Hypothesis Testing With Rank Conditions in Phylogenetics

A phylogenetic model of sequence evolution for a set of n taxa is a collection of probability distributions on the 4n possible site patterns that may be observed in their aligned DNA sequences. For a four-taxon model, one can arrange the entries of these probability distributions into three flattening matrices that correspond to the three different unrooted leaf-labeled four-leaf trees, or quartet trees. The flattening matrix corresponding to the tree parameter of the model is known to satisfy certain rank conditions. Methods such as ErikSVD and SVDQuartets take advantage of this observation by applying singular value decomposition to flattening matrices consisting of empirical data. Each possible quartet is assigned an “SVD score” based on how close the flattening is to the set of matrices of the predicted rank. When choosing among possible quartets, the one with the lowest score is inferred to be the phylogeny of the four taxa under consideration. Since an n-leaf phylogenetic tree is determined by its quartets, this approach can be generalized to infer larger phylogenies. In this article, we explore using the SVD score as a test statistic to test whether phylogenetic data were generated by a particular quartet tree. To do so, we use several results to approximate the distribution of the SVD score and to give upper bounds on the p-value of the associated hypothesis tests. We also apply these hypothesis tests to simulated phylogenetic data and discuss the implications for interpreting SVD scores in rank-based inference methods.

Real-Time Multiparameter Identification of a Salient-Pole PMSM Based on Two Steady States

Real-time multiparameter identification has been widely investigated in relation to high-performance control and fault diagnosis of salient-pole permanent magnet synchronous motors (PMSMs). However, it is rank-deficient for simultaneously estimating flux, resistance, and dq-axis inductances based on one steady state under maximum torque per ampere (MTPA) control, which will cause the ill-convergence problem in the results. This paper proposes a new method to solve the rank deficiency problem in the multiparameter identification of salient-pole PMSMs in systems where the motor working conditions do not change frequently. For this type of system, a second steady state is constructed in order to meet the full-rank conditions for multiparameter identification and minimize the torque ripple. Furthermore, in order to reduce the influence of inductance variations, a better shift direction from the first steady state to the second is ensured based on the analysis of the theoretical error. Simulation and experimental results show that the proposed method demonstrates good identification performance.

On the Generalization of Merkin’s Theorem for Circulatory Systems: A Rank Criterion

This paper provides a generalization of the celebrated Merkin theorem. It provides new results on the destabilizing effect of circulatory forces on stable potential systems. Previous results are described and discussed, and the paper uncovers a deeper understanding of the fundamental reason for the destabilization. Instability results in terms of rank conditions that deal with the potential and circulatory matrices that describe the system are obtained, thereby generalizing this remarkable theorem. These new results are compared with those obtained earlier.

Rank-Engel conditions on linear groups

We study linear groups G for which for every g in G there exists a subgroup E of G satisfying some sort of rank condition and such that for every x in G there is a positive integer m such that for all n ≥ m the repeated commutator [x, ng] lies in E. If E can always be chosen to be a Chernikov group (resp. a polycyclic-by-finite group) such G can be completely described (Wehrfritz in Adv Group Theory Appl 7:143–157, 2019; Wehrfritz in Boll Unione Mat Ital, to appear). For more general rank conditions our analyses below are complete for positive characteristics but are less so for characteristic zero.

Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer

In this work a novel chaotic system with a line equilibrium is presented. First, a dynamical analysis on the system is performed, by computing its bifurcation diagram, continuation diagram, phase portraits and Lyapunov exponents. Then, the system is applied to the problem of secure communication. We assume that the transmitted signal is an additional state. For this reason, the nonlinear system is rewritten in a rectangular descriptor form and then an observer is constructed for achieving synchronization and input reconstruction. If we assume some rank conditions (on the nonlinearities and the solvability of a linear matrix inequality (LMI)) on the system matrices then the observer synchronization can be feasible. We evaluate and demonstrate our approach with specific numerical results.