Extreme point axioms for closure spaces

A pair (X,τ) of a finite set X and a closure operator τ:2^X→2^X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space (X,τ), the extreme point operator ex:2^X→2^X is defined as ex(A)={p|p∈A,p∉τ(A-{p})}. We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectively.     

Hölder continuity of discrete Morse flows to the Alt–Caffarelli variational problem generating free boundaries

The present paper shows Hölder continuity of discrete Morse flows to a regularized Alt–Caffarelli variational functional generating free boundaries; the continuity is uniform with respect to the discrete Morse flows and the regularizations. The uniformity enables to construct Morse flows to the Alt–Caffarelli functional, which shall be dealt with in another paper.     

Power operations in elliptic cohomology and representations of loop groups

Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.

     

Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss

The problem of estimating large covariance matrices of multivariate real normal and complex normal distributions is considered when the dimension of the variables is larger than the number of samples. The Stein–Haff identities and calculus on eigenstructure for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain classes of estimators for the population covariance matrices under invariant quadratic loss functions are obtained for real and complex cases, respectively. Based on the unbiased risk estimates, shrinkage estimators which are counterparts of the estimators due to Haff [L.R. Haff, Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist. 8 (1980) 586–697] are shown to improve upon the best scalar multiple of the empirical covariance matrix under the invariant quadratic loss functions for both real and complex multivariate normal distributions in the situation where the dimension of the variables is larger than the number of samples.     

Vertices of Degree 5 in a Contraction Critically 5-connected Graph

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.     

Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss

In this paper the problem of estimating a covariance matrix parametrized by an irreducible symmetric cone in a decision-theoretic set-up is considered. By making use of some results developed in a theory of finite-dimensional Euclidean simple Jordan algebras, Bartlett's decomposition and an unbiased risk estimate formula for a general family of Wishart distributions on the irreducible symmetric cone are derived; these results lead to an extension of Stein's general technique for derivation of minimax estimators for a real normal covariance matrix. Specification of the results to the multivariate normal models with covariances which are parametrized by complex, quaternion, and Lorentz types gives minimax estimators for each model.     

Persistent hole-burning of perylene microcrystallites dispersed in PVA

A persistent hole-burning is observed in β-perylene microcrystallites, which were embedded in poly-vinyl alcohol. By laser light excitation at 22,535 cm⁻¹ and at 10 K, the hole is found at the excitation photon energy. The mechanism of the persistent hole-burning is interpreted in terms of the resolution of microcrystallites into smaller microcrystallites. This is a novel observation of the persistent hole-burning in aromatic microcrystallites. When the specimen, which includes a hole, is annealed at high temperatures, the resolved microcrystallites restore back to the old position as had been. The β-perylene microcrystallite specimen that we have grown was as small as 1.5 nm in average diameter. They are one order smaller in number of molecules included, compared to those that have been reported on aromatic microcrystallites, anthracene for example. Due to this, we were able to observe the 0-0 transition energy, which varied according as the number of molecules involved in the microcrystallites. We also observed the 0-0 absorption (excitation) spectrum, which depends on the molecular arrays in the microcrystallites. The 0-0 transition of a single molecule in poly-vinyl alcohol matrix is anticipated to be located at 22,885 cm⁻¹.     

Purification and magnetic properties of carbon nanotubes

Received: 5 January 1998     

Die protonchemischen Verschiebungen der α-Helix von Poly-L-alanin

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