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)

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)

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)

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)

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)

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)

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

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

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

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

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

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

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

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

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

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

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: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