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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5
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2answers
77 views

$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $ \lbrace a_{n}\rbrace $ be a sequence of positive terms such that $ \sum \limits_{n=1}^{\infty}a_{n} $ diverges. I am going to show that the series $$ \sum ...
2
votes
0answers
27 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
0
votes
0answers
14 views

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
0
votes
0answers
8 views

An intuitive affirmation about convex sets - normal at the boundary of a convex set

Let $\Omega_1 \subset \Omega_2$ two open bounded sets in $R^n $with $\Omega_i$, $i=1,2$ convex and with $\overline{\Omega_1} \subset \Omega_2 $. Suppose that $\partial \Omega_2$ is $C^1$. Now fix $y ...
2
votes
0answers
9 views

Parametrisation of surface

Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$. I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure ...
1
vote
1answer
8 views

Linear homotopy

Let $\lambda, \mu:[a,b]\longrightarrow X\subset\mathbb{R}^n$ paths such that the straight line $[\lambda(s),\mu(s)]$ lies in X for all $s\in[a,b]$. Set: $$\begin{array}{lccc} ...
-1
votes
0answers
7 views

Martingal Ideas -stochastic [on hold]

. Consider the martingale $Z_{n}$ in which one doubels the bet each time until a win. Thus $Z_{n}=1-2^{n}$ for $n<T_{1}$ and $Z_{n}=1$ for $n\geq T_{1}$. Show that the inequality is an equality in ...
0
votes
2answers
28 views

Finding the value of the supremum of a set

Consider the set $S = \{x \in \mathbb R: x < \frac2x\}$. Determine the value of $sup$ $S$ (if it exists). Here is my attempt: Firstly, $S = \{x \in \mathbb R: 0 < x < \sqrt 2$ $\lor$ $x ...
0
votes
1answer
27 views

Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
0
votes
0answers
13 views

Proof of Gelfand formula for spectral radius

STATEMENT: Let $A$ be a Banach algebra, then for every $x\in A$ we have $$\lim_{n\rightarrow\infty}||x^n||^{1/n}=r(x)$$ Proof: We know that $r(x)\leq \lim \inf_n||x^n||^{1/n}$, so it suffices to ...
3
votes
0answers
21 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
0
votes
0answers
21 views

Intro. analysis - proof that $x \in N_{\epsilon}a$

Define: $$N_{\epsilon}a=\{x: |x-a|< \epsilon\}$$ Show that if $\epsilon$>0 and $|x-a|<\epsilon$ then $x \in N_{\epsilon}a$ I first note that $N_{\epsilon}a =(a-\epsilon,a+\epsilon)$ then I ...
3
votes
0answers
36 views

Bachmann's construction of the real numbers

On page 44 of this book an approach to constructing the real numbers as equivalence classes of nested rational intervals is outlined and attributed to Bachmann. The outline in the book is very ...
0
votes
0answers
25 views

Arzela-Ascoli Theorem in $L^p[0,1]$

I understand the Arzela-Ascoli for $X$, compact metric space. So when $X=L^p[0,1]$, the theorem becomes the following? If $f_n\in C(L^p[0,1],L^p[0,1])$ that is uniformly bounded and equicontinuous, ...
0
votes
1answer
29 views

An ODE inequality

Suppose $Q$ is a positive smooth function of $t$ on time interval $[0,a]$, such that $$\frac{d}{dt}Q\leq 1+Q-Q^{1+b},$$ where $b$ is a positive constant. Is it true that $Q\leq ...
3
votes
0answers
75 views

Question about computing a Complicated integral

where $\beta$ is defined like this: I'm trying to prove (2.18) but i don't know how to do, i calculated the integral but i don't find anything %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT1: ...
1
vote
0answers
39 views

is the sequence $[ne]$ convergence?

Is the sequence $a_n=[ne]$ convergence or partially convergence? ($e$ is the Euler's number and the bracket mean the integer part function.)
2
votes
1answer
44 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
2
votes
2answers
62 views

Approximating solutions for the ODE $y'=\exp(y/x)$

I am currently trying to solve excercise 1-38 from Mathews and Walker. In this excercise I am asked to consider the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\exp(y/x)$$ for two ...
3
votes
1answer
26 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
0
votes
1answer
27 views

Simple inequality proof in analysis

Just need verification on whether my proof is valid. I couldn't find a straightforward way to prove this inequality directly, so I tried a proof by contradiction instead. The question: Let $a, b \in ...
2
votes
2answers
37 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
0
votes
0answers
15 views

About Lusternik-Schnirelmann category

I' studying this paper: http://www.sciencedirect.com/science/article/pii/S0022039608003744 In page 1303-1304 they defined two functions $\phi_{\varepsilon}$ and $\beta$ But i don't understand ...
0
votes
1answer
18 views

Proof of floor function identity.

Let $f(x) = \lfloor x \rfloor$ and let $l$ be the greatest integer $\le x$ How do I prove $l + 1 > x$ I see that: $x \ge \lfloor x \rfloor = l$ No complete answers, just hints
2
votes
1answer
20 views

Covering Lemma (Folland Lemma 3.15)

Lemma 3.15 from Folland's Real Analysis: Let $\mathcal{C}$ be a collection of open balls in $\mathbb{R}^n$, and let $U = \cup_{B \in \mathcal{C}}B$. If $c < m(U)$, there exist disjoint ...
1
vote
1answer
20 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
1
vote
2answers
17 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
1
vote
1answer
35 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
0
votes
0answers
13 views

find angular velocity for so that: $\exp(jt) = \exp( j(3t+\pi/3) )$ [on hold]

I have a fourier series in which there are two different arguments on the exponential function: $jt$ and $j(3t+\pi/3)$ and I have to "choose" a fitting angular velocity. It it probably easy yet it ...
1
vote
0answers
13 views

Cavalieri's principle and integrals

Let $f$ be a continuous function an $[a,b]$. Let $P\subset R^2$ be the figure under the graph of $f$ and $D \subset R^3$ a solid figure obtained by rotating a plane curve around $x$-axis. Using ...
0
votes
0answers
19 views

solve this equation $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
0
votes
1answer
26 views

Modify the Cantor pairing function

I have an infinite set of pairs $I:=\{(k,m) \mid k,m \in \mathbb{N},\quad m\geq 1, \quad 1\leq k\leq m\}$. I want to establish a bijective correspondence $\phi$ between $I$ and $\mathbb{N}$. I've ...
1
vote
0answers
26 views

Fourier transform all steps walkthrough for wave vector $k$ and $x$

Below is my walkthrough of a fourier transform. My problem is that I want to do all the similar steps for a fourier transform between position x and the wave vector k. That is working on a solution of ...
2
votes
1answer
56 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
0
votes
2answers
38 views

is this correct that if $\frac{\partial f}{\partial y}=0$ then $f$ is independent from $y$?

Suppose that $A=\{(x,y) \in \Bbb R^2 : x> 0 $ or $ y=0 \}$ and $f:A\to \Bbb R$ is an arbitary function. Prove that If $\frac{\partial f}{\partial x}=0$ then $f$ is independent from $x$ If ...
2
votes
1answer
39 views

What is the point of big Oh notation when it is used for estimation?

I'm reading a book on number theory at the moment that assumes familiarity with big Oh notation...and while I think I do understand the notation I cannot understand the point of it. For instance let ...
0
votes
1answer
25 views

Refreshing solving second order ODE

I have a boundary value problem for the following differential equation $$\frac{d^2 v}{d \chi^2} = q^2 \left( v - C \right), \; 0<\chi<S \; and \;\; v(0)=v(S)=0 $$ where $q$ and $C$ are certain ...
1
vote
2answers
21 views

Clarification from old post: Union of sigma-algebras is non sigma-algebra

I have been working on slightly different problem from one posted back in 2013 here. I followed closely the hints given by @martini there, but nevertheless I still got stuck. I am retyping the ...
1
vote
1answer
33 views

Conditions for a supremum of a set.

Suppose a function $f(x)$ is continuous on $[a, b]$ and there exists, $x_0 \in (a, b)$ such that $f(x_0) > 0$. And then define a set, $$A = \{ a \le x < x_0 \space | \space f(x) = 0 \}$$ We ...
1
vote
2answers
46 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
0
votes
1answer
12 views

from each one-third part that eliminated in construting the Cantor set pick a point, what apout the resulting set?

During constructing the cantor set, pick up a point from the one-third that eliminated. if we call the set of this points A, then what is the internal of A? is the complement of A countable?
0
votes
1answer
16 views

Solving second order nonlinear ODE given boundary condition at infinity

I am trying to solve the following differential equation $$\frac{d^2 u}{dx^2} = - \frac{d V}{du} \; \; , \;\; where \;\; \; V = \frac{1}{2}u^2 - \frac{1}{4}u^4 $$ And the given boundary conditions are ...
-3
votes
0answers
31 views

analysis/number theory study group (online) [on hold]

I plan on studying analysis from landau, rudin probably others and am looking for people (hopefully more than 1) where we could solve theorems/problems and ask each other questions. Online ...
0
votes
1answer
35 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
1
vote
1answer
33 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
0
votes
0answers
52 views

A lower bound for $\log\left( \frac{a+x^2}{b+x^2}\right)$

I am looking for a tight lower bound for $$f(x)=\log\left( \frac{a+x^2}{b+x^2} \right)$$ $x>0$ and $1<b<<a$. I didn't check for convexity analytically, but I plotted this function ...
0
votes
1answer
56 views

$ \exists c \in( a, b) \text{ such that } f(c)=\max\limits_{x \in [a, b]} f (x) $

I saw in a corrected. if We have $ f $ continuous on $ [a, b] $ with $ f (a) = f (b) $ and $ f $ differentiable left and right at $ (a,b)$ can we say that $$ \exists c \in (a,b) \text{ such that } ...
1
vote
1answer
17 views

Inequality connecting inf and liminf

Suppose $f(x,y)$ is a continuous, nonnegative-valued and bounded function on $\mathbb{R}^2$. Is the following correct? $$ \inf_{x,y} f(x,y)\le \liminf_{y\to\infty}\inf_{x}f(x,y) $$
1
vote
0answers
22 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
0
votes
4answers
61 views

2003 Putnam A-1 Help needed about sequences

Okay so for $n=1$ there is only one way. For $n=2$ you have, $1+1, 2 + 0$ for $n=3$ you have: $1+1+1, 1+ 2, 3 + 0$ three ways. So $P(n): n$ ways, we must prove the $P(n+1): n + 1$ statement is ...