Relative entropy tuples,relative u.p.e. and c.p.e. extensions

Relative entropy tuples both in topological and measure-theoretical settings, relative uniformly positive entropy (rel.-u.p.e.) and relative completely positive entropy (rel.-c.p.e.) are studied. It is shown that a relative topological Pinsker factor can be deduced by the smallest closed invariant equivalence relation containing the set of relative entropy pairs. A relative disjointness theorem involving relative topological entropy is proved. Moreover, it is shown that the product of finite rel.-c.p.e. extensions is also rel.-c.p.e.. The first author is partially supported by NCET, NNSF of China (no. 10401031) and CNRS-K.C.Wong Fellowship. The second author is supported by the national key project for basic science (973). The third author is supported by NNSF of China (no. 10401031).     

Evaluations of Parameters for the Optimization of S.S.O.R. and A.D.I. Preconditioning

The main goal of this paper is to give two ways to estimate the needed parameters in order to obtain the condition number of S.S.O.R. preconditioned matrices, namely, the algebraic matricial formulation of convexity Riesz theorem and the tridiagonal Fourier analysis. The improvement with respect to Axelsson's approach is explicitly given. Estimations of the condition number in the case of A.D.I. preconditioning is also considered.     

Isolated maximal d.r.e. degrees

There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called $(m,n)$-cupping degree and show that there exists an isolated $(2,\omega )$-cupping degree, and there exists a proper $(2,1)$-cupping degree. It helps understanding various degree structures in the Ershov Hierarchy.     

Two-point b.v.p. for multivalued equations with weakly regular r.h.s.

A two-point boundary value problem associated to a semilinear multivalued evolution equation is investigated, in reflexive and separable Banach spaces. To this aim, an original method is proposed based on the use of weak topologies and on a suitable continuation principle in Fréchet spaces. Lyapunov-like functions are introduced, for proving the required transversality condition. The linear part can also depend on the state variable x and the discussion comprises the cases of a nonlinearity with sublinear growth in x or of a noncompact valued one. Some applications are given, to the study of periodic and Floquet boundary value problems of partial integro-differential equations and inclusions appearing in dispersal population models. Comparisons are included, with recent related achievements.     

On the r.e. predecessors of d.r.e. degrees

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets) and R[d] be the class of less than d r.e. degrees in whichd is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e. degree defined by Lachlan for a d.r.e.set d is just the minimum degree in which D is r.e. Then we study for a given d.r.e. degree d class R[d] and show that there exists a d.r.e.d such that R d] has a minimum element 0. The most striking result of the paper is the existence of d.r.e. degrees for which R[d] consists of one element. Finally we prove that for some d.r.e. d R[d] can be the interval [a,b] for some r.e. degrees a,b, a b d. Received: 17 January 1996     

Optimization of the Relaxation Parameter for S.S.O.R. and A.D.I. Preconditioning

We present and compare several approaches for the optimization of the relaxation parameter both for A.D.I. and S.S.O.R. basic iteration and preconditioning conjugate gradient method. For each kind of preconditioning a detailed link between estimates of the spectral radius of the iteration matrix and of the condition number resulting from preconditioning is proposed. It allows to choose the best approach in order to obtain the optimal relaxation parameter and the corresponding optimal estimates either of the spectral radius of the iteration matrix and of the resulting condition mumber of the S.S.O.R. and A.D.I. preconditioning.     

S.A.G.B.I. bases for rings of formal modular seminvariants

We use the theory of S.A.G.B.I. bases to construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–2. Received: January 22, 1997     

The E.E.C. and the development of O.R.

The growth of the E.E.C. has brought about a change in the scale of many organizational problems and a change in the ways that they can be tackled. Accordingly it is plausible that the techniques of O.R. should develop in response to these changes. This paper examines some desirable developments and suggests ways in which they might be achieved by adapting methods devised in other fields for similar problems.     

Mathematics in the time of the pharaohs: By Richard J. Gillings. Cambridge,Mass. (M.I.T. Press), 1972. 297 p. US $25

The author's treatment of many problems and some of his hypotheses are original and elegant. He does not limit himself to describing the content of his sources but tries to reveal the paths followed by the Egyptians as well as their ideas and methods. This leads him to interesting and on the whole covinncing reconstructions and results. For example, the author concludes that there must have existed extensive addition tables that have not survived.Although the materials available today seem insufficient to explain how the Egyptians happened to use the qlumsy binary method mu multiplication and division, Gillings' discussion throws light on the problem and poses questions for further research. Careful analysis of special tables and solutions of problems and the comparison of sources enable the author in large measure to reveal the methods and ideas of Egyptian mathematicians. How the Egyptians constructed their tables still remains an unsolved problem. Because divisions sometimes began by dividing by 3, Gillings concludes that tables for 3 must have existed and his reconstruction of such a table has great interstt. His Rule G for sums and differences of fractions is original and ingenious. He strsses the importance of EMLR. He argues that the Egyptians used the concepts of arithmetic and harmonic means. He gives an interesting explanation for the origin of the Egyptian formulas for squaring the circle, though its simplicity is clear only after the result is known.Although answers are not given to all questions, the book leads the reader to search for answers, to formulate new reconstructions, and to make new hypotheses. This is its greatest merit.     

广义K.d.V.方程的数值方法

本文研究一般的广义K.d.V.方程的数值方法,给出了广义K.d.V.方程的一类半离散差分格式,证明了它们的守恒性。作者还严格证明了这类格式的广义稳定性,并由此推出收敛性。文章的最后考虑了全离散情形和两步格式。     

D.c. sets,d.c. functions and nonlinear equations

Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ⁿ into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.The revision of this paper was produced during the author's stay supported by a Sophia lecturing-research grant at Sophia University (Tokyo, Japan).     

Monotone factors of I.I.D. processes

LetB(p) andB(q) be Bernoulli shifts on {0, 1,...,d - 1}^ℤ. Ifh(p)>h(q), it is a classical theorem of Sinai that there is a factor map takingB(p) toB(q). If, in addition,p stochastically dominatesq, we can ask whether there is such a factor map ϕ which is monotone: ϕ(x) _i≤x_i for each coordinatei of almost every pointx. Here we show that there is a monotone finitary code fromB(p) toB(q) in the case whereB(q) is a shift on two symbols.     

Corporate Planning and Its Use of O.R. in the N.C.B.: A Personal View

Moving from Operational Research to Corporate Planning within one organisation is a rather modest change, which nevertheless leads to a different perspective. While still an advisory function, Corporate Planning, to be successful, has to maintain itself at the centre of a `messy' system tackling `messy' problems. Holding on to a clear strategy in such circumstances is not only difficult but vital. The success of the strategy depends on its robustness and ability to adapt to new circumstances. Recent debates in the O.R. literature have focused on the importance of robustness and the nature of uncertainty using the paradigms of rational and bureaucratic processes. This is a useful perspective from which to view our activities and the relative roles of Operational Research and Corporate Planning, either separately or together. The term `mature' O.R. is used to differentiate research that has been built up over many years in the N.C.B. from the problem solving activity often used to characterise O.R. The challenge is to prevent maturity developing into old age.     

Biased graphs. VI. synthetic geometry

A biased graph is a graph with a class of selected circles (“cycles”, “circuits”), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $\Omega$ has two natural matroids, the frame matroid $G\left(\Omega \right)$ and the lift matroid $L\left(\Omega \right)$, and their extensions the full frame matroid $G^{•}\left(\Omega \right)$ and the extended (or complete) lift matroid $L_0\left(\Omega \right)$. In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry.     

There is No Low Maximal D.C.E. Degree

We show that for any computably enumerable (c.e.) set A and any set L, if L is low and , then there is a c.e. splitting such that . In Particular, if L is low and n‐c.e., then is n‐c.e. and hence there is no low maximal n‐c.e. degree.     

The periodogram of an i.i.d. sequence

Periodogram ordinates of a Gaussian white-noise computed at Fourier frequencies are well known to form an i.i.d. sequence. This is no longer true in the non-Gaussian case. In this paper, we develop a full theory for weighted sums of non-linear functionals of the periodogram of an i.i.d. sequence. We prove that these sums are asymptotically Gaussian under conditions very close to those which are sufficient in the Gaussian case, and that the asymptotic variance differs from the Gaussian case by a term proportional to the fourth cumulant of the white noise. An important consequence is a functional central limit theorem for the spectral empirical measure. The technique used to obtain these results is based on the theory of Edgeworth expansions for triangular arrays.     

Choosing the Operational Research Programme for B.I.S.R.A

This paper examines some important problems in the evaluation and selection of research projects, with particular reference to operational research projects in a central industry research organization. A method for project evaluation is suggested which provides data suitable for use in a powerful and flexible formal procedure for project selection. In evaluating a project, the assessment of resource requirements is based on a probabilistic network plan of the project, and the estimates of benefits are derived from the marginal effects on the industry of performing or not performing the project, using discounted cash flow techniques. The procedure for project selection chooses not only the projects to be undertaken, but also the team sizes that should be used. A wide variety of constraints can be imposed on the system to reflect different aspects of management policy or other special factors. The procedures described have been implemented, and have proved well worth while.     

Estimating means from a non-I.I.D. Mixture of Poisson samples

In this paper, a family of estimators for estimating means when mixing two independent Poisson samples is proposed. This family is based on the probability-generating function of the Poisson distribution and is offered as an alternative to the maximum likelihood estimators, which have some drawbacks. These estimators include the method of moments estimators as a special limiting case.     

The kinship of E. H. Moore and R. L. Moore

Evidence is presented that E. H. Moore and R. L. Moore were seventh cousins.