Mappings of Bounded Φ-Variation with Arbitrary Function Φ

We develop the general theory of mappings of bounded -variation in the sense of L. C. Young that are defined on a subset of the real line and take values in metric or normed spaces. We single out the characterizing properties for these mappings, prove the structural theorem for them, and study their continuity properties. We obtain the existence of a geodesic path of bounded -variation between two points of a compact set with certain regularity of its modulus of continuity. The classical Helly selection principle from the theory of functions of bounded variation is generalized for mappings of bounded -variation. Under natural restrictions on the function , we show that the space of all normed space-valued mappings under consideration can be endowed with a metric. Finally, we consider the problem of existence of selections of a continuous set-valued mapping Fof bounded -variation with respect to the Hausdorff distance. We show that if (0) is finite> 0, then Fhas a continuous selection of bounded -variation; if (0) = , then Fis a constant mapping; and if (0) = 0, then, under additional assumptions on , we give examples of mappings Fwith no continuous selection and with no selection of bounded -variation.     

Spatial Behavior in Linear Theory of Thermoviscoelasticity with Voids

In the present article we study the spatial behavior of the solutions to the initial boundary value problem associated with the linear theory of thermoviscoelastic materials with voids. We prove a set of properties for an appropriate time-weighted surface power function, which allows us to obtain an idea of the domain of influence in linear thermoviscoelasticity with voids. Some spatial estimates of the Saint–Venant type, for bounded bodies, and Phragmén–Lindelöf type, for unbounded bodies, are obtained. Such estimates are characterized by time-dependent as well as time-independent decay and growth rates.     

Mappings with Three Homeomorphism Intervals That Change the Orientation of One of Them

In this paper, we consider one-to-one piecewise-homeomorphic mappings of a circle that have three homeomorphism intervals and change the orientation of one of the intervals. Moreover, it is assumed that the mappings themselves as well as their extensions to endpoints have no periodic orbits (such mappings arise in a natural way in the study of flows without periodic trajectories and separatrix connections on nonorientable surfaces). We obtain a classification (from the conjugation viewpoint) of such mappings. A key result consists in the proof of existence of a wandering interval. Also, we prove that any mapping under consideration has exactly one invariant normalized measure.     

A Complete Classification for Pairs of Real Analytic Curves in the Complex Plane with Tangential Intersection

We study the group structure of centralizers of a holomorphic mapping that is tangent to the identity. We give a complete classification for pairs of real analytic curves intersecting tangentially via the Ecalle—Voronin theory. We also classify pairs of one-dimensional holomorphic involutions that have a high-order tangency, real analytic mappings on the real line, and exceptional non-Abelian subgroups of Cerveau and Moussu. Consequently, we prove the existence of holomorphic mappings z z + O(2) that are reversible by a formal antiholomorphic mapping, but not by any (convergent) antiholomorphic one. We also prove the existence of local real analytic mappings on the real line that are not equivalent with respect to any local real analytic diffeomorphism and yet whose complexifications are holomorphically equivalent.2000 Mathematics Subject Classification. 30C35, 32B10.Research of the second author was partially supported by the NSF grant DMS-0305474.     

A note on stability in three-phase-lag heat conduction

In this note we consider two cases in the theory of the heat conduction models with three-phase-lag. For each one we propose a suitable Lyapunov function. These functions are relevant tools which allow to study several qualitative properties. We obtain conditions on the material parameters to guarantee the exponential stability of solutions. The spectral analysis complements the results and we show that if the conditions obtained to prove the exponential stability are not satisfied, then we can obtain the instability of solutions for suitable domains. We believe that this kind of results is fundamental to clarify the applicability of the models.     

An hour-ahead prediction model for heavy-tailed spot prices

We propose an hour-ahead prediction model for electricity prices that capture the heavy tailed behavior that we observe in the hourly spot market in the Ercot (Texas) and the PJM West hub grids. We present a model according to which we separate the price process into a thin-tailed trailing median process and a heavy-tailed residual process whose probability distribution can be approximated by a Cauchy distribution. We show empirical evidence that supports our model.     

Bounded Nonwandering Sets for Polynomial Mappings

In this paper, we consider a class of polynomial mappings on R ^m or C ^m which is defined by the assumption that the delay equations induced by the mappings have leading monomials in a single variable. We show that for any mapping from this class, the nonwandering set is bounded while for all unbounded orbits, some kind of monotonicity takes place. The class under consideration is proved to contain, in particular, the generalized Hénon mappings and the Arneodo–Coullet–Tresser mappings.     

A note on Carnot geodesics in nilpotent Lie groups

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct a group, with a left invariant bracket-generating distribution, for which some Carnot geodecics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, which we derive (for any Lie group) explicitly in terms of the structure constants.The first author was partially supported by an NSF Postdoctoral Fellowship, and the second author was partially supported by a Rothschild Postdoctoral Fellowship.     

Qualitative Aspects in the Coupled Theory of Thermoelastic Diffusion

In this paper we derive some qualitative results of the coupled theory of thermoelastic diffusion for anisotropic media. We establish a reciprocity relation, which involves two thermoelastic diffusion processes at different instants. We show that this relation can be used to obtain reciprocity, uniqueness and continuous dependence theorems. The reciprocity theorem avoids both the use of the Laplace transform and the incorporation of initial conditions into the equations of motion. The uniqueness theorem is derived without the positive definiteness assumption on the elastic, conductivity and diffusion tensors. We prove also that the reciprocal relation leads to a continuous dependence theorem studied on external body loads. Finally, we prove the existence of a generalized solution by means of the semigroup of linear operators theory.     

Analysis of the Impact of Nonlinear Heat Transfer Laws on Temperature Distribution in Irradiated Biological Tissues: Mathematical Models and Optimal Controls

In this paper, we consider nonlinear control problems governed by some generalized transient bioheat transfer-type models with the nonlinear Robin boundary conditions. The control estimates the blood perfusion rate, the heat transfer parameter, the distributed energy source terms, and the heat flux due to the evaporation, which affect the effects of thermal physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics in medical practices, for example, for laser surgery, photo and thermotherapy for regional hyperthermia often used in treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness, and regularity of a solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Afterwards, the optimal control problem is formulated in order to control the online temperature given by radiometric measurement. We prove that an optimal solution exists and obtain necessary optimality conditions. Some strategy for numerical realization based on the adjoint variables are provided.      

Spectral analysis of thermoelastic systems under nonclassical thermal models

We study some spectral properties of the solutions to generalized thermoelastic systems under Lord–Shulman, Green–Lindsay, and Green–Naghdi of type-II models. First, we prove that the linear operator of each model has compact resolvent and generates a C₀?semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions of the spectrum and we deduce that the spectrum-determined growth condition holds. Therefore, if the imaginary axis is not an asymptote of the spectrum, we prove that the energy of each model decays exponentially to a rate determined explicitly by the physical parameters. Finally, some simulations are given for each model to support our results.     

On regularity properties of extremal controls

We prove some regularity properties of the optimal controls for the smooth bracket generating systems with scalar control parameters, and show that the Cantor sets cannot be the sets of switching points.Partially supported by the Russian Fund for Fundamental Research and grant No. 95-01-00310a.     

Boundary Partial Stabilization of a Coupled Wave Equations on an Exterior Bounded Obstacle

We consider a boundary partial measure stabilization problem for a coupled wave equations on an exterior of bounded domain \({\Omega }=\mathbb {R}^{d}\setminus \overline {{\mathcal O}}\). With the theory of defect measure, we prove a uniformly result (exponential or polynomial) of stability in the energy space, under a geometrical control condition (BLR).     

Fundamental aspects of the Ti–H system: theoretical and experimental behaviour

We study the interaction of hydrogen with titanium in order to characterize some important microscopic and macroscopic properties of this system. It is technologically important because, among other applications, the Ti–H system is used as structural material in many applications due to the combination of two important mechanical properties, which are resistance to the corrosion and hardness. Using a calculus program based on the jellium model for the material, we obtain values of properties that are important in the determination of the macroscopic behaviour of the Ti–H system, such as the variation of the electronic density and of the induced density of states due to the presence of the hydrogen in the matrix of Ti. From an experimental point of view, we hydride a titanium matrix in order to determine the effects of the process on the properties of the material structure.The general features of these theoretical and experimental methods are discussed and the corresponding results are compared with experimental data.     

A domain of influence theorem for a natural stress–heat-flux disturbance in thermoelasticity of type-II

The aim of the present work is to establish the domain of influence theorem for a stress–heat-flux disturbance under Green and Naghdi thermoelasticity theory of type-II. We consider a mixed problem of natural type represented as stress–heat-flux disturbance in the context of Green–Naghdi thermoelasticity theory of general type. We establish a general energy identity for the problem. Then we establish the domain of influence theorem for natural stress–heat-flux disturbance in the context of Green–Naghdi model of thermoelasticity oftype-II. We prove that for a finite time t > 0, the pair of stress and the heat-flux field generates no disturbance outside a bounded domain. The domain of influence is shown to be dependent on the thermoelastic parameters.     

Type II Thermoelasticity. A New Aspect

In this paper we investigate the thermomechanical theory of thermoelasticity without energy dissipation. We consider the case where the assumptions on the constitutive tensor are different from the usual ones in the current literature. These assumptions are suitable because they correspond to situations where the material is prestressed and has initial entropy flux. When we restrict our attention to the one-dimensional problem, we first establish a condition on the coefficients to guarantee the hyperbolicity of the system. Then we prove that the problem is well posed and stable. When the hyperbolicity condition is not satisfied we prove that the problem is ill-posed. We also prove a similar result when the dimension is greater than one whenever the domain satisfies a certain condition. Though we cannot expect in general the stability of solutions in dimension greater than one, we prove the uniqueness and stability of radial solutions in the general case. To obtain the results, we need the use of a conservation law which has not previously been considered in the literature. Similar results are not known for other thermoelastic theories.     

Jumps and stochastic volatility in oil prices: Time series evidence

In this paper we examine the empirical performance of affine jump diffusion models with stochastic volatility in a time series study of crude oil prices. We compare four different models and estimate them using the Markov Chain Monte Carlo method. The support for a stochastic volatility model including jumps in both prices and volatility is strong and the model clearly outperforms the others in terms of a superior fit to data. Our estimation method allows us to obtain a detailed study of oil prices during two periods of extreme market stress included in our sample; the Gulf war and the recent financial crisis. We also address the economic significance of model choice in two option pricing applications. The implied volatilities generated by the different estimated models are compared and we price a real option to develop an oil field. Our findings indicate that model choice can have a material effect on the option values.     

Optimal control problem for deflection plate with crack

We consider a control problem where the state variable is defined as the solution of a variational inequality. This system describes the vertical displacement of points of a thin plate with the presence of crack inside [7]. As the control we define the force that originates the deection of the plate. In order to get the system of optimality for the control problem we use a penalized problem [1] and its reformation as a Lagrangian problem. We prove the existence of a Lagrange multiplier to obtain a system of optimality to the exact problem via Lagrangian. Applying the method of bounded increments [19] we get the final result that characterizes the optimal state and control.     

Spatial and temporal behavior in the theory of thermoelasticity for solids with double porosity

In this article, we study the spatial and the temporal behavior of solutions to the initial boundary value problem associated with the linear theory of thermoelastic materials with a double porosity structure. We consider two appropriate time-weighted integral measures and we deduce some exponential estimates that describe the spatial behavior of solutions. For bounded bodies, we obtain estimates of Saint-Venant type, while for unbounded bodies we deduce some alternatives of Phragmén–Lindelöf type. The temporal behavior of solutions is described using the Cesáro means of various parts of the total energy.     

Local Controllability for Linear Control Systems on Lie Groups

Abtract  In this work, we study controllability properties of linear control systems on Lie groups. Using Lie theory of semigroups, we obtain local controllability results for this type of systems. In addition, properties for the flow of the drift vector field X and for the reachable set of the system are presented. Finally, an example on the Heisenberg Lie group is considered, and its properties are proved using the theory developed. 2000 Mathematics Subject Classification. Primary 93B05; Secondary 93C25, 93B27. The authors want to thank the Fachbereich Mathematik of the Technische Universit?t Darmstadt for warm hospitality during the production of this work. The first author is pleased to thank Professor Guillermo Ferreyra and the department of mathematics at LSU for their support.