Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Software Product Lines

Abstract: It’s a fact of life that organizations love to reorganize. Reorganizations have a profound effect on the way product lines are governed. We introduce the concept of the Responsibility, Authority, and Accountability (RAA) network. An RAA network assists in the governance process of product lines for internal information systems, even in the face of massive reorganization. Armour (“Reorg Cycle” [1]) describes the pressures of reorganization to balance the “dimensions of organization” (e.g., geography, customers, product technology); we apply polarity management to balance the dimensions. Armour describes the difficulty of applying hierarchical organization charts—“single dimension management structures”—to the above “multidimensional environments”; we apply lean RAA networks to span organization charts and provide the multidimensional view needed for product lines. Armour observes that network organization approaches “do not have a good track record”; our experience is that lean, resilient RAA networks document the product line’s governance architecture. These governance architecture patterns are applied repeatedly to the strata of products in the product line. We present the governance architect’s RAA to define, monitor, and sustain governance health using these tools: polarity maps, polarity networks, RAA maps, and RAA networks.

Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions

Abstract: In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative $_{t}D_{*}^{\gamma}$, $\gamma\in(0,1)$, and the second-order space derivatives with the fractional Laplacian $-(-\Delta)^{\alpha/2}$, $\alpha\in(1,2]$. Using the matrix transfer technique proposed by Ilic et al. [M. Ilic, F. Liu, I. Turner, and V. Anh, Fract. Calc. Appl. Anal., 9 (2006), pp. 333-349], the TSFDE-2D is transformed into a time fractional differential system as $_{t}D_{*}^{\gamma}\mathbf{u} = -K_\alpha\mathbf{A}^{\alpha/2}\mathbf{u}$, where $\mathbf{A}$ is the approximate matrix representation of $(-\Delta)$. Traditional approximation of $\mathbf{A}^{\alpha/2}$ requires diagonalization of $\mathbf{A}$, which is very time-consuming for large sparse matrices. The novelty of our proposed numerical schemes is that, using either the finite difference method or the Laplace transform to handle the Caputo time fractional derivative, the solution of the TSFDE-2D is written in terms of a matrix function vector product $f(\mathbf{A})\mathbf{b}$ at each time step, where $\mathbf{b}$ is a suitably defined vector. Depending on the method used to generate the matrix $\mathbf{A}$, the product $f(\mathbf{A})\mathbf{b}$ can be approximated using either the preconditioned Lanczos method when $\mathbf{A}$ is symmetric or the $\mathbf{M}$-Lanzcos method when $\mathbf{A}$ is nonsymmetric, which are powerful techniques for solving large linear systems. We give error bounds for the new methods and illustrate their roles in solving the TSFDE-2D. We also derive the analytical solution of the TSFDE-2D in terms of the Mittag-Leffler function. Finally, numerical results are presented to verify the proposed numerical solution strategies.

Method of and system for ordering products

Abstract: In ordering products, a customer selects a product from a product list stored in a remote computer system via a communication line. A sensor detects the customer size information and transmits it to the remote computer, thus enabling the remote computer to select a product having an appropriate size. Subsequently, on a monitor display, product image is superimposed on the customer body image. The fit evaluation is automatically performed by the local computer system. Depending on the fit of the product, the customer tries different products until placing an order.