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In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free S 1 S^{1} -actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author’s previous papers.

Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52

We study the elliptic equation Δ u + k u − K u p = 0 \Delta u + ku - Ku^{p} = 0 on complete noncompact Riemannian manifolds with K K nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space R n {\mathbf {R}}^{n} and the hyperbolic space H n {\mathbf {H}}^{n} are carried out when k k is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.

If ¢ is a generic cubic rnetaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to X is an Euler product of degree 4 that has a functional equation and rnerornorphic continu.ation to the whole cornplex plane. This correspondence is obtained by convolving (b with the cubic 0-function on GL(3) in a Shirnura type RankinSelberg integral. 0. INTRODUCT[ON Suppose 0 is a metaplectic automorphic form of minimal level on the 3-fold cover of GSp(4) that is an eigenfunction of all the Hecke operators. If 0 has any non-zero Whittaker coefficients, then 0 is called generic. In this case, this paper will show tha1; there is a Dirichlet series in the Whittaker coefficients of 0 that has a formula1;ion as a degree 4 Euler product. Moreover, this Euler product has a meromorphic continuastion to the whole complex plane. This association of the Euler product with 0 will be obtained via a Shimura type JRankin-Selberg integral involving 0 and a H-function on the 3-fold cover of GL(3). Historically) the problem of associating an Euler product which has meromorphic continuation and functional equation with a metaplectic automorphic form originated with the work of Shimura [Shi]. More specifically, suppose f (z). = E a(n) qn is a holomorphic modular form of half-integral weight k/2, which is an eigenform of the Hecke operators Tp2, i.e. Tp2 f = )pf. Then via a JRankin-Selberg integral of the form f __ (0.1) | f (z) H(z) E(z, s) dz, where H(z) is a classical theta function and E(z, s) is an integral weight Eisenstein series, Shimura obtains an Euler product of the form (0.2) t| (1 _ Ap ps + pk-2-2s)-1 p The analytic continuation and functional equation of this Euler product follow from the siirlilar properties of E(z, s) in (0. 1) . Bump and Hoffstein [BH2] have subsequently extended these techniques of [Shig to GL(3) by finding a Rankin-Selberg integral of a metaplect1c automorphic form on the 3-fold cover of GL(3) which produces an Euler product of degree 3. Just as in [ShiX, this Euler product is shown to have meromorphic continuation to the whole Received by the editors November 14, 1995 and, in revised forrn, May 21, 1996. 1991 Mathematics S?lbject Classification. Prirnary llF55, llF30. @1998 Ameri( an Mathematical Society 975 This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms THOMAS GOETZE 976 complex plane and to have a functional equation under s 1-s. The integral which represents this Euler product involves the 0-function on the 3-fold cover of GL(3) over the field Q (e21rt/3) ? wllich has been studied independently by Proskurin [Pr] and Bump and HofEstein [BH12. In addition, Bump and HofEstein [BH2] have conjectured that Euler products with meromorphic continuation and functional equation may be obtained by convolving metaplectic automorphic forms on the n-fold cover of GL(r) against H-functions on the n-fold cover of GL(n). This was carried out in [BH3] in the case r-2 and n > 2. Friedberg and Wong [FrW] have also used Shimura's method to associate an Euler product to a generic metaplectic automorphic form on the double cover of the symplectic group GSp(4). They have found an integral (inspired by Novo dvorsky's GSp(4)xGL(2) convolution) involving ametaplectic automorphic form on the double cover of GSp(4), the H-function on the double cover of GL(2), and a (non-metaplectic) Eisenstein series on GL(2), that yields a degree 5 Euler product. This Euler product is shown to have meromorphic continuation and a functional equation, and furthermore it has the same local Euler factors as the L-function of an automorphic form on GSp(4). As in [Shi, BH2], the Euler product found by Friedberg and Wong is explicitly constructed from the Whittaker coeEcients of the metaplectic automorphic form. Alternatively, Flicker [Fli], Kazhdan and Patterson [KaP2], and Flicker and Kazhdan [FliKa] have used the trace formula to generalize [Shi] by showing that (in many situations) there exists a correspondence between metaplectic automorphic forms and (non-metaplectic) automorphic forms. Indeed: Shimura actually proves in [Shi] that the Euler product (0.2) is the L-function of a holomorphic integral weight modular form. In using the trace formula, however: explicit information about the interplay between the metaplectic Fourier coefficients and the corresponding L-functions (see (0.2)) is not obtained. If a generalized Shimura correspondence does exist between generic metaplectic and non-metaplectic automorphic forms, then the associated Euler products obtained by Bump-Hofistein, Friedberg-Wong, and this paper will be the L-functions of the corresponding non-metaplectic forms. There is evidence that the degree 4 Euler product obtained in this paper is the L-function of an automorphic form on GSp(4) Savin [Sa] has shown that there is an algebra isomorphism between the local Iwahori Hecke algebra of GSp(4) and the local Iwahori Hecke algebra on the 3-fold cover of GSp(4) This suggests that; if a Shimura correspondent exists in this situation, it should be an automorphic form on GSp(4)e Since there is a representation of degree 4 on the L-group of GSp(4): automorphic forms on GSp(4) will have natural L-functions with Euler products of degree 4 In this sense, having a degree 4 Euler product is consistent with Savin's results. The main results of this paper will be found in Theorems 3.1 and 5.1, which are summarized as follows: Mairl Theorem. Suppose Q is a generic metaplectic C?lsp form of minimal level on the S-fold cover of G8p(4) that is an eigenf?lnction of all the lTecke operators. Then there is a degree J E?ller prod?lct, with a meromorphic contin?ltation, which cctn be explicitly constr?lcted from the Whittaker coefficients of f0Je This association is realized as a Shim?lra type Rankin-Selberg integral of 0 against an Eisenstein series ind?lced from a f)-f?lnction on the S-fold cover of GL(3). This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the p-adic case when p is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke L-functions in separate papers. We also recover Moen’s results on when a local theta lifting from U(1) to itself does not vanish. 0. Introduction and Notation Let E/F be a quadratic extension of local fields. If (V, ( , )) is an Hermitian space over E, and (W, 〈 , 〉) is a skew-Hermitian space over E, the unitary groups G = G(W ) and G′ = G(V ) form a reductive dual pair in Sp(W), where W = V ⊗W has the symplectic form 12 trE/F ( , ) ⊗ 〈 , 〉 over F . According to a well-known result, this dual pair splits in the metaplectic cover of Sp(W), and thus has a Weil representation ω. We consider the very special case in which dimE V = dimEW = 1. In this case, G ∼= G′ = U(1) = E is compact and abelian, where E is the kernel of the norm map N : E∗ −→ F ∗. So irreducible representations of G are just characters, and the Weil representation has a direct sum decomposition

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{P\Gamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines., Comment: Sections 3-6 rewritten and restructured to fill in gaps in, streamline, and avoid repetition of arguments. To appear in Transactions of the American Mathematical Society

Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem., 29 pages; some references have been added; to appear in Trans. Amer. Math. Soc

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group W W and the Hecke algebra of double cosets of a parabolic subgroup of W . W. For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

An index formula is proved for elliptic systems of P.D.E.’s with boundary values in a simply connected region Ω \Omega in the plane. Let A \mathcal {A} denote the elliptic operator and B \mathcal {B} the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function Δ B + \Delta ^{+}_{{\mathcal {B}}} defined on the unit cotangent bundle of ∂ Ω \partial \Omega was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function Δ B + \Delta ^{+}_{{\mathcal {B}}} have invertible values. In the present paper, the index of ( A , B ) ({\mathcal {A}},{\mathcal {B}}) is expressed in terms of the winding number of the determinant of Δ B + \Delta ^{+}_{{\mathcal {B}}} .

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.’s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.’s. Introduction Let A denote an elliptic operator in Ω and let B be a boundary operator, where Ω is a bounded domain in R. In this paper several versions of the algebraic condition for the Fredholm property of (A,B) are formulated, equivalent to the Lopatinskii condition of [Lo]. The main new result is the following. A square matrix function ∆+B defined on the unit cotangent bundle of ∂Ω is constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the following condition: the function ∆+B must have invertible values. The proofs use the theory of matrix polynomials due to Gohberg, Lancaster, Rodman, and others; for instance, see [GLR], Chapter 14 in [LT], and Chapter 6 in [R]. There is a natural map defined by (A,B) 7→ A, going from the space of elliptic boundary value problems to the space of elliptic operators. Now let Aτ , 0 ≤ τ ≤ 1, be a homotopy of elliptic operators and let B0 be a boundary operator such that (A0,B0) satisfies the Lopatinksii condition. We will show that the given homotopy of elliptic operators can be lifted to a homotopy (Aτ ,Bτ ) in the space of elliptic boundary value problems satisfying the Lopatinskii condition. This result is proved in Theorem 7.3 and was motivated by the article [Ge]. The key element in the proof, i.e. the construction of the boundary operators Bτ , requires that we use pseudodifferential operators on ∂Ω and the theory of matrix polynomials mentioned above. Received by the editors August 16, 1994 and, in revised form, February 12, 1996. 1991 Mathematics Subject Classification. Primary 35J45, 35J55, 15A22.

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

An (ordinary) P P -partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur’s S S -functions. In this paper, we introduce and develop a theory of enriched P P -partitions; like ordinary P P -partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched P P -partitions given here are the tableaux associated with Schur’s Q Q -functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra. We use jet sheaves of finite order (or equivalently infinitesimal neighbourhood of finite order) and related exterior differential forms to define cohomologies for singularities of mappings and varieties. (See [2], [3],[4], [5],[6],[7] and this paper.) They are new invariances. They were first defined and studied in [2] for smooth functions. In this paper we study jet cohomology HP(QVk. 0) defined by partial differential forms with respect to y for isolated hypersurface singularities.They can be computed by linear algebra (See Theorem 1 and Theorem 2.) and can distinguish singularities whose classical cohomological invariances and Milnor numbers coincide. For example, if X is a quasi-homogeneous hypersurface with isolated singularity 0. QX , is the complex defined by ordinary exterior differential forms. HP(Q> o) = 0, p > 1, and H0(Q0o) = C. (See [1].) But the cohomology groups defined by jets can tell. For example: (1) f = x 5 + x4. Milnor number ,u = 12. If the order of jets k = 3, dimcH0(Q;3-.0) = 6. (2) f = x3 + x7. ,a = 12. If k = 3, dimcHo (Q'V3 .,0) = 7. It is interesting that the cohomological groups defined by jets are determined by higher derivatives of f. (See Theorem 1 and Theorem 2.) All results and proofs in this paper are true not only for complex holomorphic cases but also for real analytic cases. The author would like to express his gratitude to Phillip A. Griffiths for enlightening discussions with him and various supports he and the Institute for Advanced Study offered during the time the author visited IAS. The main results of this paper are as follows. Let F= _~ 1 ak)If | ka! i9xa (

A numerical Godeaux surface is a surface of general type with invariants p g = q = 0 p_g =q =0 and K 2 = 1 K^2 =1 . In this paper the moduli space of a numerical Godeaux surface with order two torsion is computed to be eight-dimensional; whether or not the moduli space of such a surface is irreducible is still unknown. The surface in this paper is constructed as one member of a four parameter family of double planes. There is a natural involution on the surface, inherited from the double plane construction, which acts on the moduli space. We show that the invariant subspace is four-dimensional and coincides with the family of double planes.

Let K K be a compact connected semi-simple Lie group, let G = K C G = K_{\mathbf C} , and let G = K A N G = KAN be an Iwasawa decomposition. To a given K K -invariant Kaehler structure ω \omega on G / N G/N , there corresponds a pre-quantum line bundle L {\mathbf L} on G / N G/N . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections O ( L ) {\mathcal O}({\mathbf L}) as a K K -representation space. We defined a K K -invariant L 2 L^2 -structure on O ( L ) {\mathcal O}({\mathbf L}) , and let H ω ⊂ O ( L ) H_\omega \subset {\mathcal O}({\mathbf L}) denote the space of square-integrable holomorphic sections. Then H ω H_\omega is a unitary K K -representation space, but not all unitary irreducible K K -representations occur as subrepresentations of H ω H_\omega . This paper serves as a continuation of that work, by generalizing the space considered. Let B B be a Borel subgroup containing N N , with commutator subgroup ( B , B ) = N (B,B)=N . Instead of working with G / N = G / ( B , B ) G/N = G/(B,B) , we consider G / ( P , P ) G/(P,P) , for all parabolic subgroups P P containing B B . We carry out a similar construction, and recover in H ω H_\omega the unitary irreducible K K -representations previously missing. As a result, we use these holomorphic sections to construct a model for K K : a unitary K K -representation in which every irreducible K K -representation occurs with multiplicity one.

Let L ^ \widehat {L} be a very ample line bundle on an n n -dimensional projective manifold X ^ \widehat {X} , i.e., assume that L ^ ≈ i ∗ O P N ( 1 ) \widehat {L}\approx i^*\mathcal {O}_{\mathbb {P}_ N}(1) for some embedding i : X ^ ↪ P N i:\widehat {X}\hookrightarrow \mathbb {P}_ N . In this article, a study is made of the meromorphic map, φ ^ : X ^ → Σ \widehat {\varphi } : \widehat {X}\to \Sigma , associated to | K X ^ + ( n − 2 ) L ^ | |K_{\widehat {X}}+(n-2)\widehat {L}| in the case when the Kodaira dimension of K X ^ + ( n − 2 ) L ^ K_{\widehat {X}}+(n-2)\widehat {L} is ≥ 3 \ge 3 , and φ ^ \widehat {\varphi } has a 1 1 -dimensional image. Assume for simplicity that n = 3 n=3 . The first main result of the paper shows that φ ^ \widehat \varphi is a morphism if either h 0 ( K X ^ + L ^ ) ≥ 7 h^0(K_{\widehat X}+\widehat L)\geq 7 or κ ( X ^ ) ≥ 0 \kappa (\widehat {X})\geq 0 . The second main result of this paper shows that if κ ( X ^ ) ≥ 0 \kappa (\widehat X)\ge 0 , then the genus, g ( f ) g(f) , of a fiber, f f , of the map induced by φ ^ \widehat \varphi on hyperplane sections is ≤ 6 \leq 6 . Moreover, if h 0 ( K X ^ + L ^ ) ≥ 21 h^0(K_{\widehat X}+\widehat L)\ge 21 then g ( f ) ≤ 5 g(f)\leq 5 , a connected component F F of a general fiber of φ ^ \widehat \varphi is either a K 3 K3 surface or the blowing up at one point of a K 3 K3 surface, and h 1 ( O X ^ ) ≤ 1 h^1(\mathcal {O}_{\widehat X})\le 1 . Finally the structure of the finite to one part of the Remmert-Stein factorization of φ ^ \widehat \varphi is worked out.

In this paper we consider a family of convex sets in R n \mathbf {R}^{n} , F = { S ( x , t ) } \mathcal {F}= \{S(x,t)\} , x ∈ R n x\in \mathbf {R}^{n} , t > 0 t>0 , satisfying certain axioms of affine invariance, and a Borel measure μ \mu satisfying a doubling condition with respect to the family F . \mathcal {F}. The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of F . \mathcal {F}. This is achieved by showing first a Besicovitch-type covering lemma for the family F \mathcal {F} and then using the doubling property of the measure μ . \mu . The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to F . \mathcal {F}.

This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current I I , of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on I I . These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to ± ∞ \pm \infty , contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165–174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.

We study the dimension of the space of homomorphisms from a given representation X of a quiver to a general representation of dimension vector 3. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as 3 increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring. In this paper we prove a number of results about hom(X, 3), the dimension of the space of homomorphisms from a fixed representation X of a quiver to a general representation of dimension vector 3. Our basic result relates this number to the dimension of a subset of a Grassmannian of submodules of X. This result is in the spirit of A. Schofield's paper on general representations of quivers [S2], and generalizes one of his results. We derive two corollaries concerning the asymptotic behaviour of hom(X, r/3), of interest in themselves, and also because of their connection with another theorem of Schofield, that the homological epimorphisms from a path algebra to a simple artinian ring are in 1-1 correspondence with indivisible Schur roots. (A homological epimorphism is a ring homomorphism R -* S with S OR S -S and Tori(S, S) = 0 for i > 0. This correspondence was announced by Schofield in a lecture in March 1995 in Krippen, Germany, but actually dates from 1991.) In fact our second corollary is already known, proved by Schofield and used in the proof of the correspondence, but his proof of the corollary involves difficult results about semi-invariants of quivers, whereas the proof here is quite elementary. Let K be an algebraically closed field, Q a finite quiver with vertex set I, and let (-,-) be the Ringel form on Z'. If 3 E N' we write RePKQW() = 17 Hom(K8('), K:(j)) a:i-j for the configuration space of representations of Q of dimension vector 3, and if y E RePKQ(/) we write Ky for the corresponding left KQ-module. If X is a finitely generated left KQ-module and y E RePKQ (/3) then Hom(X, Ky) is a finite dimensional vector space. Moreover the function y F dim Hom(X, Ky) is an upper semicontinuous function on RePKQ(/3), and it follows that the minimum value of this function, denoted hom(X,/3), is also its general value. The rank of a homomorphism X -* Ky is the dimension vector of its image. For any Ol E N' the set of homomorphisms of rank at most Ol is a closed subset of Hom(X, Ky). It follows that there is a unique maximal rank Yx,y of homomorphisms from X to Ky, Received by the editors July 21, 1995. 1991 Mathematics Subject Classification. Primary 16G20; Secondary 14M15. ?1996 American Mathematical Society

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) ∂ t u + u ∂ x u + ∂ x 3 u = f \begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*} on the interval 0 ≤ x ≤ 2 π , t ≥ 0 0\leq x\leq 2\pi , \, t\geq 0 , with periodic boundary conditions (ii) ∂ x k u ( 2 π , t ) = ∂ x k u ( 0 , t ) , k = 0 , 1 , 2 , \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*} where the distributed control f ≡ f ( x , t ) f\equiv f(x,t) is restricted so that the “volume” ∫ 0 2 π u ( x , t ) d x \int ^{2\pi }_0 u(x,t) dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f f is allowed to act on the whole spatial domain ( 0 , 2 π ) (0,2\pi ) , it is shown that the system is globally exactly controllable, i.e., for given T > 0 T> 0 and functions ϕ ( x ) \phi (x) , ψ ( x ) \psi (x) with the same “volume”, one can alway find a control f f so that the system (i)–(ii) has a solution u ( x , t ) u(x,t) satisfying \[ u ( x , 0 ) = ϕ ( x ) , u ( x , T ) = ψ ( x ) . u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) . \] If the control f f is allowed to act on only a small subset of the domain ( 0 , 2 π ) (0,2\pi ) , then the same result still holds if the initial and terminal states, ψ \psi and ϕ \phi , have small “amplitude” in a certain sense. In the case of closed loop control, the distributed control f f is assumed to be generated by a linear feedback law conserving the “volume” while monotonically reducing ∫ 0 2 π u ( x , t ) 2 d x \int ^{2\pi }_0 u(x,t)^2 dx . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

This paper deals with a reducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) acts on the formal power series ring via ( 0.1 ) (0.1) . Then I ( f ) = ( ℓ i 1 ) ⊕ ( ℓ i 2 ) ⊕ ⋯ ⊕ ( ℓ i s ) I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}}) modulo some one dimensional s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representations where ( ℓ i ) (\ell _{i}) is an irreducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representation of dimension ℓ i \ell _{i} or empty set and { ℓ i 1 , ℓ i 2 , … , ℓ i s } ⊆ { ℓ 1 , ℓ 2 , … , ℓ r } \{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\} . Unlike classical invariant theory which deals only with irreducible action and 1–dimensional representations, we treat the reducible action and higher dimensional representations succesively.

This paper deals with the problem of describing the set M M of all solutions of an equation over a free semigroup S S . The standard way to do this involves the introduction of auxiliary equations containing polynomials in natural number parameters of arbitrarily high degree. Since S S has a solvable word problem, M M must be computable. However, M M cannot necessarily be computed from the standard description of M M . The present paper shows that the only polynomials needed to describe M M are just products of one parameter by a linear combination of some other parameters. The resulting simplification of the standard description of M M clearly can be used to compute M M .

Let Ω \Omega be a non-compact complex manifold of dimension n n , ω = ∂ ∂ ¯ Ψ \omega =\partial \overline \partial \Psi a Kähler form on Ω \Omega , and K α ( x , y ¯ ) K_\alpha ( x,\overline y) the reproducing kernel for the Bergman space A α 2 A^2_\alpha of all analytic functions on Ω \Omega square-integrable against the measure e − α Ψ | ω n | e^{-\alpha \Psi } |\omega ^n| . Under the condition \[ K α ( x , x ¯ ) = λ α e α Ψ ( x ) K_\alpha ( x,\overline x)= \lambda _\alpha e^{\alpha \Psi (x)} \] F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on ( Ω , ω ) (\Omega ,\omega ) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = C n \Omega = \mathbf {C} ^n and Ω \Omega a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → + ∞ \alpha \to +\infty . This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in C n \mathbf {C}^n . Along the way, we also fix two gaps in Berezin’s original paper, and discuss, for Ω \Omega a domain in C n \mathbf {C}^n , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure | ω n | |\omega ^n| .

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\"omer., Comment: 20 pages

Modular and mock modular forms possess many striking p p -adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent p p -adic modular forms, Candelori and Castella showed this leads to p p -adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct p p -adic analogs of harmonic Maass forms of weight 0 0 with square free level. Although our approaches differ, where the two theories intersect the forms constructed are the same. However our analytic construction defines these functions on the full super-singular locus as well as on the ordinary locus. As with classical harmonic Maass forms, these p p -adic analogs are connected to weight 2 2 cusp forms and their modular derivatives are weight 2 2 weakly holomorphic modular forms. Traces of their CM values also interpolate the coefficients of half-integer weight modular and mock modular forms. We demonstrate this through the construction of p p -adic analogs of two families of theta lifts for these forms.

We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.

In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time., Comment: 17 pages, 6 figures

In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character χ \chi to squarefree modulus q q , we prove the following upper bound: | ∑ 1 ⩽ n ⩽ N χ ( n ) | ⩽ c q log ⁡ q , \begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*} where c = 1 / ( 2 π 2 ) + o ( 1 ) c=1/(2\pi ^2)+o(1) for even characters and c = 1 / ( 4 π ) + o ( 1 ) c=1/(4\pi )+o(1) for odd characters, with an explicit o ( 1 ) o(1) term. This improves a result of Frolenkov and Soundararajan for large q q . We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of log ⁡ q \log {q} as in previous approaches and is an important factor for fully explicit bounds.

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in R d \mathbb {R}^d , killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is invertible with high probability. Note that for directed graphs this was recently proved by Huang. It also gives an alternate proof of a theorem of Backhausz and Szegedy., Comment: We improved the presentation of the paper. More details are given in several proofs

Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.

In 1975, Kaplansky conjectured that a finite-dimensional semisimple Hopf algebra is necessarily involutory. Twelve years later, Larson and Radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics. The goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites, no "harpoons", and gratifyingly few "hits". Let H be a finite-dimensional Hopf algebra over a field K, and let S: H H denote its antipode. Then H is said to be involutory if S2 = idH or equivalently if S is an involution on H. In [K], Kaplansky conjectured that a semisimple Hopf algebra is necessarily involutory, and in a fundamental paper [LR2], Larson and Radford proved this assertion at least for fields of characteristic 0. Furthermore, they showed that if H is semisimple and cosemisimple and if charK = p > (dimK H)2, then H must again be involutory. A nice exposition of this material can be found in [N]. While [LR2] is reasonably brief, it depelids heavily on certain earlier results on characters of Hopf algebras as contained in [L] and [LR 1], and the combined proof is substantial. The goal of this paper is to present an efficient proof of the main results of [LR2] requiring minimal prerequisites. As will be apparent, our proof follows the same basic outline as that of Larson and Radford, and therefore we offer nothing new in this regard. However, by computing traces on certain tensor modules, we are able to obtain the necessary character formulas quite quickly, and this is where the simplifications occur. In some sense, the hero of this approach is the linear transformation T described below. If V is a left H-module, then H 0 V is well known to be H-isomorphic to H 0 VJ, where VJ is equal to V but with the trivial action determined by the counit e . This is proved quite simply by considering the map T: H( V -* H( V given by T:h ov hihi h2v for all h e H, v e V. (h) Indeed, one shows that T is invertible with inverse T-1: h X v E+ hZ X S(h2)v (h) Received by the editors July 13, 1993 and, in revised form, June 8, 1994; originally communicaated to the Proceedings of the AMS by Lance W. Small. 1991 Mathematics Subject Classification. Primary 1 6W30. Research supported in part by NSF Grants DMS-9224662 and DMS-9003374. ? 1995 American Mathematical Society

In a previous paper, it was shown that if F F is a binary form with complex coefficients having degree n ⩾ 3 n \geqslant 3 and discriminant D F ≠ 0 {D_F} \ne 0 , and if A F {A_F} is the area of the region | F ( x , y ) | ⩽ 1 \left | {F(x,y)} \right | \leqslant 1 in the real affine plane, then | D F | 1 / n ( n − 1 ) A F ⩽ 3 B ( 1 3 , 1 3 ) {\left | {{D_F}} \right |^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3}) , where B ( 1 3 , 1 3 ) B(\frac {1} {3},\frac {1} {3}) denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence { M n } \{ {M_n}\} defined by M n = max | D F | 1 / n ( n − 1 ) A F {M_n} = \max |{D_F}{|^{1/n(n - 1)}}{A_F} , where the maximum is taken over all forms of degree n n with D F ≠ 0 {D_F} \ne 0 , is decreasing, and then by showing that M 3 = 3 B ( 1 3 , 1 3 ) {M_3} = 3B(\frac {1} {3},\frac {1} {3}) . The resulting estimate, A F ⩽ 3 B ( 1 3 , 1 3 ) {A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3}) for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence { M n } \{ {M_n}\} . By appealing to the theory of hypergeometric functions, it is shown that M 4 = 2 7 / 6 B ( 1 4 , 1 2 ) {M_4} = {2^{7/6}}B(\frac {1} {4},\frac {1} {2}) and that M 4 {M_4} is attained for the form X Y ( X 2 − Y 2 ) XY({X^2} - {Y^2}) . It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that A F {A_F} is the perimeter of the polygon corresponding to F F . Based on this correspondence and a representation theorem for | D F | 1 / n ( n − 1 ) A F |{D_F}{|^{1/n(n - 1)}}{A_F} , it is conjectured that M n = D F n ∗ 1 / n ( n − 1 ) A F n ∗ {M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}} , where F n ∗ ( X , Y ) = ∏ k = 1 n ( X sin ⁡ ( k π n ) − Y cos ⁡ ( k π n ) ) F_n^*(X,Y) = \prod _{k = 1}^n \left (X \sin \left (\frac {k\pi }{n}\right ) - Y \cos \left (\frac {k\pi }{n}\right )\right ) , and that the limiting value of the sequence { M n } \{ {M_n}\} is 2 π 2\pi .

In this paper, we study the Dirichlet problem at infinity for harmonic maps between complete hyperbolic Hadamard surfaces. We will address the existence and uniqueness questions relating to the problem. In particular, we generalize results in the work of Li-Tam and Wan. 0. INTRODUCTION In a series of papers [L-T 1], [L-T 2] and [L-T 3], the first two authors considered the existence, uniqueness and boundary regularity theory for the Dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces. If we denote Em to be ther m-dimensional hyperbolic space with constant -1 sectional curvature, then using the Poincare' disk model, we can represent Hm by the unit m-disk D}m endowed with the Poincare metric ds2m = 4 ds2 where p denotes the Euclidean distance to the origin and ds2 is the Euclidean metric. The geometric compactification is given by the natural compactification of lEDm by the unit (m 1)-sphere Sm-1. Using this identification, a map from Sm-1 to S`1 can be viewed as a map from infinity of H"m to infinity of Hn. Let us summarize some of the results in [L-T 1], [L-T 2] and [L-T 3] as follows: Theorem (Li-Tam). Let q: S(m-1) -, (n-1) be a C1 map which has nowhere vanishing energy density. Then there exists a unique harmonic extension h Hm H Hn of X which is in CI(lrD"m, EDn"). Also, if X is in Ck,a(S(m-l), s(n-1)), for some 1 < k < m and some 0 < a, then h is in Ck,Y(lD"D, lEDn) for some O < y < a. We would like to mention that the existence part of this theorem for the special case when m = 2 = n and for C3 boundary maps which have nowhere vanishing energy density was independently proved by Akutagawa [A]. Using a different point of view, Wan [W] studied harmonic diffeomorphisms between hyperbolic 2-spaces via their associated Hopf differentials. By using the fact that [C-T] the Gauss map of a constant mean curvature cut in a Minkowski space-time is a harmonic map into hyperbolic space, Wan established that there is a one-to-one correspondence between the set of equivalent classes of constant mean curvature cuts and the equivalent classes of quadratic differentials in H2. Received by the editors December 12, 1994. 1991 Mathematics Subject Classification. Primary 58E20. The authors were partially supported by NSF grant #DMS9300422. c 1995 American Mathematical Society

In this paper we provide a complete classification of all hypersurfaces of Euclidean space which admit conformal deformations, other than the ones obtained through conformal diffeomorphisms of the ambient space, preserving the Gauss map. Elie Cartan, in one of his earlier papers in differential geometry, classified all Euclidean hypersurfaces, of dimension at least five, which admit conformal deformations other than the trivial ones obtained through compositions with conformal diffeomorphisms of the ambient space. When all possible conformal deformations are trivial, a hypersurface is called conformally rigid. First, Cartan proved that a hypersurface is conformally rigid if at no point does there exist a principal curvature of multiplicity at least n 2 (see [Ca], [CD2], or [Da]). Using this, he concluded that a conformally deformable hypersurface is either conformally flat or a 2-parameter envelope of spheres or planes of some very special type. Moreover, the set of all conformal deformations is either a 1parameter family or there is just one other deformation. In this paper we classify all Euclidean hypersurfaces which admit nontrivial conformal deformations preserving the Gauss map. In the isometric case, a complete classification for any codimension has been obtained by Dajczer and Gromoll [DG2], while the conformal case, but only for surfaces, is due to Vergasta [Ve2]. Theorem. Let f, g: Mn -+ Rn+1, n > 3, be conformal immersions of an ndimensional connected Riemannian manifold with the same Gauss map. Assume that on no open subset do they differ by a conformal diffeomorphism of Rn+l. Then f(Mn) is part of one of the following examples while g(Mn) is of the same type: (i) a minimal real Kaehler hypersurface, (ii) a rotation hypersurface over a plane curve, (iii) a rotation hypersurface over a minimal surface in R3. Minimal real Kaehler hypersurfaces have been completely classified by Dajczer and Gromoll in [DG2]. The assumption that f and g do not differ locally by a conformal deformation of the ambient space has been introduced in order to produce a global result. Without that assumption, our proof shows that the only other possibility is to have an open subset where, up to homothety Received by the editors December 10, 1993. 1991 Mathematics Subject Classification. Primary 53C42. ? 1995 American Mathematical Society

This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form \[ ( A ) { m i q ¨ i + ∂ V ∂ q i ( t , q ) = 0 q ( t + T ) = q ( t ) , ∀ t ∈ ℜ . ({\text {A}})\quad {\text { }}\left \{ {\begin {array}{*{20}{c}} {{m_i}{{\ddot q}_i} + \frac {{\partial V}} {{\partial {q_i}}}(t,q) = 0} \\ {q(t + T) = q(t),\quad \forall t \in \Re .} \\ \end {array} } \right . \] where q i ∈ ℜ ℓ , ℓ ⩾ 3 , 1 ⩽ i ⩽ n , q = ( q 1 , … , q n ) {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n}) and with V i j ( t , ξ ) {V_{ij}}(t,\xi ) T T -periodic in t t and singular in ξ \xi at ξ = 0 \xi = 0 Under additional hypotheses on V V , when (A) is posed as a variational problem, the corresponding functional, I I , is shown to have an unbounded sequence of critical values if the singularity of V V at 0 0 is strong enough. The critical points of I I are classical T T -periodic solutions of (A). Then, assuming that I I has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index k k grows exponentially with k k , at least when k ≡ 0 , 1 ( mod ℓ − 2 ) k \equiv 0,1( \mod \ell - 2) . The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the 3 3 -body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the 4 4 -body problem then generalized to the n n -body problem.

In this paper we give a categorical treatment of matrix Toda brackets, both in the preand post-compositional versions. Explicitly the setting in which we work is, a la Gabriel-Zisman, a 2-category with zeros. The development parallels that in the topological setting but with homotopy groups replaced by nullity groups of invertible 2-morphisms. A central notion is that of conjugation of 2-morphisms. Our treatment of matrix Toda brackets is carried forward to the point of establishing appropriate indeterminacies. 0. INTRODUCTION When P. Gabriel and M. Zisman wrote their book [3] they presented their subject (homotopy theory of simplicial sets via categories of fractions) from the most general point of view available going "back to the beginning of the theory ... in the hope of providing the reader with ... proofs which are easier or more conceptual than those already published." Their study of homotopy having begun in Chapter IV they set out in Chapter V "to give a standard and self-dual proof of various exact sequences occurring in Algebraic Topology ... the main idea is to construct an exact sequence in the category of pointed groupoids and then to reduce the study of a large class of 2-categories to the preceding one." They soon specialise to 2-categories ?1 which satisfy conditions A and B below: A. If x and y are two objects of 2', Hom sw(x, y) is a groupoid. B. There is an object o of 2' such that, for any object x of XP, Homr(o, x) and Homw(x, o) are isomorphic to the zero category Continuing in the spirit of Gabriel and Zisman the present paper is a contribution to an abstract theory of homotopy that assumes the existence of a 2-category &' satisfying only condition B. Indeed it can be regarded as sequel to a paper [8] by the third author which, in the setting of an arbitrary 2-category, studies the notion of homotopy equivalence and characterises the homotopy equivalences in various lax categories. Here the development parallels that in the topological setting but with track groups of the form 7r(EX, Y) or 7r(X, Q2Y) replaced by nullity groups of invertible 2-morphisms. A central notion is that of conjugation of 2-morphisms. This notion is available in any 2-category but does not seem to have been used previously. In fact when Received by the editors June 9, 1993. 1991 Mathematics Subject Classification. Primary 18D05, 55Q05, 55Q35. ?1995 American Mathematical Society

This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in R , C , \mathbb {R},\mathbb {C}, or R 2 {\mathbb {R}^2} ), (maps f f defined near 0 0 , such that f ( 0 ) = 0 f(0) = 0 , and f ′ ( 0 ) f’(0) is the identity). Proofs are mostly original. The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, k k th root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition 2 2 the function z ↦ z + z 2 z \mapsto z + {z^2} should be better thought of as the map ( x , y ) → ( x + x 2 − y 2 , y + 2 x y ) (x,y) \to (x + {x^2} - {y^2},y + 2xy) .

In his famous paper on modular equations and approximations to π \pi , Ramanujan offers several series representations for 1 / π 1/\pi , which he claims are derived from "corresponding theories" in which the classical base q q is replaced by one of three other bases. The formulas for 1 / π 1/\pi were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.

The results of Boyce for random Sturm-Liouville problems are generalized to random quadratic forms. Order relationships are proved between the means of eigenvalues of a random quadratic form and the eigenvalues of an associated mean quadratic form. Finite-dimensional and infinite-dimensional examples that show these are the best possible results are given. Also included are some results for a general approximation theory for random quadratic forms. 0. INTRODUCTION In [1] and [2] Boyce considered the order relationship between the means of eigenvalues for random Sturm-Liouville problems and the corresponding eigenvalues for the associated Sturm-Liouville problem with random coefficients replaced with their means. Subsequently, Kreith [7] modified Boyce's problem to obtain similar results for a stochastic initial value problem of this type. At first glance these results seem surprising. In the deterministic case, general comparison results exist for the nth eigenvalue for pairs of two quadratic forms (see [4] or [5]) and hence Sturm-Liouville problems. But why should any relationship exist if we only average the forms? The major purpose of this paper is to consider the order relations described above for the nth eigenvalue, n = 1, 2, 3, ..., for quadratic forms. Our setting is that of [5]. Thus, similar results hold for higher-order linear, selfadjoint differential systems and focal or conjugate point theory including quadratic control problems. The remainder of this paper is as follows. In Section 1 we sketch the deterministic theory needed for the remainder of this paper. In Section 2 we give our main results for order comparison. In many cases we give counterexamples to show that general results for the nth eigenvalue do not always hold. Thus, in Section 3 we consider finitedimensional examples and, in Section 4, infinite-dimensional examples including Sturm-Liouville problems. Finally, in Section 5 we present some new results for a general approximation theory for random quadratic forms. Received by the editors September 8, 1993; originally communicated to the Proceedings of the AMS by Hal L. Smith. 1980 Mathematics Subject Classification (1985 Revision). Primary 34B24; Secondary 34F05, 34CG0.

We construct a functor of spaces, Mn , and show that its multilinearization is equivalent to the nth layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type. The calculus of homotopy functors, developed by Goodwillie ([GI], [G2], [G3]), establishes that a homotopy functor, F, satisfying certain connectivity conditions, has associated to it a tower of functors, ... PF Pn-F .... These functors act like a Taylor series approximation to F in the sense that for a space, X, there is a map, PnF(X): F(X) -k PnF(X), for each n, and the connectivity of this map increases with n. This theory has been applied to the study of the functor A, Waldhausen's algebraic K-theory of spaces. In this paper, we turn our attention to the Taylor tower of the identity functor of spaces, I. The ultimate goal is to identify the Taylor tower of I and use it to study homotopy theory. In this paper, we construct a collection of symmetric functors, {Mn}, and show that the multilinearization of Mn is equivalent to the nth layer, fiber (PnI Pn_ I), of the Taylor tower of I. This construction also allows us to identify the nth derivative of I. This is a spectrum with L,,action which is equivalent to the functor fiber (PnI -k PnI) . The paper is organized as follows. In section 1 we summarize the basic results and terminology of calculus that will be used throughout the paper. In section 2 we describe the problem of finding the Taylor tower of I in more detail and state the main results. In section 3 we outline the method by which the nth derivative of a functor is determined. In section 4 we construct the functor Mn and a natural transformation, Tn , used to establish the equivalence between the multilinearization of Mn and fiber(PnF PnF) . In section 5 we determine the homotopy type of the derivatives, and in section 6 we show that T. is sufficiertly connected to induce an equivalence between the multilinearization of Mn and fiber(P,F -PIF)l. The results in this paper come from the author's thesis, written uncler the direction of Tom Goodwillie at Brown University. The author wishes to thank him for his guidance and for many insightful discussions. The author alsQ wishes to thank Randy McCarthy for his helpful suggestions during the writing of this paper. Received by the editors February 10, 1994. 1991 Mathematics Subject Clqssification. Primary 55P65. ? 1995 American Mathematical Society 0002-9947/95 $1.00 + $.25 per page

In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups Sn, of the finite symmetric groups S,n in characteristic 5 . One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. 0. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of Sn, in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and B * below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture B *, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels. In ? 1 we give a brief description of the groups Sn and their representations, leading up to Conjectures A and B * as they were formulated in [BMO]. That section also presents the background for Conjecture B as stated in [A3] and the equivalence of Conjectures B and B * is explained. Sections 2 and 3 are devoted to the proof of Conjecture B, and ?4 to the proof of Conjecture A. 1. THE CONJECTURES AND THEIR BACKGROUND For facts concerning the general representation theory of finite groups needed in the following, the reader is referred to [F, NT]. In 191 1 Schur [S1] proved that the finite symmetric groups S, have covering groups S, of order 2IS, I = 2 * n ! This means that there is an exact sequence

An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, CO,, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals. 1. A POWER SERIES REPRESENTATION FOR THE GENUS SERIES A map is a 2-cell embedding of a connected unlabelled graph ', with loops and multiple edges allowed, in a closed surface X, without boundary, which is assumed throughout to be oriented. The deletion of ' separates X into regions homeomorphic to open discs, called the faces of the map, and the number of edges bordering a face is called its degree. A map is rooted by distinguishing a mutually incident vertex, edge and face. The genus series for a class of maps is the formal generating series for the number of inequivalent maps with respect to genus, and the numbers of vertices, edges and faces. It is assumed hereinafter that maps are rooted. The general approach adopted here combines ideas of Jackson and Visentin [11] with those of 't Hooft [8] and Bessis, Itzykson and Zuber [3] who, in the above terminology, derived the genus series for diagrams akin to a class of maps by techniques from conformal field theory. Although [81 and [3] are important papers, they have remained largely inaccessible to combinatorialists because of their uncertainty about the automorphisms of these diagrams as combinatorial structures. In this paper, an explicit construction is given for an integral representation for the genus series for general maps directly from the combinatorial axiomatisation for embeddings on oriented surfaces. Moreover, we also develop methods which are extensible to vertex-regular maps (vertices have the same degree) and thence, by restriction, to quadrangulations (maps whose faces are bounded by four edges). This is done by examining dipoles (maps with two vertices) in detail. Although the argument is an algebraic one, based on the ring of formal power series, to assert that particular series belong to the ring, it is necessary to Received by the editors July 1, 1992 and, in revised form, August 15, 1993. 1991 Mathematics Subject Classification. Primary 05A1 5, 20C1 5; Secondary 57N37. (?) 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page

Hubert's Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobson's criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K,S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions K (t, S, D). we char- acterize the notion of (5, D)-algebraicity for elements a e K, and give an effective criterion for two elements a, b € K to be {S, £>)-conjugate, in the case when the (S, Z>)-conjugacy class of a is algebraic. This criterion amounts to a general Hubert 90 Theorem for division rings in the (K ,S, Z))-setting, sub- suming and extending all known forms of Hubert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.