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Abstract: Abstract Starting from an integral projective variety Y equipped with a very ample, non-special and not-secant defective line bundle \(\mathcal {L}\) , the paper establishes, under certain conditions, the regularity of \((Y \times {\mathbb {P}}^2,\mathcal {L}[t])\) for \(t\ge 2\) . The mildness of those conditions allow to classify all secant defective cases of any product of \(({\mathbb {P}}^1)^{ j}\times ({\mathbb {P}}^2)^{k}\) , \(j,k \ge 0\) , embedded in multidegree at least \((2, \ldots , 2)\) and \((\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k, \mathcal {O}_{\mathbb {P}^m\times \mathbb {P}^n\times (\mathbb {P}^2)^k} (d,e,t_1, \ldots , t_k))\) where \(d,e \ge 3\) , \(t_i\ge 2\) , for any n and m. PubDate: 2024-08-05 DOI: 10.1007/s10231-024-01493-5
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Abstract: Abstract Given a Euclidean submanifold \(g:M^{n}\rightarrow {\mathbb {R}}^{n+p}\) , Chern and Kuiper provided inequalities between \(\mu \) and \(\nu _g\) , the ranks of the nullity of \(M^n\) and the relative nullity of g respectively. Namely, they prove that 1 $$\begin{aligned} \nu _g\le \mu \le \nu _g+p. \end{aligned}$$ In this work, we study the submanifolds with \(\nu _g\ne \mu \) . More precisely, we characterize locally the ones with \(0\ne (\mu -\nu _g)\in \{p,p-1,p-2\}\) under the hypothesis of \(\nu _g\le n-p-1\) . PubDate: 2024-08-04 DOI: 10.1007/s10231-024-01492-6
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Abstract: Abstract We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs (m, n) the locus of polarised abelian surfaces of type (1, d) that contain two complementary elliptic curve of exponents m, n, denoted \(\mathcal {E}_d(m,n)\) is non-empty. We show that if d is square-free, the locus \(\mathcal {E}_d(m,n)\) is an irreducible surface (if non-empty). We also show that the loci \(\mathcal {E}_d(d,d)\) can have many components if d is an odd square. As an application, we show that for a genus 3 curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings \(f_i:C\rightarrow E_i,\ i=1,2,3\) satisfy \({{\,\textrm{lcm}\,}}(\deg (f_1),\deg (f_2))={{\,\textrm{lcm}\,}}(\deg (f_1),\deg (f_3))={{\,\textrm{lcm}\,}}(\deg (f_2),\deg (f_3))\) . PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01415-x
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Abstract: Abstract In this paper, we first study the minimality of triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms under the assumption that the shape operator is diagonalizable. Then, we prove that such nonminimal hypersurfaces have constant scalar curvature. As its applications, we estimate the constant scalar curvature and the constant mean curvature. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01422-y
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Abstract: Abstract We improve the result by Feola and Iandoli (J Math Pures Appl 157:243–281, 2022), showing that quasilinear Hamiltonian Schrödinger type equations are well posed on \(H^s({{\mathbb {T}}}^d)\) if \(s>d/2+3\) . We exploit the sharp paradifferential calculus on \({{\mathbb {T}}}^d\) developed by Berti et al. (J Dyn Differ Equ 33(3):1475–1513, 2021). PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01428-0
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Abstract: Abstract We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces the map $$\begin{aligned} L(Y,E) \rightarrow L(X, E), ~ T \mapsto T \circ \iota \end{aligned}$$ is surjective, meaning that each continuous linear map \(X \rightarrow E\) can be extended to a continuous linear map \(Y \rightarrow E\) via \(\iota \) , under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24]. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01420-0
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Abstract: Abstract We show that the set of “wild data”, meaning the initial data for which the barotropic Euler system admits infinitely many admissible entropy solutions, is dense in the \(L^p\) -topology of the phase space. PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01423-5
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Abstract: Abstract Let \(\Omega \) be a bounded, smooth domain of \({\mathbb {R}}^{N},\) \(N\ge 2.\) For \(1<p<N\) and \(0<q(p)<p^{*}:=\frac{Np}{N-p}\) , let $$\begin{aligned} \lambda _{p,q(p)}:=\inf \left\{ \int _{\Omega }\left \nabla u\right ^{p}\textrm{d}x:u\in W_{0}^{1,p}(\Omega ) \ \text {and} \ \int _{\Omega }\left u\right ^{q(p)}\textrm{d}x=1\right\} . \end{aligned}$$ We prove that if \(\lim _{p\rightarrow 1^{+}}q(p)=1,\) then \(\lim _{p\rightarrow 1^{+}}\lambda _{p,q(p)}=h(\Omega )\) , where \(h(\Omega )\) denotes the Cheeger constant of \(\Omega .\) Moreover, we study the behavior of the positive solutions \(w_{p,q(p)}\) to the Lane–Emden equation \(-{\text {div}} (\left \nabla w\right ^{p-2}\nabla w)=\left w\right ^{q-2}w,\) as \(p\rightarrow 1^{+}.\) PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01413-z
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Abstract: Abstract This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane \(\mathbb {C}^2\) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in \(\mathbb {C}^2\) , and every connected domain in \(\mathbb {C}^2\) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of \(\mathring{M}=M\setminus bM\) , and compact subset \(K\subset \mathring{M}\setminus E\) without holes in \(\mathring{M}\) , any \(\mathscr {C}^1\) embedding \(f:M\hookrightarrow \mathbb {C}^2\) which is holomorphic in \(\mathring{M}\) can be approximated uniformly on K by holomorphic embeddings \(F:M\hookrightarrow \mathbb {C}^2\) which map \(E\cup bM\) out of a given ball and satisfy some interpolation conditions. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01418-8
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Abstract: Abstract In this paper, we derive a nonlinear model for stratified arctic gyres, and prove several results on the existence, uniqueness and stability of solutions to such a model, by assuming suitable conditions for the vorticity function and the density function. The approach consists of deriving a suitable integral formulation for the problem and using fixed-point techniques. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01411-1
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Abstract: In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class \(C^2\) rather than \(C^3\) , particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper. PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01426-2
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Abstract: Abstract From a fixed acute triangular base \(\Delta ABC\) , all possible tetrahedra in three-dimensional real space are considered. The possible angles at the additional vertex P are shown to be bounded by certain inequalities, mostly linear inequalities. Together, these inequalities provide fairly tight bounds on the possible angle combinations at P. Four sets of inequalities are used for this purpose, though the inequalities in the first set are rather trivial. The inequalities in the second set can be established quickly, but do not seem to be known. The third and fourth set of inequalities are proved by studying scalar and vector fields on toroids. The first three sets of inequalities are linear in the angles at P, but the last set involves cosines of these angles. A generalization of the last two sets of inequalities is also proved, using the Poincaré–Hopf Theorem. Extensive testing of these results has been done using Mathematica and C++. The C++ code for this is listed in an appendix. While it has been demonstrated that the inequalities bound the possible combinations of angles at P, the results also reveal that additional inequalities, in particular linear inequalities, exist that would provided tighter bounds. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01416-w
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Abstract: Abstract We consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line \(\gamma =\gamma (t)\) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine. PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01424-4
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Abstract: Abstract We propose the novel spectral properties of the Neumann Laplacian in a two-dimensional bounded domain perturbed by an infinite number of compact sets with zero Lebesgue measure, so-called curved holes. These holes consist of segments or parts of curves enclosed in small spheres such that the diameters of holes tend to zero as the number of holes approaches infinity. Specifically, we rigorously demonstrate that the spectrum of the Neumann Laplacian on the perturbed domain converges to that of the original operator on the domain without holes under specific geometric assumptions and an appropriate selection of hole sizes. Furthermore, we derive sophisticated estimates on the convergence rate in terms of operator norms and estimate the Hausdorff distance between the spectra of the Laplacians. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01414-y
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Abstract: Abstract In this article, we investigate some new asymptotic behaviors of the solution for a two-component b-family equations. We first prove the persistence properties of the solution for Eq. (1.1) when the initial data decay logarithmically, algebraically at infinity with the power \(\beta \in (0,\infty )\) . Subsequently, we obtain the infinite propagation of the solution to Eq. (1.1). If the initial data satisfy certain compact condition, then the nontrivial solution u of Eq. (1.1) immediately loses compactly supported. Meanwhile, the solution u decays exponentially as \( x \rightarrow \infty \) . PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01429-z
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Abstract: Abstract Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space \((V,\Vert \cdot \Vert )\) equipped with a particular Jordan structure, called a \(\textrm{JB}^*\) -triple. We identify the horofunctions in the metric compactification of \((V,\Vert \cdot \Vert )\) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space \(V^*\) . Finally, we show that the exponential map \(\exp _0 :V \longrightarrow D\) at \(0\in D\) extends to a homeomorphism between the metric compactifications of \((V,\Vert \cdot \Vert )\) and \((D,\rho )\) , preserving the geometric structure, where \(\rho \) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of \((V,\Vert \cdot \Vert )\) . PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01419-7
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Abstract: Abstract Let G be a group, p a prime and \(H_p(G)\) the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either \(H_p(G)=1\) , \(H_p(G)=G\) , or \([G:H_p(G)]=p\) . In this paper, we prove this conjecture for finite extraspecial p-groups (where \(p>2\) ), finite minimal non-abelian p-groups and finite non-abelian p-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian p-groups which guarantee the existence of the Hughes conjecture. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01421-z
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Abstract: Abstract We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves \(R^{\bullet }\phi _{!}\mathcal {O}\) for the structure sheaf \(\mathcal {O}\) on the total space of a holomorphic fiber bundle \(\phi \) has canonical topology structures. Using the standard Čech argument we prove a density lemma for QDFS-topology on this stalks. In particular, we obtain a vanishing result for holomorphic fiber bundles with Stein fibers. Using Künnet formulas, properties of an inductive topology (with respect to the pair of spaces) on the stalks of the sheaf \(R^{1}\phi _{!}\mathcal {O}\) and a cohomological criterion for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon for the total space of holomorphic fiber bundles with (1, 0)-compactifiable fibers. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01412-0
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Abstract: Abstract For large classes of (finite and) infinite dimensional complex Banach spaces Z, B its open unit ball and \(f:B\rightarrow B\) a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in \(\partial B\) and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates \((f^n)_n\) of f. The Wolff hull generalises the concept of a Wolff point, where such a point can no longer be uniquely determined, and coincides with the Wolff point if Z is a Hilbert space. Recall that \((f^n)_n\) does not generally converge even in finite dimensions, compactness of f (i.e. f(B) is relatively compact) is necessary for convergence in the infinite dimensional Hilbert ball and all accumulation points \(\Gamma (f)\) of \((f^n)_n\) map B into \(\partial B\) (for any topology finer than the topology of pointwise convergence on B). The target set of f is $$\begin{aligned} T(f)=\bigcup _{g \in \Gamma (f)} g(B). \end{aligned}$$ To locate T(f), we use a concept of closed convex holomorphic hull, \({\text {Ch}}(x) \subset \partial B\) for each \(x \in \partial B\) and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely $$\begin{aligned} W(f) \cap {\text {Ch}}(x)\ne \emptyset \ \ \hbox {for all}\ \ x \in T(f). \end{aligned}$$ If B is the Hilbert ball, W(f) is the Wolff point, and this is the usual Denjoy–Wolff result. Our results are for all reflexive Banach spaces having a homogeneous ball (or equivalently, for all finite rank \(JB^*\) -triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional. PubDate: 2024-08-01 DOI: 10.1007/s10231-024-01427-1
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Abstract: Abstract We extend the estimates for maximal Fourier restriction operators proved by Müller et al. (Rev Mat Iberoam 35:693–702, 2019) and Ramos (Proc Am Math Soc 148:1131–1138, 2020) to the case of arbitrary convex curves in the plane, with constants uniform in the curve. The improvement over Müller, Ricci, and Wright and Ramos is given by the removal of the \({\mathcal {C}}^2\) regularity condition on the curve. This requires the choice of an appropriate measure for each curve, that is suggested by an affine invariant construction of Oberlin (Michigan Math J 51:13–26, 2003). As corollaries, we obtain a uniform Fourier restriction theorem for arbitrary convex curves and a result on the Lebesgue points of the Fourier transform on the curve. PubDate: 2024-08-01 DOI: 10.1007/s10231-023-01417-9
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