Volume 24, Issue 3

Abstract. The families of -variable fractals for , together with their natural probability distributions, interpolate between the corresponding families of random homogeneous fractals and of random recursive fractals. We investigate certain random matrices associated with these fractals and use them to compute the almost sure Hausdorff dimension of -variable fractals satisfying the uniform open set condition.

Abstract. Let be a magnetic Schrödinger operator on , , where and . In this paper, the authors establish the equivalent characterizations of the Hardy space for , defined by the Lusin area function associated with , in terms of the radial maximal functions and the non-tangential maximal functions associated with and , respectively. This gives an affirmative answer to an open problem of Xuan Thinh Duong et al. [ Ark. Mat.   44 (2006), 261–275]. The boundedness of the Riesz transforms , , from to is also presented, where is the closure of in and .

Abstract. Given a finite simplicial complex , we construct a chain-complex isomorphism from the simplicial chain complex of over endowed with the standard boundary operator (that maps any simplex onto the sum of all of its maximal subsets) to that same complex endowed with the incidence operator that maps any simplex to the sum of just all of its proper subsets. We also indicate why this became of interest in the context of the theory of phylogenetic diversity.

Abstract. In this note we study the rational homotopy types of the moduli space of representations with Borel mold for free monoid and related varieties. The moduli space has a fiber bundle structure over the configuration space in the affine space. We show that the minimal model of the moduli space with mixed Hodge structure is equivalent to the tensor product of minimal models of the configuration space and of the fiber.

Abstract. Given a quasi-modular form of nonzero weight for with algebraic Fourier coefficients, we consider the nonzero values of at the algebraic points of the moduli space . We show that these special values are transcendental numbers. We also establish the analogous result in the setting of Drinfeld quasi-modular forms.

Abstract. We give in terms of Berezin symbols some characterizations for the operators belonging to the Schatten–von Neumann classes , , which were motivated by the questions posed by Nordgren and Rosenthal [7]. Some other problems related with the multiplicative equality for Berezin symbols are also discussed. We also prove in terms of Berezin symbols a Gohberg–Krein type theorem on the weak limit of compact operators.

Abstract. In his recent investigation of a super Teichmüller space, Sachse [16], based on work of Molotkov [15], has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin–Kostant–Leites supermanifolds is equivalent to the category of finite-dimensional Molotkov–Sachse supermanifolds. Simultaneously, using the differential calculus of Bertram–Glöckner–Neeb [2], we extend Molotkov–Sachse's approach to supermanifolds modeled on Hausdorff topological super-vector spaces over an arbitrary non-discrete Hausdorff topological base field of characteristic zero. We also extend to locally base fields the `DeWitt' supermanifolds considered by Tuynman in his monograph [23], and prove that this leads to a category which is isomorphic to the full subcategory of Molokov–Sachse supermanifolds modeled on locally spaces.

Abstract. The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study the mutual inclusions of Zariski and simple Weyl chambers. Finally, we establish the fact that – even though the decompositions themselves may differ – the number of Zariski chambers always equals the number of simple Weyl chambers.

Abstract. Studying certain combinatorial properties of non-unique factorizations have been a subject of recent literature. Little is known about two combinatorial invariants, namely the catenary degree and the tame degree, even in the case of numerical monoids. In this paper we compute these invariants for a certain class of numerical monoids generated by generalized arithmetic sequences. We also show that the difference between the tame degree and the catenary degree can be arbitrary large even if the number of minimal generators is fixed.

Abstract. This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic -torus . In contrast to earlier work that gives the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of .