TheL 2 structure of moduli spaces of Einstein metrics on 4-manifolds

Partially supported by NSF Grants DMS 87-01137 and 89-01303     

TheL pDirichlet problem for small perturbations of the Laplacian

We show that Dahlberg's vanishing trace condition measuring the disagreement between the coefficients of two elliptic operators preserves harmonic measures whose logarithm belongs to VMO.     

TheL ∞-representation algebra of a foundation topological semigroup

In the present paper, for a large family of topological semigroups which includes the topological groups and discrete semigroups as elementary examples, it is shown that the representation algebraR(S) is identical to the algebra of Fourier transforms onS.     

TheL p-saturation results weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators

The aim of the present paper is to prove new equivalence results and L_p-stauration results on weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators (including results in [7]). One of the main tools and crucial estimates managing the converse results is given by a direct modified Vornorskaja theorem which uses the third order weighted modulus of smoothness. Supported by NSF of Hebei Province.     

The structure of 1-planar graphs

A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an edge with both endvertices of degrees at most 20, and we present similar results concerning bigger structures in 1-planar graphs with additional constraints.     

Weyl群的子群结构

对各种类型的不可约根系,讨论了其Ｗｅｙｌ群一类子群的极大性。     

On the structure of 1-dependent Markov chains

Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.     

On the structure of indecomposable injective modules 1

For a ring R, the following two conditions are equivalent:.

(1) If E is an indecomposable injective right R-module, then End E_R is a field (not necesarily commutative).

(2) Every co-irreducible rigtht ideal is critical.

Since (2) has been characterized ideal-theoretically, this amounts to an ideal-theoretical characterization of (1). These rings come up to the study of (QI) rings in which every quasi-injective module is injective.     

Frozen 1-RSB structure of the symmetric Ising perceptron

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1-RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.     

A structure theorem for the consecutive 1's property

A (0, 1)-matrix M has the consecutive 1's property for columns if the rows of M can be permuted so that the 1's in each column appear consecutively. A graph-theoretic approach is used to characterize matrices with this property in terms of forbidden submatrices. Graphs whose adjacency matrix has this property are also characterized.     

Effect of deformation on the supermolecular structure of PN-1 copolymer

It is shown that under the combined influence of temperature and tensile stresses the spherical structure of crosslinked PN-1 copolymer becomes ellipsoidal. The change of shape is associated with the zone of high-elastic deformation.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1129–1131, November–December, 1969.     

Quasi-Lie structure of σ-derivations of C[t±1]

Hartwig, Larsson and Silvestrov in [J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2) (2006) 314–361] defined a bracket on σ-derivations of a commutative algebra. We show that this bracket preserves inner derivations, and based on this obtain structural results providing new insights into σ-derivations on Laurent polynomials in one variable.