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Abstract: Abstract In this paper, we consider biconservative and biharmonic isometric immersions into the 4-dimensional Lorentzian space form \({\mathbb {L}}^4(\delta )\) with constant sectional curvature \(\delta \) . We obtain some local classifications of biconservative CMC surfaces in \({\mathbb {L}}^4(\delta )\) . Further, we get complete classification of biharmonic CMC surfaces in the de Sitter 4-space. We also proved that there is no biharmonic CMC surface in the anti-de Sitter 4-space. Further, we get the classification of biconservative, quasi-minimal surfaces in Minkowski-4 space. PubDate: 2024-01-29

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Abstract: Abstract We study base-point-freeness for big and nef line bundles on hyperkähler manifolds of generalized Kummer type: For \(n\in \{2,3,4\}\) , we show that, generically in all but a finite number of irreducible components of the moduli space of polarized \(\textrm{Kum}^n\) -type varieties, the polarization is base-point-free. We also prove generic base-point-freeness in the moduli space in all dimensions if the polarization has divisibility one. PubDate: 2023-11-16 DOI: 10.1007/s12188-023-00271-z

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Abstract: Abstract In this paper, we see that the hypersurfaces \(\mathcal {L}_{\pm }\) in Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Proposition 1) are neutral but not flat. Nonetheless, we find parallel almost complex structures \(\mathcal {I}_{\pm }\) suitable for Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Theorem 1) and parallel almost paracomplex structures \(\mathcal {J}_{\pm }\) suitable for Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Theorem 2). PubDate: 2023-11-13 DOI: 10.1007/s12188-023-00272-y

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Abstract: Abstract In his 2008 thesis [16] , Tateno claimed a counterexample to the Bonato–Tardif conjecture regarding the number of equimorphy classes of trees. In this paper we revisit Tateno’s unpublished ideas to provide a rigorous exposition, constructing locally finite trees having an arbitrary finite number of equimorphy classes; an adaptation provides partial orders with a similar conclusion. At the same time these examples also disprove conjectures by Thomassé and Tyomkyn. PubDate: 2023-11-01 DOI: 10.1007/s12188-023-00270-0

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Abstract: Abstract We study lines on smooth cubic surfaces over the field of p-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are 0, 1, 2, 3, 5, 7, 9, 15 or 27. We show that each of these counts is achieved. Probabilistic aspects are investigated by sampling both p-adic and real cubic surfaces from different distributions and estimating the probability of each count.We link this to recent results on probabilistic enumerative geometry. Some experimental results on the Galois groups attached to p-adic cubic surfaces are also discussed. PubDate: 2023-09-16 DOI: 10.1007/s12188-023-00269-7

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Abstract: Abstract We call a smooth irreducible projective curve a Castelnuovo curve if it admits a birational map into the projective r-space such that the image curve has degree at least 2r+1 and the maximum possible geometric genus (which one can calculate by a classical formula due to Castelnuovo). It is well known that a Castelnuovo curve must lie on a Hirzebruch surface (rational ruled surface). Conversely, making use of a result of W. Castryck and F. Cools concerning the scrollar invariants of curves on Hirzebruch surfaces we show that curves on Hirzebruch surfaces are Castelnuovo curves unless their genus becomes too small w.r.t. their gonality. We analyze the situation more closely, and we calculate the number of moduli of curves of fixed genus g and fixed gonality k lying on Hirzebruch surfaces, in terms of g and k. PubDate: 2023-06-08 DOI: 10.1007/s12188-023-00267-9

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Abstract: Abstract In 2020 S. M. Gonek, S. W. Graham and Y. Lee formulated the Lindelöf hypothesis for prime numbers and proved that it is equivalent to the Riemann Hypothesis. In this note we show that their result holds in the Selberg class of L-functions. PubDate: 2023-05-17 DOI: 10.1007/s12188-023-00268-8

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Abstract: Abstract We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra \(I{\mathfrak u}_n{:}{=}{\mathfrak u}_n < imes {\mathfrak u}_n^*\) , where \({\mathfrak u}_n\) is the Lie algebra of upper triangular \(n\times n\) matrices. Our results also extend to \(\mathfrak {gl}_{n+}^\epsilon \) , a “solvable approximation” of \(\mathfrak {gl}_n\) , as defined within. PubDate: 2023-03-16 DOI: 10.1007/s12188-023-00266-w

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Abstract: Abstract We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction. PubDate: 2023-02-24 DOI: 10.1007/s12188-023-00264-y

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Abstract: Abstract In this paper, we firstly generalize the Brunn–Minkowski type inequality for Ekeland–Hofer–Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan–Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan–Ostrover in 2012. PubDate: 2023-02-23 DOI: 10.1007/s12188-023-00263-z

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Abstract: Abstract We study the nontrivial elements in the Brauer group of a bielliptic surface and show that they can be realized as Azumaya algebras with a simple structure at the generic point of the surface. We go on to study some properties of the noncommutative Picard scheme associated to such an Azumaya algebra. PubDate: 2023-02-14 DOI: 10.1007/s12188-023-00265-x

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Abstract: Abstract We study the existence and uniqueness of renormalized solutions for initial boundary value problems of the type $$\begin{aligned} \left( {\mathcal {P}}_{b}^{1}\right) \quad \left\{ \begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))=H(u)\mu \text { in }Q:=(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$ where \(u_{0}\in L^{1}(\Omega )\) , \(\mu \in {\mathcal {M}}_{b}(Q)\) is a general Radon measure on Q and \(H\in C_{b}^{0}({\mathbb {R}})\) is a continuous positive bounded function on \({\mathbb {R}}\) . The difficulties in the study of such problems concern the possibly very singular right-hand side that forces the choice of a suitable formulation that ensures both existence and uniqueness of solution. Using similar techniques, we will prove existence/nonexistence results of the auxiliary problem $$\begin{aligned} \left( {\mathcal {P}}_{b}^{2}\right) \quad \left\{ \begin{aligned}&u_{t}-\text {div}(a(t,x,\nabla u))+g(x,u) \nabla u ^{2}=\mu \text { in }Q:=(0,T)\times \Omega ,\\&u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$ under the assumption that g satisfies a sign condition and the nonlinear term depends on both x, u and its gradient. Thus, our results improve and complete the previous known existence results for problems \(\left( {\mathcal {P}}_{b}^{1,2}\right) \) . PubDate: 2022-12-13 DOI: 10.1007/s12188-022-00262-6

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Abstract: Abstract Carmesin has extended Robertson and Seymour’s tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications. PubDate: 2022-11-29 DOI: 10.1007/s12188-022-00259-1

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Abstract: Abstract In this article we study the relation between flat solvmanifolds and \(G_2\) -geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of \(\mathsf{GL}(n,\mathbb {Z})\) for \(n=5\) and \(n=6\) . Then, we look for closed, coclosed and divergence-free \(G_2\) -structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free \(G_2\) -structure whose finite holonomy is cyclic and contained in \(G_2\) , and examples of compact flat manifolds admitting a divergence-free \(G_2\) -structure. PubDate: 2022-11-11 DOI: 10.1007/s12188-022-00261-7

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Abstract: Abstract In this paper we present a new approach to prove effective results in Diophantine approximation. This approach involves measures of local positivity of divisors combined with Faltings’s version of Siegel’s lemma instead of a zero estimate such as Dyson’s lemma. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with complex coefficients. PubDate: 2022-11-08 DOI: 10.1007/s12188-022-00260-8

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Abstract: Abstract In 2021 the authors D. Bravo, M. Lanzilotta, O. Mendoza and J. Vivero gave a generalization of the concept of Igusa-Todorov algebra and proved that those algebras, named Lat-Igusa-Todorov (LIT for short), satisfy the finitistic dimension conjecture. In this paper we explore the scope of that generalization and give conditions for a triangular matrix algebra to be LIT in terms of the algebras and the bimodule used in its definition. As an application we obtain that the tensor product of an LIT \(\mathbb {K}\) -algebra with a path algebra of a quiver whose underlying graph is a tree, is LIT. PubDate: 2022-05-24 DOI: 10.1007/s12188-022-00257-3

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Abstract: Abstract We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight \(\frac{g}{2}\) and genus g. When \(g=4\) , the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices. PubDate: 2022-04-27 DOI: 10.1007/s12188-022-00256-4

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Abstract: Abstract We already have the concept of isotropicity of a minimal surface in a Riemannian 4-manifold and a space-like or time-like surface in a neutral 4-manifold with zero mean curvature vector. In this paper, based on the understanding of it, we define and study isotropicity of a space-like or time-like surface in a Lorentzian 4-manifold N with zero mean curvature vector. If the surface is space-like, then the isotropicity means either the surface has light-like or zero second fundamental form or it is an analogue of complex curves in Kähler surfaces. In addition, if N is a space form, then the isotropicity means that the surface has both the properties. If the surface is time-like and if N is a space form, then the isotropicity means that the surface is totally geodesic. PubDate: 2022-01-03 DOI: 10.1007/s12188-021-00254-y

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Abstract: Abstract It is proved that decomposable graphs are set recognizable and that the index graph of the canonical decomposition as well as the graphs induced on the maximal autonomous sets of vertices are set reconstructible. From these results, we obtain set reconstructibility for many decomposable graphs as well as a concise description of the decomposable graphs for which set reconstruction remains an open problem. PubDate: 2022-01-03 DOI: 10.1007/s12188-021-00252-0

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Abstract: Abstract We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight. PubDate: 2021-12-14 DOI: 10.1007/s12188-021-00253-z