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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2
votes
2answers
24 views

Relating real integrability and integration over the unit disk.

Suppose we have a function $f:\mathbb{R} \rightarrow [0,\infty)$, such that $\int_0^1f < \infty$. Is it always true that $\int_\mathbb{D} f(|z|)dA < \infty$ ? $dA$ is area measure. If so, ...
13
votes
4answers
315 views

Showing $\pi/(2\sqrt3)=1-1/5+1/7-1/11+1/13-1/17+1/19-\cdots$

I am struggling to show that $$\dfrac \pi{2\sqrt3}=1-\dfrac 15+\dfrac 17-\dfrac 1{11}+\dfrac 1{13}-\dfrac 1{17}+\dfrac 1{19}-\cdots$$ by using the Fourier series $$\frac \pi2-\frac x2=\sum_1^\infty \...
3
votes
0answers
23 views

Show $f$ is locally invertible if $f = L + g$ and $|g(x)| \le M|x|^2$

Let $f, g, L: \mathbb{R}^n \to \mathbb{R}^n$ and $L$ is a linear isomorphism, and let $|g(x)| \le M|x|^2$ on $\mathbb{R}^n$ for some $M > 0$. Prove that $f$ is locally invertible at $0$, i.e. $f$ ...
0
votes
1answer
49 views

About a proof that the sequence $ x_{n+1}=\frac{x_n+1}{x_n+2}$ is Cauchy.

I have an exercise problem, which explicitly sais: Show that the sequence $\left(x_n\right)_{n=0}^{\infty}$ which is defined recursively by $$ x_{n+1}=\frac{x_n+1}{x_n+2},\space x_0=1 $$ is Cauchy ...
1
vote
0answers
15 views

Show that coefficient of homogeneous polynomial are determined by their degree and the difference of their index.

I am given the homogeneous polynomial of degree $l$: \begin{align} u(x,y,z)=\sum_{a,b}c_{ab}(x+iy)^a(x-iy)^bz^{l-a-b} \end{align} Where $0 \leq a,b \leq l$. I have to show that given $l$ and $m=a-b$, ...
1
vote
1answer
81 views

Suppose that a sequence of continuous function $(f_n)$ converges pointwise to $f$ on $[0,1]$.

Suppose that a sequence of continuous function $(f_n)$ converges point wise to $f$ on $[0,1]$. Prove that if there exists a sequence $(x_n)$ in $[0,1]$ converging to $x^*$ in $[0,1]$ such that $(f_n(...
1
vote
1answer
112 views

On a trigonometric inequality

How to prove $$\sum\limits_{k = 1}^{\infty} {\frac{{\sin kx}}{{{k^a}}}} > 0,x \in \left( {0,\pi } \right),a \in \left( {0,\frac{1}{2}} \right].$$ I have tried derivative,but it seems no use!...
0
votes
0answers
23 views

Question about proving a theorem (intro to analysis)

I am supposed to show that $f^{-1}(G\cup H)=f^{-1}(G)\cup f^{-1}(H)$ which is done by showing that $\\f^{-1}(G\cup H)\subseteq f^{-1}(G)\cup f^{-1}(H)$ and $f^{-1}(G)\cup f^{-1}(H)\subseteq f^{-1}(G\...
2
votes
1answer
32 views

Boundary value problem $y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$

A physical problem brought me to the following boundary-value problem $y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$ with $y(0)=0$ and $y(C)=-58$ for some $C>0.$ If there was no ...
2
votes
1answer
38 views

If $\mathbb{Q}$ is dense in an (Archimedean) ordered field K, is K a complete ordered field?

Let K be an ordered field. Then K contains the smallest ordered filed $\mathbb{Q}$. If $\mathbb{Q}$ is dense and proper in K, is K a complete ordered field? If $\mathbb{Q}$ is dense and proper in K, ...
1
vote
1answer
38 views

Does there exist a countable proper subset $S$ of $\mathbb{Q}$ such that $\mathbb{Q} \setminus S$ is infinite but not dense in $\mathbb{R}$?

Does there exist a countable proper subset $S$ of $\mathbb{Q}$ such that $\mathbb{Q} \setminus S$ is infinite but not dense in $\mathbb{R}$?
4
votes
3answers
95 views

Question About Definition of Almost Everywhere

I suppose I'm a bit confused about the definition in the following regard: A property holds a.e. if it holds everywhere except for a set of measure $0$. Now, if the particular property is only ...
0
votes
0answers
58 views

Failure of the Vitali Covering Lemma for open coverings

Definition: Let $E\subset\mathbb{R}$. Let $\mathcal{F}$ be a collection of non-degenerate closed intervals. If for every $\epsilon>0$ and for every $x\in E$ there exists $I\in\mathcal{F}$ with $x\...
1
vote
1answer
27 views

if $U$ is compact, how show that $[U]_+$ is compact?

Let $U$ compact in $\mathbb{R}^n$, then $[a]_+=max\{0,a\},\;a\in\mathbb{R}$; $[x]_+=([x_1]_+,...,[x_n]_+),\;x\in\mathbb{R}^n$; $[U]_+=\{[x]_+:x\in U\}$ How show that $[U]_+$ is compact? Thanks!
5
votes
1answer
66 views

Solving a system of differential equations of order 3

We have the set of differential equations $$w_{1}+\frac{d^{2}}{dt^{2}}w_{1}-3w_{2}-\frac{d}{dt}w_{2}+\frac{d^{2}}{dt^{2}}w_{2}+\frac{d^{3}}{dt^{3}}w_{2}=0$$ $$w_{1}-\frac{d}{dt}w_{1}-w_{2}+\frac{d}{...
6
votes
2answers
155 views

How to prove $\frac{n^n}{3n!}<\frac{e^n}{2}-\sum_{k=0}^{n-1}\frac{n^k}{k!}<\frac{n^n}{2n!}$

I met this problem: prove: $\displaystyle \frac{n^n}{3n!}<\frac{e^n}{2}-\sum_{k=0}^{n-1}\frac{n^k}{k!}<\frac{n^n}{2n!}$ I tried expand $e^n$ at $x=0$ then: $\displaystyle e^n=\sum_{k=0}^{n}\...
0
votes
1answer
23 views

Convergence of sequences problem

I am trying to solve this problem .But I am unable to get a method to solve it. Below is the problem. If the sequence $ a_n$ satisfies the property $A =\displaystyle \lim_{n\to \infty} (a_n$ − $ ...
1
vote
2answers
207 views

strictly increasing functions and bijection

$f $ is strictly increasing \begin{align} & \rightarrow f^{-1} \text{is strictly increasing}\tag {i} \\ & \rightarrow f^{-1} \text{is strictly decreasing}\tag {ii} \\ & \rightarrow ...
0
votes
2answers
53 views

Limit of a continuous real-valued function on $(0,\infty)$ as $x\to\infty$.

Let $f:(0,\infty)\to\mathbb R$ be a continuous function. Suppose that $f$ satisfies $$\lim_{x\to \infty}\left( f(x+1)-f(x)\right) = 0.$$ Show that $\lim_{x\to \infty }\dfrac{f(x)}{x} = 0.$ I tried ...
1
vote
1answer
59 views

Show that $\{x \in \mathbb{Q}:x \geq 0, x^2 \leq 2\}$ has no rational least upper bound.

Lets denote the least upper bound by $\alpha \in \mathbb{Q}$ and $\delta > 0$ be a small number. Now $\alpha^2 \neq 2$ because there is no such rational $\alpha$. If $\alpha^2 > 2$ then $(\...
1
vote
2answers
68 views

Another counterexample about (uniform) continuity of function of many variables

Let $\Omega\subseteq\mathbb{R}^2$, $(x_0,y_0)\in\Omega$, $f:\Omega\to\mathbb{R}$. Then $f$ is $\epsilon-\delta$ continuous at $(x_0,y_0)\in\Omega$ if, for any $\epsilon>0$, there exists a $\delta&...
0
votes
1answer
48 views

Example to prove that $C_c(\mathbb{N})$ is not a Banach space for the uniform norm?

I know The space $(C_c(\mathbb{R}), \lVert\,\cdot\,\rVert_u)$ is not complete, so it is not a Banach space. But for $X = \mathbb{N}$, why is $C_c(X)$ not a Banach space? Can a sequence of functions ...
-1
votes
1answer
65 views

Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$. [duplicate]

Find a sequence of differentiable functions $\{f_n\}$ on $[0,1]$ such that $f_n \to 0$ uniformly, but $\{f_n'(1/2)\}$ does not converge to $0$. I have tried to find an example which follows above ...
3
votes
2answers
61 views

How to prove function $f(x,y)=\frac{1}{xy}$ is not uniformly continuous?

Here I consider uniform continuity of functions in $\mathbb{R}^n$. Take a function of two variables for example. We said that $f(x,y)$ is uniformly continuous if for any $\epsilon>0$, we can find ...
1
vote
1answer
46 views

Completion of the rational numbers

I am preparing to embark on the completion of $\mathbb{Q}$ with respect to a p-adic absolute. However, I thought I should start with the completion of $\mathbb{Q}$ with respect to the usual absolute ...
1
vote
2answers
48 views

A limit problem with a background

Could you show me $$\mathop {\lim }\limits_{x \to 0} \left( {1 + \sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{{\left( {\frac{{\sin nx}}{{nx}}} \right)}^2}} } \right) = \frac{1}{2}.\tag{1}$$...
1
vote
0answers
37 views

function and derivative converging to constants implies derivative is 0

I've been working on this problem from my textbook. Suppose that $f$ is differentiable on $(a,\infty$). Show that if $f'(x) \to L$ as $x \to \infty$, then $\frac{f(x)}{x} \to L$ as $x \to \infty$. ...
1
vote
2answers
21 views

Prove that if $2^{k_1}l_1=2^{k_2}l_2$, then $k_1=k_2$ and $l_1=l_2$.

Suppose $k_1,k_2,l_1,l_2$ are natural numbers such that $l_1$ and $l_1$ are odd. Prove that if $2^{k_1}l_1=2^{k_2}l_2$, then $k_1=k_2$ and $l_1=l_2$. First, let $l_1 = 2m_1 +1$ and $l_2=2m_2+1$ ...
0
votes
1answer
34 views

Convergence of sequence of one to one correspondence sequence.

I am trying to solve this problem but am struggling with it. The problem is as below. Let $ a_n$ be a convergent sequence with $A =\displaystyle \lim_{n\to \infty} a_n$. Suppose that $f: \mathbb{N}\...
1
vote
1answer
27 views

bernoullis inequality for $x \ge -2$ by induction

Prove the generalization of the bernoulli's inequality for $x \ge -2$. I am aware of the proof for $x \ge -1$ .But I am unaware of the way to solve from -2 as x takes a negative value. i am trying to ...
2
votes
1answer
64 views

If a subspace of $X^*$ is weak*-dense, does it separate points?

Here $X$ is some normed space. I know the converse is true, but I don't know a proof for the other direction. That is, if $F\subset X^*$ is a subspace that is weak*-dense how would one show that $F$...
2
votes
1answer
27 views

differentiable function in higher dimensions

$f:\mathbb R^{n}\to\mathbb R^{k}$ is a function with $\|f(x)\| \leq \|x\|^{2}$ , $x$ is an element of $\mathbb R^{n}$. Show that $f$ is differentiable at $0$. I already found that $f(0)=0$. I thought ...
1
vote
4answers
71 views

Closed set $F$ is the boundary of any subset of $\mathbb{R}^n$

I need show that any closed subset $F\subset\mathbb{R}^n$ is the boundary of some set $A$ in $\mathbb{R}^n$. Intuition tells me to take $A=F\setminus(\mathbb{Q}^n\cap int(F))$ and $int(F)$ is the set ...
2
votes
3answers
159 views

How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$?

How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$? Edit: The limit of this sequence is already investigated ...
1
vote
0answers
26 views

Does the following inequality hold? $\sum\limits_{i=1}^a\sum\limits_{j=1}^b\sum\limits_{k=1}^c|A_iB_jC_k|\leq 1$ as $||A||=||B||=||C||=1$

Given three vectors $A\in \mathbb{R}^a$, $B\in \mathbb{R}^b$ and $C\in \mathbb{R}^c$, with $||A||=||B||=||C||=1$, where $||\cdot||$ denotes the $l_2$ norm. Then does the following inequality ...
1
vote
2answers
42 views

Change of variable in the integral

Let $f:[0,b]\times[0,b]\rightarrow\mathbb{R}$ be continuous. Prove that $$\int_0^b dx\int_0^x f(x,y) \, dy=\int_0^b dy\int_y^b f(x,y) \, dx$$ La idea es usar los teoremas: \ Regla de Leibniz, de ...
1
vote
1answer
49 views

If any bounded sequence in an ordered field K has a subsequential limit in K, is K a complete ordered field?

Let K be an ordered field. If any bounded sequence in K has a subsequential limit in K, is K a complete ordered field? (i.e. satisfying any one of the equivalent definitions of a complete ordered ...
0
votes
2answers
32 views

Proof a vector only contains -1, 0 and 1 under condition $\sum\limits_{i=1}^nv_i^2=\sum\limits_{i=1}^n|v_i|$

For a vector $v\in R^n$, if $\sum\limits_{i=1}^nv_i^2=\sum\limits_{i=1}^n|v_i|$, then can we say the elements in $v$, $v_i\in\{-1,0,1\}$ for $i=1,\cdots,n$. Thank you very much!
0
votes
0answers
65 views

Using Generalized Gauss-Laguerre Quadrature

Motivated by the paper "Analytically Pricing European-Style Options Under the Modified Black-Scholes Equation with a Spatial-Fractioonal Derivative", I am trying to use the Generalized Gauss-Laguerre ...
7
votes
1answer
356 views

Sequence converging to different limits

In a metric space, is it possible to find a sequence which converges to two different limits wrt to two different metrics? Obviously the metrics can't be equivalent.
0
votes
1answer
25 views

integral curve for $X(x,y)=(x^2+y^2)\frac{\partial}{\partial x}+(xy)\frac{\partial}{\partial y}$.

Find the integral curve for, $X(x,y)=(x^2+y^2)\frac{\partial}{\partial x}+(xy)\frac{\partial}{\partial y}$. You can use the following results (if needed) If $(x(t),y(t))$ is the integral curve $\...
1
vote
1answer
32 views

Finding smallest closed affine subspace

Given the space $H=\mathbb{L}^2(0,1)$, and the subspace $K$ of $H$ defined by: $$K=\left\{f\in H: f\geq 0\right\},$$ I would like to determine the smallest closed affine subspace of $H$ which contains ...
0
votes
2answers
116 views

Take the derivative of the norm? can I do this?

Here is a quick question to the math community. Is it possible to take the derivative of the euclidean norm? For example, if $f(x)=\|x\|$ then $f'(x)= \|1\|=1$ and $f''(x)=0.$ I know it is ...
-3
votes
4answers
50 views

Closed, open and non-empty subset of $\Bbb R^n$ equals $\Bbb R^n$ [closed]

$A$ is a subset of $\mathbb{R}^n$ closed , open and non-empty . Prove that $A = \mathbb{R}^n$.
0
votes
2answers
39 views

Compute the double integral $\int_0^\infty \int _{x/\sqrt{4kt}}^\infty y^2 e^{-y^2} dy\, dx$

I have to compute: $$\int_0^\infty \int _\frac{x}{\sqrt{4kt}} ^\infty y^2 e^{-y^2} dy\, dx, $$ where $k$ and $t$ are two constants. I want to use the fact that $\int_0 ^\infty y^2 e^{-y^2} dy dx =\...
2
votes
2answers
61 views

Closed, convex curves with minimized lengths

Let $\gamma$ be closed, smooth, strictly convex curve. It is known that we can find a special linear transformation $A\in SL(2)$ so that the length of $A\gamma$ is minimized. I am interested in ...
-3
votes
2answers
83 views

Isn't the Continuity Concept Redundant (up to definitions)? [closed]

The idea of continuity in (real) analysis—and indeed everywhere else it is used: topology, etc.—seems to me superfluous. For if, as I learnt from Stewart (Concepts of Modern Mathematics), it is based ...
2
votes
1answer
80 views

Find if $f(x) = \begin{cases} -x & x<0\\ 2 & x \geq 0 \end{cases} $ is Lebesgue integrable

I'm trying to determine if the function $$f(x) = \begin{cases} -x & x<0\\ 2 & x \geq 0 \end{cases} $$ is Lebesgue integrable with regards to the Lebesgue measure. My gut tells me that ...
0
votes
0answers
20 views

Tangent curve of a vector field

Given vector field $X=f(x,y)\frac{\partial}{\partial x}+g(x,y)\frac{\partial}{\partial y} $. We have $\frac{f}{g}(x,y)$ constant along the lines through origin. Hence $\frac{f}{g}(x,y)=k(y/x)$. How ...
0
votes
2answers
57 views

Connectedness of a hyperbola to the x-axis

Given my definition of connected as: M is connected if it contains no proper clopen subsets. And the set H as {(x,y) : xy = 1 and x,y>0 }, with the set X representing the x-axis, is the set S = X U H ...