Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.     

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β _k (Δ)⩽Σ{β _i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β _i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.) We prove an analog of the UBC for all other even-dimensional homology manifolds. Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices, . We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.     

On the role of a weak hydrogen bond OH…O in the formation of the anomalous dielectric response of crystals and polymers near 40°C

In some crystals, polymers, and gels that contain hydrogen bonds OH…O, NH…O of length 2.8–3 Å or water molecules, gigantic anomaly of dielectric permittivity (? ～ 10³–10⁶) is observed in certain circumstances at frequencies of 1–10⁶ Hz, which is accompanied by peculiarities in conductivity σ and dielectric losses tanδ. In crystals this effect appears after a sudden cooling to ?50°C and it is observed at slow heating in the range of 20–40°C. At the return temperature course from 40°C dependences ?(T), σ(T), and tanδ(T) have their usual values. Anomalies in objects that differ by their compositions are unified by their temperatures, which are all close to 40°C. Authors have made an attempt to explain the similarity of these phenomena by the features of hydrogen bonds that are present in the objects.     

Quantum Hall effect from the topological surface states of strained bulk HgTe

We report transport studies on a three-dimensional, 70-nm-thick HgTe layer, which is strained by epitaxial growth on a CdTe substrate. The strain induces a band gap in the otherwise semimetallic HgTe, which thus becomes a three-dimensional topological insulator. Contributions from residual bulk carriers to the transport properties of the gapped HgTe layer are negligible at mK temperatures. As a result, the sample exhibits a quantized Hall effect that results from the 2D single cone Dirac-like topological surface states.     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Dielectric study of natural inyoite in the frequency range from 10−1 to 107 Hz and the temperature range from −50 to 140°C

For the first time, the method of dielectric dispersion in the 10^?1–10⁷ frequency range is applied to study temperature the dependences of permittivity, conductivity, and dielectric modulus of natural inyoite in the temperature range from ?50 to 140°C. An anomalous increase in the parameters under consideration is observed at temperatures between 87 and 98°C. According to thermal gravimetrical measurements, this range is characterized by an anomaly resulting from a partial loss of crystallization water.     

Spinodal decomposition of off-critical quenches with a viscous phase using dissipative particle dynamics in two and three spatial dimensions

We investigate the domain growth and phase separation of hydrodynamically correct binary immiscible fluids of differing viscosity as a function of minority phase concentration in both two and three spatial dimensions using dissipative particle dynamics. We also examine the behavior of equal-viscosity fluids and compare our results to similar lattice-gas simulations in two dimensions.     

Face numbers and the fundamental group

We resolve a conjecture of Kalai asserting that the g ₂-number of any (finite) simplicial complex Δ that represents a normal pseudomanifold of dimension d ≥ 3 is at least as large as \(\left( {\begin{array}{*{20}{c}} {d + 2} \\ 2 \end{array}} \right)m\left( \Delta \right)\), where m(Δ) denotes the minimum number of generators of the fundamental group of Δ. Furthermore, we prove that a weaker bound, \({h_2}\left( \Delta \right) \geqslant \left( {\begin{array}{*{20}{c}} {d + 1} \\ 2 \end{array}} \right)m\left( \Delta \right)\), applies to any d-dimensional pure simplicial poset Δ all of whose faces of co-dimension ≥ 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Ψ all of whose vertex links satisfy Serre’s condition (S _r ), we establish lower bounds on h ₁(Ψ),...,h _r (Ψ) in terms of the μ-numbers introduced by Bagchi and Datta.     

Comparative estimation of the efficiency of structure formation and dispersion interactions in organic solvents of varying nature and polarity

The solubilities of n-tetracosane and dotriacontane in more than 50 solvents (aliphatic and aromatic hydrocarbons, their halo derivatives, alcohols, ethers, ketones, amines, etc.) miscible with liquid aliphatic hydrocarbons were determined at 293±1 K. The increments of the methylene group in hypothetical extraction systems n-octane — solvent and the free energies of methylene group transfer from solvent to n-octane were calculated. The values were found to vary within wide limits. For the overwhelming majority of polar and low-polar solvents, the energies are negative due to self-association of solvents according to the type of spatial structure. Positive energies are due to the closer packing of some low-polar solvent molecules (carbon tetrachloride and cyclohexane) in the structure compared to octane and to the absence of molecular self-association. The closer packing of these molecules leads to higher efficiency of dispersion interactions between the solvents and paraffins compared to octane.