Questions tagged [hypercomplex-numbers]
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15 questions
-2 votes
2 answers
311 views
Real matrix rings and associative hypercomplex numbers
Are there real matrix rings which are not hypercomplex number systems? Is there a canonical form of a real matrix ring? By a hypercomplex number system I mean a finite-dimensional, unital, associative ...
user535997
5 votes
0 answers
522 views
How to define $\mathbb{R}^\frac{1}{2}$?
The Cayley-Dickson construction generates higher-dimensional hyper complex numbers from lower-dimensional ones, producing algebras of dimension $2^n$. I want to generate an algebra of dimension $2^{-1}...
- 75
2 votes
0 answers
76 views
Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
- 131
1 vote
0 answers
55 views
Eigendecomposition of hyper-complex multiplication
There is an isomorphism between quaternions and $4\times 4$ matrices: $$ \phi: a+bi+cj+dk \longmapsto \begin{pmatrix} a&b&c&d \\ -b&a&-d&c\\ -c&d&a&-b\\ -d&-c&...
- 1,211
3 votes
0 answers
140 views
Efficient multiplication of Cayley-Dickson numbers
The question was already asked here, but doesn't have any meaningful answer, hence I'd like to re-post it. Assuming that we have an algebra with conjugation, we can use Cayley-Dickson construction to ...
- 1,211
3 votes
0 answers
507 views
Extending reals with logarithm of zero: properties and reference request
If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...
- 10.4k
4 votes
1 answer
513 views
Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?
Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this? ...
- 10.4k
6 votes
0 answers
451 views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series. If the function $f: \mathbb{C} \to \mathbb{C} $ ...
- 5,009
-5 votes
1 answer
330 views
What are the properties of 3-dimensional split-complex numbers?
I have often encountered claims that 3-dimensional numbers are impossible. But it seems to me that $\mathbb{R}^3$ with Hadamard multiplication should in fact behave quite similar to split-complex ...
- 10.4k
1 vote
1 answer
264 views
Representing split-complex numbers as intervals and related compactification
Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
- 10.4k
-5 votes
1 answer
268 views
Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]
It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
- 10.4k
1 vote
1 answer
502 views
Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
- 10.4k
0 votes
0 answers
214 views
Can the notion of algebraic closedness be generalized to the rings with zero divisors?
Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero? For instance, it seems that any polynomial of non-zero-divisor-...
- 10.4k
3 votes
2 answers
529 views
The name of special 16-dimensional hypercomplex number
Let's consider the following number: $n = n_0 + n_1\textbf{i} + n_2\textbf{j} + n_3\textbf{k}$ Where $\{1,\textbf{i},\textbf{j},\textbf{k}\}\subset\mathbb{H}$ and in turn each component can be written ...
4 votes
1 answer
788 views
Function theory of a hyperbolic variable
I've found quite a number of articles on the basics of function theory in one hyperbolic (split-complex, dual, duplex, motro,..) variable, perhaps the most notable being http://arxiv.org/PS_cache/math-...
- 41