Abstract: Abstract Let \(d,H \geqslant 2\) , \(m, u \geqslant 0\) be some integers satisfying \(m+u \leqslant d\) . Consider a set of univariate integer polynomials of degree d whose m coefficients for the highest powers of x and u coefficients for the lowest powers of x are fixed, whereas the remaining \(g=d-m-u+1\) coefficients are all bounded by H in absolute value. We show that among those \((2H+1)^g\) polynomials at most \(c d(2H+1)^{g-1}(\log (2H))^{\delta }\) are reducible over \(\mathbb Q\) , where the constant \(c>0\) depends only on two extreme coefficients (if they are fixed) and does not depend on d and H. Here, \(\delta =2\) if \(m=u=0\) ; \(\delta =1\) if only one of m, u is zero; \(\delta =0\) if none of m, u is zero. This estimate is better than the previous one in certain range of d and H. We also prove an estimate for the number of integer reducible polynomials in \(n \geqslant 2\) variables of degree \(d \geqslant 1\) in each variable and height at most \(H \geqslant 1\) . It is completely explicit in terms of n, d, H and implies that the probability for such a polynomial to be reducible tends to zero as \(\max (n,d,H) \rightarrow \infty \) . The condition \(n \geqslant 2\) is essential in the proof: despite some recent progress the problem in general remains open for \(n=1\) . PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0245-0

Abstract: Abstract In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of the main applications is in the study of lonesum matrices. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0252-1

Abstract: Abstract For an irrational \(\alpha \) , we investigate the sums \(\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) \) and \(\sum _{i=1}^n \left\{ \left( \{i \alpha \} - \frac{1}{2} \right) ^2 - \frac{1}{12} \right\} \) . We discuss exact formulae and asymptotic estimates for these sums and point out interesting geometrical properties of their graphs in the case when the continued fraction expansion of \(\alpha \) has a large isolated partial quotient. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-00273-y

Abstract: Abstract I give fully detailed proofs of two important theorems—the exact solution of the weak clique game and the compactness theorem—in the theory of positional games. Both results were published several years ago, including an outline of the proofs that explained the basic idea, but left some technical details to the reader. Unfortunately, in both cases these details turned out to be highly non-trivial. Several mathematicians asked me for help; asked me to clarify the missing details. This is why I felt obliged to write this paper. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-00280-z

Abstract: Abstract In the present note, we focus on the freeness and some combinatorial properties of line arrangements in the projective plane having only double and triple points. The main result shows that for this class of line arrangements the freeness property is combinatorially determined. As a corollary, we show that Böröczky line arrangements in the sense of Füredi and Palásti (Proc Am Math Soc 92(4):561–566, 1984), except exactly three cases, are not free. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0250-3

Abstract: Abstract We classify left-invariant Einstein-like metrics of neutral signature, over four-dimensional Lie groups. Several geometric properties such as being conformally flat, existing Ricci solitons and Walker structures are exhibited. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0262-z

Abstract: Abstract In this paper we investigate the structure of the complete lattice of principal generalized topologies, employing the notion of ultratopology. On any partially ordered set we introduce a generalized topology. The existence of an anti-isomorphism between principal generalized topologies and preorder relations on a set is proposed. After determining the very basic topological properties therein, we will show that each generalized topology has a lattice complement in principal generalized topologies. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0266-8

Abstract: Abstract Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0269-5

Abstract: Abstract We present some Tauberian conditions to recover Cesàro summability of a sequence out of the product methods of Abel and Cesàro summability of the sequence. Moreover, we generalize some classical Tauberian theorems, such as the Hardy–Littlewood theorem, the generalized Littlewood theorem for Abel summability method. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0251-2

Abstract: Abstract We give a simple construction involving partial actions which permits us to obtain an easy proof of a weakened version of L. O’Carroll’s theorem on idempotent pure extensions of inverse semigroups. PubDate: 2019-03-01 DOI: 10.1007/s10998-018-0257-9

Abstract: Abstract In this article, we study Einstein–Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic \((\kappa ,\mu )\) -manifold is Einstein or cosymplectic if it admits a closed Einstein–Weyl structure or two Einstein–Weyl structures. Next for a three dimensional compact almost \(\alpha \) -cosymplectic manifold admitting closed Einstein–Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost \(\alpha \) -cosymplectic admitting two Einstein–Weyl structures is either Einstein or \(\alpha \) -cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact K-cosymplectic manifold with a closed Einstein–Weyl structure or two special Einstein–Weyl structures is cosymplectic. PubDate: 2019-01-17 DOI: 10.1007/s10998-018-00279-6

Abstract: Abstract By examining knot Floer homology, we extend a result of Ozsváth and Stipsicz and show further infinitely many Legendrian and transversely non-simple knot types among two-bridge knots. We give sufficient conditions of Legendrian and transverse non-simplicity on the continued fraction expansion of the corresponding rational number. PubDate: 2019-01-02 DOI: 10.1007/s10998-018-00281-y

Abstract: Abstract We study Hermite interpolation problems on the exponential curve \(y=e^x\) in \(\mathbb {R}^2\) . We construct some kind of regular Hermite interpolation schemes and investigate continuity properties of interpolation polynomials with respect to interpolation conditions. PubDate: 2018-12-19 DOI: 10.1007/s10998-018-00274-x

Abstract: Abstract Truncated tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Lobachevsky–Schläfli formula neatly describes the behaviour of the volume of truncated tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every \(\ell <\ell _0\) , where \(\ell _0\) is an explicit constant, the regular truncated tetrahedron of edge length \(\ell \) maximizes the volume among truncated tetrahedra whose edge lengths are all not smaller than \(\ell \) . This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary. PubDate: 2018-12-17 DOI: 10.1007/s10998-018-00277-8

Abstract: Abstract A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of ‘reversible square’ type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities. PubDate: 2018-12-17 DOI: 10.1007/s10998-018-00275-w

Abstract: Abstract Let a finite set \(F\subset \mathbb {R}^n\) be given. The taxicab distance sum function is defined as the sum of the taxicab distances from the elements (focuses) of the so-called focal set F. The sublevel sets of the taxicab distance sum function are called generalized conics because the boundary points have the same average taxicab distance from the focuses. In case of a classical conic (ellipse) the focal set contains exactly two different points and the distance taken to be averaged is the Euclidean one. The sublevel sets of the taxicab distance sum function can be considered as its generalizations. We prove some geometric (convexity), algebraic (semidefinite representation) and extremal (the problem of the minimizer) properties of the generalized conics and the taxicab distance sum function. We characterize its minimizer and we give an upper and lower bound for the extremal value. A continuity property of the mapping sending a finite subset F to the taxicab distance sum function is also formulated. Finally we present some applications in discrete tomography. If the rectangular grid determined by the coordinates of the elements in \(F\subset \mathbb {R}^2\) is given then the number of points in F along the directions parallel to the sides of the grid is a kind of tomographic information. We prove that it is uniquely determined by the function measuring the average taxicab distance from the focal set F and vice versa. Using the method of the least average values we present an algorithm to reconstruct F with a given number of points along the directions parallel to the sides of the grid. PubDate: 2018-12-17 DOI: 10.1007/s10998-018-00278-7

Abstract: Abstract Let G be a finite group and \(G'\) its commutator subgroup. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. The monoid \({\mathcal {B}} (G)\) of all product-one sequences over G is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if G is abelian (equivalently, \({\mathcal {B}} (G)\) is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if \( G' \le 2\) if and only if \({\mathcal {B}} (G)\) is seminormal, and we study sets of lengths in \({\mathcal {B}} (G)\) . PubDate: 2018-12-17 DOI: 10.1007/s10998-018-00276-9

Abstract: Abstract A covering cycle is a closed path that traverses each edge of a graph at least once. Two cycles are equivalent if one is a cyclic permutation of the other. We compute the number of equivalence classes of non-periodic covering cycles of given length in a non-oriented connected graph. A special case is the number of Euler cycles (covering cycles that cover each edge of the graph exactly once) in the non-oriented graph. We obtain an identity relating the numbers of covering cycles of any length in a graph to a product of determinants. PubDate: 2018-12-01 DOI: 10.1007/s10998-018-0235-2

Abstract: Abstract We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior’s inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum’s problems. We formulate some open problems which involve our new examples of line arrangements. PubDate: 2018-12-01 DOI: 10.1007/s10998-017-0224-x

Abstract: Abstract In this paper, all base 10 repdigits expressible as sums of three Pell numbers are found. PubDate: 2018-12-01 DOI: 10.1007/s10998-018-0247-y