Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.     

SUPG and grad‐div stabilized finite element methods for steady weakly compressible viscous flow

The extensive application of mathematical and computational methods has become an efficient and powerful approach to the investigation and solution of many problems and processes in fluid dynamics from qualitative as well as quantitative point of view. In this work a new class of advanced numerical approximation schemes to isothermal compressible viscous flow is presented. The schemes are based on an iteration between an Oseen like problem for the velocity and a hyperbolic transport equation for the density. Such schemes seem attractive for computations because they offer a reduction to simpler problems for which highly refined numerical methods either are known or can be built from existing approximation schemes to similar equations, and because of the guidance that can be drawn from an existence theory based on them. For the generalized Oseen subproblem a Taylor–Hood finite element method is proposed that is stabilized by a reduced SUPG and grad‐div technique (cf. [1, 4]) in the convection‐dominated case. Results of theoretical investigations and numerical studies are presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behaviour of a closed‐loop thermosyphon with linear friction and viscoelastic binary fluid

Viscoelastic fluids represent a major challenge both from an engineering and from a mathematical point of view. Recently, we have shown that viscoelasticity induces chaos in closed‐loop thermosyphons even when we consider binary fluids, this is, when we consider a solute in the fluid, as water and antifreezes, for example. In this work, we consider a linear friction law, and we show that in this case with the addition of a solute to the fluid we can prove, under some conditions, chaotic asymptotic behavior for suitable geometry of the circuit and heat flux or ambient temperature functions.     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

The role of magnetic fields in the strong stabilization of a hybrid magneto‐elastic structure

In this paper, we are concerned with a model for the magneto–elastic interactions of a three‐dimensional elastic body and a two‐dimensional flexible plate, which is attached to the flat flexible part of the surface of the body. Both the solid body and the plate are permeated by magnetic fields. The mathematical model is analyzed from the point of view of existence and uniqueness and stabilization.It turns out that, in the presence of the magnetic fields in the solid and the plate, strong stabilization can be achieved under viscous damping in the plate in one direction that is determined by the nature of the primary magnetic fields in the body and the plate. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamic analysis of a fractional order prey–predator interaction with harvesting

In recent years, prey–predator models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance. In this paper, we introduce a fractional-order prey–predator model and deals with the mathematical behaviors of the model. The dynamical behavior of the system is investigated from the point of view of local stability. We also carry out a detailed analysis on the stability of equilibrium. Numerical simulations are presented to illustrate the results.     

Mathematical Analysis of Inertial Waves in Rectangular Basins with One Sloping Boundary

We consider the problem of small oscillations of a rotating inviscid incompressible fluid. From a mathematical point of view, new exact solutions to the two‐dimensional Poincaré–Sobolev equation in a class of domains including trapezoid are found in an explicit form and their main properties are described. These solutions correspond to the absolutely continuous spectrum of a linear operator that is associated with this system of equations. For specialists in Astrophysics and Geophysics, the existence of these solutions signifies the existence of some previously unknown type of inertial waves corresponding to the continuous spectrum of inertial oscillations. A fundamental distinction between monochromatic inertial waves and waves of the new type is shown: usual characteristics (frequency, amplitude, wave vector, dispersion relation, direction of energy propagation, and so on) are not applicable to the last. Main properties of these waves are described. In particular, it is proved that they are progressive. Main features of their energy transfer are described.     

A theoretical and numerical approach to some free boundary problems

Summary In this paper some free boundary problems related to the flow of fluids in porous media are studied. Using a method due to Baiocchi, for these problems we not only establish theoretical results (existence and uniqueness theorems for the solution) but at the same time develop an algorithm for the numerical approach of the solution. Such an algorithm is rigorous from a mathematical point of view and it competes very well with the ones already known both in simplicity of programming and in speed of execution. Entrata in Redazione il 18 luglio 1973. ? Laboratorio di Analisi Numerica del C.N.R. di Pavia ? and ? Università di Pavia ?. This work was supported by C.N.R. in the frame of L.A.N. at Pavia.     

An approach to benchmarking of loosely coupled low-cost navigation systems

New solutions to the navigation problem related to low-cost integrated navigation systems (INS) are often published. Since these new solutions are generally compared with ad hoc mathematical models that are not fully exposed, one cannot be sure of the relative improvements. In this work, complete mathematical model for a low-cost INS is suggested to be used as a benchmarking. As far as the authors’ knowledge, a benchmarking for low-cost INS has not been previously reported. Shown INS comprises a strapdown inertial navigation system, loosely coupled to a GPS receiver. The INS mathematical model is based upon classical navigation equations and classical sensor models, both from recognized authors. The algorithm that details the INS operation is also presented. The benchmarking is provided as an open-source toolbox for MATLAB. Additionally, this work can be taken as a starting point for new practitioners in the INS field. To validate the INS mathematical model, real-world data sets from three different Micro Electro-Mechanical Systems (MEMS) inertial measurement units (IMU) and a GPS receiver are processed. It is observed that obtained RMS errors from the three INS are coherent with the quality of corresponding MEMS IMU. This confirms that the proposed benchmarking is a suitable tool to evaluate objectively new solutions to low-cost INS.     

Mathematical Analysis of the Jin-Neelin Model of El Niño-Southern-Oscillation

The Jin-Neelin model for the El Ni$\wt{\rm n}$o--Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics. From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length $k$ and on each sub-interval to solve successively the oceanic component, using the temperature $T$ calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as $k$ tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.     

Collision of Almost Parallel Vortex Filaments

We investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda, and Damodaran and Zakharov . We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on one of the self‐similar solutions of the model, described by the authors in an earlier work. In the second part of this paper we extend this construction to the case of an arbitrary number of filaments, with polygonal symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity.© 2016 Wiley Periodicals, Inc.     

Stabilization criteria of a class of switched systems

In this paper, stabilizability property for a switched system under arbitrary switching is considered from an algebraic point of view by means of the existence of a set of block‐diagonal Lyapunov solutions with common Schur complement of certain order—or, equivalently, with common block (1,1)—for the matrix bank. It is shown that the existence of that set is equivalent to the existence of solutions for some Riccati inequalities done in terms of the blocks of matrices of the bank. In addition, we conclude that a particular class of systems with matrix bank constituted by Metzler matrices—Positive Switched Systems—are stabilizable by partial state reset.     

Fixed Point Theorems for (p,q)-Quasi-Contraction Mappings in Cone Metric Spaces

In this work, the authors introduce the concept of $(p,q)$-quasi-contraction mapping in a cone metric space. We prove the existence and uniqueness of a fixed point for a $(p,q)$-quasi-contraction mapping in a complete cone metric space. The results of this paper generalize and unify further fixed point theorems for quasi-contraction, convex contraction mappings and two-sided convex contraction of order $2$.     

Constraint propagation,relational arithmetic in AI systems and mathematical programs

This paper explores the interrelationships between methods developed in mathematical programming to discover the structure of constraint (feasibility) sets and constraint propagation over networks used by some AI systems to perform inferences about quantities. It is shown that some constraint set problems in mathematical programming are equivalent to inferencing problems for constraint networks with interval labels. This makes the inference and query capabilities associated with AI systems that use logic programming, directly accessible to mathematical programming systems. On the other hand, traditional and newer methods which mathematical programming uses to obtain information about its associated feasibility set can be used to determine the propagation of constraints in a network of nodes of an AI system. When viewed from this point of view, AI problems can access additional mathematical programming analytical tools including new ways to incorporate qualitative data into constraint sets via interval and fuzzy arithmetic.This work was partially supported by the Industrial Consortium to Develop an Intelligent Mathematical Programming System — Amoco Oil Company, General Research Corporation, Ketron Management Science, Shell Oil Company, MathPro, and US West Advanced Technologies.     

Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor

In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.     

Stability analysis of quaternion‐valued Cohen‐Grossberg neural networks

In this paper, by starting from basic quaternion algebra properties and algorithms, a quaternion‐valued Cohen‐Grossberg neural network was derived, subsequently, several new sufficient conditions are derived to ensure existence and global asymptotic stability (GAS) and global exponential stability (GES) of the equilibrium point (EP) for quaternion‐valued Cohen‐Grossberg neural networks. The obtained criteria can be checked easily in practice and have a distinguished feature from previous studies. Finally, we have numerical evidences that the mathematical system and the conclusions presented are validated.     

An upwind center difference parallel method and numerical analysis for the displacement problem with moving boundary

A nonlinear coupled mathematical system of two‐phase seepage flow displacement is discussed in this paper including an elliptic equation for the pressure and a convection‐dominated diffusion equation for the saturation. In fact, the boundary of an underground region where the fluid flows through is nonstationary. So a moving boundary should be considered. The saturation equation is convection‐dominated, therefore the method of upwind finite difference is introduced for the accurate computation. The upwind approximation could eliminate numerical oscillation and strong stability is shown. Since the computational work of saturation is larger than the pressure, the authors apply a parallel method, decomposing the whole domain into several nonoverlapping subdomains, to simplify the computation. A domain decomposition method coupled with upwind differences is presented for the saturation. The pressure equation is discretized by a five‐point center finite difference method. By using a transformation and defining new inner products and norms, error estimates in l² norm is discussed. Finally, two experimental tests are given to illustrate the efficiency and accuracy of the parallel algorithm.     

On a nonlinear flux-limited equation arising in the transport of morphogens

Motivated by a mathematical model for the transport of morphogens in biological systems, we study existence and uniqueness of entropy solutions for a mixed initial–boundary value problem associated with a nonlinear flux-limited diffusion system. From a mathematical point of view the problem behaves more as a hyperbolic system than a parabolic one.