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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1answer
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$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki

Paraphrased from Munkres' Topology: Lemma 9.2. Given a collection $\mathcal{A}$ of nonempty sets, there exists a choice function \begin{equation*} f: \mathcal{A} \to \bigcup\limits_{A \in ...
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4answers
69 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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0answers
36 views

Sphere in finit dimensional space

please How to prove that the injection $i: S^{m-1}\rightarrow \mathbb{R}^m$ is homotopic to a constant ? Where $S^{m-1}=\{x\in \mathbb{R}^m, |x|=1\}$ Thank you.
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1answer
17 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
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3answers
24 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
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1answer
72 views

Does $\int_{0}^{1}x^nf(x)\, dx=0$ imply that $f=0$ a.e. without assuming $f \in C[0,1]$?

Suppose that $f \in L^{1}[0,1]$ and $\int_{0}^{1}x^nf(x)\, dx=0$ for $n=0,1,2,\dots$ Does that imply that $f=0$ a.e.? I think that there will be a counterexample but it is hard to find out.
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1answer
25 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
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0answers
35 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
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1answer
26 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
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0answers
16 views

Find a probability of $n$ event happening from $m$ types

The question is: to find a sum $$ S=\sum\limits_{n_1+n_2+\ldots+n_m = n,\ n_i=0,1,\ldots,n} p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}, $$ where $p_i\in[0,1]$. UPDATE. This issue has no probabalistic ...
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1answer
45 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
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0answers
28 views

Variational Inequalities and how they are used?

I am doing undergrad research in this field next semester and I have never heard of this topic before. I tried wikipedia and reddit for help but nothing seems to help. I just want to know what I'm up ...
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1answer
24 views

Limit of arc-length of a curve

Let $L(f)$ denote the length of a curve $f$, if $f = \lim\limits_{n\to\infty} f_n$ then do we necessarily have that $L(f) = \lim\limits_{n\to\infty} L(f_n)$? I assume that we will have some continuity ...
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3answers
265 views

Image of open set is not open?

I'm confused by the proof that $\epsilon$-$\delta$ continuity is equivalent to open-set continuity. One can prove that a function is $\epsilon$-$\delta$-continuous if and only if the preimage of any ...
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1answer
36 views

Hint on metric space [on hold]

I want show that $(a_n):=d(x_n,y_n)$ converges, if $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are cauchy sequences in $(X,d)$. Here is what i do; From the hypothesis, $(a_n)$ is bounded, because ...
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2answers
55 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
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1answer
41 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
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1answer
41 views

Radius of convergence, prove that $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely for every $z \in \mathbb{C}$ with $|z| < R$

A proof of this is given in my lecture notes as follows: We define $R$ to be $\sup \{|z| \in \mathbb{R} : \sum |c_k z^k|$ converges $\}$ when the supremum exists. Prove that $\sum |c_k ...
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1answer
44 views

periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
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25 views

statistical analysis [on hold]

Two independent random samples of annual starting salaries for individuals with masters and bachelors degrees in business were taken and the results are shown below ...
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2answers
21 views

Proving Lipshitz continuous over a convex set with Projection Operator

Suppose a problem $$\min_{x \in \mathbb{R}^{n}} f(x)$$ subject to $x \in \Omega$ which is a closed and convex set. If $\nabla f(x)$ is Lipschitz continuous in $\Omega$, then prove that $$e(x) = x - ...
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1answer
34 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
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2answers
57 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
1
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1answer
92 views

Given $|f'(x)|\leq r<1$ show that $f(x)=x$ is unique solution

Suppose that $|f'(x)|\leq r<1, \forall x\in R$. How do I show that the equation $f(x)=x$ has a unique solution?
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1answer
31 views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write ...
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1answer
41 views

Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
2
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3answers
104 views

How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$? [duplicate]

How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$. If we let $t=\tan\theta$, then the integral becomes to ...
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0answers
17 views

Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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0answers
25 views

Inhomogeneous ODE (2nd order) - question to Laplace-transformation?

I've the following inhomogeneous second order ODE: $$a_1\cdot u(t) + a_2\cdot u'(t) + a_3\cdot u''(t) = b_1\cdot y(t) + b_2\cdot y'(t) + b_3\cdot y''(t)$$ The parameters $a_i$ and $b_i$ are ...
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3answers
61 views

Showing instablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
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0answers
23 views

Weak convergence in Lp [on hold]

got a little problem with this ex. I could use some help. Let $U := \Pi_{i=1}^d(a_i, b_i) \subset \mathbb{R}$ ($a_i < b_i$ for each $i$) and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us ...
2
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1answer
44 views

Is the polynomial a zero polynomial?

Let $p(x)$ be a polynomial over $\mathbb{R}$ with $deg[p(x)]\leqslant n$. If $p(1)=p(2)=\cdots = p(n+1)=0$, then will the polynomial be necessarily a zero polynomial? i.e., if a polynomial of degree ...
2
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2answers
93 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
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2answers
63 views

Does $\int_0^\infty |f'(x)| dx < \infty$ conclude $\lim_{x\to \infty} f(x)<\infty $

$f:[0,\infty) \to \mathbb R $ is $C^1$ and $$\int_0^\infty |f'(x)| dx < \infty$$ then can we prove that $\lim_{x\to \infty} f(x)$ exists and $$\lim_{x\to \infty} f(x)<\infty $$ My attempt: ...
5
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2answers
52 views

How prove there exsit $\xi\in (0,1)$ such $|f(\xi)|\le|f'(\xi)|$

let $f:[0,1]\to \mathbb{R}$ be a differentiable function such that $f(1)=0$, Prove that there is $\xi\in(0,1)$, such that $$|f(\xi)|\le|f'(\xi)|$$ My idea: I think we can prove there exsit ...
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2answers
26 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
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1answer
9 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
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2answers
35 views

Knowing a function while only knowing its partial derivatives?

So again we study a physics course without studying mathematics course We are in the work energy chapter , and I'd like to know if you can know the function $f(x,y,z)$ if you know all of its partial ...
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3answers
22 views

Prove of a Landau-equalities

I have to prove or disprove the following Landau-equalities: $$ O(f+g) = O(max(f,g))$$ and $$O(f-g) = O(min(f,g))$$ with $f,g: \mathbb N \to \mathbb R^+$ . To show equality of two sets, one has to ...
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1answer
25 views

A question about continuous functions

Let $f:B \longrightarrow \mathbb{R}^n$ a continuous and injective function from closed ball in $\mathbb{R^n}$ to $\mathbb{R}^{n}$. I'd like to know $f$ has to maps the boundary $\partial B$ in the ...
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1answer
36 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
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0answers
19 views

key contributions of Augustin Louis Cauchy to analysis

I am wondering if any one could organize some key contributions of Augustin Louis Cauchy to analysis properly ?
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2answers
46 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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0answers
23 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
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1answer
30 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
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2answers
57 views

If f is differentiable with a continuous derivative function, then the set of critical points of f is closed.

If f is differentiable with a continuous derivative function, then the set of critical points of f is closed. Is this a true statement? I'm kinda lost.
2
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1answer
59 views

Is sinus an unique function?

On $\mathbb{R}$, is sinus the unique $C^{\infty}$ function f with all is derivate and itself between -1 and 1 and also $ \frac{df}{dx}(0)=1 $ ?
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1answer
29 views

If a function has asymptote and the derivative does not, then its second derivative is not bounded

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be a function with second derivative everywhere in is domain. Prove that if $\lim_{x\rightarrow\infty}f(x)=b \in \mathbb{R}$ and ...
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1answer
58 views

2048 Tournament Word Problem [on hold]

Problem: There was a 2048 tournament. And after the results were counted and were announced, winners got candy. The 1st place got 2 less than a third of candy, 2nd place got 4 less than half of the ...
1
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2answers
43 views

Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$

$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...