Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings. J.-W. Oh was partially supported by BK21-Yonsei University.     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

Pull-back of currents by holomorphic maps

We define the pull-back operator, associated to a meromorphic transform, on several types of currents. We also give a simple proof to a version of a classical theorem on the extension of currents.     

On embeddability and stresses of graphs

Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K ₅-minor freeness guarantees the stress freeness. More generally, we prove that every K _r+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed. Supported by an I.S.F. grant.     

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(=  1, 2, 3) distinct principal curvatures. Dedicated to Professor Hajime Urakawa on his sixtieth birthday. H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006.     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Trifactorized locally finite groups with min-p for every prime p

A group G is trifactorized if G = AB = AC = BC with three subgroups A, B and C of G. Some structural theorems about trifactorized locally finite groups with minimum condition on p-subgroups for every prime p are proved. For instance, it is shown that G is locally supersoluble (locally nilpotent) if A and B are locally nilpotent and C is locally supersoluble (locally nilpotent). The second and the third author wish to thank the Institute of Mathematics of the University of Mainz for their excellent hospitality during the preparation of this paper. The second author is grateful to the University of Stellenbosch, South Africa, and the third author to the Deutsche Forschungsgemeinschaft (DFG) for financial assistance.     

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Vector Equilibrium Problems,Minimal Element Problems and Least Element Problems

In this paper we introduce some concepts of feasible sets for vector equilibrium problems and some classes of Z-maps for vectorial bifunctions. Under strict pseudomonotonicity assumptions, we investigate the relationship between minimal element problems of feasible sets and vector equilibrium problems. By using Z-maps, we further study the least element problems of feasible sets for vector equilibrium problems. Finally, we prove a generalized sublattice property of feasible sets for vector equilibrium problems associated with Z-maps. This work was supported by the National Natural Science Foundation of China and the Applied Research Project of Sichuan Province (05JY029-009-1). The authors thank Professor Charalambos D. Aliprantis and the referees for valuable comments and suggestions leading to improvements of this paper.     

Periods for irregular singular connections on surfaces

Given an integrable connection on a smooth quasi-projective algebraic surface U over a subfield k of the complex numbers, we define rapid decay homology groups with respect to the associated analytic connection which pair with the algebraic de Rham cohomology in terms of period integrals. These homology groups generalize the analogous groups in the same situation over curves defined by Bloch and Esnault. In dimension two, however, new features appear in this context which we explain in detail. Assuming a conjecture of Sabbah on the formal classification of meromorphic connections on surfaces (known to be true if the rank is lower than or equal to 5), we prove perfectness of the period pairing in dimension two.     

A gradient estimate in the Calabi–Yau theorem

We prove a C ¹-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C ⁰-estimate, without using a C ²-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.     

Normal Hopf subalgebras in cocycle deformations of finite groups

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = H ^σ of the Hopf algebra H = k ^G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra corresponds to a Hopf subalgebra . Our main result is an explicit necessary and sufficient condition for the normality of L′ in L. This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.      

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

Shelah’s work on non-semi-proper iterations,I

In this paper, we give details of results of Shelah concerning iterated Namba forcing over a ground model of CH and iteration of P[W] where W is a stationary subset of ω ₂ concentrating on points of countable cofinality.     

Linear fractional mappings: invariant sets, semigroups and commutativity

We study commutativity and embeddability (into continuous semi-groups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Kœnigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. Dedicated to Professor Felix E. Browder with admiration and respect