A complete set of addition laws for incomplete Edwards curves

Edwards curves were the first curves shown to have a complete addition law. However, the completeness of the addition law depends on the curve parameters and even a complete Edwards curve becomes incomplete over a quadratic field extension. This paper covers arbitrary Edwards curves and gives a set of two addition laws that for any pair of input points P₁, P₂ produce the sum P₁+P₂.     

The universe constructed from a sequence of ordinals

We prove that if where is an -sequence of ordinals then the GCH holds. Received January 1, 1993     

Transaction independence: The road to cooperative systems

The cooperation and consistency among transactions in cooperative systems suggest that traditional transaction processing paradigms need to be modified for target applications. We propose a new framework that borrows its concepts from the evolution of database management systems. Transaction Independence and Incorporation of Workflow ideas within applications are the foundations of this framework. Cooperation is gained by TCMS, an external, domain-independent framework. This paper outlines the requirements, approach, architecture, and the implementation of TCMS.     

From the neutrino to the edge of the universe

Two recent findings necessitate a closer look at the existing standard models of particle physics and cosmology. These are the discovery of neutrino oscillation, and hence a non-zero mass on the one hand and, on the other, observations of distant supernovae which indicate that contrary to popular belief, the universe would continue to expand for ever, possibly accelerating in the process. In this paper it is pointed out that relatively recent studies which indicate a stochastic, quantum vacuum underpinning and a fractal structure for space time, reconcile both of the recent observations, harmoniously.     

Fractal cosmogony: similarity of the early universe to microscopic fractal aggregates

Images of fractal aggregates condensed at the micrometer scale exhibit structures strikingly similar to the early universe, the latter as pictured by a model due to Barkana and Loeb, interpreting the data of the Sloan Digital Sky Survey and an earlier one by Hernquist, Weinberg and Katz. The condensation of matter at very small scale is thus similar to that of the early universe as far as it can be reasonably established.     

The road to equal-norm Parseval frames

The construction of equal-norm Parseval frames is fundamental for many applications of frame theory. We present a construction method based on a system of ordinary differential equations, which generates a flow on the set of Parseval frames that converges to equal-norm Parseval frames. We developed this method to address a question posed by Vern Paulsen: How close is a nearly equal-norm, nearly Parseval frame to an equal-norm Parseval frame? The distance estimate derived here can be used to substantiate numerically found, approximate constructions of equal-norm Parseval frames. The estimate is valid for a fairly general class of frames — requiring that the dimension of the Hilbert space and the number of frame vectors is relatively prime. In addition, we re-phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in Hilbert-Schmidt norm to ones which have a constant diagonal.     

The generality of the zero-one laws

We prove game-theoretic generalizations of some well-known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.     

The nozzle guide vane problem: Partitioning a heterogeneous inventory

In this paper we describe a problem which is encountered in the maintenance of gas turbine engines used in commercial and military aircraft. In particular, we address the problem of selecting a set of nozzle guide vanes, from a heterogeneous inventory of vanes, for the nozzle assembly of an engine. We formulate this problem as a partitioning problem for a heterogeneous population. Given the combinatorial complexity of the problem we develop a heuristic algorithm which is shown to be very effective in obtaining the maximum number of partitions.     

The complete closure of a graph

We define the complete closure number cc(G) of a graph G of order n as the greatest integer k ≤ 2n ? 3 such that the kth Bondy-Chvátal closure Cl_k(G) is complete, and give some necessary or sufficient conditions for a graph to have cc(G) = k. Similarly, the complete stability cs(P) of a property P defined on all the graphs of order n is the smallest integer k such that if Cl_k(G) is complete then G satisfies P. For some properties P, we compare cs(P) with the classical stability s(P) of P and show that cs(P) may be far smaller than s(P). © 1993 John Wiley & Sons, Inc.     

The complete solution to the Sylvester-polynomial-conjugate matrix equations

In this paper we propose two new operators for complex polynomial matrices. One is the conjugate product and the other is the Sylvester-conjugate sum. Then some important properties for these operators are proved. Based on these derived results, we propose a unified approach to solving a general class of Sylvester-polynomial-conjugate matrix equations, which include the Yakubovich-conjugate matrix equation as a special case. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained in terms of the Sylvester-conjugate sum, and such a proposed solution can provide all the degrees of freedom with an arbitrarily chosen parameter matrix.     

The higher reciprocity laws: An example

In Section 128 of Smith's “Report on the Theory of Numbers” (Chelsea, New York, 1965) one finds a certain theta function whose coefficients are multiplicative arithmetic functions. It is shown in this paper that this is an elementary example of the well-known connection between the higher reciprocity laws of number fields, Artin L-functions, and modular forms.     

Explicit generalized solutions to a system of conservation laws

This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.