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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2answers
24 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
votes
1answer
22 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 ...
1
vote
1answer
39 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 ...
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
9 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 ...
0
votes
0answers
10 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
votes
1answer
20 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: ...
0
votes
2answers
29 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
votes
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
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0answers
13 views

Integral inequality of a continuous function on a compact set

Let $\Omega$ be a compact subset of $\mathbb{R}^n$, and let $f:\Omega \to \mathbb{R}$ be a continuous positive function. Let $V_1$ and $V_2$ be subsets of $\Omega$. Then, I like to show that there ...
0
votes
1answer
24 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
11 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
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0answers
19 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
25 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 ...
1
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0answers
22 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
1
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0answers
18 views

Question about the following proof

Below is a proof of the theorem in Du's paper "Centrodial Voronoi Tessellations:Applications and Algorithms": I have the following questions about the proof. (1) To understand the proof better, I ...
2
votes
3answers
50 views

How to differentiate $F(x,y)=\int_x^y \sqrt{e^{tx}+3y}dt$

I want to compute $D_1f$ and $D_2f$, two partial derivatives. The only tool I have now is the fundamental theorem of calculus and chain rule. Maybe I can write $F(x,y)$ as some composition functions ...
2
votes
1answer
21 views

If a continously differentiable function has a local minimizer, can it be one to one?

Let $f$ be a continuously differentiable function defined $f : \mathbb R \to \mathbb R$ such that $f(x)$ is defined for for all $x$. Suppose $x_0$ is a local minimizer for $f$. Is $f$ one-to-one? I ...
0
votes
1answer
27 views

the $\int_1^\infty (logx)^p x^k dx $ is convergence?

Consider the $\displaystyle \int_1^\infty (logx)^p x^k dx$. by what condition on $p$ and $k$ the integral is convergence? my work: i use $logx=r$. but i can not solve the problem.
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votes
0answers
25 views

Limits curiosity.Some examples of problems encoutnered so far [on hold]

Can you guys give me some examples of the hardest mathematical limits exercises you have ever encountered?
1
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0answers
37 views

Find all solutions to integral equation

Let $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and $F:\mathbb{R}\rightarrow \mathbb{R}$ be given functions such that $\int_\mathbb{R} F(x) dx = 0$. Find all $h:\mathbb{R^2}\rightarrow \mathbb{R}$ ...
-3
votes
3answers
33 views

A problem of Schwarz derivative [on hold]

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
0
votes
0answers
24 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
1
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3answers
34 views

Is it possible to replace $\lim$ by $\sup$ here?

Let $(a_m)$ be a monotonically increasing sequence. Is it then possible that $\lim_{m\to\infty} a_m=\sup_m a_m$? I only know the fact that a bounded monotonically increasing sequence coverges to its ...
0
votes
4answers
47 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
-2
votes
1answer
31 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
0
votes
1answer
30 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
-3
votes
1answer
36 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
0
votes
1answer
24 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
0
votes
1answer
27 views

Compactness and convergence

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
2
votes
1answer
32 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
0
votes
0answers
17 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
0
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0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
2
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0answers
76 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...
2
votes
2answers
107 views

Derivative of the power tower

May somebody help me to correctly calculate the dervative of the $n$-th power tower function? $$ \begin{align} f_1(x)&=x\\ f_n(x)&=x^{f_{n-1}(x)}\\ &=x^{x^{x^{...^x}}}\text{ where ...
0
votes
0answers
15 views

Product of product-measurable function and measurable function product-measurable?

Given two measurable spaces $(\Omega, \mathcal{F}), (\Theta, \mathcal{F}_\Theta)$ and their product with the product-sigma-algebra $(\Omega \times \Theta, \mathcal{F} \otimes \mathcal{F}_\Theta)$ and ...
0
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0answers
30 views
+100

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
3
votes
2answers
46 views

Upper and Lower Darboux integral of a piecewise function $f(x)=x$ and $f(x)=0$.

Let $0<a<b$. Find the upper and lower Darboux integrals for the function $$f(x)=x$$ if $x\in[a,b]\cap\mathbb{Q}$ and $$f(x)=0$$ if $x\in[a,b]-\mathbb{Q}$. I am so lost on this problem. Any ...
-2
votes
0answers
34 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
-2
votes
0answers
68 views

Is it possible to find $\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $ without using integral or combinatory logic? [on hold]

$$\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $$ I apologize for asking the same question again, but I wanted to ask something. Is there a possibility that this problem could be ...
0
votes
1answer
25 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
2
votes
1answer
42 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
0
votes
2answers
51 views

function which is Riemann integrable

Consider $f:[-1,1]\to\mathbb{R}$, $x\mapsto \begin{cases} 1, & \text{if } x=0 \\ 0 & \text{else } \end{cases}$ I want to know why f is Riemann integrable and I tried something. First of ...
2
votes
2answers
66 views

Find the limit of $\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$

Find the limit of $$\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$$ I'm not sure how to evaluate this limit. Any hints or solutions are greatly appreciated. I ...
1
vote
0answers
21 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
votes
1answer
27 views

Example of continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ which has no point which satisfy certain conditions?

Does there exist any continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ and for which there is no pair of points $p,q\in f([0,1])$ such that $q-p=0.75$?
7
votes
3answers
139 views

What are the differences in mental skills required to master abstract algebra and analysis?? [on hold]

I had took undergraduate-level abstract algebra and analysis courses before. And I find I can do proofs in analysis faster than in abstract algebra. However some other students is opposite to me. I ...