General Minimum Lower-Order Confounding Designs with Multi-Block Variables

Blocking the inhomogeneous units of experiments into groups is an efficient way to reduce the influence of systematic sources on the estimations of treatment effects. In practice, there are two types of blocking problems. One considers only a single block variable and the other considers multi-block variables. The present paper considers the blocking problem of multi-block variables. Theoretical results and systematical construction methods of optimal blocked 2 n − m designs with N / 4 + 1 ≤ n ≤ 5 N / 16 are developed under the prevalent general minimum lower-order confounding (GMC) criterion, where N = 2 n − m .

Generic differential operators on Siegel modular forms and special polynomials

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

A Block Successive Upper-Bound Minimization Method of Multipliers for Linearly Constrained Convex Optimization

Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications, including signal processing, wireless networking, and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of first-order primal–dual algorithms called the block successive upper-bound minimization method of multipliers (BSUM-M) to solve this family of problems. The BSUM-M updates the primal variable blocks successively by minimizing locally tight upper bounds of the augmented Lagrangian of the original problem, followed by a gradient-type update for the dual variable in closed form. We show that under certain regularity conditions, and when the primal block variables are updated in either a deterministic or a random fashion, the BSUM-M converges to a point in the set of optimal solutions. Moreover, in the absence of linear constraints and under similar conditions as in the previous result, we show that the randomized BSUM-M (which reduces to the randomized block successive upper-bound minimization method) converges at an asymptotically linear rate without relying on strong convexity.

Classification of The NTEV Problems on The Commercial Building

Neutral to Earth Voltage (NTEV) is one of power quality (PQ) problems in the commercial building that need to be resolved. The classification of the NTEV problems is a method to identify the source types of disturbance in alleviating the problems. This paper presents the classification of NTEV source in the commercial building which is known as the harmonic, loose termination, and lightning. The Euclidean, City block, and Chebyshev variables for K-Nearest Neighbor (K-NN) classifying are being utilized in order to identify the best performance for classifying the NTEV problems. Then, S-Transform (ST) is applied as a pre-processing signal to extract the desired features of NTEV problem for classifier input. Furthermore, the performance of K-NN variables is validated by using the confusion matrix and linear regression. The classification results show that all the K-NN variables capable to identify the NTEV problems. While the K-NN results show that the Euclidean and City block variables are well performed rather than the Chebyshev variable. However, the Chebyshev variable is still reliable as the confusion matrix shows minor misclassification. Then, the linear regression outperformed the percentage close to a perfect value which is hundred percent.