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
52 views

$\sigma$-field and family of sets

Let $f:[0,1] \to [0,1]$ be the function $f(x)=-4x^{2} + 4x$ let $\mathcal M$ be any $\sigma$-field in $[0,1]$. Is the family of sets $\mathcal N= \{f(A): A \in \mathcal M \} $ a $\sigma$-field in ...
3
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4answers
89 views

Limit involving $(\cos x)^{1/x^4}$

I am having trouble calculating the following limit. $$\lim_{x \to 0}(\cos x)^{1/x^4}$$ In Problems in mathematical analysis by Demidovich there is a hint that in case of $1^{\infty}$ indeterminate ...
3
votes
1answer
82 views

How to find the inverse function?

For the function $f:\mathbb{N}\to\mathbb{N}$ the descrete derivative for $f$ in $n\in \mathbb{N}$ is defined as follows: $$f'(n) := f(n+1)-f(n)$$ Find the chain rule for the descrete derivative. ...
1
vote
2answers
325 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
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3answers
76 views

How to show derivatives rules?

Hey guys could someone provide me a nice solution for the rule questions...I just dont get them at all....I was able to answer the first question...but then its just abstract for me... For a total ...
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2answers
46 views

Concave or Convex of some functions?

I would like to know whether these two functions are concave or convex: 1.the first is $$ f: \mathbb{R} \to \mathbb{R} : x \mapsto (x-1)^3$$ Here I would say that the function is ist concave for $x ...
2
votes
2answers
50 views

Extending a holomorphic function

Let $D \subset \mathbb{C}$ be an open disc. Is there a function $$ f \in {\mathcal H}(\mathop D ) \cap C(\overline{D}), $$ such that, $\,\,f \notin {\mathcal H}(V)$, for every open set $V \supset ...
0
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1answer
35 views

$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$. Any hint ?
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3answers
67 views

$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$ for $f \in H(\Omega)$

Let $\Omega $ be a domain ,$\overline{D(z,r)} \subset \Omega $, $f$ holomorphic in $\Omega$. a) Show that $$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$$ where $dm$ ...
0
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1answer
120 views

How to show via induction the product rule for derivatives?

Hi I have to show by induction over the number $n$ the following product rule and I have no idea about it. Could someone provide me the proof and a nice explaination: $$(f\cdot g)^{(n)} (x)= ...
2
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2answers
55 views

I need help computing the following limit using only squeeze theorem and basic limit properties

I was able to compute the limit of the following using l'Hopital's rule, and found it to be $\frac{\pi}{4}$, but apparently there is a way to evaluate the limit using the squeeze theorem, apparently ...
4
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0answers
791 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
7
votes
1answer
388 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
3
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2answers
72 views

'uniform approximation' of real in $[0,1]$

Good evening, Prove that: For every $\varepsilon>0$, there exist an $n\in \mathbb{N}$, such that for every $x\in[0,1]$, there exist $(p,q)\in \mathbb{N^2}$, with $0\leq p\leq q\leq n$ and ...
1
vote
1answer
67 views

Polynomials with rational coefficients and maps

Problem: Let $\mathcal P $ bet the set off all polynomials in one variable with rational coefficients. For $P\in\mathcal P$ define $I(P)(x) = \int_0^xP(t)\,dt$. Show that the map $I:\mathcal ...
1
vote
2answers
40 views

understanding simple functions

Let $(X,\mathcal{M})$ be a measurable space. The definition of a simple function on a set $X$ is that it is a finite linear combination, with real coefficients, of characteristic functions of sets in ...
0
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1answer
57 views

can anybody help me with my theory homework please?

let $f(x)=\sqrt x$ for $x \ge 0$ a)find a positive $\delta$ s.t. for all $x$ in the interval $[0,\infty)$ with $0 < |x-4| < \delta$ we have $|f(x)-2| < 1$ b) find a positive $\delta$ s.t. ...
1
vote
1answer
87 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
2
votes
2answers
158 views

If an IVP does not enjoy uniqueness, then there are infinitely many solutions.

I am trying to prove than when an IVP has more than one solutions, then there exist infinitely many different solutions. I know that when the Lipschitz condition holds, there is at most one solution ...
2
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1answer
125 views

Hausdorff spaces

Let $X$, $Y$ be topological spaces. Assume that $X$ is a Hausdorff space, $D\subset X$ dense in $X$ and $f:X\to Y$ a continuous function. If $f$ when restricted to $D$ is a homeomorphism between $D$ ...
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2answers
82 views

Zeros of $e^{z}-z$, Stein-Shakarchi

This is an exercise form Stein-Shakarchi. Prove that $$f(z) = e^{z}-z$$ has infinite many zeros. What I have done : if not, by Hadamard theorem we obtain $$e^{z}-z = ...
2
votes
1answer
204 views

Biholomorphism between 2 non simply connected domains [duplicate]

Is the annulus of centre $0$ and radii $1$ and $2$ biholomorphic to the punctured disc $$\{ z \in \mathbb{C} \ | \ 0<|z| <1\}$$ ? Why ? I know the Riemann mapping theorem but here we have non ...
4
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1answer
211 views

Solve the differential equation $y'=\mathrm{e}^{-y^2}-1$

Consider the initial value problem $$ \frac{dy}{dx} = \mathrm{e}^{-y^2} - 1,\quad y(0)=0. $$ Method of Separation of Variables provides: $$ \int \frac{dy}{(e^{-y^2} - 1)} = x+c. $$ I would be ...
0
votes
1answer
92 views

$\epsilon-dense$

Let $B$ be the unit disk in the plane and $A\subset B$. I need to prove: If $A$ is not $\epsilon$-dense in $B$ then $B\setminus A$ contains a smaller disk of radius $\epsilon$?
3
votes
0answers
29 views

Order of an entire $f $ is $\limsup_{r \rightarrow + \infty} \frac{\log \log M(r)}{\log r}$ [duplicate]

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Write $M(r) = ...
2
votes
2answers
3k views

Proof of the Usual Topology on R

I'm trying to prove that the usual topology on R is in fact a topology for R. I've read through a few books on topology and most of them seem to skip over the proof and list it as "trivial". My math ...
0
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2answers
63 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...
2
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1answer
78 views

Criteria for positive semi-definiteness - zero diagonal

I am currently doing a bit of background reading on some fundamental topics in preparation for a talk, and came across a question relating to positive definiteness. It is taken from Horn and Johnson's ...
3
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1answer
114 views

Why the image of a linear map is not always a Banach space?

I have a question: Let's think about the map $T:V \rightarrow \text{ran}(T)$ and $V$ be a Banach space. Then we have that this is the same as the quotient map $[T]:V \rightarrow V/\ker(T)$ where the ...
1
vote
1answer
42 views

Laplace transform and majorant

I am looking for a majorant such that for every $t>0$ we have that for all $x>0: |x^ne^{-xt}|\le F(x)$ such that $\int_0^\infty F(x) dx < \infty$? I guess this one does not exist, but the ...
3
votes
2answers
192 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
4
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1answer
46 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
7
votes
1answer
190 views

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
1
vote
1answer
52 views

Problem in walter rudin RAC

The function $f(x)=x^2\sin(\frac{1}{x})$ if $x\neq 0$ , $f(0)=0$.Then $f$ is differentiable at every point, but $\int_0^1|f^{'}(x)|dx=\infty.$ I proved $f$ is differentiable at every point. To prove ...
0
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2answers
73 views

Exercise on differentiable function

$f\colon \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function and $m \in \mathbb{R}$ for which $f'(x)\ge m$ for every $x \in \mathbb{R}$. Let $Z= \{x\in \mathbb{R} \mid e^{\sin(x)} = f(x)\}$ ...
1
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1answer
751 views

Suprema proof: Prove $\sup(f+g)\leq \sup f+\sup g$ [duplicate]

I need help with these: 1) Show the following inequality for the supremum of functions $f:\mathbb{R}\to \mathbb{R}$ and $g: \mathbb{R}\to\mathbb{R}$ $$ \sup(f+g)(x)\leq \sup f(x)+\sup g(x) $$ 2) ...
3
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0answers
118 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
1
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1answer
27 views

Ratio of maximal to minimal jump in the set of angle multiples

Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times S^1\to\mathbb{R}$ be the distance function given by the arc length. Let $\theta\in S^1$ be an element of infinite order, that is ...
0
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1answer
93 views

A version of Rellich-Kondrachov's theorem

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$ , $k$ be positive integer, and $p \in [1,\infty)$ such that $kp < n$. Let $q\in[1,\dfrac{np}{n-kp}) $ and put $T(u) = u$ ...
0
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1answer
39 views

$f\in C^2(\mathbb{R}^n)$ with compact support, equality of derivates

Let $f\in C^2_c(\mathbb{R}^n)$ i.e. 2 times continuously differentiable with compact support. Show that $\int_{\mathbb{R}^n}\sum_{j,k=1}^n|\dfrac{\partial^2f}{\partial x_j\partial ...
0
votes
1answer
39 views

Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Let $D$ be an open subset of $\mathbb{R}^n$ , $p$ and $q$ be in $(1,\infty)$ such that $p^ {-1} +q^ {-1} = 1$. Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$ ...
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0answers
67 views

$\prod_{n}f_{n}$ converges uniformly $\Rightarrow $ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
1
vote
1answer
41 views

compute out an equation of complex integral of polynomial

Let $f(z)=\sum_{k=0}^na_kz^k$ be a polynomial with coefficients in $\mathbb{C}$. Suppose $\deg f\geq 1$. Prove for any $R>0$, \begin{equation*} \frac{1}{2\pi ...
1
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1answer
22 views

Help with inequality estimate, in $H^1$,

Given a bilinear form on $H^1 \times H^1$, where $H^1 = W^{1,2}$ \begin{align*} B[u,v] = \int_U \sum_{i,j}a^{i,j}(x)u_{x_i}v_{x_j} + \sum_ib^i(x)u_{x_i}v + c(x)uv \, \mathrm{d}x \end{align*} the book ...
1
vote
1answer
56 views

$\frac{f'(z)}{f(z)}= \sum_{n=1}^{+ \infty}\frac{f'_{n}(z)}{f_{n}(z)}$

I 've found this exercise. Let $\{f_{n}\}$ be a sequence of holomorphic functions on a given domain $\Omega$. Suppose that $\prod_{1}^{\infty}f_{n}$ converges uniformly on compact subsets of $\Omega$ ...
1
vote
1answer
63 views

Lagrange's Theorem exercise

Be $u : \mathbb{R} \rightarrow \mathbb{R}$ a $C^2$ function. Proof that exists a $x \in \mathbb{R}$ with $0<x<2$ for which $u(2)-2u(1)+u(0)=u''(x)$ Applying Lagrange's Theorem I showed: ...
6
votes
1answer
51 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
1
vote
2answers
82 views

Prove that convex subspace of $l_2$ is compact

Prove that the convex subspace of $l_2$ consisting of all sequences $\xi$ such that $$ \sum_{n=1}^{\infty} \xi_n^2 n^2 \le 1 $$ is compact. Have no idea how to proceed, any hints or suggestions?
1
vote
1answer
25 views

Average value of a density function

I want to compute an average density, but only of the yellow object on the picture (1< x^2+y^2<9) between circles and y>=0 The density of the object is a function $$p(x,y)=\frac{y}{x^2+y^2}$$ ...
4
votes
1answer
67 views

Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...