Classification of four qubit states and their stabilisers under SLOCC operations

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group SL(2,C)^4 on the Hilbert space H_4 = (C^2)^{\otimes 4}. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of SL(2,C)^4-orbits on H_4. It follows that an element of H_4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in SL(2,C)^4. We also present a complete and irredundant classification of elements and stabilisers up to the action of the semidirect product Sym_4\ltimes\SL(2,C)^4 where Sym_4 permutes the four tensor factors of H_4.

Reproducibility of mass spectrometry based metabolomics data

Background Assessing the reproducibility of measurements is an important first step for improving the reliability of downstream analyses of high-throughput metabolomics experiments. We define a metabolite to be reproducible when it demonstrates consistency across replicate experiments. Similarly, metabolites which are not consistent across replicates can be labeled as irreproducible. In this work, we introduce and evaluate the use (Ma)ximum (R)ank (R)eproducibility (MaRR) to examine reproducibility in mass spectrometry-based metabolomics experiments. We examine reproducibility across technical or biological samples in three different mass spectrometry metabolomics (MS-Metabolomics) data sets. Results We apply MaRR, a nonparametric approach that detects the change from reproducible to irreproducible signals using a maximal rank statistic. The advantage of using MaRR over model-based methods that it does not make parametric assumptions on the underlying distributions or dependence structures of reproducible metabolites. Using three MS Metabolomics data sets generated in the multi-center Genetic Epidemiology of Chronic Obstructive Pulmonary Disease (COPD) study, we applied the MaRR procedure after data processing to explore reproducibility across technical or biological samples. Under realistic settings of MS-Metabolomics data, the MaRR procedure effectively controls the False Discovery Rate (FDR) when there was a gradual reduction in correlation between replicate pairs for less highly ranked signals. Simulation studies also show that the MaRR procedure tends to have high power for detecting reproducible metabolites in most situations except for smaller values of proportion of reproducible metabolites. Bias (i.e., the difference between the estimated and the true value of reproducible signal proportions) values for simulations are also close to zero. The results reported from the real data show a higher level of reproducibility for technical replicates compared to biological replicates across all the three different datasets. In summary, we demonstrate that the MaRR procedure application can be adapted to various experimental designs, and that the nonparametric approach performs consistently well. Conclusions This research was motivated by reproducibility, which has proven to be a major obstacle in the use of genomic findings to advance clinical practice. In this paper, we developed a data-driven approach to assess the reproducibility of MS-Metabolomics data sets. The methods described in this paper are implemented in the open-source R package marr, which is freely available from Bioconductor at http://bioconductor.org/packages/marr.

An Analytic Condition for the Finite Degree-of-Freedom of Linkages and Its Computational Evaluation

The finite degree of freedom (DOF) of a mechanism is determined by the number of independent loop constraints. In this paper a method is introduced to determine the maximal number of loop closure constraints (which is independent of a specific configuration) of multi-loop linkages and is applied to calculate the finite DOF. It rests on an algebraic condition on the joint screws and the corresponding computational algorithm to determine the maximal rank of the constraint Jacobian in an arbitrary (possibly singular) reference configuration, making use of the analytic condition that minors of certain rank and their higher derivatives vanish. Unlike the Lie group methods for estimation the DOF of so-called exceptional linkages, this method does not rely on partitioning kinematic loops into partial kinematic chains, and it is applicable to multi-loop linkages. The DOF computed with this method is at least as accurate as the DOF computed with the Lie group methods. It gives the correct DOF for any (possibly overconstrained) linkage where the constraint Jacobian has maximal rank in regular configurations. The so determined maximal rank has further significance for classifying linkages as being exceptional or paradoxical, but also for detecting singularities and shaky linkages.

The Hessian Map

In this paper, we study the Hessian map $h_{d,r}$, which associates to any hypersurface of degree $d$ in ${{\mathbb{P}}}^r$ its Hessian hypersurface, and the general properties of this map and prove that $h_{d,1}$ is birational onto its image if $d\geqslant 5.$ We also study in detail the maps $h_{3,1}$, $h_{4,1}$, and $h_{3,2}$ and the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, proving that this restriction is injective as soon as $r\geqslant 2$ and $d\geqslant 3$, which implies that $h_{3,3}$ is birational onto its image. We also prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, as soon as $r\geqslant 2$ and $d\geqslant 3$.