Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models

How can large-scale twisted magnetic structures naturally emerge from buoyancy instabilities?

We consider the three-dimensional instability of a layer of horizontal magnetic field in a polytropic atmosphere where, contrary to previous studies, the field lines in the initial state are not unidirectional. We show that if the twist is initially concentrated inside the unstable layer, the modifications of the instability reported by several authors (see e.g. Cattaneo et al. (1990)) are only observed when the calculation is restricted to two dimensions. In three dimensions, the usual interchange instability occurs, in the direction fixed by the field lines at the interface between the layer and the field-free region. We therefore introduce a new configuration: the instability now develops in a weakly magnetised atmosphere where the direction of the field can vary with respect to the direction of the strong unstable field below, the twist being now concentrated at the upper interface. Both linear stability analysis and non-linear direct numerical simulations are used to study this configuration. We show that from the small-scale interchange instability, large-scale twisted coherent magnetic structures are spontaneously formed, with possible implications to the formation of active regions from a deep-seated solar magnetic field

Counting solutions from finite samplings

We formulate the solution counting problem within the framework of inverse Ising problem and use fast belief propagation equations to estimate the entropy whose value provides an estimate on the true one. We test this idea on both diluted models (random 2-SAT and 3-SAT problems) and fully-connected model (binary perceptron), and show that when the constraint density is small, this estimate can be very close to the true value. The information stored by the salamander retina under the natural movie stimuli can also be estimated and our result is consistent with that obtained by Monte Carlo method. Of particular significance is sizes of other metastable states for this real neuronal network are predicted.Comment: 9 pages, 4 figures and 1 table, further discussions adde

Evolution and characteristics of forced shear flows in polytropic atmospheres: Large and small Péclet number regimes

Complex mixing and magnetic field generation occurs within stellar interiors particularly where there is a strong shear flow. To obtain a comprehensive understanding of these processes, it is necessary to study the complex dynamics of shear regions. Due to current observational limitations, it is necessary to investigate the inevitable small-scale dynamics via numerical calculations. Here, we examine direct numerical calculations of a local model of unstable shear flows in a compressible polytropic fluid primarily in a two-dimensional domain, where we focus on determining how key parameters affect the global properties and characteristics of the resulting saturated turbulent phase. We consider the effect of varying both the viscosity and the thermal diffusivity on the non-linear evolution. Moreover, our main focus is to understand the global properties of the saturated phase, in particular estimating for the first time the spread of the shear region from an initially hyperbolic tangent velocity profile. We find that the vertical extent of the mixing region in the saturated regime is generally determined by the initial Richardson number of the system. Further, the characteristic quantities of the turbulence, i.e. typical length-scale and the root-mean-square velocity are found to depend on both the Richardson number, and the thermal diffusivity. Finally, we present our findings of our investigation into saturated flows of a ‘secular’ shear instability in the low Péclet number regime with large Richardson numbers

Shear instabilities in a fully compressible polytropic atmosphere

Shear flows have an important impact on the dynamics in an assortment of different astrophysical objects including accreditation discs and stellar interiors. Investigating shear flow instabilities in a polytropic atmosphere provides a fundamental understanding of the motion in stellar interiors where turbulent motions, mixing processes, as well as magnetic field generation takes place. Here, a linear stability analysis for a fully compressible fluid in a two-dimensional Cartesian geometry is carried out. Our study focuses on determining the critical Richardson number for different Mach numbers and the destabilising effects of high thermal diffusion. We find that there is a deviation of the predicted stability threshold for moderate Mach number flows along with a significant effect on the growth rate of the linear instability for small Peclet numbers. We show that in addition to a Kelvin-Helmholtz instability a Holmboe instability can appear and we discuss the implication of this in stellar interiors

Development of hot drawing process for nitinol tube

In recent years, Nitinol, near-equiatomic nickel-titanium alloys, have found growing applications in medical technology and joining technology, due to their special characteristics such as shape memory, superplasticity and biocompatibility. The production of Nitinol tube cost-effectively remains a technical challenge. In this paper, we describe a hot drawing process for Nitinol tube production. A Nitinol tube blank and a metal core are assembled together. The assembly is hot drawn for several passes to a final diameter. The metal core is then plastically stretched to reduce its diameter and removed from the tube. Hot drawing process has been applied to Ni50.7Ti and Ni47Ti44Nb9 alloys. Nitinol tubes of 13.6 mm outer diameter and 1 mm wall thickness have been successfully produced from a tube blank of 20 mm outer diameter and 3.5 mm thickness

A three-dimensional full Stokes model of the grounding line dynamics: effect of a pinning point beneath the ice shelf

The West Antarctic ice sheet is confined by a large area of ice shelves, fed by inland ice through fast flowing ice streams. The dynamics of the grounding line, which is the line-boundary between grounded ice and the downstream ice shelf, has a major influence on the dynamics of the whole ice sheet. However, most ice sheet models use simplifications of the flow equations, as they do not include all the stress components, and are known to fail in their representation of the grounding line dynamics. Here, we present a 3-D full Stokes model of a marine ice sheet, in which the flow problem is coupled with the evolution of the upper and lower free surfaces, and the position of the grounding line is determined by solving a contact problem between the shelf/sheet lower surface and the bedrock. Simulations are performed using the open-source finite-element code Elmer/Ice within a parallel environment. The model's ability to cope with a curved grounding line and the effect of a pinning point beneath the ice shelf are investigated through prognostic simulations. Starting from a steady state, the sea level is slightly decreased to create a contact point between a seamount and the ice shelf. The model predicts a dramatic decrease of the shelf velocities, leading to an advance of the grounding line until both grounded zones merge together, during which an ice rumple forms above the contact area at the pinning point. Finally, we show that once the contact is created, increasing the sea level to its initial value does not release the pinning point and has no effect on the ice dynamics, indicating a stabilising effect of pinning points

Inverse cascade and symmetry breaking in rapidly-rotating Boussinesq convection

In this paper we present numerical simulations of rapidly-rotating Rayleigh-B\'enard convection in the Boussinesq approximation with stress-free boundary conditions. At moderately low Rossby number and large Rayleigh number, we show that a large-scale depth-invariant flow is formed, reminiscent of the condensate state observed in two-dimensional flows. We show that the large-scale circulation shares many similarities with the so-called vortex, or slow-mode, of forced rotating turbulence. Our investigations show that at a fixed rotation rate the large-scale vortex is only observed for a finite range of Rayleigh numbers, as the quasi-two-dimensional nature of the flow disappears at very high Rayleigh numbers. We observe slow vortex merging events and find a non-local inverse cascade of energy in addition to the regular direct cascade associated with fast small-scale turbulent motions. Finally, we show that cyclonic structures are dominant in the small-scale turbulent flow and this symmetry breaking persists in the large-scale vortex motion

Kinetic study of spiramycin removal from aqueous solution using heterogeneous photocatalysis

International audienceSpiramycin macrolide antibiotic (SPM) can be photocatalytically degraded on TiO2 (anatase variety). The experiments are done in a batch reactor and the effect of some key parameters is investigated under low energy of artificial UV light. The reaction rate is affected by varying TiO2 dose, pH and SPM concentration. Under optimized conditions, a photodegradation efficiency of 98% is achieved and the SPM photodegradation follows pseudo-first order kinetics. The Langmuir–Hinshelwood (L–H) model is successfully used to fit the experimental data, indicating the dependence of the reaction rate on the chemical reaction step. The L–H model led to the determination of both reaction kinetic and adsorption/desorption equilibrium constants. In order to give an overall estimate of the by-products, chemical oxygen demand, total organic carbon, and calculated average oxidation state monitor the photodegradation proces