Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves

We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents.

Bernstein polynomials in element-free Galerkin method

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves,can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

Suitability of artificial neural networks for designing LoC circuits

he simulation of complex LoC (Lab-on-a-Chip) devices is a process that requires solving computationally expensive partial differential equations. An interesting alternative uses artificial neural networks for creating computationally feasible models based on MOR techniques. This paper proposes an approach that uses artificial neural networks for designing LoC components considering the artificial neural network topology as an isomorphism of the LoC device topology. The parameters of the trained neural networks are based on equations for modeling microfluidic circuits, analogous to electronic circuits. The neural networks have been trained to behave like AND, OR, Inverter gates. The parameters of the trained neural networks represent the features of LoC devices that behave as the aforementioned gates. This would mean that LoC devices universally compute.

Accurate Parabolic Navier-Stokes solutions of the supersonic flow around and elliptic cone

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and 4

Application of global numerical procedures to the analysis of shells

Examples of global solutions of the shell equations are presented, such as the ones based on the well known Levy series expansion. Also discussed are some natural extensions of the Levy method as well as the inherent limitations of these methods concerning the shell model assumptions, boundary conditions and geometric regularity. Finally, some open additional design questions are noted mainly related to the simultaneous use in analysis of these global techniques and the local methods (like the finite elements) to finding the optimal shell shape, and to determining the reinforcement layout.

Numerical simulation of group combustion of pulverized coal

A mathematical model for the group combustion of pulverized coal particles was developed in a previous work. It includes the Lagrangian description of the dehumidification, devolatilization and char gasification reactions of the coal particles in the homogenized gaseous environment resulting from the three fuels, CO, H2 and volatiles, supplied by the gasification of the particles and their simultaneous group combustion by the gas phase oxidation reactions, which are considered to be very fast. This model is complemented here with an analysis of the particle dynamics, determined principally by the effects of aerodynamic drag and gravity, and its dispersion based on a stochastic model. It is also extended to include two other simpler models for the gasification of the particles: the first one for particles small enough to extinguish the surrounding diffusion flames, and a second one for particles with small ash content when the porous shell of ashes remaining after gasification of the char, non structurally stable, is disrupted. As an example of the applicability of the models, they are used in the numerical simulation of an experiment of a non-swirling pulverized coal jet with a nearly stagnant air at ambient temperature, with an initial region of interaction with a small annular methane flame. Computational algorithms for solving the different stages undergone by a coal particle during its combustion are proposed. For the partial differential equations modeling the gas phase, a second order finite element method combined with a semi-Lagrangian characteristics method are used. The results obtained with the three versions of the model are compared among them and show how the first of the simpler models fits better the experimental results.

A level set approach for the analysis of flow and compaction during resin infusion in composite materials

Fluid flow and fabric compaction during vacuum assisted resin infusion (VARI) of composite materials was simulated using a level set-based approach. Fluid infusion through the fiber preform was modeled using Darcy’s equations for the fluid flow through a porous media. The stress partition between the fluid and the fiber bed was included by means of Terzaghi’s effective stress theory. Tracking the fluid front during infusion was introduced by means of the level set method. The resulting partial differential equations for the fluid infusion and the evolution of flow front were discretized and solved approximately using the finite differences method with a uniform grid discretization of the spatial domain. The model results were validated against uniaxial VARI experiments through an [0]8 E-glass plain woven preform. The physical parameters of the model were also independently measured. The model results (in terms of the fabric thickness, pressure and fluid front evolution during filling) were in good agreement with the numerical simulations, showing the potential of the level set method to simulate resin infusion

On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

A Molecular Network That Produces Spontaneous Oscillations in Excitable Cells of Dictyostelium

A network of interacting proteins has been found that can account for the spontaneous oscillations in adenylyl cyclase activity that are observed in homogenous populations of Dictyostelium cells 4 h after the initiation of development. Previous biochemical assays have shown that when extracellular adenosine 3′,5′-cyclic monophosphate (cAMP) binds to the surface receptor CAR1, adenylyl cyclase and the MAP kinase ERK2 are transiently activated. A rise in the internal concentration of cAMP activates protein kinase A such that it inhibits ERK2 and leads to a loss-of-ligand binding by CAR1. ERK2 phosphorylates the cAMP phosphodiesterase REG A that reduces the internal concentration of cAMP. A secreted phosphodiesterase reduces external cAMP concentrations between pulses. Numerical solutions to a series of nonlinear differential equations describing these activities faithfully account for the observed periodic changes in cAMP. The activity of each of the components is necessary for the network to generate oscillatory behavior; however, the model is robust in that 25-fold changes in the kinetic constants linking the activities have only minor effects on the predicted frequency. Moreover, constant high levels of external cAMP lead to attenuation, whereas a brief pulse of cAMP can advance or delay the phase such that interacting cells become entrained.

cAMP compartmentation is responsible for a local activation of cardiac Ca2+ channels by beta-adrenergic agonists.

The role of cAMP subcellular compartmentation in the progress of beta-adrenergic stimulation of cardiac L-type calcium current (ICa) was investigated by using a method based on the use of whole-cell patch-clamp recording and a double capillary for extracellular microperfusion. Frog ventricular cells were sealed at both ends to two patch-clamp pipettes and positioned approximately halfway between the mouths of two capillaries that were separated by a 5-micron thin wall. ICa could be inhibited in one half or the other by omitting Ca2+ from one solution or the other. Exposing half of the cell to a saturating concentration of isoprenaline (ISO, 1 microM) produced a nonmaximal increase in ICa (347 +/- 70%; n = 4) since a subsequent application of ISO to the other part induced an additional effect of nearly similar amplitude to reach a 673 +/- 130% increase. However, half-cell exposure to forskolin (FSK, 30 microM) induced a maximal stimulation of ICa (561 +/- 55%; n = 4). This effect was not the result of adenylyl cyclase activation due to FSK diffusion in the nonexposed part of the cell. To determine the distant effects of ISO and FSK on ICa, the drugs were applied in a zero-Ca solution. Adding Ca2+ to the drug-containing solutions allowed us to record the local effect of the drugs. Dose-response curves for the local and distant effects of ISO and FSK on ICa were used as an index of cAMP concentration changes near the sarcolemma. We found that ISO induced a 40-fold, but FSK induced only a 4-fold, higher cAMP concentration close to the Ca2+ channels, in the part of the cell exposed to the drugs, than it did in the rest of the cell. cAMP compartmentation was greatly reduced after inhibition of phosphodiesterase activity with 3-isobutyl-methylxanthine, suggesting the colocalization of enzymes involved in the cAMP cascade. We conclude that beta-adrenergic receptors are functionally coupled to nearby Ca2+ channels via local elevations of cAMP.

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.

Mathematical programming model for heat exchanger design through optimization of partial objectives

Mathematical programming can be used for the optimal design of shell-and-tube heat exchangers (STHEs). This paper proposes a mixed integer non-linear programming (MINLP) model for the design of STHEs, following rigorously the standards of the Tubular Exchanger Manufacturers Association (TEMA). Bell–Delaware Method is used for the shell-side calculations. This approach produces a large and non-convex model that cannot be solved to global optimality with the current state of the art solvers. Notwithstanding, it is proposed to perform a sequential optimization approach of partial objective targets through the division of the problem into sets of related equations that are easier to solve. For each one of these problems a heuristic objective function is selected based on the physical behavior of the problem. The global optimal solution of the original problem cannot be ensured even in the case in which each of the sub-problems is solved to global optimality, but at least a very good solution is always guaranteed. Three cases extracted from the literature were studied. The results showed that in all cases the values obtained using the proposed MINLP model containing multiple objective functions improved the values presented in the literature.

Introduction to Coding Theory for Flow Equations of Complex Systems Models

The modeling of complex dynamic systems depends on the solution of a differential equations system. Some problems appear because we do not know the mathematical expressions of the said equations. Enough numerical data of the system variables are known. The authors, think that it is very important to establish a code between the different languages to let them codify and decodify information. Coding permits us to reduce the study of some objects to others. Mathematical expressions are used to model certain variables of the system are complex, so it is convenient to define an alphabet code determining the correspondence between these equations and words in the alphabet. In this paper the authors begin with the introduction to the coding and decoding of complex structural systems modeling.