Measure-valued mass evolution problems with flux boundary conditions and solution-dependent velocities

In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of an earlier paper [J. Differential Equations, 259 (2015), pp. 10681097] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ those results as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval [0, T]. This paper is partially based on results presented in Chapter 5 of [Evolution Equations for Systems Governed by Social Interactions, Ph.D. thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there are now resolved.

A RIEMANN INTEGRAL APPROACH TO FEYNMANS PATH-INTEGRAL

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrodinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) - ''practical calculations'' - are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrodinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free. particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in ''practical'' calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

On a Limit of Perturbed Conservation Laws with Diffusion and Non-Positive Dispersion

We consider a conservation law perturbed by a linear diffusion and a general form of non-positive dispersion. We prove the convergence of the corresponding solution to the entropy weak solution of the hyperbolic conservation law.

About a Vanishing Viscosity-Capillarity Method

We consider a class of nonlinear dissipative-dispersive perturbations of the scalar conservation law @tu + div f (u) = 01 and we study the convergence of the approximated solutions to its entropy solution. In particular, we obtain conditions under which the balance between dissipation and dispersion gives rise to the convergence (by DiPerna's measure-valued solution technique).

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.

Linear stochastic differential algebraic equations with constant coefficients

We consider linear stochastic differential-algebraic equations with constant coefficients and additive white noise. Due to the nature of this class of equations, the solution must be defined as a generalised process (in the sense of Dawson and Fernique). We provide sufficient conditions for the law of the variables of the solution process to be absolutely continuous with respect to Lebesgue measure.

Continuity of set-valued maps revisited in the light of tame geometry

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result –depending on stratification techniques– holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

Criteria for hitting probabilities with applications to systems of stochastic wave equations

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k >- 1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and colored in space.

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

Fully complex-valued radial basis function networks: orthogonal least squares regression and classification

We consider a fully complex-valued radial basis function (RBF) network for regression and classification applications. For regression problems, the locally regularised orthogonal least squares (LROLS) algorithm aided with the D-optimality experimental design, originally derived for constructing parsimonious real-valued RBF models, is extended to the fully complex-valued RBF (CVRBF) network. Like its real-valued counterpart, the proposed algorithm aims to achieve maximised model robustness and sparsity by combining two effective and complementary approaches. The LROLS algorithm alone is capable of producing a very parsimonious model with excellent generalisation performance while the D-optimality design criterion further enhances the model efficiency and robustness. By specifying an appropriate weighting for the D-optimality cost in the combined model selecting criterion, the entire model construction procedure becomes automatic. An example of identifying a complex-valued nonlinear channel is used to illustrate the regression application of the proposed fully CVRBF network. The proposed fully CVRBF network is also applied to four-class classification problems that are typically encountered in communication systems. A complex-valued orthogonal forward selection algorithm based on the multi-class Fisher ratio of class separability measure is derived for constructing sparse CVRBF classifiers that generalise well. The effectiveness of the proposed algorithm is demonstrated using the example of nonlinear beamforming for multiple-antenna aided communication systems that employ complex-valued quadrature phase shift keying modulation scheme. (C) 2007 Elsevier B.V. All rights reserved.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs.