Product (mathematics)

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

Product systems over right-angled Artin semigroups

Abstract: We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid G. We show that the existing notions of product systems fit into our categorical framework, as do the k-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid G. In particular, we obtain necessary and sufficient conditions under which a collection of k 1-graphs form the coordinate graphs of a k-graph.

IncLing: efficient product-line testing using incremental pairwise sampling

Abstract: A software product line comprises a family of software products that share a common set of features. It enables customers to compose software systems from a managed set of features. Testing every product of a product line individually is often infeasible due to the exponential number of possible products in the number of features. Several approaches have been proposed to restrict the number of products to be tested by sampling a subset of products achieving sufficient combinatorial interaction coverage. However, existing sampling algorithms do not scale well to large product lines, as they require a considerable amount of time to generate the samples. Moreover, samples are not available until a sampling algorithm completely terminates. As testing time is usually limited, we propose an incremental approach of product sampling for pairwise interaction testing (called IncLing), which enables developers to generate samples on demand in a step-wise manner. Furthermore, IncLing uses heuristics to efficiently achieve pairwise interaction coverage with a reasonable number of products. We evaluated IncLing by comparing it against existing sampling algorithms using feature models of different sizes. The results of our approach indicate efficiency improvements for product-line testing.

The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

Abstract: Let $\Omega\subset\mathbb{R}^3$ be a bounded weak Lipschitz domain with boundary $\Gamma:=\operatorname{\partial}\Omega$ divided into two weak Lipschitz submanifolds $\Gamma_{\tau}$ and $\Gamma_{ u}$, and let $\varepsilon$ denote an $\mathsf{L}^{\infty}$-matrix field inducing an inner product in $\mathsf{L}^2(\Omega)$. The key result of this paper is the so-called Maxwell compactness property, i.e., the Hilbert space $\bigl\{E\in\mathsf{L}^2(\Omega):\operatorname{rot}E\in\mathsf{L}^2(\Omega),\,\operatorname{div}\varepsilon E\in\mathsf{L}^2(\Omega),\, u\times E|_{\Gamma_{\tau}}=0,\, u\cdot\varepsilon E|_{\Gamma_{ u}}=0\bigr\}$ is compactly embedded into $\mathsf{L}^2(\Omega)$. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions, and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straightfo...

Pure Spinors on Lie groups

Abstract: For any manifold M, the direct sum TM oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces.

Lifting model transformations to product lines

Abstract: Software product lines and model transformations are two techniques used in industry for managing the development of highly complex software. Product line approaches simplify the handling of software variants while model transformations automate software manipulations such as refactoring, optimization, code generation, etc. While these techniques are well understood independently, combining them to get the benefit of both poses a challenge because most model transformations apply to individual models while model-level product lines represent sets of models. In this paper, we address this challenge by providing an approach for automatically ``lifting'' model transformations so that they can be applied to product lines. We illustrate our approach using a case study and evaluate it through a set of experiments.