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 Collectanea MathematicaNumber of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 0010-075 - ISSN (Online) 2038-4815 Published by Springer-Verlag  [2351 journals]
• A necessary condition for weak maximum modulus sets of 2-analytic
functions
• Authors: Abtin Daghighi
Pages: 173 - 180
Abstract: Let $${\varOmega }\subset \mathbb {C}$$ be a domain and let $$f(z)=a(z)+\bar{z}b(z),$$ where a, b are holomorphic for $$z\in {\varOmega }.$$ Denote by $${\varLambda }$$ the set of points in $${\varOmega }$$ at which $$\left f\right$$ attains weak local maximum and denote by $${\varSigma }$$ the set of points in $${\varOmega }$$ at which $$\left f\right$$ attains strict local maximum. We prove that for each $$p\in {\varLambda }\setminus {\varSigma }$$ , \begin{aligned} \left b(p)\right =\left \left( \frac{\partial a}{\partial z} +\bar{z}\frac{\partial b}{\partial z}\right) (p)\right \end{aligned} Furthermore, if there is a real analytic curve $$\kappa :I\rightarrow {\varLambda }\setminus {\varSigma }$$ (where I is an open real interval), if a, b are complex polynomials, and if $$f\circ \kappa$$ has a complex polynomial extension, then either f is constant or $$\kappa$$ has constant curvature.
PubDate: 2018-05-01
DOI: 10.1007/s13348-017-0197-3
Issue No: Vol. 69, No. 2 (2018)

• Existence and approximation of solution to stochastic fractional
integro-differential equation with impulsive effects
• Authors: Renu Chaudhary; Dwijendra N. Pandey
Pages: 181 - 204
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: 2018-05-01
DOI: 10.1007/s13348-017-0199-1
Issue No: Vol. 69, No. 2 (2018)

• The Hilbert function of some Hadamard products
• Authors: C. Bocci; G. Calussi; G. Fatabbi; A. Lorenzini
Pages: 205 - 220
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: 2018-05-01
DOI: 10.1007/s13348-017-0200-z
Issue No: Vol. 69, No. 2 (2018)

• Group Riesz and frame sequences: the Bracket and the Gramian
• Authors: Davide Barbieri; Eugenio Hernández; Victoria Paternostro
Pages: 221 - 236
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: 2018-05-01
DOI: 10.1007/s13348-017-0202-x
Issue No: Vol. 69, No. 2 (2018)

• On recurrence coefficients of Steklov measures
• Authors: R. V. Bessonov
Pages: 237 - 248
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: 2018-05-01
DOI: 10.1007/s13348-017-0203-9
Issue No: Vol. 69, No. 2 (2018)

• Improved bounds for the regularity of edge ideals of graphs
• Authors: S. A. Seyed Fakhari; S. Yassemi
Pages: 249 - 262
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: 2018-05-01
DOI: 10.1007/s13348-017-0204-8
Issue No: Vol. 69, No. 2 (2018)

• Koszul properties of the moment map of some classical representations
• Authors: Aldo Conca; Hans-Christian Herbig; Srikanth B. Iyengar
Abstract: This work concerns the moment map $$\mu$$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $$S/(\mu )$$ , the coordinate algebra of the zero fibre of $$\mu$$ , be Koszul. The main result is that this algebra is not Koszul for the standard representation of $$\mathfrak {sl}_{n}$$ , and of $$\mathfrak {sp}_{n}$$ . This is deduced from a computation of the Betti numbers of $$S/(\mu )$$ as an S-module, which are of interest also from the point of view of commutative algebra.
PubDate: 2018-05-23
DOI: 10.1007/s13348-018-0226-x

• Extension of Pettis integration: Pettis operators and their integrals
• Authors: Oscar Blasco; Lech Drewnowski
Abstract: In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak $$^*$$ –weakly continuous linear operator F from a dual Banach space to an $$L_1$$ -space, and of its Pettis integral, understood simply as the dual operator $$F^*$$ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.
PubDate: 2018-05-19
DOI: 10.1007/s13348-018-0225-y

• Generic trilinear multipliers associated to degenerate simplexes
• Authors: Robert Kesler
Abstract: For each $$1 \le p \le \infty$$ , let $$W_{p}(\mathbb {R}) = \left\{ f \in L^p(\mathbb {R}): \hat{f} \in L^{p^\prime }(\mathbb {R}) \right\}$$ be equipped with the norm $$f _{W_{p}(\mathbb {R})} = \left \hat{f}\right _{L^{p^\prime }(\mathbb {R})}$$ . Moreover, let $$a_1,a_2 : \mathbb {R}^2 \rightarrow \mathbb {C}$$ satisfy the condition that for all $$\mathbf {\alpha } \in \mathbb {Z}_{ \ge 0}^2$$ there is $$C_{a_1, a_2,\mathbf {\alpha }}>0$$ such that for all $$\mathbf {\xi } \in \mathbb {R}^2$$ and $$j \in \{1,2\}$$ , $$\left \partial ^{\mathbf {\alpha }} a_j (\mathbf {\xi })\right \le \frac{C_{a_1, a_2,\mathbf {\alpha }}}{dist(\mathbf {\xi }, \varGamma )^{ \mathbf {\alpha } }}$$ . Our main result is that the trilinear multiplier given on $$\mathcal {S}(\mathbb {R})^3$$ by \begin{aligned} B[a_1, a_2] : (f_1, f_2, f_3) \mapsto \int _{\mathbb {R}^3} a_1(\xi _1, \xi _2) a_2(\xi _2, \xi _3) \left[ \prod _{j=1}^3 \hat{f_j} (\xi _j) e^{2 \pi ix \xi _j} \right] d\xi _1 d\xi _2 d\xi _3 \end{aligned} extends to a bounded map from $$L^{p_1}(\mathbb {R}) \times W_{p_2}(\mathbb {R}) \times L^{p_3}(\mathbb {R})$$ into $$L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb {R})$$ provided \begin{aligned} 1< p_1, p_3< \infty , \frac{1}{p_1} + \frac{1}{p_2}<1, \frac{1}{p_2} + \frac{1}{p_3}<1, 2< p_2 < \infty . \end{aligned}
PubDate: 2018-05-04
DOI: 10.1007/s13348-018-0224-z

• Duality of central Herz–Morrey–Musielak–Orlicz spaces of
variable exponents
• Authors: Fumi-Yuki Maeda; Yoshihiro Mizuta; Takao Ohno; Tetsu Shimomura
Abstract: Our aim in this paper is to establish duality of central Herz-Morrey-Musielak-Orlicz spaces $$\mathcal {H}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ of variable exponents, the space $$\underline{\mathcal {H}}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ and its complementary space $$\overline{\mathcal {H}}^{\Phi ,q(\cdot ),\omega }(\mathbf{R}^N_0)$$ by studying the associate spaces of central Herz-Morrey-Musielak-Orlicz spaces $$\mathcal {H}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ , the space $$\underline{\mathcal {H}}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ and its complementary space $$\overline{\mathcal {H}}^{\Phi ,q_1,q_2,\omega }(\mathbf{R}^N_0)$$ .
PubDate: 2018-04-20
DOI: 10.1007/s13348-018-0222-1

• Non-surjective Gaussian maps for singular curves on K3 surfaces
• Authors: Claudio Fontanari; Edoardo Sernesi
Abstract: Let (S, L) be a polarized K3 surface with $$\mathrm {Pic}(S) = \mathbb {Z}[L]$$ and $$L\cdot L=2g-2$$ , let C be a nonsingular curve of genus $$g-1$$ and let $$f:C\rightarrow S$$ be such that $$f(C) \in \vert L \vert$$ . We prove that the Gaussian map $$\Phi _{\omega _C(-T)}$$ is non-surjective, where T is the degree two divisor over the singular point x of f(C). This generalizes a result of Kemeny with an entirely different proof. It uses the very ampleness of C on the blown-up surface $$\widetilde{S}$$ of S at x and a theorem of L’vovski.
PubDate: 2018-04-17
DOI: 10.1007/s13348-018-0223-0

• Weighted inequalities for integral operators on Lebesgue and $$BMO^{\gamma }(\omega )$$ B M O γ ( ω ) spaces
• Authors: Elida V. Ferreyra; Guillermo J. Flores
Abstract: We characterize the power weights $$\omega$$ for which the fractional type operator $$T_{\alpha ,\beta }$$ is bounded from $$L^p (\omega ^p)$$ into $$L^q (\omega ^q)$$ for $$1< p < n/(n- (\alpha + \beta ))$$ and $$1/q = 1/p - (n- (\alpha + \beta ))/n$$ . If $$n/(n-(\alpha + \beta )) \le p < n/(n -(\alpha +\beta ) -1)^{+}$$ we prove that $$T_{\alpha ,\beta }$$ is bounded from a weighted weak $$L^p$$ space into a suitable weighted $$BMO^\delta$$ space for weights satisfying a doubling condition and a reverse Hölder condition. Also, we prove the boundedness of $$T_{\alpha ,\beta }$$ from a weighted local space $$BMO_{0}^{\gamma }$$ into a weighted $$BMO^\delta$$ space, for weights satisfying a doubling condition.
PubDate: 2018-04-16
DOI: 10.1007/s13348-018-0221-2

• Uniform rotundity in every direction of Orlicz function spaces equipped
with the p -Amemiya norm
• Authors: Radosław Kaczmarek
Abstract: Some conditions which guarantee that the Orlicz function spaces equipped with the p-Amemiya norm ( $$1<p<\infty$$ ) and generated by N-functions are uniformly rotund in every direction are given. Obtained result broaden the knowledge about this notion in Orlicz function spaces with the p-Amemiya norm ( $$1\le p\le \infty$$ ).
PubDate: 2018-03-26
DOI: 10.1007/s13348-018-0220-3

• Non-elementary Fano conic bundles
• Authors: E. A. Romano
Abstract: Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundles $$f:X\rightarrow Y$$ . Denoting by $$\rho _{X}$$ the Picard number of X, we investigate such contractions when $$\rho _{X}-\rho _{Y}>1$$ , called non-elementary. We prove that $$\rho _{X}-\rho _{Y}\le 8$$ , and we deduce new geometric information about our varieties X and Y, depending on $$\rho _{X}-\rho _{Y}$$ . Using our results, we show that some known examples of Fano conic bundles are elementary. Moreover, when we allow that X is locally factorial with canonical singularities and with at most finitely many non-terminal points, and $$f:X\rightarrow Y$$ is a fiber type $$K_{X}$$ -negative contraction with one-dimensional fibers, we show that $$\rho _{X}-\rho _{Y}\le 9$$ .
PubDate: 2018-03-12
DOI: 10.1007/s13348-018-0218-x

• Algebraic surfaces of general type with $$p_g=q=1$$ p g = q = 1 and genus
2 Albanese fibrations
• Authors: Songbo Ling
Abstract: In this paper, we study algebraic surfaces of general type with $$p_g=q=1$$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $$p_g=q=1, K^2=5$$ and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of $$\mathbb {P}^2$$ . We prove that these surfaces give an irreducible and connected component of $$\mathcal {M}_{1,1}^{5,2}$$ , the Gieseker moduli space of surfaces of general type with $$p_g=q=1, K^2=5$$ and genus 2 Albanese fibrations. Then by constructing surfaces with $$p_g=q=1,K^2=3$$ and a genus 2 Albanese fibration such that the number of the summands of the direct image of the bicanonical sheaf (under the Albanese map) is 2, we give a negative answer to a question of Pignatelli.
PubDate: 2018-03-10
DOI: 10.1007/s13348-018-0219-9

• A bound on the dimension of a totally geodesic submanifold in the Prym
locus
• Authors: Elisabetta Colombo; Paola Frediani
Abstract: We give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura variety of $${{\mathcal {A}}}_{g-1}$$ , contained in the Prym locus. First we give such a bound for a germ passing through a Prym variety of a k-gonal curve in terms of the gonality k. Then we deduce a bound only depending on the genus g.
PubDate: 2018-03-02
DOI: 10.1007/s13348-018-0215-0

• Mean Lipschitz spaces and a generalized Hilbert operator
• Authors: Noel Merchán
Abstract: If $$\mu$$ is a positive Borel measure on the interval [0, 1) we let $$\mathcal {H}_\mu$$ be the Hankel matrix $$\mathcal {H}_\mu =(\mu _{n, k})_{n,k\ge 0}$$ with entries $$\mu _{n, k}=\mu _{n+k}$$ , where, for $$n\,=\,0, 1, 2, \dots$$ , $$\mu _n$$ denotes the moment of order n of $$\mu$$ . This matrix induces formally the operator \begin{aligned}\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\end{aligned} on the space of all analytic functions $$f(z)=\sum _{k=0}^\infty a_kz^k$$ , in the unit disc $${\mathbb {D}}$$ . This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $$\mathcal {H}_\mu$$ on mean Lipschitz spaces of analytic functions.
PubDate: 2018-02-28
DOI: 10.1007/s13348-018-0217-y

• On generalized Littlewood–Paley functions
• 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

• Stochastic fractional perturbed control systems with fractional Brownian
motion and Sobolev stochastic non local conditions
• 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

• The Cesàro operator on Korenblum type spaces of analytic functions
• 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

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