Category Archives: math analysis

this studies generalizations of calculus

wedge product example


When bivectors are defined by

\beta^i\wedge\beta^j:=\beta^i\otimes\beta^j-\beta^j\otimes\beta^i,

so, for two generic covectors

\theta=a\beta^1+b\beta^2+c\beta^3 and \phi=d\beta^1+e\beta^2+f\beta^3,

we have the bivector

\theta\wedge\phi=(bf-ce)\beta^2\wedge\beta^3+(cd-af)\beta^3\wedge\beta^1+(ad-be)\beta^1\wedge\beta^2.

 

Otherwise,

Cf. this with the data \left(\begin{array}{c}a\\b\\c\end{array}\right) and \left(\begin{array}{c}d\\e\\f\end{array}\right) to construct the famous

\left(\begin{array}{c}a\\b\\c\end{array}\right)\times\left(\begin{array}{c}d\\e\\f\end{array}\right)=\left(\begin{array}{c}bf-ce\\cd-af\\ad-be\end{array}\right)

So, nobody should be confused about the uses of the symbol \wedge dans le calcul vectoriel XD

Advertisements

2 Comments

Filed under algebra, cucei math, differential geometry, math analysis, mathematics, multilinear algebra, what is math, word algebra

double coset counting formula


the double coset counting formula is a relation inter double cosets HaK, where a\in G and H,K subgroups in G. This is:

\#(HaK)=\frac{|H||K|}{|H\cap aKa^{-1}|}

and

\#(G/K)=\sum_a[H;H\cap aKa^{-1}]

The proof is easy.

One is to be bounded to the study of the natural map H\times K\stackrel{\phi_a}\to HaK. And it uses the second abstraction lemma.

The formula allows you to see the kinds of subgroups of arbitrary H versus K a p-SS of G, p-SS for the set of the p– Sylow subgroups.

Or, you can see that through the action H\times G/K\to G/K via h\cdot aK=haK you can get:

  • {\rm Orb}_H(aK)=\{haK\} which comply the equi-partition
  • HaK=aK\sqcup haK\sqcup...\sqcup h_taK, so \#(HaK)=m|K|, for some m\in \mathbb{N}
  • {\rm St}_H(aK)=H\cap aKa^{-1}

then you can deduce:

|G|=\sum_a\frac{|H||K|}{|H\cap aKa^{-1}|}

Now, let us use those ideas to prove the next statement:

Let G be a finite group, with cardinal |G|=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}, where each q_i are primes with q_1<q_2<...<q_t and n_i positive integers.

Let H be a subgroup of |G| of index [G:H]=q_1.

Then, H is normal.

Proof:

By employing K=H in the double coset partition, one get the decomposition:

G=HeH\sqcup Ha_1H\sqcup...\sqcup Ha_tH

So by the double coset counting formula you arrive to:

|G/H|=1+[H:H\cap a_1Ha_1^{-1}]+\cdots+[H:H\cap a_tHa_t^{-1}]

i.e.

q_1=1+\frac{|H|}{|H\cap a_1Ha_1^{-1}|}+\cdots+\frac{|H|}{|H\cap a_tHa_t^{-1}|}

From this, we get \frac{|H|}{|H\cap a_iHa_i^{-1}|}<q_1.

But |G|=q_1|H| as well |H|=|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}] so

|G|=q_1|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}], i.e.

[H:H\cap a_iHa_i^{-1}] divides |G|

Then [H:H\cap a_iHa_i^{-1}]=1. So |H|=|H\cap a_iHa_i^{-1}| for each a_i.

This implies H=H\cap a_iHa_i^{-1} and so H=a_iHa_i^{-1} for all the posible a_i, hence, H is normal.

QED.

4 Comments

Filed under algebra, categoría, category theory, fiber bundle, group theory, math, math analysis, mathematics, maths, what is math, what is mathematics

puntos críticos de una función suave en el círculo


En esta breve nota demostraremos que cada función f:S^1\to{\mathbb{R}}^1 que tenga un punto crítico aislado debe de tener otro.

Entonces supongamos que existe un punto p en S^1 talque {\rm grad}f(p)=\left[\frac{\partial f}{\partial x}|_p,\frac{\partial f}{\partial y}|_p\right]=\vec{0}, pero si elegimos la parametrización \phi:\ ]0,2\pi[\longrightarrow S^1 dada por t\longmapsto\left(\begin{array}{c}\cos(t)\\ \\ \sin(t)\end{array}\right), entonces tenemos una función g=f\circ\phi para la cual, la regla de la cadena implica que g'=f'(\phi)\phi' satisface

\frac{d g}{dt}|_{t_0}=\left[\frac{\partial f}{\partial x}|_p,\frac{\partial f}{\partial y}|_p\right]\left(\begin{array}{c}-\sin\\ \\ \cos\end{array}\right)_{|_{t_0}}

i.e.

\frac{d g(t_0)}{dt}=-\frac{\partial f(p)}{\partial x}\sin(t_0)+\frac{\partial f(p)}{\partial y}\cos(t_0)

entonces si {\rm grad}f(p)=\vec{0} tendremos \frac{d g(t_0)}{dt}=0, en otras palabras g tiene puntos críticos en t_0 y en t_0+2\pi.

Pero además g(t_0)=f\circ\phi(t_0)=f(p) tanto como

g(t_0+2\pi)=f\circ\phi(t_0+2\pi)=f\circ\phi(t_0)=f(p)

es decir g(t_0)=g(t_0+2\pi) y entonces –por el teorema de Rolle– existe t_1 en el intervalo abierto ]t_0,t_0+2\pi[ talque \frac{d g(t_1)}{dt}=0.

Pero si nos restringimos a S^1\setminus\{p\} entonces f=g\circ\phi^{-1},
y así (también por la regla de la cadena) tenemos {\rm grad}f=\frac{dg}{dt}\ {\rm grad}\ \phi^{-1} i.e.

\left[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right]=\frac{dg}{dt}\left[\frac{\partial\phi^{-1}}{\partial x},\frac{\partial\phi^{-1}}{\partial y}\right]

que evaluando en t_1 implica

\frac{\partial f(q)}{\partial x}=\frac{dg(t_1)}{dt}\frac{\partial\phi^{-1}(q)}{\partial x}=0

tanto como

\frac{\partial f(q)}{\partial y}=\frac{dg(t_1)}{dt}\frac{\partial\phi^{-1}(q)}{\partial y}=0

por lo tanto {\rm grad}f(q)=\vec{0}, donde q\neq p \Box

critical points of functions on the circle

Leave a comment

Filed under calculus on manifolds, low dimensional topology, math, math analysis

Multilinear Algebra


álgebra multilineal es como un cálculo vectorial dos o álgebra lineal tres

entonces para poder hacer cálculos en otras geometrías, inclusive muy diferentes a \mathbb{R}^n vamos viendo hacia donde tenemos que caminar: ver  (un post previo con estas ideas en mente).

RE-ENGINEERING LINEAR ALGEBRA

RE-ENGINEERING VECTOR CALCULUS

Leave a comment

Filed under cucei math, differential geometry, geometry, math, math analysis, multilinear algebra, topology

words-length and words in a group


the group is \mathbb{Z}_2*\mathbb{Z}_3=\langle a,\ b\mid a^2,\ b^3\rangle, and the picture:

Correctly predicts the next one. It is A164001 in the OEIS data-base. Dubbed “Spiral of triangles around a hexagon“. It has the generating function -(x+1)+\frac{x^2+2x+1}{1-x^2-x^3}, Why would it be? :) . This another is A000931.

Leave a comment

Filed under free group, math, sum of reciprocals

multilinear algebra 1, a synoptic view


what is math? let us discuss:

Baby Abstract Multilinear Algebra
Baby Multilinear Algebra  of Inner Product Spaces
Calculus in \mathbb{R}^n
Algebraic Differential Geometry
  • Parameterizations: curves and surfaces
  • Tangent vectors, tangent space, tangent bundle
  • Curves in \mathbb{R}^2 and \mathbb{R}^3 and on surfaces in \mathbb{R}^3
  • Surfaces in \mathbb{R}^3
    1. all classical surfaces rendered
    2. tangent space change of basis
    3. vector fields and tensor fields
    4. Christoffel’s symbols (connection coefficients)
    5. Curvatures (Gaussian, Mean, Principals, Normal and Geodesic)
  • Vector Fields, Covector Fields, Tensor Fields
  • Integration: Gauss-Bonnet, Stokes
Baby Manifolds (topological, differential, analytic, anti-analytic, aritmetic,…)
Examples: Lie groups and Fiber bundles

Leave a comment

Filed under algebra, calculus on manifolds, categoría, differential equations, differential geometry, fiber bundle, geometry, math analysis, maths, multilinear algebra, topology, what is math

duality rules!


in this post I am gonna again insist about the cruel confusion spawn by people who, I think, misunderstand certain strategic points in the grasping of tensors.

Vectors in \mathbb{R}^n are linear combinations of

e_1=\left(\begin{array}{c}1\\ 0\\ 0\\\vdots\\ 0\end{array}\right) , e_2=\left(\begin{array}{c}0\\ 1\\ 0\\\vdots\\ 0\end{array}\right) , … , e_n=\left(\begin{array}{c}0\\\vdots\\ 0\\ 0\\ 1\end{array}\right) ,

that is, vectors are expressions as v=v^1e_1+v^2e_2+\cdots+v^ne_n, or employing the Einstein-Penrose convention: v=v^se_s.

But, covectors are linear combinations of

e^1=[1,0,0,...,0] , e^2=[0,1,0,...,0] , … , e^n=[0,...,0,1] ,

which are also dubbed “projectors” due its role as a linear maps {\mathbb{R}}^n\to{\mathbb{R}}.

So a general covector is a linear combination f_se^s, using the Einstein-Penrose Convention again.

It is regrettable how many authors are merciless in the misuses of the rows and columns concepts for vectors and covectors respectively. It seems that sacrificing the difference by writing vectors as rows to gain space in the texts of linear algebra books is a big cost in the habits of understand well the matter.

This is perhaps a simple reason why it is so difficult to everybody to grasp the idea of tensors at its first steps. Well, not everybody … hehehe.

We say that duality rules ‘cuz

e^i(e_j)={\delta^i}_j

becomes a simple device to serve as a guide for a (apparently) complex calculations that arise when one consider tensors of higher rank.

Once recognized this hurdle, the constructions of a calculus à la  Cartan, using the wedge product and the exterior derivative among tensor fields is an easy cake. These techniques will allow you to free of the ugly vector calculus that rules in each bachelor science schools nowadays.

2 Comments

Filed under algebra, calculus on manifolds, cucei math, differential geometry, math analysis, multilinear algebra, tex