An intersection property for starshaped sets in the plane

Let be a family of compact starshaped sets in the plane. If every three and every two members of have a union which is connected and simply connected, then {F:F in } is simply connected and nonempty. Of course, if every three and every two members of have a starshaped union, the same result holds.Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma.     

Set systems with no union of cardinality 0 modulom

Letq be a prime power. It is shown that for any hypergraph = {F ₁,...,F _d(q–1)+1} whose maximal degree isd, there exists Ø ₀ , such that 0 (modq).     

Smoothing the function of the multiple-shooting equation

The multiple-shooting method for the solution of boundary-value problems is a modified Newton method for the solution of an equation(x) = 0, where is a special function which is differentiable in general, but may occasionally have discontinuities at some points which have to be passed during the iteration process. This is the case especially in optimal control problems and it is a severe handicap for the convergence of the Newton method which can be essentially reduced when replacing by a series of smooth functions _i dependent on the iteration process.     

Maximal Partial Spreads and Flocks

Denote by a flock of a quadratic cone of PG(3,q) by S() the spread of PG(3,q) associated with and by l the common line of the base reguli. Suppose that there are two lines not transversal to a base regulus which share the same lines of S() Then we prove that is either linear or a Kantor-Knuth semifield flock. Using this property we can extend the result of J3 on derivable flocks proving that, if a set of q + 1 lines of S() defines a derivable net different from a base regulus-net, then is either linear or a Kantor-Knuth semifield flock. Moreover if l is not a component of the derivable net, then is linear.     

Systèmes de sortie ℱ Dt prévisibles

Summary Given a random closed setM, adapted to a filtration ( _t ), we construct a local time ofM which is both ( _t ) adapted and ( _Dt ) predictable, whereD _t =inf{s>t: sM}. Similarly an exit system, both ( _t ) optional and ( _Dt ) predictable, is associated withM and with the process representing the future at each timet. The paper is motivated by the markovian case, where the general results are applied to Ray processes.     

Generation of Uniformly Closed Algebras of Functions

For a linear sublattice of C(X), the set of all real continuous functions on the completely regular space X, we denote by A() the smallest uniformly closed and inverse-closed subalgebra of C(X) that contains . In this paper we study different methods to generate A() from . For that, we introduce some families of functions which are defined in terms of suprema or sums of certain countably many functions in . And we prove that A() is the uniform closure of each of these families. We obtain, in particular, a generalization of a known result about the generation of A() when is a uniformly closed linear sublattice of bounded functions.     

Some best possible bounds concerning the traces of finite sets II

LetX be ann-element set and be a family of its subsets. Consider the family _x = {F – {x} : F } for a givenx X. We write(m, n) (m – k, n – 1), when for all with || m, there exists an elementx ofX such that| _x| m – k. We show that (m, n) (m – 10,n – 1) for allm 5n and (m, n) (m – 13,n – 1) for allm 29n/5.     

Flowchart machines

Let denote a conventional flowchart. Any algorithm can be represented by a flowchart. If action nodes in call then is a recursive flowchart. We show how to decompose arbitrary non-self-modifying programs into structure and atomic parts. We specifically give the synthesis procedure for a controller . can serve as the only sequencer in an execution of . If is recursive then is a pushdown machine, otherwise is a finite state machine. The next-state functionf and the output functiong of represent respectively all of the structure-, i.e. the programmer-oriented-, and all of the atomic-, i.e. the data-oriented-, parts of .f defines the flow or pattern of computations andg the actual transformations or operations on data. Thus we construct and analyze programs by constructing and analyzing their sequencers .     

On the uniqueness of a local martingale with a given absolute value

Summary D. Gilat has shown that any non-negative submartingale (X, .) is equal in law to the absolute value of a martingale (M, .). This result may be strenthened so that the pairs (X,.) and (¦M¦,.) are synonomous. In this paper the question of uniqueness of M is considered. Conditions on a local martingale (M, .) are found that lead to an explicit formula for the finite-dimensional distributions of M in terms of the Doob-Meyer decomposition of the local martingale X. In many cases of interest the conditions on M are unnecessary. For example, if X is the pth power of an Itô integral it is shown that (M) is unique if p> 1 but not in general if p=1.     

Shortest paths on convex hypersurfaces of Riemannian spaces

A convex hypersurface in a Riemannian space M^m is part of the boundary of an m-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in are shortest paths in M^m. For surfaces in R^m these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces M^m. Hence approximations to by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of R³) by Yu. F. Borisov are used.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 114–132, 1976.     

Superficie cubiche di\mathbb{A}_\mathbb{C}^3 a curve sottoinsieme intersezione completa

Summary In this paper we classify the algebraic cubic surfaces of the affine space is the complex field, whose algebraic curves are set-theoretic complete intersections of ; in other words surfaces such that every prime ideal of height 1in the coordinate ring [] of is the radical of a principal ideal; if is non singular in codimension 1this means that [] is semifactorial. We give the equations of such surfaces within linear isomorphisms of providing also methods by which one can construct the equations of the surfaces cutting on its curves as set-theoretic complete intersections. Moreover for each of these surfaces we determine the minimum positive number such that every algebraic curve of with multiplicity of intersection , is complete intersection of itself with another surface § 8where the results are summarized). We tackle also the problem of such a classification over algebraically closed fields k different from .

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R.     

Existence of compatible families of proper regular conditional probabilities

Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}_IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

On a construction of Thomassen

Using a construction of Thomassen [Discrete Math. 9, 91–96 (1974)] we prove that for infinitely manyn there is a family _n of triangle-free maximally non-hamiltonian graphs of ordern with | _n | exponentially inn. In particular, for everym 48 we construct such a graph; an infinite number of these provide new almost extremal examples in the sense of minimal size.     

A topological obstruction to the geodesibility of a foliation of odd dimension

Let M be a compact Riemannian manifold of dimension n, and let be a smooth foliation on M. A topological obstruction is obtained, similar to results of R. Bott and J. Pasternack, to the existence of a metric on M for which is totally geodesic. In this case, necessarily that portion of the Pontryagin algebra of the subbundle must vanish in degree n if is odd-dimensional. Using the same methods simple proofs of the theorems of Bott and Pasternack are given.     

A bijective proof of Kadell's conjecture on plane partitions

We give a weight-preserving bijection from _{r, µ} ^m to, where _{r, µ} ^m is the set of all plane partitions whose entries are m and whose entries below ther-th row form a column strict plane partition of type, and _µ ^m the set of all column strict plane partitions of type whose entries are m, and the set of all plane partitions with at mostr rows, whose entries are m. This confirms a conjecture of Kadell.     

Self-bounded controlled invariant subspaces: A straightforward approach to constrained controllability

A lattice-type structure is shown to exist in a particular subset of the set of all (A, )-controlled invariants contained in and containing , whereA denotes a linear map inR ⁿ ; , are arbitrary subspaces ofR ⁿ ; andD is an arbitrary subspace ofJ, the maximum (A, )-controlled invariant contained in . In linear system theory, this property can be used for a more direct theoretical and algorithmic approach to constrained controllability and disturbance rejection problems.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Happy families and completely Ramsey sets

Summary We use games of Kastanas to obtain a new characterization of the classC of all sets that are completely Ramsey with respect to a given happy family . We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC . We also study some cardinals that can be associated with the ideal of nowhere -Ramsey sets.Part of this research was done while the author was visiting I.V.I.C. in Caracas in September 1989. The author would like to thank Carlos Di Prisco for his hospitality.     

Ramsey families which exclude a graph

For graphsA andB the relationA(B) _r ¹ means that for everyr-coloring of the vertices ofA there is a monochromatic copy ofB inA. Forb (G) is the family of graphs which do not embedG. A familyof graphs is Ramsey if for all graphsBthere is a graphAsuch thatA(B) _r ¹ . The only graphsG for which it is not known whether Forb (G) is Ramsey are graphs which have a cutpoint adjacent to every other vertex except one. In this paper we prove for a large subclass of those graphsG, that Forb (G) does not have the Ramsey property.This research has been supported in part by NSERC grant 69-1325.