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.     

The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

Spaces betweenX and its Freudenthal compactification

Let X indicate the Freudenthal compactification of a rimcompact, completely regular Hausdorff spaceX. In this paper the spacesY which satisfyXYX are characterized. From this a characterization of whenX lies between its locally compact partL(X) and (L(X)) follows. Such spaces necessarily possess a compactification X for whichCl _X (X–X) is 0-dimensional. Conditions, including those internal toX, are provided which are necessary and sufficient for this property to hold.This research was partially supported by a grant from Moorhead State University.     

Optimal accuracy approximation of functions and their derivatives

We consider the approximation of the function (x) and its derivative '(x) on [a, b] given that (x)C _2,N, i.e., belongs to the class of functions f(x) that satisfy the conditions f(x)L, f(x_i)=y_i, i=1,,N, where L and y_i are given real numbers and x_i are the nodes of an arbitrary grid, a=x₁<x₂<<X_N=b. A solution algorithm on the class of functions C_2,L,N is proposed which has optimal accuracy with a constant not exceeding 2. A bound on the approximation error of the function and its derivative is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 57–61, 1985     

Minimum Distance Bounds for s-Regular Codes

A code C F ⁿ is s-regular provided, forevery vertex x F ⁿ, if x is atdistance at most s from C then thenumber of codewords y C at distance ifrom x depends only on i and the distancefrom x to C. If denotesthe covering radius of C and C is -regular,then C is said to be completely regular. SupposeC is a code with minimum distance d,strength t as an orthogonal array, and dual degrees ^*. We prove that d 2t + 1 whenC is completely regular (with the exception of binaryrepetition codes). The same bound holds when C is(t + 1)-regular. For unrestricted codes, we show thatd s ^* + t unless C is a binary repetitioncode.     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

Local completeness I

An infinite algebra is locally complete if its local closure is the set of all finitary operations. We give a local completeness criterion in terms of a system R of finitary relations on A such that the polymorph Pol of each R is locally incomplete and for every locally incomplete algebra A; F we have F Pol for some R. This system consists of (i) certain natural relations whose polymorphs are best possible in the sense that they are co-atoms in the lattice of locally closed incomplete algebras, (ii) five types of binary relations, (iii) one type of ternary relations and (iv) at least ternary totally reflexive and symmetric relations that are not locally central.Presented by K. A. Baker.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

Finite element approximation of a nonlinear cross-diffusion population model

Summary. We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find u_i, the population of the i^th species, i=1 and 2, such that where ji and g_i(u₁,u₂):=(_i–_iiu_i–_iju_j)u_i. In the above, the given data is as follows: v is an environmental potential, c_i, a_i are diffusion coefficients, b_i are transport coefficients, _i are the intrinsic growth rates, and _ii are intra-specific, whereas _ij, ij, are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d3. Finally some numerical experiments in one space dimension are presented.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

Inequalities for two set systems with prescribed intersections

Inequalities are presented for systems {(A _i ,B _i ):1 i m} of pairs of finite sets satisfyingA _i B _i = andA _i B _j orA _j B _i fori j.     

The distortion of Hilbert space

The unit sphere of Hilbert space, ₂, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ₂, (C _i ) _i=1 , and reals _i 0 so that a) each setC _i has nonempty intersection with every infinite dimensional closed subspace of ₂ and b) forij,xC, andyC _j , |x, y|<_{min(i, j)} E. Odell was partially supported by NSF and TARP. Th. Schlumprecht was partially supported by NSF and LEQSF.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

On asymptotically efficient estimation for a semiparametric regression model

1.IntroductionConsiderthemodelY=X"0 g(T) E,(1'1)whereX"~(xl,',xo)areexplanatoryvariablesthatenterlinearly,Pisakx1vectorofunknownparameters,Tisanotherexplanatoryvariablesthatentersinanonlinearfashion,g')isanunknownsmoothfunctionofTinR',(X,T)andeareindependent,andeistheerrorwithmean0andvariancea2.Trangesoveranondegeneratecompact1-dimensionalilltervalC*;withoutlossofgenerality,C*=[0,1].Chenl2]discussedasymptoticnormalityofestimatorsP.of0byusingpiecewisepolynthacaltoapproximateg.Speckmanls…