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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Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
2
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2answers
50 views

Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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1answer
49 views

Open map and adherence

How to prove that $$f :E\rightarrow F ~\text{is open} \Longleftrightarrow f^{-1}(\overline{A})\subset \overline{f^{-1}(A)}, \forall A\subset F$$ where $(E,\tau), (F,\theta)$ are topological spaces. ...
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1answer
26 views

Prove the following about absolutely convergent complex series

Prove that for every sequence $(a_n)_n$ of complex numbers, if the series $\sum_{n\ge 0} a_n$ is absolutely convergent, then $|\sum_{n\ge 0} a_n| \le \sum_{n \ge 0} |a_n|$. I've been given the ...
4
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3answers
88 views

How does Wolfram Alpha come up with the substitution $x = 2\sin u$? Integration/Analysis

I have to integrate $$ \int_0^2 \sqrt{4-x^2} \, dx $$ I looked at the Wolfram Alpha step by step solution, and first thing it does is it substitutes $x = 2\sin(u)\text{ and } \,dx = 2\cos(u)\,du$ ...
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1answer
76 views

How to compute $\lim_{n\to \infty} n\sin(2\pi n! e)$ [duplicate]

I want to calculate $$\lim_{n\to \infty} n\sin(2\pi n! e)$$ I have used the Stirling approximation and I think the answer is zero . But I think the limit maybe not exists. Can some one help? ...
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2answers
24 views

Continuous function positive at a point is positive in a neighborhood of that point

Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) > 0$ then there exists a $d > 0$ such that $\forall x$, $f(x) > 0$ with $|x-c| < d$. I can understand ...
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3answers
101 views

Prove exponent(m)=e^{m}

please show me how to do the third one, I just understand the 1st and 2nd, but i have no idea how to do the 3rd. thank you.
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1answer
57 views

bounded interval is bounded and connected

Can you please tell me if my proof is correct? Definition: Let $X$ be a subset of $\mathbb R$. We say that $X$ is connected iff the following property is true: whenever $x, y$ are elements in ...
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0answers
8 views

local maxima of weierstrass function

Does the weierstrass function have uncountably many local maxima on (0,1)? I don't really know how to approach this problem at all, so any help is appreciated
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2answers
97 views

If $f$ and $g$ are continuous, then max(f, g) is continuous and differentiable

If $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then $\max(f, g)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. I'm asked to either prove or disprove this ...
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0answers
49 views

Stuck on Induction Proof [duplicate]

Let $0< a_1 < b_1$ and define $a_{n+1} = \sqrt{a_nb_n}$, and $b_{n+1}$ = ${a_n + b_n}\over2$. Use induction to show that $a_n<a_{n+1}<b_{n+1}<b_n$. Also, prove $a_n$ and $b_n$ ...
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2answers
29 views

Show that it stands for each measurable $E$

Let $f$ be integrable in a space of finite measure. Show that $\forall \varepsilon >0$ $\exists \delta>0$ such that for each measurable $E$ with $\mu(E)<\delta$ we have that $$\int_E|f|d\mu ...
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2answers
100 views

Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
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2answers
49 views

Convolution integral problem

In the process of solving a certain PDE, I've arrived at a convolution integral: $$\int_{\mathbb{R}^3} G(x-y) \nabla p(y) dy$$ where $x \in \mathbb{R}^3$, $G(z)=\frac{1}{\| z \|}$ and $p(z) = ...
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3answers
86 views

proof about limits of functions

Let $f:\mathbb R \to \mathbb R$ be such that $f(x), f'(x) and f''(x)$ are all positive for each $x \in \mathbb R$. Show that $\lim_{x \to \infty} f(x)=\infty$. So $f''(x)$ is the second derivative of ...
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3answers
46 views

Prove that the polynomial $f_m(x) = x^3 − 3x + m$ never has two roots in [0, 1], for all m in R

I understand that the function never has two roots because it is only crosses the x-axis once due to it being a cubic, but I don't know how to prove it. Any help would be appreciated.
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2answers
49 views

Find two continuous, increasing functions $f,g$ such that $f(x) = g(x)$ precisely when $x \in \mathbb{Z}$

Find two continuous, increasing functions $f,g$ such that $f(x) = g(x)$ precisely when $x \in \mathbb{Z}$ I'm pretty lost as to how to go about proving this. I feel like maybe it's going to be some ...
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0answers
27 views

A bounded sequence of a complete Metric Space [closed]

Let $M$ be space of all bounded sequence, prove that $M$ is complete
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1answer
17 views

A sufficient condition that domain of solution of differential equation became $\mathbb R$

If $ f:\mathbb R^n\to \mathbb R $ be bounded and continous then differential equation $$x'=f(x)$$ has a solution with domain $\mathbb R$. outlook of proof : if the maximal domain of solution is ...
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0answers
13 views

I'm confused on how to use chi squared for the correlation between age and reaction time

I am doing my IB maths internal assessment and I am confused on how to specifically carry out chi squared with my given data. I will try to explain this quite plainly so the image is clear. I am ...
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2answers
99 views

Extremely difficult log integral, real methods only

$$\int_{0}^{1}\frac{x^2 + x\log(1-x)- \log(1-x) - x}{(1-x)x^2} dx$$ I tried this: $$M_1 = \int_{0}^{1} \frac{1}{1-x} \cdot \left(\frac{x^2 + x\log(1-x) - \log(1-x) - x)}{x^2}\right) dx$$ $$M_1 = ...
2
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1answer
14 views

Recursive sequence with Complex numbers, missing conclusion.

I am solving the following task: Let $a_1 = \sqrt{2}\sqrt{3}*i, a_{n+1} =\frac{i* a_n}{n+1}$ What can you say about the convergence of $a_n$? I already found out a lot. What i concluded so far, is: ...
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0answers
22 views

Hahn-Banach proof by extension of basis

Hahn Banach Theorem states that given a linear continuous functional $f$ on a subspace $N$ of a norm space $M$, it can be extended to a linear functional $F$ on the whole space $M$ and the norm of the ...
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142 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
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0answers
26 views

analyzing a curve in the plane

For the following function $F(x,y)$, determine whether the set $S = {(x,y): F(x,y)=0}$ is a smooth curve. Draw a sketch of $S$ near any points where $\nabla F = 0$. Near which points of $S$ is $S$ ...
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1answer
54 views

How to construct a smooth function with compact support satisfying $f(x)+f(x^{-1})=1$

How to construct a smooth function with compact support satisfying $$ f(x)+f(x^{-1})=1 $$ For example, let $$ g(x)=\left\{\begin{array}{ll} 0,&\mbox{if $x\leq 0$},\\ ...
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2answers
26 views

Entire functions and local injectivity

QUESTION: If a function, $f:\mathbb{C}\rightarrow \mathbb{C}$, is entire and it is non constant is it necessarily locally injective? That is if given some $z_0$, does there exists a disk, ...
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1answer
36 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
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1answer
24 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
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0answers
45 views

Calculus - comparison test for series [closed]

Prove that if $b_n\geq 0$ and $$\lim_{n\to\infty}\frac{a_n}{b_n} =q\in(0,+\infty)$$ then the series $\sum a_n$ is convergent iff the series $\sum b_n$ is convergent. Use this fact to determine the ...
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1answer
61 views

Homeomorphism and topology [closed]

I have to prove that $$f~\text{is a homeomorphism}\Rightarrow f~\text{is bijective and}~ \overline{f(A)}\subset f(\overline{A}), \forall A\subset E$$ Where $f:E\rightarrow F$ Help me please Thank ...
6
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3answers
615 views

Difficult general integral definite 0 to 1

$$\int_{0}^{1} \log^2(x)\cdot x^{k+1} dx$$ I tried integration by parts but it leads to an extremely complicated computation, which didnt lead me anywhere. Then I tried differentiating the beta ...
3
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1answer
55 views

Limes of $a_n = i^n$

Out of couriosity and for my understanding i want to ask: When i have the sequence $a_n = i^n$ While i is the imaginary number, i will of course have four accumulation points: $-1,1,-i,i$. So the ...
1
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1answer
38 views

Entire function that is strictly increasing on the real line

STATEMENT: Let $f$ be an entire function which maps the real line into the real line and the upper half-plane into the upper half-plane. Prove that $f$ is strictly increasing on the real line. ...
4
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0answers
103 views
+50

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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1answer
30 views

If $f\in L^1$ then is $g(x)=\int_x^{x+1} f(t)dt$ is in $C_0$?

Original questino is that for $1\le p<\infty$ if $f\in L^p$, and $g(x)=\int_x^{x+1}f(t)dt$ then, $g\in C_0$. I can prove for the case $1<p<\infty$ don't know how to for $p=1$. How can I prove ...
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2answers
52 views

How Find this limits $\lim_{n\to\infty}n(x_{n}-A)$

let equation$$e^x+x^{2n+1}=0,n=0,1,2,\cdots$$ I have prove this equation only have real root $x_{n}$, show that: $\lim_{n\to \infty}x_{n}=A$ is exsit ,and Find this value, (2):and Find ...
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2answers
31 views

Halley Method proof

$x_{n+1}$= $x_n$ - ${f(x_n)\over f'(x_n)-(f(x_n)f''(x_n)/2f'(x_n))}$ Let $m$ be a positive integer, Show that applying Halley's method to the function $f(x)= x^m -k$ gives $$x_{n+1}={(m-1)(x_n)^m + ...
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1answer
37 views

How find this Exponential constant $C$,if such this $Ax^C\le N(x)\le Bx^C$

interesting problem Let sequence $$a_{0}=x\in (0,1),a_{n}=a_{n-1}+a^3_{n-1},n=1,2,\cdots$$ and define $$N(x)=\min{\{n|a_{n}>1\}}$$ Assmue that there exsit postive constant $A,B$,and ...
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1answer
18 views

Limit of sequence definition - alternate form

We're given this: For all $N$ natural, there exists $0 < e < \frac{1}{N}$, such that for all $n > N$, $|A_n - L| < e$ Does this condition imply the series $A_n$ converges to $L$? Does ...
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1answer
46 views

Ascoli theorem's proof issues

For future reference; this is a portion of the proof of Ascoli's theorem in Anthony Knapps, Basic Real Analysis. STATEMENT: Let $(S,d)$ be a compact metric space. If $\left\{f_n\right\}$ is a ...
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1answer
16 views

Convergence of Remainder from Taylor Expansion

For a distribution function $F$ and its variance functional $T(F)$, it can be shown that the Taylor expansion of $T(F)$ at $F$ in the direction of the empirical distribution function $F_n$ gives the ...
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2answers
43 views

Why does it imply from that, that $f=0$ almost everywhere?

When $\int_G f=0, \forall$ open $ G$, we have that if $E\subset G$ then $\exists E \subset G$ such that $m(G \setminus E)< \epsilon$. So, $$\int_G f=\int_E f+\int_{G \setminus E } f$$ $\int_G ...
2
votes
1answer
48 views

Lyapunov Stability is a problem for me

Let be $ \dot{X} = F(X) $, $F \in C^1( \mathbb{R}^n)$, $P \in \mathbb{R}^n$ isolate singular point. Suppose there exists a family $S_{{r}_i}$ with $ i \in \mathbb{N}$ such that: $S_{{r}_i} = \left ...
0
votes
1answer
43 views

Determine if the series will converge uniformly

I have series $$\sum_{n=0}^{\infty} 1/(1 + n^2x^2) $$ for $x \in (0,1]$. I think this will not converge uniformly if we pick $x = 1/n$ since $x \in (0,1]$ and we can see that this will contradict the ...
0
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1answer
21 views

The second derivative of a holomorphic function

STATEMENT: Let $f:S\rightarrow S$ be a holomorphic function defined on the unit open square centered at 0. If $f(0)=0, f'(0)=1$ what is $f''(0)$? QUESTION: Can someone nudge me in the right ...
4
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0answers
208 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
0
votes
1answer
14 views

Second Order Necessary Condition for Optimality

Question: [See context below.] What would be the analog of the Thm when $f$ is only defined on, say, a domain $D\subset\mathbb{R}^n$? In that case we can't take a general ...
3
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0answers
40 views

Taylor series of $f(x) = \arctan(x)$ converges to $\arctan(x)$

I have to find out the Taylor series of $f(x) = \arctan(x)$ and prove that it converges to $f(x)$ for any $x \in (-1, 1) $. So far I determined the Taylor series to $T_f(x) = \sum ...