This paper studies the large fluctuations of solutions of scalar and finite–dimensional affine stochastic functional differential equations with finite memory, as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the partial maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploi...

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equ...

We use the double affine Hecke algebra of type GLN to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik’s quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations ac...

It is known that discrete Painlevé equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlevé equations in which the symmetries become clearly visible. We know how to obtain discrete Painlevé equations from certain rational surfaces in connection with the extended affine Weyl groups. By means of this representation, we clarify the rel...

Affine difference algebraic groups are a generalization of affine obtained by replacing equations with equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these and we establish Jordan–Hölder type theorem allows us to decompose any into almost-simple groups. also characterize via