Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds

We study the C^1,1 and Lipschitz regularity of the solutions of the degenerate complex Monge-Ampère equation on compact K?hler manifolds. In particular, in view of the local regularity for the complex Monge-Ampère equation, the obtained C^1,1 regularity is a generalization of the Yau theorem which deals with the nondegenerate case. Received: 18 April 2002 / Published online: 24 February 2003 Partially supported by KBN Grant #2 P03A 028 19     

Nontrivial solutions for Monge-Ampère type operators in convex domains

In this paper we prove existence and interior regularity of convex solutions of the Dirichlet problem for a class of Monge-Ampère type equations with right hand side vanishing on the boundary, by means of the fixed point index theory for Bakelman's weak solutions. The regularity in the interior follows from recent results of Caffarelli. Partially supported by M.U.R.S.T., Italy Partially supported by G.N.A.F.A. of C.N.R., Italy. This work was accomplished during a Humbold research stay at Karlsruhe and Heidelberg Universities.     

Boundary value problems with mixed lateral conditions for parabolic operators

Summary We study a parabolic problem in a cylinder with lateral conditions of mixed type. We get an existence, uniqueness and regularity result in dissymmetric function spaces nicely fitting the geometry of the problem.Partially supported by G.N.A.F.A. of C.N.R., Italy.Partially supported by I.N.F.N., Bologna, Italy.     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

Cyclic cohomology of etale groupoids

We extend Connes's computation of the cyclic cohomology groups of smooth algebras arising from foliations with separated graphs. We find that the characteristic classes of foliations factor through these groups. Our results also explain some results of Atiyah and Segal on orbifold Euler characteristic in the setting of cyclic homology.Partially supported by NSF grants DMS 92-03517 and DMS 89-03248.Partially supported by NSF grant DMS 92-05548.     

Differentiability and analyticity of topological entropy for Anosov and geodesic flows

Summary In this paper we investigate the regularity of the topological entropyh _top forC ^k perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].Partially supported by NSF Grant DMS85-14630     

Continuous modal control of linear multicoupled objects

A modal control method is considered in which the spectrum of the closed-loop system is continuously deformed in such a way that the spectrum of the open-loop object transforms into the desired spectrum. The algorithm of the continuous modal control is synthesized. The conditions for spectral control in the method are obtained. The approach is based on similar ideas to those in /1/, but a different class of controls is considered here. Moreover, by using the appratus of Lyapunov functions, specified in the one-parameter family of the deformed spectrum, the deviation between the required spectrum and the closed-loop system spectrum can be minimized in the Euclidean metric, in the case when the wanted spectrum cannot be obtained in the closed-loop system.     

Interactions between government and firms: a differential game approach

In this paper we study an extended version of the model described in Gradus (J. Econ. 81:1092–1109, 1989) in order to determine the optimal taxation policy of a government and its effects on the stock of capital goods growth as a result of the activity developed by firms. It is shown that, by introducing a wealth tax, there exists an optimal wealth tax rate for which the open-loop/feedback Nash/Stackelberg equilibria coincide, maximizing the payments for both agents (government and firms), so that the open-loop Stackelberg equilibrium becomes both time consistent and subgame perfect. Partially supported by MEC (Spain) Grants MTM2006-13468, BMF2002-03493 and project ‘Ingenio Mathematica (i-MATH)’ No. CSD2006-00032 (Consolider-Ingenio 2010).     

Crystalline cohomology and GL(2, ℚ)

This paper applies recent advances in crystalline cohomology to the classical case of open elliptic modular curves. In so doing control is gained over the action of inertia in the Galois representations attached to modular forms. Our aim is to study the modular Galois representations attached to automorphic forms modp of weightk≥2. We generalize to higher weightk several results which were previously accessible only in the case of weight 2 where jacobian varieties can be invoked. Additionally we reconsider Gross’s theorem on companion forms in a crystalline context. Partially supported by NSF grant DMS 90-02744. Partially supported by NSA grant MDA904-90-H-4020 and by a PSC-CUNY grant.     

Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces

We analyze the global convergence properties of some variants of regularized continuous Newton methods for convex optimization and monotone inclusions in Hilbert spaces. The regularization term is of Levenberg–Marquardt type and acts in an open-loop or closed-loop form. In the open-loop case the regularization term may be of bounded variation.     

Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.     

Locally regular coloured graphs

In this work we introduce the concept of locally regular coloured graph as a generalization to any dimension of the concept of regularity for maps on surfaces of W. Threlfall. We prove that locally regular coloured graphs can be obtained from the classical spherical, euclidean and hyperbollic tessellations. Finally we describe locally regular coloured graphs on spherical 3-manifolds.Partially supported by British-Spanish Join Research Program and DGICYT.     

Finite element analysis of pressure formulation of the elastoacoustic problem

In this paper we analyze the non symmetric pressure/displacement formulation of the elastoacoustic vibration problem and show its equivalence with the (symmetric) stiffness coupling formulation. We introduce discretizations for these problems based on Lagrangian finite elements. We show that both formulations are also equivalent at discrete level and prove optimal error estimates for eigenfunctions and eigenvalues. Both formulations are rewritten such as to be solved with a standard Matlab eigensolver. We report numerical results comparing the efficiency of the methods over some test examples. Partially supported by Xunta de Galicia (Spain) through grant No. PGIDT00-PXI20701PR and by MCYT Research Project DPI2001-1613-C02-02Partially supported by Xunta de Galicia (Spain) through grant No. PGIDT00-PXI20701PR and by MCYT Research Project DPI2001-1613-C02-02Partially supported by FONDAP in Applied Mathematics (Chile) and by MCYT Research Projects DPI2001-1613-C02-02 and BFM2001-3261-C02-02Partially supported by FONDECYT (Chile) through grant No. 1.990.346 and FONDAP in Applied Mathematics (Chile)Mathematics Subject Classification (1991):65N30     

Random projections and goodness-of-fit tests in infinite-dimensional spaces

In this paper we provide conditions under which a distribution is determined by just one randomly chosen projection. Thenweapply our results to construct goodness-of-fit tests for the one and two-sample problems. We include some simulations as well as the application of our results to a real data set. Our results are valid for every separable Hilbert space. *Partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant MTM2005-08519-C02-02. **Partially supported by grants from NSERC and the Canada research chairs program.     

A note on nonzero-sum differential games with bargaining solution

The extension of Nash's bargaining solution to differential games is discussed. It is shown that a closed-loop solution verifies very stringent necessary conditions and that an open-loop solution can present serious weakness from a normative point of view.This research has been supported by the Canada Council (S73-0935) and the Ministère de l'Education du Québec (DGES).     

On Legal Semigroups

A new class of semigroups with a two variable regularity law is introduced. These semigroups are non-regular semigroups but they are closely related to regular semigroups. The local and global structures of this class of semigroups are investigated.AMS Subject Classification (2000): 20M10Partially supported by a Chinese University of Hong Kong Direct Research grant, Hong Kong (98/99) # 2060152.Partially supported by a grant of the National Science Foundation, China.     

Large deviations in the geometry of convex lattice polygons

We provide a full large deviation principle (LDP) for the uniform measure on certain ensembles of convex lattice polygons. This LDP provides for the analysis of concentration of the measure on convex closed curves. In particular, convergence to a limiting shape results in some particular cases, including convergence to a circle when the ensemble is defined as those centered convex polygons, with vertices on a scaled two dimensional lattice, and with length bounded by a constant. The Gauss-Minkowskii transform of convex curves plays a crucial role in our approach. Partially supported by grants ININS 94-3420 and RFF1-96-01-00676. Partially supported by a US-Israel BSF grant and by the fund for promotion of research at the Technion.     

The Complete Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve. Masaaki Homma: Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS. Seon Jeong Kim: Partially supported by Korea Research Foundation Grant (KRF-2004-041-C00016)     

On the eigenvalues of matrices with given upper triangular part

We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part.Partially supported by the NSF Grant DMS-8701615-02Partially supported by the NSF Grant DMS-8802836 and United States-Israel Binational Science Foundation Grant 88-00304/I.     

Perturbation analysis of second-order cone programming problems

We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions. This is the first time such a characterization is given for a nonpolyhedral conic problem. Dedicated to R.T. Rockafellar on the occasion of his 70th birthday. Partially supported by Ecos-Conicyt C00E05.