Partial differntials with hypercomplex numbers

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Discussion Overview

The discussion revolves around the application of hypercomplex numbers in solving partial differential equations (PDEs). Participants explore the nature of constants of integration in this context and raise questions about the implications of using hypercomplex variables in mathematical transformations, particularly in relation to specific mediums like water or petroleum.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions why constants of integration in PDEs within the hypercomplex domain are not treated as arbitrary functions of x and t, suggesting they should remain constant during differentiation.
  • The same participant expresses confusion about the treatment of constants when setting t=0 and leaving x unchanged, proposing that these constants might actually be functions of x or t.
  • Another participant notes a lack of familiarity with hypercomplex numbers and mentions that the terminology has fallen out of use, indicating a potential barrier to understanding the original question.
  • A different participant provides a definition of hypercomplex numbers, specifically mentioning quaternions as an example, and references their representation as 2x2 matrices of complex numbers.
  • Further clarification is provided about hypercomplex systems being referred to as algebras in modern terminology, with a mention of Cayley numbers as another example.

Areas of Agreement / Disagreement

There is no consensus on the original question regarding the treatment of constants in hypercomplex PDEs. Participants express varying levels of understanding of hypercomplex numbers, and some uncertainty remains about the terminology and concepts involved.

Contextual Notes

Participants highlight a potential lack of clarity regarding the definitions and properties of hypercomplex numbers, which may affect the discussion. There are unresolved questions about the implications of using specific constants in the context of different mediums.

mrcuteblackie
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Thank you for your knowledge of maths and science, and for the good cooperation.
First of all, may I asked a question on solving partial differential equations using hyper complex variables?

Please can you briefly explain why the constants of integration in solving partial differential equations (x,t) within the 4D hyper-complex domain are not arbitrary functions of x,t, but are actual constants which should not be differentiated when finding derivatives.
If we have three constants for 1 initial + 2 boundary conditions, and we want to set t=0, then we must leave x as it is. Therefore the arbitrary constants are instead arbitrary functions which depend on either x or t. Therefore, I do not understand why they are not differentiated in hypercomplex variables. Or is it appropriate to take any x?

May I pop in another minor question? In Z=1x+iy+ jz+kct, I know c is the characteristic speed of the medium, what if we were dealing with water, or peroleum? Does this mean that that number was in mind when deriving this transformation? I am solving the Burgers equation, which does not require scaling the time, so why should I scale it here?
Thanks a million
 
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sorry for not answering. we may not know what hypercomplex numbers are. in an old algebra book i have from over 50 years ago, it says hypercomplex systems are called "nowadays" algebras.

so hypercomplex systems language has gone out of usage some 50-70 years ago.

i apologize, but we would answer if we understood the question better.

best regards.
 
Hypercomplex numbers are defined as extensions of the complex numbers. They are constructed in algebra: quaternions are the only ones I ever encountered - that I remember. Hamilton discovered them. Maybe that's what the OP means...

Quaternions can be represented by a 2X2 matrix of complex numbers.

I don't quite get the question, either, but that's normal. :)
 
those are called algebras "nowadays", i.e. for the last 70-80 years, they are generally not necessarily commutative rings containing another commutative ring or field in their center, such as the quaternions are an algebra over the reals.

there are others, such as the cayley numbers, but these are the only algebraic ones i think. herstein has a little section on this.
 

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