A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,I

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L ^*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + ^* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + _i L( _i , _i ), where is any ordinal, n, for every ii, _i are regular cardinals and _{i i}, and if n>0, then max({ _i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,II

A topological spaceX whose topology is the order topology of some linear ordering onX, is called aninterval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called aCO space and a space isscattered if every non-empty subspace has an isolated point. We regard linear orderings as topological spaces, by equipping them with their order topology. IfL andK are linear orderings, thenL ^*, L+K, L · K denote respectively the reverse ordering ofL, the ordered sum ofL andK and the lexicographic order onL x K (so · 2=+). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , l 0, letL(K,)=K+1+*.Theorem: Let X be a compact interval scattered space. Then X is a CO space if and only if X is homeomorphic to a space of the form +1+_{1 L(K i i), where is any ordinal, n , for every ii},_i are regular cardinals and K_ii, and if n>0, then max({K_i:i相似文献   

The order dimension of multinomial lattices

Using the notion of Ferrers dimension of incidence structures, the order dimension of multi-nomial lattices (i.e. lattices of multi-permutations) is determined. In particular, it is shown that the lattice of all permutations on ann-element set has dimensionn–1.     

Self complementary topologies and preorders

A topology on a set X is self complementary if there is a homeomorphic copy on the same set that is a complement in the lattice of topologies on X. The problem of characterizing finite self complementary topologies leads us to redefine the problem in terms of preorders (i.e. reflexive, transitive relations). A preorder P on a set X is self complementary if there is an isomorphic copy P of P on X that is arc disjoint to P (except for loops) and with the property that PP is strongly connected. We characterize here self complementary finite partial orders and self complementary finite equivalence relations.     

A compactness property of Dedekind σ-completef-rings

We prove that Dedekind -completef-rings are boundedly countably atomic compact in the language (+, –, ·,, , ). This means that whenever is a countable set of atomic formulae with parameters from some Dedekind -completef-ringA every finite subsystem of which admits a solution in some fixed productK of bounded closed intervals ofA, then admits a solution inK.Presented by M. Henriksen.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

On ηα-groups and fields

Let :=. The following are known: two -sets of power are isomorphic. Let >0. Two ordered divisible Abelian groups that are -sets of power are isomorphic, two real closed fields that are -sets of power are isomorphic. The following is shown: (1) there exist 2 nonisomorphic ordered Abelian groups (respectively ordered fields) that are -sets of power ; (2) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are -sets having the same order type.     

Binary hyperidentities of lattices

Summary An equational identity of a given type involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identity: x (y + z) = x y + x z has three variable symbols {x, y, z} and two operation symbols {+, }. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers Z; +, satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y z) = (x + y) (x + z) is not valid. By contrast, is hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 2, 2, .This research was supported by the NSERC operating grant # 8215     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

Partial Orders on Transformation Semigroups

In 1986, Kowol and Mitsch studied properties of the so-called natural partial order on T(X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T(X), ). In this paper, we extend that work to the semigroup P(X) of all partial transformations of X, compare with another natural partial order on P(X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P(X) with respect to each order.This author gratefully acknowledges the generous support of Centro de Matematica, Universidade do Minho, Portugal during his visit in May–June 2001.Received May 27, 2002; in revised form November 27, 2002 Published online May 16, 2003     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

Block elimination with one refinement solves bordered linear systems accurately

Linear systems with a fairly well-conditioned matrixM of the form , for which a black box solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the black box has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the black box is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.Dedicated to the memory of J. H. Wilkinson     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

Spanning trees with bounded degrees

Lets andk be positive integers. We prove that ifG is ak-connected graph containing no independent set withks+2 vertices thenG has a spanning tree with maximum degree at mosts+1. Moreover ifs3 and the independence number (G) is such that (G)1+k(s–1)+c for some0ck thenG has a spanning tree with no more thanc vertices of degrees+1.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.