Nonzero-sum stochastic differential games with stopping times and free boundary problems
Alain
Bensoussan;
Avner
Friedman
275-327
Abstract: One is given a diffusion process and two payoffs which depend on the process and on two stopping times ${\tau _1},{\tau _2}$. Two players are to choose their respective stopping times ${\tau _1},{\tau _2}$ so as to achieve a Nash equilibrium point. The problem whether such times exist is reduced to finding a ``regular'' solution $({u_1},{u_2})$ of a quasi-variational inequality. Existence of a solution is established in the stationary case and, for one space dimension, in the nonstationary case; for the latter situation, the solution is shown to be regular if the game is of zero sum.
Uniqueness properties of CR-functions
L. R.
Hunt
329-338
Abstract: Let M be a real infinitely differentiable closed hypersurface in X, a complex manifold of complex dimension $n \geqslant 2$. The uniqueness properties of solutions to the system ${\bar \partial _M}u = f$, where ${\bar \partial _M}$ is the induced Cauchy-Riemann operator on M, are of interest in the fields of several complex variables and partial differential equations. Since dM is linear, the study of the solution to the equation ${\bar \partial _M}u = 0$ is sufficient for uniqueness. A ${C^\infty }$ solution to this homogeneous equation is called a CR-function on M. The main result of this article is that a CR-function is uniquely determined, at least locally, by its values on a real k-dimensional ${C^\infty }$ generic submanifold ${S^k}$ of M with $k \geqslant n$. The facts that ${S^k}$ is generic and $k \geqslant n$ together form the lower dimensional analogue of the concept of noncharacteristic.
Kernel functions on domains with hyperelliptic double
William H.
Barker
339-347
Abstract: In this paper we show that the structure of the Bergman and Szegö kernel functions is especially simple on domains with hyperelliptic double. Each such domain is conformally equivalent to the exterior of a system of slits taken from the real axis, and on such domains the Bergman kernel function and its adjoint are essentially the same, while the Szegö kernel function and its adjoint are elementary and can be written in a closed form involving nothing worse than fourth roots of polynomials. Additionally, a number of applications of these results are obtained.
Extremal arcs and extended Hamiltonian systems
Frank H.
Clarke
349-367
Abstract: A general variational problem is considered; it involves the minimization of an integrand L of a very general nature. The Lagrangian L is allowed to assume the value $+ \infty$, and need satisfy no differentiability or convexity conditions. A Hamiltonian corresponding to the problem is defined via the conjugate function of convex analysis, and it is shown how one obtains necessary conditions in the form of an extended Hamiltonian system. This system is expressed in terms of certain ``generalized gradients'' previously developed by the author. A further result is given which has the feature that the principal hypotheses required, as well as the ensuing conclusions, are entirely in terms of H. This allows the treatment of classes of problems in which H is more amenable to direct analysis than L. The approach also sheds light on the relation between existence theory and the theory of necessary conditions, since the results may easily be compared with R. T. Rockafellar's recent work on existence theory, in which H also plays a central role. As an example of its application the main result is specialized to a differential inclusion problem. A specific example of its use is also given, an unorthodox optimal control problem with a discontinuous cost functional.
Boundary behavior of harmonic forms on a rank one symmetric space
Aroldo
Kaplan;
Robert
Putz
369-384
Abstract: We study the boundary behavior of 1-forms on a rank-one symmetric space M satisfying the equations $d\omega = 0 = \delta \omega $; the role of boundary is played by a nilpotent (Iwasawa) group $ \bar N$ of isometries of M. For forms satisfying certain $ {H^p}$ integrability conditions, we obtain the existence of boundary values in an appropriate sense, characterize these boundary values by means of fractional and singular integral operators on the group $\bar N$, and exhibit explicit isomorphisms between ${H^p}$ spaces of forms on M and the ordinary $ {L^p}$ spaces of functions on the group $\bar N$.
Structurally stable Grassmann transformations
Steve
Batterson
385-404
Abstract: A Grassmann transformation is a diffeomorphism on a Grassmann manifold which is induced by an $n \times n$ nonsingular matrix. In this paper the structurally stable Grassmann transformations are characterized to be the maps which are induced by matrices whose eigenvalues have distinct moduli. There is exactly one topological conjugacy class of complex structurally stable Grassmann transformations. For the real case the topological classification is determined by the ordering (relative to modulus) of the signs of the eigenvalues of the inducing matrix.
The decay of solutions of the two dimensional wave equation in the exterior of a straight strip
Peter
Wolfe
405-428
Abstract: We study an initial boundary value problem for the wave equation in the exterior of a straight strip. We assume the initial data has compact support and that the solution vanishes on the strip. We then show that at any point in space the solution is $O(1/t)$ as $t \to \infty$. This is the same rate of decay as obtains for the solution of the initial boundary value problem posed in the exterior of a smooth star shaped region. Our method is to use a Laplace transform. This reduces the problem to a consideration of a boundary value problem for the Helmholtz equation. We derive estimates for the solution of the Helmholtz equation for both high and low frequencies which enable us to obtain our results by estimating the Laplace inversion integral asymptotically.
Transversality in $G$-manifolds
M. J.
Field
429-450
Abstract: A definition of transversality is given for the category of G-manifolds (G, a compact Lie group). Transversality density and isotopy theorems are shown to hold for this definition. An example is given to show that we cannot require differential stability of intersections.
Singularities of spaces of flat bundles over complex manifolds
B.
Wong
451-461
Abstract: We use Fox differential calculus on free group to study the singularities of complex analytic varieties arising from flat bundles over complex manifolds. Criteria of regularity in terms of cohomology of the fundamental group of the underlying manifold are established.
Cyclic purity versus purity in excellent Noetherian rings
Melvin
Hochster
463-488
Abstract: A characterization is given of those Noetherian rings R such that whenever R is ideally closed ($\equiv$ cyclically pure) in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion $(A,m)$ of each local ring of R at a maximal ideal has the following two equivalent properties: (i) For each integer $N > 0$ there is an m-primary irreducible ideal $ {I_N} \subset {m^N}$. (ii) Either $\dim \;A = 0$ and A is Gorenstein or else depth $A \geqslant 1$ and there is no $P \in {\text{Ass}}(A)$ such that $\dim (A/P) = 1$ and $(A/P) \oplus (A/P)$ is embeddable in A. It is then shown that if R is a locally excellent Noetherian ring such that either R is reduced (or, more generally, such that R is generically Gorenstein), or such that Ass(R) contains no primes of coheight $ \leqslant 1$ in a maximal ideal, and R is ideally closed in S, then R is pure in S. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed. Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension $R \to S$ such that R is pure in S but $R[[T]]$ is not even cyclically pure in $ S[[T]]$.
Modular forms for $G\sb{0}(N)$ and Dirichlet series
Michael J.
Razar
489-495
Abstract: A criterion is given for a function to be a modular form for ${\Gamma _0}(N)$. It is similar to the criterion given by Weil in his 1967 Math. Ann. paper Über die Bestimmung Dirichletscher Reihen durch Funktional-gleichungen in that it involves checking that certain twists of the associated Dirichlet series satisfy functional equations. It differs in the number and type of such equations which need to be satisfied.
Efficient generation of maximal ideals in polynomial rings
E. D.
Davis;
A. V.
Geramita
497-505
Abstract: The cardinality of a minimal basis of an ideal I is denoted $ \nu (I)$. Let A be a polynomial ring in $n > 0$ variables with coefficients in a noetherian (commutative with $1 \ne 0$) ring R, and let M be a maximal ideal of A. In general $ \nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})$. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1) ${A_M}$ is not regular (valid even if A is not a polynomial ring), (2) $M \cap R$ is maximal in R and (3) $n > 1$. Equality may fail for $n = 1$, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension $ > 1$. In case $ n = 1$ and $\dim (R) = 2$ the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely, $ \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\}$. Corollary to (3) is that for R regular and $n > 1$, every maximal ideal of A is generated by a regular sequence--a result well known (for all $n \geqslant 1$) if R is a field (and somewhat less well known for R a Dedekind domain).
On the dimension of left invariant means and left thick subsets
Maria
Klawe
507-518
Abstract: If S is a left amenable semigroup, let $\dim \langle Ml(S)\rangle $ denote the dimension of the set of left invariant means on S. Theorem. If S is left amenable, then $\dim \langle Ml(S)\rangle = n < \infty$ if and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer for S countable or left cancellative. Moreover, when S is infinite, left amenable, and either left or right cancellative, we show that $ \dim \langle Ml(S)\rangle$ is at least the cardinality of S. An application of these results shows that the radical of the second conjugate algebra of ${l_1}(S)$ is infinite dimensional when S is a left amenable semigroup which does not contain a finite ideal.
Spectra and measure inequalities
C. R.
Putnam
519-529
Abstract: Let T be a bounded operator on a Hilbert space $\mathfrak{H}$ and let ${T_z} = T - zI$. Then the operators $ {T_z}T_z^\ast,{T_z}{T_t}{({T_z}{T_t})^\ast}$, and $ {T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast}$ are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let $\mathfrak{H}$ be separable and suppose that $ {T_z}{T_t}{({T_z}{T_t})^\ast} \geqslant D \geqslant 0$ for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then ${\left\Vert D \right\Vert^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text{meas}_2}\;({\sigma _p}(T))$. If T is the adjoint of the simple unilateral shift, then equality holds with $D = I - {T^\ast}T$.
On one-dimensional coupled Dirac equations
R. T.
Glassey
531-539
Abstract: The Cauchy Problem for Dirac equations coupled through scalar and Fermi interactions is considered in one space dimension. Global solutions of finite energy are shown to exist, provided that either the magnitude of the coupling constant or the $ {L_2}({R^1})$-norm of the initial data is suitably restricted.
Torsion in the bordism of oriented involutions
Russell J.
Rowlett
541-548
Abstract: In the bordism theory ${\Omega _ \ast }(Z_2^k)$ of smooth, orientation-preserving $Z_2^k$-actions all torsion has order two. Furthermore, the torsion classes inject in the unoriented theory ${N_ \ast }(Z_2^k)$, and any class represented by a stationary-point free action has infinite order. In addition, a procedure is given for producing Smith constructions in some generality.