Complex structures and the Elie Cartan approach to the theory of spinors

Each isometric complex structure on a 2$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( fermionic) degrees of freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors in the one of C. Chevalley are the associated vacua. The corresponding states are the Fock states (i.e. pure free states), therefore, none of the above terminologies is very good.Comment: 10

Serre Theorem for involutory Hopf algebras

We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$, $D \in {\mathcal C}$ such that C\ot D is semisimple, $C$ and $D$ are semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two $H$-(co)modules such that $M \otimes N$ is (co)semisimple as a $H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective $k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$) is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over $H$ the dimension of which is invertible in $k$ are Serre categories.Comment: a new version: 8 page

The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation

Representations of hom-Lie algebras

In this paper, we study representations of hom-Lie algebras. In particular, the adjoint representation and the trivial representation of hom-Lie algebras are studied in detail. Derivations, deformations, central extensions and derivation extensions of hom-Lie algebras are also studied as an application.Comment: 16 pages, multiplicative and regular hom-Lie algebras are used, Algebra and Representation Theory, 15 (6) (2012), 1081-109

Pubertal timing and body mass index gain from birth to maturity in relation with femoral neck BMD and distal tibia microstructure in healthy female subjects

Summary: Childhood body mass index (BMI) gain is linked to hip fracture risk in elderly. In healthy girls, menarcheal age is inversely related to BMI gain during childhood and to femoral neck areal bone mass density (aBMD) and distal tibia structural components at maturity. This study underscores the importance of pubertal timing in age-related fragility fracture risk. Introduction: Recent data point to a relationship between BMI change during childhood and hip fracture risk in later life. We hypothesized that BMI development is linked to variation in pubertal timing as assessed by menarcheal age (MENA) which in turn, is related to peak bone mass (PBM) and hip fracture risk in elderly. Methods: We studied in a 124 healthy female cohort the relationship between MENA and BMI from birth to maturity, and DXA-measured femoral neck (FN) aBMD at 20.4year. At this age, we also measured bone strength related microstructure components of distal tibia by HR-pQCT. Results: At 20.4 ± 0.6year, FN aBMD (mg/cm2), cortical thickness (μm), and trabecular density (mgHA/cm3) of distal tibia were inversely related to MENA (P = 0.023, 0.015, and 0.041, respectively) and positively to BMI changes from 1.0 to 12.4years (P = 0.031, 0.089, 0.016, respectively). Significant inverse (P < 0.022 to <0.001) correlations (R = −0.21 to -0.42) were found between MENA and BMI from 7.9 to 20.4years, but neither at birth nor at 1.0year. Linear regression indicated that MENA Z-score was inversely related to BMI changes not only from 1.0 to 12.4years (R = −0.35, P = 0.001), but also from 1.0 to 8.9years, (R = −0.24, P = 0.017), i.e., before pubertal maturation. Conclusion: BMI gain during childhood is associated with pubertal timing, which in turn, is correlated with several bone traits measured at PBM including FN aBMD, cortical thickness, and volumetric trabecular density of distal tibia. These data complement the reported relationship between childhood BMI gain and hip fracture risk in later lif

Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r) that one usually associates with such a group and hence with a simply-laced Coxeter-Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.Comment: 19 page

The family of quaternionic quasi-unitary Lie algebras and their central extensions

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley--Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.Comment: 17 pages, LaTe

Mutation testing on an object-oriented framework: An experience report

This is the preprint version of the article - Copyright @ 2011 ElsevierContext The increasing presence of Object-Oriented (OO) programs in industrial systems is progressively drawing the attention of mutation researchers toward this paradigm. However, while the number of research contributions in this topic is plentiful, the number of empirical results is still marginal and mostly provided by researchers rather than practitioners. Objective This article reports our experience using mutation testing to measure the effectiveness of an automated test data generator from a user perspective. Method In our study, we applied both traditional and class-level mutation operators to FaMa, an open source Java framework currently being used for research and commercial purposes. We also compared and contrasted our results with the data obtained from some motivating faults found in the literature and two real tools for the analysis of feature models, FaMa and SPLOT. Results Our results are summarized in a number of lessons learned supporting previous isolated results as well as new findings that hopefully will motivate further research in the field. Conclusion We conclude that mutation testing is an effective and affordable technique to measure the effectiveness of test mechanisms in OO systems. We found, however, several practical limitations in current tool support that should be addressed to facilitate the work of testers. We also missed specific techniques and tools to apply mutation testing at the system level.This work has been partially supported by the European Commission (FEDER) and Spanish Government under CICYT Project SETI (TIN2009-07366) and the Andalusian Government Projects ISABEL (TIC-2533) and THEOS (TIC-5906)

A planar extrapolation of the correlation problem that permits pairing

It was observed previously that an SU(N) extension of the Hubbard model is dominated, at large N, by planar diagrams in the sense of 't Hooft, but the possibility of superconducting pairing got lost in this extrapolation. To allow for this possibility, we replace SU(N) by U(N,q), the unitary group in a vector space of quaternions. At the level of the free energy, the difference between the SU(N)and U(N,q) extrapolations appears only to first nonleading order in N.Comment: 8 pages, 2 figure

A spacetime not characterised by its invariants is of aligned type II

By using invariant theory we show that a (higher-dimensional) Lorentzian metric that is not characterised by its invariants must be of aligned type II; i.e., there exists a frame such that all the curvature tensors are simultaneously of type II. This implies, using the boost-weight decomposition, that for such a metric there exists a frame such that all positive boost-weight components are zero. Indeed, we show a more general result, namely that any set of tensors which is not characterised by its invariants, must be of aligned type II. This result enables us to prove a number of related results, among them the algebraic VSI conjecture.Comment: 14pages, CQG to appea