Zeroes of slice monogenic functions

Slice monogenic functions

In this paper we offer a new definition of monogenicity for functions defined on R with values in the Clifford algebra Rn following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra Rn. We will prove a Cauchy integral formula as well as some of its conseque...

Computing zeroes of spline functions

Finely Differentiable Monogenic Functions

Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in [12] like a higher dimensional analogue of finely holomorphic functions.

Zeroes of Zeta Functions and Symmetry

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their z...

Quantum Computing and Zeroes of Zeta Functions

A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the spectral approach to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. The notion of universal, efficient quantum computation is used ...