Periodic automorphisms of takiff algebras, contractions, and θ-groups

Let be an algebraic Lie algebra and a (generalised) Takiff algebra. Any finite-order automorphism θ of induces an automorphism of of the same order, denoted . We study invariant-theoretic properties of representations of the fixed point subalgebra of on other eigenspaces of in . We use the observation that, for special values of m, the fixed point subalgebra, , turns out to be a contraction of a certain Lie algebra associated with and θ. To my teacher Supported in part by R.F.B.R. grant 06-01-72550.     

Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Let be a finite-dimensional complex reductive Lie algebra and S() its symmetric algebra. The nilpotent bicone of is the subset of elements (x, y) of whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of generated by the 2-order polarizations of invariants of . The main result of this paper is that the nilpotent bicone is a complete intersection of dimension , where and are the dimensions of Borel subalgebras and the rank of , respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of . The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

Region-Fault Tolerant Geometric Spanners

We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph on a point set P and a region F, we define to be what remains of after the vertices and edges of intersecting F have been removed. A  -fault tolerant t-spanner is a geometric graph  on P such that for any convex region F, the graph is a t-spanner for , where is the complete geometric graph on P. We prove that any set P of n points admits a -fault tolerant (1+ε)-spanner of size for any constant ε>0; if adding Steiner points is allowed, then the size of the spanner reduces to  , and for several special cases, we show how to obtain region-fault tolerant spanners of size without using Steiner points. We also consider fault-tolerant geodesic t -spanners: this is a variant where, for any disk D, the distance in between any two points u,v∈P∖D is at most t times the geodesic distance between u and v in ℝ²∖D. We prove that for any P, we can add Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size  . M.A. Abam was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 612.065.307 and by the MADALGO Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation. M. de Berg was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301. M. Farshi was supported by Ministry of Science, Research and Technology of I.R. Iran. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.     

A D’Alembert Formula for Flat Surfaces in the 3-Sphere

We find all the flat surfaces in the unit 3-sphere $\mathbb{S}^{3}We find all the flat surfaces in the unit 3-sphere that pass through a given regular curve of with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat M?bius strips of  .      

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for the boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M _b TM _b ∈WBP is bounded from to if and only if and T ^* b=0 for , where ε is the regularity exponent of the kernel of T. Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3. Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for Mathematics and Theoretic Physics.     

Groupoid and inverse semigroup presentations of ultragraph C *-algebras

Inspired by the work of Paterson on C ^* -algebras of directed graphs, we show how to associate a groupoid to an ultragraph in such a way that the C ^*-algebra of is canonically isomorphic to Tomforde’s C ^*-algebra . The groupoid is built from an inverse semigroup naturally associated to . A.E. Marrero was supported by grants from the National Science Foundation and the Sloan Foundation and by a GAANN Fellowship. Many of the results here are taken from this author’s dissertation [7]. P.S. Muhly was supported by a grant from the National Science Foundation (DMS-0355443).     

Bass Numbers of Local Cohomology Modules with Respect to an Ideal

Let be a commutative Noetherian local ring and let be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module are finite if one of the following holds:

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.     

Quantization of orbit bundles in \mathfrakg\mathfrakln* (\mathbbC) \mathfrak{g}\mathfrak{l}_n^* (\mathbb{C})

Let G be the complex general linear group and its Lie algebra equipped with a factorizable Lie bialgebra structure; let U_ħ() be the corresponding quantum group. We construct explicit U_ħ()-equivariant quantization of Poisson orbit bundles O _λ → O _μ in *.     

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions:

Formality Theorem with Coefficients in a Module

In this paper, X will denote a manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, and showed that it is formal. Denote by the DGLA of E-polyvector fields. Considering M, a module over E, we define the-module of E-polyvector fields with values in M. Similarly, we define the-module of E-polydifferential operators with values in M,. We show that there is a quasi-isomorphism of L _∞-modules over from to . Our result extends Calaque’s (and Kontsevich’s) result.