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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Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
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54 views

What is the convex-hull of the set $\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$

I know that set $$ E=\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2 $$ has infinitely many points on the line $y=x-1$, which suggests this line to be included in the upper part of the ...
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2answers
55 views

Open and connected in $R^n$ revised

I am trying to understand the following: If we have an open and connected set in $R^n$ then it can be connected with line segments parallel to the axes. I managed to prove this: If a set $U$ is open ...
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1answer
51 views

Showing a Cauchy sequence does not have a limit

Okay basically I've managed to work through parts 1 a and b (with some help) now I'm a little stuck on part c). I think I can show fn is a cauchy sequence by virtue of the fact that fn-f tends to 0, ...
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1answer
67 views

Mean Value related problem.

I'm working on a function $f : \left[a,a+h\right] \rightarrow \mathbb{R}$. I know that $f$ satisfies the conditions of the Mean Value Theorem thus I have $\theta \in \left(0,1\right)$ such that $f\...
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2answers
175 views

Field isomorphism of $\mathbb{C}$ onto itself

I am trying to find a field isomorphism of $\mathbb{C}$ onto itself other than the identity map. The isomorphism preserves the algebraic structures $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$. This ...
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63 views

How to find out the real part of this.

I have to sum this: $$S:=\cos\left(\frac{\pi}{M+1}\right)+\dots+\cos\left(\frac{M\pi}{M+1}\right)$$ Where $M$ is a given natural number. I tried with this: Since $$e^{i a}=\cos a+i\sin a$$ and ...
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47 views

What does it mean by saying that $u^n, J^n$ “$C^{\infty}$ converges” to u, J?

The question arose while reading the big book of McDuff & Salamon. Here $\Sigma$ is Riemann surface and M is compact symplectic manifold. Let $u^n(n\in \mathbb N), u : \Sigma \rightarrow M$ be J-...
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1answer
125 views

Prove that the equation has no solution

Prove that there exist infinitely many positive real numbers r such that the equation 2$^x$ +3$^y$ + 5$^z$ = r has no solution (x, y, z) $\subseteq$ Q × Q × Q. First, I prove that set S = {2$^x$ +3$^...
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208 views

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set $\|...
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3answers
109 views

Regarding limits

If $f$ is positive and differentiable in $(0,\infty)$, then I want to find the following limit. $\lim\limits_{n\to \infty}\left(\dfrac{f\left(x+\dfrac{1}{n}\right)}{f(x)}\right)^n$. I have done as ...
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1answer
46 views

Left and right continuity

I was wondering if someone can help me write down (or perhaps just check my answer) the proof for the following theorem formally. I feel that what I wrote down is too easy to be correct ... Suppose ...
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2answers
105 views

Logical Quantifiers

I am wondering if there is a reference or book that clearly translates all English forms of logical quantifiers to mathematical quantifiers. For example, when we say for any element $ x \in S$, is ...
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1answer
33 views

Why $\displaystyle\rho_{\alpha, \beta}(f)=\sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha f(x)|$ is not a norm on $\mathcal{S}(\mathbb R^n)$?

The Schwartz space $\mathcal{S}(\mathbb R^n)$ is the set of all function $f:\mathbb R^n\longrightarrow \mathbb C$ such that $\displaystyle \sup_{x\in \mathbb R^n}|x^\beta \partial^\alpha f(x)|<\...
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2answers
63 views

$\lim \sup$ of a sequence

Let $\{A_n\}$ be a sequence and $\frac{1}{R} = \lim \sup A_n$. Let $\alpha < R$. My question: Why is there $n_0\in \Bbb N $ such that $$A_n < \frac{1}{\alpha}\text{ for any } n\geq n_0$$ Thanks ...
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135 views

Use L'Hopital's Rule to Prove

Let $$f: \mathbb R\rightarrow \mathbb R$$ be differentiable, let a in $\mathbb R$. Suppose that $f''(a)$ exists. Prove that $$\lim_{h\rightarrow0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}=f''(a) $$ Suppose ...
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1answer
27 views

If T is a top-linear map what does $(T^\ast T)^{1/2}$ mean?

If $T: H_1 \rightarrow H_2$ is a continuous linear map between two Hilbert spaces, $H_1$ and $H_2$, what does the notation $(T^\ast T)^{1/2}$ mean? The book I'm reading defines $|T|$ to mean $(T^\ast ...
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2answers
45 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
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1answer
250 views

If an entire function grows slower than a polynomial, then it is a polynomial!

I was investigating the following corollary to Liouville's Theorem in Complex Analysis: if $f(z)$ is entire and $\lim_{z\rightarrow \infty}z^{-n}f(z)=0$, then $f(z)$ is a polynomial in $z$ of degree ...
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Is $dx\,dy$ really a multiplication of $dx$ and $dy$?

On the answers of the question Is $\frac{dy}{dx}$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: does $dxdy$ in the ...
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1answer
63 views

Equivalence between dual estimates

Let $H$ a Hilbert space and $X$ a measure space; let $U(t):H\to L^2(X)$ an operator defined for all real $t$. I'm considering the following estimates: $$\Vert U(t)f\Vert_{L_t^qL_x^r}\leq\Vert f\Vert_H$...
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Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in L^{1}(...
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Question in the Continuity of a function

I have this function: $$(J''(u)v,w)=(v,w)-(KN_{f'}(Ku)Kv,w)$$ for all $u,v,x\in L^2[0,1]$ such that $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$ and $Ku(t)=\int_0^1 G(t,s) u(s)ds$ $K$ is symetric, ...
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1answer
32 views

Examples of function the following three conditions

Let function f:(0,∞)→(1,∞) satisfying the following conditions: (i) f is nondecreasing; (ii) for each sequence $({x_n})⊂(0,∞),lim_{n→∞}f(x_{n})=1$ if and only if $ lim_{n→∞}x_{n}=0⁺$; (iii) there ...
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2answers
34 views

Finite vs infinite distinction in Rudin's Analysis

I'm starting to self-study Rudin's Principles of Mathematical Analysis. I'm up to the second chapter, theorem 2.24. For any collection $\{G_i\}$ of open sets, $\bigcup_i^nG_i$ is open. For ...
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1answer
37 views

asymptotic analysis

For each of the following sentences involving functions f and/or g, find a counterexample to show that it is false: What is meant by counterexamples?
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1answer
127 views

Stacked with this Problem of Calculus

I have been struggling for quite some time with the following problem and I would really appreciate some help: Consider $f(d)=\frac{(1-d)\left(1-d^{(\frac{t}{2}-1)}\right)}{(1-d)\left(1-d^{(\frac{t}{2}...
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0answers
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Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ I_{n,m}(\alpha,\sigma,\omega,r)=\int_0^...
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1answer
27 views

Applying theorems on differential functions

I'm thinking that I need to do a proof by counter example? Is it possible to use Rolle's theorem: f is cts on [a,b] and differentiable on (a,b) and f(a)=f(b), so there exists a c in (a,b) such that f'...
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1answer
98 views

Serie of functions : interchange of limit of series

Let $\{f_n\}_{n=1}^\infty$ be a sequence of real-valued functions on $\mathbb{R}$. Show that if $f_n$ is continuous for all $n \in \mathbb{N}$ and the series $\sum_{n=1}^\infty f_n$ converges ...
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2answers
67 views

Complex plane and $\mathbb{R}^2$. [duplicate]

What differences -if any- are there between the complex numbers $\mathbb{C}$ and $\mathbb{R}^2$? I am taking multi variable analysis now and I was wondering what possible changes there might be from ...
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1answer
29 views

What is $|x|$ for $x\in \mathbb T^n$?

The $n$-dimensional torus is $\mathbb T^n=\mathbb R^n/\mathbb Z^n$. Let $|x|$ be the Euclidian norm. What is $|x|$ for $x\in \mathbb T^n$?
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183 views

An analysis problem about convergence

Suppose that $f$ is a continuous function from $[a,b]$ to $[a,b]$. Let $x_0\in [a,b]$, and define by induction that $x_{n+1}=f(x_n)$. Show that $$\lim_{n \rightarrow \infty} (x_{n+1}-x_n)=0$$ implies $...
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Applications of Vito Volterra's theorem

We know from Volterra's theorem that: There cannot exist two pointwise discontinuous functions on an interval $(a,b)$ for which the continuity points of one, are the discontinuity points of the other,...
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48 views

$\lim_{ x\to 0^+}f(x)=?$

Suppose $f(x)$ is bounded on $[0,1]$, and for all $0\le x\le\frac{1}{a}$ satisfis $f(ax)=bf(x)$. ($a,b>1$) $\lim_{ x\to 0^+}f(x)=?$
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1answer
41 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants $\lambda,\...
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1answer
245 views

Proving the convergence and divergence of the p-series

We know from calculus that $\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverges if $p \in [0,1)$ and converges if $p > 1$. I want to use analysis to prove these two statements. For the case where $p > 1$...
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1answer
319 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
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30 views

Evaluating Data

I have a set of data that I need to evaluate. The data came from 15 people we asked to evaluate several project proposals using 6 different criteria. For each weighted criteria, they selected 1 of 4 ...
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1answer
70 views

find an odd differentiable function $f: \mathbb{R} \to \mathbb{R}$ s.t. $f'(x) = e^{-x^2}$

find an odd differentiable function $f: \mathbb{R} \to \mathbb{R}$ s.t. $f'(x) = e^{-x^2}$ This is for my analysis course and we are studying power series; my attempt: $e^x = \displaystyle \sum_{k=0}...
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3answers
334 views

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-...
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0answers
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If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$?

Problem: If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$? Attempt: First we note that $$ f'(z) = \sum_{n=1}^\infty n a_n z^{(n-1)} $$ so that $$ z f'(z) = \sum_{n=1}^\infty n a_n z^n $$ ...
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1answer
41 views

Showing $\lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} }{ a_n }$

Let $\sum a_n z^n$ be a complex power series. I've seen it asserted without explanation in a text that $$ \lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} ...
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1answer
80 views

Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
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2answers
232 views

Relationship Between Ratio Test and Power Series Radius of Convergence

Let $ \{a_k\} $ be a sequence of positive real numbers. Why does it hold that $$ \liminf \frac{a_{k+1}}{a_{k}} \leq \liminf (a_k)^{\frac{1}{k}}\leq\limsup (a_k)^{\frac{1}{k}} \leq \limsup \frac{a_{...
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0answers
160 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
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1answer
578 views

When can we use Fubini's Theorem?

I am using Munkres' Analysis on Manifolds textbook. Munkres defines Fubini's Theorem on rectangles and on simple regions (at least till the point that I have read). Now, according to the book, we ...
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3answers
256 views

Prove that the logarithmic mean is less than the power mean.

Prove that the logarithmic mean is less than the power mean. $$L(a,b)=\frac{a-b}{\ln(a)-\ln(b)} < M_p(a,b) = \left(\frac{a^p+b^p}{2}\right)^{\frac{1}{p}}$$ such that $$p\geq \frac{1}{3}$$ That is ...
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1answer
119 views

Convergence of a series if limit goes to 0

I am stuck on the following question. We are asked to prove the following: Assume $\lim \limits_{n \to \infty}$ $a_n$ = L 1) prove that if L>1, then $\sum_{n=1}^{\infty}$ ${1\over n^{a_n}}$ ...
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2answers
71 views

Integrals on unlimited sets

How do you evaluate this expression $$ \left| \int_{1}^{\infty} 1 \; dx - \int_{1}^{\infty} 1 \; dx \right| \quad ? $$ Using improper integral definition, this should be an indeterminate $\infty - \...