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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1answer
25 views

$f_1(x)=sin(x),\; f_{n+1}(x)=sin(f_n(x))$, $(f_n)$ converges to zero uniformly

Consider for $n\in\mathbb{N}$ the function $f_n:\mathbb{R}\to\mathbb{R}$ given by $$f_1(x)=sin(x),\; f_{n+1}(x)=sin(f_n(x)).$$ I'm stuck to prove that $(f_n)_n$ converges to $f=0$ uniformly. First ...
0
votes
0answers
15 views

Fubinu's principle on series

If one double series is absolutely convergent then $$\sum_{j\geqslant 0}\sum_{i \geqslant 0}a_{ij}=\sum_{i \geqslant 0}\sum_{j \geqslant 0}a_{ij}$$ I looked around and (probably not very good) and ...
0
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0answers
18 views

Measure of $|g| = ||g||_\infty$, with $g \in L_\infty $.

I would like to either prove or disprove that the measure of $|g| = ||g||_\infty$, with $g \in L_\infty $, is greater than zero. I'm thinking that I should use the definition of $||g||_\infty$= ...
0
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0answers
12 views

Application of the implicit function theorem on small perturbation of a function

I have a problem in numerical analysis. Assume that a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ satisfies $\frac{\partial f(u,v)}{\partial v}<-1$. Then by the implicit function theorem we ...
0
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0answers
38 views

Is there $f: U \to \mathbb{R}^{n}$ injective such that…

Let $f: U \to \mathbb{R}^{n}$ $C^{1}$ injective where $U$ is a open in $\mathbb{R}^{n}$ (so $f$ is open by invariance domain theorem). a) Is there exist $f$ such that dim $ker(df_{x}) >$ dim ...
0
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0answers
26 views

Find $\int_{-\infty}^{\infty}\frac{\sin x}{x} dx$ by integrating over a semicircle in $\mathbb{C}$.

I'm trying to find $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ by solving $$\oint_{\gamma}\frac{e^{iz}}{z}dz$$ where $\gamma$ is the upper semi-circle, with appropriate choices for the paths. My ...
0
votes
0answers
22 views

$g(x)\le f_0(x_0)$ for all $g \in B$ and $x \in A$.

I want to prove the statement : Let $A \subset \Bbb R^{n}$ be compact and let $B \subset C(A,\Bbb R^{n})$ be compact. Show that there are an $f_0 \in B$ and an $x_0 \in A$ such that $g(x)\le ...
-4
votes
2answers
147 views

Why is minimum solution example to $x^n + y^n = z^n$ comes in the form of three successive integers? [closed]

Can we prove or disprove this conjecture by elementary mathematics: If this is a true statement: $$x^n + y^n = z^n $$where $x, y, z, n$ are positive integers, then there must be a minimum integer ...
1
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2answers
57 views

Proving an Integral inequality from a given integral inequality

Problem: Let $f$ and $g$ be continuous, non-negative function on $[0, 1]$, with $$\int_{0}^{1}e^{-f(x)}dx \geq \int_{0}^{1}e^{-g(x)}dx. $$ Prove that, $$\int_{0}^{1}g(x)e^{-f(x)}dx \geq ...
0
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1answer
23 views

show that the following sequence function converges uniformly to 0 on the given set $\left\{\frac{\sin nx}{nx}\right\}$ on $[\alpha,\infty)$ …

... where $\alpha>0$. question: show that the following sequence function converges uniformly to $0$ on the given set $\displaystyle\left\{\frac{\sin nx}{nx}\right\}$ on $[\alpha,\infty)$ where ...
1
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0answers
34 views

Proving the supremum of uniform norms is 1 in a particular continuous funciton vector space

Let $$X=C_{\Bbb R}([0,1])=\{x:[0,1]\to\Bbb R\;\; |\;x \text{ is continuous}\}$$ be a vector space over $\Bbb R$ with $\|x\|_X=\|x\|_{[0,1]}=\sup_{t\in [0,1]}{|x(t)|}$. Let $$M=\Big\{x\in ...
0
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0answers
8 views

Norms of Slobodekii and classical Sobolev space

For r to be a non-negative integer, we have $$\|u\|_{W^{r,p}(\Omega)} = \left( \sum_{|\alpha|\le r} \int_\Omega |\partial^\alpha u(x)|^p dx \right)^{1/p}.$$ For $0 < \mu < 1$, let $s = r + \mu$. ...
7
votes
3answers
92 views

$\int_0^\infty {x^a\over (x^2+1)^2} dx$ where $0<a<1$. [closed]

$\int_0^\infty {x^a\over (x^2+1)^2} dx$ where $0<a<1$. I know I can use partial fraction decomposition to obtain two different integrals, but I'm not sure how to integrate them. Any solutions ...
-4
votes
1answer
29 views

Need hint on differentiable exercise [closed]

I'm looking for a sequence of continuously differentiable $f_n : [0,1] \to \mathbb R$ functions, such that $f_n \rightarrow 0$ uniformly, but $f'_n$ doesn't converge uniformly. Could someone give me ...
0
votes
1answer
23 views

Differentiability of the following function

Prove that the following function's differential is nonsigular at the origin: Define $F = (f,g):\mathbb R^2 \rightarrow \mathbb R^2$ by $f(x,y) = x$, and $ g(x,y) = y-x^2$ for $y \ge x^2$, $ g(x,y) ...
-2
votes
1answer
43 views

Does it uniformly convergence? [closed]

Does $$\sum_{n=1}^{\infty}{1 \over n}-{1\over n+x}$$ converge uniformly on any bounded interval including x?
0
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0answers
5 views

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x$?

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x=[-115/588, -95/588, 5/14]^T$? Pseudoconvexity: If $\nabla f(\bar x)^T(x-\bar x)\ge0$, then $f(x)\ge f(\bar x)$ for any $x\in \mathbb{R^3}$ (in this ...
2
votes
0answers
42 views

Continuous strictly increasing function with derivative infinity at a measure 0 set

Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)? I think there exist such ...
0
votes
0answers
23 views

Counterexample to distance between two disjoint compact manifolds always great than some positive real number

As the title says, there is a theorem that says the following: Let M,N be compact manifolds in $\mathbb R^n$ such that $M\cap N=\emptyset$. Then there exists some $\epsilon > 0$ such that for any ...
1
vote
1answer
32 views

A topology on a set $X$ is exactly determined by specifying when, for $x\in X$ and sequences $x_n \in X, x_n \to x, $ as $n \to \infty$.

A topology on a set $X$ is exactly determined by specifying when, for $x\in X$ and sequences $x_n \in X, x_n \to x, $ as $n \to \infty$. This is a statement from my functional analysis notes, but I ...
3
votes
1answer
23 views

What is the advantage (if any) of neighborhoods which are not open?

From wikipedia If $X$ is a topological space and $p$ is a point in $X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$, $$p \in U \subseteq V.$$ What is ...
1
vote
1answer
37 views

$xy \leq \frac{x^p}{p}+\frac{y^q}{q}$

I struggled to show that this is true when $x,y >0$, $p>1$ and $q=\frac{p}{p-1}$. But, I managed to show that if we assume that $x \geq 1$ the assertion is possibly false only for a finite ...
1
vote
2answers
64 views

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. [closed]

Let $f$ be analytic on the unit disk $D = \{z : |z| ≤ 1\}$ and suppose $Im(f(z)) > 0$ for $z ∈ D$ and $f(0) = i$. Prove that $|f ' (0)| ≤ 1$. For what functions do we have equality? I'm not sure ...
2
votes
0answers
28 views

Integral Representation of Brownian Motion [duplicate]

B is a Brownian motion with values in $\mathbb{R}$. I have to find a process $(F_t)_{t\in[0,T]}$ such that $X=E[X]+\int_0^T F_s dB_s$, for $X=B_T$, $X=\int_0^T B_tdt$, $X=B^2_T$, $X=B^3_T$ and find a ...
0
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0answers
9 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
0
votes
1answer
14 views

Mental block in the last step in showing $p$ is an open map

Let $p:\tilde{X}\to X$ be a covering map. Let $U$ be open in $\tilde X$. Let $y\in P(U)$. $y$ has an evenly covered open neighborhood $V$, such that $p^{-1}(V)=\coprod A_i$, where $A_i$ are disjoint ...
3
votes
1answer
27 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
0
votes
2answers
33 views

Convergence of the sequence $x_1 = 1$ and $x_{n+1} = 4 - \frac{1}{x_n}$ for all $n \geq 1$.

Let $x_1 = 1$ and $x_{n+1} = 4 - \frac{1}{x_n}$ for n $\geq 1$. Show that it converges and find its limit. Is my following proof correct? $\text{First we claim that } x_n \leq 4, \forall n. \text{ It ...
6
votes
1answer
233 views

Understanding an example of “for all” and “for some” usage in statements.

I'm reading "Analysis I" by Tao and reviewing an appendix chapter on logic. In there he gives an example on how "for all x" is usually much stronger than just saying "for some x": "$6<2x<4$ ...
0
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0answers
14 views

Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland.

This is a characterization of the Jordan decomposition of $F$ from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let $F\in BV$ be a real valued function and ...
1
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2answers
46 views

Prove if a continuous function $f$ is one-to-one, it is monotonic.

This is the converse of Prove that if function f is monotonic, then it one-to-one. Let $f:(a,b) \mapsto \mathbb{R}$ be a continuous function. Prove if $f$ is continuous in $(a,b)$, $f$ is monotonic. ...
2
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0answers
23 views

Let $F: \mathbb{R}\to \mathbb{R}$ be increasing and let $G(x)=F(x+)$. Then G is right continuous.

Let $F: \mathbb{R}\to \mathbb{R}$ be increasing and let $G(x)=F(x+)$. Then G is right continuous. I'm trying to prove this obvious statement using $\epsilon-\delta$ argument, but I can't find a ...
1
vote
1answer
29 views

how to write $\nabla(1/f)$ ???

I have to prove that $\nabla(1/f)= -(1/f^2) \nabla (f) $. I know that in general, we have $\nabla(fg)= \nabla(f) g+ \nabla(g)f$ and i have tried to write $g=1/f^2 $ which gives me $\nabla(1/f)= ...
0
votes
1answer
25 views

Problem in functional analysis: application of open mapping theorem

I'm having trouble in exercise 2.10 in Brezis' book in Functional Analysis: let $E$ and $F$ be two Banach spaces and let $T \in \cal{L}$$(E,F)$ be surjective. Let $M$ be any subset of $E$. Prove that ...
1
vote
2answers
130 views

Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} - cx^{2p} - dx^p - 1 $$ ,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove ...
0
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0answers
17 views

Convolution of two gaussian functions

I want to calculate the convolution $F * G$ of two Gaussian functions without resorting to Fouritertransforms: $F(t) := \exp(-at^2), G(t) := \exp(-bt^2) \qquad a,b>0$ But intuitively I expected ...
0
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2answers
32 views

How to prove $A$ inverse and $B$ commute

Suppose $A$ and $B$ are nonsingular matrices. Prove that if $A$ and $B$ commute,then so do $A$ inverse and $B$?
3
votes
1answer
45 views

Is this a Schwartz function?

I would like to know whether this function $f : \mathbb{R} \rightarrow \mathbb{R}$ $$f(x):=\frac{1}{\sum_{k=0}^{\infty} \frac{x^{2k}}{(k!)^2}}$$ is a Schwartz function? By applying the chain-rule ...
1
vote
1answer
18 views

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function, compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$.

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function where $\mu(E_j)\lt \infty$. I want to show that $f\in L^p$, but I don't know how to compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$. How ...
-1
votes
2answers
58 views

Determine whether or not the following series is convergent $\sum_{n=1}^\infty \frac{1}{n^n}$ [closed]

Determine whether or not the following series is convergent $$\sum_{n=1}^\infty \frac{1}{n^n}$$ What test should I use to approach this question?
0
votes
1answer
19 views

Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand. Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and ...
0
votes
2answers
22 views

Product of nonsingular and full rank matrices

Assume that $U\in\mathbb{C}^{n\times k}$ (with $k<=n$) has full column rank and $A\in\mathbb{C}^{n\times n}$ is nonsingular. Is $U^*AU$ nonsingular? Assume that $U,P\in\mathbb{C}^{n\times k}$ ...
0
votes
0answers
14 views

Fitting with initial conditions

i try to do some fits but i have ten initials conditions and i think it will be difficult to evaluate the sensitivity of my conditions. Do you know some methods which allow to know the sensitivity of ...
1
vote
1answer
53 views

Show L'Hospital limit for exponential function and power series

Given a series $$f(t):=\sum_{k=0}^{\infty} \frac{t^{2k}}{\sqrt{(k!)}},$$ then since by first term expansion we have $f(t)\ge 1+t^2$, we get that $f(t) \rightarrow \infty$ for $t \rightarrow \infty.$ ...
1
vote
1answer
41 views

Examples of complex-variable functions that fail to have a limit at some point

My notes from class have the example $\frac{\overline{z}}{z}$ as z tends to zero. That the limit does not exist is shown by exhibiting that along the $x$-axis the limit is $1$ and along the $y$-axis ...
0
votes
1answer
35 views

Proof of Jacobi triple product by taking the limit

Assume that we know $$ \prod_{k=1}^{n}(1+q^{2k-1}z)(1+q^{2k-1}z^{-1})=C_{0}+\sum_{k=1}^{n}C_{k}(z^k+z^{-k}), $$ with $$ ...
2
votes
1answer
70 views

Show by series definition of exponential function that $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty.$

There are many arguments I have seen using $\ln-$ arguments and other properties of the exponential function to show the existence of this limit $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty$. ...
1
vote
1answer
41 views

About $A=\bigcup_{n=1}^\infty A_n$, with $A_n=\{(x,y)\in\mathbb{R^2}:-e^{-x^2/n}<y\le e^{-x^2/n}\}$

I am asking this question as I would like to improve my understanding about a couple of things. Since $e^{-x^2/n}$ converges monotonically to $1$, I think we have ...
3
votes
1answer
47 views

Let $\sum a_n$ be a conditionally convergent sum of complex numbers. Can $\sum a_n z^n$ converge $\forall |z|=1$?

I'm fairly new to complex analysis, and I just thought of this problem, but I can't seem to find an easy proof, or an easy counterexample.
3
votes
2answers
38 views

Integrating over a sequence of sets $A_n$ with $\mu(A_n)\to0$

I am going through the proof of the following. Let $(X,\mu)$ be a measure space and $f\colon X\to\overline{\mathbb R}$ be a measurable function with finite integral. If $A_1,A_2,\dots$ are ...