The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Simplicial shellable spheres via combinatorial blowups

The construction of the Bier sphere for a simplicial complex is due to Bier (1992). Björner, Paffenholz, Sjöstrand and Ziegler (2005) generalize this construction to obtain a Bier poset from any bounded poset and any proper ideal . They show shellability of for the case , the boolean lattice, and thereby obtain `many shellable spheres' in the sense of Kalai (1988).

We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov (2004). We obtain `more shellable spheres' by proving the general statement that combinatorial blowups, hence stellar subdivisions, preserve shellability.

     

Disklikeness of planar self-affine tiles

We consider the disklikeness of the planar self-affine tile generated by an integral expanding matrix and a consecutive collinear digit set . Let be the characteristic polynomial of . We show that the tile is disklike if and only if . Moreover, is a hexagonal tile for all the cases except when , in which case is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of and a criterion of Bandt and Wang (2001) on disklikeness.

     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

Strict Feasibility of Variational Inequalities in Reflexive Banach Spaces

Strict feasibility is proved to be an equivalent characterization of （dual） variational inequalities having a nonempty bounded solution set, provided the mappings involved are stably properly quasimonotone. This generalizes an earlier result from finite-dimensional Euclidean spaces to infinitedimensional reflexive Banach spaces. Moreover, the monotonicity-type assumptions are also mildly relaxed.     

Biconvex sets and optimization with biconvex functions: a survey and extensions

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG).     

Comparison of radii sets,entropy, QM methods,and sampling on MM‐PBSA,MM‐GBSA,and QM/MM‐GBSA ligand binding energies of F. tularensis enoyl‐ACP reductase (FabI)

To validate a method for predicting the binding affinities of FabI inhibitors, three implicit solvent methods, MM‐PBSA, MM‐GBSA, and QM/MM‐GBSA were carefully compared using 16 benzimidazole inhibitors in complex with Francisella tularensis FabI. The data suggests that the prediction results are sensitive to radii sets, GB methods, QM Hamiltonians, sampling protocols, and simulation length, if only one simulation trajectory is used for each ligand. In this case, QM/MM‐GBSA using 6 ns MD simulation trajectories together with GB^neck2, PM3, and the mbondi2 radii set, generate the closest agreement with experimental values (r² = 0.88). However, if the three implicit solvent methods are averaged from six 1 ns MD simulations for each ligand (called “multiple independent sampling”), the prediction results are relatively insensitive to all the tested parameters. Moreover, MM/GBSA together with GB^HCT and mbondi, using 600 frames extracted evenly from six 0.25 ns MD simulations, can also provide accurate prediction to experimental values (r² = 0.84). Therefore, the multiple independent sampling method can be more efficient than a single, long simulation method. Since future scaffold expansions may significantly change the benzimidazole's physiochemical properties (charges, etc.) and possibly binding modes, which may affect the sensitivities of various parameters, the relatively insensitive “multiple independent sampling method” may avoid the need of an entirely new validation study. Moreover, due to large fluctuating entropy values, (QM/)MM‐P(G)BSA were limited to inhibitors’ relative affinity prediction, but not the absolute affinity. The developed protocol will support an ongoing benzimidazole lead optimization program. © 2015 Wiley Periodicals, Inc.     

Optimal diffuse augmented atomic basis sets for extrapolation of the correlation energy

We seek correlation-consistent diffuse-augmented double-zeta and triple-zeta basis sets that perform optimally in extrapolating the correlation energy to the one-electron complete basis set limit, denoted oAVXZ and oAV(X + d)Z. The novel basis sets are method-dependent in that they are trained to perform optimally for the correlation energy at each specific level of theory. They are shown to yield accurate results in calculating both the energy and tensorial properties such as polarizabilities while not significantly altering the Hartree-Fock energy. Quantitatively, complete basis set limit (CBS)-/(oAVdZ,oAVtZ)-extrapolated correlation energies typically outperform, by 3- to 5-fold, the ones calculated with traditional ansatzes of similar flexibility. Attaining energies of CBS/(AVtZ,AVqZ) type or better accuracy, they frequently outperform expensive raw explicitly correlated ones. Promisingly, a limited test on CBS-extrapolated energies based on conventional basis sets has shown that they compare well even with extrapolated explicitly correlated ones. Calculated atomization and dissociation energies, molecular geometries, ionization potentials, and electron affinities also tend to outperform the ones obtained with traditional Dunning's ansatzes from which the new basis sets have been determined. The method for basis set generation is simple, and there is no reason of principle why the approach could not be adapted for handling other bases in the literature.     

Constructions of Sidon spaces and cyclic subspace codes

In this paper, we firstly construct several new kinds of Sidon spaces and Sidon sets by investigating some known results. Secondly, using these Sidon spaces, we will present a construction of cyclic subspace codes with cardinality τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and minimum distance 2k−2, where τ is a positive integer. We furthermore give some cyclic subspace codes with size 2τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and without changing the minimum distance 2k−2.