Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted $L^p$ norms for the whole range of physical adiabatic exponents $\gamma\in (1, \infty)$, so that the viscosity approximate solutions satisfy the general $L^p$ compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all $\gamma\in (1, \infty)$. The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all $\gamma\in (1, \infty)$.Comment: 32 page

Quantum state transfer for multi-input linear quantum systems

Effective state transfer is one of the most important problems in quantum information processing. Typically, a quantum information device is composed of many subsystems with multi-input ports. In this paper, we develop a general theory describing the condition for perfect state transfer from the multi-input ports to the internal system components, for general passive linear quantum systems. The key notion used is the zero of the transfer function matrix. Application to entanglement generation and distribution in a quantum network is also discussed.Comment: 6 pages, 3 figures. A preliminary condensed version of this work will appear in Proceedings of the 55th IEEE Conference on Decision and Contro

X-ray diffraction to probe the kinetics of ice recrystallization inhibition

Understanding the nucleation and growth of ice is crucial in fields ranging from infrastructure maintenance, to the environment, and to preserving biologics in the cold chain. Ice binding and antifreeze proteins are potent ice recrystallization inhibitors (IRI), and synthetic materials that mimic this function have emerged, which may find use in biotechnology. To evaluate IRI activity, optical microscopy tools are typically used to monitor ice grain size either by end-point measurements or as a function of time. However, these methods provide 2-dimensional information and image analysis is required to extract the data. Here we explore using wide angle X-ray scattering (WAXS/X-ray powder diffraction (XRD)) to interrogate 100's of ice crystals in 3-dimensions as a function of time. Due to the random organization of the ice crystals in the frozen sample, the number of orientations measured by XRD is proportional to the number of ice crystals, which can be measured as a function of time. This method was used to evaluate the activity for a panel of known IRI active compounds, and shows strong agreement with results obtained from cryo-microscopy, as well as being advantageous in that time-dependent ice growth is easily extracted. Diffraction analysis also confirmed, by comparing the obtained diffraction patterns of both ice binding and non-binding additives, that the observed hexagonal ice diffraction patterns obtained cannot be used to determine which crystal faces are being bound. This method may help in the discovery of new IRI active materials as well as enabling kinetic analysis of ice growth

Flexibility and development of mirroring mechanisms

The empirical support for the SCM is mixed. We review recent results from our own lab and others supporting a central claim of SCM that mirroring occurs at multiple levels of representation. By contrast, the model is silent as to why human infants are capable of showing imitative behaviours mediated by a mirror system. This limitation is a problem with formal models that address neither the neural correlates nor the behavioural evidence directly

Spin Bose-Metal and Valence Bond Solid phases in a spin-1/2 model with ring exchanges on a four-leg triangular ladder

We study a spin-1/2 system with Heisenberg plus ring exchanges on a four-leg triangular ladder using the density matrix renormalization group and Gutzwiller variational wave functions. Near an isotropic lattice regime, for moderate to large ring exchanges we find a spin Bose-metal phase with a spinon Fermi sea consisting of three partially filled bands. Going away from the triangular towards the square lattice regime, we find a staggered dimer phase with dimers in the transverse direction, while for small ring exchanges the system is in a featureless rung phase. We also discuss parent states and a possible phase diagram in two dimensions.Comment: 4 pages, 5 figures, v3 is the print versio