in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over Fp, whose jinvariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of ...

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding–Iohara–Miki algebra and the trigonometric Feigin–Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author const...

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator deened in 4, 7]. In the preceding papers 14]-18], a class of H-matrices has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An H-matrix approximation to the operator exponent with ...

For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL = LLR, the trace of the L-operator gives a commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficie...