Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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0
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0answers
21 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
0
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0answers
28 views

Completing the Square of Quadratic Forms

I was working through a proof of a lemma that lets us determine whether a Hessian is positive definite for Mardens' Vector Calculus, page 175 Basically the lemma is if $B= \begin{bmatrix} a ...
0
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0answers
16 views

Using Jacobi-Anger Expansion to prove Bessel function property

The Jacobi-Anger expansion is represented as $$ e^{ix \cos\theta} = \sum_{n=-\infty}^{\infty} i^{n}J_{n}e^{in \theta} $$ With this known value, I'm attempting to show that $$ \int_{- \pi}^{\pi} ...
2
votes
3answers
71 views

How Do You Solve a differential equation of the form: $y'=yx+x$

How do you solve a differential equation of the form: $y'=yx+x$ In this case you cannot separate the equation indeed: $y'=yx+x \iff \dfrac{dy}{dx}=yx+x$ And I can't separate it? How does one solve ...
0
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2answers
53 views

solve the Following Differential equation: $y''=2yy'$

Solve the following equation: $y''=2yy'$ My attempt: $y''=2yy'$ integrate on both sides: $y'=\int(2yy')$ Let y'=s We have $\int 2y \dfrac{ds}{dy}dy = 2 \int yds = y^2$ Is that a way to ...
0
votes
1answer
22 views

Applied example of differential equations

In a homework, our professor gave us the following problem: An ant walks along an elastic of length $l$ with a constant speed $v$. At the same time a man pulls on the extremity of the elastic with ...
0
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0answers
32 views

explanation of thm proving an operator is compact [graduate functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand and I was wondering if anyone could help me out. The theorem is describing how to show that ...
0
votes
1answer
23 views

“Clear” reason why open sets in weak topology is unbounded

In Lax's Functional Analysis book: The open sets in the weak topology are unions of finite intersections of sets of the form $\{x:a<l(x)<b\}$. Clearly, in an infinite-dimensional space the ...
1
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1answer
19 views

Prove $\{x \in X : f(x)=g(x)+ 2 \} \in \Sigma $

Let $(X, \Sigma )$ be a measurable space and $f,g: X \rightarrow \mathbb R$ be measurable functions. Prove that $\{x \in X : f(x)=g(x)+ 2 \} \in \Sigma $. You may use the algebra of measurable ...
0
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0answers
24 views

upper bound on derivatives of a function defined on an arc

Given a smooth arc on the complex plane by $z=\cos t + 0.5 i \sin t,\; t\in[\pi/10,\pi/5] $ , and a non-analytic function $f(z) = \text{Re } z $ defined on the arc. Obviously, $f(z) = g(t) ...
2
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4answers
112 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
-1
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1answer
47 views

how can ı solve this problem? [on hold]

Suppose $f$ is in $L_1$ space of $μ$, where $μ$ is the Lebesgue measure. Prove that to each $ϵ>0$, there exists a $δ>0$ so that the Lebesgue integral of the absolute value of $f$ is less than ...
1
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0answers
39 views

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt[n]{c_n}\leq\lim\sup\sqrt[n]{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here's Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} ...
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votes
2answers
37 views

changes of integration limits

In my studies of mathematics I have encountered the following equality: $$\int_{0}^{1} \dfrac{1}{(1+x)^3}dx = \int_{2}^{1} \dfrac{1}{x^3}dx $$ What is a mathematical explanation that would explain ...
-1
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0answers
22 views

Show that step functions are dense in $L^1 (\Bbb R)$ [on hold]

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in $\Bbb R$. Assume $f \in L^1( \Bbb R)$, and prove that there is a sequence $\{g_n\}$ ...
1
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1answer
48 views

Why is The Following equality true? (limit of a sum and integrals)

I saw the following equality: $$\lim_{n\to\infty} \sum_{k=1}^{n}\left[ \frac{1}{(1+\frac{k}{n})^3}\right]\dfrac{1}{n} = \int\limits_{0}^{1} \dfrac{1}{(1+x)^3}dx$$ Why don't we divide the integral by ...
0
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1answer
32 views

Prove that if $\lim_{n \to \infty} p_n = p$ in a given metric space then the set of points $(p, p_1, p_2, …) = S$ is closed.

I'm trying to prove the following: Prove that if $\lim_{n \to \infty} p_n = p$ in a given metric space then the set of points $\{p, p_1, p_2, ...\} = S$ is closed. I tried a proof by contradiction: ...
0
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0answers
15 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
-6
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0answers
45 views

Unsolvable question in analysis? Twice differentiable function [closed]

Let $f(x)$ be twice differentiable up to the second order on $[a,b] \in \mathbb{R}$. Prove that $$ \left| \int_a^b f(x) dx - (b-a)f\left(\frac{a+b}{2}\right) \right| \leq \frac{M(b-a)^3}{24} $$ ...
0
votes
1answer
21 views

Sum of product of convergent series

Suppose $\{a_n\},\ \{c_n\}\subset\mathbb{R}$ satisfies $\lim\limits_{n\to\infty} a_n = a\in \mathbb{R}$, $\lim\limits_{n\to\infty} c_n = 1$. Prove that $$ \lim\limits_{M\to\infty} ...
1
vote
1answer
55 views

Continuous function injective over a compact set, and locally injective on each point of the set

Suppose we have a function $F: \mathbb R^n \rightarrow \mathbb R^k$ continuous over some open set $U \in \mathbb R^n$, and let compact set $K \subset U$. $F$ satisfies the following properties: 1) F ...
0
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0answers
20 views

Dinstances from closed subsets to points in $R$

Let $A$ be a non-empty closed subset of $R$, let $b$ be in $R$ prove that there is an $a$ in $A$ so that $|a-b|=inf|x-a|$. Is $a$ unique? Edited: forgot to add some details
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2answers
81 views
+50

Analytic solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations. Motivation: The source of the question is a very convinient method to create random matrices with special properties. ...
0
votes
1answer
23 views

Sufmanifold with prescribed first and second fundamental form

Is there a $2$ - dimensional submanifold $S$ of $\mathbb{R}^3$ which can be parametrized with $x : U \subset \mathbb{R}^2 \to S$ such that : $E=G=1$, $F=0$, $e=-g=1$ and $f=0$ Where $E,F,G$ and ...
3
votes
0answers
30 views

How to prove $\lim_{a \to + \infty}a^q \int_{a}^{+\infty}\frac{\sin(x)dx}{x^p}=0$ when $p>q>0$

I know a similar problem in demidovich's problem set #2357 about proving $$\lim_{x \to 0^+}x^a\int_{x}^1 \frac{f(t)}{t^{a+1}}dt$$it proves by dividing the integral into two parts and used two ...
-1
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0answers
22 views

Find supremum, infimum for $A=\{n+ \frac{x}{n} : x \in R \setminus Q, n \in N, n \le \sqrt 5, |x| \lt \sqrt5\}$

Given expression: $A=\{n+ \frac{x}{n} : x \in \mathbb{R} \setminus \mathbb{Q}, n \in \mathbb{N}, n \le \sqrt 5, |x| \lt \sqrt5\}, \mathbb{N}\setminus0 $ My goal is to find out $\sup(A)$, $\inf(A)$, ...
3
votes
1answer
47 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then ...
-2
votes
0answers
11 views

If f is differentiable twice what is $f(x + \epsilon)$ according to its Taylor Series?

Let $f$ be differentiable twice. What is $f(x+ \epsilon)$ according to Riemann's sum. Our professor wrote the following: $f(x + \epsilon) = f(x) + \epsilon f'(x) + ...
1
vote
1answer
33 views

Functions of the form $f(x) = k^x - x^k$

Let $f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = k^x - x^k$ where $k \in \mathbb{R}$ is a given constant. Currently I am thinking of positive $k$ and positive $x$ because there would be complex ...
2
votes
1answer
54 views

Sum of a polynomial with all its derivative [duplicate]

Let $$p(x)=x^n+a_1x^{n-1}+...+a_{n-1}x+a_n,$$ with $n$ is even and $p(x)>0$ for all $x\in\mathbb{R}$. Let $$q(x)=p(x)+p'(x)+..+p^{(n-1)}(x)+p^{(n)}(x).$$ Show that $q(x)>0$ for all ...
1
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0answers
16 views

Continuous function rational for every point, Cantor function

For Cantor function (https://en.wikipedia.org/wiki/Cantor_function), in my sense it is rational on every point. But it is continuous on [0,1], then such a function must be constant. What is the ...
1
vote
2answers
48 views

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$ I have expressed in the form: $$lim_{x\to a}\frac1{x^2}=\frac1{a^2}$$ ...
2
votes
1answer
35 views

Should a metric always map into $\mathbf{R}$?

Typically you see the definition of a metric as a function which maps $X\times X\to\mathbf{R},$ but does this always have to be the case? Motivating example: When you complete $\mathbf{Q}$ with the ...
0
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0answers
13 views

Darboux Sums of an n-dimensional cube

The cube $[0, 1]^n$ is admissible, and $v([0, 1]^n ) = 1$. Prove it Using Darboux sums. Hint: $L_N (\mathbb{1}[0,1]^n ) = 1$ and $U_N (\mathbb{1}[0,1]^n ) = 2^{-nN} (2^N + 2)^n$ So I get ...
-1
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1answer
13 views

$\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}=\{\lim _ {n\rightarrow \infty }\sup f_n \leq t \}$ Proof and Intuition

For a start how can I read $\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}$ ? I only seen $\lim _ {n\rightarrow \infty }\inf $ for sets and just don't understand the meaning of $\lim \inf$ for an ...
2
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0answers
15 views

Essentially bounded function which is continuously bounded Riemann integrable?

Lebesgue's criterion for Riemann integrability states: A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere. Can we change ...
1
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2answers
60 views

Using the $\epsilon$-$\delta$ definition, show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$

Using the $\epsilon$-$\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$. To what I understand of this question, is it just asking to me ...
6
votes
1answer
39 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...
1
vote
1answer
25 views

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure.

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure. I looked at the proof of this statement and it says that it follows from the fact that if $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ ...
1
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0answers
26 views

If $f$ is Schwartz, does there exist a positive Schwartz function $g$ such that $|f(x)| \leq g(x)$?

Suppose $f$ is a Schwartz function in $\mathbb{R}^n$ that takes positive and negative values. Does there exist a Schwartz function $g$ such that $|f(x)| \leq g(x)$ for all $x \in \mathbb{R}^n$? Is ...
0
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0answers
13 views

Limit of Fourier-Stieltjes transform of a complex Borel measure

Let $\mu, \nu$ be complex Borel measures on $(\mathbb{T},\mathcal{B}_{\mathbb{T}})$. Suppose $$\lim_{|n| \to \infty} \int e^{-int}d\mu(t) = 0$$ and $|\nu|$ is absolutely continuous with respect to ...
-2
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0answers
40 views

I want to find questions or problems to do math research. [closed]

This is a soft question, but I am not sure of a better forum. I am looking for a professor who can help me find a question or problem that I can independently research and try to solve in order to ...
2
votes
2answers
45 views

Showing that the set of semi-orthogonal matrices is a $C^\infty$ submanifold

For $k, n \in \mathbb{N}$ with $k ≤ n$, we define $$S_{n, k} = \{X \in \mathbb{R}^{n \times k}: X^t X = I_k\}$$ where $I_k$ is the identity matrix of rank $k$. I want to prove that $S_{n, k}$ is a ...
1
vote
1answer
29 views

Under which additional hypothesis are open maps locally injective

Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset ...
0
votes
1answer
28 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
0
votes
2answers
33 views

Solving the improper integral $1/(x^a+y^b)$

I want to discuss the convergence of this improper integral: $$\int_{1}^{\infty }dy\int_{1}^{\infty }dx \frac{1}{x^\alpha +y^\beta} \text{ with } \alpha,\beta>0$$ I know by polar coordinates that ...
0
votes
0answers
19 views

Limit Terminology

From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the ...
2
votes
3answers
47 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
0
votes
1answer
157 views

Solving a Word Problem relating to factorisation [closed]

The $\text{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\text{Ionof}(18) = \frac{18}{6} = 3$, and ...
1
vote
0answers
56 views

How to derive this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = \mathrm e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ [closed]

How to prove this identity: $\lim _{x\to a} (1 + f(x))^{\frac{1}{g(x)}} = e^{\lim_{x \to a} \frac{f(x)}{g(x)}}$ I've found references to this identity but no derivation EDIT:Here the identity only ...