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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1answer
153 views

Compact Subsets of $C[a,b]$

Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$ Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm. ...
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1answer
53 views

Prove the limit related to a recurrence

For the following sequence $\{a_n\}_{n=1}^{\infty}$, we define $a_1=\alpha\in(0,1)$, and for any $n\geq 2$, $a_{n+1}=a_n(1-a_n)$. Prove: $\lim_{n\rightarrow\infty}{na_n} = 1$.
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2answers
65 views

Is there a standard $L^2$ norm for multi-valued function $f:\mathbb R^n \to \mathbb R^n$?

Equipping $\mathbb R^n$ with the usual product Lebesgue measure, what is the standard $L^2$ norm for the function $f :\mathbb R^n \to \mathbb R^n$ define by \begin{align} f(x) &=\left(f_1(x), ...
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1answer
692 views

Showing continuity using Weierstrass M test

Prove that $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}x^4$$ defines a continuous function on $\Bbb{R}$. My proof: For any $M>0$ and $x\in [-M,M]$, $$|\sum\frac{\cos(nx)}{n^2}x^4| \le \sum ...
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3answers
94 views

Finite lebesgue Integral

Hi guys I've been trying to prove this for a very long time, if someone could help me i would appreciated very much! let $(X,S,\mu)$ be a mesurable space, if $\mu(X)$ is finite and $f$ is a mesureble ...
2
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1answer
430 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
3
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2answers
143 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
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1answer
175 views

Roots of $x^x-\tan (x)$

I conjecture, that the function $f(x)=x^x-\tan x$ has exactly one root in any of the intervals $\left[\dfrac{2n+1}{2}\pi,\dfrac{2n+3}{2}\pi\right]$ , where $n$ is a nonnegative integer. Does anyone ...
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1answer
56 views

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.Suggestion: take u to be the suitable cut off version of ...
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2answers
606 views

Limit goes to infinity, show that the f has a finite minimum.

So limit goes to infinity, and I have to show that there exists a finite infimum. how do i show this?
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1answer
84 views

Can we get an analytical solution to this equation involving the Lambert W function?

Can we get an analytical solution to the variable $t$: $$H\left(1+W\left(A\exp\left(Bt\right)\right)\right)=1+W\left(X\exp\left(Yt+Z\right)\right)$$ $W(x)$ is the Lambert W function.$A$ $B$ $X$ $Y$ ...
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0answers
111 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
2
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1answer
85 views

Laplacian on ${\bf R}^2$ and mean curvature

Consider a function $f$ on ${\bf R}^2$ whose critical point is origin. Then Gaussian curvature of graph of $f$ at origin is determinant of ${\rm Hess} \ f$ and Mean curvature is trace of ${\rm Hess} ...
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1answer
45 views

Prove: ∇⋅ϕF = ϕ∇⋅F + F⋅∇ϕ

I am asked to prove this identity using tensor notation. However, I am not sure where to even begin the problem.
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1answer
42 views

absolute and uniform convergence of a Fourier-like series

I am following stein's real analysis book and he claims that if $a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$ where $f\in L^1([-\pi,\pi])$ then $\sum_{n=-\infty}^{\infty} a_n r^{|n|}e^{inx}$ ...
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4answers
113 views

Evaluating limit making it $\frac{\infty}{\infty}$ and using L'Hopital Rule

Let $P(x)=x^n+\displaystyle\sum\limits_{k=0}^{n-1}a_kx^k$. Find $$ \lim_{x \to +\infty} ([P(x)]^{1/n}-x) $$ I know that in order to solve this problem I need to multiply it by something that will ...
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1answer
96 views

Evaluate: $A \cdot \nabla R + \nabla(A \cdot R) + A \times\nabla\times R$

How to evaluate: $A \cdot \nabla R + \nabla(A \cdot R) + A \times\nabla\times R$ where $A$ is a constant vector field while $R$ is a non-constant vector field. The only simplification I can see is ...
2
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2answers
78 views

$V(R)=Af(R\cdot B)$, where A and B are constant, prove that curl V is perpendicular to both A and B

If $V(R)$ can be expressed as $V(R)=Af(R\cdot B)$, where $A$ and $B$ are constant, prove that curl $V$ is perpendicular to both $A$ and $B$.
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2answers
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Proof of continuity of Thomae Function at irrationals.

In Thomae's Function: $$ \begin{align} t(x) = \begin{cases} 0 & \text{if $x$ is irrational}\\ \frac{1}{n} & \text{if $x = \frac{m}{n}$ where $\gcd(m,n) = 1$} \end{cases} \end{align} $$ I ...
2
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1answer
49 views

For any open subset $A\subseteq \mathbb{R}$, $\operatorname{int}(\overline{A})=A$?

In the quiz of a class in MIT OCW, there is a T/F problem : For any open subset $A \subseteq \mathbb{R}$, $\operatorname{int}(\overline{A})=A$? The hompage of the class also provided a answer, and ...
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1answer
79 views

Is a closed n-dimensional disk compact necessarily compact?

As the title asks, is a closed n-dimensional disk compact necessarily compact? I'm thinking the answer would be no. If you consider the case in $\mathbb{R}^1$ then can you define the radius to be ...
3
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1answer
55 views

Evaluating limits by subsituting special sequences, justification for that

Sometimes I saw people using transformations like $$ \lim_{x\to 0} f(x) = \lim_{n\to \infty} f(\frac{1}{n}) $$ or $$ \lim_{x \to p} f(x) = \lim_{n \to \infty} f(x + \frac{1}{n}). \quad (*) $$ I know ...
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2answers
81 views

Taylor expansion of $\log(1+ix)$

How do I obtain the Taylor expansion of $\log(1+ix)$ around $x=0$? I know how to do it for $\log(1+z)$ if $z$ is a real number. But how do I do it (formally correct) in the case of the complex ...
0
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1answer
65 views

$a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
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2answers
285 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
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0answers
48 views

inequality of some integrals of continuously differential function.

Let $f:[a,b]$→$\mathbb{R}$ be a continuously differential fuction satisfying f(a)=0. My goal is to show that $$\int_{a}^{b} |f(x)|^2 dx \le \frac{(b-a)^2}{2} \int_{a}^{b} |f'(x)|^2 dx $$ My ...
3
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1answer
144 views

Solution to a tricky inequality (math analysis)

Let $p>1$ and put $q=\frac{p}{p-1}$, so $1/p+1/q=1$. Show that for any $x>0$ and $y>0$, we have $$ xy \le \frac{x^p}{p}+\frac{y^q}{q}$$ And find where the equality holds. So far, I have ...
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1answer
31 views

how to show that $A(x)\nabla u\in L_\mathrm{loc}^{2}(\Omega) $ for $u\in H_\mathrm{loc}^{1}(\Omega)$

Let $\Omega\subset \mathbb{R}^n$ be a connected open set containing $0$, $u\in H_\mathrm{loc}^{1}(\Omega)$, $A(x)\leq C|x|^{-1+\epsilon}$, where $\epsilon$ is small, and we also have $$ \|\nabla ...
0
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1answer
33 views

Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?

Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$ converge to $f(x)$? I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
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1answer
175 views

Integrals Regulated functions

stuck on an example for this question, Give an example of a regulated function $f \colon [a,b] \to \mathbb{R}$ with the properties that $\forall x \in [a,b] f(x) \ge 0 , \int_a^b f = 0$ and there is ...
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1answer
18 views

Using Taylor's series

Using Bayes Theorem I have solved a problem to the equation P(Dc|-) = (0.98-0.98p)/(0.98-0.93p) between the interval [0,0.1]. Show (e.g., by means of Taylor series) that in this interval the P(Dc|−) ...
2
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3answers
77 views

Minimum of set $\{\frac{m}{n} + \frac{4n}{m}\}$

We have the following set: $\mathcal{A} = \{ \frac{m}{n} + \frac{4n}{m};\ \ m, n \in \mathbb{N} \} $ Attempting to prove that the set's minimum is 4 yields: $$\frac{m}{n}+\frac{4n}{m} = \frac{m^2 + 4 ...
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1answer
21 views

Addition of distributions in statistics

Is it possible to add distributions? I've worked out "Say that you are given ten identical coins for which you assume Beta(4,4) prior distribution on the unknown probability θ of any of the coins ...
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0answers
100 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
2
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1answer
126 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
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2answers
159 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
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2answers
296 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
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2answers
86 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
2
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1answer
182 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
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2answers
58 views

Why is this set closed?

Let $(X,d)$ be a metric space. Let $a \in X$ and $r \ge 0$. Define: $E_r(a) = \{b \in X : d(a,b) \le r\}$ I want to show that $E_r(a)$ is closed. Here's what I know: $E_r(a)$ is closed if every ...
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2answers
624 views

Prove open set is not closed

The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY. Open set: A set $U$ is open if for every $a$ belonging to $U$, ...
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0answers
50 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
4
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2answers
94 views

$\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$

Let $f$ be a continuous real-valued function defined on an open subset $U$ of $\mathbb{R}^n$. Show that $\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$ I known $f(x)$ is open ...
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1answer
75 views

Natural Analysis: a possible new field?

I have studied real analysis for two years now, yet I often find that when applying it, (say to finding a path of shortest time) I often have to get solutions without closed forms (i.e. defined point ...
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1answer
195 views

Uniform closure of an algebra $\Rightarrow$ uniformly closed algebra

In PMA, Rudin's book, there is the following theorem (7.29): Let $B$ be the uniform closure of an algebra $A$ of bounded functions. (Here, an algebra means a family of function satisfying that it is ...
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1answer
63 views

Differentiability of scalar function

Let $f:\mathbb R→\mathbb R$ be a continuous function, with $f(0)=0$. Let $F(x,y)=xf(y)+yf(x)$. Analize if $F$ is differentiable at the origin.$$$$ I've proved that ...
2
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1answer
45 views

If $A$ is null set, then $\int\limits_A f dm = 0 $

Define $ \int_E f dm = \sup Y(E, f) $ where $ Y(E,f) = \{ \int_E \phi : 0 \leq \phi \leq f \} $ $\phi$ is simple Suppose $A$ is a null set. We show $Y(A, f) = \{ 0 \}$. Pick $x \in Y(A, f)$. So, we ...
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0answers
69 views

How prove this series $\sum_{n=1}^{\infty}a_{n}$ converges

Question: let $E$ is a point set on $(-\infty.+\infty)$,and let $x_{0}$ is a limit point of $E$(maybe $x_{0}=\pm \infty$ possible),if the series $\sum_{n=1}^{\infty}U_{n}(x)$ converges uniformly ...
2
votes
1answer
539 views

Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
1
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1answer
229 views

Show that $f$ is constant?!

Problem: Suppose $f$ is a non-vanishing continuous function on $\bar{\mathbb{D}}$(closure of unit disk) that is holomorphic in $\mathbb{D}$. Prove that if $$|f(z)|=1, \mbox{whenever} ...