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The main theme is to discuss the process of coefficient extension for modules, and its reverse, descent. For example, consider a module M over a ring R and let R⟶R′ be an extension of rings. Then extending coefficients from R to R′ on M means that one passes from M to the tensor product M⊗ R R′, viewed as an R′-module. The chapter starts with the construction of tensor products of general type for modules and their morphisms. Special attention is payed to so-called flat and even faithfully flat extensions. These are particularly well adapted to the formation of kernels, cokernels, and images of module morphisms under the process of coefficient extension. The next step is to look at module properties that are maintained under coefficient extension, or descent. For this to work well, the extension R⟶R′ is assumed to be flat, and even faithfully flat in the descent case. Finally, Grothendieck’s descent theory is studied for modules and their morphisms.

Different first order representations of high order linear PDEs, depending on which notion of the degree of a polynomial is chosen, are possible. Two notions of degree, namely, the total degree and multi-degree, are standard and the most natural. In our recent article, a straightforward way was provided to convert a system of high order constant coefficient linear PDEs into a system of first order PDEs based on the notion of total degree. In the present article, an analogous development is carried out in the multi-degree setting.

Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensor-product structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order O ( p 9 ) to O ( p 7 ) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same O ( p 7 ) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected O ( p 7 ) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements.

By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as L ∞ -modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if 0 → Y → X → Z → 0 is an exact sequence of L ∞ -modules and homomorphisms, where Y and Z are function spaces, and V is another function space, then the tensorized sequence 0 → Y ⊗ V → X ⊗ V → Z ⊗ V → 0 is exact too.

The tensor product of groups is one of the most important concepts and indispensable tools in the theory of abelian groups. They compete in importance with homomorphism groups, but their features are totally different.