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.     

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).     

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.     

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.     

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.     

Graph Minors. XX. Wagner''s conjecture

In a previous paper (J. Combin. Theory 48 (1990) 255) we showed that for any infinite set of (finite) graphs drawn in a fixed surface, one of the graphs is isomorphic to a minor of another. In this paper we extend that result in two ways:

• we generalize from graphs to hypergraphs drawn in a fixed surface, in which each edge has two or three ends, and

• the edges of our hypergraphs are labelled from a well-quasi-order, and the minor relation is required to respect this order.

This result is another step in the proof of Wagner's conjecture, that for any infinite set of graphs, one is isomorphic to a minor of another.     

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).     

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.     

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     

论高斯

高斯(C.F.Gauss,1777—1855)生于德国Brunswick,1795—98求学于Gttinsen大学。1807—55任天文观测台台长及Gttingen大学教授。 大家都知道高斯的名言:“数学是科学的女王,算术是数学的女王”这里“算术”是在古希腊人的意义下理解的,指的初等数论,而区别于近代的解析数论。     

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     

The mathematics of F. J. Almgren, Jr.

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a more-or-less chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.     

Gradient-constrained minimum networks. I. Fundamentals

In three-dimensional space an embedded network is called gradient-constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. A gradient-constrained minimum Steiner tree T is a minimum gradient-constrained network interconnecting a given set of points. In this paper we investigate some of the fundamental properties of these minimum networks. We first introduce a new metric, the gradient metric, which incorporates a new definition of distance for edges with gradient greater than m. We then discuss the variational argument in the gradient metric, and use it to prove that the degree of Steiner points in T is either three or four. If the edges in T are labelled to indicate whether the gradients between their endpoints are greater than, less than, or equal to m, then we show that, up to symmetry, there are only five possible labellings for degree 3 Steiner points in T. Moreover, we prove that all four edges incident with a degree 4 Steiner point in T must have gradient m if m is less than 0.38. Finally, we use the variational argument to locate the Steiner points in T in terms of the positions of the neighbouring vertices.     

On a problem of S.L. Sobolev

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.     

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.     

I.I.D. STATISTICAL CONTRACTION OPERATORS AND STATISTICALLY SELF－SIMILAR SETS

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.     

On a theorem of M. Deuring and I. R. Šafarevič

Let F/K be a field of algebraic functions of one variable, K algebraically closed of characteristic p≠0. Let E/K be a cyclic extension of F/K of degree ?, a prime not necessarily different from p. Let ξ, ρ denote the ?-ranks of the null class groups of E and F respectively. If E/F is ramified, Deuring proved that ξ=?ρ+(t?δ)(??1) where t is the number of ramified primes and δ is 1 or 2 according as ? equals p or not. ?afarevi? proved the same relation in the unramified case for ?=p where ξ, ρ denote the ranks of the Hasse-Witt matrices. Subrao gave a unified proof for ?=p. Rosen and Sullivan have proved the theorem in special cases. In this paper, using galois cohomology, Deuring's method of proof is modified so as to give a proof of the theorem in all cases.     

Editorial: Math. Log. Quart. 1/2007

Dear Readers, A happy and successful 2007 to all of you. I wish to draw your attention again to an enlargement of the Editorial Board of our journal. We are pleased to welcome three new members: Ramon Jansana (Barcelona), Klaus Meer (Odense), and Andrzej Roslanowski (Omaha). I would like to thank the outgoing member, Peter Koepke (Bonn), for the work he has done in supporting MLQ. Also, it is more than justified that the name of our Editorial Assistant, who always faithfully compiles the final layout, can now be seen at the cover. All readers are once more encouraged to contact the Editorial Office or a member of the Editorial Board with any suggestions concerning the contents or the editorial work of our journal. In particular, proposals for issues devoted to special subjects or conferences are highly welcome. In 2007 there will be two such issues: one will be dedicated to “Trends in Constructive Mathematics” and a further one will deal with “Computability and Complexity in Analysis”. It should finally be mentioned that our publishing house is celebrating its 200th anniversary in 2007. During its whole history, Wiley has grown and evolved and has always been successful in serving the needs of its customers. So we can be sure to have a good background for the further development of our journal. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)