Showing papers on "Prime (order theory) published in 1999"

Issues in Using Function Approximation for Reinforcement Learning

Abstract: Reinforcement learning techniques address the problem of learning to select actions in unknown, dynamic environments. It is widely acknowledged that to be of use in complex domains, reinforcement learning techniques must be combined with generalizing function approximation methods such as artificial neural networks. Little, however, is understood about the theoretical properties of such combinations, and many researchers have encountered failures in practice. In this paper we identify a prime source of such failures—namely, a systematic overestimation of utility values. Using Watkins’ Q-Learning [18] as an example, we give a theoretical account of the phenomenon, deriving conditions under which one may expected it to cause learning to fail. Employing some of the most popular function approximators, we present experimental results which support the theoretical findings.

Proving in zero-knowledge that a number is the product of two safe primes

Abstract: We present the first efficient statistical zero-knowledge protocols to prove statements such as: - A committed number is a prime. - A committed (or revealed) number is the product of two safe primes, i.e., primes p and q such that (p - 1)=2 and (q - 1)=2 are prime. - A given integer has large multiplicative order modulo a composite number that consists of two safe prime factors. The main building blocks of our protocols are statistical zero-knowledge proofs of knowledge that are of independent interest. We show how to prove the correct computation of a modular addition, a modular multiplication, and a modular exponentiation, where all values including the modulus are committed to but not publicly known. Apart from the validity of the equations, no other information about the modulus (e.g., a generator whose order equals the modulus) or any other operand is exposed. Our techniques can be generalized to prove that any multivariate modular polynomial equation is satisfied, where only commitments to the variables of the polynomial and to the modulus need to be known. This improves previous results, where the modulus is publicly known. We show how these building blocks allow to prove statements such as those listed earlier.

Linear Groups with Orders Having Certain Large Prime Divisors

Abstract: In this paper we obtain a classification of those subgroups of the finite general linear group GLd (q) with orders divisible by a primitive prime divisor of qe − 1 for some . In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups. 1991 Mathematics Subject Classification: primary 20G40; secondary 20C20, 20C33, 20C34, 20E99.

PRIME—toward process-integrated modeling environments: 1

Abstract: Research in process-centered environments (PCEs) has focused on project management support and has neglected method guidance for the engineers performing the (software) engineering process. It has been dominated by the search for suitable process-modeling languages and enactment mechanisms. The consequences of process orientation on the computer-based engineering environments, i.e., the interactive tools used during process performance, have been studied much less. In this article, we present the PRIME (Process Integrated Modeling Environments) framework which empowers method guidance through process-integrated tools. In contrast to the tools of PCEs, the process-integrated tools of PRIME adjust their behavior according to the current process situation and the method definitions. Process integration of PRIME tools is achieved through (1) the definition of tool models; (2) the integration of the tool models and the method definitions; (3) the interpretation of the integrated environment model by the tools, the process-aware control integration mechanism, and the enactment mechanism; and (4) the synchronization of the tools and the enactment mechanism based on a comprehensive interaction protocol. We sketch the implementation of PRIME as a reusable implementation framework which facilitates the realization of process-integrated tools as well as the process integration of external tools. We define a six-step procedure for building a PRIME-based process-integrated environment (PIE) and illustrate how PRIME facilitates change integration on an easy-to-adapt modeling level.

An algorithm for the k -error linear complexity of sequences over GF ( p m ) with period p n , p a prime

Abstract: An algorithm is given for the k-error linear complexity of sequences over GF(pm) with period pn, p a prime. The algorithm is derived by the generalized Games?Chan algorithm for the linear complexity of sequences over GF(pm) with period pn and by using the modified cost different from that used in the Stamp?Martin algorithm for sequences over GF(2) with period 2n. A method is also given for computing an error vector which gives the k-error linear complexity.

On the number of types of finite dimensional Hopf algebras

Abstract: For an odd prime p we construct an infinite class of non-isomorphic Hopf algebras of dimension p 4 over an infinite field containing primitive p-th roots of unity, answering in the negative a long standing conjecture of Kaplansky.

The Cohomology of SL(3, ℤ[1/2])

Abstract: We compute the cohomology of SL(3,Z[1/2]) with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod 2 congruence subgroups of SL(3,Z) with coefficients in Fp for p > 2. Finally we compute the cohomology of GL(3,Z[1/2]).

Perfect forms and the Vandiver conjecture

Abstract: Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi reduction theory of quadratic forms.

Normal bases via general Gauss periods

Abstract: Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive $r$th root of unity, one obtains under certain conditions a normal basis for $mathbb F_q^n$ over $ F_q$, where $r$ is a prime and $nk=r-1$ for some integer $k$. We generalize this construction by allowing arbitrary integers $r$ with $nk=\varphi(r)$, and find in many cases smaller values of $k$ than is possible with the previously known approach

Independent locally-finite intersections of localizations

Abstract: Let D be an integral domain and letF be a set of prime ideals of D. We say that D is an F-IFC domain if D = P2F DP , where the intersection has nite character and for P;Q2F there is no nonzero prime ideal contained in P\Q. Examples ofF-IFC domains include h-local do- mains (F = Max(D)), Noetherian domains in which grade-one primes have height one, and independent rings of Krull type. Using star operations we give several characterizations ofF-IFC domains.

On Certain Exponential Sums and the Distribution of Diffie–Hellman Triples

Abstract: Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let e>0 be fixed. Then uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.

Prime spectra of quantized coordinate rings

Abstract: This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them The first part of the paper is expository We survey a number of classes of quantized coordinate rings, as well as some related algebras that share common properties, and we record some of the basic properties known to occur for many of these algebras, culminating in stratifications of the prime spectra by the actions of tori of automorphisms As our main interest is in the generic case, we assume various parameters are not roots of unity whenever convenient In the second part of the paper, which is based on joint work with E S Letzter in [The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras (to appear in Trans Amer Math Soc)], we offer some support for the conjecture above, in the form of an axiomatic basis for the observed stratifications and their properties At present, the existence of a suitable supply of normal elements is taken as one of the axioms; the search for better axioms that yield such normal elements is left as an open problem

The Powers Behind the Prime Minister: The Hidden Influence of Number Ten

Abstract: How many people know what lies on the other side of the door to 10 Downing Street, or indeed, who built the building? Providing a portrait of the centre of a British Prime Minister's power, this book contains revelatory material on the private behaviour of many of the 20th century's political leaders, as well as an anlysis of how successive Prime Ministers have wielded their power within these 18th-century walls.

Two-generator two-groups of class two and their nonabelian tensor squares

Abstract: The nonabelian tensor square G[otimes ] G of a group G is generated by the symbols g[otimes ] h, g,h ∈ G, subject to the relations gg\prime\otimesh=(^gg\prime\otimes^gh)(g\otimesh) and g\otimeshh\prime-(g\otimesh)(^hg\otimes^hh\prime),$g,g\prime,h,h\prime \in G , where . The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J.-L. Loday in [4] and [5], extending ideas of J.H.C. Whitehead in [10]. The topic of this paper is the classification of 2-generator 2-groups of class two up to isomorphism and the determination of nonabelian tensor squares for these groups.

Speeding Fermat's factoring method

Abstract: A factoring method is presented which, heuristically, splits composite n in O(n 1/4+ ∈) steps. There are two ideas: an integer approximation to √(q/p) provides an O(n 1/2+ ∈) algorithm in which n is represented as the difference of two rational squares; observing that if a prime m divides a square, then m 2 divides that square, a heuristic speed-up to O(n 1/4+ ∈) steps is achieved. The method is well-suited for use with small computers: the storage required is negligible, and one never needs to work with numbers larger than n itself.

Enumerating p-Groups

Abstract: We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.

On a diophantine inequality involving prime numbers (III)

Abstract: Let 1 0 such that for each real numberN>N(c) the inequality $$|p_1^c + p_2^c + p_3^c - N|< N^{ - \tfrac{1}{c}(\tfrac{{11}}{{10}} - c)} \log ^{c_1 } N$$ is solvable in prime numbersp 1,p 2,p 3, wherec 1 is some absolute positive constant.

PRIME—Toward Process-Integrated Modeling Environments

Abstract: Research in process-centered environments (PCEs) has focused on project management support and has neglected method guidance for the engineers performing the (software) engineering process. It has been dominated by the search for suitable process-modeling languages and enactment mechanisms. The consequences of process orientation on the computer-based engineering environments, i.e., the interactive tools used during process performance, have been studied much less. In this article, we present the PRIME (ProcessIntegrated Modeling Environments) framework which empowers method guidance through process-integrated tools. In contrast to the tools of PCEs, the process-integrated tools of PRIME adjust their behavior according to the current process situation and the method definitions. Process integration of PRIME tools is achieved through (1) the definition of tool models; (2) the integration of the tool models and the method definitions; (3) the interpretation of the integrated environment model by the tools, the process-aware control integration mechanism, and the enactment mechanism; and (4) the synchronization of the tools and the enactment mechanism based on a comprehensive interaction protocol. We sketch the implementation of PRIME as a reusable implementation framework which facilitates the realization of process-integrated tools as well as the process integration of external tools. We define a six-step procedure for building a PRIME-based process-integrated environment (PIE) and illustrate how PRIME facilitates change integration on an easy-to-adapt modeling level. The PRIME framework is a result of the Project Ja 445/5 and PO 607/1 “Prozesintegration von Modellierungsarbeitsplatzen” funded by the Deutsche Forschungsgemeinschaft. The PRIME framework has been used to implement two prototypical process-integrated environments within the ESPRIT Reactive Long Term Research 21.903 Project CREWS (funded by the European Community) and the Collaborative Research Center (SFB) 476 IMPROVE (funded by the Deutsche Forschungsgemeinschaft).