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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The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in C^{\infty}_{c}(\mathbb{R}), $$ ...
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1answer
11 views

Proof that a set that has supremum property implies it has infimum property

I really have no clue where to start with this proof: Suppose S is an ordered field and satisfies supremum property. Show that S has the infimum property as well. Any help would be great.
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1answer
12 views

Correctness of Proof that the Archimedean Property of Reals is equivalent to lim $1/n$ as n tends to infinity

Here's what I have gotten so far: The Archimedean Property states 1) For every $\epsilon$ >0 there is a positive integer n s.t. $1/n$< $\epsilon$ and 2) For every positive number c there is a ...
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1answer
13 views

Question about the Chebyshev Inequality.

Let $p_1 < p_2 <\dots < p_n$ be the $n$ first primes listed in crescent order. Using the Chebyshev Inequality (for $x$ sufficiently large) $$0.92\leq \frac{\pi(x)\log x}{x}\leq 1.11,$$ How ...
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Bertrand's Postulate and and Chebyshev Inequality

Let $\theta(x) = \sum_{p\leq x}\log p$ and $\pi(x) = |\{p\leq x:p\text{ is prime}\}|$. Using Abel's formula, one can prof the following $$\pi(x) = \frac{\theta(x)}{\log x} + ...
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12 views

Sobolev space of p-forms on a Riemannian mamifold

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$. Let $(U';\varphi =x^1,\cdots, x^d)$ be a chart of M, $U\subset\subset U'$ be an open set of $U'$. $A^p(M)$ denotes the set of smooth ...
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11 views

Unique extrema of sum of monotonically increasing and decreasing functions on an interval

If I have two functions f and g defined on the interval [0, 1] with both f and g non negative (i.e. f(x), g(x) >= 0) f(x) is monotonically increasing, while g(x) is monotonically increasing. and ...
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11 views

Given a Riemann Integrable function f, calculate the values of A,B,C [on hold]

Given a Riemann Integrable function f, calculate the values of A,B,C Any help will be thankful. Thanks!
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9 views

Expected value and General distribution Function

Given an iid random variable, the expected value is usually defined as follow: $E[T]=\int_0^\infty t f(t) dt $ where $f(t)$ is the pdf of $T$. On a book I found this definition (without any other ...
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12 views

Sups and Infs Proof

Show that the $\sup(-a_n)=-\inf(a_n)$. I can see this intuitively, but as for the infinite case I don't understand how to show it. Can someone lead me in the right direction?
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1answer
16 views

Adding integrals with different domains

Suppose I have two integrals $$ \int_{\Omega_1} f \, \, d \eta$$ and $$ \int_{\Omega_2} g \, \, d \eta$$ how would I define the sum of these two integrals? Is it possible? I want something of the ...
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7 views

Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
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Let Sn={$x\in Rn+1$; $\langle\ x,x\rangle$=1} a sphere n-dimensional. [on hold]

I study Metric Spaces and I have this problem. Let $\Bbb S^n=\{x\in \Bbb R^{n+1}; \langle x,x\rangle=1\}$ be a $n$-dimensional sphere. The projective space with dimension $n$ is the set $\Bbb P^n$, ...
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Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
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36 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
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1answer
24 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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17 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
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5 views

Finding the extremal curve satisfying a variable endpoint

Below is a question I am trying to solve, and my attempt. $\int_0^T \frac{\dot{x}^2}{t^3} \mathrm{d} t$, where $x(0)=1 $ and $x(T)$ lies on the curve Transversal condition: $$f-(\dot{c} ...
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1answer
31 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
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30 views

Cauchy product proof

I'm searching for a proof of the Cauchy product that states: If the series $\sum a_n$, $\sum b_n$ and $\sum c_n$ converge to $A$, $B$ and $C$, and $c_n = a_0b_n+\cdots a_nb_0$ then $C = AB$ All the ...
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1answer
50 views

Analysis Problem - Showing statements are true.

Prove the following. Let $\{A_n \}_{n \in \mathbb{N}}$ and $\{ B_n\}_{n \in \mathbb{N}}$ be sequences of sets with $$ A_1 \subset A_2 \subset A_3 \dots \subset A_n \dots $$ $$ B_1 \subset B_2 \subset ...
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1answer
19 views

Measure defined in an atypical way

I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ). Let $\Omega \subset ...
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2answers
38 views

Stuck on proving a statement

Prove the following Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of sets with $$A_1 \subset A_2 \subset A_3 \dots $$ and $B \subset \bigcup_{n = 1}^{\infty} A_n$. If for every infinite subset $E$ ...
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1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
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18 views

Properties of normal set

For any closed subset $F$ of $X\subset \mathbb{R}^{n}$, we define the normal set $\mathcal{N}(F)$of $F$ as follows: if there exists $f\in C^2(X)$, and $x_0\in F$, such that $$ df(x_0)\neq 0;\\ ...
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1answer
20 views

completely monotone sequence

Is the sequence $(\frac{1}{n^2})$ completely monotone$^*$? $*$A sequence $(a_n)$ is completely monotone iff $(-1)^k(\Delta^k a)_n\geq 0$, where $(\Delta a)_n=a_{n+1}-a_n.$
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Cesaro means converges in first mean to 0.

I would appreciate any suggestions to prove the following statement If $\{ X_i: i=1,..\}$ is a sequence of independent uniformly integrable (U.I) random variables ...
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44 views

Prove that $C[a,b]$ with inner product $\langle f,g\rangle:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space.

Prove that $C[a,b]$ with inner product $<f,g>:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space. Now the norm induced by the inner product is \begin{align} ...
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1answer
15 views

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$. Is it sufficient to take $x=(1,0)$, $y=(0,1)$ in $\mathbb{C}^2$ and just showing that \begin{align} ...
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21 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
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1answer
25 views

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent:

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent: a)$f$ is continuous in $D$ b)If $O\subset \Bbb R^m$$f$ is an open set, then there exists an open set $G\subset ...
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1answer
14 views

Period of two equal functions

I'm dealing with a problem here. We know that two functions are the same if they have the same domain and codomain. Let's say we have given the functions $f$ ang $g$ where ...
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1answer
21 views

Proving the integral of a discontinuous function

Let $y_n$ be a monotone decreasing sequence with $\lim_{n\to\infty}y_n=0$. Define the function $f:\left[0,1\right]\to\mathbb{R}$ by $$ f(x)= \begin{cases} y_n &\text {there ...
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40 views

For a natural number $n$ and numbers $a$ and $b$ such that $a \geq b \geq 0$, prove that $a^n-b^n \geq nb^{n-1}(a-b)$

I tried to do an induction proof and I've played around with it for about an hour and haven't really gotten anywhere. For my base case, I let $n=1$ and got $(a-b) \geq (a-b)$, however when I tried to ...
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1answer
55 views

An open set in $\mathbb{R^n}$ is connected if and only if it is path connected

Here is a proof I found on the internet but cannot understand a part of it which is highlighted. I hope someone can help me understand this. Thanks in advance
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21 views

Reference for the following equation

Can someone suggest me references about the following equation $$u_t+A\cdot\nabla u=i\Delta u$$ with $A$ a smooth vector field.
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47 views

How find this limit $\lim_{x\to 0}\left(\frac{a^x-x\ln{a}}{b^x-x\ln{b}}\right)^{\frac{1}{x^2}}$

Find this limit $$\lim_{x\to 0}\left(\dfrac{a^x-x\ln{a}}{b^x-x\ln{b}}\right)^{\frac{1}{x^2}}$$ I think $$e^{\lim_{x\to 0}\dfrac{\ln(a^x-x\ln{a})-\ln{(b^x-x\ln{b})}}{x^2}}$$ I can use L'Hôpital's ...
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1answer
27 views

Decay of Fourier Coefficients implies Holder Continuity?

This is an exercise problem. I got stuck here and would like to get a hint. The problem is Suppose $f$ is continuous and $2\pi$-periodic, and $|\hat{f}(n)|\leq |n|^{-3/2}$ for all non-zero ...
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2answers
51 views

Good analysis texts

I'm looking for a good introductory text to analysis, or, more specifically, a text that puts calculus on a much more rigorous ground. I've just finished a year of calculus at my local university, ...
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36 views

Question about a paragraph in the book complex analysis by Ahlfors.

By $C_1$, we denote family of circles passing through $a,b$ and by $C_2$ we denote family of Appolonius circles with limit point $a,b$. In section $3.5$ entitled Families of circles, in one paragraph ...
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2answers
82 views

Formula for $\sum\cos(\pi kt)/(1+k^2)$

Is there an explicit formula for the sum $$F = \sum_{k=0}^\infty \frac{1}{1+k^2} \cos(\pi k t)$$ This is the green function for the operator $1 + \Delta$ on the circle.
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1answer
39 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
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63 views

Prove that if $f(x+y)=f(x)f(y)$ and f is continuous in $ x=0$, then it is continuous in all its domain

Prove that, for $f:\Bbb{R} \rightarrow \Bbb{R}$, if $f(x+y)=f(x)f(y)$ and f is continuous at $x=0,$ then it is continuous in all $\Bbb{R}$. I haven't figured out how to prove this. What would you ...
3
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1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
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176 views

Evaluating a series to order “three halves”

In doing some calculations related to one-dimensional Brownian Motion confined to a finite interval, I have come across functions such as $$ f(t) = \sum_{n=1}^\infty\frac{\exp(-n^2t)}{n^4}. $$ I ...
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1answer
40 views

Oscillation of a Function

Let $f\colon (a,b)\rightarrow \mathbb{R}$ be function. For a non-empty subset $T$ of $(a,b)$, define $\Omega(f,T)=\sup\{|f(x)-f(y)|\colon x,y\in T \}$, and the oscillation function from $(a,b)$ to ...
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when does bijective map exist for any pair of rational function?

Let me ask kind of different questions than former ones. Given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{, and }\frac{P_3(y_1,y_2,\dots,y_n)}{P_4(y_1,y_2,\dots,y_n)}$$ where $P_i$ ...
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1answer
20 views

Prove that Hölder condition in $\Bbb R^n$ implies continuity

$f:I\subset \Bbb R^n \rightarrow \Bbb R^m$ is said to be Hölder continuous if $\exists$ $\alpha>0$ and $M>0$ such that $\|{f(x)-f(y)}\| \leq M\|x-y\|^\alpha$, $ \forall x,y \in I$, ...
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2answers
18 views

How to find point or interval of intersection in nested interval

so I am give a problem which sets sequences $a_n$=1-1/n and $b_n$= 1+1/n and the interval $I_n$ As [$a_n$,$b_n$]. How would you find the intersection of these two sequences by nested interval theorem ...
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0answers
24 views

Why can't the general solution of separable first order ODE cross the stationary solution?

For example, if we have the following Cauchy problem: $y'=y^2-4, y(0)=0$ In class, our professor told us that $y=-2,2$ are the two stationary solutions, but how could it be, since our initial point ...