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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4
votes
1answer
133 views

How find this function $f(x)\equiv 0,x\in R$?

Let $f:\mathbb R \to \mathbb R$ have the properties (1): for any prime number $p$ and any real number $x$, $$\sum_{j=0}^{p-1}f\left(x+\dfrac{j}{p}\right)=0$$ (2): there exist real ...
0
votes
2answers
36 views

A problem in real valued function on compact set. [duplicate]

If $f$ be a real valued continuous function defined on $[0,2]$ such that $f(0)=f(2),$ then prove that there exist a $ x \in [0,1]$ such that $f(x)=f(x+1).$ I tried in the following way, Since $f$ ...
2
votes
1answer
146 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
4
votes
1answer
90 views

Find this sum $S$ using Real-analysis methods only

$$S = \sum_{k=1}^{\infty}\frac{2H_k}{(k+1)(k+2)^3}$$ I have tried a lot and failed, any help is appreciated. $H_k$ is the harmonic number. Thanks (real method only please)
3
votes
0answers
62 views

Dual Radon transform: different conventions?

I am having a hard time trying to understand apparently two different definitions of the dual Radon Transform. I am reading simultaneously the book "Mathematics of computerized tomography", by Frank ...
6
votes
2answers
564 views

A difficult inequality involving complex numbers

Suppose that $z_1,\ldots,z_n$ are complex numbers with the property that there is some constant $C$ such that $$\big|z_1^r+\cdots+z_n^r\big|\leqslant C$$ for all integers $r\geqslant0$. Show that ...
0
votes
1answer
45 views

Sequence that converges as for the norm but not almost everywhere

How can I find a sequence that converges as for the norm but doesn't converge almost everywhere, in some space $L^p$ ?? Could you give me some hints ??
1
vote
0answers
40 views

Multiplying two sums?

(Real-analysis only) I will admit, I have posted a question similar to this, but this question's aim is to ask how to multiply the sum and integrate it. $\displaystyle \log^2(x) = ...
10
votes
3answers
130 views

How find this limits $\lim_{n\to\infty}\left(\sin{\frac{\ln{2}}{2}}+\sin{\frac{\ln{3}}{3}}+\cdots+\sin{\frac{\ln{n}}{n}}\right)^{1/n}$

Find this limit $$\lim_{n\to\infty}\left(\sin{\dfrac{\ln{2}}{2}}+\sin{\dfrac{\ln{3}}{3}}+\cdots+\sin{\dfrac{\ln{n}}{n}}\right)^{1/n}$$ My idea:use $$x=e^{\ln{x}}$$ so we only find $$\lim_{n\to ...
0
votes
2answers
50 views

Long term behavior of the solution of $u'=e^{-u}-u$

Consider the autonomous differential equation $$\left\{\begin{matrix}u'&=&e^{-u}-u&=:&f(u)\\u(0)&=&u_0&\in&\mathbb{R}\end{matrix}\right.$$ How can we analyze the *long ...
7
votes
1answer
273 views

Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem ...
2
votes
0answers
69 views

Evaluate $\int_{0}^{1} \log^2(x)\log^2(1-x)\,dx$ using non-elementary methods. [duplicate]

The task is to evaluate, using the specific given hint, and only real-analysis methods, no complex analysis is allowed in the problem. $$\int_0^1 \log^2(x)\log^2(1-x)\, dx$$ The hint given is: $$ ...
2
votes
0answers
40 views

question about property of $L^p$ Lipschitz space

$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that $\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p ...
2
votes
1answer
63 views

Continuous function that has limit at infinity is uniformly continuous (another viewpoint)

I know how to prove that, given a continuous $f:[0,\infty) \rightarrow \mathbb{R}$ such that $\displaystyle \lim _ {x \rightarrow \infty} f(x)=L$, then $f$ is uniformly continuous (by means of taking ...
3
votes
0answers
62 views

An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
0
votes
1answer
37 views

How to restore a function from its Fourier transform on the imaginary axis?

Let $f$ be a `very good' function on the real line; say, infinitely differentiable and compactly supported. We are given its Fourier transform on the imaginary axis: $$g(x)=\int_{\mathbb ...
0
votes
2answers
76 views

How can an interval be an open ball?

I have been asked this question, but I do not understand it. "For what values of $\alpha$ and $\beta$ are the sets $(\alpha, \beta)$, $[\alpha, \beta)$, $(\alpha, \beta]$ and $[\alpha, \beta]$ open ...
1
vote
1answer
56 views

How to prove convergence of $\sum_{n=1}^\infty \log(1 + 1/n^2)$

I need to see whether $$\sum_{n=1}^\infty \log\left(1 + \frac{1}{n^2}\right) $$ is converging or diverging. It is pretty obvious that this series is converging, and I believe I can proof it with the ...
0
votes
1answer
38 views

Proving a function converges to 0

For $h \in C[-1,1]$ defined by $h(x) = (1+x)^b - 1 - bx$ for $b>1$, show that $nh(-\frac{1}{n}) \to 0$ as $n \to \infty$. I tried expanding it as a binomial series but didn't know where to go ...
1
vote
1answer
27 views

If $f(\cdot,y)$ is measurable and $f(x,\cdot)$ is continuous, $\{x:|f(x,y)-f(x,0)| \leq \epsilon, \; \;\forall y <\delta\}$ is measurable

Suppose $\mu(X) < \infty$ and $f : X \times [0,1] \rightarrow \mathbb{C} $ is a function such that $f(\cdot,y)$ is measurable for each $y \in [0,1]$ and $f(x,\cdot)$ is continuous for each $x ...
2
votes
1answer
141 views

Continuity of eigenvalues and spectral radius for a general matrix

Given a general matrix $A(t), t>0$, with real entries, I would like to know if the eigenvalues of $A(t)$ are continuous functions of $t$. These eigenvalues may be real or complex. What about the ...
4
votes
1answer
74 views

Understand this Fourier transform $\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$

I found the equation $$\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$$ in a 'physics' textbook and I just don't understand what this equation tries to tell me. Is there anybody who ...
1
vote
1answer
42 views

Calculate sum of$ (1-2^n) / 3^n$

I feel incredibly stupid right now, but I am not able to calculate the sum of this infinite series... $\displaystyle \sum_{n=0}^{+\infty} (1-2^n) / 3^n$ I've gone through the first terms, as all the ...
9
votes
1answer
260 views

Evaluate $\int_{-1}^{1} \exp(x+e^{x})\,dx$

Evaluate $$\int_{-1}^{1} \exp({x+e^{x}})\,dx$$ where $\exp(x)=e^x$. Can anyone give me any tips on where to start with this? I've tried doing it be substitution, with $ u=e^x$ and ended up needing ...
0
votes
2answers
26 views

A problem in vector valued function

Let $f:R^2 \to R^2$ be defined by $f(x,y)=(x+y,xy).$ I intend to show that inverse image of each element in $R^2$ under f has at most two elements. that is the possibilities for the number of ...
0
votes
5answers
88 views

Limit of derivatives and continuous

Suppose f is a function on $\;(0,1)\;$ contained in class $\;C^1$ If $$\lim_{x\to 1^-} f '(x) \;\;\text{ exists, then}\;\;\lim_{x\to 1^-} f(x)\;\;\text{ exists}\;\;?$$ Probably, it is true but how ...
1
vote
1answer
45 views

Linearisation of a system of equations - answer check

Find all of the critical points for the following nonlinear system. $$\begin{pmatrix}\dot{y}_1 \\ \dot{y}_2\end{pmatrix}=\begin{pmatrix}-y_1+ y_2 - 2\\ y_1 -y_1y_2^2\end{pmatrix}$$ and then use ...
1
vote
2answers
79 views

elementary properties of closure

Let X be arbitrary subset of R, then $$X\subset \overline X$$ proof by contradiction: let $x \in X$ and suppose X not a subset of its closure then for every $y\in X$, $|x-y|> \epsilon $ where ...
0
votes
2answers
156 views

How to evaluate $\int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx}$

How to evaluate the following integral? $$ \int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx} $$ Unlike this example, according to maple, the solution does not contain sine ...
1
vote
1answer
25 views

show series converges by root test

Let $x$ be a real number with $|x|< 1$, and $q$ be a real number. Show that the series $\sum\limits_{n=1}^\infty n^qx^n$ is absolutely convergent, and that $\lim\limits_{n\to \infty} n^qx^n= 0$ My ...
0
votes
2answers
68 views

Prove that the limit of a real sequence (if it exists) can't be a complex number

I was wondering whether a sequence, all of whose terms are real numbers can converge (if convergence is at all possible) to a complex number, or more generally to any other number field. ...
2
votes
3answers
41 views

limit point of $\frac{1}{m}\sin(m)$ for $m \geq1$

how do I show that 0 is the only limit point of $\frac{1}{m}\sin(m)$ for $m \geq1$ integer? It is clear that 0 is a limt point since it is the limit of this sequence, but I cannot prove that there are ...
1
vote
1answer
24 views

Existence of Limit iff $x',x'' > X, |f(x')-f(x'')| < \epsilon$

I was given a theorem in class regarding uniform continuity that does not appear in my textbook. It says that $$\lim_{x \to \infty} f(x) = a \iff \text{ for all } x',x'' > X, |f(x')-f(x'')| ...
0
votes
2answers
40 views

Inverses / Bijections

Let $f:A\to B$, and $g:B\to A$ such that $$ g(f(a))=a \ ,\ \forall\ a \in A, $$ and $$ f(g(b))=b\ ,\forall\ b \in B. $$ Does this mean that $f,g$ are inverses and bijections? Bests
1
vote
0answers
30 views

Poisson Distribution Research Question in R

I am working on doing a Poisson distribution based upon the number of potholes and accidents on a given road. The problem I currently have is that I am basing a general linear model off of only the ...
6
votes
2answers
85 views

Path Connectedness and fixed points

We have the following given to us, Let $α, β \colon [0, 1] \to [0, 1]$ be (not necessarily continuous) functions such that $α(x) ≤ β(x)$, for all $x ∈ [0, 1]$. The set $K = \{\,(x, y); α(x) ≤ y ≤ ...
1
vote
2answers
44 views

Why is {(0,1), (1,2)} an antisymmetric relation?

This is my relation: $R= \{(0,1), (1,2)\}$ I know it is not transitive, because: $$ 0R1 \wedge 1R2 \Rightarrow 0R2,$$ but this is false Now I want to check, if it is antisymmetric. How can I write ...
2
votes
2answers
27 views

Find $\inf$ and $\sup$

Find $\inf$ and $\sup$ of $A=\left\{ \dfrac{2013}{1+\epsilon+\epsilon^{-1}}: \epsilon \in (0,1)\right\}$ . Check if $A$ has the biggest element and the smallest element.
0
votes
1answer
47 views

Relation between sum and integral

I have an exercise (from physics) where I am supposed to show $$\sum_{k'<k_f} \frac{1}{|k-k'|^2} = C \left( \frac{1}{2} + \frac{1-(\frac{k}{k_f})^2}{4 \left( \frac{k}{k_f} \right) } ln |\frac{1 + ...
2
votes
1answer
32 views

How to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$

I am solving an exercise in my textbook. After some steps, I need to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$, where $\alpha\neq 0,0<\beta<1$, in order to ...
1
vote
1answer
41 views

Can $[x, x+2\pi]$ cover the real line?

I'm trying to prove that $\sin x$ is uniformly continuous, but we are not allowed to use the MVT or unit circle. I proved that $\sin x$ is uniformly continuous on $[0, 2\pi]$, and we know that $\sin ...
1
vote
0answers
73 views

How prove there exsit $f''(\xi)=\dfrac{1}{8}$

Let $f$ be twice differentiable on $[-1,1]$,and such that $$\min_{x\in[0,1]}f(x)=-1,f(1)=f(-1)=0$$ show that: there exists $\xi\in[-1,1]$ such that $$f''(\xi)=\dfrac{1}{8}$$ My idea: let ...
5
votes
2answers
158 views

Showing that $ \int_0^1 x^{2n}f(x) dx = 0 $ implies $f = 0$

This is my question: Show that if $f \in C[0,1]$ satisfies $ \int_0^1 x^{2n}f(x) dx = 0 $, then $f$ is the zero function. Note: I am aware that a similar question to this has been asked on maths ...
1
vote
1answer
38 views

Locally bounded additive function is linear

Let $T:\mathbb{R}^k\to\mathbb{R}^m$ is additive, i.e. $T(X_1+X_2)=T(X_1)+T(X_2)$. $T$ is locally bounded. Prove that $T$ is a linear transformation. I know there is a relation between boundedness, ...
1
vote
1answer
28 views

Maximum of k real numbers

Let $a_1, a_2, \cdots, a_k$ are real numbers. Prove that $$\sum_{1 \leq m \leq k} a_{m} - \sum_{1 \leq m < n \leq k } \min (a_{m},a_{n}) +\sum_{1 \leq m < n <p \leq k } \min ...
3
votes
2answers
64 views

Check if the sequence is convergent?

Check if the sequence $a_n=\sqrt[n]{\sum_{k=1}^{n}(2-\frac{1}{k})^k}$ is convergent? How to do this?
2
votes
5answers
134 views

Prove $ \frac {x_1 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$

I am studying computer science in the first term. I have to proof the following inequality: $$ \frac {x_1 + \cdots+ x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$$ $x$ can be any positive real ...
1
vote
1answer
35 views

Does this funcion define a norm on $\mathbb{C}^n$?

Let $m$ and $n$ be two given positive integers. And, let $f \colon \mathbb{C}^n \to \mathbb{R}$ be defined as follows: $$ f(x_1, x_2, \ldots, x_n) \colon= \left( \sum_{i=1}^n \sqrt[m]{|x_i|} ...
1
vote
1answer
71 views

Is open proper map surjective?

I want to know the relationship between "proper" and "surjective", is open proper map surjective? Or it needs more condition to imply surjection? For example, the map is a homogeneous complex map. ...
0
votes
1answer
21 views

Verification of Proof that given x<0 and F'(x)=1if x$\geq$0 and F'(x)=0 if x<0 there exists no differentiable function F:R->R

My instructor wants me to show that if such a function $F$ exists it would necessarily be continuous, constant on any $x< 0$ and not including $0$, and of the form $F(x)=A+Bx$ for $x\in [0, ...