Small Periodic Solutions Generated by Sublinear Terms

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.     

Liouville Theorems for Generalized Harmonic Functions

Each nonzero solution of the stationary Schrödinger equation u(x)–c(r)u(x)=0 in R ⁿ with a nonnegative radial potential c(r) must have certain minimal growth at infinity. If r ² c(r)=O(1), r, then a solution having power growth at infinity, is a generalized harmonic polynomial.     

Sets that force recurrence

We characterize those subsets of the positive integers with the property that, whenever a point in a dynamical system enters a compact set along , contains a recurrent point. We do the same for uniform recurrence.

     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

     

Synthesis of a giant 222 carbon graphite sheet

In this paper we present the synthesis and characterization of the so far largest polycyclic aromatic hydrocarbon (PAH), containing 222 carbon atoms or 37 separate benzene units. First a suitable three-dimensional oligophenylene precursor molecule is built up by a sequence of Diels-Alder and cyclotrimerization reactions and then planarized in the final step by oxidative cyclodehydrogenation to the corresponding hexagonal PAH. Structural proof is based on isotopically resolved MALDI-TOF mass spectra and electronic characteristics are studied by UV/Vis spectroscopy.     

Highly sulfurated heterocycles via dithiiranes and trithietanes as key intermediates

2,2,4,4-Tetramethyl-3-thioxocyclobutanone (8b) easily reacts with gaseous chlorine to yield the stable alpha-chloro sulfenyl chloride 10. The same product was obtained when 8b was treated either with phosphorus pentachloride (PCl(5)) or sulfuryl chloride (SO(2)Cl(2)) in CCl(4) solution. Sulfur dichloride (SCl(2)) reacts with 8b to give the alpha-chloro thiosulfenyl chloride 12 along with an almost equimolar amount of the trisulfide 13b. The less reactive disulfur dichloride (S(2)Cl(2)) was shown to react slowly with 8b and the symmetrical tetrasulfide 15 was found as the exclusive product. The pure thiosulfenyl chloride 12 added to adamantanethione (8c) yielded the unsymmetrical trisulfide 13c. When 12 was treated with thioacetic acid, the acetylated trisulfide 17 was formed in high yield. "Unzipping" reactions with the acetylated disulfide 16 and trisulfide 17 with morpholine in THF at -40 degrees C led to the formation of mixtures of two sulfur-rich heterocycles identified as the pentathiepane 6b and the hexathiepane 7b. A mixture of analogous products was obtained when alpha-chloro sulfenyl chloride 10 was treated with sodium sulfide in anhydrous THF at -40 degrees C. The formation of 6b and 7b is believed to occur via the intermediate dithiirane 1b and/or the isomeric thiosulfine 2b. In the case of 17 the reaction starts probably with the formation of a nonisolable tetrathiane 18b as presented in Scheme 5.