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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2answers
135 views

Implicit function theorem problem

I have the function $$(x-2)^3y+xe^{y-1}=0$$ And I have to see if $y$ can be described as a function of $x$ around (1,1). The implicit function theorem can't be applied in this case. What should I ...
3
votes
1answer
96 views

Smooth function composed with sobolev function vanishes at 0

Let $\Omega$ be a bounded domain with sufficiently smooth boundary. Let $u \in W^{1, 2}_{0}(\Omega)$ and $F \in C^{\infty}(\mathbb{R} \rightarrow \mathbb{R})$ such that $F(u(x)) = 0$ for almost every ...
0
votes
1answer
28 views

Semi-Inner Product question

I have a semi-inner product space question, that is to prove that $u(0,y)=0$ from this property $u(ax+by,z)=au(x,z)+bu(y,z)$ and we assume that $a=0$. Also, note that $x,y,$ and $z$ are vectors and ...
5
votes
3answers
239 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
0
votes
1answer
170 views

Show that function is partially differentiable

I have the following function: $$F: \mathbb{R}^2 \rightarrow \mathbb{R}, ~~ (x,y) \rightarrow xy\frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \ne 0$ and $F(0,0) = 0$. I want to show that $F$ is partially ...
5
votes
1answer
600 views

Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function

So I'm working a problem that states: A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the ...
1
vote
2answers
53 views

Prove that for $c \geq 1 \lim_{n \rightarrow \infty} \sqrt[n]{c} = 1$ using Bernoulli's inequality

My approach was: Since $c \geq 1 \Rightarrow \sqrt[n]{c} \geq 1 \Rightarrow \sqrt[n]{c}-1 \geq 1$. Therefore I need to find an $N\in \mathbb{N}$ such that $\forall n \geq N: |\sqrt[n]{c}-1|= ...
1
vote
0answers
42 views

CW structure on $P^n \mathbb{R}$(projective space)

I am supposed to show that $P^n \mathbb{R} = e_0 \cup\cdots\cup e_n$, where $e_i$ is an $i$-dimensional cell. I also know that there is a quotient map $S^n \rightarrow S^n/\tilde - = P^{n-1} ...
2
votes
1answer
77 views

Why $(X,d)$ is a complete $\mathbb{R}$-tree?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = ...
1
vote
1answer
22 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
39 views

Operator compact on $H^1 (0,\pi)$

Consider the operator $K\colon H^1(0,\pi)\to H^1(0,\pi)$ defined by duality (Riesz. Theorem) as $$ \langle K\phi,\psi\rangle = \int_{0}^{\pi}{\phi(x)\psi(x)\,dx} $$ for all $\psi \in H^1(0,\pi)$, ...
2
votes
1answer
87 views

Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$?

I want to decide whether or not $f_n(x)=nx^n(1-x)$ is uniformly convergent or not. I have shown that $\lim_{n\to\infty} f_n(x)=0$ for $x \in [0,1]$. Now $f_n(0)=f_n(1)=0$. And in $(0,1)$, we have ...
2
votes
1answer
120 views

What is a CW complex

In a lecture, I have written down the following definition for CW complexes. $X= \bigcup_i \{e_i\}$ and the $\{e_i\}$ form a partition. Furthermore $e_i$ is homeomorphic to $B(x,1)\subset ...
2
votes
1answer
52 views

How to embed a total ordering into the real field.

Let $(S,<_S)$ be a total ordering with $card(S)\leq card(2^{\aleph_0})$. Does there exist a subset $A$ of the real numbers such that $(A,<_A)$, being a total ordering, is isomorphic to ...
0
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1answer
44 views

For which value of $b$ the function $f(x)=b \sin(x)-x$ is one-one?

I draw the graphs and the correct answer is $[0,1]$ the domain of $b$ for which $f$ is one-one. But i don't know how to prove this. Please somebody help me.
1
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3answers
45 views

Question about sequences

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and suppose there exists $K \in \mathbb{R}$ with $0 \leq K <1$ such that for all $x,y \in \mathbb{R}$ with $x \neq y$. $$|f(x)-f(y)|<K|x-y|$$ ...
12
votes
1answer
145 views

When is $\mathbb{Z}[\alpha]$ dense in $\mathbb{C}$?

Let $\alpha$ be a nonreal algebraic number. I'm interested in conditions that imply that $\mathbb{Z}[\alpha]$ is dense in $\mathbb{C}$. I'm particularly interested algebraic integer $\alpha$. This is ...
0
votes
1answer
18 views

Existence of a local mininum

Let $F: R^{2} \rightarrow R$ be a differentiable function such that $\lim_{|v| \to \infty}f(v) = +\infty$. Then F has a local minimum. Hint: Think of the Weierstrass Theorem. I'm trying to solve it, ...
0
votes
1answer
109 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
1
vote
1answer
41 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
2
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3answers
150 views

Does $f(x)\,dx$ denote multiplication of $f(x)$ by $dx$? [duplicate]

In the integral form $\int \! f(x) \, \mathrm{d}x$ does $f(x)\,\mathrm{d}x$ can be seen as a multiplication of $f(x)$ and $\mathrm{d}x$?
0
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1answer
39 views

The Existence of $f \in C_{c}^{\infty}$ with $\sup_{x} |\partial^{\alpha}f(x)|<M$where $M$ is independent of $\alpha$

I hope to know the existence of the function $f \in C_{c}^{\infty}$ such that $$\sup_{x} |\partial^{\alpha}f(x)|<M$$ where $M$ is independent of $\alpha$. If that function exists, I hope to know ...
2
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0answers
39 views

How to prove this inequation

Suppose functions $$f(t),g(t),h(t)$$are defined on$(0,+\infty)$,satisfying: (1)$f(t),g(t),h(t)$ are positive and real-valued; (2)$g(t)\le f(t)+\int_0^tg(s)h(s)ds,t\ge0$; (3)$f'(t)\ge0$; ...
2
votes
1answer
37 views

outer measure for definition question

I'm reading Pugh's "Real Mathematical Analysis" and just started the chapter on Lebesgue Theory. I guess you'd have to a copy available to answer this question, but on page 374, he defines the outer ...
5
votes
1answer
122 views

An inequality of $L^p$ norms of linear combinations of characteristic functions of balls

Let $1<p<\infty$. Let $(a_n)_{n=1}^\infty$ be a sequence of nonnegative real numbers and $\{B_{r_i}(x_i)\}_{i=1}^\infty$ be a sequence of open balls in $\mathbb{R}^n$. Prove that there exists ...
5
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2answers
165 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
0
votes
1answer
39 views

Do we have such kind of estimates?

Let $0<a_0\leq a(x)$ be a smooth function on $\mathbb{T=[0,2\pi]}$, and $a(0)=a(2\pi)$, then whether it holds that $$ \int_{\mathbb{T}}a(x)|\partial_x\phi|^2 dx\geq ...
6
votes
3answers
170 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
2
votes
2answers
136 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
0
votes
2answers
73 views

How to prove $\theta$ which $\in(0,1)$ is unique & $\theta$ is related with $x$ &$\lim_{x\rightarrow+\infty}\theta=1$ [closed]

Show that for any $x\gt 0$ there is a $θ=θ(x)\in(0,1)$ satisfying $$ \int_{0}^{x}e^{t^2}\,dt=xe^{\theta x^2} $$ and asymptotically, the parameter $θ$ satisfies $\lim_{x\to+\infty}θ(x)=1$. The ...
0
votes
2answers
34 views

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
3
votes
4answers
82 views

To prove:$f(x)$ is continuous but not uniform continuous on $(0,+\infty)$.

where$$f(x)=\sum_{n=1}^{\infty}ne^{-nx}$$ Is there any good ways to deal with this kind of functions?
1
vote
2answers
263 views

if $f_{n}(x)=f(f_{n-1}(x))$then $f_{10}(x)=x,x\in [0,1]$

Define the function $f:[0,1]\to[0,1]$ by the following. $$f(x)=\begin{cases} x+\dfrac{1}{2},&0\le x\le\dfrac{1}{2}\\ 2(1-x),&\dfrac{1}{2}<x\le 1. \end{cases}$$ Let $f_1(x)=f(x)$ and ...
0
votes
2answers
33 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
2
votes
0answers
69 views

Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

Does anybody know whether this limit here is zero, if we assume that $np_n \to \lambda$ for $n \to \infty$? $$ \mbox{So, do we have}\quad \lim_{n \to \infty}\sum_{k=0}^{n}{1 \over k!}\, ...
1
vote
2answers
51 views

To calculate the limit :

$$\lim_{n\rightarrow\infty}{n^2}(\arctan\frac{a}{n}-\arctan\frac{a}{n+1})$$ I used the formula $\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$,but it just doesn't work. Waiting for your help...
0
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2answers
44 views

Does this sequence converge to zero?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$, such that $\forall k \in \mathbb{N}: \lim_{n \rightarrow \infty} f(n,k) = 0$. Is it then true, that $\sum_{k=0}^n \frac{f(n,k)}{k!}$ converges to zero ...
1
vote
1answer
97 views

How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$

show that $$I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$$ follow is my methods: let $$x^2=t$$ then ...
1
vote
1answer
80 views

Show that topologies are the same

I just read a proof where it was said that if for each element in the topology 2 we find an element in topology 1 that is contained in this set and vice versa, then they are the same. How do I see ...
4
votes
1answer
52 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
1
vote
1answer
55 views

a question about $L^p$ functions on domains in Euclidean spaces

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $f\in L^p(\Omega)$, $1\leq p<\infty$. Define $||f||_{p,\Omega}=\inf\{||f-a||_p: a\in\mathbb{R}\}$. Prove that there exists $a\in\mathbb{R}$ such ...
1
vote
2answers
45 views

Analysis on using Unconventional underlying fields

I'm curious if people study analysis while using fields that are not $\mathbb{R}$. I remember seeing a post about doing analysis on $\mathbb{Q}$, but $\mathbb{Q}$ is not complete! Mostly I'm ...
4
votes
0answers
38 views

Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
3
votes
1answer
87 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
0
votes
1answer
63 views

Definition of $\limsup$

please tel me what is the definition of $$\limsup_{|u|\rightarrow\infty}\frac{2F(t,u)}{|u|^2}<\lambda$$ using $\varepsilon$ Please Thank you
1
vote
1answer
41 views

Why is this the possible Taylor series???

I am looking at an exercise at which it is asked to find the Taylor series of $f(x)=\log(1+x)$, $\xi=0, x \in (-1,1)$ $$f'(x)=(1+x)^{-1}$$ $$f''(x)=-1 \cdot (1+x)^{-2}$$ $$f'''(x)=2 \cdot ...
1
vote
2answers
60 views

Convergence of an infinite Riemann sum to an integral

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be smooth, bounded, uniformly continuous, and $|f(x)| \leq 1/|x|^{N}$ for any $N$. Then is it true that $$\frac{1}{n}\sum_{k = -\infty}^{\infty}f(k/n) ...
1
vote
2answers
40 views

How to prove a limit?

I saw at the solution of an exercise that $$\frac{|x|^{n+1}}{(n+1)!} \to 0, \text{ when n } \to +\infty$$ But,how can I show that it is actually like that?
3
votes
0answers
186 views

Prove that $\prod\limits_i(1+2\alpha_{i})\prod\limits_j(1-2\beta_{j})<\prod\limits_i(1+2x_{i})\prod\limits_j(1-2y_{j})$

Let $m,n\in N^{+}$ and $i=1,2,\ldots,n,\;j=1,2,\ldots,m\,$ and $\,x_{i},\alpha_{i},y_{j},\beta_{j}$ be real numbers such that $$0\le x_{i}<\alpha_{i}<\dfrac{1}{2},\qquad0\le ...
1
vote
1answer
44 views

Find the rule of a sequence

I have a sequence $\{x(n), n=0,1,2,\ldots\}$ as follows: $x(0) = 1$ $x(1) = 1- e^{-a}$ $x(2) = \dfrac 12(1 - 4e^{-a} + 3e^{-2a})$ $x(3) = \dfrac{1}{6}(1-12e^{-a}+27e^{-2a}-16e^{-3a}) $ $x(4) = ...