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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  [2467 journals]
• Disjoint strong transitivity of composition operators

Abstract: Abstract A Furstenberg family $$\mathcal {F}$$ is a collection of infinite subsets of the set of positive integers such that if $$A\subset B$$ and $$A\in \mathcal {F}$$ , then $$B\in \mathcal {F}$$ . For a Furstenberg family $$\mathcal {F}$$ , finitely many operators $$T_1,...,T_N$$ acting on a common topological vector space X are said to be disjoint $$\mathcal {F}$$ -transitive if for every non-empty open subsets $$U_0,...,U_N$$ of X the set $$\{n\in \mathbb {N}:\ U_0 \cap T_1^{-n}(U_1)\cap ...\cap T_N^{-n}(U_N)\ne \emptyset \}$$ belongs to $$\mathcal {F}$$ . In this paper, depending on the topological properties of $$\Omega$$ , we characterize the disjoint $$\mathcal {F}$$ -transitivity of $$N\ge 2$$ composition operators $$C_{\phi _1},\ldots ,C_{\phi _N}$$ acting on the space $$H(\Omega )$$ of holomorphic maps on a domain $$\Omega \subset \mathbb {C}$$ by establishing a necessary and sufficient condition in terms of their symbols $$\phi _1,...,\phi _N$$ .
PubDate: 2022-11-23

• Multi-Rees algebras of strongly stable ideals

Abstract: Abstract We prove that the multi-Rees algebra $${\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)$$ of a collection of strongly stable ideals $$I_1, \ldots , I_r$$ is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of $${\mathcal {R}}(I_1 \oplus \cdots \oplus I_r)$$ based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of $${\mathcal {R}}(I_1 \oplus I_2)$$ where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.
PubDate: 2022-11-22

• Lefschetz properties for jacobian rings of cubic fourfolds and other
Artinian algebras

Abstract: Abstract In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.
PubDate: 2022-11-19

• The moduli space of quasistable spin curves

Abstract: Abstract We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.
PubDate: 2022-11-18

• Spacelike mean curvature flow solitons, polynomial volume growth and
stochastic completeness of spacelike hypersurfaces immersed into pp-vave
spacetimes

Abstract: Abstract Our purpose in this paper is to study some geometric properties of spacelike hypersurfaces immersed into a pp-wave spacetime, namely, a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold, we obtain sufficient conditions which guarantee that a complete noncompact spacelike hypersurface with polynomial volume growth is either totally geodesic, maximal or 1-maximal. As a consequence, we establish nonexistence results concerning such spacelike hypersurfaces. Next, using a weak form of the Omori–Yau maximum principle, we get uniqueness and nonexistence results for stochastically complete spacelike hypersurface with constant mean curvature. Finally, we establish the notion of spacelike mean curvature flow soliton in pp-wave spacetimes and we provide some geometric conditions that allow us to guarantee how close a complete spacelike mean curvature flow soliton is to a totally geodesic immersion.
PubDate: 2022-11-15

• Correction to: Capacity and stability on some Cegrell classes of
m − subharmonic functions

PubDate: 2022-11-02

• On weighted compactness of commutators of square function and semi-group
maximal function associated to Schrödinger operators

Abstract: Abstract Let $$\Delta$$ be the Laplacian operator on $${\mathbb{R}}^n$$ and V be a nonnegative potential satisfying an appropriate reverse Hölder inequality. The Littlewood–Paley square function g associated with the Schrödinger operator $$L=-\Delta +V$$ is defined by: \begin{aligned} g(f)(x)=\Big (\int _{0}^{\infty }\Big \frac{d}{dt}e^{-tL}(f)(x)\Big ^2tdt\Big )^{1/2}. \end{aligned} In this paper, we show that the commutators of g are compact operators on $$L^p(w)$$ for $$1<p<\infty$$ if $$b\in {\rm{CMO}}_\uptheta (\uprho )$$ and $$w\in A_p^{\uprho ,\uptheta }$$ , where $${\rm{CMO}}_\uptheta (\uprho ) ({\mathbb{R}}^n)$$ denotes the closure of $$\mathcal{C}_c^\infty ({\mathbb{R}}^n)$$ in the $$\mathrm{BMO}_\uptheta (\uprho )$$ topology and $$A_p^{\uprho ,\uptheta }$$ is a weighted class which is more larger than Muckenhoupt $$A_p$$ weight class. An extra weight condition in a previous weighted compactness result is removed for the commutators of the semi-group maximal function defined by $$\mathcal{T}^*(f)(x)=\sup _{t>0} e^{-tL}f(x) .$$
PubDate: 2022-11-01

• The symmetrization map and $$\Gamma$$ -contractions

Abstract: Abstract The symmetrization map $$\pi :{\mathbb{C}}^2\rightarrow {\mathbb{C}}^2$$ is defined by $$\pi (z_1,z_2)=(z_1+z_2,z_1z_2).$$ The closed symmetrized bidisc $$\Gamma$$ is the symmetrization of the closed unit bidisc $$\overline{{\mathbb{D}}^2}$$ , that is, \begin{aligned} \Gamma = \pi (\overline{{\mathbb{D}}^2})=\{ (z_1+z_2,z_1z_2)\,:\, z_i \le 1, i=1,2 \}. \end{aligned} A pair of commuting Hilbert space operators (S, P) for which $$\Gamma$$ is a spectral set is called a $$\Gamma$$ -contraction. Unlike the scalars in $$\Gamma$$ , a $$\Gamma$$ -contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded operators. We characterize all $$\Gamma$$ -contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a $$\Gamma$$ -contraction $$(S,P)=(T_1+T_2,T_1T_2)$$ for a pair of commuting bounded operators $$T_1,T_2$$ , no real number less than 2 can be a bound for the set $$\{ \Vert T_1\Vert ,\Vert T_2\Vert \}$$ in general. Then we prove that every $$\Gamma$$ -contraction (S, P) is the restriction of a $$\Gamma$$ -contraction $$({{\widetilde{S}}}, {{\widetilde{P}}})$$ to a common reducing subspace of $${{\widetilde{S}}}, {{\widetilde{P}}}$$ and that $$({{\widetilde{S}}}, {{\widetilde{P}}})=(A_1+A_2,A_1A_2)$$ for a pair of commuting operators $$A_1,A_2$$ with $$\max \{\Vert A_1\Vert , \Vert A_2\Vert \} \le 2$$ . We find new characterizations for the $$\Gamma$$ -unitaries and describe the distinguished boundary of $$\Gamma$$ in a different way. We also show some interplay between the fundamental operators of two $$\Gamma$$ -contractions (S, P) and $$(S_1,P)$$ .
PubDate: 2022-10-25

• Morse index bounds for minimal submanifolds

Abstract: Abstract In this paper, we study the Morse index of closed minimal submanifolds immersed into general Riemannian manifolds. Using the strategy developed by Ambrozio et al. (J Differ Geom 108(3):379–410, 2018) and under a suitable constrain on the submanifold, we obtain that the Morse index of the submanifold is bounded from below by a linear function of its first Betti’s number, as conjectured by Schoen and Marques-Neves. We also present many Riemannian manifolds and a sufficient condition to get the cited linear lower bound.
PubDate: 2022-10-21

• Some remarks on the geometry of the tautological model

Abstract: Abstract S. Brivio has established a linkage between the $$\Theta$$ divisor of a vector bundle E, over a smooth complex curve, and the geometry of the tautological model of E. In this paper we slightly generalize her results by considering other values for the rank and the degree of E.
PubDate: 2022-10-05

• A note on the BMO and Calderón–Zygmund estimate

Abstract: Abstract In this note, we give a simple proof of the pointwise BMO estimate for Poisson’s equation. Then the Calderón–Zygmund estimate follows by the interpolation and duality.
PubDate: 2022-09-19

• Correction to: Characteristic polyhedra of singularities without
completion: part II

PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00326-5

• Sparse domination results for compactness on weighted spaces

Abstract: Abstract By means of appropriate sparse bounds, we deduce compactness on weighted $$L^p(w)$$ spaces, $$1<p<\infty$$ , for all Calderón–Zygmund operators having compact extensions on $$L^2({\mathbb {R}}^n)$$ . Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $$A_p$$ class.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00333-6

• Smooth determinantal varieties and critical loci in multiview geometry

Abstract: Abstract Linear projections from $$\mathbb {P}^k$$ to $$\mathbb {P}^h$$ appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of sufficiently many correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in $$\mathbb {P}^k$$ containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical projective varieties.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00329-2

• Monomial ideals and the failure of the Strong Lefschetz property

Abstract: Abstract We give a sharp lower bound for the Hilbert function in degree d of artinian quotients $$\Bbbk [x_1,\ldots ,x_n]/I$$ failing the Strong Lefschetz property, where I is a monomial ideal generated in degree $$d \ge 2$$ . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00324-7

• A new concept of smoothness in Orlicz spaces

Abstract: Abstract In a 2015 article Cuenya and Ferreyra defined a class of functions in $$L^p$$ -spaces, denoted by $$c_n^p(x)$$ . The class $$c_n^p(x)$$ contains the class of $$L^p$$ -differentiability functions, denoted by $$t_n^p(x)$$ , introduced in a 1961 article by Calderón-Zygmund. A more recent paper by Acinas, Favier and Zó introduced a new class of functions in Orlicz spaces $$L^\Phi$$ , called $$L^\Phi$$ -differentiable functions in the present article. The class of $$L^\Phi$$ -differentiable functions is closely related to the class $$t_n^p(x)$$ . In this work, we define a class of functions in $$L^\Phi$$ , denoted by $$c_n^{\Phi }(x)$$ . The class $$c_n^{\Phi }(x)$$ is more general than the class of $$L^{\varPhi}$$ -differentiable functions. We prove the existence of the best local $$\Phi$$ -approximation for functions in $$c_n^{\varPhi }(x)$$ and study the convexity of the set of cluster points of the set of best $$\Phi$$ -approximations to a function on an interval when their measures tend to zero.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00331-8

• A generalized Grobman–Hartman theorem for nonautonomous dynamics

Abstract: Abstract The purpose of this note is to extend the recent generalized version of the Grobman–Hartman theorem established by Bernardes Jr. and Messaoudi from an autonomous to nonautonomous dynamics. More precisely, we prove that any sufficiently small perturbation of a nonautonomous linear dynamics that admits a generalized exponential dichotomy is topologically conjugated to its linear part. In addition, we prove that under certain mild additional conditions, the conjugacy is in fact Hölder continuous.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00327-4

• Structure of the Lipschitz free p-spaces
$${\mathcal{F}}_p({\mathbb{Z}}^d)$$ F p ( Z d ) and
$${\mathcal{F}}_p({\mathbb{R}}^d)$$ F p ( R d ) for 0       p ≤ 1

Abstract: Abstract Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for $$0<p\le 1$$ over the Euclidean spaces $${\mathbb{R}}^d$$ and $${\mathbb{Z}}^d$$ . To that end, we show that $${\mathcal{F}}_p({\mathbb{R}}^d)$$ admits a Schauder basis for every $$p\in (0,1]$$ , thus generalizing the corresponding result for the case $$p=1$$ by Hájek and Pernecká (J Math Anal Appl 416(2):629–646, 2014, Theorem 3.1) and answering in the positive a question that was raised by Albiac et al. in (J Funct Anal 278(4):108354, 2020). Explicit formulas for the bases of $${\mathcal{F}}_p({\mathbb{R}}^d)$$ and its isomorphic space $${\mathcal{F}}_p([0,1]^d)$$ are given. We also show that the well-known fact that $${\mathcal{F}}({\mathbb{Z}})$$ is isomorphic to $$\ell _{1}$$ does not extend to the case when $$p<1$$ , that is, $${\mathcal{F}}_{p}({\mathbb{Z}})$$ is not isomorphic to $$\ell _{p}$$ when $$0<p<1$$ .
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00322-9

• Jacobian syzygies, Fitting ideals, and plane curves with maximal global
Tjurina numbers

Abstract: Abstract First we give a sharp upper bound for the cardinal m of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve C. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve C, in terms of its degree d and of the minimal degree $$r\le d-1$$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for m is also attained. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). We conjecture that such curves exist for any pair (d, r), and that, in addition, they may be chosen to be line arrangements when $$r\le d-2$$ . This conjecture is proved for degrees $$d \le 11$$ .
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00325-6

• On the Davis-Wielandt shell of an operator and the Davis-Wielandt index of
a normed linear space

Abstract: Abstract We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space $$\mathcal {X}$$ . We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on $$\mathcal {L}(\mathcal {X})$$ which is equivalent to the usual operator norm on $$\mathcal {L}(\mathcal {X})$$ . We introduce the Davis-Wielandt index of a normed linear space and compute its value explicitly in case of some particular polyhedral Banach spaces. We also present a general method to estimate the Davis-Wielandt index of any polyhedral finite-dimensional Banach space.
PubDate: 2022-09-01
DOI: 10.1007/s13348-021-00332-7

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