All Questions
0
votes
0answers
2 views
Must a complex power series *fail* to be convergent somewhere on its circle of convergence?
My textbook (Saff & Snider, pg. 256) asserts so, but I can't seem to find its proof of the claim. On the other hand, a uniMelb lecture slide I'm cross-referencing claims that a power series is ...
0
votes
1answer
6 views
Any way to simplify this gcd totient function
I have the following expression
$$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$
$a,b$ are known positive integers. Is there any way to rephrase this or simplify it?
0
votes
0answers
5 views
Coset of W containing $v$ is a subspace of $V$ iff $v \in W$
I would like if someone could look over my proof. It feels odd to me.
Let $W$ be a subspace of a vector space $V$ over a field $F$. Prove that $v + W = \{v + w \mid w \in W\}$ is a subspace of $V$ ...
3
votes
0answers
11 views
The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-(\frac{-ab}{p})$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime,
the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $p-(\frac{-ab}{p})$.
I got a hint that I have to use ...
2
votes
1answer
10 views
Any arbitrary closed smooth curve bounds a orientable surface?
I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ?
I have no ...
1
vote
4answers
37 views
Difference of two powers of $3$ divisible by $2011$
How to prove that there exists two powers of $3$ that differ by a number that is divisible by $2011$?
2
votes
1answer
15 views
Uniform convergence and uniform boundedness
I try to understand a demonstration from a book, but I have a problem with a line.
We have the series
$$u(x,t) = \sum_{k=0}^\infty \frac{g^{(k)}(t)}{(2k)!}x^{2k} \qquad (*)$$
where
$$g(t) = \left\{ ...
1
vote
0answers
20 views
Closed form for a summation
Is there a 'closed form' for the following summation?
$$S_k=\sum_{i=0}^{m-1}\binom{mk-ik}{k}(mk-i)$$
I evaluated that $S_1=\frac{m(m+1)(2m+1)}{6}$ and $S_2=\frac{m(m+1)(7m^2-1)}{6}$.
0
votes
0answers
18 views
Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.
Okay, so the question is the following:
Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that
\begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 ...
0
votes
0answers
10 views
Resolvent of a restriction of a dual operator
Let me begin with some well known theory. Let $ A\colon X \supset D(A) \to X $ be a linear, densely defined (that is $\overline{D(A)} = X$) operator in a Banach space $ X $. Assume that for all $ ...
2
votes
3answers
49 views
Why integrating the area of the square doesn't give the volume of the cube?
I had a calculus course this semester and I studied that the integration of the area gives the size (volume):
$$V = \int\limits_a^b {A(x)dx}$$
But this doesn't seem to work with the square. Since ...
0
votes
1answer
15 views
Convergence in $L^p$ and $L^q$ - multiplication
We have: $X_n \rightarrow X$ in $L^p$ and $Y_n \rightarrow Y$ in $L^q$. Moreover $p,q>1$ are such that $\frac{1}{p} + \frac{1}{q} =1$. Prove that $X_nY_n \rightarrow XY$ in $L^1$. Please, can you ...
2
votes
1answer
29 views
$n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
I need to show that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
could any one give me a hint?
2
votes
1answer
31 views
Finding the maximum and minimum of $f$ on a set $Q$
I stumbled across a proposed task that I'm unable to solve.
We have a function $f:\mathbb{R^2}\to\mathbb{R}$ defined as:
$$
f(x,y):=x^2+xy+y^2+x+y+1
$$
The task is to "find the maximum and minimum" ...
2
votes
1answer
13 views
Finite ultraproduct
I stucked when trying to prove:
If $A_\xi$ are domains of models of first order language and $|A_\xi|\le n$ for $n \in \omega$ for all $\xi$ in index set $X$ and $\mathcal U$ is ultrafilter of $X$ ...
2
votes
2answers
20 views
How to calculate the norm of an ideal?
Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly.
For example, I know that the norm ...
1
vote
1answer
49 views
which of the followings are positive definite:
Suppose $A,B$ are $n\times n$ positive definite. Then which of the followings are positive definite:
$A+B$
$ABA$
$A^2+I$
$AB$
1
vote
0answers
14 views
Fredholm operators and Compact operators
Suppose $X$ be an infinite dimensional Banach space. How to prove that:
$A$ and $B$ are two Fredholm operator on $X$, if
$\mathrm{index}(A)=\mathrm{index}(B)$, then there exists an invertible ...
0
votes
0answers
7 views
Some questions on invariant factor decomposition of modules
I am trying to understand the following example but have no clue why is $N=\operatorname{im}(\varphi)$ and why is ...
-3
votes
0answers
29 views
Uniform probability question
Anyone here that can solve this challenging question that I have?
Let $U \sim U[a,b]$. Suppose $X = U$ and $Y = \frac{1}{2} U$.
Find $P(X \le x, Y \le y)$ for $-\infty \le x, y \le + \infty$.
0
votes
0answers
16 views
Example on Correspondences
Give an example of correspondences F: X $\rightarrow$ Y, G: Y $\rightarrow \mathbb{R}^s$ such that F and G are closed, but (G o F) is not, if any, where $ \varnothing \neq X \subset\mathbb{R}^m, ...
1
vote
0answers
27 views
Probability exponential distribution question
Could anyone please help me answer these questions? Or a little hint as to how I can answer them? It's for my assignment that's due tomorrow.. Really appreciate if anyone could help!
Consider ...
6
votes
4answers
59 views
Open Cover / Real Analysis [duplicate]
I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel).
I'm ...
0
votes
0answers
12 views
The invertible Toeplitz operators on H^2 space
Suppose $ϕ$ be a real-valued function. I want to prove that the Toeplitz operator $T_ϕ$ is invertible if the constant function $1$ is in the range of $T_ϕ$.
There is a function $f\in H^2$ such that ...
1
vote
2answers
29 views
what is the general step to solve $f(x)\pm f(g(x))=m(x)$
I try with a simple example
I put $$f(x)=2x$$
and $$f(2x)=4x$$
so
$$f(x)-f(2x)=-2x$$
and I try to solve the last equation
$$f(x)-f(2x)=-2x$$
by put $$f(x)=e^{mx}$$
then the solution ...
0
votes
0answers
108 views
Is computability theory a joke? [duplicate]
by N.J Wildberger Set Theory: Should You Believe?
I read the book, find some very idea shocking me. The author just destroys everything I had learned from computer science's courses.
Look at last ...
2
votes
1answer
38 views
Subgroups of $\mathbb{Z}^k$ of finite index $n$
I want to describe all subgroups in $\mathbb{Z}^k$ of finite index $n$.
I have solved it for the case $k=2$. In $\mathbb{Z}^2$, each subgroup of index $n$ corresponds to a matrix $\left( ...
0
votes
4answers
33 views
rewriting equation in terms of $y$
From Stewart, Precalculus, 5th ed, P98, Q.45
$$x^2+xy+y^2=4$$
how can I re-write this equation in terms of $y$? I want to put this equation into graphing software but don't know to put $y$ on one ...
0
votes
0answers
8 views
what is the derivative of Berthelot combining rule
Is anyone knows the derivative of Berthelot combining rule
$\epsilon_{ij}$ = $\sqrt {\epsilon_{ii} \epsilon_{jj}}$
Thanks in advance
1
vote
0answers
22 views
show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]
I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
0
votes
0answers
10 views
Clamped B-spline: repeat knots or control points
When we want a B-spline that reaches its first and last control points (clamped B-spline or open uniform B-spline), we can play on the multiplicity of the first and last knot of the knot vector $u = ...
1
vote
1answer
22 views
Divergence of $\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$ for $x>1$
How can we show that the series
$$\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$$
diverges for $x>1$ ?
The book gives the following hint:
consider
$$\sum_{k=1}^\infty\sum_{n=M_{k-1}+1}^{M_k}\frac{1}{(\ln ...
1
vote
0answers
17 views
Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)
Got a quick question from a past exam paper.
If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
0
votes
1answer
20 views
The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$
Suppose $x$ is invertible in the unital Banach algebra $A$.
How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
3
votes
0answers
25 views
algebraic or homotopical proof for Kakutani fixed point theorem
As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
5
votes
0answers
57 views
Help solving this Diophantine equation
For a problem that I'm working on, I need to solve this Diophantine equation:-
$ -2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all DISTINCT positive integers, and $a, b, c \notin$ { 2, 9, ...
2
votes
1answer
26 views
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
0
votes
1answer
21 views
Spectrum of T in $B(\ell^2)$
Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows:
$$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$
What is spectrum of T in $B(\ell^2)$?
-1
votes
0answers
23 views
How to detect ellipse [closed]
I tried to detect the rim of the this cup
I've tried this but the detection result is not quite good. The reason might be that ChanVeseBinarize function can't separate the rim and the body. Therefore ...
0
votes
1answer
23 views
Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
4
votes
0answers
23 views
Set of maximal ideals such that localization not integrally closed
Let $A$ be a Noetherian domain that is also finitely generated as a $k$ - algebra over a field $k$. Form the quotient field of $A$ (which I denote $K$) and consider the integral closure $\overline{A}$ ...
3
votes
1answer
38 views
Do the adjoint functor theorems usefully dualise?
The special and general adjoint functor theorems exist to construct left adjoints to particular functors given certain conditions on them. However, I've not been able to find much mention – at least, ...
2
votes
2answers
37 views
which of the followings are true for bijective functions
which of the followings are true:-
1. There is a continuous bijection from $\mathbb{R}^2\to \mathbb{R}$.
2. There is a bijection between $\Bbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$.
Can somebody ...
0
votes
1answer
24 views
Tutte's 1947 proof and paper on a family of cubical graphs
It is known that if a graph is connected, cubic, simple and $t$-transitive, then $t \le 5$. A proof is given in [Biggs, Algebraic Graph Theory, Chapter 18], and this result is due to [Tutte, ``A ...
3
votes
1answer
70 views
Career in Number Theory?
I am about to get my B.S. in Mathematics, and I will be applying for PhD in pure mathematics next year, with future plans of teaching and doing research. Over the past year, I have developed a great ...
0
votes
2answers
106 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
1
vote
0answers
40 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
0
votes
1answer
31 views
Rolling dices and simple problem
I'm facing the following problem.
Let's say I have N dices in a hand. I need to calculate how much time I should roll my dices to make all of them equal selected (pre-defined) number. Each time when ...
0
votes
0answers
19 views
Question about Lambert W function
I'm looking for a series for $W_0(x)$ for x $\in [\frac{-1}{e},\infty [$ but every time i found only for $x\in [\frac{-1}{e}, \frac{1}{e}]$
and what about a series for $W_-1(x)$
if it is no series ...
0
votes
1answer
12 views
question on five number summary & quantile.
i know that in five number summary :
25% of a data set lies between Min & 1st quartile.
50% of a data set lies between Min & 2nd quartile, that is, Median.
75% of a data set lies between ...
