On the complexity of the Leibniz hierarchy

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.     

Some classes of abstract simplicial complexes motivated by module theory

In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a left-module on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring.Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes.More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families.Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work.     

Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh

In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby bypassing the obstacle of standard finite element discretizations on arbitrary mesh. The velocity and pressure are approximated by the practical significative lowest equal-order virtual element space pair (Xh,Qh) which does not satisfy the discrete inf-sup condition. Combined with the penalty method, the error estimation is proved rigorously. Numerical results on the 3D polygonal mesh illustrate the theoretical results and effectiveness of the proposed method.     

关于凸曲面的几个定义的关系

本文讨论了凸曲面的几种定义及其关系，发现有的定义是局部凸的定义，有的是整体凸的定义，有的则对于局部凸和整体凸都适合，最后给出了各种定义之间互推的证明，对于局部凸和整体凸定义之间不能推证的，则说明了原因．     

Novel Cu–Te Clusters Synthesised from tBuTeSiMe3: [Cu18Te6(TetBu)6(PPh2Et)7], [Cu19Te6(TetBu)7(PEt3)8], [Cu27Te15(PiPr2Me)12] and [Cu58Te32(PtBu2nBu)14]

The reactions of CuCl and tBuTeSiMe₃ in the presence of phosphine ligands result in the formation of four new Cu/Te cluster complexes, [Cu₁₈Te₆(TetBu)₆(PPh₂Et)₇], [Cu₁₉Te₆(TetBu)₇(PEt₃)₈], [Cu₂₇Te₁₅(PiPr₂Me)₁₂] and [Cu₅₈Te₃₂(PtBu₂ nBu)₁₄], which have been structurally characterized by single crystal structural analysis. The former two clusters show a layer-type tellurium frameworks in which the copper atoms are asymmetrically spread. The latter two clusters possess a tellurium framework in a body-centered Te₁₄-Frank-Kasper polyhedron or a Te₂₈ polyhedron with four interstitial tellurium atoms and belong to mixed-valence Cu/Te compounds.     

RANDOM EDGE can be exponential on abstract cubes

We prove that RANDOM EDGE, the simplex algorithm that always chooses a random improving edge to proceed on, can take a mildly exponential number of steps in the model of abstract objective functions (introduced by Williamson Hoke [Completely unimodal numberings of a simple polytope, Discrete Appl. Math. 20 (1988) 69-81.] and by Kalai [A simple way to tell a simple polytope from its graph, J. Combin. Theory Ser. A 49(2) (1988) 381-383.] under different names). We define an abstract objective function on the n-dimensional cube for which the algorithm, started at a random vertex, needs at least exp(const·n^1/3) steps with high probability. The best previous lower bound was quadratic. So in order for RANDOM EDGE to succeed in polynomial time, geometry must help.     

Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations

     

Local minima of lattices and vertices of Klein polyhedra

We prove that each vertex of a Klein polyhedron of a lattice is a local minimum.     

Approximation of local convoluted semigroups

We extend the Trotter-Kato theorem on C₀-semigroups to local convoluted semigroups on dual spaces and apply these results to the general Banach space setting. Compared to known results we obtain weaker convergence assumptions on the resolvent.