3 Quaternion Representation 5 3.1 Axis-Angle to Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Quaternion to Axis-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Quaternion to Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Matrix to Quaternion . . . . . . . . . . ....

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We concl...

We determine the Shimura modular curve X0(3) and the Jacobian of the Shimura modular curve X1(3) associated with the congruence subgroups Γ0(3), Γ1(3) of the (2, 3, 7) triangle group. This group is known to be arithmetic and associated with a quaternion algebra A/K ramified at two of the three real places of K = Q(cos 2π/7) and at no finite primes of K. Since the rational prime 3 is inert in K,...

Let C1 ⊗F C2 be the tensor product of two composition algebras over a field F with char(F ) 6= 2. R. Brauer [7] and A. A. Albert [1], [2], [3] seemed to be the first mathematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general situation by B. N. Allison [4], [5], [6] and to biquaternion algebras over rings by Knus [12]....

A method of reducing general quaternion functions of first degree, i.e., linear quaternion functions, to quaternary canonical form is given. Linear quaternion functions, once reduced to canonical form, can be maintained in this form under functional composition. Furthermore, the composition operation is symbolically identical to quaternion multiplication, making manipulation and reduction of sy...

Let H be the real quaternion algebra and Hm×n denote set of all m×n matrices over H. For A?Hm×n, we by A? n×m matrix obtained applying ? entrywise to transposed AT, where is a non-standard involution A?Hn×n said ?-skew-Hermicity if A=?A?. In this paper, provide some necessary sufficient conditions for existence ?-skew-Hermitian solution system equations with four unknowns AiXi(Ai)?+BiXi+1(Bi)?=...

Recurrent Neural Networks (RNNs) are known for their ability to learn relationships within temporal sequences. Gated Unit (GRU) networks have found use in challenging time-dependent applications such as Natural Language Processing (NLP), financial analysis and sensor fusion due capability cope with the vanishing gradient problem. GRUs also be more computationally efficient than variant, Long Sh...

We show that the completed Hecke algebra of $p$ -adic modular forms is isomorphic to continuous automorphic for units quaternion ramified at and $\infty$ . This gives an affirmative answer a question posed by Serre in 1987 letter Tate. The proof geometric, lifts mod argument due Serre: we evaluate identifying quaternionic double-coset with fiber Hodge–Tate period map, extend functions off using...

Similar sublattices of the root lattice A4 are possible [9] for each index that is the square of a non-zero integer of the form m + mn − n. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q( √ 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given ...

In this paper we consider space-time codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also point out that lattice codes based on Q-central division algebras satisfy this optimality criterion. As...