Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2 ^n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4 ^m 2 ⁿ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

Maximal <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following two domains of definition: , or , where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on L ^p (Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and , the Laplacian with domain D ₂(Δ) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that proves that, in general, the Dirichlet–Laplacian with domain D ₁(Δ) is not even a closed operator. The main results of this paper are taken from the author’s Ph.D. thesis, written at the TU Darmstadt under the supervision of Prof. M. Hieber. The author wishes to thank Prof. Hieber for his guidance, encouragement and support in the last few years. Many thanks also go to Prof. C. E. Kenig for his hospitality and many ruitful discussions on the subject during a 1-year stay at the University of Chicago.     

Geometric dual formulation for first-derivative-based univariate cubic <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> splines

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based -smooth univariate cubic L ₁ splines. An L ₁ spline minimizes the L ₁ norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L ₁ spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.     

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

Quasi-<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm orthogonal Galerkin expansions in sums of Jacobi polynomials

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) ^α (1 + x) ^β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L ^p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.     

Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> (<Emphasis Type="Italic">d</Emphasis> = 2, 3)

In this paper, we address the finite element method and discontinuous Galerkin method for the stochastic scattering problem of Helmholtz type in ℝ ^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided to examine our theoretical results. The first author is supported by NSF under grand number 0609918 and AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154, the second author is supported by NSFC(10671082, 10626026, 10471054), and the third author is supported by NNSF (No. 10701039 of China) and 985 program of Jilin University.     

The distance between homotopy classes of <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-valued maps in multiply connected domains

Certain Sobolev spaces of S ¹-valued functions can be written as a disjoint union of homotopy classes. The problem of finding the distance between different homotopy classes in such spaces is considered. In particular, several types of one-dimensional and two-dimensional domains are studied. Lower bounds are derived for these distances. Furthermore, in many cases it is shown that the lower bounds are sharp but are not achieved. The first author’s work of was supported in part by NSF grant 0503887. The second author’s research of was supported by G.S. Elkin research fund.     

On Levi-flat hypersurfaces with given boundary in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let S ⊂ ℂ ⁿ be a compact connected 2-codimensional submanifold. If n ⩾ 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in ℂ ⁿ of a Levi-flat (2n−1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point. For n = 2 many results are known since the 1980’s and a new result with a very technical hypothesis is announced. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.     

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

The <Emphasis Type="Italic">Q</Emphasis> Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.     

The existence of global attractors for 2D Navier-Stokes equations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> spaces

In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in Hk（Ω,R2） for any k ≥ 1, which attracts any bounded set of Hk（Ω,R2） in the H^k-norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong＇s conclusion.     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153     

Slightly larger than a graph<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebra

To certain higher rank Cuntz algebras including the classical cases we trace a certain partial isometryU in its strong closure. AdjoiningU to we obtain a kind of uniqueness property for this largerC ^*-algebra. Its explanation is not entirely “Cuntz’s uniqueness argumentation”. The author is supported by the Austrian Research Foundation (FWF) project S8308.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

Rearrangement Invariant Continuous Linear Functionals on Weak <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We show that in the dual of Weak L¹ the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L¹(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L¹ contains a norm closed weak* dense ideal which is lattice isometric to an ℓ¹-sum of spaces of type C(K). Helmut H. Schaefer in memoriam     

Matrix decomposition algorithms for the <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-quadratic finite element Galerkin method

Explicit expressions for the eigensystems of one-dimensional finite element Galerkin (FEG) matrices based on C ⁰ piecewise quadratic polynomials are determined. These eigensystems are then used in the formulation of fast direct methods, matrix decomposition algorithms (MDAs), for the solution of the FEG equations arising from the discretization of Poisson’s equation on the unit square subject to several standard boundary conditions. The MDAs employ fast Fourier transforms and require O(N ²log N) operations on an N×N uniform partition. Numerical results are presented to demonstrate the efficacy of these algorithms.     

On the ideal (<Emphasis Type="Italic">v</Emphasis><Superscript>0</Superscript>)

The σ-ideal (v ⁰) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v ⁰) to the family of Ramsey null sets. To describe add(v ⁰) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v ⁰) = add(v ⁰) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v ⁰) = ω ₁ implies that (v ⁰) has the ideal type (c, ω ₁, c).      

Universally <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-bad arithmetic sequences

We present a modified version of Buczolich and Mauldin’s proof that the sequence of square numbers is universally L ¹-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes. Furthermore, we show that any subsequence of the averages taken along these sequences is also universally L ¹-bad.