In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and crypto-systems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...

In this paper, an ASIC suitable for cryptography applications based on modular arithmetic techniques, is presented. These applications, such as for example digital signature (DSA) and public key encryption and decryption (RSA), use, as basic operation, the modular exponentiation. This ASIC works as a coprocessor with a special set of instructions specialized on dealing with high accuracy integers, as well as on the rapid evaluation of modular multiplications and exponentiations. The algorithm, the hardware architecture, the design methodology and the results are described in detail. 1. Introduction Security has become a key issue in the world of electronic communication. Besides how fast data are transmitted, the security of these data through the communication channel arises as one of the most important problems. Though, the time overhead due to data encryption and decryption should not impose a bottleneck in the communication process. Public key cryptography (RSA), as well as othe...

In [6]–[9] the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series ∞∑ akzk at the origin and vice versa, implementations of which exist in Mathematica [19], (s. [9]), Maple [12] (s. [4]) and Reduce [5] (s. [13]). One main part of this procedure is an algorithm to derive a homogeneous linear differential equation with polynomial coefficients for the given function. We call this type of ordinary differential equations simple. Whereas the opposite question to find functions satisfying given differential equations is studied in great detail, our question to find differential equations that are satisfied by given functions seems to be rarely posed. In this paper we consider the family F of functions satisfying a simple differential equation generated by the rational, the algebraic, and certain transcendental functions. It turns out that F forms a linear space of transcendental functions. Further F is closed under multiplication and under the composition with rational functions and rational powers. These results had been published by Stanley who had proved them by theoretical algebraic considerations. In contrast our treatment is purely algorithmically oriented. We present algorithms that generate simple differential equation for f +g, f ·g, f ◦r (r rational), and f ◦x p/q (p, q ∈ IN0), given simple differential equations for f, and g, and give a priori estimates for the order of the resulting differential equations. We show that all order estimates are sharp. After finishing this article we realized that in independent work Salvy and Zimmermann published similar algorithms. Our treatment gives a detailed description of those algorithms and their validity. k=0 1 1 Simple

Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges' function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov's) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment, the closed representation of de Branges' function system is derived. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the Lowner differential ...

We describe a generalization of equivalence between constraint sets, called weak equivalence. This new equivalence relation takes into account that not all variables have the same function in a constraint set and therefore distinguishes between restriction variables and intermediate variables. We explore the properties of weak equivalence and its underlying notion of weak implication with an axiomatic approach. In addition a complete set of axioms for weak implication is presented. With examples derived from the declarative rule language RL we show the applicability of weak equivalence to constraint solving. 1

: We give an overview of an approach on special functions due to Truesdell, and show how it can be used to develop certain type of identities for special functions. Once obtained, these identities may be verified by an independent algorithmic method for which we give some examples. 1 The F-equation and F-functions Truesdell [11] studied solutions of the functional equation @ @z F (z; ff) = F (z; ff + 1) (1) satisfying the initial condition F (z 0 ; ff) = \Phi(ff) (2) where F is a function of the two variables z and ff. Here z is assumed to be a real or complex variable, ff is such that either ff = ff 0 + k (k 2 IN 0 ) ; or ff = ff 0 + k (k 2 ZZ) ; or ff ff 0 (ff 2 IR) ; or Re ff ff 0 (ff 2 C) ; (3) (ff 0 may equal \Gamma1), and \Phi is a given function of ff. Equation (1) is called the F -equation, and we call a solution of the F-equation that satisfies the initial condition (2) an F -function corresponding to the initial function \Phi. Truesdell showed how a functional equat...

We explore the postulates of string no-scale supergravity in the context of free-fermionic string models. The requirements of vanishing vacuum energy, flat directions of the scalar potential, and stable no-scale mechanism impose strong restrictions on possible string no-scale models, which must possess only two or three moduli, and a constrained massless spectrum. All soft-supersymmetrybreaking parameters involving untwisted fields are given explicitly and those involving twisted fields are conjectured. This class of models contain no free parameters, i.e., in principle all supersymmetric particle masses and interactions are completely determined. A computerized search for free-fermionic models with the desired properties yields a candidate SU(5) \Theta U(1) model, and evidence that all such models contain extra (10,10) matter representations that allow gauge coupling unification at the string scale. Our candidate model possesses a novel assignment of supersymmetry breaking scalar mas...

Quantifying the errors introduced by signal sampler imperfections is of interest to people who are doing frequency domain measurements. The phase-distortion introduced by the sampler is hard to quantify and is usually neglected. In this article upper bounds for this phase-distortion error are derived which are based upon simple assumptions, namely that the sampler weighting function is strictly positive, that it is limited in time and that the function has only one local maximum. The theoretical limits are applied in order to specify the accuracy of a calibration procedure for broadband sampling oscilloscopes.

We characterize CP violation in the SU(2) × U(1) model due to an extra vectorlike quark or sequential family, giving special emphasis to the chiral limit mu,d,s =0. In this limit, CP is conserved in the three generation Standard Model (SM), thus implying that all CP violation is due to the two new CP violating phases whose effects may manifest either at high energy in processes involving the new quark or as deviations from SM unitarity equalities among imaginary parts of invariant quartets (or, equivalently, areas of unitarity triangles). In our analysis we use an invariant formulation, independent of the choice of weak quark basis or the phase convention in the generalized Cabibbo-Kobayashi-Maskawa matrix. We identify the three weakbasis invariants, as well as the three imaginary parts of quartets B1−3 which, in the chiral limit, give the strength of CP violation beyond the SM. We find that for an extra vector-like quark |Bi | ≤10 −4, whereas for an extra sequential family |Bi | ≤10 −2. PACS: 11.30.Er, 12.15.Ff, 12.60.-i, 14.80.-j 1

In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i. e. they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the generalized hypergeometric functions, and therefore most of the special functions that can be found in mathematical dictionaries. The types of identities for which we present verification algorithms cover differential equations, power series representations, identities of the Rodrigues type, hypergeometric representations, and algorithms containing symbolic sums. The current implementations of special functions in existing Computer Algebra systems do not meet these high standards as we shall show in examples. They should be modified, and we show results of our implementations. 1