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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9 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
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0answers
10 views

Homeomorpic spaces

We define the two following sets: $ E_1 =\{ (L,v) \in \mathbb{R}P^n \times \mathbb{R}^{n+1} \mid v \in L \ or \ v=0\} $ and $A=([0,1], \mathbb{R})/\sim $ . Where $\sim$ is the equivalence ...
-1
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0answers
13 views

Which of the following are proper patches. (Showing that an inverse of a mapping is continuous)

In which of the following cases is the mapping $\mathbf{x}:\mathbb{R^2} \to \mathbb{R^3}$ a proper patch? (a)$\mathbf{x}(u,v)=(u,uv,v)$ (b)$\mathbf{x}(u,v)=(u^2,u^3,v)$ ...
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2answers
27 views

Can be $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$ uniform convergence?

We have $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$. I don't know how we can find the pointwise convergence...This sequence can be a uniform convergence? and explain your argument.
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0answers
11 views

Expanding a multivariable function [on hold]

How can I expand a function $$ f(x,y) = \dfrac{x+y}{x^2 + y^2} \ln(1+xy) $$ to have a total differential everywhere on $\mathbb{R}^2$?
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1answer
18 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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0answers
10 views

Directional derivative of a multivariable function

How can I find a directional derivative of a function $$ f(x,y,z) = \sqrt[3]{x^3 +y^3 + z^3} $$ in every direction in a point $[0,0,0]$? Does this function have a total differencial in that point?
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1answer
7 views

Finding a tangent plane [on hold]

How can I find a tangent plane to this anuloid $$ A=\{(x;y;z)|(x^2+y^2+z^2 + 12)^2 - 64(x^2 + y^2) = 0\} $$ in a certain point, for example in $[0, 3, \sqrt{3}]$?
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1answer
17 views

Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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0answers
18 views

Efficiently compute the bases functions of the Signal [on hold]

Let there be 4 1D signals such that \begin{cases} x(t)=4\sin(10\pi t) \\ y(t)=8\cos(20\pi t) \\ z(t)=16\sin(30\pi t) \\ m(t)=x(t) +y(t)-z(t) \end{cases} Is there a way to compute ...
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0answers
12 views

Limits of Taylor POlynomials over $k$-tuples?

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to $\textbf{0}$ that the points \begin{equation*} ...
2
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0answers
23 views

If $ds$ is not a differential form, can I make sense of its intuitive notation somehow?

I understand that a line element is not actually a differential form but a $1-density$, my question is: is the notation $ds^2 = dx^2 + dy^2$ formal in any way? Can it be interpreted as outer or tensor ...
2
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3answers
429 views

Prove that the following is a constant function

Let $f : R \rightarrow R $ $\lvert f(x)-f(y) \rvert \le (x-y)^2, \forall x,y \in R $ Any sort of help is appreciated! I know I am not suppose to ask for the entire solution, so I will ask for ...
0
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0answers
16 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
2
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1answer
37 views

Some relation between parallel vector field and Jacobi field along a geodesic

Cross posted from my question: http://mathoverflow.net/questions/204097/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found ...
0
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1answer
21 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
0
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1answer
28 views

What is $\sum_{i,k=1}^n x_k^2x_i^2$?

I stumbled over the double sum $$\sum_{i,k=1}^n x_k^2x_i^2$$ and was wondering whether this is anything that can be expressed in terms of the euclidean norm? Maybe it is even $||x||^4$ but I am not ...
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0answers
13 views

Characterisation of continuous functions whose support is compact

I saw this question some time ago but I can't remember where. Basically the question ask to find the set of all continuous functions $f:\mathbb{K} \to \mathbb{R}$ whose support $\text{supp}(f) := ...
5
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4answers
72 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
2
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1answer
32 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
0
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0answers
18 views

Boundary points of a set M [on hold]

A boundary point of a set M is defined to be any point P in the metric space X such that every neighborhood $O_E(P)$ of P(={x $\epsilon$ C = p(x,p) < $E$}) for all $E$ > 0, contains at least 1 ...
0
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0answers
11 views

Uniform Convergence of a sequence of polynomials to $e^x$

Show that there exists no sequence of polynomials $P_n(x)$ converging to $e^x$ on $\mathbb R$ uniformly. This is pretty standard but I have come up with a proof of my own, and have not gone ...
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5answers
94 views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
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2answers
19 views

Infinite Intersection of Open Sets [on hold]

Give an example of an infinite intersection of open sets which is not open.
0
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0answers
23 views

Stephen Abbott “Understanding Analysis”: is my proof correct?

I am self studying Stephen Abbott's Understanding Analysis and I just want to ask if my proof on exercise 1.3.4 is correct. The question is: Assume that $A$ and $B$ are nonempty, bounded above, and ...
0
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3answers
28 views

Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
1
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1answer
48 views

How to solve equation $ x=W(a+bx^{n})+1 $?

How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function? thanks
0
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1answer
11 views

Does a bounded countably infinite union of sets with volume have volume?

If $ A_1, A_2,...$ are sets with volume and $A= \cup_{i=1}^\infty A_i$ is a bounded set, must $A$ have volume? This was a homework problem that we went over in class, and if I remember correctly the ...
0
votes
1answer
30 views

What is the difference between uniform convergence and dominate convergence theorem?

I saw that both have aim to change limit with integral... that's the part that interests me most. I saw in some cases where we couldn't use uniform convergence, we use dominate convergence theorem to ...
1
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3answers
54 views

How to show there exists $E$ such that $E \cap K_n$ is dense for every $n$?

Let $\Omega$ be a region (nonempty connected open subset of the complex plane). Let $K_n$ be a sequence of compact sets whose union is $\Omega$, such that $K_n \subset \mathring{K_{n+1}}$ (the ...
0
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1answer
28 views

$\left( 1 - \frac{1}{n} \right)\left( 1 - \frac{2}{n} \right) \cdot … \cdot \left( 1 - \frac{k-1}{n} \right) = \frac{n!}{n^k r! (n-k-r)!}$

I'm trying to understand a proof in "Interpolation and Approximation by Polynomials" by Phillips. Let me quote (page 253): "For $k\geq 1$ we begin with $$B_{n+k}^{(k)}(f;x)=\frac{(n+k)!}{n!} ...
1
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1answer
24 views

Show that all solutions remain in the interval for all time

I really have no idea on how to get started with these, there's no similar example in my book. Do I need to compute $\frac{dy}{dx}$? Any help would be greatly appreciated. Maybe there's just some ...
2
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1answer
28 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
2
votes
1answer
48 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...
0
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1answer
24 views

Did Spivak leave out Jordan-measurability too in his definition of partition of unity?

This is a continuation of these two questions that are asking the same thing as each other: An application of partitions of unity: integrating over open sets. Is this definition missing some ...
2
votes
1answer
41 views

If a sequence $f(x_n)$ goes to its minimum, will $x_n$ go to the point at which $f$ achieve the minimum?

I have a continuous function $f$ that is defined on a compact set. And $f(x_0)$ is its minimum. If I have a sequence $x_n$ such that $f(x_n)\to f(x_0)$, how can I show that $x_n\to x_0$? I tried ...
2
votes
1answer
42 views

If $\ell_1$ embeds into $X$ a separable Banach space, can $X^*$ be separable?

First let's defined embedding: $Y$ embeds into $X$, where $X$ and $Y$ are normed spaces, if there exists a 1-to-1 linear map from $Y$ into $X$ that is bicontinuous. Suppose that $\ell_1$ embeds ...
0
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1answer
13 views

Linear Operator Boundedness

a) Show that the linear operator $L_2: \ell^2 \to \ell^2$ defined by $L_2(\langle x_1,x_2,\ldots,x_n,\ldots\rangle) = \langle(1+1/2)x_1, (1+ 1/2 + 1/4)x_2, \ldots, ...
-1
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0answers
10 views

Example of a metric space and Countable Infinite Collection [on hold]

Give a concrete example of a metric space (x,p) and a countable infinite collection $M_i$ (i=1,2,...) of nowhere dense subsets of X which illustrate the truth of Baire's Theorem: A complete metric ...
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0answers
11 views

Decomposition of a measure

Let μ be the Lebesgue-Stieltjes measure on R corresponding to the distribution function, F where F(x) = 0 if x<0 x+1 if 0<= x<1 2x+3 if 1<= x<2 8 if ...
2
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2answers
24 views

Help establishing a bound on the Fourier coefficients of a bounded $2\pi$ periodic function that is discontinous at the end points?

This is from a practice midterm, and I'm having trouble with the first part. Suppose $f$ is a $2\pi$-periodic function that is continuous and differentiable on the interval $[-\pi, \pi]$, but has jump ...
1
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2answers
19 views

base b expansion of real numbers

This is a problem in Zygmund's analysis book. It is intuitively very straightforward. However, I could not give a rigorous proof. I hope someone can show me how to prove this rigorously. Problem: ...
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2answers
33 views

Find image of complex set:

Find image of set: $$ \{ z \in C : 0 \le Im (z), 0 \le Re(z) \}$$ and $$f(z)=\frac{i-z}{i+z}$$ I caclulate $ w=\frac{i-z}{i+z} $ and then $z=\frac{i(1-w)}{w+1}$ and don't know what to do next... I ...
4
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2answers
27 views

Is power set of a power set of a set equal to the power set of the same set?

I have to decide whether this statement is true, I think it is not. Since the power set of a set with cardinality $n$, will have $2^n$ subsets, however the power set of this set will include the ...
1
vote
1answer
29 views

Image of Möbius transformation

What's the image of the first quadrant $Rez\ge0$ and $Imz\ge0$ under transformation $f(z)=(i-z)/(i+z)$? I know that real axis is mapped to the unit circle, $f(0+i*0)=1$ and $f(\infty)=-1 $.
1
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0answers
13 views

inverse function theorem on manifolds

suppose there are two 3-manifolds(consider them as orthogonal matrices $SL(2,\mathbb R)$), and there is $F:SL(2,\mathbb R)\to SL(2,\mathbb R)$, given by $F(A)=A^3$. Can we apply inverse function ...
3
votes
0answers
20 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
2
votes
1answer
26 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
3
votes
2answers
18 views

Increasing/Decreasing intervals of a parabola

I am being told to find the intervals on which the function is increasing or decreasing. It is a normal positive parabola with the vertex at $(3,0).$ The equation could be $y = (x-3)^2,$ but my ...
2
votes
0answers
23 views

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...