Authors:Alexandru Dimca Abstract: We give lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in \(\mathbb {P}^3\) is improved and applications to deformation theory of nodal surfaces are given. PubDate: 2017-03-17 DOI: 10.1007/s11565-017-0278-y

Authors:Poornapushkala Narayanan Abstract: Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\) , the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\) , and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\) . We show that any dominating component of \({\mathscr {W}}^1_{d}( L' )\) corresponds to \(\mu _{L'}\) -stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in L \) and a base-point free \(g^1_d\) on C, say (A, V), we study the \(\mu _L\) -semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the \(g^1_d\) , we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\) -semistable. PubDate: 2017-03-10 DOI: 10.1007/s11565-017-0277-z

Authors:Alberto Alzati; Riccardo Re Abstract: In this note we give a different proof of Sacchiero’s theorem about the splitting type of the normal bundle of a generic rational curve. Moreover we discuss the existence and the construction of smooth monomial curves having generic type of the normal bundle. PubDate: 2017-03-04 DOI: 10.1007/s11565-017-0276-0

Authors:Myong-Hwan Ri Abstract: In this paper we show that a Leray–Hopf weak solution u to 3D Navier–Stokes initial value problem is smooth if there is some \(\alpha \in {{{\mathbb {R}}}}, \alpha \ne 0,\) such that \(\alpha u_3+(-\Delta )^{-1/2}\omega _3\) is suitably smooth, where \(\omega =\text {curl}\,u\) . PubDate: 2017-03-01 DOI: 10.1007/s11565-017-0274-2

Authors:Edoardo Sernesi Abstract: We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blow-up X of the projective plane along their singular points. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The proof makes essential use of the geometry of X. PubDate: 2017-02-20 DOI: 10.1007/s11565-017-0275-1

Authors:Filippo F. Favale; Roberto Pignatelli Abstract: By a theorem of Reider, a twisted bicanonical system, that means a linear system of divisors numerically equivalent to a bicanonical divisor, on a minimal surface of general type, is base point free if \(K^2_S \ge 5\) . Twisted bicanonical systems with base points are known in literature only for \(K^2=1,2\) . We prove in this paper that all surfaces in a family of surfaces with \(K^2=3\) constructed in a previous paper with G. Bini and J. Neves have a twisted bicanonical system (different from the bicanonical system) with two base points. We show that the map induced by the above twisted bicanonical system is birational, and describe in detail the closure of its image and its singular locus. Inspired by this description, we reduce the problem of constructing a minimal surface of general type with \(K^2=3\) whose bicanonical system has base points, under some reasonable assumptions, to the problem of constructing a curve in \({\mathbb {P}}^3\) with certain properties. PubDate: 2017-02-10 DOI: 10.1007/s11565-017-0273-3

Authors:Chhaya Singhal; G. S. Srivastava Abstract: In the present paper, we obtain the characterization of various growth parameters of an entire function F(s) represented by Laplace–Stieltjes transformation in terms of the rate of decrease of \(E_n ( {F,\beta } ),\) where \(E_n ( {F,\beta } )\) represents the error in approximating the function F(s) by exponential polynomials. PubDate: 2017-02-09 DOI: 10.1007/s11565-017-0272-4

Authors:A. Taghavi; V. Darvish; H. M. Nazari; S. S. Dragomir Abstract: In this paper, we prove some singular value inequalities for sum and product of operators. Also, we obtain several generalizations of recent inequalities. Moreover, as applications we establish some unitarily invariant norm and trace inequalities for operators which provide refinements of previous results. PubDate: 2017-01-31 DOI: 10.1007/s11565-017-0271-5

Authors:Paltin Ionescu; Francesco Russo Abstract: This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016). PubDate: 2017-01-27 DOI: 10.1007/s11565-017-0270-6

Authors:Marian Aprodu; Yeongrak Kim Abstract: In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles. PubDate: 2017-01-21 DOI: 10.1007/s11565-017-0269-z

Authors:Henrique F. de Lima; Márcio S. Santos Abstract: We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Émery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space. PubDate: 2016-11-16 DOI: 10.1007/s11565-016-0268-5

Authors:Lucian Bădescu Abstract: The aim of this note is to prove, in the spirit of a rigidity result for isolated singularities of Schlessinger see Schlessinger (Invent Math 14:17–26, 1971) or also Kleiman and Landolfi (Compositio Math 23:407–434, 1971), a variant of a rigidity criterion for arbitrary singularities (Theorem 2.1 below). The proof of this result does not use Schlessinger’s Deformation Theory [Schlessinger (Trans Am Math Soc 130:208–222, 1968) and Schlessinger (Invent Math 14:17–26, 1971)]. Instead it makes use of Local Grothendieck-Lefschetz Theory, see (Grothendieck 1968, Éxposé 9, Proposition 1.4, page 106) and a Lemma of Zariski, see (Zariski, Am J Math 87:507–536, 1965, Lemma 4, page 526). I hope that this proof, although works only in characteristic zero, might also have some interest in its own. PubDate: 2016-11-15 DOI: 10.1007/s11565-016-0267-6

Authors:Zoltán Finta Abstract: We introduce a sequence of Stancu type integral operators involving q-Beta functions, and we establish some direct results, which include a Korovkin type theorem and error estimations in terms of the modulus of continuity and the Lipschitz type maximal function, respectively. We also define the limit operator of our sequence of operators, and we obtain quantitative approximation theorems for it. PubDate: 2016-09-12 DOI: 10.1007/s11565-016-0257-8

Authors:Basudeb Dhara; Mohd Arif Raza; Nadeem Ur Rehman Abstract: Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U, C be the extended centroid of \(R,\, F\) and G be two nonzero generalized derivations of R and \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C which is not central valued on R. If $$\begin{aligned} {[}F(u)u, G(v)v]=0 \end{aligned}$$ for all \(u,v\in f(R)\) , then there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \([a, b]=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R. PubDate: 2016-08-18 DOI: 10.1007/s11565-016-0255-x

Authors:Jeremy Berquist Abstract: The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollár, Reid, and others, beginning in the 1980s with the introduction of the Minimal Model Program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied by using a resolution of singularities (or a semi-resolution), and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, it has been assumed that a variety X is has a \({\mathbb {Q}}\) -Cartier canonical class: some multiple \(mK_X\) of the canonical class is Cartier. In particular, this divisor can be pulled back under a resolution \(f: Y \rightarrow X\) by pulling back its local sections. Then one has a relation \(K_Y \sim \frac{1}{m}f^*(mK_X) + \sum a_iE_i\) . It is then the coefficients of the exceptional divisors \(E_i\) that determine the type of singularities that belong to X. It might be asked whether this \({\mathbb {Q}}\) -Cartier hypothesis is necessary in studying singularities in birational classification. de Fernex and Hacon (Compos Math 145:393–414, 2009) construct a boundary divisor \(\Delta \) for arbitrary normal varieties, the resulting divisor \(K_X + \Delta \) being \({\mathbb {Q}}\) -Cartier even though \(K_X\) itself is not. This they call (for reasons that will be made clear) an m-compatible boundary for X, and they proceed to show that the singularities defined in terms of the pair \((X,\Delta )\) are none other than the singularities just described, when \(K_X\) happens to be \({\mathbb {Q}}\) -Cartier. Thus, a wider context exists within which one can study singularities of the above types. In the present paper, we extend the results of de Fernex and Hacon (Compos Math 145:393–414, 2009) still further, to include demi-normal varieties without a \({\mathbb {Q}}\) -Cartier canonical class. Our main result is that m-compatible boundaries exist for demi-normal varieties (Theorem 1.1). This theorem provides a link between the theory of singularities for arbitrary demi-normal varieites (whose canonical class may not \(\hbox {be } {\mathbb {Q}}\) -Cartier), that theory being developed in the present paper, and the established theory of singularities of pairs. PubDate: 2016-08-02 DOI: 10.1007/s11565-016-0251-1

Authors:Kazuaki Taira Abstract: This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. Our degenerate boundary conditions include as particular cases the isothermal condition (Dirichlet condition) and the adiabatic condition (Neumann condition). We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. More precisely, we give sufficient conditions for our semilinear boundary value problems to have three positive solutions, which suggests that the bifurcation curves are S-shaped. PubDate: 2016-07-29 DOI: 10.1007/s11565-016-0254-y

Authors:Sandro Zagatti Abstract: We study the Dirichlet problem for Hamilton–Jacobi equations of the form $$\begin{aligned} {\left\{ \begin{array}{ll} H(x, \nabla u(x)) = 0 &{}\quad \text {in} \ \Omega \\ u(x)=\varphi (x) &{}\quad \text {on} \ \partial \Omega , \end{array}\right. } \end{aligned}$$ without continuity assumptions on the hamiltonian H with respect to the variable x. We find a class of Caratheodory functions H for which the problem admits a (maximal) generalized solution which, in the continuous case, coincides with the classical viscosity solution. PubDate: 2016-07-18 DOI: 10.1007/s11565-016-0252-0

Authors:Yuji Liu Abstract: This paper is concerned with a boundary value problem of second order singular differential equations on whole line. Sufficient conditions to guarantee existence and non-existence of positive solutions are established. Our results improve some theorems in known papers but the methods used are different. We give two examples to illustrate main theorems. PubDate: 2016-06-08 DOI: 10.1007/s11565-016-0246-y

Authors:Alexandre Laugier; Manjil P. Saikia; Upam Sarmah Abstract: In this article, based on ideas and results by Sándor (J Inequal Pure Appl Math 2:Art. 3, 2001; J Inequal Pure Appl Math 5, 2004), we define k-multiplicatively e-perfect numbers and k-multiplicatively e-superperfect numbers and prove some results on them. We also characterize the k- \(T_0T^*\) -perfect numbers defined by Das and Saikia (Notes Number Theory Discrete Math 19:37–42, 2013) in details. PubDate: 2016-06-06 DOI: 10.1007/s11565-016-0248-9