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In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

We characterize directs sums of twists of symmetric powers of the universal quotient bundle over the Grassmannian of lines. We use a method that could be used for analogue results on any arbitrary variety, and that should give stronger results than using the standard technique of Beilinson's spectral sequence.

The starting point of this work is that the class of evolution algebras over a fixed field is closed under tensor product. This arises questions about the inheritance of properties from the tensor product to the factors and conversely. For instance nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures so we achieve a complete classification of four-dimensional perfect evolution algebras arising as tensor product of two-dimensional ones. We find that there are 4-dimensional evolution algebras which are the tensor product of two nonevolution algebras., 18 pages, 2 tables

Nowadays one of the main focuses of the Spanish University system is achieving the active learning paradigm in the context of its integration into the European Higher Education Area. This goal is being addressed by means of the application of novel teaching mechanisms. Among a wide variety of learning approaches, the present work focuses on peer review, understood as a collaborative learning technique where students assess other student’s work and provide their own feedback. In this way, peer review has the overarching goal of improving the student learning during this process. Peer review has been successfully applied and analyzed in the literature. Indeed, many authors also recommend improving the design and implementation of self and peer review, which has been our main goal. This paper presents an empirical study based on the application of peer review assessment in different higher education BSc and MSc courses. In this way, six courses from different studies at the University of Malaga in Spain are subject to the application of peer review strategies to promote student learning and develop cross-wise skills such as critical thinking, autonomy and responsibility. Based on these experiences, a deep analysis of the results is performed, showing that a proper application of the peer review methodology provides reliable reviews (with close scores to the ones from the teacher) as well as an improvement in the students’ performance.

In the tensor space $\mathrm{Sym}^d {\mathbb R}^2$ of binary forms we study the best rank $k$ approximation problem. The critical points of the best rank $1$ approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank $k$ approximation problem lie in the same hyperplane., 9 pages

Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f. The critical rank-one tensors for f lie in a linear subspace $$H_f$$ H f , the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space $$H_f$$ H f . This is the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $$H_f$$ H f is spanned by the complex critical rank-one tensors. Since f itself belongs to $$H_f$$ H f , we deduce that also f itself is a linear combination of its critical rank-one tensors.