Authors:J. J. Vásquez Pages: 23 - 50 Abstract: We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\) . We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0185-2 Issue No:Vol. 88, No. 1 (2018)

Authors:Humio Ichimura Pages: 51 - 66 Abstract: Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\) . We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \) , and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \) , let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\) -extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0186-1 Issue No:Vol. 88, No. 1 (2018)

Authors:Hidenori Katsurada Pages: 67 - 86 Abstract: We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m). PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0178-1 Issue No:Vol. 88, No. 1 (2018)

Authors:Lucas Dahinden Pages: 87 - 96 Abstract: The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0180-7 Issue No:Vol. 88, No. 1 (2018)

Authors:Ngoc Phu Ha Pages: 163 - 188 Abstract: In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2 1)\) . The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2 1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \) -modular category of previous authors. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0181-6 Issue No:Vol. 88, No. 1 (2018)

Authors:Christopher Deninger Pages: 189 - 192 Abstract: It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0179-0 Issue No:Vol. 88, No. 1 (2018)

Authors:Lucio Cadeddu; Maria Antonietta Farina Pages: 193 - 199 Abstract: In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\) ) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0183-4 Issue No:Vol. 88, No. 1 (2018)

Authors:D. Huybrechts Pages: 201 - 207 Abstract: We observe that derived equivalent K3 surfaces have isomorphic Chow motives. The result holds more generally for arbitrary surfaces, as pointed out by Charles Vial. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0182-5 Issue No:Vol. 88, No. 1 (2018)

Authors:Michael Müger Pages: 209 - 216 Abstract: Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for \(f:[1,\infty )\rightarrow {\mathbb R}\) non-negative and non-decreasing we prove \(f(x)-x=O(x^\gamma )\) with \(\gamma <1\) under certain assumptions on f. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0187-0 Issue No:Vol. 88, No. 1 (2018)

Authors:Lucia Alessandrini Pages: 217 - 245 Abstract: This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications. PubDate: 2018-04-01 DOI: 10.1007/s12188-018-0193-x Issue No:Vol. 88, No. 1 (2018)

Authors:Anneleen De Schepper; N. S. Narasimha Sastry; Hendrik Van Maldeghem Abstract: The original version of this article unfortunately contained a mistake in the author’s name N. S. Narasimha Sastry. The corrected name is given above. PubDate: 2018-02-09 DOI: 10.1007/s12188-018-0192-y

Authors:Ljuben Mutafchiev Abstract: Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\) . The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series. PubDate: 2018-02-06 DOI: 10.1007/s12188-018-0191-z

Authors:Anneleen De Schepper; N. S. Narasimha Sastry; Hendrik Van Maldeghem Abstract: A symplectic polarity of a building \(\varDelta \) of type \(\mathsf {E_6}\) is a polarity whose fixed point structure is a building of type \(\mathsf {F_4}\) containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type \(\mathsf {F_4}\) ). In this paper, we show in a geometric way that every building of type \(\mathsf {E_6}\) contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type \(\mathsf {F_4}\) fully embedded in the natural point-line geometry of \(\varDelta \) arises from a symplectic polarity. PubDate: 2018-01-16 DOI: 10.1007/s12188-017-0190-5

Authors:Neil Hindman; Imre Leader; Dona Strauss Pages: 275 - 287 Abstract: Suppose that we have a finite colouring of \(\mathbb R\) . What sumset-type structures can we hope to find in some colour class' One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with \(X+X\) (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with \(X+X\) monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums. PubDate: 2017-10-01 DOI: 10.1007/s12188-016-0166-x Issue No:Vol. 87, No. 2 (2017)

Authors:Wilfried Imrich; Simon M. Smith Pages: 289 - 297 Abstract: This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G, with \(\aleph _0 \le {\text {Aut}}(G) < 2^{\aleph _0}\) and subdegree-finite automorphism group, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof. PubDate: 2017-10-01 DOI: 10.1007/s12188-016-0167-9 Issue No:Vol. 87, No. 2 (2017)

Authors:Péter Komjáth Pages: 337 - 341 Abstract: We consider the infinite form of Hadwiger’s conjecture. We give a(n apparently novel) proof of Halin’s 1967 theorem stating that every graph X with coloring number \(>\kappa \) (specifically with chromatic number \(>\kappa \) ) contains a subdivision of \(K_\kappa \) . We also prove that there is a graph of cardinality \(2^\kappa \) and chromatic number \(\kappa ^+\) which does not contain \(K_{\kappa ^+}\) as a minor. Further, it is consistent that every graph of size and chromatic number \(\aleph _1\) contains a subdivision of \(K_{\aleph _1}\) . PubDate: 2017-10-01 DOI: 10.1007/s12188-016-0170-1 Issue No:Vol. 87, No. 2 (2017)

Authors:W. Mader Pages: 409 - 419 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: 2017-10-01 DOI: 10.1007/s12188-016-0173-y Issue No:Vol. 87, No. 2 (2017)

Authors:Victor Alexandrov Abstract: We choose some special unit vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) in \({\mathbb {R}}^3\) and denote by \({\mathscr {L}}\subset {\mathbb {R}}^5\) the set of all points \((L_1,\ldots ,L_5)\in {\mathbb {R}}^5\) with the following property: there exists a compact convex polytope \(P\subset {\mathbb {R}}^3\) such that the vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal \({\mathbf {n}}_k\) is equal to \(L_k\) for all \(k=1,\ldots ,5\) . Our main result reads that \({\mathscr {L}}\) is not a locally-analytic set, i.e., we prove that, for some point \((L_1,\ldots ,L_5)\in {\mathscr {L}}\) , it is not possible to find a neighborhood \(U\subset {\mathbb {R}}^5\) and an analytic set \(A\subset {\mathbb {R}}^5\) such that \({\mathscr {L}}\cap U=A\cap U\) . We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces. PubDate: 2017-12-11 DOI: 10.1007/s12188-017-0189-y

Authors:Hansjörg Geiges; Christian Lange Abstract: We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers \(\alpha _1^0,\alpha _2^0\) , we give an algorithmic construction of a Seifert fibration \(L(p,q)\rightarrow S^2(\alpha \alpha _1^0 ,\alpha \alpha _2^0 )\) , where the natural number \(\alpha \) is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models. PubDate: 2017-10-23 DOI: 10.1007/s12188-017-0188-z