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
votes
2answers
136 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
0
votes
2answers
54 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...
0
votes
1answer
29 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
0
votes
1answer
55 views

Analogue of differentiation for sequences?

I remember learning (2 semester calculus for engineers) about all the below ones, but nothing that fits in place of the question mark. Is there anything nontrivial? ...
1
vote
1answer
92 views

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Suppose $x_n\rightharpoonup x$ in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ with the weak* topology, in other words, $\forall f\in L^{1}([0,T];L^2(\mathbb{T}^2))$ we have $$\lim_{n \to\infty} \int _0 ^T ...
0
votes
1answer
245 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
8
votes
3answers
478 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
0
votes
1answer
52 views

How do you formally prove $\limsup\limits_{n\rightarrow\infty}(|na_n|^{\frac{1}{n}}) = \limsup\limits_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}})$

My definition of $\limsup$ is it is the supremum of the accumulation points of a sequence (i.e the supremum of the limits of all possible subsequences of a sequence). So if: ...
0
votes
2answers
47 views

Question about limit and continuity

I have that $u_0>0$ , $u_n=u_n^+-u_n^{\raise{1pt}{-}}$ and $u\mapsto u^{±}$ is continuous if $u_n\rightarrow u_0$ why we have that $u_n^+\rightarrow u_0$ and $u_n^{\raise{1pt}{-}}\rightarrow 0 $ ...
0
votes
1answer
69 views

Lengths of curves - Arc length

If $\gamma\colon [a,b] \rightarrow \mathbb{R}^d$ is a differentiable curve we usually define $$ \text{length}(\gamma) := \int_a^b \|\dot{\gamma}(t)\|_2\, \text{d}t,$$ where $\|\cdot\|_2$ is the ...
0
votes
2answers
58 views

If $F$ is a field, show the following function is a permutation

Let $F$ be a field. Show that the function $a\rightarrow a^{-1}$ is a permutation of $F\{0_F\}$ So I know that if it is indeed a permutation, then it is one-to-one and onto. Also, For every $a$,$b$ ...
0
votes
1answer
22 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
2
votes
1answer
46 views

strengthen the condition of convergence in measure of sequence of functions

Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative measurable functions on a measurable set $E$. (1). Suppose for any $\epsilon>0$, $$\sum_{n=1}^\infty \mu\{x\in E: ...
2
votes
0answers
66 views

Showing an inner product space is complete

I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2: Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner ...
0
votes
1answer
37 views

Convergence of $\sum_{n=1}^{\infty} 2^{n} |a_{2^{n}} | $

Let $\sum_{n=1}^{\infty} a_{n}$ be a series of nonzero real number with $\sup\{\frac{a_{n+1}}{a_{n}}\}_{n \in \mathbb{N}} \leq 1$. (a) If $\sum_{n=1}^{\infty} a_{n}$ absolutely converges, does ...
0
votes
1answer
37 views

Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
1
vote
1answer
41 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
0
votes
0answers
59 views

Functions which can not be integrated via Riemann.

I am looking for some (possibly exotic) functions which can not be integrated via Riemann integration but can be integrated via Lebesgue. I am aware of the rational indicator function, i.e., $$ f(x) = ...
2
votes
2answers
73 views

Is the image of this operator on $\ell^2$ closed?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
2
votes
0answers
154 views

Stone-Weierstrass Approximation Theorem for Vector-Valued Functions?

A classical result in analysis is the Stone-Weierstrass approximation theorem: Theorem. Let $X$ be a compact Hausdorff topological space and $(C(X,\mathbb C),\|\cdot\|_{\infty})$ denote the space ...
0
votes
1answer
37 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
57 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
0
votes
1answer
87 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
1
vote
1answer
59 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
0
votes
1answer
73 views

Small question about calculus

I have this lemme from this paper: "Multiplicity results for quasi-linear problems A.Ayoujil, A.R. El Amrouss, 2008" We consider the truncated problem $$(\mathcal ...
0
votes
2answers
33 views

Prove that $F$ is constant on $S$.

If $F'(x;y)=0$ for every $x$ in an open convex set $S$ and for every $y$ in $\mathbb{R^n}$, prove that $F$ is a constant on $S$, where $S\subset\mathbb{R^n}$. Somewhere I need to define a function ...
0
votes
1answer
93 views

the area of the image under a specific holomorphic function of the unit disk

Let $f(z)=z^3+\frac{z^2}{2}$. Let $D$ be the unit disk in $\mathbb{C}$. How to compute $$ Area(f(D))? $$ In the case that $f:D\to \mathbb{C}$ is injective, \begin{align*} Area(f(D))&= \int_D ...
1
vote
4answers
82 views

If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
4
votes
0answers
279 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
2
votes
2answers
54 views

power series representation in terms of another

Hi how to do the following: Given $f(z) = \sum c_n z^n$ How to express $\sum c_{3n} z^{3n}$ in terms of $f(z)$ Thanks a lot!
1
vote
1answer
83 views

surface area of the graph of a convex function

I started out with the following question: Say $\Omega$ is a nice bounded domain in $\mathbb{R}^{n-1}$. (One can imagine it being a unit ball in $\mathbb{R}^{n-1}$.) Let $f:\Omega\rightarrow ...
0
votes
1answer
157 views

Rudin: A compact metric space $K$ has a countable base, therefore $K$ is separable.

Hi this is a problem from Rudin's Princ. of Mathematics. I was hoping someone could check this part of my proof for the following question, comments would be very appreciated!: $25.$ Prove that ...
0
votes
1answer
58 views

Cauchy sequence in $\mathbb{Q}_p$ implies its p-absolute value is cauchy in $R$

Actually, I don't understand why $\{ a_{n}\} \in \mathbb{Q}_{p}$ is cauchy implies $|a_{n}|_{p} \in \mathbb{R}$ is cauchy. Could anyone give me a hint for understand this?
0
votes
1answer
32 views

Transformation of quadratic form

I've got the following quadratic form $T(x_1,x_2)=x^TQx$, with $$ x=\begin{pmatrix}x_1\\x_2\end{pmatrix}, Q=\begin{pmatrix}\frac{1}{2}(m_1+m_2)L_1^2 & \frac{1}{2} ...
1
vote
2answers
117 views

Fundamental groups and some properties

I have some basic questions about fundamental groups that came up when I tried to prove a few things: I am sorry that they are kind of informal questions, but I could not find any answers to them in ...
3
votes
1answer
112 views

Basic Analysis Help. Open & Closed Sets; Topology

I need some help! First, let me say that I am Mathematics major, as I'm a senior in college finishing up my undergraduate work with the hope of going to graduate school for Mathematics in the future; ...
0
votes
1answer
26 views

Fourier transforms similar $\Rightarrow$ functions similar?

I am wondering if there is a theorem that states something like the following? If $$\big|\;\tilde f(\omega)-\tilde g(\omega)\,\big| < \varepsilon\qquad \forall\omega$$ then there exists a ...
1
vote
1answer
91 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
48
votes
7answers
6k views

What is integration by parts, really?

Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher ...
0
votes
2answers
55 views

showing that $f(x,y)$ is continuous at $(0,0)$

Let $$f(x,y) = \begin{cases} 0, & \text{if $y \le 0$, $y \ge x^2$ } \\[2ex] 1, & \text{if $0 \lt y \lt x^2$ } \\ \end{cases}$$ Show that $f(x,y) \to 0$ as $(x,y) \to (0,0)$ along any ...
2
votes
2answers
88 views

solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$ |f(t,x)|\leq \phi(t)|x|. $$ Suppose ...
1
vote
1answer
46 views

inequalities concerning integration and measure

Let $f$ be a non-negative function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} f=1$. Let $p\in(0,1)$. Let $E$ be any measurable subset of $\mathbb{R}^n$. Prove that $$ \int _E f^p\leq ...
0
votes
2answers
226 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
0
votes
1answer
65 views

Questions about van Kampen's theorem.

I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions: It was said that the images of the ...
3
votes
2answers
111 views

Solving the differential equation $x^3y''+2x^2y-6xy = 0$

First question on here, so I hope I'm doing this right. I've been reading up on differential equations lately and have now stumbled upon one that I have no idea how to solve. $x^3y''+2x^2y-6xy = 0$ ...
4
votes
1answer
390 views

ODE with additional term

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ I don't ...
0
votes
1answer
18 views

Injetivity of a function

Let f: U $\rightarrow R^{m}$ differentiable in $U \in R^m$. If $|f(x)|$ is constant in $U$, then for all $x \in U$ $, f'(x)$ is not injective. Hint: Derive the function $||f(x)||^{2}$. And verify ...
4
votes
1answer
57 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
1
vote
1answer
96 views

Show that there exists a sequence $\{x_n\}$ such that $f'(x)\to f'(c)$

Suppose that $f'(x)$ exists for all $x \in (a,b)$. Let $c \in (a,b)$, show that there exists a sequence $\{x_n\}$ in $(a,b)$ with $x_n \neq c$ and $x_n \to c$ such that $f'(x_n) \to f'(c)$ This is a ...
0
votes
1answer
60 views

Is Fredholm operator always a closed map?

Let $f:E\rightarrow F$ be a Fredholm operator between Banach spaces, then should $f$ always be a closed map? If this is not the case, then is it true that $f$ always maps a closed linear subspace to a ...