More Efficient Algorithm for Ordered Tree Inclusion

Given two ordered trees and , the tree inclusion problem is to determine whether it is possible to obtain from by deleting nodes. Recently, this problem has been recognized as an important primitive in query processing for structured text databases. In this paper we present anO(|leaves()| ||) time andO(|leaves()|min(depth(), |leaves()|)) space algorithm for ordered tree inclusion, by means of a sophisticated bottom-up-matching strategy. Our algorithm improves the previous best one (Kilpeläinen, 1992, Ph.D. thesis, Dept. Computer Science, Univ. Helsinki) that requiresO(|| ||) time andO(||min(depth(), |leaves()|)) space.     

Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext¹_A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H². One condition involves an extension problem for the Hankel operator with symbolB,Γ_B, but viewed as a module map from H²into (W; B). The group Ext¹_A()(, (W; B))=(0) precisely whenΓ_Bextends to a module map from L²into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓ_B and the operator corona problem.     

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space . Specifically, we prove that the Brownian motion belongs to for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces , and indicate the endpoint large deviation estimates.     

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism which splits , where is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over .     

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Second order, multi-point problems with variable coefficients

In this paper, we consider the eigenvalue problem consisting of the equation where and λ∈R, together with the multi-point boundary conditions where m^±?1 are integers, and, for i=1,…,m^±, , , with , . We show that if the coefficients are sufficiently small (depending on r), then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients are not sufficiently small, then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case (r≡1), but the variable coefficient case has not been considered previously (apart from the existence of ‘principal’ eigenvalues).Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for the existence of general solutions and also of nodal solutions—these results rely on the spectral properties of the linear problem.     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

Let be the open non-cuspidal locus of the modular curve associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on for suitably chosen n. In this note, we give a genus formula for the modular curves and we give three new solutions to the class number one problem using the modular curves for n=16,20,21. These are the only such modular curves of genus ?2 that had not yet been exploited.     

A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → [0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

The Schur sufficiency condition for boundedness of any integral operator with non-negative kernel betweenL₂-spaces is deduced from an observation, Proposition 1.2, about the central role played byL₂-spaces in the general theory of these operators. Suppose (Ω, , μ) is a measure space and thatK: Ω×Ω→[0, ∞) is an ×-measurable kernel. The special case of Proposition 1.2 for symmetrical kernels says that such a linear integral operator is bounded onanyreasonable normed linear spaceXof -measurable functions only if it is bounded onL₂(Ω, , μ) where its norm is no larger. The general form of Schur's condition (Halmos and Sunder “Bounded Integral Operators onL₂-Spaces,” Springer-Verlag, Berlin/New York, 1978) is a simple corollary which, in the symmetrical case, says that the existence of an -measurable (not necessarily square-integrable) functionh>0μ-almost-everywhere onΩwithimplies thatKis a bounded (self-adjoint) operator onL₂(Ω, , μ) of norm at mostΛ. When (Ω, , μ) isσ-finite, we show that Schur's condition is sharp: in the symmetrical case the boundedness of onL₂(Ω, , μ) implies, for anyΛ>‖‖₂, the existence of a functionh∈L₂(Ω, , μ) which is positiveμ-almost-everywhere and satisfies (*). Such functionshsatisfying (*), whether inL₂(Ω, , μ) or not, will be calledSchur test functions. They can be found explicitly in significant examples to yield best-possible estimates of the norms for classes of integral operators with non-negative kernels. In the general theory the operators are not required to be symmetrical (a theorem of Chisholm and Everitt (Proc. Roy. Soc. Edinburgh Sect. A69(14) (1970/1971), 199–204) on non-self-adjoint operators is derived in this way). They may even act between differentL₂-spaces. Section 2 is a rather substantial study of how this method yields the exact value of the norm of a particular operator between differentL₂-spaces which arises naturally in Wiener–Hopf theory and which has several puzzling features.     

Multiplicatively badly approximable numbers and generalised Cantor sets

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that for all α∈R. We show that with the additional factor of the statement is false. Indeed, our main result implies that the set of α for which is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.     

Regularity of solutions for an integral system of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials:(1) whereThis system includes many known systems as special cases, in particular, when and γ=2, system (1) reduces to(2) The solutions (u,v) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs which comprises the well-known Lane–Emden system and Yamabe equation.We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

Sequences that omit a box (modulo 1)

Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in [0,1d⁾, we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S₁,…,Sd be strictly increasing in N. Define to be the set of x in [0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vr−ur→∞ and1+vr−ur?C#(Br). Kaufman has shown that is countable whenever S₁,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.     

Unconditional constants and polynomial inequalities

If P is a polynomial on R^m of degree at most n, given by , and P_n(R^m) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on P_n(R^m), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.