On the connectedness of self-affine attractors

Let T = T(A, D) be a self-affine attractor in defined by an integral expanding matrix A and a digit set D. In the first part of this paper, in connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers . It is shown that in and , for any integral expanding matrix A, T(A, D) is connected. In the second part, we study connectedness of Pisot dual tiles, which play an important role in the study of -expansions, substitutions and symbolic dynamical systems. It is shown that each tile of the dual tiling generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. We even give a simple necessary and sufficient condition of connectedness of the Pisot dual tiles of degree 4. Detailed proofs will be given in [4]. Received: 2 March 2003     

On the structure of rings which are sums of two subrings

We study the structure of rings that are sums of two subrings one of which is nil and the other reduced. Our main results concern the problem whether in this situation the nil subring must be an ideal.Received: 13 July 2001     

Stable closed characteristics on partially symmetric convex hypersurfaces

Let n be a positive integer and P=diag(−I_n−κ,I_κ,−I_n−κ,I_κ) for some integer κ∈[0,n]. In this paper, we prove that for any convex compact smooth hypersurface Σ in R²ⁿ with n?2 there always exists at least one closed characteristic on Σ which possesses at least 2n−4κ Floquet multipliers on the unit circle of the complex plane, provided Σ is P-symmetric, i.e., x∈Σ implies Px∈Σ.     

The flow due to a rough rotating disk

Von Kármáns problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given.     

Electrical characterization of strontium tartrate single crystals

The a.c. and d.c. conductivity of SrC₄H₄O₆·3H₂O are measured and are found to lie between usual conductivities of semiconductor and insulator. Temperature dependence of d.c. conductivity shows intrinsic conduction, which is confirmed by the slope of versus data. Due to application of thermal energy, noticeable conductivity peaks imply liberation of water molecules during dehydration and the formation of strontium oxalate. The conductivity plot has a nature similar to the intrinsic-to-extrinsic transition found in normal semiconductors. There occurs Efros hopping conduction in our samples.     

Chemical diffusion in calcium titanate

This work reports the gas/solid equilibration kinetics for the O₂/CaTiO₃ system. The electrical conductivity measurement was applied for monitoring the kinetics in the ranges of temperature 973-1323 K and oxygen partial pressure 10 Pa-72 kPa. It was found that the gas/solid equilibration kinetics for the polycrystalline CaTiO₃ specimen in the above experimental conditions is determined by bulk diffusion rather than by grain boundary conditions. The obtained data of the electrical conductivity vs. time were used for the determination of the chemical diffusion coefficient as a function of temperature at low and high p(O₂), respectively:

(1)     

I.I. Rabi: The Two Cultures and the Universal Culture of Science

I set forth and discuss I.I. Rabis views on the nature of science and society, focusing specifically on Rabis views on science and government, science and education, and science and religion. I also explore the influence of Rabi and C.P. Snow on each other. In the 1960s, Rabi set forth his mature and boldest positions, advocating science as the center of education and science as the replacement for religion. His positions culminate in science as the universal culture. I highlight Rabi not only as a scientist and public servant, but also as a public intellectual actively engaged with and connected to major issues of his time.Michael A. Day is Professor of Physics at Lebanon Valley College. He holds doctorates in both physics and philosophy.     

Chiral polymers containing a carbohydrate moiety: synthesis

This paper reports the preparation of poly(acetal-ethers) by two alternative synthetic pathways (alkaline and acid-catalyzed conditions). Polycondensations of methyl 2,3:4,6-di-O-salicylidene-alpha-D-mannopyranoside (3) (mixtures of endo-H and exo-H dioxolan-2-yl-diastereomers) with 1,4-dibromobutane (4) (method I a and I b) were performed in solvents (DMF, butyl acetate/DMF, DMSO) and were catalyzed by K2CO3/KI or KOH. A similar polymer (6) was formed by the reaction of methyl alpha-D-mannopyranoside (1) and 1,4-bis(2-formylphenoxy)butane (7), catalyzed by p-toluenesulfonic acid (method II). Regardless of conversion or initial comonomer feed ratios, the composition of the polycondensates depended on the reaction conditions, leading to the formation of macrocyclic [1 + 1] (5) and [2 + 2] compounds, which were macromolecules with diverse molecular weights and optical properties. The regioselective polycondensation was examined by 1H NMR spectroscopy of selected polymers. In the case of 5-membered cyclic acetal units, mixtures of the endo-H and exo-H dioxolan-2-yl system, diastereomers were formed in the polymer chain. The macrocycles and linear oligomers were identified by NMR and electrospray mass spectrometry (ESI-MS). Thermodynamically controlled reactions for making macrocycles as well as oligomers in the absence of templates are also discussed.     

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology of any smooth symplectic resolution X?V/G (multiplicative McKay correspondence). We prove further that if is an irreducible Weyl group and , then no smooth symplectic resolution of V/G exists unless G is of types .