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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-1
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1answer
34 views

Proof that it is not uniformly convergent on R

Prove that the series $$\sum_{n=1}^\infty 2^n \sin \left( \frac{x}{3^n} \right)$$ is not uniformly convergent on R.
1
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2answers
19 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 ...
0
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3answers
34 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
votes
0answers
9 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
30 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
2answers
18 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
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0answers
14 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
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0answers
15 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
30 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
31 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 ...
2
votes
0answers
31 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_{x\to\infty} ...
3
votes
0answers
12 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 $Lip (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $Lip(\alpha, L_2(0,1))$ is defined as the set of all functions ...
0
votes
0answers
24 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 ...
1
vote
0answers
31 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
2
votes
2answers
52 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
0
votes
3answers
37 views

Contraction-like mapping without fixed point?

If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies: $$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$ $$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$ for all $n= 1,2,\dots$, ...
3
votes
3answers
53 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
0
votes
0answers
13 views

Two different definitions of “scale invariance”

I found the following definition of "scale invariance" from a book (http://books.google.co.kr/books/about/Critical_Phenomena_in_Natural_Sciences.html?id=rQSIZVOQfWYC&redir_esc=y) A function ...
0
votes
1answer
14 views

A simple question related to One-to-One function and linear operator

I was stuck in one line derivation about the linear operator-related question: Suppose $T$ is linear operator maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. and let $c>0$ be constant. If for all ...
0
votes
1answer
17 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
2
votes
1answer
25 views

Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
5
votes
1answer
50 views

Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
1
vote
1answer
28 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
2
votes
4answers
56 views

Show the subset $A$ of $\mathbb{R}^n$ is compact

Show the subset $$A = \{(x_1, . . . , x_n) ∈ \mathbb{R}^n| −1 ≤ x_1 ≤ x_2 ≤ · · · ≤ x_n ≤ 1\} \subset \mathbb{R}^n $$ is compact, and show the function $$\left\{\begin{array}{}f : A → ...
0
votes
1answer
22 views

Uniform continuity of the function: [on hold]

Test the uniform continuity of the function $f(x)=x^{2/3}\log x$ where $x$ belongs to $(0, \infty)$.
0
votes
1answer
26 views

How can I show that if a set is bounded, then it's contained in a k-cell?

The set is a bounded subset of R (under the Euclidean metric), and a k-cell is a set of points {x_1...x_k} such that a_j < x_j < b_j for j=1...k. Any ideas on how to show this?
2
votes
1answer
32 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
1
vote
2answers
26 views

Is every Lebesgue measurable function bounded on a set of positive measure

Let $f$ be a Lebesgue measurable function from $[0,1]\to\mathbb{R}$. Let $\mu$ be Lebesgue measure. Does there exist a measurable set $B$ with $\mu(B)>0$ and an $M>0$ such that for all $x\in B$, ...
0
votes
1answer
20 views

Constructing an outer measure on a collection of subsets

Let $X$ be the set of three elements $\{a,b,c\}$ . On the collection of subsets $C = \{\{ \emptyset \} , \{a\} , \{a,b\}\}$ Define the set function $m: C → [0,∞]$ by $m( \emptyset ) = 0 $, ...
1
vote
2answers
38 views

Short proof using continuity and set conclusion

I'm new to uni math and in my most recent assignment I got stuck trying to proof the following: Let $a,b \in \mathbb{R}$ and $a<b$. Suppose $\space f:[a,b] \rightarrow \mathbb{R}$ be continuous. ...
0
votes
3answers
30 views

Find couples of complex numbers

I found this exercise, given: $$u=|z|+|u|$$ and $$z=|u|+1$$ (it is a system I don't how to write it in latex from) I have to find the couples of complex numbers $u,z$ that comes from the two equation. ...
0
votes
1answer
23 views

Finite measure and a measure $>0$?

This is a problem from one of the analysis qualifying exams in my school Let $\mu$ be a $\sigma$-finite measure on $(X,F)$ with $\mu(X) = \infty$. Show that for every $C > 0$, there exists an $E ...
0
votes
0answers
13 views

Find a matrix and a vector using partial derivative and system of matrices.

Let $f(x)$:=[$f_1(x),...,f_d(x)]^T$ and suppose that |$\frac{\partial^2 f_i(x)}{\partial x_j \partial x_k}|$$\le$K for all $i,j,k$=1,...,d and $x\in\Re^2$. Show how to define an $dxd$ matrix $J(y)$ ...
1
vote
1answer
12 views

integration concening Fourier transfom variable and space variable

We define the short time Fourier transform as follows: $$V_{g}f(x,w)=\int_{\mathbb R} f(t)g(t-x)e^{-2\pi itw} dt, (x,w \in \mathbb R).$$ (We may assume that $f$ and $g$ nice functions so that every ...
0
votes
1answer
15 views

Quadratic spline and quadratic interpolation

I am trying to understand what is the difference between quadratic spline and quadratic interpolation. Thank you for any help and advice.
0
votes
1answer
45 views

Irrational numbers in [0,1] [on hold]

Why iirational numbers in interval [0,1] can't be countable union of closed sets?
1
vote
1answer
42 views

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) [on hold]

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) - (continuous functions)
0
votes
0answers
17 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
0
votes
2answers
25 views

continuity of functions on intervals

Suppose that $f : (a,b) \to \mathbb R$ is continuous. Then, there is a continuous $g : [a,b] \to \mathbb R$ such that $g(x) = f(x)$ for all $x \in (a,b)$. That is, a function defined and continuous on ...
1
vote
1answer
22 views

Continuity of $f(x)=(xI-A)^{-1}$?

Let $A\in \mathbb{C}^{n\times n}$ and $I_n$ be an identity matrix. If $z\in \mathbb{C}$ is not a eigenvalue of $A$, then $f(x)=(xI-A)^{-1}$ is a continuous function at $z$. Is that correct?
-2
votes
1answer
31 views

Limits and ranges of functions

There is no function $f : \mathbb R \to \mathbb R$ that is continuous on $\mathbb R$ and with range equals to $[-2,5) \cup (-7,-4]$. True or false? If true, prove. If false, give counterexample. I ...
0
votes
3answers
30 views

Functions of sequences and convergence

(a) If $f$ is continuous on $[0,\infty)$ and {$x_n$} is a sequence in $(0,\infty)$ such that {$f(x_n)$} diverges to $\infty$, then $\lim_{n \to \infty} x_n = \infty$. (b) If $f$ is continuous on ...
0
votes
2answers
24 views

continuity and sequences

If $f$ is continuous on $[a,b]$ and {${x_n}$} is a sequence in $(a,b)$, then {$f$(${x_n}$)} has a convergent subsequence. True or False? If true, prove. If false, give a counterexample. I'm guessing ...
0
votes
1answer
42 views

Continuous functions on closed and open intervals

(a) If f is continuous on $[a,b]$ and $f(x) > 0$ for all $x \in [a,b]$, then there is some constant $C > 0$ such that $f(x) \geq C$ for all $x \in [a,b]$. (b) If f is continuous on $(a,b)$ and ...
1
vote
1answer
26 views

Prove $f:[a,b] \rightarrow \mathbb{R}$ is integrable and has zero content.

In my text book, it denotes the upper Riemann sum and lower Riemann sum as follow. $$S_Pf = \sum_{i=1}^n \sup\{f(x_i) : x_i \in [x_{i} - x_{i-1}]\}*(x_i - x_{i-1})$$ is the upper Riemann sum and ...
1
vote
2answers
42 views

I.V.T Continuity proof

If $f$ is defined on $[a,b]$ and has the property that, for any $k$ between $f(a)$ and $f(b)$, there is some $c \in (a,b)$ such that $f(c) = k$, then $f$ must be continuous on $[a,b]$. True or False? ...
0
votes
0answers
12 views

Show that a set is smooth curves

I have a question from my text book about smoothness of a curve. My text book defines the smoothness of a curve as follow. A set $S \subset \mathbb{R}^2$ is smooth curve if $S$ is connected and ...
-1
votes
0answers
24 views

Prove limx->infinity n(e^-n) converges to 0 using an epsilon proof. [on hold]

Hint Using the inequality: $e^{-n}<n^{-a}\qquad (a>0)$
1
vote
0answers
17 views

counting function for minima of continuous function?

Given an absolutely continuous function $f(x)$ with $x\in[a,b]$ and $f(x)\geq 0$ (e.g. a signal pack), I am trying to deduct analytically another function $g(x)$ which counts (similar a step ...
0
votes
1answer
18 views

Jordan's Decomposition Theorem in Real Analysis

Define $f(x)$ = $\sin x$ on $[0, 2\pi]$. Find two increasing functions $h$ and $g$ for which $f = h - g$ on $[0, 2\pi$]. I started this problem by using $f(x)=[f(x)+TV(f[0,x])]-TV(f[0,x])$. And I ...