Mathematical Modeling of Product Rating: Sufficiency, Misbehavior and Aggregation Rules

Many web services like eBay, Tripadvisor, Epinions, etc, provide historical product ratings so that users can evaluate the quality of products. Product ratings are important since they affect how well a product will be adopted by the market. The challenge is that we only have {\em "partial information"} on these ratings: Each user provides ratings to only a "{\em small subset of products}". Under this partial information setting, we explore a number of fundamental questions: What is the "{\em minimum number of ratings}" a product needs so one can make a reliable evaluation of its quality? How users' {\em misbehavior} (such as {\em cheating}) in product rating may affect the evaluation result? To answer these questions, we present a formal mathematical model of product evaluation based on partial information. We derive theoretical bounds on the minimum number of ratings needed to produce a reliable indicator of a product's quality. We also extend our model to accommodate users' misbehavior in product rating. We carry out experiments using both synthetic and real-world data (from TripAdvisor, Amazon and eBay) to validate our model, and also show that using the "majority rating rule" to aggregate product ratings, it produces more reliable and robust product evaluation results than the "average rating rule".Comment: 33 page

Quaternion Electromagnetism and the Relation with 2-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary 'i' should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotationComment: Version published in journal Universe (2019

Neutron Stars with Bose-Einstein Condensation of Antikaons as MIT Bags

We investigate the properties of an antikaon in medium, regarding itas a MIT bag. We first construct the MIT bag model for a kaon with$\sigma^*$ and $\phi$ in order to describe the interaction of$s$-quarks in hyperonic matter in the framework of the modifiedquark-meson coupling model. The coupling constant $g'^{B_K}_\sigma$in the density-dependent bag constant $B(\sigma)$ is treated as afree parameter to reproduce the optical potential of a kaon in asymmetric matter and all other couplings are determined by usingSU(6) symmetry and the quark counting rule. With various values ofthe kaon potential, we calculate the effective mass of a kaon inmedium to compare it with that of a point-like kaon. We thencalculate the population of octet baryons, leptons and $K^-$ and theequation of state for neutron star matter. The results show thatkaon condensation in hyperonic matter is sensitive to the $s$-quarkinteraction and also to the way of treating the kaon. The mass andthe radius of a neutron star are obtained by solving theTolmann-Oppenheimer-Volkoff equation.Comment: 14 figure

Radial basis function classifier construction using particle swarm optimisation aided orthogonal forward regression

We develop a particle swarm optimisation (PSO) aided orthogonal forward regression (OFR) approach for constructing radial basis function (RBF) classifiers with tunable nodes. At each stage of the OFR construction process, the centre vector and diagonal covariance matrix of one RBF node is determined efficiently by minimising the leave-one-out (LOO) misclassification rate (MR) using a PSO algorithm. Compared with the state-of-the-art regularisation assisted orthogonal least square algorithm based on the LOO MR for selecting fixednode RBF classifiers, the proposed PSO aided OFR algorithm for constructing tunable-node RBF classifiers offers significant advantages in terms of better generalisation performance and smaller model size as well as imposes lower computational complexity in classifier construction process. Moreover, the proposed algorithm does not have any hyperparameter that requires costly tuning based on cross validation

Electro-optic scanning of light coupled from a corrugated LiNbO3 waveguide

Light diffracted from a grating output coupler in a Ti-diffused LiNbO3 waveguide is scanned electro-optically. Using a coupling length of 2.5 mm in our arrangement we have demonstrated a scanning capability of one resolved spot per 3 V/µm applied field

Sparse kernel density estimation technique based on zero-norm constraint

A sparse kernel density estimator is derived based on the zero-norm constraint, in which the zero-norm of the kernel weights is incorporated to enhance model sparsity. The classical Parzen window estimate is adopted as the desired response for density estimation, and an approximate function of the zero-norm is used for achieving mathemtical tractability and algorithmic efficiency. Under the mild condition of the positive definite design matrix, the kernel weights of the proposed density estimator based on the zero-norm approximation can be obtained using the multiplicative nonnegative quadratic programming algorithm. Using the -optimality based selection algorithm as the preprocessing to select a small significant subset design matrix, the proposed zero-norm based approach offers an effective means for constructing very sparse kernel density estimates with excellent generalisation performance