Optimal control problems for distributed parameter systems in Banach spaces

We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf ₀ (u) defined in a metric spaceV subject to the constraintf(u) Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL ¹ andL spaces.This work was supported in part by the National Science Foundation under Grant DMS-9001793.     

Foliations with a Kupka component on algebraic manifolds

We consider codimension one holomorphic foliations in complex projective manifolds of dimension at least 3, having a compact Kupka component and represented by integrable holomorphic sections of the bundleTM ^*L, whereL denotes a very ample holomorphic line bundle. We will show that, if the transversal type is not the radial vector field andH ¹ (M,) = 0, then the foliation has a meromorphic first integral.Supported by Conacyt: 3398-E 9307     

Uniqueness results for optimization problems with prescribed rearrangement

In this paper we study the case of equalities in some comparison results for L ¹-norm or L -norm of the solutions of Dirichlet elliptic problem or Hamilton-Jacobi equations. We show that equalities are achieved only in spherically symmetric situations.Work partially supported by MURST (40%).     

The generalised wold decomposition for subnormal operators

Subnormal operatorsS with the spectrum of the minimal normal extension contained in the boundary of (S) are studied. Under certain geometric assumptions it is shown that (up to unitary equivalence)S is the orthogonal sum of a normal operator and of the multiplication by the independent variablez on the Hardy spaceH ² [E] of a certain flat unitary bundleE over the interior of (S). This extends the results of Abrahamse and Douglas [1], [2].     

The Exterior Cube L-Function for GL(6)

We construct a Rankin Selberg integral to represent the exterior cube L function L(,³,s) of an automorphic cuspidal module of GL₆( _F ) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(,³,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(,³,1/2).     

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice L of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-ech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; X ^Y denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of L ^P is determined, where P is a poset and L a bounded distributive lattice.     

Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

Spectral estimates of Cauchy's transform inL 2(Ω)

We prove that the singular numbers of the Cauchy transform onL ²(D) are asymptotically , whiles _n (C _| L _a ² (D))1/n (whereL _a ² (D) is the subspace of analytic functions inL ²(D)). Also, the singular numbers of the logarithmic potential onL ²(D) are asympoticallys _n (L)1/n, whiles _n(L _{|L a} ² (D))1/n ². Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL _L ² () intoL ²() where and are rotation-invariant measures on .The author was partly supported by a grant from the national Science Foundation.     

Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted intoL _loc estimates, following theLip convergence theory developed by Tadmor et al. Comparisons between the local truncation error and theL _loc -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].     

On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems

Summary We consider a (possibly) vector-valued function u: R^N, Rⁿ, minimizing the integral , 2-2/(n*1)<p<2, whereD _i u=u/x _i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D₁ u,..., D_n–1 u L^p/(p-1) and D_nu L²; this result allows us to get existence of second weak derivatives: D(D₁ u),...,D(D_n–1u)L² and D(D_n u) L _p.This work has been supported by MURST and GNAFA-CNR.     

Applications of Riemannian and Einstein–Weyl Geometry in the Theory of Second-Order Ordinary Differential Equations

We consider some properties of four-dimensional Riemannian spaces whose metric coefficients are associated with the coefficients of second-order nonlinear differential equations, and we study the properties of three-dimensional Einstein–Weyl spaces related to the dual equations b = g(a,b,b), where the function g(a,b,b) satisfies a special partial differential equation.     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Polylogarithmic functional equations: A new category of results developed with the help of computer algebra (MACSYMA)

The interrelation of polylogarithmic functional equations and certain numerical results, known as ladders, is discussed, and leads to a consideration of three new, single-variable functional equations at the second order. Two of these families each contain six leading terms whose interrelationship constitutes a constraint on the integration process, but the third has only a single leading term with no such constraints. It is shown how this functional equation can be integrated to the third order, and the process reduced to an algorithm — actually a sequence of instructions — for incorporation into a computer program for symbolic manipulation. The procedure utilizes results from Kummer's equations to cancel out, in sequence, terms which do not vanish, or do vanish, with the variablez. Arguments are all of the form ±z ^p (1–z) ^q (1+z) ^r , and the process is algebraicized by using a (p,q,r,s) notation (withs=±1) to represent such terms. Application of the procedure leads to an integration to the fourth and fifth orders, the latter exhibiting 55 transcendental terms. The first step for the transition to the sixth order can also be achieved but the subsequent steps are frustrated by the restricted forms that the Kummer equations take at the fifth order — it is not possible to create the needed equations in a form which vanishes withz; this corresponding to the elimination of the (5) constant in the extension of the numerically determined ladders to the sixth and higher orders. The existence of the higher-order ladders strongly suggests functional equations af these orders, but the present process has not yet been successful in finding them. The new equations have, however, produced ladders that were inaccessible from Kummer's equations, and had heretofore been only obtainable numerically, up to the fifth order. The method which was developed should be capable of generalization to other systems of equations characterized by the appearance of arguments with recurrent factors. Some new feature, however, will need to be determined before the barrier to the sixth order can be breached.     

The resolvent problem for the stokes system in exterior domains: An elementary approach

We consider the Stokes system with resolvent parameter in an exterior domain: under Dirichlet boundary conditions. Here Ω is a bounded domain with C² boundary, and [λ??\] ? [∞, 0], ν >0. Using the method of integral equations, we are able to construct solutions ( u , π) in L^p spaces. Our approach yields an integral representation of these solutions. By evaluating the corresponding integrals, we obtain L^p estimates that imply in particular that the Stokes operator on exterior domains generates an analytic semigroup in L^p.     

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Numerically positive divisors on algebraic surfaces

Numerically positive line bundles on a complex projective smooth algebraic surfaceS are studied. In particular for any such line bundleL Pic(S) we prove the following facts: (i)g(L) 0 and (ii)L is ample ifg(L) 1,g standing for the arithmetic genus. Some applications are discussed. We also investigate numerically positive non-ample line bundlesL withg(L)=2.     

Parallel Computation of Invariant Measures

Let S:[0,1][0,1] be a nonsingular transformation and let P:L ¹(0,1)L ¹(0,1) be the corresponding Frobenius–Perron operator. In this paper we propose a parallel algorithm for computing a fixed density of P, using Ulam's method and a modified Monte Carlo approach. Numerical results are also presented.