Product (mathematics)

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

Integration analysis of product decompositions

Abstract: This paper describes a methodology for the analysis of product design decompositions The technique is useful for developing an understanding of the "system engineering" needs which arise because of complex interactions between components of a design This information can be used to define the product architecture and to organize the development teams The method involves three steps: 1) decomposition of the system into elements, 2) documentation of the interactions between the elements, and 3) clustering the elements into architectural and team chunks By using this approach, development teams can better understand the complex interactions within the system, thus simplifying the development process for large and complex projects arranged in chunks The choice of product architecture has broad implications for product performance, product change, product variety, and manufacturability Product architecture is also strongly coupled to the firm's development capability, manufacturing specialties, and product strategy Selecting the proper architecture of the product is an extremely influential decision which must be made during the concept development and system-level design phases of the project; the architecture defines the sub-systems upon which the team will work for the bulk of the development effort In product development, analysis of the product decomposition provides valuable insight into the structure of the problem and the choice of architecture The integration analysis presented in this paper considers the interactions which occur between the elements of the decomposition The building blocks (called chunks) which result from integration analysis can be used to define the product architecture and to structure the development teams Examples of architecture and team structure can be found in any highly engineered product In the automobile industry, development programs include hundreds or thousands of team members It would be impractical to design the entire vehicle at once (too complex); nor would it be possible to develop the thousands of components one at a time (too slow) The vehicle is decomposed into a few major systems: body, powertrain, chassis, interior, climate control, electrical, and trim Each of these major systems is in turn decomposed into a large number of sub-systems, resulting in hundreds of interconnected pieces with names like: passenger restraint system, fuel delivery system, remote entry system, etc Finally, these sub-systems are decomposed into component parts which are designed and tested individually and together The decomposition of the vehicle into sub-systems and components facilitates the rapid development of the individual pieces, yet this strategy does not address the needs for integration of the components' functions during the development process

Pseudo-Hermitian Representation of Quantum Mechanics

Abstract: A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...

Generative Image Modeling Using Style and Structure Adversarial Networks

Abstract: Current generative frameworks use end-to-end learning and generate images by sampling from uniform noise distribution. However, these approaches ignore the most basic principle of image formation: images are product of: (a) Structure: the underlying 3D model; (b) Style: the texture mapped onto structure. In this paper, we factorize the image generation process and propose Style and Structure Generative Adversarial Network (\({\text {S}^2}\)-GAN). Our \({\text {S}^2}\)-GAN has two components: the Structure-GAN generates a surface normal map; the Style-GAN takes the surface normal map as input and generates the 2D image. Apart from a real vs. generated loss function, we use an additional loss with computed surface normals from generated images. The two GANs are first trained independently, and then merged together via joint learning. We show our \({\text {S}^2}\)-GAN model is interpretable, generates more realistic images and can be used to learn unsupervised RGBD representations.

Inner Product Modules Over B ∗ -Algebras

Abstract: This paper is an investigation of right modules over a B*algebra B which posses a B-valued "inner product" respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a W*-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and W*-algebras. The extension of an inner product module over B by a B*-algebra A containing B as a *-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.