All questions with tag [math: analysis]


Problem connecting Topology and Algebra via Analysis

Let $C(X):=$ Set of all complex/real valued continuous functions. If $X$ is compact then all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$. Is it true that: If all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$ then $X$ is compact.
2022-07-25 17:47:10

convergence of power series

We have given a power series $\sum\limits_{n=0}^\infty a_nx^n$ and $a_0\ne0$. Suppose the power series has radius of convergence $R>0$. Then we may assume that $a_0=1.$ Explanation: It's $$\sum\limits_{n=0}^\infty a_nx^n=a_0\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$$ But why do $\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$ and $\sum\...
2022-07-25 17:46:47

If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$

For $f:\mathbb{R}^n \to \mathbb{R}^m$, if $f(x + y) = f(x) + f(y)$ for then for rational $c$, how would you show that $f(cx) = cf(x)$ holds? I tried that for $c = \frac{a}{b}$, $a,b \in \mathbb{Z}$ clearly $$ f\left(\frac{a}{b}x\right) = f\left(\frac{x}{b}+\dots+\frac{x}{b}\right) = af\left(\frac{x}{b}\right) $$ but I can't seem to finish it, a...
2022-07-25 17:46:29

Why Jordan measure is undefined?

$R^2$, $A=\{(x,\;y)\in R^2\colon 0\leqslant x\leqslant 1,\;0\leqslant y\leqslant 1\}$. Take into consideration $X=A\cap Q^2$. Why for $X$, $m_e X=1,\;m_i X=0,\;m_e X\neq m_i X$? Specifically i curious about why internal action equals to $0$.
2022-07-25 17:44:35

Exercise: continuity of a function of two variables

Consider a continuous function $\phi: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}_{\geq 0}$ and a locally bounded function $\psi:\mathbb{R}^n \rightarrow \mathbb{R}^m$. So we study functions of the kind $\phi(x,\psi(y))$ for $x,y \in \mathbb{R}^n$. Prove the following proposition. Fixed $\bar x \in \mathbb{R}^n$, for any $\epsilon&am...
2022-07-25 17:20:44

A metric space problem

Possible Duplicate: ยข Let $(X,d)$ be a statistics room and also $A,B$ be 2 distinctive shut embeded in $X$ such that $dist(A,B)=0$. Does it indicate $A\cap B=\emptyset$?
2022-07-25 17:19:52

Convex function with linear grow?

I'm seeking a continual, purely raising, purely convex function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $f(0)=0$, and also such that $$ \lim_{x \rightarrow\infty} \frac{f(x)}{x} \leq c $$ for some $c \in \mathbb{R}_\geq 0$. Pointers?
2022-07-25 17:17:10

power series estimate (convergence)

Let $f(x)=\sum\limits_{n=0}^\infty a_nx^n$ a power series and $f(0)\ne0$. (w.l.o.g. $f(0)=1$) Suppose the power series has radius of convergence $r>0$. A power series is continuous in her convergence interval. So there is a $\delta\in]0,r[$ so that for $|x|<\delta$ it's $|a_1x+a_2x^2+\dots|<1$. My Questions: why is $|a_1x+a...
2022-07-25 17:17:03

To prove $f(x)\rightarrow \infty$ with a "home made" strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ for all $ x\rightarrow 1^+$ In order to show this statement I show that no matter how big you choose $N\in \mathbb{R}$, you can always find a $\delta >0$ so $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} = 1+\frac{1}{2^{x}}+\cdo...
2022-07-25 17:16:04

Non-finite series implies product is zero

Given $0 \le y_n \le 1$ and $\sum_{n \in \mathbb{N}} y_n = \infty$, how can we show $\prod_{n=1}^\infty (1 - y_n) = 0$?
2022-07-25 17:07:58

To prove $f(x)\to\infty$ with an "Oresme" strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ when $ x\rightarrow 1^+$ My first approach (which failed) is here: To prove $f(x)\rightarrow \infty$ with a "home made" strategy I think the confusing part is the "$x\rightarrow 1^+$". I have now argued for the statement, b...
2022-07-25 17:06:41

$f(x) \rightarrow \infty $ when $x\rightarrow 1^+$

I want to prove that $f(x) \rightarrow \infty $ when $x\rightarrow 1^+$. My tactic is to prove that no matter how big you choose a $N\in \mathbb{R}$, you can always find a $\delta>0$ so the following statement is true: $ f(x) > N $ for all $x\in \left] 1, 1+\delta \right[$ Is my technique correct?
2022-07-25 17:00:35

Cauchy sequence alternative definition.

Is it possible to define a Cauchy sequence as follows? Let $(X,d)$ be a metric space and $(x_{n})_{n\in \mathbb{N}}$ be a sequence in it. Then $(x_{n})_{n\in \mathbb{N}}$ is Cauchy iff $\lim_{(j,k)\to (\infty,\infty) }d(x_{j},x_{k})=0$. Thanks. Note: By $\lim_{(j,k)\to (\infty,\infty) }d(x_{j},x_{k})=0$, I mean the standard definition of the lim...
2022-07-25 16:41:02

Continuity of Parametric Integral

Consider a continuous function $f: X \times Y \rightarrow \mathbb{R}_{} \geq 0$, where $X \subset \mathbb{R}^n$ is compact, and $Y \subseteq \mathbb{R}^m$ is closed. Define $\hat{f}:X \rightarrow \mathbb{R}_{\geq 0}$ as the parametric integral $$ F(x) \ := \ \int_Y f(x,y) dy $$ Assume that $X$ is such that $\sup_{x \in X} F(x) < \infty ...
2022-07-25 13:31:01

Redefining Outer Lesbegue Measure on $\Bbb{R}^{d}$ From Closed Cubes to Rectangles.

UPDATE: I added an answer based off the hints provided by copper.hat. It may, however, need some adjustment. I'm trying to solve another question from Stein and Shakarchi's analysis text. Basically, I'm trying to prove that $m_{\star}(E)=m_{\star}^{R}(E)$ for every $E\subset\mathbb{R}^{d}$ where $m_{\star}$ is the exterior measure taken with cl...
2022-07-25 13:28:51

Smoothness of Fourier series

In a book from differential equations I found the following theorem, without proof and references: Let functions $f, g: R \rightarrow R$ be continuous and $2\pi$-periodic and let $m\in N$. Assume that $$\frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos nx+b_n \sin nx),$$ $$\frac{A_0}{2}+\sum_{n=1}^\infty (A_n \cos nx+B_n \sin nx)$$ be Fourier series of ...
2022-07-25 13:28:33

About the remainder of Taylor expansion and Riemann-Liouville integral

Integral kind of Taylor development resembles this: $$f(x)=\sum_{i=0}^k\frac{f^{(i)}(a)}{i!}(x-a)^i+\int_a^x\frac{f^{(k+1)}(t)}{k!}(x-t)^kdt$$ Riemann - Liouville indispensable is $$I^{\alpha}f=\frac{1}{\Gamma(\alpha)}\int_a^x{f(t)(x-t)^{(\alpha-1)}}dt$$ Q1: The indispensable kind rest of Taylor development is specifically $I^{k+1}f^{(k+1)}$. Wh...
2022-07-25 13:27:21

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. $\mathcal{L}_X(s)=\int_0^\infty e^{-s t} d F(t)$ Can one conclude immediately that, as $s \to 0$, $\log \mathcal{L}_X(s) \approx -s \mathbb E X + o(s^2) $ ? If n...
2022-07-25 13:22:10

properties of a real analytic function

If there are a distance $r>0$ and also constants $M,C\in\mathbb R$ for all $y\in U$ with $$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb N_0^n$$ after that $f\in C^\infty(U)$ is actual analaytic. Yet I do not have any kind of suggestion just how to confirm this. I feel in one's...
2022-07-25 13:19:23

inequality in a differential equation

Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that pleases $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean standard. According to Temam is publication paragraph 2.2 on web page 32 number (2.10), inequality (1) indicates $$|u(t)|^2\le|u(0)|^2\exp(-t)+1-\exp(-t),\tag{2}$$but I do n...
2022-07-25 13:16:00