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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69 views

What is the name of this theorem (analysis)

Please Can you remind me with the name of this theorem and its reference book Assume that $f : \mathbb{R} \to \mathbb{R} $ is continuous and integrable function over $\mathbb{R}$ then we've ...
3
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1answer
93 views

Embedding 2nd countable, zero-dimensional Hausdorff space in the Cantor space

The Cantor Space $2^{\mathbb N}$ is the space of all infinite $0$-$1$-sequences with the metric $d(x,y) = 0$ for $x=y$ or $d(x,y) = 1/k$ where $k$ is the least integer such that $x_k \ne y_k$. Now I ...
2
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0answers
47 views

Set of critical points of polynomial: why finite

Let $P$ be a $C^\infty$ real constant coefficients polynomial defined on $\mathbb{R}^n$ and let $Z(P)$ be its set of critical values, that is $Z(P)=P(\{\xi\in\mathbb{R}^n: \nabla P (\xi)=0\})$. I read ...
3
votes
2answers
138 views

calculation of Stefan's constant

In the calculation of Stefan's constant one has the integral $$J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx$$ which according to Wikipedia is equal to $\frac{\pi^4}{15}$. In this page of ...
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1answer
42 views

tangent functions

let $A,B \subset \Bbb K $ $f:A \rightarrow \Bbb K $ $ g:B \rightarrow \Bbb K $ $ a \in \Bbb K $ We say f is tangent to g if $ a \in int (A \cap B)$ and $ \forall \epsilon > 0 \exists \delta ...
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1answer
40 views

prove that the following interesting problem

Let $L^{\infty}=L^{\infty}(m)$, where $m$ is the lebesgue measure on [0,1].Show that there is no non zero bounded linear functional $\Lambda$ on $L^{\infty} $ that is 0 on $C([0,1])$, and that ...
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1answer
38 views

convergence, but not almost uniform convergence.

Can anyone help me with finding an example of function convergence, which is not an almost uniform convergence?
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2answers
75 views

Series $\sum_{n=1}^{ + \infty}\sum_{k = 1}^{+ \infty} \frac{1}{(\sqrt{k^{2}+n^{2}})^{1+\epsilon}}$

Is the series $$ \sum_{n = 1}^{\infty}\sum_{k = 1}^{\infty} {1 \over \left(k^{2} + n^{2}\right)^{\left(1+\epsilon\right)/2}} $$ convergent for $\epsilon > 0$ ? I don't know how to manage the double ...
2
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3answers
73 views

All convergent sequences are bounded confusion

We proved that all convergent sequences are bounded. However, when proving the following: If $x_n$ converges to $x$ and $y_n$ converges to $y$, then $\dfrac{x_n}{y_n}$ converges to $\dfrac{x}{y}$, ...
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2answers
131 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
2
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1answer
51 views

curl free fields are gradient fields.

I am supposed to show that a curl free field $f:\mathbb{R}^3\rightarrow \mathbb{R}^3$ (such that $\nabla \times f=0$) is always a gradient field of some potential $\phi$. A hint was given by saying ...
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5answers
104 views

Prove $\exp(x) \geq 1+x \forall x \in \mathbb{R}$

I've managed to prove the statement for $x \geq 0 $ and $x \leq -1$ but I can't manage to construct a proof for $ -1 < x < 0 $ My lecture done it by proving $ \exp(x) - (1+x) = \displaystyle ...
-1
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2answers
63 views

Power series with radius of convergence 2 that diverges at both -2 and 2?

I'm looking for a real power series that has radius of convergence 2 but diverges at both 2 and -2. Any idea? Thank you!
3
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1answer
179 views

Finding a radius of convergence of power series

I have to find the radius of convergence of some power series but I find myself in trouble for three of them : the series are $\sum2^kx^{k!}$ $\sum\sinh(k)x^k$ $\sum\sin(k)x^k$. For the first ...
0
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1answer
87 views

Function whose discontinuity points are a prefixed $F_\sigma$ set in $\mathbb{R}$.

I have been reading Carothers' book on real analysis and I found the following question on page 130: If E is an $F_\sigma$ set in $\mathbb{R}$, is $E=D(f)$ for some $f:\mathbb{R}\rightarrow ...
1
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1answer
72 views

Ratio of coefficients for Laurent series expansions [duplicate]

Let $f$ be analytic in the disk $D(0,2)$ except for a pole of order $1$ at $z=1$, and let $$f(z)=\sum_{k=0}^\infty a_k z^k$$ be the series expansion for $f$ in the disk $D(0,1)$. Prove that ...
25
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6answers
1k views

How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?

I'd like to find out why \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align} I tried to rewrite it into a geometric series \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = ...
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2answers
154 views

Prove a metric space is compact, if every infinite subset in it has a limit point.

This is an exercise in W. Rudin's book. Actually my question is, what is an open cover of a metric space? Since an open set is embedded in a certain metric space, how can it cover those points which ...
1
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1answer
60 views

show $\int g\log (g/f)$ is $0$ only if $g=f$ almost everywhere

Question: Suppose that $f$ and $g$ are two probability density functions, show that $\int g\log (g/f)$ is always non-negative and equals to $0$ $\it only\ if$ $\ g=f$ almost everywhere. I have ...
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1answer
109 views

continuous function $h(x)=\sup\{f(t)\colon t\le x\}$

Assuming $f:[a,b]\to\mathbb{R}$ is continuous, how do you prove that the following function is continuous? $$h(x)=\sup\{f(t): t\leq x\}$$
3
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4answers
397 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
2
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1answer
116 views

Non second countable space which has limit point not expressable as the limit of a sequence

Theorem: If a topological space $(X, \tau)$ has a countable base (2nd countable), then for every $Y \subseteq X$ its closure $cl(Y)$ is the set of limit points of sequences $(x_j)_{j\in \mathbb N}$ ...
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1answer
48 views

Does modifying metric of a space change the limit of a sequence in it?

Im reading Rudin's Principles of Mathematical Analysis and get this idea. In my opinion, doing it will remove or add some limit points in the space because this modification may change the radius of ...
5
votes
2answers
598 views

$f$ is a real function and it is $\alpha$-Holder continuous with $\alpha>1$. Is $f$ constant?

Maybe this is a well know result, however, I could not find it. Before stating it, let me write here a well know result (at least for me) Assume that $\Omega\subset\mathbb{R}^N$ is a open domain ...
6
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5answers
332 views

Puzzled by $\displaystyle \lim_{x \to - \infty} \sqrt{x^2+x}-x$

I am preparing for the next Semester and therefore review a few of my Analysis I limits, I have found this example in C.T. Michaels Analysis I: Compute $ \displaystyle \lim_{x \to - \infty} ...
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0answers
51 views

A “complementary” topology

If $(X, \tau)$ is an Alexandrov topology then arbitrary intersection of open sets are open, and likewise arbibtrary unions of closed sets are closed, so we can define a topology $(X, \tau')$ as $$ U ...
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0answers
48 views

Premetrics, where are they useful?

Wikipedia defines a premetric as a function $d : X\times X \to \mathbb R$ such that $d(x,y) \ge 0$ $d(x,x) = 0$. For me these axioms are so weak that I am wondering where they are used, do you ...
0
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1answer
31 views

Bounding an $L^2$ function (with Cubic…)

Suppose we have a function $f(x) \in H^1(R)$. Is it possible to bound $\int f^4 dx \leq \|f\|_{H^1(R)} ^3$. Likewise, it is possible to say $\int f^3 dx \leq \|f\|_{H^1(R)} ^3$? Thanks.
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2answers
90 views

How prove this $f'(c)=2c(f(c)-f(0))$ [duplicate]

let $f(x)$ is continuous on $[0,1/2]$, and derivative on $(0,1/2)$,such $$f'(1/2)=0$$ show that there exsit $c\in(0,1/2)$, such $$f'(c)=2c(f(c)-f(0))$$ My try: let ...
2
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2answers
113 views

Example of a Lipschitz function $f$ on $[0,\infty ]$ such that its square $f^{2}$ is **NOT** Lipschitz?

Question: Can anyone come up with an example of a Lipschitz function $f$ on $[0,\infty ]$ such that its square $f^{2}$ is NOT Lipschitz? Thanks!
4
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1answer
120 views

A strange limit

I have to compute the following limit $$\lim_{x\to \infty} \frac{x\log(1+\tan(8x))}{6^{x^2}-1}.$$ Is that a problem if $\log(1+\tan(8x))$ is not always defined? Why not?
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0answers
73 views

The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
2
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3answers
239 views

Gamma function can be extended as a meromorphic function

Let $$\Gamma (z)= \int_{0}^{\infty} e^{-t}t^{z-1}dz$$ for $\Re z\gt 0$ Can be extended as a meromorphic function to the entire complex plane with simple poles at non positive integers. How can I ...
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2answers
165 views

Example of continuous function that isn't uniformly continuous and isn't 1/x

I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and ...
1
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0answers
83 views

What is a kernel? [duplicate]

It seems the term 'kernel' pops up all over the place and has different meanings everywhere. Is there a unifying feature to all things called 'kernel' in math that would better help me understand what ...
0
votes
1answer
30 views

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$

Find an integral expression for $\Gamma'(z)$ for $\Re z\gt 0$ I know the result. But I dont know how to show this step by step.
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4answers
427 views

Prove that if all the vertices of a graph have degree 3, then the graph must have a cycle

Hello can you help me to prove this. The hint for the problem is: Think of what it means for a graph to have no cycles. So I believe this will be a contrapositive proof, but still could not do it.
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1answer
160 views

Raabe's test, logarithm test, Bertrand test

Raabe's test, logarithm test and Bertrand test are the most commonly used criterion in calculus. The relationship between them is quite interesting. Here is how: $\sum\limits_{n=1}^\infty a_n$, ...
1
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1answer
39 views

Stronger condition then ultrametric condition on metric space

A metric space $(X,d)$ is called an ultrametric space if it is a metric space and fulfills the stronger triangle inequality (see Wikipedia) $$ d(x,y) \le \max\{ d(x,z), d(z, y) \}. $$ Examples are ...
0
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1answer
80 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
1
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1answer
68 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
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1answer
272 views

Clopen sets in $\mathbb{R}^n$

Are there any further clopen sets in $\mathbb{R}^n$ besides $\mathbb{R}^n$ and the empty set? $\mathbb{R}^n$ shall carry the topology that is induced by the canonical metric. So far, I could not find ...
3
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0answers
67 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
2
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3answers
53 views
1
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2answers
121 views

Analysis Open and Closed Sets

I have the following question and i'm not sure how to go about proving whether sets are open closed, or both. Which of the following sets are open and which are closed? $1)\ [1,2] \cup [3,4] \ in \ ...
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2answers
118 views

Is this set open or closed or neither of both in $\Bbb{R}$?

I'm solving this problem in Rudin's Principles of Mathematical Analysis: and came up with a idea that is $\{p \in \Bbb{Q} \mid 2 < p^2 < 3 \}$ open or closed in $\Bbb{R}$ or neither of both? ...
2
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3answers
214 views

Limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$

I have to find the limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$ as $x\to\infty$. I was studying this function finding its real graph but to do this I need to know where the function goes as $x$ ...
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1answer
200 views

Understanding Stokes' theorem

Stokes' theorem( here I am only talking about the special $\mathbb{R}^3$ case) contains a line integral $\int_{\partial S} \langle f, \tau \rangle ds$. (Actually, I would be confident if somebody ...
7
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1answer
113 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
1
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1answer
135 views

left sided limit of a monotonic function

Definition: Let $X,Y\subset\mathbb R$ and $f \colon X\to Y$. Let $x$ be a limit point of $X\cap(-\infty,x)$. If for all sequences $x_n\in X\cap(-\infty,x)$ it holds $f(x_n)\to y$, then $\lim_{x\to ...