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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2 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[0, a \right]^2\right)$ as follows: $$u_{n,m}(x,y)=\frac{2}{a}\sin\left(\frac{n\pi}{a}x\right)\sin\left(\frac{m\pi}{a}x\right), \forall ...
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1answer
28 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$$\epsilon$$A$ and $b$$\epsilon$$B$. Prove $inf(A+B)=inf(A)+inf(B)$ My attempt: Since $A$ and ...
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3answers
20 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 ...
0
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2answers
41 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 ...
0
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1answer
32 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 ...
4
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30 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
8 views

Does symmetric decreasing rearrangement of a smooth function preserves smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
2
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0answers
40 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 ...
1
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1answer
10 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 ...
2
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1answer
29 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
38 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
40 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
14 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
24 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
25 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
29 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 ...
2
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1answer
43 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
18 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
29 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 ...
2
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2answers
95 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 − ...
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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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0answers
19 views

<html5> draw circle by arc and triangle [on hold]

I want if click the retacgle, draw a triangle around the circle. source code like this... but, triangle was not good each of positions.. How can i draw a triangle around the circle like attached ...
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0answers
47 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 ...
7
votes
2answers
135 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: ...
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3answers
44 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, ...
1
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1answer
50 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?
4
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2answers
88 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) ...
2
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1answer
29 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 ...
2
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0answers
44 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
13 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
43 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?
1
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1answer
23 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
37 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), ...
0
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0answers
32 views

estimation of an integration?

How to integrate the following expression, given $\int \rho(x)dx=M>0$, $\rho\geq 0$, and $\omega$ is a function? \begin{equation} \int \frac{x_1y_2-x_2y_1}{2\pi|x-y|^2}\rho(x)\omega(y)dxdy ...
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0answers
64 views
+50

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 ...
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0answers
20 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
31 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 $L$, ...
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0answers
15 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
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1answer
22 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
35 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 ...
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2answers
33 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 $g \in L_{p'}(E)$, ...
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1answer
33 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 ...
2
votes
1answer
21 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
25 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
31 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) = ...
0
votes
1answer
74 views

How to show the convergence of this infinite series: $\frac{x}{1+x}- \frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}\dots$

My series is $$ \frac{x}{1+x}-\frac{x^2}{1+x^2}+ \frac{x^3}{1+x^3}-..... $$ Given: $0<x<1$ I see that my nth term is $(-1)^{n+1} (\frac{x^n}{1+x^n})$ My approach was to use Dirichlet's test. ...
1
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1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
4
votes
1answer
50 views

Prove that the series is convergent: 1- (1+1/3)/2 + (1+1/3+1/5)/3-…

I can see that this is an alternating series with the $n$-th term $$(-1)^{n+1}\frac{1+\frac13+\frac15+\cdots+ \frac{1}{2n-1}}{n}.$$ What test can I apply to show that it converges? Also, it ...
1
vote
1answer
31 views

Need my data to fit ANOVA…

I am doing some research on constructed wetlands. I have four wetland as follows. Wetland A - Gravels Wetland B - Gravels + Plant Wetland C - Biochar + Plant Wetland D - Biochar + Gravels + Plants ...
0
votes
1answer
34 views

Proving differentiability using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...