Authors:Vincenzo Di Gennaro; Davide Franco Pages: 75 - 86 Abstract: Abstract In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class \(\theta \in Hom_{D(Y)}(R\pi _*\mathbb {Q}_X, \mathbb {Q}_Y)\) , compatible with pullback and with restriction to the complement of the singularity. PubDate: 2017-05-01 DOI: 10.1007/s11565-016-0253-z Issue No:Vol. 63, No. 1 (2017)

Authors:Steven L. Kleiman Pages: 147 - 158 Abstract: Abstract We prove a projection formula, expressing a relative Buchsbaum–Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula. PubDate: 2017-05-01 DOI: 10.1007/s11565-016-0250-2 Issue No:Vol. 63, No. 1 (2017)

Authors:G. Lusztig Pages: 159 - 167 Abstract: Abstract We show that each Weyl group which enters in the generalized Springer correspondence carries a natural weight function. PubDate: 2017-05-01 DOI: 10.1007/s11565-016-0249-8 Issue No:Vol. 63, No. 1 (2017)

Authors:Giuseppe Pareschi Pages: 185 - 199 Abstract: Abstract The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of generic vanishing. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop. PubDate: 2017-05-01 DOI: 10.1007/s11565-016-0256-9 Issue No:Vol. 63, No. 1 (2017)

Authors:David S. Tartakoff Abstract: Abstract We study the regularity of Gevrey vectors for Hörmander operators $$\begin{aligned} P = \sum _{j=1}^m X_j^2 + X_0 + c \end{aligned}$$ where the \(X_j\) are real vector fields and c(x) is a smooth function, all in Gevrey class \(G^{s}.\) The principal hypothesis is that P satisfies the subelliptic estimate: for some \(\varepsilon >0, \; \exists \,C\) such that $$\begin{aligned} \Vert v\Vert _{\varepsilon }^2 \le C\left( (Pv, v) + \Vert v\Vert _0^2\right) \qquad \forall v\in C_0^\infty . \end{aligned}$$ We prove directly (without the now familiar use of adding a variable t and proving suitable hypoellipticity for \(Q=-D_t^2-P\) and then, using the hypothesis on the iterates of P on u, constructing a homogeneous solution U for Q whose trace on \(t=0\) is just u) that for \(s\ge 1,\) \(G^s(P,\Omega _0) \subset G^{s/\varepsilon }(\Omega _0);\) that is, $$\begin{aligned}&\forall K\Subset \Omega _0, \;\exists C_K: \Vert P^j u\Vert _{L^2(K)}\le C_K^{j+1} (2j)!^s, \;\forall j\\&\quad \implies \forall K'\Subset \Omega _0, \;\exists \tilde{C}_{K'}:\,\Vert D^\ell u\Vert _{L^2(K')} \le \tilde{C}_{K'}^{\ell +1} \ell !^{s/\varepsilon }, \;\forall \ell . \end{aligned}$$ In other words, Gevrey growth of derivatives of u as measured by iterates of P yields Gevrey regularity for u in a larger Gevrey class. When \(\varepsilon =1,\) P is elliptic and so we recover the original Kotake–Narasimhan theorem (Kotake and Narasimhan in Bull Soc Math Fr 90(12):449–471, 1962), which has been studied in many other classes, including ultradifferentiable functions (Boiti and Journet in J Pseudo-Differ Oper Appl 8(2):297–317, 2017). We are indebted to M. Derridj for multiple conversations over the years. PubDate: 2017-08-29 DOI: 10.1007/s11565-017-0293-z

Authors:Hocine Gabsi; Abdelouaheb Ardjouni; Ahcene Djoudi Abstract: Abstract In this work we use some mixed techniques of the Mawhin coincidence degree theory and fixed point theorem to prove the existence of positive periodic solutions of delay systems. As a consequence, we offer existence criteria and sufficient conditions for existence of periodic solutions to the systems with feedback control. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved. PubDate: 2017-08-02 DOI: 10.1007/s11565-017-0292-0

Authors:P. N. Agrawal; Pooja Gupta Abstract: Abstract The purpose of the present paper is to introduce a Kantorovich modification of the q-analogue of the Stancu operators defined by Nowak (J Math Anal Appl 350:50–55, 2009). We study a local and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Further A-statistical convergence properties of these operators are investigated. Next, a bivariate generalization of these operators is introduced and its rate of convergence is discussed with the aid of the partial and complete modulus of continuity and the Peetre‘s K-functional. PubDate: 2017-07-01 DOI: 10.1007/s11565-017-0291-1

Authors:A. Hosseini Abstract: Abstract Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\) , where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \) . By applying this conclusion, we offer a characterization of left \(\sigma \) -centralizers, generalized derivations and generalized \(\sigma \) -derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi- \(\sigma \) -derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero. PubDate: 2017-06-21 DOI: 10.1007/s11565-017-0290-2

Authors:Abderrahim Aslimani; Imad El Ghazi; Mohamed El Kadiri; Sabah Haddad Abstract: Abstract We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here. PubDate: 2017-06-20 DOI: 10.1007/s11565-017-0289-8

Authors:Meenu Goyal; P. N. Agrawal Abstract: Abstract In this paper, we introduce the Bézier variant of the Jakimovski–Leviatan–Păltănea operators based on Appell polynomials. We establish some local results, a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and also study the rate of convergence for the functions having a derivative of bounded variation for these operators. PubDate: 2017-06-08 DOI: 10.1007/s11565-017-0288-9

Authors:Tongkeun Chang; Kyungkeun Kang Abstract: Abstract We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in \({\mathbb {R}}^n_+ \times (0,T)\) . We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier–Stokes equations in a half-space in dimension three. PubDate: 2017-06-06 DOI: 10.1007/s11565-017-0287-x

Authors:M. S. Mahadeva Naika; B. Hemanthkumar; S. Chandankumar Abstract: Abstract In his second notebook, Ramanujan recorded total of seven P–Q modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained for higher orders. As a consequence, several values of quotients of theta-function are evaluated. The cubic singular modulus is evaluated at \(q=\exp (-2\pi \sqrt{n/3})\) for \(n\in \{5k, 1/5k, 5/k, k/5\}\) , where \(k\in \{4,7,16\}\) . PubDate: 2017-05-22 DOI: 10.1007/s11565-017-0286-y

Authors:Nipen Saikia; Chayanika Boruah Abstract: Abstract We derive a new special case C(q) of a general continued fraction recorded by Ramanujan in his Lost Notebook. We give a representation of the continued fraction C(q) as a quotient of Dedekind eta-function and then use it to prove modular identities connecting C(q) with each of the continued fractions \(C(-q)\) , \(C(q^{2})\) , \(C(q^{3})\) , \(C(q^{5})\) , \(C(q^{7})\) , \(C(q^{11})\) , \(C(q^{13})\) and \(C(q^{17})\) . We also prove general theorems for the explicit evaluation of the continued fraction C(q) by using Ramanujan’s class invariants. PubDate: 2017-05-12 DOI: 10.1007/s11565-017-0283-1

Authors:Alexandru Dimca Abstract: 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:Edoardo Sernesi Abstract: 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: 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:Marian Aprodu; Yeongrak Kim Abstract: 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:Lucian Bădescu Abstract: 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