound for $A\cdot B$. The first statement is very easy to prove: since for every $a\in A$ and every $b\in B$, one has $a\...
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3 months ago
Question $\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbers
> Problem. Suppose $A$ and $B$ are two subsets of positive real numbers. In addition, assume that $A$ and $B$ are both bounded. Show that $$ (\sup A)(\sup B) = \sup A\cdot B$$ where the set of the right-hand side is defined as $$ A\cdot B = \\{ ab\mid a\in A, b\in B\\} $$ The problem abo...
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3 months ago
Answer A: Why is $ \int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty$?
Since, $\displaystyle \lim{x\to 0}\frac{\sin x}{x}=1$, the singularity of the integral is not at $x=0$. On the other hand, one can rewrite the integral as $ \int{0}^\infty\frac{1}{x}\cdot |\sin(x)|\ dx\. $ It suffices to analyze the sum $$ \int0^\pi\frac{\sin(x)}{x}\ dx+\int{\pi}^{N\pi}\frac{1...
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3 months ago
Question Why is $ \int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty$?
> Question: Why is $$ \int0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty\quad ? $$ There are several other ways to state the fact in the question depending on the contexts. For examples: - The function $\displaystyle f(x)=\frac{\sin(x)}{x}$ is not absolutely integrable on $0,\infty)$. ...
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3 months ago
Answer A: Universal property of quotient spaces
I figured out an answer after posting the question for a while. I would like to record it here. The problem with the mentioned argument is that one attempted to prove the continuity of $g$ at one point $[x]$ using merely the continuity of $f$ at $x$ instead of using the global continuity of $f$ ...
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about 1 year ago
Question Universal property of quotient spaces
A typical textbook theorem about quotient space is as follows: >Theorem (Gamelin-Greene Introduction to Topology p.106): Let $f$ be a continuous function from a topological space $X$ to a topological space $Y$. Let $\sim$ be an equivalence relation on $X$ such that $f$ is constant on each equiva...
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about 1 year ago
Question Criterion in terms of the bases for determining whether one topology is finer than another
When topologies are given by bases, one has a useful criterion in terms of the bases for determining whether one topology is finer than another: > Lemma 13.3. (Munkres's Topology p.81) Let $\mathscr{B}$ and $\mathscr{B}^{\prime}$ be bases for the topologies $\mathcal{T}$ and $\mathcal{T}^{\prime...
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about 1 year ago
Question Proving $|{\bf R}^{\bf R}|=|2^{\bf R}|$ using the Schroeder-Bernstein Theorem
Let $A$ be the set of all functions from ${\bf R}$ to ${\bf R}$ and $B$ the power set of ${\bf R}$. Then $|A|=|B|$. This is a well-known result in set theory. A quick search on Google returns answers in various places explicitly using cardinal arithmetic. Question: Can one prove the above res...
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about 1 year ago
Question Extra "bar" in the post
GitHub issue: https://github.com/codidact/qpixel/issues/802 I have experimented with various things but cannot get rid of an extra "bar" in my recent answer (now edited by putting the expression in another separate line to avoid the bar) on the main site: > snapshot of the mentioned post ...
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about 1 year ago
Answer A: What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ?
> How do you symbolize "the line integral reduces to an ordinary single integral in this case"? $\int^ba f(x {\color{goldenrod}{, 0)}} \, dx = \int^ba f(x) \, dx $? NO. For any function with two variables, $f(x,y)$, if you fix the value of $y$ at $0$, then $$x\mapsto f(x,0)$$ gives you a fun...
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about 1 year ago
Question Example of $f:[0,1]\to\mathbf{R}$ with $\lim_{a\to 0^+}\int_a^1f(x)dx=L $ for some real number $L$ but $\int_0^1|f(x)|dx=\infty $
In the Wikipedia article on improper integrals, the function $f(x)=\frac{\sin x}{x}$ gives an example that is improperly integrable: $$ \lim{N\to\infty}\int0^N f(x)dx=\frac{\pi}{2} $$ but not absolutely integrable: $$ \int0^\infty|f(x)|dx=\infty $$ I am looking for such an example for funct...
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over 1 year ago
Question Finding the limit $ \lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $
> Let $\lfloor x \rfloor$ be the maximum integer $n\le x$. Find the limit $$ \lim{x\to 0^+}e^{1/x}\sum{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $$ I do not have an idea how to approach this problem except for a few observations. I don't have much progress from here: - The limit is of the ...
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over 1 year ago
Answer A: How can 3/1 ≡ 1/(1/3), when left side features merely integers, but right side features a repetend?
> ... it's impossible to measure and cut anything physical at a repetend. This is incorrect. Given a segment with $1$ unit length, one can easily construct, "physically", a segment with a length of $1/3$ units. In fact, given any line segment, there are many ways to trisect it. See for instance t...
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over 1 year ago
Question Why can't we conclude the extrema property of a function from its quadratic approximation when the discriminant is zero?
Suppose $f:\mathbf{R}^2\to\mathbf{R}$ is a smooth function and $P=(0,0)$ is a critical point of $f$. The second-derivative test is inconclusive when the discriminant at $P$ is zero: $$f{xx}(0,0)f{yy}(0,0)-(f{xy}(0,0))^2=0\ .$$ For simplicity, assume further that $f(0,0)=0$ and $f{xx}(0,0)\ne 0$....
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over 1 year ago
Answer A: Proving that $p\mid (p+1776)$ if $p$ is a prime and $p(p+1776)$ is a perfect square
Inspired by Derek Elkins's excellent answer, I have the following proof. By the assumption, we have $p(p+1776)=k^2$ for some integer $k$ and thus $p\mid k^2$. Then, Euclid's lemma implies that $p\mid k$. It follows that $k=mp$ for some integer $m$ and thus $(mp)^2=k^2=p(p+1776)$, which implies by...
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over 1 year ago
Question Proving that $p\mid (p+1776)$ if $p$ is a prime and $p(p+1776)$ is a perfect square
Problem: Suppose $p$ is a prime number and $p(p+1776)$ is a perfect square. Prove that $p\mid (p+1776)$. From the assumption of the problem, $p(p+1776)=k^2$ for some positive integer $k$. This does not help much. Intuitively, one can write $p(p+1776)=p^2m^2$ for some integer $m$ due to the fundam...
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over 1 year ago
Question Given two angles of a triangle, finding an angle formed by a median
> Problem: Suppose in $\triangle ABC$, $\angle BAC = 30^\circ$ and $\angle BCA = 15^\circ$. Suppose $BM$ is a median) of $\triangle ABC$. Show that $\angle MBC=\angle BAC$. > Suppose in $\triangle ABC$, $\angle BAC = 30^\circ$ and $\angle BCA = 15^\circ$. Suppose $BM$ is a median of $\triangle ABC$....
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over 1 year ago
Question If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension?
It is known that $\mathbf{R}$, as a vector space over the field of real numbers, has the dimension $1$. I know that $\mathbf{Q}$ is also a field. Question: If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension?
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over 1 year ago
Question The gcd of powers in a gcd domain
Question: $\def\gcd{\operatorname{gcd}}$Let $R$ be a gcd domain. Does it always hold that $\gcd(x^m,y^m)=\gcd(x,y)^m$? Context. If the ring is a Bezout domain, then we can apply this method. However, a gcd domain may not be a Bezout domain. I don't know how I can go on.
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over 1 year ago