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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Continuity of the variation

Let $f:[a, b]\rightarrow \mathbb{R}$ be bounded variation and continuous. Then why is it true that the partial variations on intervals $[a, c]$ are continuous in $c$?
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2answers
99 views

$\int_0^1|f_n|^3\leq 1\Rightarrow \int_E |f_n|<\varepsilon$ when $|E|$ is small

Let $f_n\colon [0,1]\to\mathbb{R}$ be Lebesgue measurable with $$\int_0^1|f_n|^3\leq 1 \mbox{ for all } n.$$ Show that for all $\varepsilon>0$ there exists $\delta>0$ so that if $E\subset[0,1]$ ...
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3answers
2k views

Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
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2answers
2k views

If $f$ is Lebesgue measurable on $[0,1]$ then there exists a Borel measurable function $g$ such that $f=g$ ae?

If $f:[0,1]\to\mathbb{R}$ is Lebesgue measurable then there exists a Borel measurable function $g:[0,1]\to\mathbb{R}$ such that $f=g$ a.e.?
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1answer
40 views

Convergence weakly to measure?

Let $ w_e (x)=\frac {\partial^2}{\partial x_1^2}\sqrt {x_1^2+e}.$ Show that $ w_e $ converges weakly as $e\to 0$ in the dual of $ C (\bar {B_1}) $ to measure $\mu $ I am that dual $ C (B) $ is borel ...
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0answers
190 views

Doubling measure is absolutely continuous with respect to Lebesgue

Let $\mu$ be a fixed finite measure on $\mathbb R$. We say that $\mu$ is doubling if there exists a constant $C>0$, such that for any two adjacent intervals $I=[x−h,x]$ and $J=[x,x+h]$, ...
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4answers
152 views

$\int f_k\to 0 $ but $f_k $ does not converge to $0 $ ae, where $ f_k $ is defined in $[0, 1] $

Give an exemple, in [0, 1], of a sequence of functions $ f_k $ such that $||f_k||_ 1=\int |f|_k \to 0 $ but $ f_k $ does not converge to $0 $ a.e.
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1answer
67 views

Another question about showing that a point is not an accumulation point of a given set

Let $C = \{ (\frac 1n , \frac mn) \in \mathbb{R}^2 : m,n \in \mathbb Z , n \neq 0 \} $. I'm trying to argue that each point not on the $y$-axis is not an accumulation point of $C$. Is this ...
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1answer
105 views

Use the secant method to approximate $\sin^{-1}(0.1)$ correct to 3 decimal

Use the secant method to approximate $\sin^{-1}(0.1)$ correct to $3$ decimal places starting with $x_0 = 0$ and $x_1 = 0.1$.
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2answers
84 views

Compute $\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$

How do I compute this integral? $$\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$$
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3answers
122 views

continuity of the derivative under certain conditions

I am working on this exercise in a book which asks to prove that $f$ is differentiable if $f$ is continuous and that $\lim \limits_{x\rightarrow x_0} f'(x)$ exists. I know that this is easy to show ...
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2answers
75 views

To argue that a point is not an accumulation point of a given set

I want to show that $\mathbb Z^2$ has no accumulation points in $\mathbb R^2\backslash\mathbb Z^2$. Is this argument correct? In particular, have I correctly invoked the density property of $\mathbb ...
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1answer
59 views

If a set in a general metric space consistes entirely of isolated points, can it still have any accumulation points in its complement?

It seems not in $\mathbb R ^n$ (correct?), but how about in a general metric space? On the other hand, I'm not so sure about my claim above regarding $\mathbb R^n$: surely you can have points outside ...
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2answers
76 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
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1answer
74 views

$\int_{B} f_n \phi \rightarrow 0$ if the Weak-$L^p$ norm of $f$ tends to zero?

Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then by some $\phi \in C^\infty_0(B)$ applying ...
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2answers
244 views

Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...
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1answer
212 views

Does the frontier of an open set have measure zero (in $\mathbb{R}^n$)?

When studying weak border conditions (in Sobolev Spaces), the usual motivation for the weak meaning of inequalities is that the frontier of most open sets in $\mathbb{R}^n$ has zero (Lebesgue) ...
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1answer
96 views

what are real and imaginary part of this expression

I have $M:=\sqrt{\frac{a\cdot(b+ic)}{de}}$ and all variables $a,b,c,d,e$ are real. Now I am looking for the real and imaginary part of this, but this square root makes it kind of hard.
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1answer
270 views

Prove this infinite sequence space is compact.

Let space $S=\{1,2,3,...,N\}^{\infty}$, with $\sigma$ in S represented by $\sigma=\sigma_1\sigma_2\sigma_3...$, which is an infinite string and $\sigma_i\in\{1,2,3,...,N\}$. Define the metric on this ...
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3answers
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Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
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1answer
557 views

Is a $L^p$ function almost surely bounded a.e.?

I just have a quick question related to $L^p$ spaces. Any help is greatly appreciated. Is it true that if a function $f$ belongs to $L^p$ space, absolute value of $f$ raise to the power of $p$ is ...
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0answers
59 views

convolution inequality on R

Let $\nu$ be a complex Radon measure on $\mathbb{R}$ such that $$ \int_{\mathbb{R}} \check{\overline{f}}*f\ d\nu\geq 0 $$ for any complex continuous function $f$ with compact support, where ...
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1answer
67 views

Is there a closed representation of these two integrals?

The first one is: $$ \int _R^{\infty} k_l(\alpha r) \frac{r^{l+2}}{z^{l+1}} dr$$ $$ \int _R^{\infty} k_l(\alpha r) \frac{z^{l+2}}{r^{l-1}} dr$$ where $R > 0$, $k_l$ is the l-th modified ...
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0answers
70 views

Simplify this double series

if I have a double sum and I have an expression like $$ \sum_{l=0}^{\infty} \sum_{l'=0}^{\infty} g(l)f(l') \frac{1+\cos(\pi(l+l'))}{1+l+l'},$$where g and f are some functions. The thing is: I could ...
3
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2answers
70 views

Prove approximation given by the physicist Max Born

In an old book about optics, I have found a nice approximation, that for large l one has: $$P_l(\cos(\theta)) \sim \sqrt{\frac{2}{l \pi \sin(\theta)}} \sin \left((l+\frac{1}{2}) \theta + ...
2
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0answers
56 views

Property of solutions of uniformly elliptic PDE, true?

Let $u:\mathbb{R}^n\rightarrow \mathbb{R}$ be a positive solution in $\mathbb{H}^1$ of an uniformly elliptic PDE in an open cube of side lenght 4 centered at the origin. Is it true that there exists a ...
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1answer
173 views

minimization of L^2 norm of the second derivative of a probability density

I have a question: Let $\rho$ denote a probability density function defined on $[0,1]$. It is twice-differentiable and has a continuous second derivative. Denote by $M$ the set of all such functions ...
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2answers
95 views

How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?

Let $f:[0,+\infty)\longrightarrow R^{+}\bigcup\{0\}$ be a continous and for any $x\in[0,+\infty)$ the sequence $\{f(x+n)\}$ converges to zero,prove that $$\lim_{x\to+\infty}f(x)=0$$ I think ...
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1answer
76 views

Null sets on Daniell integral

Can someone help me clarify the definition of null sets related to Daniell integrals. The book I'm using is Introduction to analysis and integration theory by Philips. The definition that I'm provided ...
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1answer
49 views

Determine the area of $\phi (A) $

Let $ A=[0, 1] \times[0, 1] $. Let $ h $ be a continous function on $\mathbb{R}$ and let $\phi $ be defined by $$\phi (x, y)=(x+h (x+y), y-h (x+y))$$ Determine the area of $\phi (A)$.
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61 views

Roots of a polynomial plus a logistic equation

I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form: $$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$ where $P(z)$ is a ...
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2answers
67 views

Convergence of polynomials

Consider $x\in [0,1]$. I want to know if, in the limit as $n\rightarrow\infty$, the following remains a polynomial: $$\sum_{k=1}^{n}\frac{x^k}{k!}.$$ For any finite n, this is a polynomial. But will ...
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1answer
121 views

Why are rational singletons nowhere dense on the real line?

I'm trying to understand this using the definition that the interior of the closure must be non-empty for a nowhere dense set. Thanks
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2answers
1k views

When is the infimum of the sum of two sets equal to the sum of their infima?

When is the following true? $A$ and $B$ are subsets of real numbers. I don't say that $A$ and/or $B$ are closed: $$\inf (A + B) = \inf (A) + \inf(B)$$ When is there a strict inequality in between? ...
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1answer
112 views

How to find extrema points

I took notes in my class on finding extrema points, but I don't understand what I wrote in my notebook. I need to learn their solution methods before my exam tomorrow (this is not homework or ...
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3answers
112 views

A non-constant, increasing function $f$ such that $f(b)=\int_a^bf$

Is there a non-constant, increasing function $f\colon A\to B$, where $A,B\subset\mathbf{R}$ such that $$f(b)=\int_a^bf(x)\;\mathrm{d}x$$ for $a,b\in{A}$ with $a<b$.
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1answer
138 views

An estimate for $\ln(1+f(x))$ using Taylor expansion

A crucial skill for every aspiring analyst (like myself) is confidence in estimation - knowing when, where, and how to use tools like Big-and-little-O to gain quick upper bounds. I'm trying to push ...
2
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1answer
173 views

Sublinear functional as supremum of linear functionals

Given a sublinear functional on a Vector space $V$, is it possible to write it as supremum of family of linear functionals?
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1answer
286 views

Continuous functions as regulated functions: a property.

In Differential and Integral by Paul Lorenzen (1971) pag. 148, I read ... every continuous function is trivially approximable by step functions that have no jump at a given arbitrary point .... All ...
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1answer
88 views

A question about the concept of tangent plane from William Wade's book

This question is from William Wade's book 11.6.9 page: 435. I have the book's solution manual. That's, I have the question's answer. But, the answer is complicated accourding to me. I dont ...
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2answers
236 views

Conversion of sum of series into product form

Show that the following series and product are equivalent: $$ \sum_{n=1}^\infty \left[ \dfrac{1}{n(n+1)} \right] = \dfrac{1}{2} \prod_{n=2}^\infty \left[ 1+\dfrac{1}{n^2-1} \right] $$ Thought of ...
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1answer
61 views

Showing the existance of the functions $u,v,s,t$ by Implicit function theorem.

I am studying from William Wade's introduction to analysis book the question 11.6.5 at page 434 Question: The given nonzero numbers $x_0, y_0, u_0, v_0, s_0, t_0$ which is simultanuously equations ...
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1answer
1k views

Proof for Strong Induction Principle

I am currently studying analysis and I came across the following exercise. Proposotion 2.2.14 Let $m_0$ be a natural number and let $P(m)$ be a property pertaining to an arbitrary natural ...
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0answers
61 views

If someone asked, and if I do t understand its soution, then, how do i understand? Do I not have aright to ask again? [duplicate]

First of all, I searched the question, and someone asked, I found its solution. But I think, that solution is not clear enough. Forvexample, there, the reason why the integral is zero is not ...
2
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2answers
173 views

What is the definition of the norm

Let $x$ and $y$ be in $ \mathbb{R}^{n}$. I know from the definition of norm that $\|x\|=\sqrt{\sum_{1}^{n}x_{i}^{2}}$. Can anyone tell me what will be the norm of $\|x-y\|$? Is it ...
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2answers
2k views

Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
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1answer
278 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
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1answer
77 views

General solutions of the equation $ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0$

What is a general solution of the equation $$ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0 \textrm{ for } x,y \in \mathbb R, h>0, $$ with unknown function $f: \mathbb R^2 \rightarrow \mathbb R$? ...
5
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3answers
118 views

What are the differences between the Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$ and the standard Lebesgue measure on $[0,1]^n$?

There are no Lebesgue measure on infinite dimensional Banach space. However, there is a Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$. What are the differences between this measure and the ...
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0answers
158 views

Has this Principal Component Analysis (PCA) been done correctly?

I have a set of 3D data points, indicated by the blue color in the picture below. I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...