Sharp inequalities, the functional determinant, and the complementary series
Thomas P.
Branson
3671-3742
Abstract: Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions $2$, $4$, and $6$ for the functional determinants of operators which are well behaved under conformal change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on ${S^2}$, and in the standard conformal classes on ${S^4}$ and ${S^6}$. The ${S^2}$ results are due to Onofri, and the $ {S^4}$ results to Branson, Chang, and Yang; the ${S^6}$ results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of $ {\text{S}}{{\text{O}}_0}(m + 1,1)$, and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on ${S^6}$, we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding $ L_1^2({S^6})\hookrightarrow {L^3}({S^6})$ for section spaces of trace free symmetric two-tensors.
Orthogonal calculus
Michael
Weiss
3743-3796
Abstract: Orthogonal calculus is a calculus of functors, similar to Goodwillie's calculus. The functors in question take finite dimensional real vector spaces (with an inner product) to pointed spaces. Prime example: $F(V) = BO(V)$, where $O(V)$ is the orthogonal group of $V$. In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of Stiefel-Whitney classes. Similarly, second derivatives in the orthogonal calculus reproduce and generalize much of the theory of Pontryagin classes.
Coexistence states and global attractivity for some convective diffusive competing species models
Julián
López-Gómez;
José C.
Sabina de Lis
3797-3833
Abstract: In this paper we analyze the dynamics of a general competing species model with diffusion and convection. Regarding the interaction coefficients between the species as continuation parameters, we obtain an almost complete description of the structure and stability of the continuum of coexistence states. We show that any asymptotically stable coexistence state lies in a global curve of stable coexistence states and that Hopf bifurcations or secondary bifurcations only may occur from unstable coexistence states. We also characterize whether a semitrivial coexistence state or a coexistence state is a global attractor. The techniques developed in this work can be applied to obtain generic properties of general monotone dynamical systems.
Banach spaces with the $2$-summing property
A.
Arias;
T.
Figiel;
W. B.
Johnson;
G.
Schechtman
3835-3857
Abstract: A Banach space $ X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $\ell _\infty ^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $\ell _\infty ^3$ and their duals.
Automorphism group schemes of basic matrix invariants
William C.
Waterhouse
3859-3872
Abstract: For $3 \leqslant k < n,\quad {\text{let}}\quad {E_k}(X)$ be the polynomial in ${n^2}$ variables defined by ${\text{det}}(X + \lambda I) = \sum {{E_k}(X){\lambda ^{n - k}}}$. Let $R$ be a ring containing a field of characteristic $p \geqslant 0$. If $p$ does not divide $n - k + 1$, the invertible linear transformations on matrices preserving ${E_k}(X)$ up to scalars are (in essence) just the obvious ones arising from scaling, similarities, and transposition. If the power ${p^s}$ dividing $n - k + 1$ is greater than $k$, then we have these elements times maps of the form $ X \mapsto X + f(X)I$. When smaller powers ${p^s}$ divide $n - k + 1$, the group scheme is like the first with an infinitesimal part of the second. One corollary is that every division algebra of finite dimension $ {n^2} > 4$ over its center carries a canonical cubic form that determines it up to antiisomorphism.
On spectral geometry of minimal surfaces in $\bold C{\rm P}\sp n$
Yi Bing
Shen
3873-3889
Abstract: By employing the standard isometric imbedding of $C{P^n}$ into the Euclidean space, a classification theorem for full, minimal, $2$-type surfaces in $C{P^n}$ that are not $\pm$ holomorphic is given. All such compact minimal surfaces are either totally real minimal surfaces in $C{P^2}$ or totally real superminimal surfaces in $ C{P^3}$ and $C{P^4}$. In the latter case, they are locally unique. Moreover, some eigenvalue inequalities for compact minimal surfaces of $C{P^n}$ with constant Kaehler angle are shown.
Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture
Satoshi
Naito
3891-3919
Abstract: Generalized Kac-Moody algebras were introduced by Borcherds in the study of Conway and Norton's moonshine conjectures for the Monster sporadic simple group. In this paper, we prove the Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras under a certain mild condition, by using a generalization (to the case of generalized Kac-Moody algebras) of Jantzen's character sum formula. Our (main) formula generalizes the celebrated result for the case of Kac-Moody algebras, and describes the characters of irreducible highest weight modules over generalized Kac-Moody algebras in terms of the "extended" Kazhdan-Lusztig polynomials.
Focusing at a point and absorption of nonlinear oscillations
J.-L.
Joly;
G.
Métivier;
J.
Rauch
3921-3969
Abstract: 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.
Cyclic Sullivan-de Rham forms
Christopher
Allday
3971-3982
Abstract: For a simplicial set $X$ the Sullivan-de Rham forms are defined to be the simplicial morphisms from $X$ to a simplicial rational commutative graded differential algebra (cgda)$\nabla$. However $\nabla$ is a cyclic cgda in a standard way. And so, when $X$ is a cyclic set, one has a cgda of cyclic morphisms from $X$ to $\nabla$. It is shown here that the homology of this cgda is naturally isomorphic to the rational cohomology of the orbit space of the geometric realization $\left\vert X \right\vert$ with its standard circle action. In addition, a cyclic cgda $ \nabla C$ is introduced; and it is shown that the homology of the cgda of cyclic morphisms from $X$ to $\nabla C$ is naturally isomorphic to the rational equivariant (Borel construction) cohomology of $\left\vert X \right\vert$.
$L\sb 2(q)$ and the rank two Lie groups: their construction in light of Kostant's conjecture
Mark R.
Sepanski
3983-4021
Abstract: This paper deals with certain aspects of a conjecture made by B. Kostant in 1983 relating the Coxeter number to the occurrence of the simple finite groups $L(2,q)$ in simple complex Lie groups. A unified approach to Kostant's conjecture that yields very general results for the rank two case is presented.
$L\sp p$ spectra of pseudodifferential operators generating integrated semigroups
Matthias
Hieber
4023-4035
Abstract: Consider the $ {L^p}$-realization $ {\text{O}}{{\text{p}}_p}(a)$ of a pseudodifferential operator with symbol $a \in S_{\rho ,0}^m$ having constant coefficients. We show that for a certain class of symbols the spectrum of $ {\text{O}}{{\text{p}}_p}(a)$ is independent of $p$. This implies that $ {\text{O}}{{\text{p}}_p}(a)$ generates an $N$-times integrated semigroup on ${L^p}({\mathbb{R}^n})$ for a certain $N$ if and only if $ \rho ({\text{O}}{{\text{p}}_p}(a)) \ne \emptyset$ and the numerical range of $ a$ is contained in a left half-plane. Our method allows us also to construct examples of operators generating integrated semigroups on $ {L^p}({\mathbb{R}^n})$ if and only if $p$ is sufficiently close to $2$.
Classifying spaces and Dirac operators coupled to instantons
Marc
Sanders
4037-4072
Abstract: Let $M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $SU(l)$ instantons on principal bundles over $ {S^4}$ with second Chern class equal to $k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $l$ increases relative to $k$. This "coupling" procedure produces maps ${\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $l$ such maps recover Kirwan's $[$K$]$ homotopy equivalence $M(k,SU) \simeq BU(k)$. We also compute, for any $ k$ and $l$, the image of the homology map ${({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $Sp(l)$ instantons.
Branched circle packings and discrete Blaschke products
Tomasz
Dubejko
4073-4103
Abstract: In this paper we introduce the notion of discrete Blaschke products via circle packing. We first establish necessary and sufficient conditions for the existence of finite branched circle packings. Next, discrete Blaschke products are defined as circle packing maps from univalent circle packings that properly fill $D = \{ z:\left\vert z \right\vert < 1\}$ to the corresponding branched circle packings that properly cover $D$. It is verified that such maps have all geometric properties of their classical counterparts. Finally, we show that any classical finite Blaschke product can be approximated uniformly on compacta of $ D$ by discrete ones.
On permutations of lacunary intervals
Kathryn E.
Hare;
Ivo
Klemes
4105-4127
Abstract: Let $\{ {I_j}\}$ be an interval partition of the integers and consider the Littlewood-Paley type square function $S(f) = {(\sum {\left\vert {{f_j}} \right\vert^2})^{1/2}}$ where $ {\hat f_j} = \hat f\chi {I_j}$. We prove that if the lengths $\ell ({I_j})$ of the intervals ${I_j}$ satisfy $\ell ({I_{j + 1}})/\ell ({I_j}) \to \infty$, then $ {\left\Vert {S(f)} \right\Vert _p} \approx {\left\Vert f \right\Vert _p}$ for $1 < p < \infty$. As these intervals need not be adjacent, such partitions can be thought of as permutations of lacunary intervals. This work generalizes the specific partition considered in a previous paper [H2]. We conjecture that it suffices to assume $ \ell ({I_{j + 1}})/\ell ({I_j}) \geqslant \lambda > 1$, and we also conjecture a necessary and sufficient condition.
Transition time analysis in singularly perturbed boundary value problems
Freddy
Dumortier;
Bert
Smits
4129-4145
Abstract: The paper deals with the boundary value problem $\varepsilon \ddot x + x\dot x - {x^2} = 0$, with $ x(0) = A,x(T) = B$ for $ A,B,T > 0$ and $\varepsilon > 0$ close to zero. It is shown that for $T$ sufficiently big, the problem has exactly three solutions, two of which reach negative values. Solutions reaching negative values occur for $T \geqslant T(\varepsilon ) > 0$ and we show that asymptotically for $\varepsilon \to 0,\quad T(\varepsilon ) \sim - {\text{ln}}(\varepsilon )$, $ {\text{i}}{\text{.e}}{\text{.}}\quad {\text{li}}{{\text{m}}_{\varepsilon \to 0}} - \frac{{T(\varepsilon )}} {{{\text{ln(}}\varepsilon {\text{)}}}} = 1$. The main tools are transit time analysis in the Liénard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems.
Mean value property and subdifferential criteria for lower semicontinuous functions
Didier
Aussel;
Jean-Noël
Corvellec;
Marc
Lassonde
4147-4161
Abstract: We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.