Condensation of Eigen Microstate in Statistical Ensemble and Phase Transition

In a statistical ensemble with $M$ microstates, we introduce an $M \times M$ correlation matrix with the correlations between microstates as its elements. Using eigenvectors of the correlation matrix, we can define eigen microstates of the ensemble. The normalized eigenvalue by $M$ represents the weight factor in the ensemble of the corresponding eigen microstate. In the limit $M \to \infty$, weight factors go to zero in the ensemble without localization of microstate. The finite limit of weight factor when $M \to \infty$ indicates a condensation of the corresponding eigen microstate. This indicates a phase transition with new phase characterized by the condensed eigen microstate. We propose a finite-size scaling relation of weight factors near critical point, which can be used to identify the phase transition and its universality class of general complex systems. The condensation of eigen microstate and the finite-size scaling relation of weight factors have been confirmed by the Monte Carlo data of one-dimensional and two-dimensional Ising models.Comment: 9 pages, 16 figures, accepted for publication in Sci. China-Phys. Mech. Astro

Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity

We introduce two new "degree of complementarity" measures: supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with foundations - i.e. level 0 of the hierarchies - resting exactly on submodular and subadditive functions, respectively. We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy: Every monotone set function of supermodular degree d has supermodular width at most d, and there exists a supermodular-width-1 function over a ground set of m elements whose supermodular degree is m-1. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the supermodular degree to the supermodular width. We also establish almost matching information-theoretical lower bounds for these two well-studied fundamental maximization problems over set functions. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides not only a natural notion of complementarity, but also an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We also provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy

First-principles study on the effective masses of zinc-blend-derived Cu_2Zn-IV-VI_4 (IV = Sn, Ge, Si and VI = S, Se)

The electron and hole effective masses of kesterite (KS) and stannite (ST) structured Cu_2Zn-IV-VI_4 (IV = Sn, Ge, Si and VI = S, Se) semiconductors are systematically studied using first-principles calculations. We find that the electron effective masses are almost isotropic, while strong anisotropy is observed for the hole effective mass. The electron effective masses are typically much smaller than the hole effective masses for all studied compounds. The ordering of the topmost three valence bands and the corresponding hole effective masses of the KS and ST structures are different due to the different sign of the crystal-field splitting. The electron and hole effective masses of Se-based compounds are significantly smaller compared to the corresponding S-based compounds. They also decrease as the atomic number of the group IV elements (Si, Ge, Sn) increases, but the decrease is less notable than that caused by the substitution of S by Se.Comment: 14 pages, 6 figures, 2 table

Finite Element Analysis of Composite Frames in Wheelchair under Upward Loading

The finite element analysis is adopted in this primary study. Using the Tsai-Wu criterion and delamination criterion, the stacking sequence [45/04/-454/904]s is the final optimal design for the wheelchair frame. On the contrary, the uni-directional laminates, i.e. [9013]s, [4513]s and [-4513]s, are bad designs due to the higher failure indexes