Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids

Given a set of points and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing . We establish that our algorithm is polynomial for fixed ε. In addition, the algorithm returns a small core set , whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing . Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate. This work was supported in part by the National Science Foundation through CAREER Grants CCF-0643593 and DMI-0237415.     

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.     

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions , the corresponding vector optimization problem has a solution.     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with . A sufficient condition on a cyclotomic scheme that guarantee the inclusion where is a finite field with elements, is given. I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Riesz and Poisson-Jensen Representation Formulas for a Class of Ultraparabolic Operators on Lie Groups

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . Investigation supported by University of Bologna. Funds for selected research topics.     

Symmetric groups and expander graphs

We construct explicit generating sets S _n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.     

Dimension of the Torelli group for Out(F<Subscript>n</Subscript>)

Let be the kernel of the natural map Out(F_n)→GL_n(ℤ). We use combinatorial Morse theory to prove that has an Eilenberg–MacLane space which is (2n-4)-dimensional and that is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of is equal to 2n-4 and recovers the result of Krstić–McCool that is not finitely presented. We also give a new proof of the fact, due to Magnus, that is finitely generated.     

Point Counting in Families of Hyperelliptic Curves

Let E _Γ be a family of hyperelliptic curves defined by , where is defined over a small finite field of odd characteristic. Then with in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve by using Dwork deformation in rigid cohomology. The time complexity of the algorithm is and it needs bits of memory. A slight adaptation requires only space, but costs time . An implementation of this last result turns out to be quite efficient for n big enough. H. Hubrechts is a Research Assistant of the Research Foundation–Flanders (FWO–Vlaanderen).     

Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator $\mathcal {L}We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator of a general symmetric Markov process. An illuminating concrete example for is , where D is a bounded Euclidean domain in is the Laplace operator in D with zero Dirichlet boundary condition and is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to is a Lévy process that is the sum of Brownian motion in and an independent symmetric (2s)-stable process in killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated It? formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)]. The research of T.-S. Zhang is supported by the British EPSRC.     

Existence of Solutions of a Vector Optimization Problem with a Generic Lower Semicontinuous Objective Function

We study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of lower semicontinuous bounded-from-below objective functions . We establish the existence of a G _δ everywhere dense subset ℱ of such that, for any objective function belonging to ℱ, the corresponding minimization problem possesses a solution.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

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     

Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism , such that is the identity. This answers a question by Thurston (see [11]). Mathematics Subject Classification (2000)  Primary 20H10, 37F30     

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).     

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E ^*, a multifunction and a multifunction Γ:X→2 ^X , find a point such that , . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous. The authors express their sincere gratitude to the referees for helpful suggestions and comments. This research was partially supported by a grant from the National Science Council of Taiwan, ROC. B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.