Regular function of quaternion variable

Because I explore calculus over division ring I put attention on different papers related to quaternion calculus. These papers mainly discuss definition of regular function. According to definition regular function is function which satisfy to differential equation


^∂f/_∂x0+i^∂f/_∂x1+j^∂f/_∂x2+k^∂f/_∂x3=0

Recently Daniel Alayon-Solarz sent me interesting example of regular function.

f(x₀+x₁i+x₂j+x₃k)=x₀-x₁i+x₂j+x₃k

Higgs boson

The time when we see the answer on question does Higgs boson exist is close. I think that independently on answer physics will face new questions to answer.

If we will not find the Higgs boson then we need to reconsider model which predicted them.

More question will arise if we do find Higgs boson. The problem here is that Higgs boson is responsible for mass. Or to be more particular, Higgs boson is responsible for inertial mass. However we know two masses. We know inertial mass and we know gravitational mass. And these masses are equal. We yet not know the mechanism of this equality. Discovery of Higgs boson will refresh this question. If different interactions create inertial and gravitational masses, then third interaction should exist. And the third interaction is responsible for equality of masses. However if some interaction determines equality of inertial and gravitational masses, there is posibility when this equality will be broken.

How it may happens nobody knows. I can assume one of possibilities. Few years ago I explored tidal forces. I came to conclusion that these forces as fundamental as gravitational. I cannot tell which forces are primary. Do they interact one with another. Or one force is the reason of another. Feature research will show the answer on this question.

Derivative of function of division ring

When we learn derivative in calculus, we know that derivative at point is number. When we move to vector function we understand that derivative is linear operator. This allows use Jacobian as expression for derivative. However in this case we again have option to separate derivative and increment of argument.

When we start to work with function of division ring (quaternion valued function for instance) we expect something similar. However it does not work or makes set of differentiable functions very limitted. Derivative of function of division ring is not number, but additive map and it is easy to see that this is the Gateaux derivative. When we learn derivative of function of division ring we cannot separate derivative itself and increment of argument.

For instance derivative of function f(x)=axb has form

∂f(x)(h)=ahb

Derivative of function f(x)=x² has form

∂f(x)(h)=xh+hx

Derivative of function f(x)=x^-1 has form

∂f(x)(h)=-x^-1hx^-1

Additive map of division ring

I created this blog to share and discus ideas related to my research in geometry.

Recently one question arised. What happens when in geometry that I know to put division ring instead of field. When I started research in 2005 it was not clear how far I can go. However I discovered that linear algebra has a lot of common, at least when we work with linear space or try to solve system of linear equations.

I met first severe problems when decided to work with tensor product. I saw that non-commutativity may cause strong problems. Moving to calculus I realised that there exists new type of map between vector spaces. This map holds sum of vectors, however it does not hold product of vector over scalar. This is why I called such map additive.

When I returned back to linear algebra additive map helped me to define tensor product of vector spaces. Because we can consider division ring as vector space we can define tensor product of division rings. Tensor product of division rings is not division ring and tensor product of vector spaces is not vector space. However this module has basis over tensor product of rings.

Quaternion algebra is an example of division ring.

You can read my papers in arXiv

eprint arXiv:math.GM/0701238 Lectures on Linear Algebra over Division Ring, 2007

eprint arXiv:0812.4763 Introduction into Calculus over Division Ring, 2008