On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

Integrated Uncertainty/Disturbance Suppression Based on Improved Adaptive Sliding Mode Controller for PMSM Drives

Permanent magnet synchronous motors (PMSMs) have attracted great attention in the field of electric drive system. However, the disturbances caused by parameter mismatching, model uncertainty, external load and torque ripple seriously weaken the control accuracy. The traditional adaptive sliding mode control (ASMC) methodology can address slow-varying uncertainties/disturbances whose frequencies are located at the bandwidth of the filter used to design the adaptive law well; however, it has been barely discussed with respect to the periodic situation. In this paper, we extend the ASMC arrangement to periodic case to suppress the torque ripple by using a series-structure resonant controller. Firstly, a typical SMC is designed to force the tracking error of speed to converge to zero and obtain a certain capacity to disturbance. Then, the improved adaptive law is incorporated to estimate the lumped disturbance and torque ripple. The improved adaptive law is enhanced by embedding the resonant controller, which can obtain a better estimating result for torque ripple with repetitive feature. Finally, simulation and experimental results with PI, SMC and proposed methods are compared to verify the effectiveness of the developed controller.

On the Lyapunov–Perron reducible Markovian Master Equation

We consider an open quantum system in [Formula: see text] governed by quasiperiodic Hamiltonian with rationally independent frequencies and under the assumption of Lyapunov–Perron reducibility of the associated Schrödinger equation. We construct the Markovian Master Equation and the resulting CP-divisible evolution in the weak coupling limit regime, generalizing our previous results from the periodic case. The analysis is conducted with the application of projection operator techniques and concluded with some results regarding stability of solutions and existence of quasiperiodic global steady state.

Global bifurcation of solitary waves for the Whitham equation

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

Derivation of Darcy’s law in randomly perforated domains

We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average distance between the balls is of size $$\varepsilon $$ ε and their average radius is $$\varepsilon ^{\alpha }$$ ε α , $$\alpha \in (1; 3)$$ α ∈ ( 1 ; 3 ) . We prove that, as in the periodic case (Allaire, G., Arch. Rational Mech. Anal. 113(113):261–298, 1991), the solutions converge to the solution of Darcy’s law (or its scalar analogue in the case of Poisson). In the same spirit of (Giunti, A., Höfer, R., Ann. Inst. H. Poincare’- An. Nonl. 36(7):1829–1868, 2019; Giunti, A., Höfer, R., Velàzquez, J.J.L., Comm. PDEs 43(9):1377–1412, 2018), we work under minimal conditions on the integrability of the random radii. These ensure that the problem is well-defined but do not rule out the onset of clusters of holes.

On infinite groups in which all abelian subgroups are locally cyclic

The structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Beyond periodic revivals for linear dispersive PDEs

We study the phenomenon of revivals for the linear Schrödinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast to the case of the linear Schrödinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrödinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast to the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.

Solving patients with rare diseases through programmatic reanalysis of genome-phenome data

Reanalysis of inconclusive exome/genome sequencing data increases the diagnosis yield of patients with rare diseases. However, the cost and efforts required for reanalysis prevent its routine implementation in research and clinical environments. The Solve-RD project aims to reveal the molecular causes underlying undiagnosed rare diseases. One of the goals is to implement innovative approaches to reanalyse the exomes and genomes from thousands of well-studied undiagnosed cases. The raw genomic data is submitted to Solve-RD through the RD-Connect Genome-Phenome Analysis Platform (GPAP) together with standardised phenotypic and pedigree data. We have developed a programmatic workflow to reanalyse genome-phenome data. It uses the RD-Connect GPAP’s Application Programming Interface (API) and relies on the big-data technologies upon which the system is built. We have applied the workflow to prioritise rare known pathogenic variants from 4411 undiagnosed cases. The queries returned an average of 1.45 variants per case, which first were evaluated in bulk by a panel of disease experts and afterwards specifically by the submitter of each case. A total of 120 index cases (21.2% of prioritised cases, 2.7% of all exome/genome-negative samples) have already been solved, with others being under investigation. The implementation of solutions as the one described here provide the technical framework to enable periodic case-level data re-evaluation in clinical settings, as recommended by the American College of Medical Genetics.

The structure of entrance laws for time-inhomogeneous Ornstein–Uhlenbeck processes with Lévy noise in Hilbert spaces

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein–Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalizing corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin’s seminal work “Sufficient statistics and extreme points” in Ann. Probab. 1978, which contains a purely measure theoretic generalization of the classical analytic Krein–Milman and Choquet Theorems. As an application, we obtain an easy uniqueness proof for [Formula: see text]-periodic entrance laws in the general periodic case. A number of further applications to concrete cases are presented.

Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case

It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.