Tagged Questions

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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1answer
22 views

Showing that a multivariable limit doesn't exist

Showing that a multivariable limit exists With the same set up as the above question, I need to show that $$ \lim_{(x,y) \to (0,0)} f(x,y) \text{ does not exist.} $$ I need to show that, given any ...
0
votes
1answer
24 views

What about the convergence of the geometric mean sequence of the terms of a given convergent sequence?

Let $(a_n)$ be a sequence of positive real numbers such that $$ \lim_{n\to} a_n = a.$$ Let $b_n \colon= \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdots a_n}$ and $c_n \colon= a_n^{a_n}$. Then what can we say ...
2
votes
1answer
21 views

Periodic solutions of this systems

I need to prove that the system of differential equations $$ \dot x = y \\ \dot y = 1+x^2-(1-x)y $$ doesn't contain periodic solutions. I know the Bendixon criteria (that is to have div no sign ...
0
votes
1answer
33 views

Suppose $f(x)$ is continuous on $[0,1]$,differential in$(0,1)$

$f(0)=0,f(1)=1$ $k_{1},k_{2},...k_{n}>0$, and $k_{1}+k_{2}+...+k_{n}=1$ Prove that there are $n$ different number $t_{1},t_{2},...,t_{n}$ ,such that $$\sum_{i=1}^n k_{i}/f'(t_{i})=1$$
0
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0answers
14 views

Approximation of Simple functions by Step functions

In this question, all measures are Lebesgue. Let $ E $ be a measurable set. Is it true that for every $ \epsilon > 0 $, there is a step function $ \gamma : \mathbb{R} \longmapsto \mathbb{R} $ ...
0
votes
2answers
43 views

How to calculate this limit?

Let $$a_n \colon= \frac{1}{\sqrt[n]{n!}}$$ for $n = 1, 2, 3, \ldots$. Then how to decide about the convergence or otherwise of the sequence $(a_n)$? And if this sequence IS convergent, then how to ...
1
vote
2answers
36 views

How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$?

Let $\mathbb{R}$ be the set of all real numbers under the usual metric $d$ defined as follows: $$d(x,y) \colon= |x-y|$$ for all $x$, $y$ in $\mathbb{R}$, and let $\mathbb{Q}$ be the set of all ...
1
vote
1answer
5 views

how to show that the set where Real part of an analytic function vanish contains arcs

Consider a function $f$ given by $f(z)=z^3g(z)$, where $g(0)\neq 0$. Then clearly $z=0$ is a zero of function $f$ of multiplicity $3$. Let $A$ be the set given by $\{z: \Re f=0\}$. How do i show that ...
0
votes
1answer
11 views

Strictly convex if and only if derivative strictly increasing?

Suppose $f$ is a real-valued function that is differentiable on an open interval $I$. It is well-known that $f^{\prime}$ is increasing on $I$ if and only if $f$ is convex on $I$. Is the following ...
0
votes
1answer
7 views

Different domains for (apparently) equivalent functions

Let's look at: $f_1(x)=ln(x^2-4)$ $f_2(x)=ln(x-2)+ln(x+2)$ Every high school student can tell they are the same, but the first is defined only for $\{x<-2\}\cup\{x>2\}$, and the latter is ...
1
vote
1answer
15 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
1
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0answers
5 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
1
vote
1answer
31 views

Showing that a multivariable limit exists

Consider the function $f\colon \mathbb{R^2} \rightarrow \mathbb{R}$ defined on all of $\mathbb{R^2}$ by $f(x,y) = \left\{ \begin{array}{lr} 1, & \text{if } (x,y) \in A\\ 0, ...
-1
votes
1answer
28 views

Show that the $\|f\|_2\le\|f\|_\infty$ in $X=C[0,1] $ [on hold]

In $X=C[0,1] $, show that $\|f\|_2 \le \|f\|_\infty$.
1
vote
2answers
22 views

$p$-norm on $\mathbb{R}^n$ question

How I can show that $$\lim_{p \to \infty} \|x\|_{p} = \max\{|x_1|, \; |x_2|, \; \cdot ,\; |x_n|\}$$ if $\mathbb{R}^n$ has the p-norm? $p > 1$ of course. Has anyone done this or know how to? I'm ...
1
vote
1answer
16 views

Is the closure of an open connected set polygonally connected?

In an arbitrary metric space, I know that connected does not imply polygonally connected. However, in $\mathbb{R}^n$, an open connected set is connected iff it is polygonally connected. Is the last ...
3
votes
1answer
29 views

A function $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ with $\varphi(x)=x,$ $\|\varphi(y)-x\|\leq K\|y-x\|^\alpha$ for $\alpha>1, K>0$

If we have a function $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ with $\varphi(x)=x,$ $\|\varphi(y)-x\|\leq K\|y-x\|^\alpha$ for $K>0,$ and we define $\varphi^1:=\varphi, ...
0
votes
0answers
20 views

Finding an integral for a given Riemann Sum

Take the Riemann sum: $= \displaystyle \lim_{m\to\infty} \frac{1}{m}\sum_{x=1}^{m} me^{-x}$ How can someone convert that into an integral? We know $\Delta(x) = \frac{1}{m}$. So $me^{-x}$, is the ...
1
vote
3answers
19 views

$C_n:=A_n\cap (A_1\cup\cdots\cup A_{n-1})^c$ pairwise disjoint?

Let $\Omega$ be a set and $A_1,\ldots\in Pot(\Omega)$. Why are the sets $C_n:=A_n\cap (A_1\cup\cdots\cup A_{n-1})^c$ pairwise disjoint? I've tried to write it like $C_n\cap ...
3
votes
2answers
34 views

What can we say about the rate of growth of a function growing faster than all polynomials?

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the following: $$ \forall k \in \mathbb{N} \hspace{5pt} \lim_{t \rightarrow \infty} \frac{t^k}{f(t)} = 0.$$ Can we deduce a stronger growth ...
0
votes
4answers
56 views

How to show that $\sum_{n=1}^{\infty}(-1)^n \sin(a/n)$ converges but not absolutely.

How to show that, for any fixed constant $a\in(0,1)$, the series $$\sum_{n=1}^{\infty}(-1)^n \sin\frac{a}{n}$$ is convergent yet not absolutely convergent. My idea is to express sin(x) as series but ...
0
votes
0answers
9 views

Find a right inverse of a map with gauss brackets.

I am having a composition of two maps: $$ f:\mathbb{R}->\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}->\mathbb{\mathbb{N}},g(x)=\lfloor x\rfloor $$ $$h=g\circ f:\mathbb{R}->\mathbb{N_0}$$ ...
1
vote
1answer
15 views

Find relations on the real number: transitive and/or antisymmetric

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$ antisymmetric and transitive antisymmetric and NOT transitive NOT antisymmetric ,but ...
5
votes
1answer
30 views

A problem on the sum of the reciprocals of two derivatives

If $f(x)$ is continuous in the closed interval $[a,b]$ and differentiable in the open interval $a<x<b$, and if $f(a)=a$, $f(b)=b$, prove there exist points $x_1$ and $x_2$ with ...
0
votes
2answers
20 views

On converging and diverging sequences and their respective arithmetic mean

I'm working on a problem set which was given by our analysis lecturer (a) Let $(a_n)$ be a convergent sequence with limit $a$. show that the arithmetic mean $$s_n := \frac{1}{n} \sum_{k=1}^n ...
1
vote
0answers
19 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
2
votes
2answers
83 views

Finding the infinite sum of $e^{-n}$ using integrals

I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle ...
0
votes
1answer
14 views

Absolute value of function in Sobolev Space

Assume $1<p<\infty$ and that $\Omega$ is bounded. Now I would like to prove that $u\in W^{1,p}$ implies that $|u|\in W^{1,p}$ but I have no idea on how do this. Could anyone help?
0
votes
1answer
41 views

Partitions of $[0,1]$

Trying to test my understanding of analysis, today I came up with two questions, that will probably look obvious (be patience, because I am self-thaught). Anyway, here they are: 1) Is possible to ...
3
votes
1answer
23 views

Existence of unique solution on $(-\delta,\delta)$ for $f(x)=1+x+\displaystyle\int^x_0\sin(tf(t))dt$

The following was a question previously given in a test at my university: Show that there exists some $\delta>0$ for which there is a unique continuous function ...
4
votes
1answer
49 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
2
votes
1answer
14 views

Prove that there is x, c < x < b, such that f(x) > f(c)

If $f : [a,b] \to R$ is differentiable at c, a < c < b and $f'(c) > 0$, prove that there is some x, c < x < b, such that $f(x) > f(c)$. I'm not totally sure where to begin with ...
0
votes
2answers
31 views

How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $

I'm trying to resolve this integral $$ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $$ I tried with polar coordinates: $$ x = r\cos{\theta} \\ ...
0
votes
0answers
27 views

Does an interval have a countinuous boundary

Does an interval on the real line, e.g. $(0,1)\subset \mathbb{R}$ have a $C^1$ boundary or does the boundary have to be connected, e.g. $B(0,1)$ where the boundary is the disc $D(0,1)$.
0
votes
0answers
9 views

contractive sequence

Consider the polynomial $p(x)=x^3-7x+2$ a.Find a contractive sequence $(x_n)$ that approximates the root $0<x_0<1$ of p. (I was thinking of writing $x=(x^3+2)/7$ b.Determine the minimal number ...
0
votes
1answer
24 views

Convergence of a sequence of linearly independent vectors in normed space

In an infinite dimensional normed vector space is it possible to find a sequence ${v_n}$ of linearly independent vector (so the sequence is a set of linearly independent vectors) each has norm 1 such ...
0
votes
1answer
40 views

Prove that B is vector subspace [on hold]

In $ℝ^∞$ we've got B={$\{x_n\}^{\infty}_{n=0}| \exists K\in~ (0,\infty) ∀n\in \mathbf{N}, | x_n|\leq K n^{-1}$}. Prove that B is vector subspace.
1
vote
1answer
33 views

Banach fixed-point theorem and Picard's theorem, exam revision

As the title suggests this is for exam revision so answers and not just hints are okay. I've got the current two problems I'm doing: $\mathbf{1}.$ Show that there exist a unique continuous function ...
2
votes
0answers
12 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
0
votes
1answer
25 views

Banach spaces, partial sums converge?

Let $(X,||.||)$ be a Banach space. Suppose the sequence $(x_n) \subset X$ is such that $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb{R}$. Prove that $S_n = x_1+x_2+...+x_n$ the "Partial sums" ...
3
votes
2answers
40 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
1
vote
1answer
30 views

Practical convergence in $C^{\infty}_c$

Let $C^{\infty}_c$ be the space of $C^{\infty}$ functions with compact support in $\mathbb{R}$ with the usual topology derived by the convergence in infinity norm in every $C^{k}_c$. I would like to ...
0
votes
1answer
17 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
0
votes
1answer
32 views

Proper definite of riemann integral (limit version)

I am sort of confused. Suppose we are given the series, $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$ How can this be written as an integral, and what would the variable ...
0
votes
1answer
13 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
2
votes
2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...
0
votes
0answers
6 views

Visualizing construction of a certain function

Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\ 1) $f$ fixes the poles $(0,0,\pm1)$.\ 2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\ 3) $F$ is ...
0
votes
1answer
25 views

Is the square root of the absolute functiom differentiable at x = 0?

I've been trying to solve the problem below for hours but so far I haven't managed to find a solution. Help would really be appreciated. Thanks a lot! Problem Show whether or not the function ...
1
vote
1answer
36 views

Is there a function $f \gt 0$ such that $\int f dx=0$?

If $$f \ge 0$$, then $$\int f dx \ge 0$$. There is a function such that $$f \gt 0$$ but $$\int f dx=0$$?
1
vote
1answer
36 views

A sequence of truncates of $f$

If $f$ is measurable and $A>0$ then the truncation $f_{A}$ defined by: $$f_{A}(x)=\begin{cases} f(x)&\text{if $\left | f(x) \right |\leq A$}\\ A&\text{if $ f(x)> A ...