Authors:Ryan R. Martin; Shanise Walker Abstract: Abstract The \(\mathscr {N}\) poset consists of four distinct sets W, X, Y, Z such that \(W\subset X, Y\subset X\) , and \(Y\subset Z\) where W is not necessarily a subset of Z. A family \({{\mathscr {F}}}\) , considered as a subposet of the n-dimensional Boolean lattice \(\mathscr {B}_n\) , is \(\mathscr {N}\) -free if it does not contain \(\mathscr {N}\) as a subposet. Let \(\mathrm{La}(n, \mathscr {N})\) be the size of a largest \(\mathscr {N}\) -free family in \(\mathscr {B}_n\) . Katona and Tarján proved that , where and is the size of a single-error-correcting code with constant weight \(k+1\) . In this note, we prove for n even and \(k=n/2, \mathrm{La}(n, \mathscr {N}) \geqslant {n\atopwithdelims ()k}+A(n, 4, k)\) , which improves the bound on \(\mathrm{La}(n, \mathscr {N})\) in the second order term for some values of n and should be an improvement for an infinite family of values of n, depending on the behavior of the function . PubDate: 2017-04-06 DOI: 10.1007/s40879-017-0139-3

Authors:Maxime Fairon Abstract: Abstract This paper aims at setting out the basics of \(\mathbb {Z}\) -graded manifolds theory. We introduce \(\mathbb {Z}\) -graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential \(\mathbb {Z}\) -graded manifolds and algebraic structures. PubDate: 2017-03-27 DOI: 10.1007/s40879-017-0138-4

Authors:Adrien Dubouloz Abstract: Abstract We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non-algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non-isomorphic exotic affine 3-spheres parametrized by the closed points of S. PubDate: 2017-03-22 DOI: 10.1007/s40879-017-0136-6

Authors:Jan-Hendrik de Wiljes Abstract: Abstract We introduce the uniform coprime hypergraph of integers \(\mathrm{CHI}_k\) which is the graph with vertex set \({\mathbb {Z}}\) and a -hyperedge exactly between every \(k+1\) elements of \({\mathbb {Z}}\) having greatest common divisor equal to 1. This generalizes the concept of coprime graphs. We obtain some basic properties of these graphs and give upper and lower bounds for the clique number of certain subgraphs of \(\mathrm{CHI}_k\) . PubDate: 2017-03-15 DOI: 10.1007/s40879-017-0137-5

Authors:Emily Cliff Abstract: Abstract We introduce stacks classifying étale germs of pointed n-dimensional varieties. We show that quasi-coherent sheaves on these stacks are universal \(\mathscr {D}\) - and \(\mathscr {O}\) -modules. We state and prove a relative version of Artin’s approximation theorem, and as a consequence identify our stacks with classifying stacks of automorphism groups of the n-dimensional formal disc. We introduce the notion of convergent universal modules, and study them in terms of these stacks and the representation theory of the automorphism groups. PubDate: 2017-03-09 DOI: 10.1007/s40879-017-0135-7

Authors:Dino Lorenzini Abstract: Abstract Néron models were introduced by André Néron in his seminar at the IHÉS in 1961. This article gives a brief survey of past and recent results in this very useful theory. PubDate: 2017-03-01 DOI: 10.1007/s40879-017-0134-8

Authors:Jean-Louis Colliot-Thélène Abstract: Résumé Sur un corps de fonctions d’une variable sur le corps des complexes, l’approximation forte hors d’un ensemble fini non vide de places vaut pour tout espace homogène d’un groupe semi-simple simplement connexe. En particulier l’approximation forte hors d’un ensemble fini non vide de places vaut pour les quadriques affines lisses de dimension au moins 2 sur un tel corps. L’approximation forte hors d’un ensemble fini de places ne vaut pas pour le groupe multiplicatif. PubDate: 2017-02-28 DOI: 10.1007/s40879-017-0133-9

Authors:Igor Burban; Yuriy Drozd; Volodymyr Gavran Abstract: Abstract We develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra. PubDate: 2017-02-23 DOI: 10.1007/s40879-017-0128-6

Authors:Matthias Kunik Abstract: Abstract The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have simple main terms, whereas the deviations are determined by a sequence of polygonal functions \(f_n\) . In a former paper we obtained a limit function for \(n \rightarrow \infty \) which describes an asymptotic scaling property of functions \(f_n\) in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive new representation formulas for \(f_n\) and related functions which give much better remainder term estimates. We also combine these results with those from our previous papers in order to prove that the sequence of functions \(f_n\) converges pointwise to zero. PubDate: 2017-02-13 DOI: 10.1007/s40879-017-0132-x

Authors:Shabnam Akhtari; Jeffrey D. Vaaler Abstract: Abstract We call a unit \(\beta \) in a finite Galois extension \(l/\mathbb {Q}\) a Minkowski unit if the subgroup generated by \(\beta \) and its conjugates over \(\mathbb {Q}\) has maximum rank in the unit group of l. Minkowski showed the existence of such units in every Galois extension. We give a new proof of Minkowski’s theorem and show that there exists a Minkowski unit \(\beta \in l\) such that the Weil height of \(\beta \) is comparable with the sum of the heights of a fundamental system of units for l. Our proof implies a bound on the index of the subgroup generated by the algebraic conjugates of \(\beta \) in the unit group of l. If k is an intermediate field such that \(\mathbb {Q}\subseteq k \subseteq l\) , and \(l/\mathbb {Q}\) and \(k/\mathbb {Q}\) are Galois extensions, we prove an analogous bound for the subgroup of relative units. In order to establish our results for relative units, a number of new ideas are combined with techniques from the geometry of numbers and the Galois action on places. PubDate: 2017-02-10 DOI: 10.1007/s40879-017-0131-y

Authors:Masoud Bayrami; Mahmoud Hesaaraki Abstract: Abstract We study the optimal value of p for solvability of the problem 1 Here \(\lambda ,\alpha >0\) , \(p>1\) , f is a non-negative measurable function and , \(N\geqslant 3\) , is an open bounded domain with smooth boundary such that \(0 \in \mathrm{\Omega }\) . We find the critical threshold exponent \(p_{+}(\lambda ,\alpha )\) for solvability of (1) and show that if , \(1<p<p_{+}(\lambda ,\alpha )\) and for some sufficiently small \(c_0>0\) , then there exists a solution as a limit of solutions to approximating problems. Moreover, for \(p \geqslant p_{+}(\lambda ,\alpha )\) we show that a complete blow-up phenomenon occurs. PubDate: 2017-02-03 DOI: 10.1007/s40879-017-0129-5

Authors:Paolo Cascini; Hiromu Tanaka Abstract: Abstract We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata–Viehweg vanishing. PubDate: 2017-01-25 DOI: 10.1007/s40879-016-0127-z

Authors:Raffaele Marcovecchio Abstract: Abstract Two linear forms, \(\sigma _n \zeta (5)+\tau _n \zeta (3)+\varphi _n\) and \(\sigma _n\zeta (2)+\tau _n/2\) , with suitable rational coefficients \(\sigma _n,\tau _n,\varphi _n\) , are presented. As a byproduct, we obtain an identity between simple and double binomial sums, where the simple sum is the value of a terminating well-poised Saalschützian \({}_4 F_3\) series. This complements a recent note of the author on two linear forms: \(\alpha _n \widetilde{\zeta }(4)+\beta _n \widetilde{\zeta }(2)+\gamma _n\) , based on an identity of Paule–Schneider, and \(\alpha _n\zeta (2)+\beta _n\) , coming from the Apéry–Beukers construction. PubDate: 2017-01-23 DOI: 10.1007/s40879-016-0126-0

Authors:Shimon Garti Abstract: Abstract Let \(\kappa \) be any regular cardinal. Assuming the existence of a huge cardinal above \(\kappa \) , we prove the consistency of for every ordinal \(\tau <\kappa ^{++}\) . Likewise, we prove that is consistent when \(\mathscr {A}\) is strongly closed under countable intersections. PubDate: 2017-01-04 DOI: 10.1007/s40879-016-0125-1

Authors:Tomokazu Onozuka Abstract: Abstract We prove three results on the a-points of derivatives of the Riemann zeta function. The first result is a formula of the Riemann–von Mangoldt type; we estimate the number of a-points of derivatives of the Riemann zeta function. The second result is on certain exponential sum involving a-points. The third result is an analogue of the zero density theorem. We count the a-points of derivatives of the Riemann zeta function in \(1/2-(\log \log T)^2/\log T<\mathrm{Re}\, s<1/2+(\log \log T)^2/\log T\) . PubDate: 2016-12-20 DOI: 10.1007/s40879-016-0124-2

Authors:Olena Karlova; Volodymyr Mykhaylyuk Abstract: Abstract We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_n:X\rightarrow Z\) with an additional condition (which is a weakening of the uniform convergence at each point) in the following cases: (a) X is stratifiable and Z is locally convex equiconnected; (b) X is stratifiable with \(\dim X<\infty \) and Z is equiconnected; (c) X is stratifiable with \(\dim X=0\) . We show that for a hereditarily Baire space X, a compact space Y and a metric space Z every separately continuous map is a pointwise limit of a sequence of continuous maps which is uniformly convergent to f with respect to each variable at every point of in cases (a)–(c); (d) X is a perfect zero-dimensional compact space; and (e) X is the Sorgenfrey line. PubDate: 2016-12-05 DOI: 10.1007/s40879-016-0123-3

Authors:Yuri G. Zarhin Abstract: Abstract We study abelian varieties over finitely generated fields K of characteristic zero, whose \(\ell \) -adic Tate modules are isomorphic as Galois modules for all primes \(\ell \) . PubDate: 2016-11-30 DOI: 10.1007/s40879-016-0122-4

Authors:Andrei Pajitnov Abstract: Abstract Let \(\phi :M\rightarrow M\) be a diffeomorphism of a \(C^{\infty }\) compact connected manifold, and X its mapping torus. There is a natural fibration \(p:X\rightarrow S^1\) , denote by the corresponding cohomology class. Let \(\lambda \in {\mathbb {C}}^*\) . Consider the endomorphism \(\phi _k^*\) induced by \(\phi \) in the cohomology of M of degree k, and denote by \(J_k(\lambda )\) the maximal size of its Jordan block of eigenvalue \(\lambda \) . Define a representation \(\rho _\lambda :\pi _1(X)\rightarrow {\mathbb {C}}^*\) ; \(\rho _\lambda (g)=\lambda ^{p_*(g)}\) ; let \(H^*(X,\rho _\lambda )\) be the corresponding twisted cohomology of X. We prove that \(J_k(\lambda )\) is equal to the maximal length of a non-zero Massey product of the form \(\langle \xi , \ldots , \xi , a\rangle \) where \(a\in H^k(X,\rho _\lambda )\) (here the length means the number of entries of \(\xi \) ). In particular, if X is a strongly formal space (e.g. a Kähler manifold) then all the Jordan blocks of \(\phi _k^*\) are of size 1. If X is a formal space, then all the Jordan blocks of eigenvalue 1 are of size 1. This leads to a simple construction of formal but not strongly formal mapping tori. The proof of the main theorem is based on the fact that the Massey products of the above form can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of \(\phi ^*\) . PubDate: 2016-11-10 DOI: 10.1007/s40879-016-0121-5

Authors:Steven Rayan Abstract: Abstract For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse–Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of \(\text{ A }\) -type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of \(\text{ A }_1\) . For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are coprime, thereby verifying conjectural lowest Betti numbers coming from high-energy physics. PubDate: 2016-11-10 DOI: 10.1007/s40879-016-0120-6

Authors:Shavkat Ayupov; Karimbergen Kudaybergenov Abstract: Abstract We prove that a von Neumann algebra M is abelian if and only if the square of every derivation on the algebra S(M) of measurable operators affiliated with M is a local derivation. We also show that for general associative unital algebras this is not true. PubDate: 2016-10-24 DOI: 10.1007/s40879-016-0118-0