Hexagonal → cholesteric phase transition of DNA molecules in liquid-crystalline dispersion particles

To assess the character of packing of double-stranded (ds) DNA molecules in liquid-crystalline dispersion particles formed by phase exclusion of DNA molecules from aqueous salt solutions of poly(ethylene glycol), the circular dichroism spectra of these dispersions at different temperatures have been compared. It has been shown for the first time that heating dispersion particles with the hexagonal packing of ds-DNA molecules is accompanied by the hexagonal → cholesteric phase transition. This result can be described using the notion of quasi-nematic layers composed of orientationally ordered adjacent ds-DNA molecules in the structure of dispersion particles; these layers can be packed in two ways dictating their hexagonal or cholesteric spatial structure.     

Bounding volume by systoles of 3-manifolds

We prove a new systolic volume lower bound for non-orientablen-manifolds, involving the stable 1-systole as well as the codimension-1systole with coefficients in ₂. As an application, we provethat Lusternik–Schnirelmann category and systolic categoryagree for non-orientable closed manifolds of dimension 3, extendingour earlier result in the orientable case. Finally, we provethe homotopy invariance of systolic category.     

A non-Abelian analogue of Whitney’s 2-isomorphism theorem

We give a non-Abelian analogue of Whitney’s 2-isomorphism theorem for graphs. Whitney’s theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an Abelian object, we consider a mildly non-Abelian object, the 2-truncation of the group algebra of the fundamental group of the graph considered as a subalgebra of the 2-truncation of the group algebra of the free group on the edges. The analogue of Whitney’s theorem is that this is a complete invariant of 2-edge connected graphs: let G, G′ be 2-edge connected finite graphs; if there is a bijective correspondence between the edges of G and G′ that induces equality on the 2-truncations of the group algebras of the fundamental groups, then G and G′ are isomorphic.     

Volume Contents

Volume Contents

Volume Contents     

Free products of kω-topological groups with normal amalgamation

It is shown in this paper that if A is a closed normal subgroup of k_ω-topological groups G and H, then the free product of G and H with A amalgamated, G1_AH, exists, is Hausdorff and indeed a k_ω-group.