Product (mathematics)

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

Product‐centered processing: Manufacture of chemical‐based consumer products

Abstract: A systematic framework for product-centered processing is useful particularly in developing chemical-based consumer product manufacturing processes. The objective is to provide directions and guidelines toward the development of a process for manufacturing a product with the desired performance in reduced time and effort. The product performance, represented by several quality factors, is related to product ingredients and structural attributes, as well as the process flowsheet and operating conditions. The procedure consists of five steps. First, the product functionality, form, and packaging are defined. Second, relevant quality factors are identified. Third, necessary ingredients are selected and product microstructure is determined. Fourth, the manufacturing process is designed in light of the desired product properties. Limitations on achievable product quality are also identified. Finally, the product and process are evaluated with the help of experimental data. The framework is illustrated using industrial examples, including the production of dry toner, laundry detergent, shampoo, and cosmetic lotion.

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates

Abstract: In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.

Mixable shuffles, quasi-shuffles and Hopf algebras

Abstract: The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota---Baxter algebras.

Coupled cells with internal symmetry: II. Direct products

Abstract: We continue the study of arrays of coupled identical cells that possess both global and internal symmetries, begun in part I. Here we concentrate on the `direct product' case, for which the symmetry group of the system decomposes as the direct product of the internal group and the global group . Again, the main aim is to find general existence conditions for symmetry-breaking steady-state and Hopf bifurcations by reducing the problem to known results for systems with symmetry or separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and -axial subgroups of the direct product and to relate them to axial and -axial subgroups of the two groups and . We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where and . In particular we show that for Hopf bifurcation the case n = 4 modulo 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n.

Macro level product development using design for modularity

Abstract: Modular design is an engineering methodology that will organize and structure a complex product, process or system into a set of distinct sub-systems and components that may be developed independently of each other and then assembled together. Modular design aims at identifying independent and standard units that could be used to create a variety of products. A structured methodology is proposed for identifying components that can be developed in parallel. In the proposed methodology the needs and product functional requirements are first established. The product is then decomposed based on its functional and physical characteristics. Next, a similarity index is introduced to measure the associativity between the basic components. Finally, a clustering technique is used to integrate the basic components into design modules based on their similarity index.