The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shim...

In this paper we study generators of t-norms on the lattice LI of closed subintervals of the unit interval. Starting from a minimal set of axioms that any addition and multiplication operator on LI must fulfill, we investigate the properties of the additive and multiplicative generators on LI obtained using these arithmetic operators. We also investigate under which conditions the generators on...

We show that the free propagation and interactions of all fundamental particle fields can be derived from arithmetic summations over a suitably parametrized null-cone. Our approach represents a considerable simplification of the path integral formalism and leads to essentially identical results as those obtained from conventional procedures, but without using i) differential operators, ii) the ...

We address the problem of formally verifying arithmetic instructions of microprocessors implemented by microprograms that contain loops. We try to avoid theorem proving techniques using a new symbolic representation: Binary Moment Diagrams (*BMDs). In order to use *BMDs for verifying sequential circuits as well as micro-programs, we extend this representation and deene several bit-vector level ...

This paper presents a method, based on the formalism of affine recurrence equations, for the synthesis of digital circuits exploiting parallelism at the bit-level. In the initial specification of a numerical algorithm, the arithmetic operators are replaced with their yet unscheduled (schedule-free) binary implementation as recurrence equations. This allows a bit-level dependency analysis yieldi...

It is well-known that for Hilbert space linear operators 0 ≤ A and 0 ≤ C, inequality C ≤ A does not imply C ≤ A. We introduce a strong order relation 0 ≤ B≪ A, which guarantees that C ≤ BAB for all 0 ≤ C ≤ B, and that C ≤ A when B commutes with A. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia–Kittaneh as well as the Kantorovich constant of A are mentioned.

Multiplicative Binary Moment Diagram (*BMD) is a new representation that inherits from BDD. *BMDs map Boolean variables to integer values and so they are well-suited to handle a hierarchical veri-cation methodology. In this paper we experiment this representation to verify sequential arithmetic circuits. We extend *BMDs and provide an algebra that includes main operators commonly used at the bi...

Stack filters are a class of non-linear, digital filters whose behaviour is defined by Positive Boolean Functions rather than the multiply-accumulate, arithmetic operators that are widely used in conventional digital filters. The implementation of stack filters that can be adapted in real-time is feasible using dynamically reconfigurable FPGAs. This paper reports on efforts to map adaptive stac...

We study the number of registers required for evaluating arithmetic expressions formed with any set of unary and binary operators. Our approach consists in a singularity analysis of intervening generating functions combined with a use of (complex) Mellin inversion. We illustrate it first by rederiving the known results about binary trees and then extend it to the fully general case of unaj-bina...

Due to the technological advances in recent years, paper scientific documents are used less and less. Thus, the trend in the scientific community to use digital documents has increased considerably. Among these documents, there are scientific documents and more specifically mathematics documents. In this context, we present our own dataset of handwritten mathematical symbols composed of 10,379 ...