This paper is devoted to deformation theory of graded Lie algebras over Z or Zl with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations...

In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard affine arithmetic, and (ii) modified affine arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified affine arithmetic is not onl...

quantum-dot cellular automaton (qca) is a novel nanotechnology with a very different computational method in compared with cmos, whereas placement of electrons in cells indicates digital information. this nanotechnology with specifications such as fast speed, high parallel processing, small area, low power consumption and higher switching frequency becomes a promising candidate for cmos technol...

A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid the excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and the central limit theorem, with a much tighter bounding r...

units (adders, ALUs, multipliers, dividers) are essential to fault-tolerant computer designs. Some researchers based early design schemes for such units on arithmetic residue codes.1 Others proposed parity prediction schemes for the same purpose.2 These schemes compute the output operand’s parity as a function of the operator’s internal carries and of the input operands’ parities. The basic dra...

Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the proposed algorithms for the arithmetic Fourier transform are surveyed. A new arithmetic transform for computing the discrete Hartley transform is introduced:...

While connected arithmetic discrete lines are entirely characterized, only partial results exist for the more general case of arithmetic discrete hyperplanes. In the present paper, we focus on the 3-dimensional case, that is on arithmetic discrete planes. Thanks to arithmetic reductions on a vector n, we provide algorithms either to determine whether a given arithmetic discrete plane with n as ...

While connected arithmetic discrete lines are entirely characterized by their arithmetic thickness, only partial results exist for arithmetic discrete hyperplanes in any dimension. In the present paper, we focus on 0-connected rational arithmetic discrete planes in Z. Thanks to an arithmetic reduction on a given integer vector n, we provide an algorithm which computes the thickness of the thinn...

A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid the excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and the central limit theorem, with a much tighter bounding r...