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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0
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0answers
11 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
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1answer
10 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
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0answers
17 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
19 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
23 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
18 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: ...
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0answers
13 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
25 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 ...
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0answers
5 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
38 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) ...
2
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3answers
52 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
33 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
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1answer
27 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
21 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
112 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 $(x_n)$ ...
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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
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0answers
25 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
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2answers
21 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
91 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
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0answers
45 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 ...
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0answers
17 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
2answers
30 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
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0answers
18 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
59 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
61 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
24 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
46 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
15 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?
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0answers
15 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
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5answers
42 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
81 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.
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2answers
26 views

Continuous increasing bounded function, derivative

Is it true that a differentiable (and hence continuous) increasing bounded function $f:\mathbb{R} \to \mathbb{R}$ has derivative $f'$ that must go to zero as $x \to \infty$. If it is, could someone ...
1
vote
3answers
47 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
0
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0answers
10 views

Proving a certain function involving the Riemann-Zeta function is non-increasing

Show that $ f(x) = \frac{\zeta(x -2)}{\zeta(x-1)} \qquad x > 3, $ where $\zeta$ is the Riemann-Zeta function, is non-increasing. My attempt was to use $\zeta(s) = \frac{1}{\Gamma(s)} ...
1
vote
1answer
39 views

Continuity of function and topology

I have this exercice $E=\{a,b,c,d\}$ with the topology $\tau=\{\emptyset, \{a\},\{a,b\},\{a,b,c\},E\},$ and the space $F=\{x,y,z,w\}$ with the topology $\theta=\{\emptyset.\{y\},\{y,z,w\},F\}$ I ...
1
vote
3answers
30 views

$f$ is continuous at $x_0=0$ if and only if $f$ is continous $\forall x\in X$?

Let $f$ be a linear functional on a normed space $(X, \|\cdot\|)$. Prove that $f$ is continuous at $x_0=0$ if and only if $f$ is continuous at every $x\in X$. I understand that the $\Leftarrow$ is ...
1
vote
0answers
16 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
1
vote
0answers
18 views

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

$X(t)$ is a fundamental matrix of linear differential equation $x'=A(t)x$ where $A(t)$ is a periodic matrix with period $T$ . Show that there exist a non-singular matrix like $C$ such that for ...
2
votes
3answers
41 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
0
votes
1answer
36 views

Convergent squence in topology

Please, I consider this topological space $(E,\tau)$ where $\tau=\{G\subset E, ~\text{card}~ (E\setminus G)~\text{countable}\}\cup\{\emptyset\}$ How to prove that a sequence $(x_n)$ is convergent in ...
3
votes
0answers
45 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
4
votes
0answers
21 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
0
votes
0answers
29 views

Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now ...