Referential Doyle-Ericksen formulae for the Eshelby tensor in non-linear elasticity

This article introduces referential Doyle-Ericksen formulae for the various Eshelby tensors in the context of the covariant theory of non-linear elasticity. In light of these formulae, reciprocal equilibrium equations involving stress and energy-momentum are discussed for material bodies embedded in smooth manifolds.     

The tensor product of f-algebras

In this paper we prove that the Fremlin tensor product of two f-algebras can be endowed with an f-algebra structure and satisfies an appropriate universal property. In particular, the Riesz tensor product of C(X) and C(Y), where X and Y are topological spaces, is an f-subalgebra of C(X × Y).     

The stress-energy tensor for biharmonic maps

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map. S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).     

The matrix capacity of a tensor

In 1990, D. Coppersmith and S. Winograd published an estimate of the amount of arithmetic operations necessary for the multiplication of square matrices n?×?n, which equals O(n ^2.3755). In this article, we make a systematization of the theoretical instruments that were used by D. Coppersmith and S. Winograd for their estimate. The improved estimate O(n ^2.373) is one of the results of this systematization.     

The tensor product of function semimodules

Given two domains of functions with values in a field, the canonical map from the algebraic tensor product of the vector spaces of functions on the two domains to the vector space of functions on the product of the two domains is well known to be injective, but not generally surjective. By constructing explicit examples, we show that the corresponding map for semimodules of semiring-valued functions is in general not even injective. This impacts the formulation of topological quantum field theories over semirings. We also confirm the failure of surjectivity for functions with values in complete, additively idempotent semirings by describing a large family of functions that do not lie in the image.     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

The structure of certain rigid tensor categories

We consider rigid tensor categories over a field of characteristic zero in which some exterior power of each object is zero. To cite this article: P. O'Sullivan, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The diffusion equation for the concentration tensor

The purpose of this paper is to show what changes the tensor nature of the concentration introduces in the solution of an equation in comparison with the solution for a scalar quantity. The subsequent analysis of the latter may serve as the basis for experimental determination of the relations between the different components of the concentration tensor (or the chemical potential tensor).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 36–39.     

The Ricci tensor of almost parahermitian manifolds

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi–Civita connection. The formula uses the intrinsic torsion of an underlying \(\mathrm {SL}(n,\mathbb {R})\)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakähler version of the Goldberg conjecture and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics.