Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Inverse Conductivity Problem on Riemann Surfaces

An electrical potential U on a bordered Riemann surface X with conductivity function σ>0 satisfies equation d(σ d ^c U)=0. The problem of effective reconstruction of σ from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: U| _bX ↦ σ d ^c U| _bX is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. Novikov (Funkc. Anal. Ego Priloz. 22:11–22, 2008) for simply connected X. We apply for this new kernels for on the affine algebraic Riemann surfaces constructed in Henkin (, 2008).      

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287–349, [2005]] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), [2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, [1985]] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, [1987]] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, [1965], [1974]] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245–246, [1976]] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, [1976]]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951–976, [1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.      

An identity of Andrews and the Askey-Wilson integral

Andrews (Adv. Math. 41:137–172, 1981) derived a four-variable q-series identity, which is an extension of the Ramanujan ₁ ψ ₁ summation. In this paper, we shall give a simple evaluation of the Askey-Wilson integral by using this identity. The author was supported by the National Science Foundation of China, PCSIRT and Innovation Program of Shanghai Municipal Education Commission.     

Global error bounds for the extended vertical LCP

A new necessary and sufficient condition for the row -property is given. By using this new condition and a special row rearrangement, we provide two global error bounds for the extended vertical linear complementarity problem under the row -property, which extend the error bounds given in Chen and Xiang (Math. Program. 106:513–525, 2006) and Mathias and Pang (Linear Algebra Appl. 132:123–136, 1990) for the P-matrix linear complementarity problem, respectively. We show that one of the new error bounds is sharper than the other, and it can be computed easily for some special class of the row -property block matrix. Numerical examples are given to illustrate the error bounds. The work was in part supported by a Grant-in-Aid from Japan Society for the Promotion of Science, and the National Natural Science Foundation of China (10671010).     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

On Convergence Results for Weak Efficiency in Vector Optimization Problems with Equilibrium Constraints

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997).     

Dirichlet Problem at Infinity for $\mathcal A$-Harmonic Functions

We study the Dirichlet problem at infinity for -harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x ₀∈M(∞) to be -regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x ₀ in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.      

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

The Unfolding of Equivariant Bifurcation Problems with Two Types of State Variables in the Presence of Parameter Symmetry

The unfolding of equivariant bifurcation problems with two types of state variables under the action of group K（Г, △） is discussed by using DA-algebraic tools. One of the main results is the equivariant versal unfolding theorem.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

An exact Turán result for the generalized triangle

Let Σ _k consist of all k-graphs with three edges D ₁, D ₂, D ₃ such that |D ₁ ∩ D ₂| = k − 1 and D ₁ Δ D ₂ ⊆ D ₃. The exact value of the Turán function ex(n, Σ _k ) was computed for k = 3 by Bollobás [Discrete Math. 8 (1974), 21–24] and for k = 4 by Sidorenko [Math Notes 41 (1987), 247–259]. Let the k-graph T _k ∈ Σ _k have edges

Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;_Ω=w ²Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds ²=w(z)^-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;_Ω ², with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;_Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ _ω∈G (1-|ω0|²) ^s for G the covering group of the uniformization map of Ω and 0<s<1. Received: August 21, 2000?Published online: October 30, 2002 RID="*" ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.