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Questions tagged [analysis]

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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2answers
21 views

Convergence of series and limit

Let $y_{k\in N}$ $\subset R^2 $. How do I find out whether this complicated series converges or not and find its limit?
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1answer
23 views

Prove if the function is Riemann Integrable based on its upper and lower sums

Let f be defined as: f(x) = \begin{cases} x & \text{ if } x\in\mathbb{Q} \\\\ 0 & \text{ if }x\notin\mathbb{Q} \end{cases} Is f Riemann Integrable on [0,1]? Prove it. We know that:...
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1answer
9 views

Lebesgue-Stieltjes integral with a directed and absolutely continuos cash-flow

Let $Z$ be a directed cash-flow (i.e. right-continuous, monotone increasing function $Z:[0,\infty) \rightarrow [0,\infty)$, furthermore let Z be absolutely continuous (i.e there exists a measurable ...
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0answers
13 views

Lebesgue-Stieltjes integral of a measurable function

Let let $N$ be a natural number, $Z$ a directed cash flow (i.e. a right continuous, monotone increasing function $Z:[0,\infty) \rightarrow [0,\infty)$, furthermore there exists $0 \leq t_{1} < \...
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2answers
11 views

Use Induction to prove the following generalisation of the triangle inequality:

If n ≥ 2 is a natural number and a1, . . . , an are real numbers, then |a1 +···+an|≤|a1|+···+|an|. Im stuck on where to start and what direction to go with this question? any help would be useful ...
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1answer
18 views

prove that an isosceles triangle has the largest area

Inside this angle, alpha is place in a segment of length whose ends are on the sides of the angle so that the area of ABC was maximum
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0answers
28 views
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0answers
15 views

Can the existence of a Schauder basis help in the verification of the convergence of a sequence of operators?

I was wondering if the existence of a Schauder basis helps in the verification of the convergence of a sequence of operators. The following is straightforward: Let $(T_n), T\in \mathcal{L}(X)$, ...
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1answer
17 views

Determining tangent space

Let $g(x,y)=\sqrt{y-x}$ defined on $\{(x,y):y>x\}$ and let manifold $M=\{(x,y,z):z=g(x,y)\}$. I want to determine the tangent space in $(x,y,z)=(2,6,*)$ I usually solved this sort of tasks using ...
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2answers
40 views

Computing a double integral: $\int _0^{\frac{1}{4}}\int _{\sqrt{x}}^{\frac{1}{2}}\:\frac{\cos\left(\pi y\right)}{y^2}~\mathrm dy~\mathrm dx$

I've been trying to solve this integral for the past few hours but to no avail: $$\int _0^{\frac{1}{4}}\int _{\sqrt{x}}^{\frac{1}{2}}\:\frac{\cos\left(\pi y\right)}{y^2}~\mathrm dy~\mathrm dx$$ I ...
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0answers
8 views

taylor expansion of scalar fields

Starting of with electrodynamics I have to compute the taylor expansion around $\vec{r} = 0$ of $\psi (\vec{r}) = |\vec{r} - \vec{r_0}|^{\frac{3}{2}}$ where $\vec{r_0}$ is a constant vector up to ...
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0answers
15 views

Question on integral equality using the average value of a function

Let $\Omega \subset \mathbb R^3$ be an open,bounded and connected set with boundary $\Gamma =\partial \Omega$. Consider also $u \in L^2(0,T;H^1(\Gamma))\cap H^1(0,T;H^1(\Gamma)^{\ast})$. It holds ...
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1answer
48 views

Riemann Sums of Improper Integrals on Unbounded Domain

Assume that $f:[0,\infty)\rightarrow[0,\infty)$ is decreasing (nonincreasing), continuous, and in $L^{1}[0,\infty)$, it is not hard to see that the following holds: \begin{align*} \lim_{n\rightarrow\...
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5answers
90 views

$\sum_{n=1}^{\infty} \left\{ e-(1+\frac{1}{n})^n \right\}$. is this converge or diverge

$$\sum_{n=1}^{\infty} \left\{ e-\left(1+\frac{1}{n}\right)^n \right\}$$ Is this converge or diverge series .It is a series with positive terms ,but none of test of positive term series is seems to ...
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2answers
52 views

Are there any general guidelines for proving limits of multivariable functions?

Today I was trying to prove that $$\lim_{(x, y) \to (0, 0)}\dfrac {x^2y^2}{x^2+y^2} = 0 $$ I got really lucky because the AM-GM inequality directly applies here to give us $$\dfrac {x^2y^2}{x^2+y^2}...
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1answer
40 views

$\sum_{n=1}^\infty{a_n}$ converges $\iff$ $\sum_{n=1}^\infty{b_n}$ converges.

I am trying to prove the following theorem: If $\sum_{n=1}^\infty{a_n}$ and $\sum_{n=1}^\infty{b_n}$ are positive term series and $L\in (0,\infty)$ such that $\lim_{n\to\infty}{\frac{a_n}{b_n}}=L$, ...
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0answers
14 views

How do you solve this balance scale question? [on hold]

Oh nice! Some balance scales! Remember those? [you can also think of them as see-saws] The triangles and boards don't weigh anything. The numbers, like 2 and 5, represent weights. The 2 is 2 ...
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0answers
23 views

understanding an estimation of the perimeter of sets, $|P(\hat{V})-P(V)|\le P(B_r(x))$

Let $V$ be a Borel set in $\mathbb{R}^n$ such that the Lebesgue measure of $V$, $|V|$, satisfies $|V|\approx |B_1(0)|$, but $|V|\neq |B_1(0)|$ (i.e. $|V|$ is slightly greater or less than $|B_1(0)|$). ...
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1answer
33 views

Why is $f(x)=1/\sqrt{|x|} $ local integrable and $f^2$ not?

Why is $f(x)=1/\sqrt{|x|} $ local integrable and $f^2$ not? I think i have to show this $$\int_\mathbb{R}|f(x)| d\lambda<\infty$$ But how i calculate with it?
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1answer
39 views

Lebesgue measure of intersection of sets

Let $A,B\subset\mathbb{R}^n$ two Borel sets with finite and positive Lebesgue measure such that $|A|=|B|$, where $|A|$ denotes the Lebesgue measure of $A$, and such that $A\triangle B \subset\subset C(...
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3answers
43 views

Evaluate lim $a_n$, $a_n=\frac{3^n}{n!}$.

I've come here as I need some help. I have to find $\lim_{x\to \infty}a_{n}$, where $a_{n}=\frac{3^n}{n!}$ I've tried to use Squeeze theorem ,but I couldn't find another two strings that are ...
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1answer
20 views

$f(u)$ has twice continuous Derivative in $[1,+\infty),f(1)=-1,f'(1)=3/2,w=(x^2+y^2+z^2)f(x^2+y^2+z^2)$

$f(u)$ has twice continuous derivative in $[1,+\infty)$, $$f(1)=-1,\ f'(1)=3/2,\ w=(x^2+y^2+z^2)f(x^2+y^2+z^2)$$such that $$\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}+\frac{\...
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0answers
32 views

The set of points reached exactly $n$ times is measurable

Let $p:X \to Y$ be a measurable surjection and assume that for each $y \in Y$ the set $p^{-1}(x)$ is at most countable. Define $Y_n$ (for $n \in \mathbb{N} \cup \{ \infty \}$) to be the set of those $...
2
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0answers
33 views

Generalized Telescoping Series

Goog Morning Everyone, I'd like to ask the following exercise that my professor gave, that i think it has more theory behind it than expected it. The exercise ask to prove what the following ...
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2answers
53 views

Is this an 'acceptable proof'?

Let $f$ be a function such that$f\in C^1$ and $-2<f'(x)<-1 \forall x \in \mathbb{R}$ , prove that $lim_{x\rightarrow +\infty}f(x)=-\infty$ Using MVT on an Interval $I=[0,x]$, $x>0$ $\frac{f(...
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1answer
46 views

Is $f(x) = \frac{1}{\sqrt{1+x^{2}}}$ surjective?

The domain was given to be $\mathbb{R}$. I calculated that the range was $(0,1]$ and that for every $y$ I could find an $x$ in that domain since $x = \sqrt{\frac{1}{y^{2}}-1}$. I therefore thought $f(...
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0answers
48 views

Showing $\sum_{i=1}^nq_i\sqrt{p_i}+q_0$ is irrational for $q_0,…,q_n\in\mathbb{Q}$, $q_1,…,q_n\not=0$, and $p_i$ distinct prime

In my analysis class these questions have kept coming up over and over again about sums of radicals of primes being irrational. I wanted to just prove the general case so I never had to worry again. ...
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0answers
14 views

What is the range of koebe function? [on hold]

I don't know about the behaviour of koebe function k(z)=z/(1-z)^2.
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0answers
22 views

Bounds on a logarithmic function

Let $\alpha_i$, $i=1,\dots,n$, and $k$ be positive real numbers and consider the following function: $$ f(k_1,\dots,k_n) = \sum_{i=1}^n \log(1 + k_i\alpha_i) $$ where $k_i:=\max\{0,\mu-1/\alpha_i\}$ ...
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0answers
22 views

The range of a sequence in a metric space is a closed set?

I'm stuck in a step in the proof of the theorem: In any metric space $(S, d)$ every compact subset $T$ is complete. Let $\{ x_n \}$ be a Cauchy sequence in $T$ and let $A = \{ x_1, x_2, ... \}$ ...
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3answers
72 views

Having trouble showing a function is continuous: $x/(1+|x|)$

I am having trouble showing that a function is continuous using the $\epsilon, \delta$ definition. My function $f:\mathbb{R}\rightarrow(-1,1)$ is defined as $f(x) = \frac{x}{1+|x|}$. Looking at the ...
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0answers
20 views

approximate identities for convolution with measure

Convolution makes $L^1(\mathbb{R}^n, \mathbb{C})$ into an associative algebra that has no identity, but that does have an "approximate identity" in the sense that for any sequence $\varphi_1, \...
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2answers
52 views

Is this function always differentiable?

I need some help with the following exercise: Assume $A \subset \mathbb{R}$ is a compact set. Is the following function always differentiable? $f: \mathbb{R}^{n+1} \rightarrow \mathbb{R}, \quad f(...
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1answer
54 views

Uniform convergence of $\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$ [on hold]

How can i prove uniform convergence on $E=[0, \frac{\pi}{2}]$ ? $$\sum^\infty _{n=1} {x e^{-nx}\cos(nx)}$$
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1answer
57 views

Smooth and not analytic

Can someone show me, without reference to Taylor series, why a complex function can be smooth but not analytic? I do not understand it intuitively or visually either. I would like an explanation ...
3
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1answer
28 views

perimeter of sets. How do $P(E,\Omega)$ and $P(E\cap \Omega,\mathbb{R}^n)$ relate?

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $E$ a Borel set. The relative perimeter of $E$ w.r.t. $\Omega$ is defined as $$P(E,\Omega)=\sup\left\{\int_{\Omega}\chi_E(x) \mathrm{div}\boldsymbol{\...
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0answers
11 views

Compute the Fourier series (example)

Determine the Fourier series of the function $f\colon [-\pi,\pi]\to\mathbb{R}$ given by $f(x):=\lvert x^3\rvert+12$. First of all, this surely means to find the Fourier series of the function which ...
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1answer
30 views

Showing that $3z+3xy^2-x^3=0$ defines a submanifold of $ \mathbb{R}^3$

$M \subseteq \mathbb{R}^3$ $$M=\{(x,y,z) \in \mathbb{R}^3:f(x,y,z)=0\}$$ with $$f(x,y,z):=3z+3xy^2-x^3$$ I want to show that $M$ is a manifold and determine its dimension. $f:\mathbb{R}^3 \to \...
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3answers
60 views

Taylor series for $\sqrt{1+\sqrt{x}}$ around $x=0$

I want a Taylor series expansion of $\sqrt{1+\sqrt{x}}$ around $x=0$. I have two doubts here and these are as follows: I see that the first and higher order derivatives of the above function blows up ...
2
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1answer
58 views

Where is misunderstanding dA = dx * dy?

Using infinitesimals from $ A(x, y) = x * y $ I have $ dA = A_x * dx + A_y * dy $ solving it for $ dA $, I have $ dA = y * dx + x * dy $ which is a mistake. Where in my thinking way is that mistake? I ...
3
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0answers
38 views

Limit distribution equal to Dirac delta

This is the problem 6.19 from the book Distributions and Operators, Gerd Grubb. I already have done parts (a) and (b). The part (a) of this problem is proving that for $r\in(0,1]$, the sequence $$\{\...
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1answer
50 views

$\lim \frac{x}{\sin x} = +\infty$ or $-\infty $ as $ x \rightarrow (n\pi) , n \neq 0$ is a wrong statement.

$$\lim \frac{x}{\sin x} = +\infty \textrm{ or } -\infty $$ as $ x \rightarrow (n\pi) , n \neq 0$ is a wrong statement. My professor told me this and he told me that the correct statement to write is:...
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0answers
32 views

Borsuk–Ulam theorem proof using Brouwer degree

I wonder if Borsuk–Ulam theorem (if $f:\mathbb{S}^n\rightarrow\mathbb{R}^n$ is continuous, then exists $x_0\in\mathbb{S}^n$ such that $f(x_0)=f(-x_0)$) can be sucesfully proved by using the Brouwer ...
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0answers
40 views

Fatou's Lemma proof from Royden 4th

I'm having a hard time understanding the proof presented by Royden, Fitzpatrick in Real Analysis 4th edition. The proof is described as follows... Fatou's Lemma Let $\{f_n\}$ be a sequence of ...
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1answer
17 views

Linear stability analysis on a simple pendulum

So I have a simple pendulum (rod has no weight, point mass, no frictional forces) and I’m measuring the angle theta from the downward vertical, hence I have the governing equation $$\ddot{\theta}+sin(\...
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1answer
30 views

Royden “Real Analysis”, 3rd edition, chapter 4, exercise 9

Let $\langle f_n\rangle$ be a sequence of nonnegative measurable functions on $(-\infty,\infty)$ such thta $f_n\rightarrow f$ a.e., and suppose that $\int f_n\rightarrow\int f<\infty$. Then for ...
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0answers
15 views

Hessian matrix and analysis

I've been trying to solve this problem for a long time, but I don't know exactly how to start: Translation: M is square dxd, we want to show that f belongs to C^2 in R^d. After we want to calculate ...
2
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1answer
32 views

If $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well? [duplicate]

If $\{s_n\}$ is a sequence of positive real numbers such that $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well?
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1answer
20 views

Related to Hausdorff-young inequality: p,q>=1. If exists C, $\|\hat f\|_{q}\le C\| f\|_{p}$ for all $f\in L^p(\mathbb R^n)$, is 1/p+1/q=1?

Hausdorff-young inequality: If $1\le p\le2, f\in\mathbb L^p(\mathbb R^n)$, then $\|\hat f\|_{q}\le \| f\|_{p}$, where $1/p+1/q=1$. Here is a question: If $1\le p,q\le\infty$ (that is, p can be ...
1
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0answers
16 views

Find harmonic function vanishing on the boundary and with a specific bound

Find all harmonic functions $u$ in a half-plane $H$ so that $u=0$ on $\partial H$ and $\vert u(x)\vert \le \vert x\vert$ in $H$. This domain is not bounded. If it's bounded, then after using the ...