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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28 views

Proving the existence of z 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
votes
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
votes
0answers
44 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 $$ ...
-3
votes
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
vote
1answer
25 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 $U\subseteq\mathbb{R}^n$ ...
2
votes
2answers
27 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
votes
1answer
21 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 $$ ...
0
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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
votes
1answer
24 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 $M \subseteq \mathbb{K}^n$ ...
0
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0answers
11 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 ...
1
vote
1answer
15 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)}, ...
1
vote
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 $$g(c)\int_a^bf(x)\,dx=f(c)\int_a^b g(x)\,dx.$$ ...
-1
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0answers
36 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
vote
1answer
20 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).
1
vote
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 ...
1
vote
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 ...
5
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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 ...
-2
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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 ...
1
vote
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
vote
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. ...
0
votes
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 ?
5
votes
2answers
128 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
votes
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 ...
1
vote
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
votes
1answer
26 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 ...
1
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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
votes
2answers
30 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
votes
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 ...
0
votes
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
votes
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
votes
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!
-1
votes
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}) ...
-2
votes
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
votes
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
votes
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 ...
-3
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.
0
votes
0answers
19 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise if $f:[0,1]\to \mathbb R$ is only piecewise continuous (but ...
0
votes
0answers
34 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
0
votes
2answers
24 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
0
votes
1answer
39 views

Bring a proof for the fundamental theorem of calculus.

If $f\in \mathscr{R}$ on $[a,b]$ and if there is a differentiable function $F$ on $[a,b]$ such that $F'=f$, then $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$