Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)

We prove converse and smoothness theorems of polynomial approximation in weightedL_pspaces with norm ‖fW‖_Lp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.     

A pattern sequence approach to Stern’s sequence

Suppose that w∈1{0,1}^∗ and let a_w(n) be the number of occurrences of the word w in the binary expansion of n. Let {s(n)}_n?0 denote the Stern sequence, defined by s(0)=0, s(1)=1, and for n?1, In this note, we show that where denotes the complement of w (obtained by sending 0?1 and 1?0) and [w]₂ denotes the integer specified by the word w∈{0,1}^∗ interpreted in base 2.     

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.     

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.     

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

A new characterization of projections of quadrics in finite projective spaces of even characteristic

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, α, or q lines through p intersecting L. An example of such a geometry with α=2 is the following well known geometry . Let Q_n+1 be a nonsingular quadric in a finite projective space , n≥3, q even. We project Q_n+1 from a point r∉Q_n+1, distinct from its nucleus if n+1 is even, on a hyperplane not through r. This yields a partial linear space whose points are the points p of , such that the line 〈p,r〉 is a secant to Q_n+1, and whose lines are the lines of which contain q such points. This geometry is fully embedded in an affine subspace of and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.     

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 .     

Transcendence of Binomial and Lucas' Formal Power Series

The formal power series[formula]is transcendental over (X) whentis an integer ≥ 2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn = ∑n_ipⁱis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over _p(X), the field of rational functions over the Galois field _p, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over (X).     

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.     

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.     

Best Approximation of Functions like |x| exp(−A|x|)

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?_λ, α(x)=|x|^λ exp(−A|x|^−α). In particular, it is shown thatE(?_λ, α, _n, L_p(−1, 1))∼n^{−(2λp+αp+2)/2p(1+α)}×exp(−(1+α⁻¹)(Aα)^1/(1+α) cos απ/2(1+α) n^α/(1+α)), whereE(?_λ, α, _n, L_p(−1, 1)) denotes best polynomial approximation of?_λ, αinL_p(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?_0, 2, _n, L_∞(−1, 1)), has been posed by S. N. Bernstein.     

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.     

Asymptotic Enumeration of Convex Polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice ^dtaking at least one step in every dimension. A polygon on ^dis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n, d}of length 2nwithd(n)→∞ is asymptoticallywherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r²/sinh² r). The convergence is uniform over alld?ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d⁻¹)^2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.     

Quadratic Gauss Sums

Letpbe a prime integer andmbe an integer, not divisible byp. LetKbe the splitting field ofX^m−1 over the prime field _p. Solving the Gauss sums problem of ordermin characteristicpmeans determining Gauss sums of all multiplicative characters ofKof order dividingm. Our aim is to solve this problem when the subgroup 〈p〉 is of index 2 in (/m)^*.     

A model containing both the Camassa-Holm and Degasperis-Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the H¹-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space Hs with is established under the assumptions u₀∈Hs and ‖u0xL^∞_‖<∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with is also developed.     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

Boxicity of line graphs

The boxicity of a graph H, denoted by , is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper we show that for a line graph G of a multigraph, , where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, . For the d-dimensional hypercube Q_d, we prove that . The question of finding a nontrivial lower bound for was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800].The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).     

Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Newman proved for the classical Thue–Morse sequence, ((−1)^s(n))_n≥0, that for all N∈N with real constants satisfying c₂>c₁>0 and λ=log3/log4. Coquet improved this result and deduced , where F(x) is a nowhere-differentiable, continuous function with period 1 and η(N)∈{−1,0,1}. In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum a Coquet-type formula exists for every r∈{0,1,2} if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.     

Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤p≤∞. Here Q is a d-parallelepiped in R^d with sides parallel to the coordinate axes. We consider the error of best approximation of a function f by algebraic polynomials of fixed degree at most r_i−1 in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.