Showing papers in "Canadian Journal of Mathematics in 2000"
TL;DR: In this paper, the existence of solutions with multiple spikes both on the boundary and in the interior of the mean curvature function was proved and it was shown that the boundary spikes and the interior spikes usually have different scales of error estimation.
Abstract: We consider the problem where is a bounded smooth domain in is a small parameter and is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as approaches zero, at multiple critical points of the mean curvature function . It is also proved that this equation has multiple interior spike solutions which concentrate, as , at sphere packing points in . In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.
157 citations
TL;DR: In this article, it was shown that for a Grothendieck category and a complex in there is an associated localization endofunctor in, which is idempotent (in a natural way) and that the objects that go to 0 by are those of the smallest localizing subcategory of that contains the complex.
Abstract: In this paper we show that for a Grothendieck category and a complex in there is an associated localization endofunctor in . This means that is idempotent (in a natural way) and that the objects that go to 0 by are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of that contains . As applications, we construct -injective resolutions for complexes of objects of and derive Brown representability for from the known result for , where is a ring with unit.
131 citations
TL;DR: In this paper, the authors present reiteration theorems for real interpolation methods involving broken-logarithmic functors, where the resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced.
Abstract: We present "reiteration theorems" with limiting values� = 0 and� = 1 for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in (D).
109 citations
TL;DR: In this article, it was shown that the stabilizer of every point of an algebraic group X is isomorphic to a semidirect product U⋊ A of a unipotent group U and a diagonalizable group A.
Abstract: Let G be an algebraic group and let X be a generically free G-variety. We show that X can be trans- formed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety Xwith the following property: the stabilizer of every point of Xis isomorphic to a semidirect product U⋊ A of a unipotent group U and a diagonalizable group A. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus trans- formation.
105 citations
TL;DR: For a reductive p-adic group G, the authors compute the supports of the Hecke algebras for the K-types for G lying in a certain frequently-occurring class.
Abstract: For a reductive p-adic group G, we compute the supports of the Hecke algebras for the K-types for G lying in a certain frequently-occurring class. When G is classical, we compute the intertwining between any two such K-types.
69 citations
TL;DR: In this article, the polynomes F ∈ C{Sτ }[Y ] a coefficients dans l’anneau de germes de fonctions holomorphes au point special d’une variete torique affine.
Abstract: Nous etudions les polynomes F ∈ C{Sτ }[Y ] a coefficients dans l’anneau de germes de fonctions holomorphes au point special d’une variete torique affine. Nous generalisons `a ce cas la parametrisation classique des singularites quasi-ordinaires. Cela fait intervenir d’une part une generalization de l’algorithme de Newton-Puiseux, et d’autre part une relation entre le polyedre de Newton du discriminant de F par rapport a Y et celui de F au moyen du polytope-fibre de Billera et Sturmfels [3]. Cela nous permet enfin de calculer, sous des hypotheses de non degenerescence, les sommets du polyedre de Newton du discriminant a partir de celui de F, et les coefficients correspondants a partir des coefficients des exposants de F qui sont dans les aretes
de son polyedre de Newton
47 citations
TL;DR: In this paper, the authors study K-orbits in G/P where G is a complex connected reductive group, P is a parabolic subgroup and K ⊆ G is the fixed point subgroup of an involutive automorphism.
Abstract: We study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism �. Generalizing work of Springer, we parametrize the (finite) orbit set K\ G/P and we determine the isotropy groups. As a conse- quence, we describe the closed (resp. affine) orbits in terms of �-stable (resp. �-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle L on G/B having nonzero global sections, we show that the restriction map resX : H 0 (G/B,L)→ H 0 (X,L) is surjective and that H i (X,L) = 0 for i ≥ 1. Moreover, we describe the K-module H 0 (X,L). This gives information on the restriction to K of the simple G-module H 0 (G/B,L). Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal K-types.
37 citations
TL;DR: In this paper, a complex moduli space for SU(2) moduli spaces has been constructed, in the sense that one can obtain Mh from P by symplectic reduction, andMh fromP by a complex quotient.
Abstract: There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and ex- tended by Bhosle and Ramanathan to other groups), between the symplectic variety Mh of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group G, with fixed conjugacy classes h at the punctures, and a complex varietyMh of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For G = SU(2), we build a symplectic variety P of pairs (representations of the fundamental group into G, "weighted frame" at the puncture points), and a corresponding complex varietyP of moduli of "framed parabolic bundles", which encompass respectively all of the spaces Mh,Mh, in the sense that one can obtain Mh from P by symplectic reduction, andMh fromP by a complex quotient. This allows us to explain certain features of the toric geometry of the SU(2) moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.
33 citations
TL;DR: In this paper, it was shown that the cyclic cohomology group of order zero HC 0 (Bθ) is generated by the nine canonical modules of rotation algebra.
Abstract: Let Aθ denote the rotation algebra—the universal C�-algebra generated by unitaries U, V satisfying VU = e2πiθUV, whereθ is a fixed real number. Letσ denote the Fourier automorphism of Aθ defined by U 7→ V, V 7→ U 1, and let Bθ = Aθ⋊σ Z/4Z denote the associated C�-crossed product. It is shown that thereis a canonical inclusion Z 9 → K0(Bθ) foreachθ given by nine canonical modules. The unboundedtrace functionals of Bθ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero HC 0 (Bθ) whenθ is irrational. The Chern characters of the nine modules—and more impor- tantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense Gδ set of values ofθ one has K0(Bθ)∼ Z 9 and is generated by the nine classes constructed here.
28 citations
TL;DR: In this paper, explicit evaluations of the symmetric Euler integral are obtained for some particular functions, which are related to duplication formulae for Appell's hypergeometric function which give reductions of in terms of more elementary functions for arbitrary with and for with arbitrary.
Abstract: Explicit evaluations of the symmetric Euler integral are obtained for some particular functions . These evaluations are related to duplication formulae for Appell’s hypergeometric function which give reductions of in terms of more elementary functions for arbitrary with and for with arbitrary . These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at independent randomtimes with uniformdistribution on , then the broken line approximation to the bridge obtained from these values has a total variation whose mean square is .
24 citations
TL;DR: In this paper, the authors revisited Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887, and gave a proof of Russell's main theorem and indicated the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields.
Abstract: In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. MotivatedbyRussell'stheorem,westateandproveitscubicanaloguewhichallowsustoconstructRussell-type modular equations in the theory of signature 3.
TL;DR: In this paper, a finite union of finite quotients of the affine building of the group over the field of -adic numbers is constructed and the spectrum of its underlying graph is estimated.
Abstract: We will construct a finite union of finite quotients of the affine building of the group over the field of -adic numbers . We will view this object as a hypergraph and estimate the spectrum of its underlying graph.
TL;DR: In this article, an automorphic realization of the global minimal representation of quaternionic exceptional groups using the theory of Eisenstein series is presented, and used for the study of theta correspondences.
Abstract: Abstract We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences.
TL;DR: Jacquet module methods are used to study the problem of classifying discrete series for the classical $p$ -adic groups $\text{Sp}(2n,F)\,\text{ and}\, \text{SO}\, (2n+1,F)$ .
Abstract: In this paper, we use Jacquet module methods to study the problem of classifying discrete series for the classical -adic groups .
TL;DR: In this paper, the authors give a characterization of bounded plurisubharmonic functions by using their complex monge-amp measures, which implies a both necessary and sufficient condition for a positive measure to be complex Monge-Amp` ere measure of some bounded plurinomial function.
Abstract: We give a characterization of bounded plurisubharmonic functions by using their complex Monge- Ampmeasures. This implies a both necessary and sufficient condition for a positive measure to be complex Monge-Amp` ere measure of some bounded plurisubharmonic function.
TL;DR: In this article, a simple C�-algebra is constructed for which the Murray-von Neumann equivalence classes of Pro-Jections, with the usual addition, induced by addition of orthogonal projections, form the additive semi-group.
Abstract: A simple C�-algebra is constructed for which the Murray-von Neumann equivalence classes of pro- jections, with the usual addition—induced by addition of orthogonal projections—form the additive semi- group
TL;DR: In this paper, an argument in Galois cohomology of a kind first used by D.J. Kazhdan in the connected case is presented, and the trace formula in invariant form for all connected reductive groups and certain disconnected ones.
Abstract: J. Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D. Kazhdan in the connected case.
TL;DR: In this article, it was shown that an unbounded linear operator affiliated with a semidefinite von Neumann algebra is a bounded self-adjoint linear operator from and, where is a symmetric operator space associated with.
Abstract: We study estimates of the type 1 where is an unbounded linear operator affiliated with a semifinite von Neumann algebra is a bounded self-adjoint linear operator from and , where is a symmetric operator space associated with . In particular, we prove that belongs to the non-commutative -space for some , provided belongs to the noncommutative weak -space for some . In the case and , we show that this result continues to hold under the weaker assumption . This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.
TL;DR: In this paper, an approach for reducing the dimension of the cokernel of the natural map to the case that is ample is presented. Butler et al. showed that the problem of determining the modules in minimal free resolutions of fat point subschemes can be solved in an algebraically closed ground field.
Abstract: Let be a divisor on the blow-up of at general points and let be the total transform of a line on . An approach is presented for reducing the computation of the dimension of the cokernel of the natural map to the case that is ample. As an application, a formula for the dimension of the cokernel of is obtained when , completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes . All results hold for an arbitrary algebraically closed ground field .
TL;DR: In this article, it was shown that there are infinitely many short intervals containing considerably more elements of the sum of two squares than expected, and infinitely many irregular intervals with considerably fewer than expected.
Abstract: LetS denote the set of integers representable as a sum of two squares. SinceS can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected thatS has many properties in common with the set of prime numbers. In this paper we exhibit "unexpected irregularities" in the distri- bution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements ofS than expected, and infinitely many intervals containing considerably fewer than expected.
TL;DR: In this paper, a complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given, and it is shown that the spaces and have a unique symmetric structure.
Abstract: A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that and are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on , the only spaces with the Dunford-Pettis property are and , where denotes the norm closure of in . It is also proved that all Banach dual spaces of and have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces and have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Kothe-Bochner spaces.
TL;DR: In this article, a unified formulation is given to various generalizations of the classical numerical range including the congruence numerical range, q-numerical range and von Neumann range.
Abstract: A unified formulation is given to various generalizations of the classical numerical range including the c-numerical range, congruence numerical range, q-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized. Received by the editors March 12, 1998; revised October 14, 1999. Research of the first author was partially supported by an NSF grant. AMS subject classification: 15A60, 17B20.
TL;DR: In this article, a modified version of the "Brownian snake" whose lifetime is driven by a path-dependent stochastic equation is presented, which is a representation of super-processes.
Abstract: We study the "Brownian snake" introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itˆ o's formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.
TL;DR: In this article, it was shown that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on B.
Abstract: arusson Abstract. Let Y be an infinite covering space of a projective manifold M in P N of dimension n≥ 2. Let C be the intersection with M of at most n− 1 generic hypersurfaces of degree d in P N. The preimage X of C in Y is a connected submanifold. Letbe the smoothed distance from a fixed point in Y in a metric pulled up from M. LetO�(X) be the Hilbert space of holomorphic functions f on X such that f 2 eis integrable on X, and defineO�(Y) similarly. Our main result is that (under more general hypotheses than described here) the restrictionO�(Y)→O�(X) is an isomorphism for d large enough. This yields new examples of Riemann surfaces and domains of holomorphy in C n with corona. We con- sider the important special case when Y is the unit ball B in C n , and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on B. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to B.
TL;DR: In this paper, the nonlinear Sturm-Liouville equation is considered subject to boundary conditions and the boundary conditions depend on the independent variable alone, while depends on boundary conditions.
Abstract: The nonlinear Sturm-Liouville equation is considered subject to the boundary conditions . Here and and are functions depending on the independent variable alone, while depends on . Results are given on existence and location of sets of bifurcating from the linearized eigenvalues, and for which has prescribed oscillation count, and on completeness of the in an appropriate sense.
TL;DR: In this article, the moduli space of representations of the fundamental group of a Rie- mann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp�, whereis in the fundamental alcove of a Lie algebra.
Abstract: This paper treats the moduli spaceMg,1(�) of representations of the fundamental group of a Rie- mann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp�, whereis in the fundamental alcove of a Lie algebra. We construct natural line bundles overMg,1(�) and exhibit natural homology cycles representing the Poincar´ e dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank k and degree d.
TL;DR: In this paper, the authors describe the embedded resolution of an irreducible quasi-ordinary surface singularity, which results from applying the canonical resolution of Bierstone-Milmanto (V, p).
Abstract: We describe the embedded resolution of an irreducible quasi-ordinary surface singularity (V, p) whichresults fromapplying the canonicalresolution ofBierstone-Milmanto (V, p). We showthatthis process depends solely on the characteristic pairs of (V, p), as predicted by Lipman. We describe the process explicitly enough that a resolution graph for f could in principle be obtained by computer using only the characteristic pairs.
TL;DR: For affine Kac-Moody algebras, the level 3 exceptionals occur for B 2 ∼ = C(1) 2 and D (1) 7 as discussed by the authors, while the level 2 exceptionals are related to the lattice invariants of affine u(1).
Abstract: The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of SL2(Z), and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra B r and D (1) r all of these at level k ≤ 3. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the B r , D (1) r level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for B 2 ∼ = C (1) 2 and D (1) 7 . The B2,3 and D7,3 exceptionals are cousins of the E6-exceptional and E8-exceptional, respectively, in the A-D-E classification for A (1) 1 , while the level 2 exceptionals are related to the lattice invariants of affine u(1). Received by the editors April 1, 1999. AMS subject classification: Primary: 17B67; secondary: 81T40.
TL;DR: In this article, a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, R) (that is SO(4) acting on symmetric 4×4 matrices).
Abstract: InthefirstpartofthispapergeneralizationsofHesselink's q-analogofKostant'smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, R) (that is SO(4) acting on symmetric 4× 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.
TL;DR: In this paper, the essential spectrum of the discrete Laplacian of an infinite graph was studied and a new quantity for the spectrum was introduced, in terms of which new lower bound estimates of the spectrum and also upper bound estimates when the infinite graph is bipartite.
Abstract: In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.