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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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v ...
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1answer
44 views

Metric topology: boundaries proof

I am stuck on what seems like a completely intuitive proof. A is a subset of X. ( + for disjoint union) I need to show, first, that (i) Closure of S = interior of s + boundary of S Then I am ...
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2answers
19 views

Help me correct this proof (Metric Space Topology)

Question: Let $X$ be a topological space. Prove if each point in $X$ is open, then each point in $X$ is closed. Proof: Suppose $\{x\}$ is open for $∀ x∈X$. Pick some arbitrary $x=x_1$. Clearly, ...
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2answers
61 views

Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective

Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective. $f$ is continuous and bounded below by $m>0$. Also $a$ belongs to $\mathbb{R}$ (reals).
2
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1answer
59 views

$f \in Aut (\mathbb{D})$ with two fixed points is the identity

I have been working on this homework problem for my complex analysis class for some time to no avail. The questions asks for me to show that for $f \in Aut (\mathbb{D})$ such that $f(z_1) = z_1$ and ...
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0answers
33 views

Time series analysis ARCH($m$) model

Consider a stationary ARCH($m$) model $a_t=\sigma_t\epsilon_t$, where $\sigma^2=\alpha_0+\alpha_1a^2_{t-1}+\cdots+\alpha_ma^2_{t-m}$. V. Consider an ARCH(2) model. Write down the predictor of ...
2
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0answers
46 views

How can I justify combining the limit of a limit into a single limit?

I'm trying to prove this formula for the second derivative of a $f:\, \mathbb{R} \to \mathbb{R}$: $$ \begin{align*} f''(a) &= \lim_{h \to 0} \frac{f'(a)-f'(a-h)}{h} \\ ...
3
votes
2answers
88 views

Show that $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$

Problem: I need to show that the power series $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$. I tried to prove it by contradiction by assuming that diverges for finitely ...
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1answer
72 views

Show f(x) = sin(1/x), for x does not equal 0, is differentiable for nonzero real numbers.

Show f(x) = sin(1/x), for x does not equal 0, is differentiable for nonzero real numbers. I was wondering if this would be enough to show the the previous statement: Let c<0 => c does not equal ...
3
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1answer
37 views

Find critical points of sin(x*y)

so I got this homework problem that I was having trouble with. The problem is: Let $f(x, y) = \sin(xy)$ defined on all of $\mathbb{R}^2$. Find the critical points of $f$ and classify them as local ...
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1answer
30 views

Prove central symmetry in (0.5;0.5) if f(1-x)=1-f(x)

I would like to prove that if $f:\left[0,1\right]\to\left[0,1\right]$ such that and $f\left(1-x\right)=1-f\left(x\right)$, then $f$ has a central symmetry at $\left(0.5,0.5\right)$. This is ...
2
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0answers
88 views

Proving differentiability

I'm trying to do Spivak's Calculus on Manifold excersise 2-4. It goes as follows: Let $g$ be a continuous real valued function on the unit circle $\{x\in\mathbb{R}^2:||x||=1\}$ such that ...
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4answers
76 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
0
votes
1answer
35 views

Is there a ratio (root) test for complex valued sequences?

I am reading a book about complex analysis (Complex Analysis by Ian Stewart/David Tall). Can we use the ratio test (or the n-th root test) for complex valued sequences as well? If so, how can this ...
2
votes
3answers
49 views

function has partial derivatives but is not differentiable

Can you write me an example of function which has partial derivatives but is not differentiable? How could I create and prove the function like that?
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2answers
52 views

Infinite multidimensional limits

I cannot find the definition of limit of a function of several variables involving infinity such as $(x,y) \rightarrow (0,-\infty)$
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1answer
46 views

Domains of Lipschitz class are domains of type A.

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. We say that $\Omega$ is of type $A$ if there exists a constant, $A$, such that \begin{equation} |\Omega\cap B_{\rho}(x_0)|\geq A\rho^n ...
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0answers
34 views

support of continuous function- counter example

I want to find two functions such that $sup(f)\cup sup(g)\nsubseteq sop(f+g)$, I´ve already found functions such that $sup(f)\cap sup(g)\nsubseteq sup(fg)$, but I have problems with the first one. Can ...
3
votes
1answer
113 views

Can the rational numbers be specified as an ordered field with <order property>?

In other words, (the opposite of my question is) does there exist an ordered field which is isomorphic as (as an ordered SET) to $\mathbb{Q}$? If not, does there exist an order property which ...
0
votes
1answer
36 views

Sets A and C with m-1 Elements

If A is a set with m elements and C is a set with one element, then A-C is a set with m-1 elements. What is a proof for this statement?
1
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1answer
28 views

Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
0
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0answers
27 views

How to find $\alpha_{k} \uparrow \infty $ such that $\sum |\alpha_{k}a_{k}|< \infty$.

let $\sum |a_{k}|< \infty$. How to show that there exists a sequence $\alpha_{k} \uparrow \infty $ such that $\sum |\alpha_{k}a_{k}|< \infty$. Could you please help me with this question.
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1answer
105 views

An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
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2answers
20 views

Proof Regarding Infinite Sets

If A is an infinite set and B is a finite set prove that A-B is an infinite set.
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1answer
69 views

The Baire space $\mathscr{N}$ is separable

Given is the Baire space $\mathscr{N}$. The elements are functions (or sequences) $f : \mathbb{N} \to \mathbb{N}$ and the metric $d$ is given by $d(f, g) = \frac{1}{k}$ if $f(i) = g(i)$ for all $1 ...
2
votes
1answer
75 views

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+...)$ as $x\rightarrow \infty$ I started by ...
4
votes
3answers
136 views

How can I check the convergence of the sequence? Does it diverge?

How can I check the convergence of the sequence $\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+\cdots+\frac{n}{\sqrt{n^2+n}}$? I think that it diverges,because it is bounded below from ...
1
vote
1answer
44 views

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$

Let $f_n \in C([0,2014]) $ Show, that if $f_n \rightrightarrows f$ and for all n $\int_{[0,2014]} ff_n dl_1=0$ then $ f\equiv 0$ I have no idea how to start the exercise like that.
0
votes
1answer
119 views

Cardinality of subsets of real numbers

Let A $\subset \mathbb{R}$ be a countable subset of $\mathbb{R}$. Prove that $\mathbb{R}$ and $\mathbb{R}$ \ A have the same cardinality. Use We proved that if X is in finite and B = {x1} ...
0
votes
3answers
36 views

can I find the limit in this way?

I have to check if the sequence $b_{n}=\frac{n+cos(n^2)}{n+sin(n)}$ converges.I thought that I could find it like that: $$-1 \leq sin(n) \leq 1 \Rightarrow n-1 \leq n+sin(n) \leq 1+n \Rightarrow ...
3
votes
2answers
50 views

Proving an integral using a series

If $f:(0,1]\rightarrow \textbf {R}$ is defined by $f(x)=2nx$ for $\frac{1}{n+1}\leq x \leq \frac 1n$ and $n$ is a natural number, assuming that $\sum_{k=1}^{k=\infty}1/k^2=\pi^2/6$, show that ...
3
votes
2answers
122 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set ...
2
votes
1answer
92 views

Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$.

It is given that $a_{n}$ is a positive and decreasing sequence. Show that if $\sum_{n=1}^{\infty}a_{n}$ converges, $\lim_{n \to \infty}na_{n}=0$. That's what I tried.Could you tell me if it is right?? ...
0
votes
1answer
51 views

How to ensure extreme? — using Extreme Value Theorem

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem? For example, this multivariate case: $$ ...
0
votes
1answer
63 views

Is the function $f(x)=[x]\sin(πx)$ continuous?

I have a question..Is the function $f(x)=[x]\sin(πx)$ continuous? How can I check it??
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vote
1answer
36 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
4
votes
1answer
507 views

If $f$ is twice differentiable and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
2
votes
0answers
88 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
6
votes
1answer
58 views

How to determine measure from the integral equation?

Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...
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vote
2answers
112 views

Finding the $n$th term of some sequences given by its few first terms

Does anyone know how to find the $n$th term of the following sequences: $(1)\ $ $1,0,0,0,1,0,0,0,1,0,0,0,1,....\\$ $(2)\ $1,0,0,0,-1,0,0,0,1,0,0,0,-1,....\ $ $(3)\ $ $1,0,-1/2,0,1/3,0,-1,4,0,...\\$ ...
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1answer
77 views

Radius of Convergence of $\sum\ z^{n!}$

Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number? I tried to use the definition: $\frac{1}{R}=Limsup|\frac{a_n+1}{a_n}|$, but I ...
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votes
1answer
45 views

How do I expand absolute values?

If we have this expression: $$f = uu-\left( u + \frac{\partial u}{\partial x} \delta x \right) \left( u + \frac{\partial u}{\partial x} \delta x \right)$$ we can expand it to this: $$f = u^2-\left( ...
0
votes
1answer
83 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
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0answers
30 views

A question about Helly's Selection Principle

Im reading Chapter13 of Carothers' Real Analysis, 1ed talking about functions of bounded variation. Here is a proof, See. {$f_n^{(k)}$($x_k$)} is a little like nest interval of Bolzano-Weierstrass ...
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1answer
47 views

$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ integration

$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ I have no idea how to do it:( What is more I don't know how to start it... Any ideas? I will be pretty grateful. I tried some tricks but no one works.
3
votes
1answer
113 views

Transpose of continuous operator bounded below is surjective

Suppose $T: X\rightarrow Y$ is a continuous linear transformation between 2 normed vector spaces such tha $\|Tx\| \geq C\|x\|$ for some $C > 0$. Why must the transpose $T^{\ast}: Y^{\ast} ...
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0answers
19 views

convex function - prove

Using a convex function prove: for all $x \ge -1:$ $2\sqrt{x+1} \le e^x+1$ so: $f(x)=e^x+1-2\sqrt{x+1}$ $f''(x)=e^x+\frac{1}{2}( \frac{1}{2}) ^\frac{3}{2}$ is it correct? What I should do next? ...
2
votes
1answer
38 views

Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...
0
votes
2answers
40 views

Help-limits with integral part

Can you help me to find the limits? $$\lim_{x \to 0^{+}}\frac{x}{a}\left[\frac{b}{x}\right] , \quad \lim_{x \to 0^{+}}\frac{b}{x}\left[\frac{x}{a}\right], \quad a,b>0$$ And what happens when $x ...
1
vote
2answers
35 views

Check if the sequence converges?

Given the sequence $$\frac{1+2+...+n}{n+2}-\frac{n}{2}$$ I am asked to check if it converges. How can I do this? One way is to check if the sequence is bounded and monotonic, right? But how could we ...