Showing papers in "Transactions of the American Mathematical Society in 1983"

Viscosity solutions of Hamilton-Jacobi equations

Abstract: Publisher Summary This chapter examines viscosity solutions of Hamilton–Jacobi equations. The ability to formulate an existence and uniqueness result for generality requires the ability to discuss non differential solutions of the equation, and this has not been possible before. However, the existence assertions can be proved by expanding on the arguments in the introduction concerning the relationship of the vanishing viscosity method and the notion of viscosity solutions, so users can adapt known methods here. The uniqueness is then the main new point.

F-Purity and Rational Singularity.

Abstract: We investigate singularities which are F-pure (respectively F-pure type). A ring R of characteristic p is F-pure if for every R-module M, 0 M ? R M ? 'R is exact where 1R denotes the R-algebra structure induced on R via the Frobenius map (if r E R and s E 1R, then r s = rPs in 1R). F-pure type is defined in characteristic 0 by reducing to characteristicp. It is proven that when R = S/I is the quotient of a regular local ring S, R is F-pure at the prime ideal Q if and only if (I[PI: I) ? Q[P]. Here, J[P] denotes the ideal {aP I a E J}. Several theorems result from this criterion. If f is a quasihomogeneous hypersurface having weights (r1,...,r) and an isolated singularity at the origin: (1) 17 1ri > I implies K[X1,. X]/(f) has F-pure type at m = (X1,. X). (2) 1n I ri < I implies K [ X1 .Xn ]/(f) does not have F-pure type at m. (3) 1%= 1ri = I remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying F-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the F-pure locus in the maximal spectrum of K [ X1, . Xn ]/I, where K is a perfect field of characteristic P, is Zariski open. An important conjecture is that R/fR is F-pure (type) should imply R is F-pure (type) whenever R is a Cohen-Macauley, normal local ring. It is proven that Ext1( 1R, R) = 0 is a sufficient, though not necessary, condition. A local ring (R, m) of characteristic p is F-injective if the Frobenius map induces an injection on the local cohomology modules H' (R) H' (1R). An example is constructed which is F-injective but not F-pure. From this a counterexample to the conjecture that R/fR is F-pure implies R is F-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic 0 version of the conjecture. 0. Introduction. Let R be a ring of characteristic p and let 'R denote the ring R viewed as an R-module via the Frobenius map F(r) = rP. R is F-pure if for every R-module M, 0 -R ? M -4'R ? M is exact. A notion of F-pure type is then defined in characteristic 0 by reduction to characteristicp. F-pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is Cohen-Macaulay [3]. It has also been demonstrated that F-purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology [1]. Received by the editors March 30, 1982 and, in revised form, August 3, 1982. 1980 Mathematics Subject Classification. Primary 13H99. ?1983 American Mathematical Society 0002-9947/82/0000-0829/$05.25

On lexicographically shellable posets

Abstract: Lexicographically shellable partially ordered sets are studied. A new recursive formulation of CL-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are CL-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and CL-shellability.

Elementary first integrals of differential equations

Abstract: It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.

On the interpretation of whitney numbers through arrangements of hyperplanes, zonotopes, non-radon partitions, and orientations of graphs

Abstract: The doubly indexed Whitney numbers of a finite, ranked partially ordered set L are (the first kind) w,, -- E{(x', xJ): x', xl E L with ranks i, j} and (the second kind) WJ the number of (x', x') with x' < xJ. When L has a 0 element, the ordinary (simply indexed) Whitney numbers are w, = wo, and W, = Wo, - W,,. Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of k-dimen- sional faces for any k, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope P inside the visible boundary as seen from a distant point on a generating line of P. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley's theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly q sources (generalizing Renyi's enumera- tion of permutations with q "outstanding" elements). The number of totally cyclic orientatiofis of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph's having a unique source.

Some examples of square integrable representations of semisimple p-adic groups

Abstract: We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple p-adic group.

Rotation Hypersurfaces in Spaces of Constant Curvature

Abstract: Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the helicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space H3 ; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in H3.

Systems of conservation laws with invariant submanifolds

Abstract: Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatogra- phy, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coinci- dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of 2 X 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide. Introduction. Systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, nonlinear wave motion in elastic strings, as well as in multicomponent chromatography (1,4,5,6,9,11,12). These systems have many interesting features. The Riemann problem for these equations can be explicitly solved in the large, and wave interactions have a simplified structure, even in the presence of a nonconvex flux function. For this reason, these systems represent some of the few examples for which the Cauchy problem has been solved for arbitrary data of bounded variation. Also, hyperbolic degeneracies appear in each of these systems. In the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves. This phenomenon turns out to be a special case of the general phenomenon of a submanifold of R" being invariant for weak solutions of a given system of n conservation laws. Invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations; i.e., when the concentrations C, of any of the species vanish, the system must reduce to a lower order system that expresses the conservation of the remaining species. Thus C, = 0 defines a manifold which is invariant in the sense that solutions that start on the manifold, remain there for all time. Moreover, such an invariant manifold of dimension n — 1 is the boundary of an «-dimensional invariant region in the sense of Chueh, Conley and Smoller (14). In the case of

Systems of fixed point sets

Abstract: Let G be a compact Lie group. A canonical method is given for constructing a C-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as C-EilenbergMac Lane spaces and Postnikov towers. 0. Introduction. Let G be a compact Lie group and X a G-space. The equivariant homotopy theory of X is reflected to a remarkable extent in its system of fixed point sets, defined as a functor from a certain category 0G to Top, the category of topological spaces. (Our spaces will be compactly generated weak Hausdorff; they may or may not be equipped with a basepoint, depending on the context.) These functors, or systems, have considerable technical advantages over G-spaces; it is easy to apply most homotopy theoretic constructions to them, whereas in many cases it is unclear how to proceed for G-spaces. It is the purpose of this paper to present a canonical way of recovering from any system a G-space which preserves all the homotopy theoretic structure of the system. This allows us to give easy equivariant versions of some standard topological constructions such as Eilenberg-Mac Lane spaces and Postnikov towers, and to simplify other equivariant constructions.1 1. Statements of the main theorems. Throughout, G is a fixed compact Lie group. Definitions. The category of canonical orbits, written 0G, is a topological category with discrete object space \0G\ = (G/77: 77 a closed subgroup of G} and morphisms the G-maps, topologized by requiring the natural bijection (•) Hoxn0ciG/H,G/K)^{G/K)H to be a homeomorphism. By an Oc-space we shall mean a continuous contravariant functor from 0G to Top; these functors form the objects of a topological category in the usual manner. We will also consider GG-rings, Oc-groups, etc., defined similarly. Definition. Let A1 be a G-space. The fixed point set system of X, written $X, is an GG-space defined as follows: $ *( G/77) = X", Received by the editors October 30, 1981 and, in revised form, April 26, 1982. 1980 Mathematics Subject Classification. Primary 57S10, 55N25.

The initial trace of a solution of the porous medium equation

Abstract: Let u = u(x, t) be a continuous weak solution of the porous medium equation in Rd X (0, TI for some T > 0. We show that corresponding to u there is a unique nonnegative Borel measure p on Rd which is the initial trace of u. Moreover, we show that the initial trace p must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then Ix 2 as Ix ) 00. Introduction. Let u = u(x, t) be a nonnegative solution of the equation of heat conduction (1) ~~~~~~au/at = Au in ST Rd X (0, T] for some T > 0. A consequence of the Widder representation theorem is the existence of a unique nonnegative Borel measure p such that (2) lim (x, t)(p(x) dx = (x)p(dx) for all test functions q E CO(Rd) [1]. That is, every nonnegative solution u of (1) has a unique Borel measure p as initial trace in the sense of (2). The measure p is a-finite and satisfies the growth condition (3) Lde-2/4Tp(d{) 1 is a constant. In addition, we show that the initial trace p must satisfy a growth condition analogous to (3) which limits the amount of mass which p can place at I x I = + x . Specifically, there exists a constant C > 0 depending only on d and m such that (5) f p(dx) ? C{rK/(m-l)T-l/(m-l) + Td/2UK/2 (0, T)} {x: IxI 0 and let g(x, t) = (47t)-d/2e_lX12/41 It is known [1] that the function u(x, t) _f g(x (, t)p(d() is a nonnegative solution of (1) in ST whose initial trace is p. Recently Benilan, Crandall and Pierre [4] have constructed solutions of the porous medium equation (4) whose initial traces are measures which satisfy (5). This extends some earlier work by Kalashnikov [7] for measures with smooth densities on R. Moreover, it shows that the condition (5) is not only necessary but also sufficient for existence of solutions of (4). Even more recently Dahlberg and Kenig [9] have proved uniqueness for the solutions constructed in [4]. 1. Comparison principles. Throughout this paper we will be dealing with continuous weak solutions of the porous medium equation (1.1) au/at =A (Um) in ST XRd X (0, T] for some T > 0. We always assume that m > 1. Specifically, a function u = u(x, t) is said to be a continuous weak solution of (1.1) in ST if u is nonnegative and continuous in ST and satisfies the integral identity (1.2) ffAqRdx (um p+ U-at) dx dt =| uP dx-f ucp dx for all Ti such that 0 0, and v satisfies the integral identity (1.5) ff(vlA9 + v a) dxdt Tfj gnl aLids dt +fv p dx-fgcp dx Br aIB t=T IB t=T for all p E C' 0(F) n C2'1(F) with 9p = 0 on aBr X [Tr, T2]. In (1.5), a/av denotes the exterior normal derivative on aBr. There is an a priori comparison principle for solutions of (1.4). This is the content of the next result, which is simply a restatement of Theorem 12 of [3]. PROPOSITION 1. 1 (LOCAL COMPARISON PRINCIPLE). For i = 1 and 2, let v, denote a solution of (1.4) with boundary values g,. If g1 g on ap F, then v in F. Proposition 1.1 is proved in [3] for x E R, but there is no difficulty in extending the proof for x E Rd. A proof for x E Rd is also given in [9]. The local comparison principle, Proposition 1.1, can be applied to continuous weak solutions of (1.1) since, as we show next, every such solution is also a solution of (1.4) with boundary values g = ulapr. PROPOSITION 1.2. Let u be a continuous weak solution of (1.1) is ST for some T > 0. For any cylinder r of the form (1.3), u is' a weak solution of (1.4) in F with boundary values g =uap. PROOF. It suffices to show that the weak solutions of (1.1) satisfy (1.5). Fix t E Rd, r E R+, 0 C T 2, (d 2)(r q)d-2(rE)d-2 We d-2 d-2 (r ) _-(re)d As a distribution on Rd, A4 is a signed measure. Specifically, (A4?n)(dx) WfJ Sr-q(dI x 1)r-3(dI X I)} dw, where d, denotes the surface element on the unit ball in Rd and 8x(d I x 1) denotes the Dirac measure concentrated at I x I= A. Similar expressions can be derived for d = I and 2. Let ky(x) be a C' symmetric averaging kernel with support in BY(O) and define 4Q,(x ky(x )4e( ) dt.

Spaces of complex null geodesics in complex-Riemannian geometry

Abstract: The notion of a conlplex-Rienlanmianl 11-mcanifold, meaning a complex )1-manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point, is briefly developed. It is shown that, provided ii > 4, every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal-bundle type in an associated space of comple.x lIull geodesics. This relationship between local geometry and global complex analysis is stable under deformations. Introduction. The present work provides a general framework analogous to (but distinct from) Penrose's twistor correspondence (Penrose [15], Atiyah et al. [2]) for the study of real-analytic pseudo-Riemannian geometry on manifolds whose dimension exceeds three, in which points of the original manifold correspond to compact complex submanifolds (quadrics) in an associated complex manifold. The natural intermediate step in the construction is to consider complexification of the geometry, leading to the notion of a complex-Riemannian manifold (implicitly present, for instance, in the work of Penrose). In fact, it is really the conformal structure that is of fundamental importance, and so, having briefly introduced holomorphic metrics in Chapter I, we proceed to establish the fundamental facts of holomorphic conformal geometry in Chapter II. Finally, in Chapter III we develop our generalized " twistor" correspondence. A brief historical note is in order. A special case of this correspondence (where the complex-Riemannian manifold is complex Euclidean 4-space) proved to be useful for the study of Yang-Mills fields (Issenberg et at. [7], Witten [18]) several years ago. In the wake of this discovery, the present author first sketched his ideas on the subject in 1979 in the informal Twistor Newsletter of the Oxford Mathematical Institute; a more fully developed version became the author's Ph. D. thesis (LeBrun [10]), which forms the basis of much of the present, more refined, paper. (Several results are completely new, however, such as the theorems in 111.4, 5.) In dimension four, some recent results have shown these notions to be of genuine physical interest; within the last year, a beguiling theory of the Dirac (Manin [13]) and wave (LeBrun [11]) operators, fitting neatly within this general framework, has emerged. For the present, however, we will examine only the big picture, relegating this (arguably more beautiful) restricted case to treatment elsewhere. Received by the editors March 10, 1982. 1980 Mathematics Subject Classification. Primary 32G10, 53A30; Secondary 32D15, 53C15, 53C22, 83C99. (D1983 American Mathematical Society 0002-9947/82/00001440/$06.75

Neighborhoods of algebraic sets

Abstract: In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighbor- hoods will be defined as level sets of polynomial or semialgebraic functions. Introduction. Let M be an algebraic set in real n-space R", and let X be a compact algebraic subset of M containing its singular locus, if any. An algebraic neighborhood of A' in M is defined to be a_1(0, 8), where o > 0 is sufficiently small and a: M -> R is a proper polynomial function for which a > 0 and a_1(0) = X. Such an a will be called a rug function. Occasionally we will need rational or analytic a, but this is not a significant generalization. Algebraic neighborhoods always exist. The curve selec- tion lemma is used to prove uniqueness ; anyone familiar with (Milnor 2) will recognize the technique. Since uniqueness is a crucial result, and since there are a few trouble- some small points, this proof is given in detail. The uniqueness theorem shows that the "link" of a singularity of an algebraic set M is independent of the embedding of M in its ambient space; I have been unable to find a proof of this result in the litera- ture. In addition, an algebraic neighborhood of a nonsingular X in M is shown to be a tubular neighborhood in the sense of differential topology. This material is in §1. In §2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space. As an application, it is shown that when M is an affine algebraic set with projective completion M, then the complement in M of a large ball centered at the origin of affine space is an alge- braic neighborhood of the intersection of M with the hyperplane at infinity. In §3, the theory is generalized to the case where M is a semialgebraic subset of R" and X is a compact semialgebraic subset of M. (For example, X could be a non- isolated singular point of M.) A semialgebraic neighborhood of X in M is defined to be a-1(0, o), where o > 0 is sufficiently small and a: M -> R is a proper semialgebraic function for which a > 0 and a_1(0) = X. Again these neighborhoods exist and are unique ; this is shown by mimicking the proof in the algebraic case, using semialgebraic

Applications of q-lagrange inversion to basic hypergeometric series

Abstract: A family of g-Lagrange inversion formulas is given. Special cases include quadratic and cubic transformations for basic hypergeometric series. The g-analogs of the so-called "strange evaluations" are also corollaries. Some new Rogers- Ramanujan identities are given. A connection between the work of Rogers and Andrews, and ^-Lagrange inversion is stated.

Graphs with relations, coverings and group-graded algebras

Abstract: The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced. The object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra. The linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology, group theory or combinatorics. In the representation theory of Artin algebras the association to each algebra of a finite directed graph, called the quiver of the algebra, has been a useful tool. The reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra. §1 gives a slight extension of these concepts to finitely generated algebras. The main emphasis of the paper is to show that the theory of coverings of graphs with relations, introduced by C. Riedtmann (9) and expanded by P. Gabriel (2), and the theory of group-graded algebras are essentially the same. Although the original connection between coverings and gradings was inspired by the similarity of results for Z-graded Artin algebras (3,4) and P. Gabriel's announced results (2), the context of this paper is more general and deals with all finitely generated algebras over a field. We associate to each such algebra a finite directed graph which we still call a quiver of the algebra. We show that for each regular covering T of a quiver T0 of an algebra A, with certain prescribed restrictions, we get a G-grading of the algebra A, where G is the automorphism group of the covering Y over ro. Conversely, given a certain type of G-grading of A, where G is a group, we construct a regular covering Y of the quiver T0 of the algebra such that G is isomorphic to the automorphism group of T over ro. Furthermore, if Y is a regular covering of T0 with automorphism group G, we show that the category of representations of Y satisfying a certain set of relations is equivalent to the category of finite-dimensional graded G-modules. We

Approximation by smooth multivariate splines

Abstract: : One of the important properties of univariate splines is that in most senses smooth splines approximate just as well as do piecewise polynomials on the same mesh. This report shows this to be untrue in the multivariate setting. In particular, it details the cost in approximating power one may have to pay for the luxury of a smooth piecewise polynomial approximant. In an extreme case, piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or = RHo continuous will fail to approximate certain smooth functions at all (as the grid goes to zero) unless RHo is kept below (r-3)/2. During the analysis of approximation on a certain regular triangular grid, a novel kind of bivariate B-spline is introduced. This B-spline, in contrast to the established multivariate B-spline derived from a simplex, can be made to have all its breaklines in a given regular grid. This makes it a prime candidate for use in the construction of smooth multivariate piecewise polynomial approximation, and its properties will be explored further.

Meromorphic functions that share four values

Abstract: An old theorem of R. Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Mbbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.

Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution

Abstract: The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if / is the momentum mapping for a Hamiltonian torus action on a symplectic manifold M and Q is the fixed point set of an antisymplectic involution of M leaving / invariant, then /((?) — f(M) = a convex polytope. Also it is proved that the coordinate functions of /are tight, using \"half-turn\" involutions of Q.

Strongly Cohen-Macaulay schemes and residual intersections

Abstract: This paper studies the local properties of closed subschemes Y in Cohen-Macaulay schemes X such that locally the defining ideal of Y in X has the property that its Koszul homology is Cohen-Macaulay. Whenever this occurs Y is said to be strongly Cohen-Macaulay in X. This paper proves several facts about such embeddings, chiefly with reference to the residual intersections of Y in X. The main result states that any residual intersection of Y in X is again Cohen-Macaulay. Introduction. Our purpose in this paper is to investigate a property of a closed subscheme Y in a Noetherian Cohen-Macaulay scheme X, which we call strong Cohen-Macaulayness. This is a local property which we first describe for a local ring. Let X = Spec(R), where R is a Cohen-Macaulay local ring, and let Y = Spec(R/I)

Sign changes in harmonic analysis on reductive groups

Abstract: Let G be a connected reductive group over a field F. In this note the author constructs an element e(G) of the Brauer group of F. The square of this element is trivial. For a local field, e(G) may be regarded as an element of { ?I} and is needed for harmonic analysis on reductive groups over that field. For a global field there is a product formula. The purpose of this note is to provide a cohomological interpretation of the sign changes in harmonic analysis on reductive groups over local fields that are caused by inner twistings. The best understood example occurs in the work of Shelstad [8] on inner twistings of real groups. Theorem 6.3 of that paper gives the following character identity: -() = ( 1)q(G')-q(G)X( ) Here G is a connected reductive group over R, G' is its quasi-split inner form, X., and X. are stabilized characters of matched tempered L-packets of G'(R) and G(R), and y' and y are matched regular semisimple elements of G'(R) and G(R). The sign change in the character identity is ()q(G')-q(G) where q(G) is one half of the dimension of the symmetric space attached to G (more precisely, attached to the simply connected cover of the derived group of G). Although q(G) may be only half integral, the difference q(G') q(G) is always integral. It is reasonable to expect an analogous character identity for groups G over a nonarchimedean local field F. We can determine what the sign change must be by considering the Steinberg characters of G(F) and G'(F), since these should be related by the character identity. Let r(G) denote the F-rank of the derived group of G; the Steinberg character of G(F) has value ( I)r(G) on the elliptic regular set of G(F), and hence the sign change in the character identity must be (_l)r(G')-r(G). There are two groups for which the character identity has been proved: the multiplicative group of a central division algebra of dimension d2 over F for d = 2 (see Proposition 15.5 on p. 484 of [4]) and d = 3 (see Theorem 1 of [1]). Later in this paper we will use a cohomological construction to define a sign e(G) = 1 for any connected reductive group G over a local field. Let G' denote a quasi-split inner form of G. We will show that e(G) = (_l)r(G')-r(G) when the base Received by the editors May 10, 1982. 1980 Mathematics Subject Classification. Primary 22E50. Partially supported by the National Science Foundation under Grant MCS 78-02331. ?1983 American Mathematical Society 0002-9947/82/0000-11 93/$03.25

Existence of infinitely many solutions for a forward backward heat equation

Abstract: Let 4 be a piecewise linear function which satisfies the condition s40(s) 2 cs2, c > 0, s E R, and which is monotone decreasing on an interval (a, b) c R+. It is shown that for f E C2[0, 11, with max f' > a, there exists a T > 0 such that the initial boundary value problem ut =00-."x)x UX(OS t)-U,(ls t) = , U(. 0)-f, has infinitely many solutions u satisfying 11Ul u x u x U, 1l 2 6 c(f, 4)) on [0, 1I X [0, T]. 0. Introduction. Consider the initial boundary value problem Ut = (U.,x (X, t) E= [0, 1] X [0, T], (1) ~~~U.(?' t) = uj(l t) = 0, t E [0, T], u(x,0) = f(x), x E [0, 1]. If 4 is strictly monotone increasing with 4)' > c > 0, (1) has a unique solution which is, roughly speaking, as smooth as the function o. On the other hand, if 4)' -c 0. This assumption allows 4 to have monotone decreasing parts (e.g. the model cubic et)(S) = 1S3 S2 2s 4)s~3-42 +2s). A natural and interesting question is whether problem (1) has a solution if the usual assumption 4' > 0 is replaced by the weaker coercivity condition (c). Under this hypothesis J. Bona, J. Nohel and L. Wahlbin [BNW, HN] obtained several a Received by the editors April 2, 1982 and, in revised form, May 14, 1982. 1980 Mathematics Subject Classification. Primary 35K55, 35K65.

Uniformly exhaustive submeasures and nearly additive set functions

Abstract: Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that co (or any E.-space) is a T;space, i.e. cannot be realized as the quotient of a nonlocally convex F-space by a one-dimensional subspace.

Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators

Abstract: We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators, max¡^k 0.

Multi-invariant sets on tori

Abstract: Given a compact metric group G, we are interested in those semigroups L of continuous endomorphisms of G, possessing the following property: The only infinite, closed, :-invariant subset of G is G itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization-for the case of finitedimensional tori-of those commutative semigroups with the aforementioned property.

Mixed Hodge Structures

Abstract: With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's mixed Hodge structure on cohomology of complex algebraic varieties is described.