Volume 24, Issue 4

Abstract. Let R be a ring and Q be a finite quiver, and let be the number of vertices of Q . Let be the class of representations of Q by right R -modules with local endomorphism ring and R -module homomorphisms. The endomorphism ring of a representation has at most n maximal right ideals, all of which are also left ideals, and the isomorphism class of M is determined by n invariants. The main theorem of this paper states that a finite direct sum of representations in is unique up to n permutations of m elements. Let . A non-directed graph associated to M is introduced and is shown to determine the unique decomposition of M into indecomposable representations. This class of representations is shown to generalize the known classes of modules for which a theorem analogous to the case of our main theorem holds.

Abstract. In this paper, various logarithmically improved regularity criteria for the 3D MHD equations are established in terms of the velocity field and pressure respectively. These regularity criteria can be regarded as log in time improvements of the standard Serrin's criteria established before.

Abstract. We show that every tilting module of projective dimension one over a ring R is associated in a natural way to the universal localization at a set of finitely presented modules of projective dimension one. We then investigate tilting modules of the form . Furthermore, we discuss the relationship between universal localization and the localization given by a perfect Gabriel topology . Finally, we give some applications to Artin algebras and to Prüfer domains.

Abstract. We consider complex projective schemes defined by quadratic equations and satisfying a technical hypothesis on the fibres of the rational map associated to the linear system of quadrics defining X . Our assumption is related to the syzygies of the defining equations and, in particular, it is weaker than properties N 2 , and K 2 . In this setting, we show that the degree d of is bounded by a function of its codimension  c , whose asymptotic behavior is given by , thus improving the obvious bound . More precisely, we get the bound . Furthermore, if X satisfies property or , we obtain the better bound . Some classification results are also given when equality holds.

Abstract. We revisit Hardy's inequality in the scope of regular Dirichlet forms following an analytical method. We shall give an alternative necessary and sufficient condition for the occurrence of Hardy's inequality. A special emphasis will be given for the case where the Dirichlet form under consideration is strongly local, extending therefore some known results in the Euclidean case.

Abstract. Let X be a Banach function space over a nonatomic probability space and let be a filtration. We consider two Banach spaces of martingales and that are associated with X and . One of our results is a characterization of Banach function spaces X such that for every filtration .

Abstract. We consider a class of ordinary differential equations describing one-dimensional quasi-periodically forced systems in the presence of large damping. We give a fully constructive proof of the existence of response solutions, that is quasi-periodic solutions which have the same frequency vector as the forcing. This requires dealing with a degenerate implicit function equation: we prove that the latter has a unique solution, which can be explicitly determined. As a by-product we obtain an explicit estimate of the minimal size of the damping coefficient.

Abstract. In this work we investigate complex Leibniz superalgebras with characteristic sequence and nilindex , where , n and m () are dimensions of even and odd parts, respectively. Such superalgebras with the condition were completely classified. In the present paper, we prove that in the case the Leibniz superalgebras have nilindex less than . Thus, we obtain the classification of Leibniz superalgebras with characteristic sequence and nilindex .

Abstract. Let be a graph. The (unlabeled) configuration space of n points on is the space of n -element subsets of . The n -strand braid group of , denoted , is the fundamental group of . This paper applies the methods of discrete Morse theory to the spaces . We describe how to compute presentations for , where n is an arbitrary natural number and is an arbitrary finite connected graph. Particular attention is paid to the case , and many examples are given.

Abstract. We show that the holomorph of the free group on two generators satisfies the Farrell–Jones Fibered Isomorphism Conjecture. As a consequence, we show that the lower K -theory of the above group vanishes.

Abstract. The starting point for this work was the question whether every finite group G contains a two-generated subgroup H such that , where denotes the set of primes dividing the order of G . We answer the question in the affirmative and address the following more general problem. Let G be a finite group and let be a property of G . What is the minimum number t such that G contains a t -generated subgroup H satisfying the condition that ? In particular, we consider the situation where is the set of composition factors (up to isomorphism), the exponent, the prime graph, or the spectrum of the group G . We give a complete answer in the cases where is the prime graph or the spectrum (obtaining that in the former case and t can be arbitrarily large in the latter case). We also prove that if is the exponent of G , then t is at most four.