Authors:Katsumi Akahori Pages: 1 - 16 Abstract: Abstract Let L be a special very ample line bundle of degree \(\text{ deg }(L)\) on a curve X with sufficiently high genus g. We prove that L is normally generated if \(\text{ deg }(L) \ge 2g-3-6h^1(L) (h^1(L) \ge 2)\) and X is not a double covering of a curve. Furthermore we also show that L of \(\text{ deg }(L) = 2g-4-6h^1(L)\) is normally generated if X is not a double or a triple covering a curve. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0119-4 Issue No:Vol. 87, No. 1 (2017)

Authors:M. Castrillón López; P. M. Gadea; I. V. Mykytyuk Pages: 17 - 22 Abstract: Abstract Two explicit expressions of the canonical 8-form on a Riemannian manifold with holonomy group \(\mathrm {Spin}(9)\) have been given: one by the present authors and another by Parton and Piccinni. The relation between these two expressions is obtained. Moreover, it is shown that they are different only from a combinatorial viewpoint. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0124-7 Issue No:Vol. 87, No. 1 (2017)

Authors:Tao Yang; Xuan Zhou; Juzhen Chen Pages: 23 - 38 Abstract: Abstract Based on a pairing of two regular multiplier Hopf algebras A and B, Heisenberg double \(\mathscr {H}\) is the smash product \(A \# B\) with respect to the left regular action of B on A. Let \(\mathscr {D}=A\bowtie B\) be the Drinfel’d double, then Heisenberg double \(\mathscr {H}\) is a Yetter–Drinfel’d \(\mathscr {D}\) -module algebra, and it is also braided commutative by the braiding of Yetter–Drinfel’d module, which generalizes the results in Semikhatov (Commun Algebra 39, 1883–1906, 2011) to some infinite dimensional cases. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0125-6 Issue No:Vol. 87, No. 1 (2017)

Authors:Shunsuke Yamana Pages: 43 - 59 Abstract: Abstract We prove a functional equation of Siegel series associated to nondegenerate semi-integral skew Hermitian forms over quaternion algebras over nonarchimedean local fields of characteristic not 2. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0127-4 Issue No:Vol. 87, No. 1 (2017)

Authors:Á. Figula; K. Strambach Pages: 61 - 68 Abstract: Abstract We show that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the group of its left translations is classical. We give a particular simple general construction for proper loops such that the compact group of their left translations is direct product of at least 3 factors. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0128-3 Issue No:Vol. 87, No. 1 (2017)

Authors:Horst Alzer; Man Kam Kwong Pages: 69 - 82 Abstract: Abstract In 1953, Turán published the inequality $$\begin{aligned} \sum _{k=1}^n \frac{1\cdot 3 \cdots (2k-1)}{2\cdot 4 \cdots 2k}\cos (kx)>-1 \quad {(n\in \mathbf {N}; \, 0<x<\pi )}. \end{aligned}$$ We prove a refinement of this result and offer inequalities for the corresponding sine polynomial. Moreover, we present an application to the theory of absolutely monotonic functions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0129-2 Issue No:Vol. 87, No. 1 (2017)

Authors:Andrea Santi Pages: 83 - 103 Abstract: Abstract We apply the general theory of codimension one integrability conditions for G-structures developed in Santi (Ann Mat Pura Appl 195:1463–1489, 2016) to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0136-3 Issue No:Vol. 87, No. 1 (2017)

Authors:Dmitri Akhiezer; Boris Kazarnovskii Pages: 105 - 111 Abstract: Abstract We consider the eigenfunctions of the Laplace operator \(\Delta \) on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue \(\lambda \) of \(\Delta \) we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals \(\lambda ^{n/2}\mathrm{vol}\,M\) , the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0138-1 Issue No:Vol. 87, No. 1 (2017)

Authors:Shin-ichiro Mizumoto Pages: 113 - 134 Abstract: Abstract For three Siegel modular forms of degrees \(n_1, n_2\) and \(n_1+n_2-1\) respectively, we attach a certain Dirichlet series which has a meromorphic continuation to the whole complex plane and satisfies a functional equation. We also show some algebraicity property of its special values. For the proof we use a Rankin-Selberg type integral involving a pullback of Siegel-Eisenstein series of degree \(2n_1+2n_2-1\) . In some special cases our series coincides with known Dirichlet series associated with automorphic L-functions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0139-0 Issue No:Vol. 87, No. 1 (2017)

Authors:Robert Laterveer Pages: 135 - 144 Abstract: Abstract The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen–Vial and Fu–Tian–Vial on multiplicative Chow–Künneth decompositions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0140-7 Issue No:Vol. 87, No. 1 (2017)

Authors:Bernhard Heim Abstract: Abstract In this paper the image of the Saito–Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to \(\Gamma _0(N)\) . We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level \(N=1\) . PubDate: 2017-02-22 DOI: 10.1007/s12188-017-0177-2

Authors:W. Mader Abstract: Abstract It is proved that every non-complete, finite digraph of connectivity number k has a fragment F containing at most k critical vertices. The following result is a direct consequence: every k-connected, finite digraph D of minimum out- and indegree at least \(2k+ m- 1\) for positive integers k, m has a subdigraph H of minimum outdegree or minimum indegree at least \(m-1\) such that \(D - x\) is k-connected for all \(x \in V(H)\) . For \(m = 1\) , this implies immediately the existence of a vertex of indegree or outdegree less than 2k in a k-critical, finite digraph, which was proved in Mader (J Comb Theory (B) 53:260–272, 1991). PubDate: 2017-02-06 DOI: 10.1007/s12188-016-0173-y

Authors:Reinhard Diestel Abstract: Abstract We show that an arbitrary infinite graph can be compactified by its \({\aleph _0}\) -tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its \({\aleph _0}\) -tangles. The \({\aleph _0}\) -tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The \({\aleph _0}\) -tangles that are ends are precisely the limits of principal ultrafilters.The \({\aleph _0}\) -tangles that correspond to a highly connected part, or \({\aleph _0}\) -block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations. PubDate: 2017-01-18 DOI: 10.1007/s12188-016-0163-0

Authors:Béla Bollobás; Alex Scott Abstract: Abstract A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\) , \(B\in A^{(r-1)}\) is in \(\mathcal H\) . Erdős and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs. PubDate: 2017-01-09 DOI: 10.1007/s12188-016-0162-1

Authors:Matthias Kriesell Abstract: Abstract For each positive integer k, we give a finite list C(k) of Bondy–Chvátal type conditions on a nondecreasing sequence \(d=(d_1,\dots ,d_n)\) of nonnegative integers such that every graph on n vertices with degree sequence at least d is k-edge-connected. These conditions are best possible in the sense that whenever one of them fails for d then there is a graph on n vertices with degree sequence at least d which is not k-edge-connected. We prove that C(k) is and must be large by showing that it contains p(k) many logically irredundant conditions, where p(k) is the number of partitions of k. Since, in the corresponding classic result on vertex connectivity, one needs just one such condition, this is one of the rare statements where the edge connectivity version is much more difficult than the vertex connectivity version. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. We prove that any sublist equivalent to C(k) has length at least p(k), where p(k) is the number of partitions of k, which is in contrast to the corresponding classic result on vertex connectivity where one needs just one such condition. Furthermore, we demonstrate how to handle other types of edge-connectivity, such as, for example, essential k-edge-connectivity. Finally, we informally describe a simple and fast procedure which generates the list C(k). Specialized to \(k=3\) , this verifies a conjecture of Bauer, Hakimi, Kahl, and Schmeichel, and for \(k=2\) we obtain an alternative proof for their result on bridgeless connected graphs. The explicit list for \(k=4\) is given, too. PubDate: 2017-01-05 DOI: 10.1007/s12188-016-0171-0

Authors:William T. Trotter Abstract: Abstract This paper continues a recent resurgence of interest in combinatorial properties of a poset that are associated with graph properties of its cover graph and order diagram. The following two theorems appearing in a 1977 paper of Trotter and Moore have played important roles in motivating this more modern research: (1) The dimension of a poset is at most 3 when its cover graph is at tree; (2) The dimension of a poset is at most 3 when the poset has a zero and its order diagram is planar. Although the underlying ideas lay dormant for more than 30 years, the first of these two results has become the base case for recent results bounding the dimension of a poset in terms of (a) the tree-width of its cover graph, and (b) the maximum dimension of its blocks. The second result is the base case for bounding the dimension of a planar poset in terms of the number of minimal elements. Continuing with this line of research, we show that every poset whose cover graph is a tree is a circle order, i.e., it has a representation as a family of circular disks in the Euclidean plane partially ordered by inclusion. PubDate: 2017-01-05 DOI: 10.1007/s12188-016-0176-8

Authors:Guus Regts; Alexander Schrijver; Bart Sevenster Abstract: Abstract We characterize the virtual link invariants that can be described as partition function of a real-valued R-matrix, by being weakly reflection positive. Weak reflection positivity is defined in terms of joining virtual link diagrams, which is a specialization of joining virtual link diagram tangles. Basic techniques are the first fundamental theorem of invariant theory, the Hanlon–Wales theorem on the decomposition of Brauer algebras, and the Procesi–Schwarz theorem on inequalities for closed orbits. PubDate: 2017-01-04 DOI: 10.1007/s12188-016-0175-9

Authors:Bernd Ammann; Nadine Große Abstract: Abstract We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As a conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries. PubDate: 2016-12-19 DOI: 10.1007/s12188-016-0159-9

Authors:Toshiyuki Kikuta; Shoyu Nagaoka Abstract: Abstract The mod p kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime p. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod p kernel of the theta operator in the case of Hermitian modular forms of degree 2. PubDate: 2016-11-29 DOI: 10.1007/s12188-016-0141-6