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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1answer
31 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
1
vote
1answer
26 views

$s := \sup(\{a_n\mid n \in \mathbb{N}\})$ and $s =\limsup_{n \to ∞} a_n$ at the same time

$(a_n)_{n=0}^∞$ is a sequence in $\mathbb{R}$, $ s := \sup(\{a_n\mid n \in \mathbb{N}\})$ and $s \in \mathbb{R}$. Proof the following: If $a_n \ne s$ for all $n \in \mathbb{N}$, then $s = \limsup_{n ...
1
vote
1answer
30 views

Convergence radius and is a series convergent in the ends of that radius

Find the convergence radius of $$\sum_{n=1}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^{1/3}(5x)^n$$ I find that the convergence radius is $1/5$, but how does one check if the series is ...
0
votes
1answer
36 views

Computing the derivative of a function defined by critical points

Let $[a,b]\subset \mathbb{R}$ and $f_\lambda:[a,b]\rightarrow \mathbb{R}$ be a set of continuous functions which depend continuously on $\lambda\in[c,d]\subset \mathbb{R}$ (by that I mean that I can ...
8
votes
3answers
260 views

Why do we use $\mathbb{R}$?

Since there are holes in the $\mathbb{Q}$ we have constructed $\mathbb{R}$ in order to fill in these holes. But I was wondering why we don't use $\mathbb{C}$ or some other number system that is even ...
3
votes
3answers
37 views

Inverse function of $f(x,y,z) = (xy-z^2, x+z)$?

How do you determine the inverse function $f^{-1}: \mathbb{R}^2 \to \mathbb{R}^3$ of $f: \mathbb{R}^3 \to \mathbb{R}^2 , f(x,y,z) = (xy-z^2, x+z) $ ? Or to put it into a bigger context: ...
2
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1answer
38 views

Exercise 8.1 in Brezis' Functional Analysis

Consider the function $$u(x) = \frac{1}{(1+x^2)^{\frac{\alpha}{2}}} \frac{1}{\ln(2+ x^2)} \qquad\; x\in \mathbb{R}$$ with $0<\alpha<1$. Check that $u\in W^{1,p}(\mathbb{R})$ for all $p\in ...
3
votes
1answer
71 views

How to continue $f \in C ^\infty[a;b]$ to $f \in C ^\infty (\mathbb {R })$?

Is there a general way to extend a smooth function on a closed interval $[a,b]$ to one that is defined on the entire $\mathbb R$? It is not OK to reflect this function in points $a$ and $b$, and then ...
-1
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1answer
41 views

Is the function $f(x)$ positive in $[0,1]$ where $ a, b>0$

Is the function $$ f(x) = \frac {e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x)+\frac{3}{\pi} \sin(\pi x)\right)$$ positive in $[0,1]$ where $ a, b>0$
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2answers
34 views

What to know about convergence of integrals

According to the values of p>0 examine the convergence of the integral: $$\int_0^{+\infty} \dfrac{\ln(1+2x^{3p})}{(x+x^2)^{4p}\arctan(x)^{1/2}}dx$$ I didn't find a good explanation about this kind of ...
0
votes
1answer
12 views

Integral of derivative of rational map on unit disk

Let $f:D \rightarrow D$ be a surjective rational map of the unit disk of degree $n$. Prove that $$\iint_D |f'(x+iy)|\:\mathrm{d}x\:\mathrm{d}y\leq \pi \sqrt{n}.$$ Attempt: We know that rational maps ...
1
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1answer
54 views

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
2
votes
1answer
43 views

No analytic continuations of $\sum_{n = 1}^\infty \frac{a_n}{z - e^{i \pi \lambda n}}$ beyond unit disk.

I am working on the following complex analysis problem: Let $\lambda$ be irrational and let $\left\{a_n\right\} \subset \mathbf{C}$, $\sum |a_n| < \infty$ and $a_n \neq 0$. Letting $D = ...
0
votes
2answers
31 views

When the standard methods of testing convergence don't work

For instance, $\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)}$ is convergent,but which are the possible and most common methods for proving such series(for which we can't use d'Alembert's and Cauchy's ...
3
votes
1answer
100 views

Iteration with bounded converging sequence as input

Consider the sequence of vectors $( x_k )_{k=0}^{\infty}$ such that $x_k \in X \subset \mathbb{R}^n$ for all $k$, where $X$ is compact, and $x_k \rightarrow \bar{x} \in X$. Consider a vector $y_0 \in ...
7
votes
2answers
101 views

To prove that $\sum\limits_{n=1}^{\infty}\frac{n^{n-1}}{n!e^n}=1$

$\sum\limits_{n=1}^{\infty}\frac{n^{n-1}}{n!e^n}=1?$ I have noticed that $\lim\limits_{n\to\infty}{\frac{1}{\sqrt[n]\frac{n^{n-1}}{n!}}}=1/e$, so the series ...
2
votes
1answer
45 views

Little-o, Big-O and differentiation

The functions $f,g, h: \mathbb{R} \rightarrow \mathbb{R}$ have the origin 0 as an internal point of their domain. Prove that if $f = \mathcal{O}(x^{k})$, $f = \mathcal{o}(x^{k-1})$ Prove that if $g ...
0
votes
1answer
21 views

Convergence radius

I know the Cauchy Hadamard equation to calculate the convergence radius of a power series $$\sum_{n=0}^{\infty} a_n x^n$$ Is there a way to generalize this for series of the form ...
1
vote
1answer
15 views

Showing that $|f(a+v) - f(a) - T\dot\,v| \leq \sup_{0<t<1} |f'(a+tv) - T|\dot\,|v|$

Question: Let $f: U \to \mathbb R^n$ continuous at the open subset $U \subset \mathbb R^m$, and consider the line segment $[a, a+v ] \subset U$. If $f$ is diferentiable at all points $(a, a+v)$ ...
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0answers
7 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
0
votes
1answer
20 views

Convexity of $f(x) = \sum_i x_i^p$ for $x_i \ge 0, p \ge 1$

Let $x = (x_1,\ldots, x_n)$ be nonnegative real numbers and $p \ge 1$, then the function $f(x) = \sum_{i=1}^n x_i^p$ is convex, the following proof is wrong, or not? I have it from here, but then ...
1
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1answer
36 views

Folland's Real Analysis Exercise 1.22a

The exercise states: Let $(X, \mathcal{M}, \mu)$ be a measure space, $\mu^*$ the outer measure induced by $\mu$ according to (1.12), $\mathcal{M}^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, ...
2
votes
1answer
42 views

Failure of Newton-Leibniz formula

Suppose that $f : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable but $f \notin C^1 ( \mathbb{R} )$ . It means that $f'$ exist but it is not continuous. Question 1 Is function $f'$ locally ...
-1
votes
1answer
15 views

Get Rank from two Ranks

i'm a math-noob and looking for a way to get the rank of two ranks. For example 26939 customers total Rank 1 is 10 from total volume of sales grouped by each customer Rank 2 is 470 from total ...
2
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0answers
62 views

a Bound for functions in $L^p$ after convolution with a $G_\lambda$ almost a heat Kernel

The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 ) We consider the operator $G_\lambda$ $$G_\lambda ...
1
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3answers
33 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
3
votes
1answer
27 views

Finding all accumulation points of the a set.

I am working on homework for my intro to analysis class and I was assigned a problem to find all accumulation points of the set $S=\{x\mid x\in[0,1]$ and $x$ is rational$\}$. I hodge podged a solution ...
0
votes
1answer
43 views

Wave equation- solution extincts in time

I am currently dealing with the following task: Let $u$ be a solution to the wave equation on $\mathbb{R}^3 \times (0,\infty)$ with initial conditions $u(x,0)=g, u_t(x,0)=h$ where $g,h \in ...
0
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0answers
18 views

Simple question about Sobolev Spaces

I'm studying Sobolev Spaces and my lecturer told us that $H^1_0(U)=H^1(U)$. Is this true for all $U\subseteq \mathbb{R}^n$? Even $U=\mathbb{R}^n$? And why exactly is this true?
0
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0answers
4 views

On coercive functions

Let $\Omega$ be a two-dimensional index set, $M=|\Omega|,$ and $q>0$. Suppose $(v^k)$ is a sequence in $R^M$ and $\|v^k\| = 1.$ Let matrix $K\in R^{M\times M}$ and $\nabla$ is the discrete gradient ...
0
votes
2answers
50 views

Monotone convergence of functions ant theor asymptotic power series

consider a sequence of functions $f_n:(0,\infty)\rightarrow\mathbb{R}$ which are positive and monotone, i.e. $$0< f_1\leq f_2\leq....\leq f_n\leq f_{n+1}...$$ Now let us assume we know the ...
0
votes
3answers
39 views

Find the Maclaurin Series of a function

How to find the Maclaurin series of the function $$f(x)=\frac{1}{(9-x^2)^2}$$ I guess we are gonna use derivatives but i have no idea how the final answer should be formed.
0
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1answer
34 views

Find the area of the parts formed from the parabola $2y=2-x^2$ separating the circle $x^2+y^2 \leq 5$?

Find the area of the parts formed from the parabola $2y=2-x^2$ separating the circle $x^2+y^2 \leq 5$? I know the standard method when we are given equations,but what we do when we have inequalities?
0
votes
2answers
30 views

Expansion coefficients of an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ for $\sum|c_n|^2$ to converge.

Expansion coefficients with respect to an orthonormal basis must satisfy $c_n n^{1/2}\to 0$ as $n\to \infty$ in order that $\sum|c_n|^2$ may converge. Is this true or false? Give a proof or ...
1
vote
1answer
26 views

Are some of the conditions of the alternating series test neccesary?

The alternating series test says that if $a_n$ is a sequence such that \begin{align} (a) a_n\geq 0\\ (b) a_n\rightarrow 0\\ (c) a_{n+1}<a_n, \end{align} then the series $$\sum\limits_{n=1}^\infty ...
2
votes
0answers
22 views

Bias induced by splitting a log sum into independent log [closed]

Does anyone know the introduced bias ($\epsilon$) when a log-sum is split into a sum-log $$\log \left(\sum_{k=1}^N a_k\right) = \sum_{k=1}^N \log(a_k) + \epsilon$$ Many thanks for your help!
0
votes
2answers
33 views

Bounded sequence with subsequences all converging to the same limit means that the sequence itself converges to the same limit.

I have a question regarding one exercise in Stephen Abbotts' Understanding Analysis. The question is: Assume $(a_n)$ is a bounded sequence with the property that every convergent subsequence of ...
1
vote
0answers
68 views

Set of measure zero - Analysis in $\mathbb{R}^n$

Why every set of measure zero has emptiness? I.e, why $m(A)=0$ implies $\operatorname{int}(A)=\emptyset$, where $A \subset \mathbb{R}^n$. So I assumed that $m(A)=0$. By definition, $A \subset ...
1
vote
1answer
38 views

Point-wise bounded and equicontinuous sequence of functions has a uniformly convergent subsequence

Problem We have a sequence $(f_n)$ of continuous functions on a compact metric space K. It is also given that $(f_n)$ is point-wise bounded and equicontinuous. Now show that $(f_n)$ has a ...
1
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0answers
20 views

Hypergeometric differential equation with nonlinearity

I have come across a problem involving a hypergeometric differential equation (http://mathworld.wolfram.com/HypergeometricDifferentialEquation.html) with a nonlinear term added as in ...
-1
votes
1answer
31 views

(x,y) can be expressed as a differentiable function of (w,z) [closed]

Consider equations $$ \exp(w) + x = y + \exp(z)$$ and $$\cos(w) + \sin(x) + \tan(y) = z + 1$$ Show that near $(x, y,w, z) = (0, 0, 0, 0)$, $(x, y)$ can be expressed as a differentiable function of ...
0
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0answers
16 views

Help with $\int_{0}^{T} I_{0}(2\sqrt{ y x})\cos(tx)dx$

i am trying to do the integral : $$\int_{0}^{T} I_{0}(2\sqrt{ y x})\cos(tx)dx$$ Where $I_{0}(\cdot)$ is the modified Bessel function if the first kind of the zeroth order.
4
votes
1answer
30 views

Dimension of a certain $L^p$ quotient space.

Define $L^p_0 := \{ f \in L^p : \int f = 0 \}$. I am trying to calculate the dimension of the cokernel of the inclusion operator $i:L^p_0 \to L^p$. That is, I am trying to calculate $$\dim ( L^p / ...
1
vote
0answers
14 views

Eliminate singularity from a PDE system

I need some help with the theory of PDE systems, which I am not very familiar with. Assume $f$ and $g$ are two functions of $x,y$ in the real plane. I have the equation ...
10
votes
6answers
148 views

$\lim_{n\rightarrow \infty} n \sum^\infty_{k=n} \frac1{2k(2k+1)}=\frac14$?

This isn't a homework problem, just something that came up while I was studying measure theory. It is well known that the limit of the tails of any convergent series goes to 0. However, the problem ...
4
votes
0answers
66 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
0
votes
2answers
23 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
0
votes
1answer
13 views

Inequality involving gradient and subgradient

I stumbled across a proof where I don't get an inequality. It goes as: $f$ is log-concave and $\nabla f(x)$ exists almost everywhere. Then we have $$\ln f(x) - \ln f(y) \leq \left<g(y), x-y ...
4
votes
1answer
81 views

Does $\int_1^\infty\sin (\frac{\sin x}{x})\mathrm d x$diverge or not?

Does $\int_1^\infty\sin (\frac{\sin x}{x})\mathrm d x$diverge or not? If it converges, does it converge conditionally or absolutely? I guess that it converges conditionally, also,I think it may be ...
1
vote
0answers
30 views

Understanding the role of parameters in an equation in order to fit data

I have an equation for the reflection coefficient of a certain circuit (I can give more context if you are interested), and it is given by $$\Gamma(\omega) = \frac{8 J_0^2 (-2 i \gamma +i \kappa +4 ...