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 Collectanea Mathematica   [SJR: 0.805]   [H-I: 11]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 0010-075 - ISSN (Online) 2038-4815    Published by Springer-Verlag  [2352 journals]
• Godement resolution and operad sheaf homotopy theory
• 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)

• $$\ell ^2(G)$$ ℓ 2 ( G ) -linear independence for systems generated by
dual integrable representations of LCA groups
• 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)

• The Halphen cubics of order two
• 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)

• Boundary oscillations of harmonic functions in Lipschitz domains
• 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)

• Perturbed Kirchhoff-type p -Laplacian discrete problems
• 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)

• Local Bézout theorem for Henselian rings
• 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)

• Tests for injectivity of modules over commutative rings
• 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)

• De Branges functions of Schroedinger equations
• 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)

• Best rank k approximation for binary forms
• Authors: Giorgio Ottaviani; Alicia Tocino
Abstract: 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

• The Cesàro operator on Korenblum type spaces of analytic functions
• Authors: Angela A. Albanese; José Bonet; Werner J. Ricker
Abstract: 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

• Improved bounds for the regularity of edge ideals of graphs
• Authors: S. A. Seyed Fakhari; S. Yassemi
Abstract: 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

• On recurrence coefficients of Steklov measures
• Authors: R. V. Bessonov
Abstract: 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

• Group Riesz and frame sequences: the Bracket and the Gramian
• 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

• Sharp inequalities for one-sided Muckenhoupt weights
• 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

• The Hilbert function of some Hadamard products
• 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

• Existence and approximation of solution to stochastic fractional
integro-differential equation with impulsive effects
• 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

• Theta divisors and the geometry of tautological model
• 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

• Convergence and stability analyses of hierarchic models of dissipative
second order evolution equations
• 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

• Minimal generating sets of lattice ideals
• 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

• Nash multiplicities and resolution invariants
• 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

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