3
votes
2answers
250 views

Partition of Unity question

I am starting to read the book "Differential Forms in Algebraic Topology" by Bott and Tu. In the proof of the exactness of the Mayer - Vietoris sequence (Proposition 2.3, page 22 - 23) a partition ...
6
votes
3answers
120 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
0
votes
1answer
43 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
0
votes
0answers
5 views

generating orthogonal parity check matrix, from a random generator matrix

I have a matrix G of dimension 13x20. It is a full rank matrix. It is not in the standard form of a generator matrix. Now for the parity check matrix 'H', I need a standard representation H=[-P;I]. ...
1
vote
6answers
96 views

Solution of $(x^2 + y^2)\ dx -2xy\ dy$ = 0

Solve $(x^2 + y^2)dx -2xydy = 0$ The answer is $x^2 - y^2 = Cx$ I've tried the following methods but I'm not getting the answer : Variable Separable (n/a) Homogenous Differential Equation ...
0
votes
0answers
6 views

$K_0$ of a ring via idempotents

As it is described here, the group $K_0(R)$ of a unital ring can also be described in termes of conjucation classes of idempotents. In this text it is shown that the semigroup of isomorphism classes ...
0
votes
1answer
21 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
0
votes
1answer
151 views

Derivative of Cholesky decomposition

I would like to compute the derivative of the Cholesky decomposition, for example I have a matrix 2 x 2 R = 1 rho rho 1 where rho is a parameter, now I compute the Cholesky decomposition of ...
1
vote
1answer
20 views

What is this 2-D array of numbers called?

Define $c(n,r)$ ($n\in\Bbb N;r\in\Bbb Z$) by setting $c(0,-1)=-1$, $c(0,0)=1$, and $c(0,r)=0$ otherwise, with all further $c(n, r)$ given recursively by $$c(n+1,r)=rc(n,r-1)-(r+1)c(n,r)$$(rather in ...
1
vote
1answer
51 views

Connection between harmonic functions, Bochner Laplacian and Ricci curvature

I stumbled upon the following claim in a paper: "We write the (Bochner) Laplacian in suffix notation: $\Delta_B = \nabla ^k \nabla_k$". after this statement, the following is written: ($M$ is a ...
1
vote
0answers
17 views

Q($\sqrt[3]{2}$) - Unique Factorisation Domain?

I am considering the set of "integers" of the from $$ a+b\sqrt[3]{2} + c\sqrt[3]{4} $$ where $a,b,c$ are integers. It is easy to show this field is closed under addition and multiplication. I then ...
2
votes
6answers
36 views

Ratio and algebric problem.

If $\frac{(a+b)}{b}=\frac{(b+c-a)}{(c+a-b)}=\frac{(2a+b+2c)}{(a+b+c)}$ then prove that $6a=4b=3c$.
7
votes
4answers
21k views

Proving that the number of vertices of odd degree in any graph G is even

I'm having a bit of a trouble with the below question Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$. ...
0
votes
1answer
20 views

$SL(n,\mathbb R)$ diffeomorphic to $SO(n) \times \mathbb R^{n(n+1)/2-1}$?

Question : How to show $SL(n,\mathbb R)$ diffeomorphic to $ SO(n) \times \mathbb R^{n(n+1)/2-1}$? Also, how to show $SL(n,\mathbb C)$ diffeomorphic to $ SU(n) \times \mathbb R^{n^2-1}$? I have ...
0
votes
0answers
15 views

About definition of Sobolev spaces

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a open subset and $dx$ be a Lebesgue measure on $U$. We often define \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
-1
votes
0answers
9 views

About Imre Hermann's book 'Parallelismes'

Apparently, in his book Parallelismes, Imre Hermann discusses Hilbert, Brouwer en Russell from the viewpoint of psycho-pathology. Does anyone know whether the entire book is about ...
3
votes
1answer
32 views

$f_n$ converges uniformly to f but $f^2_n$ fails to converge uniformly to $f^2$

Consider functional sequence $f_{n}$ which is differentiable on $\left(a,b\right)$. Find an example of $f_n$ converging uniformly to f on $\left(a,b\right)$ such that $f^2_n$ fail to converge ...
1
vote
2answers
59 views

Proving the an expression is larger than a simplified quadratic

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$ Any explanation/answer would be extremely helpful. Thanks : )
1
vote
3answers
87 views

All simple modules are projective $\Rightarrow$ semisimple [duplicate]

Let $A$ be a finite dimensional algebra over a field $K$. It is clear that if $A$ is semisimple, then every simple module is projective. Does the converse hold ? It seems false, but I can't find a ...
0
votes
1answer
46 views

Find constant $c < 1$, such that Fibonacci number $F(n) \le 2^{cn}$ for every $n \ge 0$

I have an outline for solution, but I am afraid that it's not mathematically rigorous at all. Would you be so kind to point problems with this solution if any? 1) I looked up in Wikipedia that $F(n)$ ...
0
votes
1answer
32 views

Continuous linear operator T at a point T is then continued

The problem is the next If T is continuous at a single point, it is continuous, without using that T is continuous iff T is bounded. I tried this result as follows If T is continuous at a single ...
0
votes
0answers
11 views

Total derivative of a complex function using Wirtinger derivatives

The mathworld.wolfram page on Cauchy-Riemann Equations states that given a complex function $f(z) = f(x,y) = u(x,y) + iy(x,y)$ has the derivative: $$\frac {df} {dz} = \frac {\partial f} {\partial x} ...
0
votes
0answers
11 views

Standard notation for the set of children of a node in a rooted tree

In graph theory, given a rooted tree $T$ and a node $a \in V(T)$, is there a standard way to refer to the set of all children of $a$? I have seen $CHILDREN_T(a)$ being used, but this seem quite clumsy ...
3
votes
3answers
74 views

Prove $\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$ using $\delta-\epsilon$ definition of limit

I need to prove that the $$\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$$ using $\delta-\epsilon$ definition of limit. Now, I have started with a discussion, saying that what we want is that if $\left| x - ...
9
votes
6answers
1k views

Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
3
votes
1answer
59 views
+50

Steepest descent method with movable maximum

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral $$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$ where $C$ is some contour in the ...
3
votes
8answers
74 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
0
votes
0answers
6 views

How to find maximal dimension abelian subalgebra in finite Lie Algebra?

Is there any well known algorithm how to find maximal dimension abelian subalgebra in finite dimension Lie Algebra? If there is a built-in routine in some computer algebra system, it is the most ...
0
votes
0answers
18 views

Prove or disprove that $φ_v:u\mapsto \langle\mathcal A u,v\rangle$ is in $V^*$

Let us consider a linear and continuous operator on a Hilbert space $V$, $\mathcal A:V\rightarrow V$, such that: $$\|\mathcal A u\|\leq M \|u\|, \ \ \forall u\in V, M>0$$ and now consider ...
1
vote
0answers
24 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
2
votes
1answer
318 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
0
votes
0answers
10 views

Measure of variation(?) of multidimensional polynomial function

I have a multidimensional function $$\mathbf{f}(x) = [f_0(x), ... , f_N(x)]$$ where $f_n$ are real-valued trigonometric polynomials. I want to measure how much $\mathbf{f}(x)$ varies over some ...
1
vote
2answers
64 views

Is the area of $T(\Omega)=|\det A|\,(\text{area of }\Omega)$?

We are given that $\Omega$ is a parallelogram in $\mathbb{R}^3$ and $\left\{ T(\vec{x}) = A\vec{x} \mid \mathbb{R}^3 \mapsto \mathbb{R}^3\right\}$ is a linear transformation. From the definition of ...
0
votes
0answers
23 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}x^n\left(\frac{n}{S_{n-1}}-1\right)$$ I need to show that ...
0
votes
0answers
12 views

multi objective optimization

Suppose we want to maximize two positive bounded objectives. A usual approach for this aim is to maximize a weighted sum of these two objectives. Now, my question is why not to maximize the product ...
0
votes
0answers
9 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
1
vote
1answer
46 views

Alternative Proof of the Extreme Value Theorem

I have proven the Boundedness Theorem for continuous functions and would now like to prove the Extreme Value Theorem; that is, show that the upper bound is indeed attained for continuous functions. I ...
0
votes
1answer
23 views

Two parameters inequality problem

Find $p \gt 0$ for which $f(x) =x\ln p - p\ln x$ is $\ge0$ for $x \gt 0$ They also say $p$ is real number.
-4
votes
3answers
25 views
11
votes
8answers
1k views

Why the group $\langle x,y\mid x^2=y^2\rangle $ is not free?

$G= \langle x,y\mid x^2=y^2\rangle $. I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
1
vote
1answer
5 views

Does $\beta \eta$ reduction preserve free variables?

It seems to be a know fact that if $M$, $N$ are $\lambda$-terms, and $M \twoheadrightarrow_{\beta\eta} N$, then $fv(N) \subseteq fv(M)$. My problem is: is it true that if $M ...
0
votes
3answers
30 views

Problem about $\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$

Consider $$\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$$ A. sum of the series is bounded on its set of convergence B. sum of the series is continuous on its set of convergence The correct ...
4
votes
1answer
68 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
1
vote
1answer
22 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
0
votes
0answers
13 views

Matrix representation of another matrix

Let $\mathbf{c}\in \mathbb{R}^n$ and $\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...
0
votes
2answers
64 views

Non-linear regression fit

I'm trying to fit my data to the following equation: $$ Y = A(1-2e^{bx}) $$ What I tried to do was transform the equation to a linear form via the following steps: \begin{align*} & A-Y = ...
0
votes
1answer
21 views

basic notions of measure theory: differences?

Could you help me differentiating the following notions of measure theory: law, probability, probability density, probability measure, probability distribution, distribution, distribution function. ...
2
votes
2answers
31 views

How do you multiply this

How can you multiply these ordinal numbers: $(\omega+1)(\omega+1)(\omega2+2)$ I tried and have gotten to this: $(\omega^2+1)(\omega2+2)$ Is that the correct way, or did i made a mistake?
0
votes
0answers
10 views

Complement of Conditional Probability

I'm currently reading this paper Censored Exploration and the Dark Pool Problem and have difficulties in understanding the following simple equality: Let $S$ be a positive integer random variable. ...
28
votes
6answers
2k views

Why is the last digit of $n^5$ equal to the last digit of $n$?

I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ...

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