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iF(y)=( where G c R* is a bounded domain, y: G -» R , y'(x) = (dyydx), and F: M -> R is continuous. Here M denotes the linear space of real N X k matrices. We suppose throughout that K > 2, N > 2. In [7] F is called T-conυex if there exists a convex function /, defined on R, r = (t) ~ 1, such that F(p) = f(τ(p)) for all/? e M, where τ(p) denotes the minors of p of all orders j , 1 y uniformly on G with supx χGG\yj(x) ~ yj(x)\ < C < oo for ally. (Equivalently, if G has sufficiently regular boundary then IF is lsc if and only if IF is sequentially weak* lower semicontinuous on the Sobolev space W(G; R).) A consequence of [7? Theorem 3.6] is that IF lsc implies F polyconvex; that this conclusion is false was pointed out implicitly by Morrey [4, p. 26]. Morrey's remark is based on an example due to Terpstra [8] of a quadratic form

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics

In a previous paper the authors defined the growth rate of the tunnel number of knots, an invariant that measures that asymptotic behavior of the tunnel number under connected sum. In this paper we calculate the growth rate of the tunnel number of m-small knots in terms of their bridge indices.

For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups, or covering groups whose complex dual group is of adjoint type. In the second part of the paper, we consider covering groups of certain simply-connected groups and give necessary and sufficient condition for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character such that its associated theta representation is distinguished., Comment: Retitled

A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of \'Alvarez L\'opez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties., Comment: Minor corrections; the term `tame' changed to `stable'

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GLn /μm, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form G = (GLn1 × · · · ×GLnr )/C , where C is a central subgroup of GLn1 × · · · ×GLnr . Equivalently, we are interested in the essential dimension of a generic r-tuple (A1, . . . , Ar) of central simple algebras such that deg(Ai) = ni and the Brauer classes of A1, . . . , Ar satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n1, . . . , nr are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GLn1 × · · · ×GLnr )/C. This description and its corollaries are used throughout the paper.

When non-trivial local structures are present in a topological space $X$, a common approach to characterizing the isomorphism type of the $n$-th homotopy group $\pi_n(X,x_0)$ is to consider the image of $\pi_n(X,x_0)$ in the $n$-th \v{C}ech homotopy group $\check{\pi}_n(X,x_0)$ under the canonical homomorphism $\Psi_{n}:\pi_n(X,x_0)\to \check{\pi}_n(X,x_0)$. The subgroup $\ker(\Psi_n)$ is the obstruction to this tactic as it consists of precisely those elements of $\pi_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(\Psi_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(\Psi_n)$ is equal to the $n$-th Spanier group of $X$. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that $\Psi_{n}$ is an isomorphism., Comment: 21 pages, 2 figures

This paper is a follow-up to (MT13). In that paper we categorified the affine q-schur algebra b(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification o f Uq(bln) (KL09, KL11, KL10). In this paper we categorify b(n,n) for n � 3, using an extension of the aforementioned quotient.

In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions less than ten., 31 pages

In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $\Theta^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $\Theta$ is the classical Jacobi theta function., Comment: Significant changes have been made. Several proofs are corrected. New results and authors are added

We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity., v1 has been divided into two papers: the present article and "Bridge trisections and Seifert solids," which will be posted simultaneously; 29 pages, 11 figures

This paper studies some geometric aspects of moduli of curves Mg, using as a tool the deformation formula of holomorphic one-forms. Quasi-isometry guarantees the L 2 convergence of deformation of holomorphic one-forms, which is a kind of global result. After giving the period map a full expansion, we can also write out the Siegel metric, curvature and second fundamental form of a nonhyperelliptic locus of Mg in a quite detailed manner, while gaining some understanding of a totally geodesic manifold in a nonhyperelliptic locus. This paper is a complement to our joint paper [Liu et al. 2012b] with Kefeng Liu, and explores more applications of the deformation formula of holomorphic oneforms to some problems related to moduli spaces of Riemann surfaces, including the full expansion of the period map, the Siegel metric and its curvature formulae, the second fundamental form of Torelli space’s nonhyperelliptic locus, and also a global result about the deformation of holomorphic one-forms. We start with the Kuranishi coordinate of the Teichmuller space Tg of Riemann surfaces of genus g and the deformation formula of holomorphic one-forms. t/, whose construction is contained in Section 2. The key points of the deformation formula lie in Theorem 2.1. To be more precise, on the Kuranishi family$VX!1 with a Riemann surface $ 1 .0/D X0 as its central fiber and a global holomorphic one-form of the central fiber 2 H 0 .X0; 1 X0 /, the deformation formula of holomorphic one-forms emerges as

PROOFS - PAGE NUMBERS ARE TEMPORARY PACIFIC JOURNAL OF MATHEMATICS Vol. , No. , 2013 dx.doi.org/10.2140/pjm.2013..101 SYMMETRIC REGULARIZATION, REDUCTION AND BLOW-UP OF THE PLANAR THREE-BODY PROBLEM R ICHARD M OECKEL AND R ICHARD M ONTGOMERY We carry out a sequence of coordinate changes for the planar three-body problem, which successively eliminate the translation and rotation symme- tries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regu- larization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while empha- sizing the role of the shape sphere. By using homogeneous coordinates to describe Hamiltonian systems whose configurations spaces are spheres or projective spaces, we are able to take a modern, global approach to these familiar problems. We also show how to obtain the reduced and regularized differential equations in several convenient local coordinates systems. 1. Introduction and history The three-body problem of Newton has symmetries and singularities. The reduction process eliminates symmetries thereby reducing the number of degrees of freedom. The Levi-Civita regularization eliminates binary collision singularities by a nonin- vertible coordinate change together with a time reparametrization. The McGehee blow-up eliminates the triple collision singularity by an ingenious polar coordi- nate change and another time reparametrization. Each process has been applied individually and in various combinations to the three-body problem, many times. In this paper we apply all three processes globally and systematically, with no one body singled out in the various transformations. The end result is a complete flow on a five-dimensional manifold with boundary. We focus attention on the geometry of the various spaces and maps appearing along the way. At the heart of this paper is a beautiful degree-4 octahedral covering map of the shape sphere, branched over the binary collision points (see Figure 4 on page 151). This map Research supported by NSF grant DMS-1208908. MSC2010: primary 37N05, 70F07, 70G45; secondary 53A20, 53CXX. Keywords: celestial mechanics, three-body problem, regularization.

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. As requested by earlier referees of this series of papers, in this third part, we shall first give an updated description of the geodesic principles and the classification of compact almost homogeneous Kahler manifolds of cohomogeneity one. Then, we shall give a proof of the equivalence of the geodesic stability and the negativity of the integral in the first part. Finally, we shall address the relation of our result to Ross‐ Thomas version of Donaldson’s K-stability. One should easily see that their result is a partial generalization of our integral condition in the first part. And we shall give some further comments on the Fano manifolds with the Ricci classes. In Theorem 14, we give a result of Nadel type. We define the strict slope stability. In our case, it is stronger than Ross‐Thomas slope stability. We strengthen two Ross‐Thomas results in Theorems 15 and 16. The similar proofs of the results other than the existence for the type II cases are more complicated and will be done elsewhere.

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Furthermore, we obtain corollaries connecting the $ 3 $ rank of the class number of the real quadratic field $ \mathbb{Q}(\sqrt{m}) $ to the $ 3 $ divisibility of the number of certain quadratic residues modulo $ m $. Secondly, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain distribution results of $ m $ modulo any prime $ p $ that properly divides $ n+1 $. Our result implies that the primes $ m $ for which $ n $ is a primitive root belong to one of two equivalence classes modulo any prime $ p $ as above., Comment: Some Corollaries added

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.

In this paper we show that if G is a primitive subgroup of Sn that is not large base, then any irredundant base for G has size at most 5 log n. This is the first logarithmic bound on the size of an irredundant base for such groups, and is best possible up to a multiplicative constant. As a corollary, the relational complexity of G is at most 5 log n+1, and the maximal size of a minimal base and the height are both at most 5 log n. Furthermore, we deduce that a base for G of size at most 5 log n can be computed in polynomial time. Publisher PDF

In this paper, by introducing the notion of "\textit{distributive constant}" of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces in projective varieties. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically degenerate mappings and for the families of hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex disc into a project variety, and for meromorphic mappings on a complete K\"{a}hler manifold are also given. For the last aim, we will prove a Schmidt's subspace theorem for an arbitrary families of homogeneous polynomials, which is the counterpart in Number theory of our second main theorem. Our these results are generalizations and improvements of all previous results., Comment: Some typos are cleared. arXiv admin note: text overlap with arXiv:1712.05698

In a previous paper, we classified all Lie bialgebras structures of Hamiltonian type. In this paper, we give an explicit formula for the quantization of Hamiltonian-type Lie algebras.

In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new supertrace gives rise to a non-zero bilinear form on a space of invariant tensors of a Lie superalgebra of type I. The results of this paper are completely classical results in the theory of Lie superalgebras but surprisingly we can not prove them without using quantum algebra and low-dimensional topology.

We give a multimensional version of the p-adic heat equation, and show that its fundamental solution is the transition density of a Markov process. 1. Introduction. In recent years p adic analysis has received a lot of attention due to its applications in mathematical physics, see e.g. [1], [2], [4], [5], [16], [17], [18], [22], [28] and references therein. One motivation comes from statistical physics, in particular in connection with models describing relaxation in glasses, macromolecules, and proteins. It has been proposed that the non exponential nature of those relaxations is a consequence of a hierarchical structure of the state space which can in turn be put in connection with p adic structures ([4], [5], [22]). In [4] was demostrated that the p-adic analysis is a natural basis for the construction of a wide variety of models of ultrametric di¤usion constrained by hierarchical energy landscapes. To each of these models is associated a stochastic equation (the master equation). In several cases this equation is a p-adic parabolic equation of type: > : @u(x;t) @t + a(Au)(x; t) = f(x; t); x 2 Q n p ; t 2 (0; T ]; u(x; 0) = '(x); (1.1) where a is a positive constant, A is pseudo-di¤erential operator, and Qp is the …eld of p-adic numbers. The simplest case occurs when n = 1 and A is the Vladimirov operator: (D ') (x) = F 1 !x j j p Fx! '(x) ; > 0; where F is the Fourier transform. The fundamental solution of (1.1) is density transition of a timeand space-homogeneous Markov process, that is consider the p adic counterpart of the Brownian motion (see [18], [28]). It is relevant to mention that in the case n = 1, the fundamental solution of (1.1) when A = D (also called the p adic heat kernel) has been studied extensively, see e.g. [6], [11], [12], [14], [18], [28]. A natural problem is to study the initial value problem (1.1) in the n-dimensional case. Recently, the second author considered Cauchy’s problem (1.1) when (A') (x) = F 1 !x jf ( )j p Fx! '(x) ; > 0; 2000 Mathematics Subject Classi…cation. Primary 35R60, 60J25; Secondary 47S10, 35S99. Key words and phrases. Parabolic equations, Markov processes, p-adic numbers, ultrametric di¤usion. 1 2 J. J. RODRIGUEZ-VEGA AND W. A. ZUNIGA-GALINDO here f ( ) is an elliptic homogeneous polynomial in n variables, and the datum ' is a locally constant and integrable function. Under these hypotheses it was established the existence of a unique solution to Cauchy’s problem (1.1). In addition, the fundamental solution is a transition density of a Markov process with space state Qp (see [29]). In this paper we study Cauchy’s problem (1.1) when A is the Taibleson pseudodi¤erential operator which is de…ned as follows: D T' (x) = F 1 !x max 1 i n j ijp Fx! '(x) ! ; > 0: (1.2) Recently Albeverio, Khrennikov, and Shelkovich studied D T in the context of the Lizorkin spaces [3]. We prove existence and uniqueness of the Cauchy problem (1.1-1.2) in spaces of increasing functions introduced by Kochubei in [19], see Theorem 1. We also associate a Markov processes to equation the fundamental solution (see Theorem 2). These results constitute an extension of the corresponding results in [18], [28]. Let us explain the connection between the results of this paper and those of [29]. There are in…nitely many homogeneous polynomial functions satisfying jf ( )jp = max 1 i n j ijp d ; for any 2 Qp ; here d denotes degree of f (c.f. Lemmas 14-15). Hence the pseudo-di¤erential operators considered here are a subclass of the ones considered in [29]. However, the function spaces for the solutions and initial data are completely di¤erent. In this paper the initial datum and the solution to Cauchy problem (1.1-1.2) are not necessarily bounded, neither integrable, but in [29] are. Finally, our results can be extended to operators of the form (A') (x) = a0(x; t)(D T')(x) + n X k=1 ak(x; t)(D k T ')(x) + b(x; t)'(x); (1.3) > 1, 0 < 1 < : : : < n < ; where the ak(x; t) ,and b(x; t) are bounded continuous functions, using the techniques presented in [18]-[20]. These results will appear later elsewhere. 2. Preliminary Results As general reference for p-adic analysis we refer the reader to [25] and [28]. The …eld of p-adic numbers Qp is de…ned as the completion of the …eld of rational numbers Q with respect to the non-Archimedean p-adic norm j jp which is de…ned as follows: j0jp = 0; if x 2 Q , x = p ab with a, b integers coprime to p, then jxjp = p . The integer = (x) is called the p-adic order of x, and it will be denoted as ord (x). We use the same symbol, j jp, for the p-adic norm on Qp. We extend the p-adic norm to Qp as follows: kxkp := max 1 i n jxijp , for x = (x1; : : : ; xn) 2 Q n p . Note that kxkp = p min1 i nford(xi)g. p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 3 Any p-adic number x 6= 0 has a unique expansion of the form

This paper is a contribution to the ongoing project of Gorenstein, Lyons, and Solomon to produce a complete unified proof of the classification of finite simple groups. A part of this project deals with classification and characterization of bicharacteristic finite simple groups. This paper contributes to that particular situation.

A pseudofree group action on a space X is one whose set of singular orbits forms a discrete subset of its orbit space. Equivalently ? when G is finite and X is compact ? the set of singular points in X is finite. In this paper, we classify all of the finite groups which admit pseudofree actions on S2 × S2. The groups are exactly those that admit orthogonal pseudofree actions on S2 ×S2 ? R3 × R3, and they are explicitly listed. This paper can be viewed as a companion to a preprint of Edmonds, which uniformly treats the case in which the second Betti number of a four-manifold M is at least three.

The aim of this paper is to introduce a concrete notion of multiplicity for invariant algebraic curves in polynomial vector fields. In fact, we give several natural definitions and show that they are all equivalent to our main definition, under some ?generic? assumptions. In particular, we show that there is a natural equivalence between the algebraic viewpoint (multiplicities defined by extactic curves or exponential factors) and the geometric viewpoint (multiplicities defined by the number of algebraic curves which can appear under bifurcation or by the holonomy group of the curve). Furthermore, via the extactic, we can give an effective method for calculating the multiplicity of a given curve. As applications of our results, we give a solution to the inverse problem of describing the module of vector fields with prescribed algebraic curves with their multiplicities; we also give a completed version of the Darboux theory of integration that takes the multiplicities of the curves into account. In this paper, we have concentrated mainly on the multiplicity of a single irreducible and reduced curve. We hope, however, that the range of equivalent definitions given here already demonstrates that this notion of multiplicity is both natural and useful for applications.

Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M$, a submanifold $C$ and an integrable smooth distribution $D \subseteq TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution., Comment: 31 pages

Several very interesting results connecting the theory of abelian ideals of Borel subalgebras, some ideas of D. Peterson relating the previous theory to the combinatorics of affine Weyl groups, and the theory of discrete series are stated in a recent paper (\cite{Ko2}) by B. Kostant. In this paper we provide proofs for most of Kostant's results extending them to $ad$-nilpotent ideals and develop one direction of Kostant's investigation, the compatible discrete series., Comment: AmsTex file, 27 Pages; minor corrections; to appear in Pacific Journal of Mathematics

We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic γ. For a Riemannian or a non-spacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. In the general semi-Riemannian case, under a certain nondegeneracy assumption on the conjugate points, this number is equal to an algebraic count of their multiplicities. In this paper we reprove some results that were incorrectly stated by Helfer in 1994, where the occurrence of degeneracies was overlooked; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian geodesics, in which some new phenomena appear.

Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro-p abelian extension, and by L' the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N'/k∞) be the pro-p abelian extension generated by all p-power roots of all units (resp. p-units) of k∞. In the previous paper, we proved that the Zp-torsion subgroup of the odd part of the Galois group Gal(N ∩ L/k∞) is isomorphic, over the group ring Z p [Gal(k/Q)], to a certain standard subquotient of the even part of the ideal class group of k∞. In this paper, we prove that the same holds also for the Galois group Gal(N'∩L'/k∞).

Given an n-tuple {b 1 ,...,b n } of self-adjoint operators in a finite von Neumann algebra M and a faithful, normal tra-cial state r on M, we define a map Ψ from M to R n by Ψ(a) = (τ(a),τ(b 1 a),...,τ(bna)). The image of the positive part of the unit ball under Ψ is called the spectral scale of {b 1 ,...,b n } relative to r and is denoted by B. In a previous paper with Nik Weaver we showed that the geometry of B reflects spectral data for real linear combinations of the operators {b 1 ,...,b n }. For example, we showed that an exposed face in B is determined by a certain pair of spectral projections of a real linear combination of the bi's. In the present paper we extend this study to faces that are not exposed. In order to do this we need to introduce a recursive method for describing faces of compact convex sets in R n . Using this new method, we completely describe the structure of arbitrary faces of B in terms of {b 1 ,...,b n } and τ. We also study faces of convex, compact sets that are exposed by more than one hyperplane of support (we call these sharp faces). When such faces appear on B, they signal the existence of commutativity among linear combinations of the operators {b 1 ,...,b n }. Although many of the conclusions of this study involve too much notation to fit nicely in an abstract, there are two results that give their flavor very well. Theorem 6.1: If the set of extreme points of B is countable, then N = {b 1 ,...,b n } is abelian. Corollary 5.6: B has a finite number of extreme points if and only if N is abelian and has finite dimension.

Let v be a valuation of the quotient field of a noetherian local domain R. Assume that v is centered at R. This paper studies the structure of the value semigroup of v, S. Ideals defining toric varieties can be defined from the graded algebra K[T] of cancellative commutative finitely generated semigroups such that T n (-T) = {0}. The value semigroup of a valuation S need not be finitely generated but we prove that S n (-S) = {0} and so, the study in this paper can also be seen as a generalization to infinite dimension of that of toric varieties. In this paper, we prove that K[S] can be regarded as a module over an infinitely dimensional polynomial ring A v . We show a minimal graded resolution of K[S] as A v -module and we give an explicit method to obtain the syzygies of K[S] as A v -module. Finally, it is shown that free resolutions of K[S] as A v -module can be obtained from certain cell complexes related to the lattice associated to the kernel of the map A v → K[S].

Many applied problems resulting in hyperbolic conservation laws are nonstrictly hyperbolic. As of yet, there is no comprehensive theory to describe the solutions of these systems. In this paper, a proof of existence is given for a class of nonstrictly hyperbolic conservation laws using a proof technique first applied by Glimm to systems of strictly hyperbolic conservation laws. We show that Glimm's scheme can be used to construct a subsequence converging to a weak solution. This paper necessarily departs from previous work in showing the existence of a convergent subsequence. A novel functional, shown to be equivalent to the total variation norm, is defined according to wave interactions. These interactions can be bounded without any assumptions of strict hyperbolicity.

A nonnegative number d1, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number 0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of 0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos.

In this paper we make a contribution to the MargulisPlatonov conjecture, which describes the normal subgroup structure of algebraic groups over number fields. We establish the conjecture for inner forms of anisotropic groups of type An. We obtain information on the commuting graph of nonabelian finite simple groups, and consequently, using the paper by Segev, 1999, we obtain results on the normal structure and quotient groups of the multiplicative group of a division algebra.

It is known that there is no nonconstant bounded harmonic map from the Euclidean space R n to the hyperbolic space H m . This is a particular case of a result of S.-Y. Cheng. However, there are many polynomial growth harmonic maps from R 2 to H 2 by the results of Z. Han, L.-F. Tam, A. Treibergs and T. Wan. One of the purposes of this paper is to construct harmonic maps from R n to H m by prescribing boundary data at infinity. The boundary data is assumed to satisfy some symmetric properties. On the other hand, it was proved by Han-Tam-Treibergs-Wan that under some reasonable assumptions, the image of a harmonic diffeomorphism from R 2 into H 2 is an ideal polygon with n + 2 vertices on the geometric boundary of H 2 if and only if its Hopf differential is of the form odz 2 where o is a polynomial of degree n. It is unclear whether one can find explicit relation between the coefficients of o and the vertices of the image of the harmonic map. The second purpose of this paper is to investigate this problem. We will explicitly demonstrate some families of polynomial holomorphic quadratic differentials, such that the harmonic maps from R 2 into H 2 with Hopf differentials in the same family will have the same image. In proving this, we first study the asymptotic behaviors of harmonic maps from R 2 into H 2 with polynomial Hopf differentials odz 2 . The result may have independent interest.

This is the first of a series of four papers describing the finitely generated modules over all commutative noetherian rings that do not have wild representation type (with a possible exception involving characteristic 2). This first paper identifies the wild rings, in the complete local case. The second paper describes the finitely generated modules over the remaining complete local rings. The last two papers extend these results by dropping the "complete local" hypothesis.