Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
2
votes
1answer
28 views
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is ...
4
votes
1answer
67 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
0
votes
0answers
17 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
0
votes
3answers
37 views
Finding sequence in a set $A$ that tends to $\sup A$
I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed ...
4
votes
1answer
68 views
A question on limsup
Let $a_n>0$. Prove that $$\varlimsup_{n\to\infty}n\left(\frac{1+a_{n+1}}{a_n}-1\right)\geq 1.$$
I argue by contradiction. If it is not ture, then $$\exists\ N,\ \forall\ n\geq N, ...
2
votes
2answers
49 views
alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$
I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$
I know the necessary and sufficient conditions are:
The ...
2
votes
2answers
83 views
Lipschitz continuous
Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds
$|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
1
vote
0answers
40 views
Extending a rational entry matrix to an orthogonal matrix.
Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
4
votes
3answers
106 views
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
...
6
votes
1answer
86 views
Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?
"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any ...
3
votes
2answers
70 views
Limit: $\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$
How can I find the following limit?
$$\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$$
It's a limit of type $\displaystyle 1^{\infty}$ and if I note with $\displaystyle ...
2
votes
3answers
84 views
Show that $c$ is closed in $l^{\infty}$
Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$
$$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
16
votes
6answers
252 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
1
vote
2answers
29 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
0
votes
1answer
42 views
What is the Sobolev Lemma?
In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult.
The ...
3
votes
1answer
75 views
Formula for Sum of Logarithms $\ln(n)^m$
As you know $\sum_{n=1}^k \ln(n) =\ln(k!)$ is there a formula for $\sum_{n=1}^k \ln(n)^m$?
0
votes
2answers
18 views
Some algebraic inequalities with the binomial theorem.
I am working on proving the following limits.
1), $\lim_{n \to \infty} \sqrt[n]{n} = 1$
2), If $p >0$ and $\alpha \in \Bbb R$, then $\lim_{n \to \infty} {n^{\alpha}\over{(1+p)^n}} =0$
...
2
votes
1answer
16 views
Proof of a theorem with upper/lower limits.
Theorem: If $s_n \le t_n$ for all $n$ greater than a fixed integer $N$, then $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$$
I would like to prove this and it would be nice if ...
3
votes
1answer
24 views
I'm having trouble with a definition of the upper and lower limits, and a theorem that follows it.
The following is the definition.
Let $\{s_n\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ such that $s_{n_{k}}\rightarrow{x}$. This set $E$ contains all subsequential limits, ...
0
votes
0answers
34 views
Proving that $\bigotimes_{i=1}^n \cal{B}_{X_i} = \cal{B}_{X}$
Theorem: Given separable metric spaces $X_1,\ldots,X_n$ and $X=\prod_{i=1}^n X_i$, where $X$ has the product metric $d(f,g)=\sqrt{d_1 (f(1),g(1))^2 +\cdots + d_n (f(n),g(n))^2}$. Then ...
0
votes
2answers
44 views
Relationship between sobolev spaces
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
2
votes
0answers
34 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
0
votes
0answers
25 views
How to show that the partial derivatives exist
In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
1
vote
1answer
30 views
sectionally/piecewise continuous functions
Assume
$f$
and
$g$
are two piecewise continuous functions on an interval
$(
a
,
b
)$
containing the point
$t_0$
. Assume further that
$f$
has a jump discontinuity at
$t_0$
while
$g$
is
continuous at
...
2
votes
3answers
88 views
What does this mean: there exist an integer N such that $n\ge N$?
I'm reading Rudin's, Principles of Mathematical Analysis, and I keep tripping over this phrase. Usually the phrase by that it implies a some equation with n being the index, subscript, of a point. My ...
0
votes
1answer
54 views
Sequence version of L'Hospital's Rule
Consider two sequences $A_n$ and $B_n$ such that $B_n$ is monotonically decreasing and both $A_n$ and $B_n$ tend to zero.
Now let us consider the limits ...
4
votes
1answer
67 views
About Lusin's condition (N)
We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin's condition (N) provided
$$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$
where $m$ stands for the Lebesgue measure on ...
0
votes
2answers
56 views
Definition of metastability
I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
1
vote
0answers
22 views
closest point property of subset of Hilbert space - what are the conditions for existence of inf?
I'm proving the closest point property of a subset of a Hilbert space, ie:
$$H$$
is a Hilbert space with a norm generated by the inner product and so on.
$$h\in H$$
is a point in H
$$M\subset H$$
M ...
1
vote
1answer
29 views
Small question about derivative
how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$
Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$
Please help me
Thank you
0
votes
1answer
33 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
1
vote
1answer
22 views
double integrals and iterated integrals
Give an example (if any) for a non-integrable function $f:\mathbb{R\times R}$ $\to$
$\mathbb{R}$ with domain in $[0,1]^2$ such that both iterated integrals exists(i.e. in both order of integration).
...
3
votes
1answer
55 views
Oscillation and Hölder continuity
I am studying a proof of a theorem. And I have the following situation in the proof:
Consider $\Omega$ is a bounded open set of $\mathbb R^n$ and $u: \Omega \to \mathbb R$ is a function satisfying:
...
3
votes
0answers
39 views
Problem of understanding
Please , can someone help me to understand this part of text please
What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$"
and please how to ...
3
votes
0answers
31 views
SVD, infinite matrices and normal operators from a function
I'm trying to understand the behavior the Singular Value Decomposition on a deeper level, and why it might give a particular result. Take the function
$$
f(x,y) = \frac{1}{(1+2x+y)^2}
$$
and ...
3
votes
1answer
53 views
Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$
Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$
I tried WA; it does not return a function.
0
votes
0answers
19 views
Volume of cone and volume of set
Let A be a bounded set in $\mathbb{R}^n\times\mathbb{R}$ and denote the cone over it by C. How do you show that between the (n+1)-dimensional volume of C, vol(C), and the volume of A, vol(A), we have ...
3
votes
1answer
30 views
Finding the maximal $t$ satisfying a family of inequalities
Given $c \in (0,1)$, find the maximal positive $t$ satisfying the following:
$$\forall n \in \{1,2,\ldots \}: 1+\frac{c}{n+(1-c)} \le \left(1+\frac{1}{n+t}\right)^{c}$$
My progress thus far:
A ...
1
vote
1answer
26 views
Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1
The task is as follows:
Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial
z}{\partial y}$ (evaluated at $x$) * $\frac{\partial y}{\partial x}$
(evaluated at $z$) * $\frac{\partial ...
1
vote
2answers
57 views
Is this function from $[-1,1] \rightarrow \mathbb{R} \cup \infty$ continuous?
just a short question. So I was wondering about functions from compact sets into
$\mathbb{R} \cup \{+\infty\}$.
Let's say we have a function $f : [-1,1] \rightarrow \mathbb{R} \cup \{+\infty\}$,
...
0
votes
0answers
17 views
intersections of two algebraic curves
Consider the following two algebraic curves on $\Bbb{R}^2$. $x$ and $y$ are variables.
\begin{align*}
\left(x^2 p_{-1,-1} + x p_{0,-1} + p_{1,-1}\right) y^2 + \left(x^2 p_{-1,0} + x p_{0,0} + p_{1,0} ...
4
votes
1answer
54 views
Counterexamples in Double Integral
I need to:
$a.$ Give an example of function $f:\mathbb{R\times R}$ $\to$
$\mathbb{R}$ with domain in $[0,1]^2$ so that double integral exists but the function is not Riemann integrable.
$b.$ Give ...
1
vote
3answers
90 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
1
vote
1answer
26 views
Difference between Rician distribution and Gaussian distribution
could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated
Thank ...
0
votes
1answer
30 views
Radius of convergence - ratio test for power series/real numbers
Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers.
For example, find radius of convergence of these....
$\sum_{n=0}^{\infty}(-1)^n ...
0
votes
0answers
36 views
product of piecewise continuous functions
I know that the product of 2 continuous functions on $[a,b]$ is continuous. This also holds for 2 piecewise continuous functions on $[a,b]$, the product is piecewise continuous on $[a,b]$.
What ...
1
vote
1answer
34 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
1
vote
2answers
67 views
Sequence of Functions Converging to 0
I encountered this question in a textbook. While I understand the intuition behind it I am not sure how to formally prove it.
Define the sequence of functions $(g_n)$ on $[0,1]$ to be $$g_{k,n}(x) = ...
1
vote
0answers
47 views
Taylor Expansion of Power Series
Suppose that $\space f:[0,1]\rightarrow \mathbb{R}$ is real analytic and that its power series expansion is:
$\\ f(x)=\sum\limits_{n=0}^\infty a_nx^n$
Prove that there exists an $x_0\epsilon (0,x)$ ...
