Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

learn more… | top users | synonyms (1)

0
votes
2answers
15 views

Something not working out for me in the continuity definition

I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ...
1
vote
4answers
31 views

$s_1 = 1$ and $s_{n+1}=(\frac{n}{n+1})s_n^2$ monotonically decreases?

Hi I came across this question on page 65 of Elementary Analysis by Kenneth A.Ross. I am given that $s_1 = 1$ and I need to show that $s_{n+1} = (\frac{n}{n+1}) s_n^2$ monotonically decreases. I'm ...
0
votes
1answer
18 views

Function continuous Uryson's lemma?

when we proved Uryson's lemma we checked that the function $f:X \rightarrow [0,1]$, where $X$ is a $T_4$ space, i continuous by checking whether $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open. $f$ is ...
1
vote
4answers
83 views

Antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$

What is the antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$ ? $\displaystyle\int t^2e^{-\frac{1}{2}t^2}\,dt=\displaystyle\int{t}_{}te^{-\frac{1}{2}t^2}\,dt=-te^{-\frac{1}{2}t^2}\Big|_{?}^?+\int ...
0
votes
0answers
12 views

Heat equation for x approaching infinity

I'm trying to solve a PDE problem given to me, but I'm stuck. It's all about heat equation applied to the groud temperature. We assume that the temp in the ground is function of time $t$ and depth ...
0
votes
1answer
7 views

A question on an asymptotic combinatorial expasion

Suppose we are given $(\lambda a + \bar{\lambda}b+O(\lambda^2))^{n}$, where $0 < \lambda < 1$ and $\bar{\lambda} := 1-\lambda$; also, $0 < a,b < 1$. $O(\cdot)$ is the traditional Big-Oh ...
1
vote
2answers
36 views

Prove that f(x) is uniformly continuous

Show that $f(x)=x^2cos(\frac{1}{x^2})$ on (0,1). Let $\epsilon > 0$ and $x,y \epsilon (0,1)$. $| f(y) - f(x)| = |y^2cos(\frac{1}{y^2}) - x^2cos(\frac{1}{x^2})|$. After this I do not know how to ...
0
votes
1answer
14 views

Limit points of a real function.

Given a real number $a$ and an no empty compact set $K$, get a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set of the limit point of $f$ in $a$ is $K$. Can anyone give a hint to ...
3
votes
5answers
376 views

How to solve for a variable that is only in exponents?

Hi there. I've got this equation: $$\left(\frac{3}{5}\right)^x+\left(\frac{4}{5}\right)^x=1$$ How can I find the $x$? Thanks for helping!
1
vote
1answer
36 views

$\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
0
votes
0answers
5 views

Limite of the distance of the iteration to each set

Let $C_1$, $C_2$ convex and closed sets such that the intersection is noempty. I want to show that the iteration $x^{k+1}=f(x^k)$ generated by the function $f: \mathbb{R^n} \to \mathbb{R^n}$ defined ...
1
vote
0answers
19 views

Exercise regarding normal matrices and their spectrum

I hope could get a few hints to this exercise Let $T\in M_n (\mathbb{C})$ be a normal matrix. Let $\lambda \in \sigma(T)$, where $\sigma(T)$ is the spectrum of $T$. Argue that $1_{\{\lambda\}}\in ...
0
votes
0answers
17 views

Approximation of a discontinuous function

I have a function from $R^n$ to $R$ such that $f_i(x_1,...,x_n)$ has a value of $1$ if $x_i$ is strictly above the $m$ th order statistics of $x_1,...,x_n$ $0$ if $x_i$ is strictly below the $m$ ...
1
vote
1answer
20 views

Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ...
0
votes
1answer
19 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
-1
votes
0answers
19 views

cardinality of connected subset [on hold]

Let X be a connected subset of real numbers. if every element of x is irrational then the cardinality of x is (a) infinite (b) countably infinite (c)2 ...
2
votes
1answer
27 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
1
vote
2answers
62 views

a question about limit, I am struggling with this!

Suppose that {$a_n$}is a sequence of positive numbers.For each n which is a natural number,let $b_n$=($a_1+a_2+......a_n$)/n,prove that $\sum b_n$ diverges to $+\infty$. This question is my homework, ...
0
votes
1answer
33 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
1
vote
0answers
21 views

Biorthogonal functionals continuous?

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
0
votes
0answers
14 views

On reconciling different definitions of the $\nabla$ operator in curvilinear coordinates

Note: This questions was originally asked in iMechanica. The main confusion appears to be on whether Christoffel symbols should appear in the divergence of a field expressed in curvilinear ...
1
vote
0answers
28 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
1
vote
1answer
24 views

when can you extend a map from a Hilbert basis?

Suppose $H$ and $K$ are Hilbert spaces and $H$ has Hilbert basis $h_i$. What is a necessary and sufficient condition for elements $k_i$ of K so that $h_i \mapsto k_i$ extends to a continuous linear ...
1
vote
2answers
39 views

Minimum value of a function

For $x \in [0, 5]$, let $$f(x) = \sum_{i = 1}^{5}\frac{1}{|x - i|}.$$ Why is $$f(x) \geq 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = f(0)?$$ This of course is true if one simply plots ...
0
votes
1answer
27 views

Where to go with continuity?

I've started to see continuity at university. The lecturer has given the $\delta-\epsilon$ definition of continuity, but we've seen applications of those only to silly examples of functions: ...
0
votes
1answer
11 views

how can I find the limit-pointwise convergence

Let $f_n(x)=\left\{\begin{matrix} 0, & x<\frac{1}{n+1} \text{ or } \frac{1}{n}<x\\ \sin^2(\frac{\pi}{x}),&\frac{1}{n+1} \leq x \leq \frac{1}{n} \end{matrix}\right.$ Show that $(f_n)$ ...
2
votes
0answers
35 views

Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
0
votes
1answer
21 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
0
votes
0answers
23 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
0
votes
0answers
22 views

find a minimum of a function

let $F(x)= \frac{2}{nx} + L^{2}e^{\frac{\alpha}{x^{r}}}$ for given $ n,L,\alpha$ show that $x^*=argminF(x) $ where $ x^*=\frac{\alpha^{\frac{1}{r}}}{(ln(n))^{\frac{1}{r}}}$ i know that F is convex ...
0
votes
1answer
35 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
0
votes
0answers
17 views

ratio test for convergence of series, different versions

In lecture we had the ratio test: Let $(a_k)$ be a sequence in $\mathbb{K} \in\{ \mathbb{R}, \mathbb{C}\}, a_k\not= 0$ for all $k \ge k_0$, where $k_0\in \mathbb{N}$. (I) If there is a $q\in (0,1)$, ...
1
vote
0answers
29 views

Radius of convergence and uniqueness

I was reviewing some fundamental concepts of real analysis, and the following exercise popped up: Let there be a power series: $\sum\limits_{k=0}^\infty a_k \mid x - 3 \mid $ Such that for $\ x = ...
0
votes
0answers
21 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
0
votes
0answers
22 views

If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

First I'd like to say that although this question was asked before (here) and is from the same text, the answer used methods that were not introduced in the text. Let $P_n(x)$ be a polynomial of ...
1
vote
1answer
24 views

Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$. Now it is obvious that the graph reaches a maximum at $x=e$ but how ...
0
votes
0answers
6 views

convergence double series in Hilbert spaces

Let $(a_{ij})$ be sequence in Hilbert spaces H. What is sufficient condition convergence $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$ ? Whether $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$ = ...
3
votes
1answer
112 views

How was this sequence discovered?

Let $N$ be a positive integer and consider the following rational sequence for $n \ge 0$: $$ a_{n+1} = \frac{N a_n + N}{a_n + N}, a_0 \in \Bbb{Q}. $$ If $-\sqrt{N} < a_0 < \sqrt{N}$, then ...
1
vote
1answer
29 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
2
votes
0answers
16 views

On the uniform convergence of generalized integral

Is the integral $$ \int_{1}^{\infty} e^{-yx^2}\sin{y}dx.$$ uniformly convergent in $y \in [0,\infty]$? Why or why not?
2
votes
0answers
25 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
0
votes
0answers
27 views

Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
2
votes
1answer
37 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
1
vote
1answer
15 views

Performance Index for sports [on hold]

I am trying to create a performance index weighted of strength and speed values. I have the equation working fine for the strength index. Ex. ...
1
vote
1answer
37 views

Uncountability of a nonmeasurable set

As per the Vitali's theorem, every measurable set of positive measure has a subset which is nonmeasurable. Which proceeds by defining a rational equivalence, followed by using the axiom of choice on ...
4
votes
2answers
73 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
2
votes
2answers
53 views

Constructing $\mathbb{R}$ from $\mathbb{Q}$ and showing $\mathbb{Q}$ is dense in $\mathbb{R}$

This is a very long, multi-part problem that we were told to figure out by any means possible. There are no limits on getting help or finding answers online. I haven't had much luck at all solving ...
0
votes
0answers
16 views

bounded and compact orthonormal basis for L^2(R) [on hold]

Whether there is a bounded and compact support orthonormal basis for $L^2(\mathbb{R})$ such that $(B_jK_j)$ is contained in $L^1(\mathbb{R})$ (where for every j, $B_j$ is $esssup e_j$ and $supp e_j$ ...
1
vote
1answer
28 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
0
votes
0answers
27 views

Is it a sufficient and necessary condition?

Let $f_n \to f$ uniformly in $[a,b]$. If each $f_n$ is integrable in $[a,b]$,then $f$ is integrable in $[a,b]$ and $\int_a^b f_n(x) dx \to \int_a^b f(x) dx$. If $\int_a^b f_n(x) dx \nrightarrow ...