Authors:Beatriz Rodríguez González; Agustí Roig Pages: 301 - 321 Abstract: Abstract We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric \(\Omega \) -spectra, filtered dg algebras, operads and operad algebras. PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0171-5 Issue No:Vol. 68, No. 3 (2017)

Authors:Ivana Slamić Pages: 323 - 337 Abstract: Abstract Let T be a dual integrable representation of a countable discrete LCA group G acting on a Hilbert space \(\mathbb H\) . We consider the problem of characterizing \(\ell ^2(G)\) -linear independence of the system \(\mathcal B_{\psi }=\{T_{g}\psi :g\in G\}\) for a given function \(\psi \in \mathbb H\) in terms of the bracket function. The characterization theorem is obtained for the case when G is a uniform lattice of the p-adic Vilenkin group acting by translations and a partial answer is given for the case when \(\mathcal B_{\psi }\) is the Gabor system. PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0175-1 Issue No:Vol. 68, No. 3 (2017)

Authors:Thomas Bauer; Brian Harbourne; Joaquim Roé; Tomasz Szemberg Pages: 339 - 357 Abstract: Abstract For each \(m\geqslant 1\) , Roulleau and Urzúa give an implicit construction of a configuration of \(4(3m^2-1)\) complex plane cubic curves. This construction was crucial for their work on surfaces of general type. We make this construction explicit by proving that the Roulleau–Urzúa configuration consists precisely of the Halphen cubics of order m, and we determine specific equations of the cubics for \(m=1\) (which were known) and for \(m=2\) (which are new). PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0172-4 Issue No:Vol. 68, No. 3 (2017)

Authors:P. Mozolyako Pages: 359 - 376 Abstract: Abstract Let u(x, y) be a harmonic function in the halfspace \({\mathbb {R}}^n\times {\mathbb {R}}_+\) that grows near the boundary not faster than some fixed majorant w(y). Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the law of iterated logarithm (LIL). We extend this result to a class of Lipschitz domains in \({\mathbb {R}}^{n+1}\) . In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight w. The growth rate of such Bloch function depends on w and, for slowly increasing w, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour. PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0177-z Issue No:Vol. 68, No. 3 (2017)

Authors:Shapour Heidarkhani; Giuseppe Caristi; Amjad Salari Pages: 401 - 418 Abstract: Abstract The present paper is an attempt to the study of multiplicity results of solutions for a class of perturbed p-Laplacian discrete problems of Kirchhoff-type. Indeed, we will use variational methods for smooth functionals, defined on the reflexive Banach spaces in order to achieve the existence of at least three solutions for the problems. Moreover, assuming sign conditions on the nonlinear terms, we will prove that the solutions are non-negative. Finally, by presenting two examples, we will ensure the applicability of our results. PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0180-4 Issue No:Vol. 68, No. 3 (2017)

Authors:M. Emilia Alonso; Henri Lombardi Pages: 419 - 432 Abstract: In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and \(\mathfrak {m}\) -adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature. PubDate: 2017-09-01 DOI: 10.1007/s13348-016-0184-0 Issue No:Vol. 68, No. 3 (2017)

Authors:Anders Björn; Jana Björn; Juha Lehrbäck Pages: 229 - 241 Abstract: Abstract We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted \(\mathbf {R}^n\) and in metric spaces, primarily under the assumptions of an annular decay property and a Poincaré inequality. In particular, if the measure has the 1-annular decay property at \(x_0\) and the metric space supports a pointwise 1-Poincaré inequality at \(x_0\) , then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at \(x_0\) . This generalizes the known estimate for the usual variational capacity in unweighted \(\mathbf {R}^n\) . We also characterize the 1-annular decay property and provide examples which illustrate the sharpness of our results. PubDate: 2017-05-01 DOI: 10.1007/s13348-016-0178-y Issue No:Vol. 68, No. 2 (2017)

Authors:Lars Winther Christensen; Srikanth B. Iyengar Pages: 243 - 250 Abstract: Abstract It is proved that a module M over a commutative noetherian ring R is injective if \(\mathrm {Ext}_{R}^{i}((R/{\mathfrak p})_{\mathfrak p},M)=0\) holds for every \(i\geqslant 1\) and every prime ideal \(\mathfrak {p}\) in R. This leads to the following characterization of injective modules: If F is faithfully flat, then a module M such that \({\text {Hom}}_R(F,M)\) is injective and \({\text {Ext}}^i_R(F,M)=0\) for all \(i\geqslant 1\) is injective. A limited version of this characterization is also proved for certain non-noetherian rings. PubDate: 2017-05-01 DOI: 10.1007/s13348-016-0176-0 Issue No:Vol. 68, No. 2 (2017)

Authors:A. Baranov; Y. Belov; A. Poltoratski Pages: 251 - 263 Abstract: Abstract We characterize the Hermite–Biehler (de Branges) functions E which correspond to Schroedinger operators with \(L^2\) potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances. PubDate: 2017-05-01 DOI: 10.1007/s13348-016-0168-0 Issue No:Vol. 68, No. 2 (2017)

Authors:Yu-Xia Liang; Lei Xia Pages: 265 - 278 Abstract: Abstract Given a locally compact group G, the disjoint supercyclicity of finite weighted translations with the same translation part generated by an aperiodic element acting on \(L^p(G)\) , \(1\le p<\infty \) , was investigated. PubDate: 2017-05-01 DOI: 10.1007/s13348-016-0164-4 Issue No:Vol. 68, No. 2 (2017)

Authors:Daniel Vera Pages: 279 - 299 Abstract: Abstract Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hörmander definition with x-dependent parameter \(\delta \) for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO. PubDate: 2017-05-01 DOI: 10.1007/s13348-016-0167-1 Issue No:Vol. 68, No. 2 (2017)

Authors:Davide Barbieri; Eugenio Hernández; Victoria Paternostro Abstract: Abstract Given a discrete group and a unitary representation on a Hilbert space \(\mathcal {H}\) , we prove that the notions of operator Bracket map and Gramian coincide on a dense set of \(\mathcal {H}\) . As a consequence, combining this result with known frame theory, we can recover all previous Bracket characterizations of Riesz and frame sequences generated by a single element under a unitary representation. PubDate: 2017-07-28 DOI: 10.1007/s13348-017-0202-x

Authors:Paul Hagelstein; Ioannis Parissis; Olli Saari Abstract: Abstract Let \(A_\infty ^+\) denote the class of one-sided Muckenhoupt weights, namely all the weights w for which \(\mathsf {M}^+:L^p(w)\rightarrow L^{p,\infty }(w)\) for some \(p>1\) , where \(\mathsf {M}^+\) is the forward Hardy–Littlewood maximal operator. We show that \(w\in A_\infty ^+\) if and only if there exist numerical constants \(\gamma \in (0,1)\) and \(c>0\) such that $$\begin{aligned} w(\{x \in \mathbb {R} : \, \mathsf {M}^+ \mathbf 1 _E (x)>\gamma \})\le cw(E) \end{aligned}$$ for all measurable sets \(E\subset \mathbb R\) . Furthermore, letting $$\begin{aligned} \mathsf {C_w ^+}(\alpha ){:}{=}\sup _{0<w(E)<+\infty } \frac{1}{w(E)} w(\{x\in \mathbb R:\,\mathsf {M}^+ \mathbf 1 _E(x)>\alpha \}) \end{aligned}$$ we show that for all \(w\in A_\infty ^+\) we have the asymptotic estimate \(\mathsf {C_w ^+}(\alpha )-1\lesssim (1-\alpha )^\frac{1}{c[w]_{A_\infty ^+}}\) for \(\alpha \) sufficiently close to 1 and \(c>0\) a numerical constant, and that this estimate is best possible. We also show that the reverse Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of \(A_\infty ^+\) . Our methods also allow us to show that a weight \(w\in A_\infty ^+\) satisfies \(w\in A_p ^+\) for all \(p>e^{c[w]_{A_\infty ^+}}\) . PubDate: 2017-06-17 DOI: 10.1007/s13348-017-0201-y

Authors:C. Bocci; G. Calussi; G. Fatabbi; A. Lorenzini Abstract: Abstract In this paper we address the Hadamard product of not necessarily generic linear varieties, looking in particular at its Hilbert function. We find that the Hilbert function of the Hadamard product \(X\star Y\) of two varieties, with \(\dim (X), \dim (Y)\le 1\) , is the product of the Hilbert functions of the original varieties X and Y. Moreover, the same result is obtained for generic linear varieties X and Y as a consequence of our showing that their Hadamard product is projectively equivalent to a Segre embedding. PubDate: 2017-06-15 DOI: 10.1007/s13348-017-0200-z

Authors:Renu Chaudhary; Dwijendra N. Pandey Abstract: In this paper, we study a stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space. Using a finite dimensional subspace, semigroup theory of linear operators and stochastic version of the well-known Banach fixed point theorem is applied to show the existence and uniqueness of an approximate solution. Next, these approximate solutions are shown to form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is then a solution of the original problem. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. In the last, we have given an example to illustrate the applications of the abstract results. PubDate: 2017-05-20 DOI: 10.1007/s13348-017-0199-1

Authors:Sonia Brivio Abstract: Abstract Let E be a stable vector bundle of rank r and slope \(2g-1\) on a smooth irreducible complex projective curve C of genus \(g \ge 3\) . In this paper we show a relation between theta divisor \(\Theta _E\) and the geometry of the tautological model \(P_E\) of E. In particular, we prove that for \(r > g-1\) , if C is a Petri curve and E is general in its moduli space then \(\Theta _E\) defines an irreducible component of the variety parametrizing \((g-2)\) -linear spaces which are g-secant to the tautological model \(P_E\) . Conversely, for a stable, \((g-2)\) -very ample vector bundle E, the existence of an irreducible non special component of dimension \(g-1\) of the above variety implies that E admits theta divisor. PubDate: 2017-05-16 DOI: 10.1007/s13348-017-0198-2

Authors:Shohei Nakamura; Yoshihiro Sawano Abstract: Abstract In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the \(A_q\) -theory by Muckenhoupt and Wheeden does not suffice. We also discuss the boundedness of the commutators. The difference between the maximal operator and the singular integral operators will be clarified. Further, the property of the sharp maximal operator is investigated. PubDate: 2017-02-23 DOI: 10.1007/s13348-017-0193-7

Authors:Serge Nicaise Abstract: We perform some hierarchical analyses of dissipative systems. For that purpose, we first propose a general abstract setting, prove a convergence result and discuss some stability properties. This abstract setting is then illustrated by significant examples of damped (acoustic) wave equations for which we characterize the family of reduced problems. For each concrete problems the decay of the energy is discussed and the density assumption is proved. PubDate: 2017-02-08 DOI: 10.1007/s13348-017-0192-8

Authors:Hara Charalambous; Apostolos Thoma; Marius Vladoiu Abstract: Abstract Let \(L\subset \mathbb {Z}^n\) be a lattice and \(I_L=\langle x^{\mathbf {u}}-x^{\mathbf {v}}:\ {\mathbf {u}}-{\mathbf {v}}\in L\rangle \) be the corresponding lattice ideal in \(\Bbbk [x_1,\ldots , x_n]\) , where \(\Bbbk \) is a field. In this paper we describe minimal binomial generating sets of \(I_L\) and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of \(I_L\) . As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices. PubDate: 2017-01-10 DOI: 10.1007/s13348-017-0191-9

Authors:A. Bravo; S. Encinas; B. Pascual-Escudero Abstract: Abstract The Nash multiplicity sequence was defined by Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities. Moreover, this indicates that these invariants from constructive resolution are intrinsic to the variety since they can be read in terms of its space of arcs. This result is a first step connecting explicitly arc spaces and algorithmic resolution of singularities. PubDate: 2017-01-04 DOI: 10.1007/s13348-016-0188-9