Ruthenium(II)-catalyzed regio- and stereoselective hydroarylation of alkynes via directed C-H functionalization

The ruthenium-catalyzed hydroarylation of alkynes with benzamides proceeds regio- and stereoselectively through a directed C-H bond cleavage. Preliminary mechanistic investigations indicate that the reaction involves amide-directed ortho-metalation, carbometalation of alkyne, and protonolysis. Similarly, phenylazoles also add to alkynes regioselectively.     

Optoelectronic oscillator using a LiNbO3 phase modulator for self-oscillating frequency comb generation

Optical frequency comb generation by using a novel optoelectronic oscillator (OEO) is proposed and demonstrated with the emphasis placed on self-oscillating operation. In the OEO, a wideband LiNbO3 phase modulator is driven with a large-amplitude radio-frequency (RF) feedback signal to generate a deeply phase-modulated light wave; accordingly, an optical frequency comb with a bandwidth greater than the RF signal is generated by self-oscillation. Although it generates multifrequency components, the OEO exhibits characteristics of a single-mode oscillator. Its operation is stable and self-starting. An optical frequency comb with a 120 GHz bandwidth and 9.95 GHz frequency spacing was successfully generated by self-oscillation at a single frequency.     

A verified method for bounding clusters of zeros of analytic functions

In this paper, we propose a verified method for bounding clusters of zeros of analytic functions. Our method gives a disk that contains a cluster of m   zeros of an analytic function f(z)$f(z)$. Complex circular arithmetic is used to perform a validated computation of n  -degree Taylor polynomial p(z)$p(z)$ of f(z)$f(z)$. Some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p(z)$p(z)$. A validated computation of an upper bound for Taylor remainder series of f(z)$f(z)$ and a lower bound of p(z)$p(z)$ on a circle are performed. Based on these results, Rouché's theorem is used to verify that the disk contains the cluster of zeros of f(z)$f(z)$. This method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function. Numerical examples are presented to illustrate the efficiency of the proposed method.     

A non-D-space with large extent

We proved that ^?+ implies the existence of a non-D-space whose all closed subspace F satisfies e(F)=L(F). The existence of such a space under MA+¬CH or PFA is also discussed.     

Lower Semimodular Types of Lattices: Frankl's Conjecture Holds for Lower Quasi-Semimodular Lattices

We introduce a measure of how far a lattice L is from being lower semimodular. We call it the lower semimodular type of L. A lattice has lower semimodular type zero if and only if it is lower semimodular. In this paper we discuss properties of the measure and we show that Frankl's conjecture holds for lower quasi-semimodular lattices: if a lattice L is lower quasi-semimodular then there is a join-irreducible element x in L such that the size of the principal filter generated by x is at most (|L|− 1) /2. Revised: July 2, 1997     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

The Giry monad is not strong for the canonical symmetric monoidal closed structure on Meas

We show that the Giry monad is not strong with respect to the canonical symmetric monoidal closed structure on the category Meas of all measurable spaces and measurable functions.     

Excess of a Lattice

 A pair (x,y) of elements in a lattice is mismatching if x is join-irreducible, y is meet-irreducible, and xy. The excess of a lattice L is defined by ex(L):=|L|− min{|V _x |+|I _y |:(x,y) is mismatching}, where V _x (I _y ) is the principal filter (ideal) generated by x(y), respectively. In the first half of this paper, it is shown that a lattice has excess zero (one) if and only if it is isomorphic to a Boolean lattice (one of a chain of length two, a diamond, and a pentagon), respectively. In the second half, we show that for each relatively complemented lattice which is not Boolean, its size is at most five times its excess. Moreover we determine extremal configurations. Received: September 2, 1999 Final version received: July 25, 2000 Acknowledgments. The author thanks the referee for constructive comments. Present address: Bureau of Human Resources, National Personnel Authority, 1-2-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8913, Japan. e-mail: abetet@jinji.go.jp     

CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X. (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d_∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α,β∈∂X with α≠β, (1) lim sup{d_∂X(gα,gβ)∣g∈G}>0 and (2) lim inf{d_∂X(gα,gβ)∣g∈G}=0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.     

Products of tight continuous functions

Geometriae Dedicata -