Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

Cloud for Gaming

Cloud for Gaming refers to the use of cloud computing technologies to build large-scale gaming infrastructures, with the goal of improving scalability and responsiveness, improve the user's experience and enable new business models.Comment: Encyclopedia of Computer Graphics and Games. Newton Lee (Editor). Springer International Publishing, 2015, ISBN 978-3-319-08234-

Electronic density of states derived from thermodynamic critical field curves for underdoped La-Sr-Cu-O

Thermodynamic critical field curves have been measured for $La_{2-x}Sr_{x}CuO_{4+\delta}$ over the full range of carrier concentrations where superconductivity occurs in order to determine changes in the normal state density of states with carrier concentration. There is a substantial window in the $H-T$ plane where the measurements are possible because the samples are both thermodynamically reversible and the temperature is low enough that vortex fluctuations are not important. In this window, the data fit Hao-Clem rather well, so this model is used to determine $H_c$ and $\kappa_c$ for each temperature and carrier concentration. Using N(0) and the ratio of the energy gap to transition temperature, $\Delta (0)/k_BT_c$, as fitting parameters, the $H_c vs T$ curves give $\Delta (0)/k_BT_c \sim 2.0$ over the whole range of $x$. Values of N(0) remain rather constant in the optimum-doped and overdoped regime, but drops quickly toward zero in the underdoped regime.

Complements of hypersurfaces, variation maps and minimal models of arrangements

We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants

Anisotropic photo-induced magnetism of a RbjCok[Fe(CN)6]l⋅nH2ORb_j Co_k [Fe(CN)_6]_l \cdot n H_2 O thin film

A magneto-optically active thin film of Rb$_{j}$Co$_{k}$[Fe(CN)$_{6}$]$_{l}$ $\cdot$$n$H$_{2}$O has been prepared using a sequential assembly method. Upon irradiation with light and at 5 K, the net magnetization of the film increased when the surface of the film was oriented parallel to the external magnetic field of 0.1 T. However, when the surface of the film was perpendicular to the field, the net magnetization \emph{decreased} upon irradiation. The presence of dipolar fields and the low-dimensional nature of the system are used to describe the orientation dependence of the photo-induced magnetization. The ability to increase or decrease the photo-induced magnetization by changing the orientation of the system with respect to the field is a new phenomenon that may be useful in future device applications.Comment: 10 pages, 2 figures, 1 tabl