Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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1answer
33 views

Small question about convergence

I have a small question: if i have that $$\int_0^{+\infty}p(t)|u'_n(t)-u'(t)|^2dt\rightarrow 0$$ is it true that $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt $$ ...
0
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0answers
50 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
2
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2answers
48 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
3
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2answers
69 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
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0answers
23 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
2
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1answer
86 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
0
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1answer
43 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
2
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1answer
29 views

How to proof the limit is convergent for arbitrary initial state?

$\{a_n\},\{b_n\},\{c_n\},\{d_n\}$ is series. And $d_n=c_{n-1},c_n=b_{n-1},b_n=a_{n-1},a_n=b_{n-1}+c_{n-1}$ how to proof for any $a_0,b_0,c_0,d_0$ belong to $Z^+$, $\lim\limits_{n\rightarrow \infty} ...
0
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0answers
21 views

Fuel efficiency word problem

I am writting a paper for school and came across this fact, A single Boeing 747 spends an avergae of 32 minutes per flight taxiing, departing, and landing. During this time, it produces 87 kilograms ...
1
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1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
3
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0answers
41 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
2
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1answer
87 views

Can't understand a step in the advanced calculus book by thomas P. Dence

On page 9 of the Advanced calculus book by Thomas P. Dence he defines the set $S_1 = \{x\in \Bbb Q: x\geq 0, x^2 \leq 5\}$ and he said that the supremum $U$ is less than $u=3$ and that $U$ is ...
0
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0answers
37 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
2
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0answers
34 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
3
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1answer
59 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
0
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1answer
19 views

Find a weak derivative of this function

Let $B(0,1)$ be the open unit ball in Euclidean space $\mathbb R^2$ and $(a_n)_{n=1}^\infty$ be a dense subset of $B(0,1)$. I wish to show for fixed $s\in (0,1)$ the function $$ ...
1
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1answer
21 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
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0answers
21 views

Initial value problem with intermediate value

The Picard Lindelöf theorem I know always assumes that we specify the value at the left end of the time-interval Picard Lindelöf. Is it true that $x'(t) = f(x(t))$ has a unique solution, in an open ...
0
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1answer
34 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq ...
2
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1answer
52 views

Generalizing the Monotone Subsequence theorem

In proving the Bolzaono-Weierstrass theorem, one proves the lemma that every infinite real sequence has a(n infinite) monotone subsequence. In all of the proofs I've seen so far, this is done by ...
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1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
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1answer
23 views

Part limit in Complex Analysis

Can somebody explain to me why this limit: $lim(n(\sqrt[n]{r} - 1))$ converges to $logr$ when $n \rightarrow \infty$? Thanks a lot! This was a part of a limit in Complex Analysis asking to show ...
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0answers
14 views

why the set of ring two-sided ideals coincides with the set of algebraic two-sided ideals in $B(H)$

Why does the set of ring two-sided ideals coincide with the set of algebraic two-sided ideals in $B(H)?$
4
votes
2answers
46 views

Show that f is measurable

Let $U$ be a open Set of $\mathbb{R} \times [0,\infty]$ and let f be defined as $$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$ How can I show that $f$ is ...
1
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1answer
37 views

A direct application of inverse function theorem

Let $f:U\longrightarrow \mathbb{R}^n$ a function with $U\subset \mathbb{R}^n$ open, $f$ injective of class $C^1$ (i.e. continuous with the first derivate continuous) such that $\forall x\in U$ the ...
2
votes
3answers
86 views

Intuitive Numerical Analysis Texts

Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ...
4
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2answers
154 views

Show a function is monotonically decreasing.

Show that $f(x)=\dfrac{\sin x}{x}$ is monotonically decreasing on $[0,\frac{\pi}{2}]$ I'm trying to show that $f'(x)\leq0$ to show it's monotonically decreasing. So $f'(x)=\dfrac{x\cos x-\sin ...
4
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1answer
75 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
1
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2answers
85 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty)$ be a nonnegative, ...
2
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0answers
47 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
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1answer
19 views

Does convergence of $S_{n!}=\sum{1}^{n!}a_{k}$ implies $a_{k}$ approaches to zero?

**Let $\{{a_n}\}$ be a sequence . Define $ S_{n}=\sum_{1}^{n}a_{k}$. Does convergence of $\{{S_{n!}}\}$ implies $\lim{a_{n}}=0$ as $n\rightarrow\infty$. **
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0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
3
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6answers
128 views

Constructive proof of Euler's formula

In most textbooks on the subject I have seen, Euler's formula (by which I mean $e^{ix}=\cos(x)+i\sin(x)$) is proved by applying either differential equations or the power series of sine and cosine. ...
2
votes
2answers
75 views

How find the function $f(x)$ such $\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$

let $f(x)$ is Continuous function on $[0,\pi]$,and for infinite positive integer $n$ such $$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$$ Find the $f(x)$? I think the answer is $f(x)=c$?,But maybe have ...
0
votes
1answer
46 views

A problem with the density of sin (N) [duplicate]

Actually I can prove the fact that $\sin(\mathbb{Z})$ is dense in $[-1,1]$ using the result that "any non trivial subgroup of the additive group of $\mathbb{R}$ is either cyclic or is dense in ...
1
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1answer
63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
1
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3answers
80 views

How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
2
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3answers
93 views

Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
3
votes
2answers
47 views

Every interval in real numbers has a rational and irrational

How can you prove the following: where $a\not= b$, show every $[a,b]$ of R has a rational and irrational number. The context for my question is as follows. In my intro calculus class, we were showing ...
2
votes
3answers
90 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
3
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2answers
44 views

Test for convergence $\sum_{n = 2}^\infty \frac{1}{(n+1)\ln^2(n+1)}$ [duplicate]

Test for convergence $$\sum_{n = 2}^\infty \frac{1}{(n+1)\ln^2(n+1)}$$ Here's my attempt! I decided to use the integral test for this. $$\frac{1}{(n+1)\ln^2(n+1)} > \frac{1}{(n+1)^2\ln^2(n+1)}$$ ...
1
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2answers
85 views

Test for convergence $\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$

Test for convergence $$\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$$ So far I attempted to use the ratio test, but I'm stuck on what to do after. ...
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1answer
19 views

Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets?

Let $U$ be an open subset in $\mathbb R^n$. Then there exists a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and ...
0
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1answer
32 views

Need a formula to calculate rating, based on 3 factors?

I have multiple users.... millions actually, each user has two values: "Talk About Theft" and "Stole Something". "Talk About Theft" - 1 to Infinity "Success Rate" - 0 to Infinity I need to have ...
5
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0answers
132 views

Why is the derivative of the translates of a measure measurable?

Let G be a topological group and X a measure space. Let $G \times X \rightarrow X$ be a measurable group action, $\mu$ a $\sigma$-finite measure on $X$, and $g\mu$ (for any $g \in G$) the measure ...
5
votes
1answer
94 views

$E \subseteq [0, 1]$, $m(E) > 0$. Show that there are $\alpha$ and $\beta$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

This was originally a proof verification question, but I have since moved the proof to an answer as discussed on meta. I still welcome comments on the proof as well as any alternative proofs. ...
0
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1answer
31 views

Volume of a solid in R3

How can I find the volume of this field? : $$ G=\{\left. (x,y,z) \, \right| \, x^2+y^2+z^2 \le 16 \wedge 1 \le x+y+z \le 2\}. $$ Can anybody help me? Thanks.
2
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0answers
38 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
0
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0answers
30 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
0
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0answers
36 views

Question about the Continuity of the derivative of a function

I have that $$\begin{cases}-(p(t) u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\end{cases}$$ where $f:[0,+\infty)\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and $\displaystyle\frac1p\in ...