We thus introduce and study the problem of fair two-dimensional division wherein the allotted pieces must be of some restricted two-dimensional geometric shape(s), particularly squares and fat rectangles.
Furthermore, the corresponding valuation ring (mathcal O _mathcal{D }) has a unique maximal 2-sided ideal (mathfrak P _mathcal{D }) (the valuation ideal), and the quotient (overline{mathcal{D }} = mathcal O _mathcal{D }/mathfrak P _mathcal{D }) is a finite-dimensional division (but not necessarily central) algebra, called the residue algebra, over the residue field (overline{mathcal{K }}).
Whereas vitality and unease bear some resemblance with dimensions of arousal and valence, respectively, sublimity might be more specific to the aesthetic domain, and the traditional bi-dimensional division of affect does not seem sufficient to account for the whole range of music-induced emotions.
Let (D) be a finite-dimensional central division algebra over a field (K).
Now let (D) be a finite-dimensional central division algebra over (K) such that ([D] in _nmathrm{Br}(K)).
For a finite-dimensional central division algebra (A) over a field (K), we let ([A]) denote the corresponding class in the Brauer group (mathrm{Br}(K)) of (K).
We note that for (G = mathrm{SL}_{1, D}), where (D) is a finite-dimensional central division (K -algebra, only maximal separable subfields of (D) give rise to maximal (K -algebra (G).
The quaternions were invented by Sir William Rowan Hamilton in 1843 and form a four-dimensional normalized division algebra over the real numbers i.e. a quaternion is defined by four real values, usually noted (q1, q2, q3, q4).
We recall that given a finite-dimensional central division algebra (mathcal D ) over a field (mathcal K ) which is complete with respect to a discrete valuation (v), the valuation (v) uniquely extends to a discrete valuation (tilde{v}) of (mathcal D ) (cf. [27, Chap. 12, §2], [33]).
Suppose (mathcal D _1) and (mathcal D _2) are two finite-dimensional central division (mathcal K )-algebras of degree (n) prime to (mathrm{char} k), and, for (i = 1, 2), let (mathcal E _i) be the center of the residue algebra (overline{mathcal{D }}_i).
