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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1answer
41 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
27 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: ...
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0answers
10 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
36 views

Unsolvable question in analysis? Twice differentiable function [on hold]

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
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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
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1answer
31 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
18 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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0answers
30 views

Analytic solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations. Mathematica can give me solutions up to certain sizes of the linear system, but I would like to have it for arbitrary size ...
0
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1answer
18 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
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0answers
19 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 ...
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0answers
20 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 $\sup(A)$, $\inf(A)$, ...
2
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0answers
25 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 ...
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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
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1answer
32 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 ...
1
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0answers
37 views

Sum of a polynomial with all its derivative

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
15 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
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2answers
43 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
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1answer
34 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 ...
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0answers
9 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 ...
0
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1answer
12 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 ...
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0answers
11 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 ...
2
votes
2answers
54 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 ...
5
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1answer
35 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
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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
22 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 ...
-3
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0answers
38 views

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

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
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2answers
39 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
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1answer
27 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 ...
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1answer
23 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
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2answers
32 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
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0answers
18 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
45 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
79 views

Solving a Word Problem relating to factorisation [on hold]

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 ...
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0answers
47 views

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

How to prove this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ I've found references to this identity but no derivation This identity is ...
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0answers
25 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...
2
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1answer
29 views

Order of remainder term in Taylor series approximation

I'm having trouble verifying a bound on the remainder term of a Taylor series approximation. I have a $C^\infty$ function $f$ of compact support. Using the two-term Taylor series for $f$ centered at ...
0
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1answer
12 views

If D_m is the mth Dirichlet kernel, $||D_m||_1\to\infty$ as $m\to\infty$

I was working on this problem for my own studying but am stuck on how to solve it. Let $D_m$ be the $m$th Dirichlet kernel. Show that $||D_m||_1\to\infty$ as $m\to\infty$. Anything would help. Thanks
0
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0answers
22 views

Proving a Weierstrass function is not differentiable anywhere

I'm working with the function $$f(x) = \sum_{n=0}^{\infty}\frac{\cos(3^nx)}{2^n}, \quad x \in \mathbb{R}.$$ I'm attempting to prove that it's not differentiable anywhere. I have no clue where to start ...
-1
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2answers
46 views

Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
0
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2answers
21 views

$f:(X,d_X)\rightarrow (Y,d_Y)$ isometry then $f(x)=g(x)$ for $x\in\Omega\subset X$ dense, $g: X\rightarrow Y$?

Let $(X,d_X)$ be a metric space and $\Omega\subset X$ be dense. Let $(Y,d_Y)$ be a complete metric space and $f:\Omega\rightarrow X$ such that $d_Y(f(x_1),f(x_2))=d_X(x_1,x_2)$. Why do we get a map ...
0
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0answers
18 views

Inverse function theorem as consequence of Implicit function theorem

I'm using Rudin, and in its proof of the implicit function theorem, it uses the inverse function theorem. I've heard that you can prove the inverse function theorem as a consequence of the implicit ...
1
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1answer
31 views

If X,Y are both bounded and closed, does this imply that X+Y is bounded and closed?

Define $X+Y$ to be the set of elements $x+y$, with $x$ in $X$ and $y$ in $Y$. If $X$ and $Y$ are bounded and closed is $X+Y$ bounded and closed? I think that showing X+Y is bounded is fairly ...
1
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0answers
13 views

unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...
1
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2answers
41 views

How to prove if $\int^{\infty}_{0}f(x)dx$ a converges, then there is increasing sequence $x_n$, $\lim_{n \to \infty}f(x_n)=0$

I tried to prove it directly, but examples like $\sin(x^{2})$ makes it impossible to find the proper subsequence $x_{n}$; I also tried proving by contraposition, but the converse negative statement ...
-1
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0answers
17 views

ill-posed problem [on hold]

What are the reference to read nonlinear ill-posed problem? what is necessary and sufficient condition to the nonlinear ill-posed problem. Linear case it is available I want for nonlinear case.
0
votes
0answers
4 views

Metric space of non empty closed bounded parts of $R$ with the Hausdorff metric

Consider the metric space of non empty closed bounded parts of $R$ with the Hausdorff metric. For n $\in N_{0}$ and $F_{n} = \{0,1/n,2/n,3/n, ..., 1\}$ i am wondering if $(F_{n})_{n}$ is convergent? ...
-1
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0answers
35 views

Analysis and algebra [on hold]

I'd like to know if there exist a field of the theoretical math that really combines analysis and algebra. Some people say that Model theory combines those two subjects but I personally want ...
0
votes
1answer
17 views

Controlling the size of a function

Consider the function $$f(\delta,r)=\frac{2e^{-\delta r} }{r}\sinh\left(r/2\right)$$ with $\delta >0$ Show that $\exists r >0 \text{ such that }f(\delta,r)<1$ My attempt: \begin{align*} ...
1
vote
1answer
47 views

$f(x)=\frac{x}{x}$ is continuous at $0$?

$x$ is divided by $x$. Thus, $f(x)=1$ when $x \neq 0$. However, at $0$ can we consider $f(x)$ as $1$? More specifically, do we have to define a rational function as a reduced form?