The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic

By RCA₀, we denote a subsystem of second order arithmetic based on ⁰₁ comprehension and ⁰₁ induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilberts Nullstellensatz in RCA₀.Mathematics Subject Classification (2000): Primary 03F35; Secondary 03B30, 12D10     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

Chains of baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered setB ₁(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB ₁(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2^ω is embeddable intoB ₁(ℝ) iff ℒ does not contain a copy of ω₁ or ω _* ¹ . We present aZFC example of a linear order of size 2^ω showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness. Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620. The second author's research for this paper was partially supported by NSERC of Canada.     

BMO Solvability and the <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

The dimension of suborders of the Boolean lattice

We consider the order dimension of suborders of the Boolean latticeB _n . In particular we show that the suborder consisting of the middle two levels ofB _n dimension at most of 6 log₃ n. More generally, we show that the suborder consisting of levelss ands+k ofB _n has dimensionO(k ² logn).The research of the second author was supported by Office of Naval Research Grant N00014-90-J-1206.The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Edge-pancyclicity and Hamiltonian connectivity of twisted cubes

The twisted cube TQn is a variant of the hypercube Qn. It has been shown by Chang, Wang and Hsu [Topological properties of twisted cube. Information Science, 113, 147-167 （1999）] that TQn contains a cycle of every length from 4 to 2^n. In this paper, we improve this result by showing that every edge of TQn lies on a cycle of every length from 4 to 2^n inclusive. We also show that the twisted cube are Hamiltonian connected.     

Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations

We show that theL ^p norms, 0<p<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectivelyA _∞ with respect to the corresponding harmonic measure or normalized harmonic measure. Both authors are supported by NSF     

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

A Parallel Algorithm for the Estimation of the Global Error in Runge–Kutta Methods

The object of this work is the estimate of the global error in the numerical solution of the IVP for a system of ODE's. Given a Runge–Kutta formula of order q, which yields an approximation y _n to the true value y(x _n ), a general, parallel method is presented, that provides a second value y _n ^* of order q+2; the global error e _n =y _n –y(x _n ) is then estimated by the difference y _n –y _n ^*. The numerical tests reported, show the very good performance of the procedure proposed. A comparison with the code GEM90 is also appended.     

Random coin tossing

Summary. A sequence of heads and tails is produced by repeatedly selecting a coin from two possible coins, and tossing it. The second coin is tossed at renewal times in a renewal process, and the first coin is tossed at all other times. The first coin is fair (Prob(heads)=1/2), and the second coin is known either to be fair, or to have known biasθ∈(0,1] (Prob(heads) ). Letting u _k := Prob (There is a renewal at time k), we show that if ∑ _{k =0} ^∞ u _k ²=∞, we can determine, using only the sequence of heads and tails produced, if the second coin had bias θ or 0. If , we show that this is not possible. Received: 20 November 1996 / In revised form: 20 February 1997     

Extensions of homogeneous polynomials on c 0(l 2 i )

We show that a 2-homogeneous polynomial on the complex Banach space c ₀ l ₂ ⁱ ) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c ₀(l ₂ ⁱ ). The second author was supported by FAPESP, Brazil, Research Grant 01/04220-8.     

Higher Order Asymptotic Theory for Discriminant Analysis in Exponential Families of Distributions

This paper deals with the problem of classifying a multivariate observation X into one of two populations Π₁: p(x; w⁽¹⁾) S and Π₂: p(x; w⁽²⁾) S, where S is an exponential family of distributions and w⁽¹⁾ and w⁽²⁾ are unknown parameters. Let ; be a class of appropriate estimators ( ⁽¹⁾, ⁽²⁾) of (w⁽¹⁾, w⁽²⁾ based on training samples. Then we develop the higher order asymptotic theory for a class of classification statistics D = [ | = log{p(X; ⁽¹⁾)/p(X; ⁽²⁾)}, ( ⁽¹⁾, ⁽²⁾) ;]. The associated probabilities of misclassification of both kinds M( ) are evaluated up to second order of the reciprocal of the sample sizes. A classification statistic is said to be second order asymptotically best in D if it minimizes M( ) up to second order. A sufficient condition for to be second order asymptotically best in D is given. Our results are very general and give us a unified view in discriminant analysis. As special results, the Anderson W, the Cochran and Bliss classification statistic, and the quadratic classification statistic are shown to be second order asymptotically best in D in each suitable classification problem. Also, discriminant analysis in a curved exponential family is discussed.     

The limit of a Stanley–Wilf sequence is not always rational, and layered patterns beat monotone patterns

We show the first known example for a pattern q for which is not an integer, where S_n(q) denotes the number of permutations of length n avoiding the pattern q. We find the exact value of the limit and show that it is irrational, but algebraic. Then we generalize our results to an infinite sequence of patterns. We provide further generalizations that start explaining why certain patterns are easier to avoid than others. Finally, we show that if q is a layered pattern of length k, then L(q)(k-1)² holds.     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

Estimating the Length of a Diagnostic Test for Some Nonrepeating Functions

For a given nonrepeating function f(x ₁, ... , x _n) that essentially depends on all its variables we estimate the length of the diagnostic test on the set of all nonrepeating functions arbitrarily dependent on the variables x ₁, ... , x _n. Previously the author has shown that the corresponding Shannon function is of order (n ²). In this article nonrepeating functions f(x ₁, ... , x _n) are constructed for which the length of the minimal test increases superlinearly but not faster than n ²/2.     

Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap

For a second order differential operator A \mathcal{A} _ε  = −div g(x/ε)∇ + ε ⁻²p(x/ε) in L ₂(ℝ ^d ) with periodic coefficients and small parameter ε > 0 we study an approximation of the resolvent of A \mathcal{A} _ε at a point close to an edge of an inner gap in the spectrum of A \mathcal{A} _ε . Under certain regularity conditions, we construct an approximation (with a first order corrector taken into account) for the resolvent with error estimate of order O(ε ²). We show that a proper effective operator and a proper corrector are associated to each (regular) edge of the gap. Bibliography: 14 titles.     

Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.