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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51 views

Rudin Principles of mathematical analysis p307

"For if $ A=\bigcup A^{'}_{n}$ with $A^{'}_{n} \in M_F(\mu)$, write $A_1=A^{'}_{1} $, and $$ A_n=(A^{'}_1\cup ...\cup A^{'}_n)-(A^{'}_n \cup ... \cup A^{'}_{n-1})$$ $(n=2,3,4,...)$. Then $$ A=\...
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1answer
75 views

Laplace transform quick answer check :) using second shift theorem

I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$" Mine gives me $e^{-...
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2answers
1k views

An example of set with a countably infinite set of accumulation points

I have to give An example of set with a countably infinite set of accumulation points, and I say: We can consider the set or real numbers and we take an arbitrary real number $x$ then the interval $(...
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1answer
67 views

proof about sigma-algebra

The question is that, $f$ is a function mapping $\Omega$ to another space $E$ with a $\sigma$-algebra $\varepsilon$. Let $\mathbf{A}= \{A\subset\Omega : \text{there exists } B \in \varepsilon \text{ ...
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2answers
40 views

Convergence of a sequence in absolut value. [duplicate]

I need to prove this: If $a_{n}$ converges to $A$, then $|a_{n}|$ converges to $|A|$. And I have this: $a_{n} \rightarrow A$ then, given $\epsilon>0$ there exists $N \in J$ such that $$|a_{n}-A|...
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1answer
82 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
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1answer
48 views

Monotonicity of some function

I have a function $f: \mathbb R^n \rightarrow \mathbb R$ and nonzero $y\in \mathbb R^n$ such that $$ f(z+ty) \leq f(z) \textrm{ for all } t\geq 0, z\in \mathbb R^{n}. $$ Is it then function $t \...
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1answer
451 views

O Notation and taylor series

Wolframalpha tells me that the Taylor series of the exponential function is $1 + x + \frac{x^2}{2}+ O(x^3).$ Taylor series I just don't get this big O there, shouldn't this be a small o?
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1answer
44 views

additive subgroup of real numbers with non empty interior

G is an additive subgroup of real numbers with a nonempty interior.Then G is all the real numbers.what is the exact proof?
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1answer
81 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= \sin\left(...
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1answer
215 views

Clarification needed: $\inf (A+B) = \inf A + \inf B$

Let $A$ and $B$ be nonempty bounded subsets of $R$ and let $A+B$ be the set of all sums $a+b$ where $a\in A$ and $b\in B$. Prove $\inf(A+B)=\inf(A)+\inf(B)$ My attempt: Since $A$ and $B$ are ...
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3answers
36 views

How prove $\lim_{x\to 0,y\to 0}\frac{2y^2x}{y^4+x^3}=0$

prove or disprove $$\lim_{x\to 0,y\to 0}\dfrac{2y^2x}{y^4+x^3}=0?$$ consider $$x^3+y^4>x^4+y^4,(x,y)\to (0,0)$$ so $$0\le |\dfrac{2y^2x}{x^3+y^4}|\le\dfrac{2y^2x}{x^4+y^4}|\le |\dfrac{2y^2x}{2x^...
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2answers
101 views

Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in $F$ ...
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1answer
48 views

Convergence of a sequence.

Well I have to prove that $\{ a_{n}\}$ converges to $A$ iff $\{a_{n} -A\}$ converges to zero, and I have: $\Rightarrow]$ We suppose that $\{ a_{n}\}$ converges to $A$, then by definition, given $\...
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1answer
91 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
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0answers
67 views

Loss of derivatives

In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved. Can someone clarify to me (maybe with an example) the meaning of this expression? For ...
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1answer
24 views

Inverse Laplace Transform Table, Absolution of Form

Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...
4
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1answer
156 views

composition of $L^{p}$ functions

Suppose $f, g\in L^{p}(\mathbb R), (1\leq p < \infty).$ For simplicity, let us assume that, $g,f:\mathbb R\to \mathbb R$ so that composition of $f$ and $g$, namely, $f\circ g(x)= f(g(x)); (x\in \...
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0answers
64 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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5answers
259 views

How to show if $a$$\leq$$b_1$, for every $b_1>b$, then $a$$\leq$$b$ where a,b$\epsilon$R?

Not positive on the proper approach to this problem. My first thought: $a $ $\leq$ $b_1$ means either $a=b_1$ or $a<b_1$. Should it broken up into cases? Second attempt: Assume, to the ...
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1answer
33 views

Typo in Cafferelli-Silvestre?

I am reading the 2008 paper by Caffarelli and Silvestre on Regularity theory for fully nonlinear integro-differential equations, and something is puzzling me. On the third page, an operator $L$ is ...
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1answer
37 views

Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$ (*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0. $$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
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1answer
85 views

Show that there exists an $\sum$-measurable simple function $\phi$ such that: $\int |f-\phi| d\mu <\epsilon$

Problem: Let $f \in L(X;\Sigma)$ where $L(X;\sum)$ is the set of integrable functions that can be written as $f=f^{+}-f^{-}$ where $\int f^{+} d\mu < \infty $ and $\int f^{-} d\mu < \infty $ ...
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2answers
67 views

Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n a_j\overline{b}_j$...
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1answer
362 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
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1answer
28 views

Elements in a convex set, regarding distance

I am looking at a much bigger proof than this, but this step is bugging me a lot. Suppose that $\Omega$ is a convex set and that $x,y\in \Omega$ are two elements in the set such that $|x-y|<r_1 + ...
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0answers
40 views

Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$.

Let $f$ be a holomorphic function on the unit disc $\{z : |z| < 1\}$ satisfying $f(0) = 0$ and $Ref(z) ≤ A$ for some positive number $A > 0.$ Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$. Not sure how ...
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2answers
116 views

Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below

I have read the following (http://www.math.harvard.edu/~elkies/Misc/sol8.html) but I dont understand the last part of the solution: For positive $x<1$, consider the alternating sum $$S(x) = x − x^...
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1answer
20 views

How do people show limit in the past?

How do people express the idea of limit before using $\varepsilon-N$ and $\varepsilon-\delta$ to express?
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64 views

existence of a positive root

Consider the polynomial $$ P(\omega)=\omega^8+\phi_7\omega^7+\phi_6\omega^6+\phi_5\omega^5+\phi_4\omega^4+\phi_3\omega^3+\phi_2\omega^2+\phi_1\omega+\phi_0 $$ with real coefficients. Assuming that $\...
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2answers
182 views

Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: $1-\frac12-\frac14-\frac16-\frac18+\frac13-\frac{1}{10}-\frac{1}{12}-\frac{1}{14}-\frac{1}{16}+...
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3answers
54 views

Convergence of series $\sum$$u_n$= $\sum$$\frac{n! x^n}{(n+1)^n}$

My series is $$1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots$$ My approach: $$u_n= \frac{n! x^n}{(n+1)^n}$$ So, $$u_{n+1}= \frac{(n+1)! x^{n+1}}{(n+2)^{n+1}}$$ So, $$\lim_{n\to\infty}\...
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1answer
54 views

Test for convergence of the series.

$$\frac{1}{1 \cdot 3}\ + \frac{2}{3 \cdot 5}\ + \frac{3}{5 \cdot 7}\ + \cdots$$ What is the $nth$ term here and what test should I use?
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2answers
150 views

Why is there a 'missing' $1$ in the Euler–Mascheroni constant?

It is easy to show that: $$ \sum_{k=1}^n \frac{1}{k} > \ln(n+1), $$ but the Euler–Mascheroni constant is defined as: $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)...
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1answer
58 views

Surjective function from a set of funtions to itself

Let a function be defined by $f \longmapsto f'$ acting from the set of all polynomials to itself. I am asked if this is surjective. I would like to think it isn't, but I'm in doubt how I should ...
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0answers
69 views

Derivative nonlinear Schrodinger equation

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I ...
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1answer
52 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied $$\...
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3answers
74 views

Is this positive term series convergent?

My series is: $\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$ I see my $nth$ term is $\frac{1}{1+2^{-n}}$ How do I test for its convergence?
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1answer
36 views

Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?
4
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1answer
88 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), &...
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1answer
115 views

Is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and $f:M\times\mathbb{R}^k\...
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0answers
42 views

Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
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1answer
185 views

Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
1
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1answer
35 views

Limit problem using sequential criteria for limits

$$\lim(n+n^2\log \frac{n}{n+1})= \frac12$$ How? In the text book it is simply said that this happens by Sequential criteria of limits. I don't get it.
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1answer
94 views

Projection of a set $G_\delta$

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps $G_\delta$ sets to Borel sets? i.e. If $A=\cap_n^\infty A_n$ with $A_n$ open sets, then $\pi(A)$ is ...
3
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3answers
163 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
0
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1answer
79 views

Common subdifferentials of convex function

Let $f: \mathbb R^n \rightarrow \mathbb R$ be a convex function. By a subdifference of $f$ in $x\in \mathbb R^n$ we mean an $h\in \mathbb R^n$ such that $f(x) \geq f(p)+<x-p,h>$ for all $x\...
1
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1answer
131 views

If $\{f_n\}$ and $\{g_n\}$ be uniformly convergent sequences of bounded functions on S, then $\{f_ng_n\}$ is uniformly convergent on S.

If $\{f_n\}$ converges uniformly to $f$ and $\{g_n\}$ converges uniformly to $g$, does it mean $\{f_ng_n\} $ will converge uniformly to $fg$? I am absolutely stuck on this. Please help.
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3answers
42 views

About the convenient rewriting of formulas of sequences?

I'm reading Courant's Introduction to Calculus and Analysis. In the introduction, he shows some examples of limits of sequences, the sequence in question is: $$a_n=\frac{n^2-1}{n^2+n+1}$$ Then he ...
1
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0answers
42 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...