Deterministic Annealing: A Variant of Simulated Annealing and its Application to Fuzzy Clustering

Deterministic annealing (DA) is a deterministic variant of simulated annealing. In this chapter, after briefly introducing DA, we explain how DA is combined with the fuzzy c-means (FCM) clustering by employing the entropy maximization method, especially the Tsallis entropy maximization. The Tsallis entropy is a q parameter extension of the Shannon entropy. Then, we focus on Tsallis-entropy-maximized FCM (Tsallis-DAFCM), and examine effects of cooling functions for DA on accuracy and convergence. A shape of a membership function of Tsallis-DAFCM depends on both a system temperature and q. Accordingly, a relationship between the temperature and q is quantitatively investigated

Strategyproof matching with regional minimum and maximum quotas

This paper considers matching problems with individual/regional minimum/maximum quotas. Although such quotas are relevant in many real-world settings, there is a lack of strategyproof mechanisms that take such quotas into account. We first show that without any restrictions on the regional structure, checking the existence of a feasible matching that satisfies all quotas is NP-complete. Then, assuming that regions have a hierarchical structure (i.e., a tree), we show that checking the existence of a feasible matching can be done in time linear in the number of regions. We develop two strategyproof matching mechanisms based on the Deferred Acceptance mechanism (DA), which we call Priority List based Deferred Acceptance with Regional minimum and maximum Quotas (PLDA-RQ) and Round-robin Selection Deferred Acceptance with Regional minimum and maximum Quotas (RSDA-RQ). When regional quotas are imposed, a stable matching may no longer exist since fairness and nonwastefulness, which compose stability, are incompatible. We show that both mechanisms are fair. As a result, they are inevitably wasteful. We show that the two mechanisms satisfy different versions of nonwastefulness respectively; each is weaker than the original nonwastefulness. Moreover, we compare our mechanisms with an artificial cap mechanism via simulation experiments, which illustrate that they have a clear advantage in terms of nonwastefulness and student welfare

Deterministic Annealing Approach to Fuzzy C-Means Clustering Based on Entropy Maximization

This paper is dealing with the fuzzy clustering method which combines the deterministic annealing (DA) approach with an entropy, especially the Shannon entropy and the Tsallis entropy. By maximizing the Shannon entropy, the fuzzy entropy, or the Tsallis entropy within the framework of the fuzzy c-means (FCM) method, membership functions similar to the statistical mechanical distribution functions are obtained. We examine characteristics of these entropy-based membership functions from the statistical mechanical point of view. After that, both the Shannon- and Tsallis-entropy-based FCMs are formulated as DA clustering using the very fast annealing (VFA) method as a cooling schedule. Experimental results indicate that the Tsallis-entropy-based FCM is stable with very fast deterministic annealing and suitable for this annealing process

Optimization of electrolysis conditions for ti film electrodeposition from lif-licl eutectic molten salt

PRiME 2020, Honolulu, USA, October 4-9, 2020.The optimum conditions for electrodepositing compact, smooth, and adherent Ti films in LiF–LiCl–Li₃TiF₆ at 823 K were investigated. The Li₃TiF₆ was formed in-situ in the melt via comproportionation reaction between Li₂TiF₆ and Ti powder. The solubility of Li₃TiF₆ was confirmed to be higher than 7.1 mol% by cyclic voltammetry and ICP-AES measurement. Galvanostatic electrolysis was conducted on Ni plate substrates at various concentrations of Li₃TiF₆ (0.55, 2.6, 7.1 mol%) and cathodic current density (50–1200 mA cm⁻²). Ti films with smoother surface were obtained at higher Li₃TiF₆ concentration and lower current density. In the present study, Ti films having the smoothest surface were obtained at 7.1 mol% of Li₃TiF₆ and 50 mA cm⁻²

Effects of Temperature, Ti(III) Ion Concentration, and Current Density on Electrodeposition of Ti Films in LiF-LiCl Melt

The effects of temperature, Ti(III) ion concentration, and current density on the electrodeposition of Ti films were investigated in the eutectic LiF–LiCl melt at 823–973 K. The Ti(III) ions were prepared by adding Li₂TiF₆ and Ti metal to the melt. The diffusion coefficients of Ti(III) were 1.4, 1.8, 2.3, and 3.2 × 10⁻⁵ m² s⁻¹, at 823, 873, 923, and 973 K, respectively. Galvanostatic electrolysis was conducted at 823–973 K. The surface roughness (Sa) of the Ti films decreases with decreasing temperature. Thus, the electrodeposition of Ti films was conducted at the lowest temperature of 823 K with various Li3TiF6 concentrations (0.55–7.1 mol%) and cathodic current densities (50–1200 mA cm⁻²). The Sa was lower at higher Ti(III) ion concentrations and lower current densities. The smoothest Ti films with a Sa of 1.23 μm and a thickness of 10 μm were obtained at a cathodic current density of 50 mA cm⁻² and Li₃TiF₆ concentration of 7.1 mol%