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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Proving that when $\alpha$ is a root of order $r$ in a polynomial, $y(x)=x^k e^{\alpha x}$ satisfies the same polynomial differential equation.

Solution to (c) I'm having difficulty following the solution. First of all, why can the summations be interchanged in the second line of the solution? I mean, $\sum_{l=0}^n$ goes inside $\sum_{s=...
4
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2answers
70 views

Solving Inner Product Equations

I'm trying to solve an exercise from Cheney's Analysis for Applied Mathematics. Let $X$ be a normed linear space with $a,b,c\in X$ taken as fixed vectors, and consider the equation $x+\langle x, a\...
4
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1answer
61 views

How do I prove that $\,\left\lvert\,\int f \,dg\, \right\rvert \leq \int\left\lvert f \right\rvert \,d\left\lvert g\right\rvert$?

Let $\;g:[a,b]\rightarrow \mathbb{C}\,$ be a function of bounded variation. Let $\;f:[a,b]\rightarrow \mathbb{C}\,$ be a function Riemann-integrable along $g$. Define $\,\alpha\left(x\right) = V_a^x ...
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1answer
36 views

Topological Homogeonity of $\mathbb R^n$

I wish to show that for every two points $x, y$ in the open unit ball $B_1(0)$ of $\mathbb R^n$ there exists a homeomorphism of the closed unit ball $\overline{B_1(0)}$ which maps $x$ to $y$ and which ...
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4answers
73 views

show that a function $A$ such that $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ not necessary has a fix point $ (Ax\neq x \space \forall x )$

I don´t know an example wich $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ is not sufficient for the existence of a fixed point . can anybody help me? please
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1answer
84 views

Showing that $f^{(n)}(0)=0$ for $f(x)=e^{-1/x^2}$ if $x\neq 0$ and $f(0)=0$.

Define $f$ as follows: $f(x)=e^{-1/x^2}$ if $x\neq 0$ and $f(0)=0$. Show that $f^{(n)}(0)$ is continuous for all $x$ and $f^{(n)}(0)=0$. $n=1,2,\dots$. To show this, I have shown that for $x\neq 0$, ...
2
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1answer
391 views

What does “arg inf” mean?

I noticed this term on this post. But the term arg inf is not clearly defined.
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0answers
43 views

General Formula for finding the point P on a line, so that the sum of distances between P and k random points will be minimal

Let's say we have a total of $k$ points in $\Bbb{R}^2$: $$ P_1 = (u_1, v_1),\ldots,P_k = (u_k, v_k) $$ We also have a line $G$ defined by the formula: $$ G(x) = \lambda x + \mu $$ I need to find a ...
6
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189 views

Another conditionl leading to irrationality of $\sum _{k=1}^ \infty \dfrac 1{n_k}$?

If $\{n_k\}$ is a strictly increasing sequence of positive integers such that $\lim \inf _{k \to \infty} n_k ^{1/2^k} >1$ and $\lim _{k \to \infty} n_k^{1/2^k}$ does not exist , then is it true ...
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1answer
28 views

$x^Ty\leq 1, \forall\text{ y with }||y||_2 = 1 \iff ||x||_2\leq 1$

I came across this: This follows from Cauchy-Schwarz inequality: $x^Ty\leq 1, \forall\text{ y with }||y||_2 = 1 \iff ||x||_2\leq 1$ where $x\in\mathbb{R}^n$ When I try to do it myself, this ...
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1answer
25 views

Analytic function on intersection of two open sets

We are given two open subsets $\Omega_1$ and $\Omega_2$ both open and convex subsets of $\mathbb R^n$. if $f_1$ and $f_2$ are (real) analytic in $\Omega_1$ and $\Omega_2$. Show that $f_1=f_2$ in $\...
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0answers
49 views

Hard convergence

Let $(a_n)$ be a sequence of complex numbers. I need to prove that $$ \sum_{n=0}^\infty\frac{|a_n|^2}{n+1}\log^2(n+1)<\infty\Rightarrow \sum_{n=0}^\infty\frac{1}{n+1}\left|\sum_{k=0}^\infty\frac{...
6
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1answer
90 views

Prove that the closed unit ball is closed directly.

I'm trying to prove the following theorem, and I'm not sure my proof holds. Theorem. Let $(X,d)$ be a metric space, $p\in X$, and $r >0$. Then $$E = \left\{x\in X\ |\ d(x,p)\leq r\right\}$$ is ...
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1answer
49 views

Nonmeasurable sets.

I'm studying Lebesgue integration with Frank Jones's book that goes step by step and I found a proposed problem about nonmeasurable sets that says: Prove that if A and B are subsets of $\mathbb{R}^n$ ...
3
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2answers
43 views

Proof of divergence in analysis

I aim to show that the sequence $x_n := n^2 - 10n $ diverges to $+\infty$ by using the definition of divergence (i.e. for a given $M \in \mathbb{R}$, there exists $N$ such that $n \geq N$ implies $x_n ...
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1answer
33 views

Error in proof of convergent sequences?

A friend presented the following proof that sequence converges iff every subsequence converges to the same limit. Something seems awry, though I didn't know how to assure him. His proof of the ...
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0answers
20 views

Elementary limit theorems in analysis

I aim to show that $\lim_{n\to\infty} n2^{-n} = 0$. I just trying to prove that it equals zero by using the Squeeze Theorem. Here is my reasoning: Suppose $n \in \mathbb{R}$. Since $2^n > n$, by ...
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1answer
28 views

Line Integrals and Parametrizations

I want to find: $$\int_{\gamma}\frac{\mathrm{d}z}{z}$$ where $\gamma$ is the curve with orientation as follows (the circle and the square are centered at the origin) I'm not sure on how to find ...
2
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1answer
53 views

Strongly continuous semigroup

Given the following PDE $$\left\{\begin{array}{lll}u'(t)&=&A(u(t)) + f(t)\ \mathrm{for\ all}\ t \geq 0\ ,\\ u(0)&=&u_0\\\end{array}\right.$$ with $D(A) = H^2(0,1) \cap H_0^1((0,1))$ ...
2
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1answer
68 views

If $E \subset \mathbb{R}$ is uncountable, there exists $x$ so that both $E \cap (x, \infty)$ and $E \cap (-\infty,x)$ are uncountable.

Given $E \subset \mathbb{R}$, show that there exists $x$ so that both $E \cap (x, \infty)$ and $E \cap (-\infty,x)$ are uncountable. A very simple statement that seems to be eluding me.. Here ...
3
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1answer
38 views

Define a particular function

Does anybody know how you can define a function $\eta \in C_c^1(B_R(0))$ such that $\eta = 1$ on $B_{\frac{R}{2}}(0)$ cause I need such a function in a particular proof, so I would really like to know ...
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2answers
71 views

limit $ \lim_{x\rightarrow3}\left(\frac{x+1 - \sqrt{5x+1}}{\ln(\frac{x}{3})}\right) $

I am trying to find the limit of $$ \lim_{x\rightarrow3}\left(\frac{x+1 - \sqrt{5x+1}}{\ln(\frac{x}{3})}\right) $$ the function is continuous for all $ x \in \mathbb{R^+} $ except for $ x = 3 $ since ...
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2answers
73 views

Derivates of function and limits

I wonder if it's true $$\lim_{n \rightarrow \infty} \frac{d}{dx}f_n(x)=\frac{d}{dx} \lim_{n \rightarrow \infty} f_n(x)$$ When $\displaystyle f_n(x)$ is a sequence of function.
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0answers
57 views

Is there a name for the set theory usually described in analysis textbooks?

It seems to me that in most analysis textbooks, from calculus to abstract analysis, except for AC, the set theory described there has very little to do with ZFC. You would never see anything like $2\...
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1answer
72 views

Showing a function increasing

I was solving exercises from the book Real Analysis by Carothers. I am having trouble in solving Problem 36 of Exercise of chapter 2 on page 33. I am actually very poor in producing $\epsilon-\delta$ ...
4
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1answer
167 views

Unusual way to define a class of functions?

I am working on a problem and I came to a point where the problem has to be considered within a class of functions. But this class cannot be defined just in terms of the properties of each individual ...
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2answers
39 views

Show the Set is Connected

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. Show the set $$A:= \{\alpha \in \mathbb{R}: \exists \{x_n\}_{n \in \mathbb{N}} \subset \mathbb{R} \, \, \text{ with} \, \, \lim_{n \to \infty} f(x_n) ...
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1answer
85 views

Inequality between Hausdorff measure and spherical Hausdorff measure

I have a doubt on spherical Hausdorff measure. Given $k, \delta \in (0,\infty)$, the $\delta$-Hausdorff premeasure is defined for $E\subset \mathbb R^n$ as: $$\mathcal H^k_\delta(E):=\inf\{\sum_j\...
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1answer
26 views

What's the $\sigma$-algebra generated by a function?

I know that $\sigma(\mathcal{G})$ for a family of sets $\mathcal{G}$ is the smallest $\sigma$-algebra which contains $\mathcal{G}$. I'm reading some online notes, and ran into the term $\sigma(T)$ ...
2
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1answer
69 views

On the Princeton Lectures

I've heard nothing but good things about the Princeton Lectures in Analysis and was looking to start reading them. I just have a question for anyone who's read them before. Do they have to be read ...
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1answer
46 views

Showing that two topologies give rise to the same topology

Suppose that for a metric space $(X,d)$ we have another metric $p$ forming the metric space $(X,p)$. Both of these will give rise to two topologies $\tau_{1}$ and $\tau_{2}$. I want to show that $\...
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2answers
68 views

How would you interpret the following statement involving “a.e.”?

Here is an edited fragment from an exercise: Let $(X, \mathcal A, \mu)$ be a measure space, $(f_n)$ be a sequence of such and such functions. If $f(x)= \lim f_n(x)$ exists for almost every $x\in X$...
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1answer
19 views

A modulus inequality involving a minimum

When studying Fourier analysis, I have come across inequalities of the form $$ |\hat{K}_j(\xi)|\leq \min (|2^j\xi|^{-a},|2^j\xi|) $$ where we have the dilation operator $K_j(x)=2^{-jn}K(2^{-j}x),~j\...
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2answers
43 views

Are the Elementary Binary Operations Analytic?

Are the elementary binary operations of addition, multiplication, and exponentiation -- taken as multivariate functions over the real numbers -- analytic? That is, $f(a, b) = a + b$ and so forth. Does ...
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0answers
83 views

Every Continuous Function Attains Maximum Implies Compact

Let $K$ be a compact metric space, and let $A \subset K$. Prove that $A$ is compact if and only if, for every continuous function $f:K \to \mathbb{R}$, there exists a $q \in A$ so that $$f(q) = \max_{...
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3answers
196 views

Proving that if $f'=cf$ on some interval then $f=ke^{cx}$ on the interval.

Suppose that on some interval the function $f$ satisfies $f'=cf$ for some number $c$. (a) Assuming that $f$ is never $0$, prove that $|f(x)|=le^{cx}$ for some number $l\gt 0$. It follows that $f(x)=...
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0answers
29 views

To show a subset, which is union of open sets, is proper in $\mathbb R$

I was solving exercises from the book Real Analysis by Carothers. I am having trouble in solving a part of Problem 58 of Exercise of chapter 4 on page 59. I have managed to solve the other parts of ...
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1answer
49 views

Is max(a, b) Analytic?

Wikipedia briefly references the concept of an analytic function in several variables, but I don't have any reference to this idea in any of my analysis texts at hand: https://en.wikipedia.org/wiki/...
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38 views

Simple question on the linearity of a dynamical system

Take a continuous-time dynamical system $\Sigma=(\mathbb{T},\mathbb{W},\mathfrak{B})$ with $\mathbb{T}=\mathbb{R}$, $\mathbb{W}=\mathbb{R}$ and all sinusoidal signals with period $2\pi$. i.e. $w:\...
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0answers
38 views

Let $X$ and $Y$ be finite sets. Then $X \times Y$ is finite and $|X \times Y| = |X| \times |Y|$.

$X$ is finite with cardinality $n$ and $Y$ is finite with cardinality $m$. Lets now assume that $|X \times Y| > |X| \times |Y| = nm$, then there music exist a subset of $Z \subset |X \times Y|$ ...
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3answers
73 views

Limit of general solution of ordinary differential equation

The equation is $$t\dot{x}+(1+\alpha t)x=t$$ with $\alpha\in\mathbb{R}$ and $x(t)$ the general solution. For what $\alpha$ does the limit $\lim_{t\rightarrow\infty}x(t)$ exist and what is this limit?
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2answers
42 views

Double integral over a single variable

How to reduce $\int_0^t\Big(\int^s_0y(r)dr\Big)ds$ to $\int_0^t\Big(\int^t_ry(r)ds\Big)dr$ using Fubini's theorem?
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1answer
132 views

Fourier series over non-symmetric interval

Consider the function $$H:[-3,2] \to[2,2] $$ \begin{cases} -2 & -3\leq x < \frac{-1}{2} \\ 2x & \frac{-1}{2}\leq x< \frac{1}{2} \\ 2 & \frac{1}{2}\leq x \leq 2 ...
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1answer
56 views

The importance of the Van der Corput lemma in analysis and beyond

The Van der Corput lemma states the following: Introduce the following oscillatory integral $$ I(a,b)=\int^{b}_{a}e^{ih(t)}dt. $$ Then $(1)$ if $|h'(t)|\geq \lambda>0$ and $h'$ is monotonic, then ...
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2answers
26 views

How to prove a subspace?

Let X be a Banach space and let M be a subset of X. Then M is itself a Banach space (using the norm from X) if and only if M is closed subspace. How to prove (=>) to show that M is closed subspace? ...
0
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1answer
36 views

Definition of time global solution for PDE heat

Thank you for your kindness!! In detail, i consider the following problem \begin{equation} \begin{cases} \partial_{t}u(x,t)=\Delta u(x,t),\ (x,t)\in[0,1]\times (0,\infty),\\ u(0,t)=u(1,t)=0,\ [0,\...
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0answers
40 views

The $p=\infty$ case for an $L^2$ convolution operator on $\mathbb{R}^n$

Let $T$ be a convolution operator on $L^2(\mathbb{R}^n)$, suppose $K$ is a tempered distribution in $\mathbb{R}^n$ that coincides with a locally integrable function on $\mathbb{R}^n\setminus \{0\}$. ...
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0answers
21 views

Can we show, that $\langle\left[\nabla^2f(z+th)-\nabla^2f(z)\right]h,h\rangle=o(\|h\|)$?

Let $f\in C^2(\mathbb R^n)$. Taylor's theorem yields $$f(z+h)=f(z)+\langle\nabla f(z),h\rangle+\frac 12\langle\underbrace{\left[\nabla^2f(z+th)\color{blue}{-\nabla^2f(z)}\right]h}_{=:d(h)},h\rangle\...
1
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1answer
16 views

Why is $\int_0^1\left[\nabla^2f(x+td)-\nabla^2f(x)\right]d\;dt=\mathcal o\left(\left\|d\right\|\right)$?

Let $f\in C^2(\mathbb R^n)$ and $x\in\mathbb R^n$. By the mean value theorem, $$\nabla f(x+d)=\nabla f(x)+\nabla^2f(x)d+\underbrace{\int_0^1\nabla^2f(x+td)-\nabla^2f(x)\;dt}_{=:\;I(d)}\;d\;\;\;\text{...
1
vote
1answer
20 views

Exist a theorem about the radius of convergence of a sequence of functions?

I would like to know if there exists a theorem regarding the radius of convergence of a sequence of functions. E.g. I have a sequence of functions of the form $f_{n}(x) = \sum_{k=0}^{\infty} a_{k,n}x^{...