The structure of the optimal income tax in the quasi-linear model

Existing numerical characterizations of the optimal income tax have been based on a limited number of model specifications. As a result, they do not reveal which properties are general. We determine the optimal tax in the quasi-linear model under weaker assumptions than have previously been used; in particular, we remove the assumption of a lower bound on the utility of zero consumption and the need to permit negative labor incomes. A Monte Carlo analysis is then conducted in which economies are selected at random and the optimal tax function constructed. The results show that in a significant proportion of economies the marginal tax rate rises at low skills and falls at high. The average tax rate is equally likely to rise or fall with skill at low skill levels, rises in the majority of cases in the centre of the skill range, and falls at high skills. These results are consistent across all the specifications we test. We then extend the analysis to show that these results also hold for Cobb-Douglas utility.

QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments

QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments.

On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Linear and nonlinear generalized Fourier transforms

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.

Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

On the convergence of the no response test

The no response test is a new scheme in inverse problems for partial differential equations which was recently proposed in [D. R. Luke and R. Potthast, SIAM J. Appl. Math., 63 (2003), pp. 1292–1312] in the framework of inverse acoustic scattering problems. The main idea of the scheme is to construct special probing waves which are small on some test domain. Then the response for these waves is constructed. If the response is small, the unknown object is assumed to be a subset of the test domain. The response is constructed from one, several, or many particular solutions of the problem under consideration. In this paper, we investigate the convergence of the no response test for the reconstruction information about inclusions D from the Cauchy values of solutions to the Helmholtz equation on an outer surface $\partial\Omega$ with $\overline{D} \subset \Omega$. We show that the one‐wave no response test provides a criterion to test the analytic extensibility of a field. In particular, we investigate the construction of approximations for the set of singular points $N(u)$ of the total fields u from one given pair of Cauchy data. Thus, the no response test solves a particular version of the classical Cauchy problem. Also, if an infinite number of fields is given, we prove that a multifield version of the no response test reconstructs the unknown inclusion D. This is the first convergence analysis which could be achieved for the no response test.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Predictive control using feedback linearization based on dynamic neural models

This paper presents a hybrid control strategy integrating dynamic neural networks and feedback linearization into a predictive control scheme. Feedback linearization is an important nonlinear control technique which transforms a nonlinear system into a linear system using nonlinear transformations and a model of the plant. In this work, empirical models based on dynamic neural networks have been employed. Dynamic neural networks are mathematical structures described by differential equations, which can be trained to approximate general nonlinear systems. A case study based on a mixing process is presented.

Similarity solutions of the shallow water equations

A one-dimensional shock (bore) reflection problem is discussed for the two-dimensional shallow water equations with cylindrical symmetry. The differential equations for a similarity solution are derived and solved numerically in conjunction with the Rankine-Hugoniot shock relations.

On the existence of extreme waves and the Stokes conjecture with vorticity

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface.