International Journal of Mathematics for Industry

This is an up-to-date introduction to and overview of the Minimum Description Length (MDL) Principle, a theory of inductive inference that can be applied to general problems in statistics, machine learning and pattern recognition. While MDL was originally based on data compression ideas, this introduction can be read without any knowledge thereof. It takes into account all major developments since 2007, the last time an extensive overview was written. These include new methods for model selection and averaging and hypothesis testing, as well as the first completely general definition of MDL estimators. Incorporating these developments, MDL can be seen as a powerful extension of both penalized likelihood and Bayesian approaches, in which penalization functions and prior distributions are replaced by more general luckiness functions, average-case methodology is replaced by a more robust worst-case approach, and in which methods classically viewed as highly distinct, such as AIC versus BIC and cross-validation versus Bayes can, to a large extent, be viewed from a unified perspective.

Grain-oriented silicon steel is mainly used as the core material of transformers, and it is manufactured by applying secondary recrystallization. The driving force of this process is the grain boundary energy, based on the nature of the grain boundary, which is determined by coincidence site lattice (CSL) relations. CSL relations are determined by the arrangement of lattice points in three-dimensional space and have already been shown mathematically by using advanced mathematics. However, their derivation processes are abstract, making them difficult for material engineers to understand. Therefore, in this study, a derivation of CSL relations is attempted in order to enable material engineers to easily understand the derivation. This study contributes to industrial mathematics by helping material engineers understand the essence of the mathematical method in order to use it appropriately. Specifically, a derivation method for coincidence relations is proposed using the hexagonal lattice (in the case of an axial ratio of $\sqrt{8/3}$) as an example that avoids the need for advanced mathematics. This method involves applying the scale rotation of a quaternion, and it is thus named the quaternion-matrix method. The matrix specifying the $\mathrm{\Sigma }N$ coincidence relation of a certain lattice system is expressed by a similarity transformation using the matrix comprising its primitive translation vectors and is given as the following transformation matrix: $H_i=E_i^{-1}{E}_R{E}_i$. Based on the rational number property of the transformation matrix elements, the following formula is derived: $N=N_0^2+2N_1^2+6N_2^2+3N_3^2$, $N_i\in {\mathbb{N}}^{+}$, $N:\mathrm{\Sigma }$ value. Here, ($N_0{N}_1{N}_2{N}_3$) is specified by the integrality (lattice point) and irreducibility (unit cell) among the elements of $H_i$, and the quaternion for the CSL formation is thus derived. Finally, based on the polar form of this quaternion, the coincidence relation can be derived.

In this paper, as data, ellipsoids in a color coordinate called the Commission Internationale de l’Eclairage (CIE)-Lab system are given as data for 19 colors. Each ellipsoid is a region where all points are visually recognized as the same color at the center of the coordinate system. Our aim here is to predict the shape of an ellipsoid whose center is given by a new color. We proposed two prediction methods of positive definite matrices determining ellipsoids.

The first one is a nonparametric method with Gaussian kernel. The prediction is provided as a weighted sum of positive definite matrices corresponding to 19 ellipsoids in the training data. The second one is to use a matrix-valued regression model applied to a logarithm of positive definite matrices where explanatory variables are three elements of color centers. The best result was obtained by the nonparametric methods with three bandwidth parameters. The log normal regression had a weaker performance, but even so the model estimation was easily carried out.

In this paper, we propose a key exchange protocol using multivariate polynomial maps whose security relies on the hardness in finding a solution to a certain system of nonlinear polynomial equations. Under the hardness assumption of solving the system of equations, we prove that our protocol is secure against key recovery attacks by passive attackers if the protocol is established honestly.

Two-dimensional flow is considered in the complex plane. We discuss Blasius’ formula in a perfect fluid through stochastic complex integrals. This formula is also investigated in a viscous fluid. We mention the theorems corresponding to Green’s formulae last.

In many spatio-temporal data, their spatial variations have inherent global and local structures. The spatially continuous dynamic factor model (SCDFM) decomposes the spatio-temporal data into a small number of spatial and temporal variations, where the spatial variations are represented by the factor loading (FL) functions. However, the FL functions estimated by the maximum likelihood or maximum $L_2$ penalized likelihood capture global structures but do not capture local structures. We propose a method for estimating the spatially multiscale FL functions using a sparse penalty. To overcome the problems of existing sparse penalties, we propose the adaptive graph lasso (AGL) penalty. The method with the AGL penalty eliminates redundant basis functions contained in the FL functions, and leads to the FL functions having global and localized structures. We derive the EM algorithm with block coordinate descent that enables us to maximize the AGL penalized log-likelihood stably. Applications to synthetic and real data show that the proposed modeling procedure accurately extract not only the spatially global structures but also spatially local structures, which the $L_2$ penalized estimation do not extract.

Topological indices are numeric quantities that transform chemical structure to real number. Topological indices are used in QSAR/QSPR studies to correlate the bioactivity and physiochemical properties of molecule. In this paper, some newly designed neighborhood degree-based topological indices named as neighborhood Zagreb index ($M_N$), neighborhood version of Forgotten topological index ($F_N$), modified neighborhood version of Forgotten topological index ($F_N^{\ast }$), neighborhood version of second Zagreb index ($M_2^{\ast }$) and neighborhood version of hyper Zagreb index ($H{M}_N$) are obtained for Graphene and line graph of Graphene using subdivision idea. In addition, these indices are compared graphically with respect to their response for Graphene and line graph of subdivision of Graphene.