Solvability of the\bar \partial problem withC ∞ regularity up to the boundary on wedges of ℂNproblem withC ∞ regularity up to the boundary on wedges of ℂN

For awedge W of ?^N, we introduce an intrinsic condition of weakq-pseudoconvexity which can be expressed in terms ofq-subharmonicity both of a defining function or an exhaustion function. Under this condition we prove solvability of the $\bar \partial $ system for forms withC ^∞ (W)-coefficients of degree ≥q+1. Our method relies on theL ²-estimates by Hörmander. ForC ^∞(W) solvability we refer to Hörmander (if ?W∈C ²), and to Zampieri (for general wedgesW). ForC ^∞ (W) solvability and with ?W∈C ², we 025 refer to Dufresnoy (ifq=0), Michel (if the number of negative Levieigenvalues of ?W is constant), and finally Zampieri (for more generalq-pseudoconvexity).     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Let Ω ? ? ⁿ be an open set and X(Ω) be any rearrangement invariant function space close to L ^q (Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ? u ? f from W ¹ X to W ¹ X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.     

Relaxation of multiple integrals in subcritical Sobolev spaces

The study of lowersemicontinuity properties in W^1·p of multiple integrals with quasiconvex integrands withq growth is undertaken forp >q ? 1. In the case where the integrand is not quasiconvex, the absolutely continuous part of the relaxed functional is analyzed.     

q-Deformation of W（2, 2） Lie Algebra Associated with Quantum Groups

The q-deformation of W (2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W (2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W (2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W (2, 2) Lie algebra in the q → 1 limit.     

On reducible Heegaard splittings

Let V∪S W be a reducible Heegaard splitting of genus g = g(S)≥2.For a maximal prime connected sum decomposition of V∪S W,let q denote the number of the genus 1 Heegaard splittings of S2×S1 in the decomposition,and p the number of all other prime factors in the decomposition.The main result of the present paper is to describe the relation of p,q and dim(C V∩CW).     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Topology of Sobolev mappings III

We establish a necessary and sufficient topological condition for maps that are in W^1,p(M, N) to be connected by continuous paths in W^1,p(M, N) to maps in W^1,q(M, N), q > q ≥ 1. We also obtain a necessary and sufficient topological condition under which W^1,q(M, N) is strongly dense in W^1,p(M, N). Several results concerning sequential weak density of smooth (or W^1,q(M, N) maps in W^1,p(M, N) are also proven. © 2003 Wiley Periodicals, Inc.     

Pointwise estimates of potentials for higher-order capacities

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)?W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)?kf(x)∈W _p ^m (D)?W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.     

Nonlinear gradient estimates for elliptic equations of general type

We study a very general model of nonvariational elliptic equations of p-Laplacian type. We prove the W ^1,q -estimates for each q > p regarding such a nonlinear elliptic equation in divergence form under the assumption that for each point and for each sufficiently small scale the nonlinearity is suitably close to a vector of the gradient in BMO space. In the same spirit our results can be easily extended to Orlicz spaces as well.     

A Duality Method in Prediction Theory of Multivariate Stationary Sequences

Let W be an integrable positive Hermitian q × q–matrix valued function on the dual group of a discrete abelian group G such that W^–1 is integrable. Generalizing results of T. Nakazi [N] and of A. G. Miamee and M. Pourahmadi [MiP] for q = 1 we establish a correspondence between trigonometric approximation problems in L²(W) and certain approximation problems in L²(W^–1). The result is applied to prediction problems for q–variate stationary processes over G , inparticular, to the case G = ℤ.     

Linearized Elasticity as Γ-Limit of Finite Elasticity

Linearized elastic energies are derived from rescaled nonlinear energies by means of Γ-convergence. For Dirichlet and mixed boundary value problems in a Lipschitz domain Ω, the convergence of minimizers takes place in the weak topology of H ¹(Ω,R ⁿ ) and in the strong topology of W ^1,q (Ω,R ⁿ ) for 1≤q<2.     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

We consider a class of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^?∞?, which is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a ε???δ domain as defined by Jones and the fractal set is not totally disconnected. We compare two notions of trace on Γ^?∞? for functions in W ^1,q (Ω): the classical one, see for instance the book by Jonnson and Wallin, 1984, using the strict definition of a function at a point of $\overline{\Omega}$ , and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide μ-almost everywhere on Γ^?∞?. As a corollary, we characterize the critical number $\bar q$ for which for all $q<\bar q$ (resp. $q > \bar q$ ) there is a (resp. no) continuous extension operator from W ^1,q (Ω) to W ^1,q (?²).     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

Widths and distributions of values of the approximation functional on the sobolev spaces with measure

We obtain asymptotic values for the integraln-widths of Sobolev classesW ₂ ^r equipped with Gaussian measure in theL _q -norm.     

Boundary behavior of Orlicz-Sobolev classes

It is proved that homeomorphisms of the Orlicz-Sobolev class W _loc ^{1, φ} can be continuously extended to the boundaries of some domains if the function φ defining this class satisfies a Carderón-type condition and the outer dilatation K _f of the mapping f satisfies the divergence condition for integrals of special form. In particular, the result holds for homeomorphisms of the Sobolev classes W _loc ^1,1 with K _f ∈ L _loc ^q for q > n ? 1.     

Principal components in the nonnormal case: The test of equality of Q roots

The limiting distribution of the likelihood ratio statistic W_q for testing the hypothesis of equality of q characteristic roots of a covariance matrix is studied in the case of nonnormal populations. It is shown, both theoretically and empirically, that the limiting distribution of W_q is not robust to departures from normality and that W_q cannot be used for nonnormal populations with long tails. For the class of elliptical populations the limiting distribution of W_q can be expressed in close form. A corrected test statistic, W_q^*, is proposed that can be used in principal component analysis when sampling from elliptical populations.