The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n n random matrices within spectral intervals of the order O(n ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermi...

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form a(s)e−b(s) for a simply related to a Painlevé transcendent and b its anti-derivative. ...

Any rotation of a 3-dimensional object can be performed by three consecutive rotations over Euler angles. Intrinsic rotations produce the same result as extrinsic rotations in transformation. Euler rotations are the movement obtained by changing one of the Euler angles while leaving the other two constant. These Euler rotations are applied in a simple two-axis gimbals set mounted on an automoti...

Let B be an undefined quaternion algebra over Q. Following the explicit chacterization of some Eichler orders in B given by Hashimoto, we define explicit embeddings of these orders in some local rings of matrices; we describe the two natural inclusions of an Eichler order of leven Nq in an Eichler order of level N . Moreover we provide a basis for a chain of Eichler orders in B and prove result...