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 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
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1answer
17 views

How to find the limit points of the set $\{\ a+\alpha\ b \ \ | a,b \in \mathbb Z, \ \alpha \in \mathbb R -\mathbb Q\}$

How to find the limit points of the set $\{\ a+\alpha\ b \ \ |\ a,b \in \mathbb Z \ \text{and}\ \alpha \in \mathbb R -\mathbb Q\}.$ limit point: A point $x$ is said to be a limit point of a non ...
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0answers
14 views

Equality and infimum

I have this: Can someone explain me how we can do this $$\inf_{u\Sigma_D}\frac{||u||^p_{D}}{|u|^N_{p^*,D}}=\inf_{u\in ...
2
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2answers
40 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
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0answers
40 views

Proving the existence of a sequence such that

I am trying to prove that there exists a sequence, for example: $$ f(n) = n! $$ (or we can select any sequence we need to prove the existence of just one), with the following property: edit: for ...
2
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1answer
26 views

Radon-Nikodem Derivative of a purely nonatomic Borel Measure

If $\mu$ is a purely non-atomic Borel measure on a topological space $X$ then must its density be a continous function to $\mathbb{R}$? My intuition says yes because all my counterexamples are not ...
2
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0answers
48 views

Finding the maximum of two functions with complicated formulas

Let $$ f(\omega)=1+\frac{m(a+\omega^2)}{a^2+\omega^2}+\alpha\left(\frac{a^2+\omega^2-ma}{a^2+\omega^2}\right)\cos(\omega\tau)+\frac{\alpha m\omega}{a^2+\omega^2}\sin(\omega\tau)\;, $$ and $$ ...
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0answers
29 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
1
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1answer
27 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
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2answers
28 views

A Chain of Subsets of $\mathbb{R}$ Without any Good Countable Subchain

Consider which $\bigl{(} A_i \bigr{)}_{i\in I}$ is a chain of subsets of $\mathbb{R}$. We say that a countable chain like $\bigl{(} B_n \bigr{)}_{n\in \mathbb{N}}$ is good if : for every $n\in ...
3
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1answer
22 views

Connections between Cesaro summation and Borel summation of series

Let $\sum_{n=0}^\infty x_n$ be a given series of numbers, let $S_n=\sum_{k=0}^n x_k$, $n=0,1,2,...$, let $g\in \mathbb R$. We say that this series is convergent to $g$ in the sense of Cesaro if $$ ...
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0answers
14 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
4
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1answer
28 views

Revolving a $k$-manifold around an axis gives a $(k+1)$-manifold

I want to solve the following problem from M. Spivak's Calculus on Manifolds: Let $\mathbb{K}^n=\{x \in \mathbb{R}^n:x^1=0 \text{ and }x^2>0,\dots,x^{n-1}>0\}$. If $M \subseteq \mathbb{K}^n$ ...
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1answer
16 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
2
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0answers
33 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
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1answer
17 views

modulation-translation operator continuous in $L^{p}$ norm?

We put, $T_{y}f(x):=f(x-y), \ (x, y\in \mathbb R^{n}).$ It is well-known that $\|T_yf-f\|_{L^{p}} \to 0$ as $y\to 0$ for $1\leq p <\infty.$ Next we put, $M_tT_yf(x):= f(x-ty) e^{i t (x\cdot y)}, ...
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1answer
36 views

Prove or disprove regarding continuity of $f$ and $g$

Prove or disprove: Let, $f,g:[a,b]\to \mathbb R$ be continuous in $[a,b]$ and are non-zero at any point. There exists $c\in [a,b]$ such that $$g(c)\int_a^bf(x)\,dx=f(c)\int_a^b g(x)\,dx.$$ ...
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37 views

The Coin-Exchange Problem (Application of the Residue Theorem) [on hold]

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
1
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1answer
21 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
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2answers
58 views

Find functions $f$ and $\alpha$ such that $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist?

Find functions $f$ and $\alpha$ such that the improper Riemann-Stieltjes integral $\int_1^{\infty}|f|d\alpha$ converges, but $\int_1^{\infty}fd\alpha$ does not exist? I'm really not sure how to start ...
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1answer
28 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
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0answers
65 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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4answers
32 views

Proof $log_{r} a = log_r s \cdot log_s a $ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
2
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0answers
36 views

How to prove that $\|x+y\|_p<\|x\|_p+\|y\|_p$? [duplicate]

Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for ...
3
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1answer
33 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
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0answers
17 views

Second derivative and Hessian-Matrix

Suppose $f$ is twice differentiable. Why is $Df(a)[v] + \frac12 D^2f(a)[v,a] = \frac 12 \langle H_f(a) v, v \rangle $ ? $Df(a)[v]$ is the multidimensional derivative of $f$ at point $a$ in ...
4
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2answers
34 views

On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
1
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1answer
19 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
3
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1answer
25 views

About convergence in norm of the Fourier Transform

Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that: \begin{equation} \lim_{R \rightarrow \infty}\big\|S_{R}\,f - f\big\|_{p} = 0 \end{equation} for $1<p<\infty$, where ...
2
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1answer
20 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
1
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1answer
27 views

Problem with the Definition of contractible set

I have this definition of contractible set: we say that $A\subset X$ is contractible in $X$ if there exists a continuous function $\eta:[0,1]\times A\rightarrow X$ such that $\eta(0,x)=x, \forall ...
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2answers
43 views

Finding a choice for Epsilon for open/closed set proofs

I'm studying the proofs for open/close sets by using the following definition: I'm having problems to understand the proofs. The proofs sounds pretty straightforward: just choose a value for ...
1
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1answer
28 views

Fourier Analysis Help - $\mathcal L^2$

Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per. ...
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1answer
26 views

If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
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2answers
130 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
2
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1answer
52 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
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0answers
13 views

FTC and Points of differentiability

Just having a little bit of difficulty understanding the solution to the following problem: Let $f$ be defined as follows, $$f(t) = \left\{\begin{array}{ll}0, & \quad t<0, \\ t, & \quad ...
4
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1answer
27 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
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0answers
23 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
2
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2answers
31 views

How to find the length between 2 points given a pivot

I am not great at math but I have done the previous steps to my problem. This is the last step where I need to find out the distance between C,D. I am writing a program that will output this ...
0
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0answers
24 views

Help in finding a paper on nonlinear Schrodinger equations [on hold]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
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1answer
21 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
2
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1answer
41 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
3
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1answer
63 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
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1answer
26 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
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2answers
52 views

A question related to Montel's Theorem

Given $c>0$, there exist $\varepsilon > 0,$ such that, whenever $\{a_n\} \subset \mathbb C$ and $\sum_{n=1}^{\infty}\lvert a_n\rvert \le c\,$ implies that $$ \sup_{\frac{1}{2}\leq x\leq1}\left|1 ...
2
votes
1answer
38 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
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1answer
33 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
0
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0answers
27 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
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votes
1answer
22 views

Show, directly from the definition, that the following series is convergent. [on hold]

Using the definition of a convergent series, how do you show that the series $\sum_{n=1}^{\infty} (\frac{-2}3)^n $ converges.