In this paper we derive a generalizing concept of $G$-norms, which we call $G$-sets, which is used to characterize minimizers of non-differentiable regularization functionals. Moreover, the concept is closely related to the defnition of slopes as published in a recent book by Ambrosio, Gigli, Savaré. A paradigm of regularization models fitting in this framework is robust bounded variation regularization. Two essential properties of this regularization technique are documented in the literature and it is shown that these properties can also be achieved with metric regularization techniques.

This article presents an analysis of the dynamics of a bidimensional oscillator, which has been proposed as a simple model for the vocal fold motion at phonation. The model is capable of producing an oscillation with physiologically realistic values for the parameters. A simple extension of the model using even-powered polynomials in the damping factor is proposed, to permit the occurrence of an oscillation hysteresis phenomenon commonly observed in voice onset-offset patterns. This phenomenon appears from the combination of a subcritical Hopf bifurcation where an unstable limit cycle is produced, with a fold bifurcation between limit cycles, where the unstable limit cycle coalesces and cancels with a stable limit cycle. The results are illustrated with phase plane plots and bifurcation diagrams obtained using numerical continuation techniques.

We consider Navier-Stokes equations coupled to nonlinear Fokker-Planck equations describing the probability distribution of particles interacting with fluids. We describe relations determining the coefficients of the stresses added in the fluid by the particles. These relations link the added stresses to the kinematic effect of the fluid's velocity on particles and to the inter- particle interaction potential. In equations of type I, where the added stresses depend linearly on the particle distribution density, energy balance requires a response potential. In equations of type II, where the added stresses depend quadratically on the particle distribution, energy balance can be achieved without a dynamic response potential. In unforced energetically balanced equations, all the steady solutions have fluid at rest and particle distributions obeying an uncoupled Onsager equation. Systems of equations of type II have global smooth solutions if inertia is neglected.

A mathematical model describing supply chains on a network is introduced. In particular, conditions on each vertex of the network are specified. Finally, this leads to a system of nonlinear conservation laws coupled to ordinary differential equations. To prove the existence of a solution we make use of the front tracking method. A comparison to another approach is given and numerical results are presented.

In this paper, we consider minimization of a real-valued function $f$ over $\bold R\sp {n+1}$ and study the choice of the affine normal of the level set hypersurfaces of $f$ as a direction for minimization. The affine normal vector arises in affine differential geometry when answering the question of what hypersurfaces are invariant under unimodular affine transformations. It can be computed at points of a hypersurface from local geometry or, in an alternate description, centers of gravity of slices. In the case where $f$ is quadratic, the line passing through any chosen point parallel to its affine normal will pass through the critical point of $f$. We study numerical techniques for calculating affine normal directions of level set surfaces of convex $f$ for minimization algorithms.

In a recent paper, we analyzed the {$L\sp 2$}-well posedness of an initial boundary value problem (ibvp) for the two-dimensional system of the linear elasticity under the uniform Kreiss- Lopatinskii condition. The present work is devoted to studying the analog of this problem in the three-dimensional case, when the Majda-Osher's analysis cannot be applied. The well-posedness is achieved by constructing an everywhere smooth non-degenerate dissipative Kreiss symmetrizer of the ibvp: this is done by adapting to the present situation the techniques already implemented for the two-dimensional linear elasticity. Compared with the latter case, some further technical difficulties have to be accounted for.

In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite dimensional analogue of the Langevin SDE used in finite dimensional sampling. Here the theory is developed for conditioned Gaussian processes for which the resulting SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A companion paper studies the nonlinear case, building on the linear analysis provided here.

We solve the Smoluchowski equation for steady state solutions of rigid nematic polymers and suspensions under imposed elongational flow, magnetic or electric fields, respectively. Under the three imposed fields, we show that (1) the Smoluchowski equation can be cast into a generic form, (2) the external field must parallel to one of the eigenvectors of the second moment tensor in steady states, and (3) the steady state solution of the Smoluchowski equation (probability density function or simply pdf) is of the Boltzmann type parameterized by material parameters and two order parameters governed by two algebraic-integral equations. Then, we present a complete bifurcation diagram of the order parameters with respect to the material parameters by solving the algebraic-integral equations. The stability of the pdf solutions is inferred from the minimum of the free energy density. The solution method is extended to dilute solutions of dipolar, rigid nematic polymers under an imposed electric field. The first moment of the steady state pdf is shown to be parallel to the external field direction at sufficiently strong permanent dipole or relatively weak dipole-dipole interaction. In this case, the steady solution of the Smoluchowski equation is parameterized by one order parameter and material parameters in the Boltzmann form. Otherwise, the first moment is not necessarily parallel to the external field direction.

In this work we study the existence of time periodic weak solution for the three dimensional Vlasov-Maxwell system with boundary conditions. The main idea consists of using the mass, momentum and energy conservation laws which allow us to obtain a priori estimates in the case of a star-shaped bounded spatial domain. We start by constructing time periodic smooth solutions for a regularized system. The existence for the Vlasov-Maxwell system follows by weak stability under uniform estimates. These results apply for both classical and relativistic cases and for systems with several species of particles.

Most efficient adaptive mesh methods employ a few strategies, including local mesh refinement (h-adaptation), movement of mesh nodes (r-adaptation), and node reconnection (c-adaptation). Despite its simplicity, node reconnection methods are seldom analyzed apart from the other adaptation methods even in applications where severe restrictions are imposed on topological operations with a mesh. However, using only node reconnection the discretization error can be significantly reduced. In this paper, we develop and numerically analyze a new c-adaptation algorithm for mimetic finite difference discretizations of elliptic equations on triangular meshes. Our algorithm is based on a new error indicator for such discretizations, which can also be used for unstructured general polygonal meshes. We demonstrate the efficiency of our new algorithm with numerical examples.