Authors:Beatriz Rodríguez González; Agustí Roig Pages: 301 - 321 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: 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: 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: 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: 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:Lisa Nicklasson Abstract: In this paper we classify the monomial complete intersections, in two variables, and of positive characteristic, which has the strong Lefschetz property. Together with known results, this gives a complete classification of the monomial complete intersections with the strong Lefschetz property. PubDate: 2017-11-09 DOI: 10.1007/s13348-017-0209-3

Authors:H. Al-Qassem; L. Cheng; Y. Pan Abstract: We study the \(L^{p}\) boundedness of certain classes of generalized Littlewood–Paley functions \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\) . We obtain \(L^{p}\) estimates of \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\) with sharp range of p and under optimal conditions on \(\Phi \) . By using these estimates along with an extrapolation argument we obtain some new and improved results on generalized Littlewood–Paley functions. The approach in proving our results is mainly based on proving vector-valued inequalities and in turn the proof of our results (in the case \(\lambda =2)\) provides us with alternative proofs of the results obtained by Duoandikoetxea as his approach is based on proving certain weighted norm inequalities. PubDate: 2017-10-24 DOI: 10.1007/s13348-017-0208-4

Authors:Kerboua Mourad; Ellaggoune Fateh; Dumitru Baleanu Abstract: This paper investigates the approximate controllability for Sobolev type stochastic perturbed control systems of fractional order with fractional Brownian motion and Sobolev fractional stochastic nonlocal conditions in a Hilbert space, A new set of sufficient conditions are established by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach’s fixed point theorem. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory. PubDate: 2017-10-20 DOI: 10.1007/s13348-017-0207-5

Authors:Giorgio Ottaviani; Alicia Tocino Abstract: In the tensor space \({{\mathrm {Sym}}}^d \mathbb {R}^2\) of binary forms we study the best rank k approximation problem. The critical points of the best rank 1 approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank k approximation problem lie in the same hyperplane. As a consequence, every binary form may be written as linear combination of its critical rank 1 tensors, which extends the Spectral Theorem from quadratic forms to binary forms of any degree. In the same vein, also the best rank k approximation may be written as a linear combination of the critical rank 1 tensors, which extends the Eckart–Young theorem from matrices to binary forms. PubDate: 2017-10-11 DOI: 10.1007/s13348-017-0206-6

Authors:Angela A. Albanese; José Bonet; Werner J. Ricker Abstract: The spectrum of the Cesàro operator \(\mathsf {C}\) , which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fréchet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, never nuclear. Some consequences concerning the mean ergodicity of \(\mathsf {C}\) are deduced. PubDate: 2017-09-26 DOI: 10.1007/s13348-017-0205-7

Authors:S. A. Seyed Fakhari; S. Yassemi Abstract: Let G be a graph with n vertices, let \(S={\mathbb {K}}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \({\mathbb {K}}\) and let I(G) denote the edge ideal of G. For every collection \({\mathcal {H}}\) of connected graphs with \(K_2\in {\mathcal {H}}\) , we introduce the notions of \({{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)\) and \({{\mathrm{min-match}}}_{{\mathcal {H}}}(G)\) . It will be proved that the inequalities \({{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)\) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\) . Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality \(\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])\) holds. PubDate: 2017-09-19 DOI: 10.1007/s13348-017-0204-8

Authors:R. V. Bessonov Abstract: A measure \(\mu \) on the unit circle \(\mathbb {T}\) belongs to Steklov class \({\mathcal {S}}\) if its density w with respect to the Lebesgue measure on \(\mathbb {T}\) is strictly positive: \(\mathop {\mathrm {ess\,inf}}\nolimits _{\mathbb {T}} w > 0\) . Let \(\mu \) , \(\mu _{-1}\) be measures on the unit circle \({\mathbb {T}}\) with real recurrence coefficients \(\{\alpha _k\}\) , \(\{-\alpha _k\}\) , correspondingly. If \(\mu \in {\mathcal {S}}\) and \(\mu _{-1} \in {\mathcal {S}}\) , then partial sums \(s_k=\alpha _0+ \ldots + \alpha _k\) satisfy the discrete Muckenhoupt condition \(\sup _{n > \ell \geqslant 0} (\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{2s_k})(\frac{1}{n - \ell }\sum _{k=\ell }^{n-1} e^{-2s_k}) < \infty \) . PubDate: 2017-08-14 DOI: 10.1007/s13348-017-0203-9

Authors:Davide Barbieri; Eugenio Hernández; Victoria Paternostro 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: 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: 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: 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: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: 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