Complete weights andv-peak points of spaces of weighted holomorphic functions

We examine the geometric theory of the weighted spaces of holomorphic functions on bounded open subsets ofC ⁿ ,C ⁿ ,H _v (U) and , by finding a lower bound for the set of weak^*-exposed and weak^*-strongly exposed points of the unit ball of and give necessary and sufficient conditions for this set to be naturally homeomorphic toU. We apply these results to examine smoothness and strict convexity of and . We also investigate whether is a dual space. The second author was supported by MCYT and FEDER Project BFM2002-01423.     

The information-based complexity of approximation problem by adaptive Monte Carlo methods

In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MWpr,α(Td), 1 < p < ∞, in the norm of Lq(Td), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem.     

A fast algorithm for numerical solutions to Fortet's equation

A fast algorithm for computation of default times of multiple firms in a structural model is presented. The algorithm uses a multivariate extension of the Fortet's equation and the structure of Toeplitz matrices to significantly improve the computation time. In a financial market consisting of M1 firms and N discretization points in every dimension the algorithm uses O(nlogn·M·M!·N^M(M-1)/2) operations, where n is the number of discretization points in the time domain. The algorithm is applied to firm survival probability computation and zero coupon bond pricing.     

Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”-and “ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h ^p + τ ^q ). The estimates hold true even in the hyperbolic case when ɛ = 0.     

Nonisospectral flows on semiinfinite unitary block Jacobi matrices

It is proved that if the spectrum and the spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix J(t) vary appropriately, then the corresponding operator J(t) satisfies the generalized Lax equation , where Φ(gl, t) is a polynomial in λ and with t-dependent coefficients and is a skew-symmetric matrix. The operator J(t) is analyzed in the space ℂ ⊕ ℂ² ⊕ ℂ² ⊕ …. It is mapped into the unitary operator of multiplication L(t) in the isomorphic space , where . This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation. A procedure that allows one to solve the corresponding Cauchy problem by the inverse-spectral-problem method is presented. The article contains examples of block difference-differential lattices and the corresponding flows that are analogs of the Toda and the van Moerbeke lattices (from the self-adjoint case on ℝ) and some notes about the application of this technique to the Schur flow (the unitary case on and the OPUC theory). Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 521–544, April, 2008.     

On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space on the n-dimensional Heisenberg group ℍ ⁿ = ℂ ⁿ × ℝ, where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of ℍn. Suppose that the subelliptic gradient is gloablly L ^p integrable, i.e., is finite. We prove a Poincaré inequality for f on the entire space ℍ ⁿ . Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of under the norm of

8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.     

Remarks on dispersionless KP,KdV, and 2D gravity

Summary We use the SDiff(2) framework of Takasaki and Takebe and the (L, M) program (L is the Lax operator andMω=ω_λ) to show that =semiclassical limit ofM is , where ( ) are action angle variables in the Gibbons-Kodama theory of Hamilton-Jacobi type for dispersionless KP. We also show is the semiclassical limit ofWxW ⁻¹ (W is the gauge operator), whereG=WxW ⁻¹ is a quantity studied by the author in an earlier paper in connection with symmetries. We give then a semiclassical version of the Jevicki-Yoneya action principle for 2D gravity, where again arises in calculations, and this yields directly the Landau-Ginsburg equation that corresponds to the semiclassical limit of an integrated string equation. For KdV we also show how inverse scattering data are connected to Hamiltonians for dispersionless KdV. We also discuss Hirota bilinear formulas relative to the dispersionless hierarchies and establish various limiting formulas.     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

The maximal Fejér operator on real Hardy spaces

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H ¹, which may be considered over and . We also draw corollaries for the corresponding Hardy spaces over ² and ². This revised version was published online in August 2006 with corrections to the Cover Date.     

Low rank Tucker-type tensor approximation to classical potentials

This paper investigates best rank-(r ₁,..., r _d ) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝ ^d . Super-convergence of the best rank-(r ₁,..., r _d ) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r ₁,..., r _d ) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example and with x, y ∈ ℝ ^d . Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.      

Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

Consider the nonparametric regression model , whereg is an unknown function to be estimated on [0, 1], are the fixed design points in the interval [0, 1] and is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth _π ^* be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh _π ^* , the rate of convergence and the weak consistency of the median cross-validated estimate are obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Foundation of Education of China.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

Vive la différence III

We show that, consistently, there is an ultrafilter on ω such that if N _n^ℓ = (P _n^ℓ ∪ Q _n^ℓ, P _n^ℓ, Q _n^ℓ, R _n^ℓ) (for ℓ = 1, 2, n < ω), P _n^ℓ ∪ Q _n^ℓ ⊆ ω, and are models of the canonical theory t ^ind of the strong independence property, then every isomorphism from onto is a product isomorphism. The first version of this work done in 93; First typed: May 1993. This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509     

Regularity of solutions of Poisson’s equation in multiplier spaces

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ¹(ℝ ^d ) → (Ḣ⁻¹(ℝ ^d )) is a complex-valued distribution on ℝ ^d , satisfy the regularity property D ^k u ∈ (Ḣ¹ → Ḣ⁻¹) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.      

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let denote the class of quasinilpotent quasiaffinities Q in such that Q ^* Q has an infinite dimensional reducing subspace M with Q ^* Q| _M compact. It was known that if every quasinilpotent operator in has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in . The purpose of this paper is to provide sufficient conditions for elements in to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.