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Register a new userThe unimodality of the Ehrhart $delta$-polynomial of the chain polytope of the zig-zag poset
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We prove the unimodality of the Ehrhart $delta$-polynomial of the chain polytope of the zig-zag poset, which was conjectured by Kirillov. First, based on a result due to Stanley, we show that this polynomial coincides with the $W$-polynomial for the zig-zag poset with some natural labeling. Then, its unimodality immediately follows from a result of Gasharov, which states that the $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ are unimodal.
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We wish to renew the discussion over recent combinatorial structures that are 3-uniform hypergraph expanders, viewing them in a more general perspective, shedding light on a previously unknown relation to the zig-zag product. We do so by introducing a new structure called triplet structure, that maintains the same local environment around each vertex. The structure is expected to yield, in some cases, a bounded family of hypergraph expanders whose 2-dimensional random walk converges. We have applied the results obtained here to several known constructions, obtaining a better expansion rate than previously known. Namely, we did so in the case of Conlons construction and the $S=[1,1,0]$ construction by Chapman, Linal and Peled.
Static magnetic susceptibility chi, ac susceptibility chi_{ac} and specific heat C versus temperature T measurements on polycrystalline samples of In2VO5 and chi and C versus T measurements on the isostructural, nonmagnetic compound In2TiO5 are reported. A Curie-Wiess fit to the chi(T) data above 175 K for In2VO5 indicates ferromagnetic exchange between V^{4+} (S = 1/2) moments. Below 150 K the chi(T) data deviate from the Curie-Weiss behavior but there is no signature of any long range magnetic order down to 1.8 K. There is a cusp at 2.8 K in the zero field cooled (ZFC) chi(T) data measured in a magnetic field of 100 Oe and the ZFC and field cooled (FC) data show a bifurcation below this temperature. The frequency dependence of the chi_{ac}(T) data indicate that below 3 K the system is in a spin-glass state. The difference Delta C between the heat capacity of In2VO5 and In2TiO5 shows a broad anomaly peaked at 130 K. The entropy upto 300 K is more than what is expected for S = 1/2 moments. The anomaly in Delta C and the extra entropy suggests that there may be a structural change below 130 K in In2VO5.
SrTm$_2$O$_4$ has been investigated using heat capacity, magnetic susceptibility, magnetization in pulsed fields, and inelastic neutron scattering measurements. These results show that the system is highly anisotropic, has gapped low-energy dispersing magnetic excitations, and remains in a paramagnetic state down to 2K. Two theoretical crystal field models were used to describe the single-ion properties of SrTm$_2$O$_4$without any optimization procedures; a standard point-charge model and a Density Functional Theory (DFT) based model that uses Wannier functions. The DFT model was found to better describe the system at low energy by predicting a singlet ground state for one Tm site and a doublet for the second Tm site and anisotropy of second site Tm dominating the anisotropy of the system. Additionally, muon spin rotation/relaxation ($mu^+$psr) spectra reveal oscillations, typically a sign of long-range magnetic order. We attribute these observations to lattice distortion induced by muon implantation, causing renormalization of the gap size.
Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990s.
The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.
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