Questions tagged [hypercomplex-numbers]

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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}...
Giulio Binosi's user avatar
1 vote
0 answers
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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&...
Oleksandr  Kulkov's user avatar
3 votes
0 answers
86 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 ...
Oleksandr  Kulkov's user avatar
3 votes
0 answers
369 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 ...
Anixx's user avatar
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2 votes
1 answer
286 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? ...
Anixx's user avatar
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6 votes
0 answers
365 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} $ ...
Max Muller's user avatar
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-4 votes
1 answer
223 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 ...
Anixx's user avatar
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0 votes
1 answer
157 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-...
Anixx's user avatar
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-5 votes
1 answer
196 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} &...
Anixx's user avatar
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1 vote
1 answer
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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\...
Anixx's user avatar
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0 votes
0 answers
168 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-...
Anixx's user avatar
  • 9,306
3 votes
2 answers
472 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 ...
Мікалас Кaрыбутоў's user avatar
4 votes
1 answer
691 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-...
HeWhoHungers's user avatar