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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2
votes
1answer
24 views

Automorphisms of the upper half plane

STATEMENT: Suppose $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ are two pairs of three distinct points on the real axis with$$x_1<x_2<x_3 \;\;\;\;\text{and} \;\;\;\;\;y_1<y_2<y_3$$ Prove that ...
2
votes
1answer
21 views

Proof about outer measure. For an interval $I$, $|I|_e=v(I)$?

My question is when proving $|I|_e \ge v(I)$, why cannot I conclude from $S={I_k}_{k=1}^\infty$ is a cover of $I$, then $v(I)\le \sigma(S)$, so $v(I)\le inf \sigma(S)=|I|_e$? Why do we need the ...
0
votes
0answers
15 views

Show that for a sequence $p_{n}$ of real numbers, $\limsup p_{n} < +∞$ iff $p_{n}$ is bounded above.

Show that for a sequence ${p_{n}}$ of real numbers, $\limsup {p_{n}} < +∞$ iff ${p_{n}}$ is bounded above. My Partial Proof: $\Leftarrow$ Given ${p_{n}}$ is bounded above, prove $\limsup {p_{n}} ...
1
vote
2answers
28 views

counter-example: aboslute convergence => convergence in incomplete vector space

Is the following statement true? Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$ and $\sum_{k=0}^\infty \lVert x_k\rVert$ convergent. Then $\sum_{k=0}^\infty x_k$ is also ...
5
votes
1answer
41 views

How prove this limits is exsit $\displaystyle\lim_{n\to\infty}x_{n}$

let $f:[a,b]\to [a,b]$ be Continuous function,Assmue that sequence $\{x_{n}\}(n\ge 0)$ such $$x_{0}=x,x_{1}=f(x_{0}),x_{2}=f(x_{1}),\cdots,x_{n+1}=f(x_{n}),\forall n\in N^{+}$$ and ...
1
vote
0answers
14 views

higher moments of a r.v., combinatorical problem

I'm studying the book of Rick Durrett, I want to understand the proof of the Erdös Kac central limit theorem, so I also need to understand the Lindeberg-Feller theorem: for every $n \in \mathbb{N}$ ...
0
votes
3answers
12 views

corresponding system of equation of the given solution space

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following: Consider the vectors in $R^4$ defined by ...
2
votes
3answers
37 views

Strongly convergent to zero in $L^2$ but $H^1$ norm not vanishing

Let $\Omega$ be some open, bounded, smooth subset of $\mathbb{R}^n$. I'm wondering whether it is possible for a sequence of functions $f_n:\Omega \rightarrow \mathbb{R} $ to be strongly convergent to ...
0
votes
3answers
61 views

$\lim_{n\to\infty} \dfrac{a^n}{n!} = 0$ [duplicate]

Show that for any a in $\mathbb{R}$ $$\lim_{n\to ∞} \frac{a^n}{n!} = 0. $$ Hint: There exists a $n\in\mathbb{N}$ such that $n > |a|.$ I really do not know how to begin here with the proof and ...
0
votes
0answers
17 views

Answer to Plane Trigonometry Ex XLIX Q16?

If $\alpha, \beta, \gamma, \cdots$ be the roots of the equation $sin(mx) - nx cos(mx) = 0$ prove that $\tan^{-1}\frac{x}{\alpha} + \tan^{-1}\frac{x}{\beta} + \cdots + \tan^{-1}\frac{x}{v} = 0$. The ...
2
votes
0answers
32 views

Number of zeros of solution of differential equation

Assume $a>0$ , $b>0$ and there exists a non-zero function $\phi(t)$ such that is the solution of $y''+(a+bcos(2t))y=0$ and on $(-\pi/2,\pi/2)$ has $2n$ zero. prove that $(2n-1)^2\le a+b $ ...
1
vote
1answer
33 views

Weak-* convergence in Sobolev spaces

Let's consider a sequence $\{f_n\}_n$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$. What does it mean that $\{f_n\}_n$ converges weakly-* in $L^\infty(0,T;H^1(\mathbb{R}^n))$?
0
votes
0answers
19 views

Circuit to state space model [on hold]

I have this circuit and I must represent it in state space. I know the process because I have done similar ones but having trouble with this one. I have found the Kirchhoff equtations for the 2 loops ...
-1
votes
1answer
18 views

Comput Spectrum of Idempotent

Let A be a unital banach algebra and a in A if a is idmepotent and a do not equal to 0 and 1 then the spectrum of a = {0,1}??
3
votes
0answers
21 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
1
vote
1answer
20 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
1
vote
1answer
25 views

function of three variable is even than $f(a,b,c)=f(|a|,|b|,|c|)$

I used in the proof of Hlawka's Inequality you can find the link here Hlawka's Inequality that's if i have function of three variable is even in each variable, so that : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...
1
vote
1answer
18 views

Volume of a body bounded by planes

I'm just after the lecture about Fubini's theorem. And I "don't feel" how to do some exercises. Here is an example: What is the volume of the body bounded by: the graph of the function ...
0
votes
1answer
20 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: ...
0
votes
0answers
14 views

Find an Upper bound of absolute value (triangle equality application)

Given the functions f(x) and g(x), how can I find a bound for the absolute value \begin{equation} \|f(x)-g(x)-2\| \end{equation} is it correct to say $\|f(x)-g(x)-2\|\leq ...
1
vote
1answer
28 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...
0
votes
0answers
9 views

SEnsitivity Indices are non zero

I am trying to compute the sensitivity indices (SI) of a function using Monte Carlo simulation. I had written a matlab code that perform the computation directly and just return the final answer of my ...
0
votes
2answers
40 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
3
votes
3answers
57 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
0
votes
1answer
36 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
1
vote
1answer
28 views

\lim_{n\to ∞}c_n * a_n = 0

Let $(a_n)$ be a sequence in R that converge to 0 and $(c_n)$ be a bounded sequence. Show that $$\lim_{n\to ∞}c_n * a_n = 0$$. Obviously $\lim_{n\to ∞}c_n * a_n $ = $\lim_{n\to ∞}c_n * \lim_{n\to ...
0
votes
2answers
22 views

Prove that Recurvisv limits are equal [on hold]

Prove that: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} a_{n-1}$ Ideas?!?
4
votes
4answers
120 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
3
votes
1answer
40 views

Is the limit function $f$ continuous if $f_n(x_n)\to f(x)$? [duplicate]

Let $I\subset\mathbb{R}$ be an interval and let $(f_n)$ be a sequence of continuous real-valued functions on $I$. Consider the following statements: $f_n\to f$ uniformly; For every sequence $(x_n)$ ...
1
vote
2answers
43 views

What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
3
votes
0answers
29 views

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point ...
2
votes
2answers
23 views

Visualizing Balls in Ultrametric Spaces

I've been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one ...
-1
votes
3answers
93 views

Prove that $a\lt c$ for every $c\gt b \implies a\le b$ [on hold]

Show that given $a,b,c \in \Bbb R$, if $a\lt c$ for every $c\gt b$, then $a\le b$. I have no idea.
3
votes
0answers
49 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
2
votes
1answer
21 views

Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset ...
1
vote
0answers
19 views

Weak convergence in $l^p$-space [on hold]

Let $1<p<\infty$, $x_n=(x_n^{(j)})_j\in l^p$ for $n\in \mathbb N$ and $x=(x^{(j)})_j\in l^p$. Show that $$x_n\rightharpoonup x\iff \forall j\in \mathbb N:x_n^{(j)}\rightarrow x^{(j)}, ...
3
votes
1answer
23 views

$L^\infty(S^1)$ is not separable

Let $S^1$ be the unit circle and $L^\infty(S^1)$ the space of measurable functions $f:S^1\to\mathbb{C}$ such that $\|f\|_\infty<\infty$. (In fact $L^\infty(S^1)$ consists of equivalence classes of ...
1
vote
1answer
33 views

$\forall x,y\in \mathbb{R}\quad |\sqrt{|x|}-\sqrt{|y|}|\leq\sqrt{|x-y|}\leq\sqrt{|x|}+\sqrt{|y|}$.

$\forall x,y\in \mathbb{R}\quad |\sqrt{|x|}-\sqrt{|y|}|\leq\sqrt{|x-y|}\leq\sqrt{|x|}+\sqrt{|y|}$. i tired we want to prove for all $x,y\in \mathbb{R}\quad ...
0
votes
1answer
44 views

Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
0
votes
0answers
20 views

Primes in two arithmetical progression

For each $x\geq 1$, let $\mathcal{P}$ be the collection of all prime numbers and $$z(x) = \left|\left\{n\in\mathbb{N}:(n\leq x)\wedge\exists k,l\in\mathbb{N}\;\exists p,q\in\mathcal{P}\big((1+3n = ...
5
votes
1answer
61 views

$\forall\ x,y,z\in \mathbb{R}$ Show that: $|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$

$\forall\ x,y,z\in \mathbb{R}$ Show that: $$|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$$ i tired, i notice that $x,y,z$ plays a symmetrical role in the inequality notice also that ...
0
votes
1answer
66 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
0
votes
0answers
29 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
0
votes
1answer
24 views

Show that $F_1$ is a continuous linear functional in the normed space $(C[0,1],\|\cdot\|_\infty)$?

$(C[0,1],\|\cdot\|_\infty)$ and $$F_1(f)=\int_{\frac{1}{2}}^{\frac{4}{3}} f(t) dt$$. Show that $F_1$ is a continuous linear functional.
5
votes
1answer
47 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
0
votes
1answer
16 views

Support of a tempered distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
0
votes
0answers
17 views

A Question about fundamental matrix of system $x'=A(t)x$

Assume in linear system $x'=A(t)x$ the coefficient matrix $A(t)$ is a periodic matrix with period $T$ and $A(-t)=-A(t)$ . If $X(t)$ be a fundamental matrix for $x'=A(t)x$ such that $X(0)=I$ then show ...
1
vote
5answers
44 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
0
votes
1answer
28 views

the limit superior of a sequence exists iff the limit inferior of all subsequences of the sequence exist?

The question is nearly the same as the title, that is, the limit superior of a sequence (of real numbers) exists (can be infinity)iff the limit superior of all subsequences of the previously ...
-1
votes
1answer
83 views

Proof that it is not uniformly convergent on R [on hold]

Prove that the series $$\sum_{n=1}^\infty 2^n \sin \left( \frac{x}{3^n} \right)$$ is not uniformly convergent on R.