On the existence of resolvable K4 − e designs

In this article, we settle a problem which originated in 4 regarding the existence of resolvable (K₄ ? e)‐design. We solve the problem with two possible exceptions. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 502–510, 2007     

Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

The first eigenvalue of the p‐Laplace operator under powers of mean curvature flow

In this paper, we show the following main results. Let (Mⁿ,g(t)), t ∈ [0,T), be a solution of the unnormalized H^k ? flow on a closed manifold, and λ_1,p(t) be the first eigenvalue of the p‐Laplace operator. If there exists a nonnegative constant ε such that in M × [0,T) and in M × [0,T),then λ_1,p(t) is increasing and the differentiable almost everywhere along the unnormalized H^k ? flow on [0,T). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

K4−‐factor in a graph

Let G be a graph of order 4k and let δ(G) denote the minimum degree of G. Let F be a given connected graph. Suppose that |V(G)| is a multiple of |V(F)|. A spanning subgraph of G is called an F‐factor if its components are all isomorphic to F. In this paper, we prove that if δ(G)≥5/2k, then G contains a K₄^?‐factor (K₄^? is the graph obtained from K₄ by deleting just one edge). The condition on the minimum degree is best possible in a sense. In addition, the proof can be made algorithmic. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 111–128, 2002     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications

Through a Hermitian‐type (skew‐Hermitian‐type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240 :199–205), where A is Hermitian (skew‐Hermitian), we show how to find a Hermitian (skew‐Hermitian) matrix X such that the matrix expressions A ? BX ± X^*B^* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X^*B^* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ²V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least‐squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A? ?_b,cl (χ,^bΩ^1/2) (cf. [22], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C* – algebra. On the other hand, the symbolic structure of A can be extended continuously to the C* – algebra B generated by A as a subalgebra of ζ(σ^bL²(χ, ^bΩ^1/2)) by a generalization of a method of Gohberg and Krupnik. Furthermore, A is densely embedded in a Fréchet algebra A ? B which is a ?* – algebra in the sense of Gramsch [9, Definition 5.1], reflecting also smooth properties of the original algebra A.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1)

It has been long conjectured that the crossing number of C_m × C_n is (m?2)n, for all m, n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m(m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of C_m × C_n is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004     

More results on perfect (3,3,6k + 4; 6k − 2)‐threshold schemes

A (2,3)‐packing on X is a pair (X, ), where is a set of 3‐subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of . In this article, we shall construct a set of 6k ? 2 disjoint (2,3)‐packings of order 6k + 4 with K_1,3 ∪ 3kK₂ or G₁ ∪ (3k ? 1)K₂ as their common leave for any integer k ≥ 1 with a few possible exceptions (G₁ is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson ( 22 ). © 2006 Wiley Periodicals, Inc. J Combin Designs     

Index sets for ω‐languages

An ω‐language is a set of infinite sequences (words) on a countable language, and corresponds to a set of real numbers in a natural way. Languages may be described by logical formulas in the arithmetical hierarchy and also may be described as the set of words accepted by some type of automata or Turing machine. Certain families of languages, such as the languages, may enumerated as P₀, P₁, … and then an index set associated to a given property R (such as finiteness) of languages is just the set of e such that P_e has the property. The complexity of index sets for 7 types of languages is determined for various properties related to the size of the language.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic analysis of unsteady neutron transport equation

Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The major difficulty is the lack of regularity in the boundary layer with geometric correction. Our contribution relies on a detailed analysis of asymptotic expansions inspired by the compatibility condition and an intricate L^2m ? L^∞ framework, which yields stronger remainder estimates.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].