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Abstract: Abstract We define a notion of extrinsic black hole in pure Lovelock gravity of degree k which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null 2k-mean curvature in Euclidean space \({\mathbb {R}}^{n+1}\) , \(2\le 2k\le n-1\) . We then combine a regularity argument with a rigidity result by Araújo and Leite (Indiana University Mathematics Journal pp. 1667–1693, 2012) to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge et al. (Advances in Mathematics 266: 84–119, 2014), a local rigidity result for the Lovelock-Schwarzschild solutions. PubDate: 2021-10-05 DOI: 10.1007/s00574-021-00279-0

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Abstract: Abstract Let \(\mathbb {F}_{q^n}\) be a finite field with \(q^n\) elements, and let \(m_1\) and \(m_2\) be positive integers. Given polynomials \(f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]\) with \(\deg (f_i(x)) \le m_i\) , for \(i = 1, 2\) , and such that the rational function \(f_1(x)/f_2(x)\) satisfies certain conditions which we define, we present a sufficient condition for the existence of a primitive element \(\alpha \in \mathbb {F}_{q^n}\) , normal over \(\mathbb {F}_q\) , such that \(f_1(\alpha )/f_2(\alpha )\) is also primitive. PubDate: 2021-10-02 DOI: 10.1007/s00574-021-00277-2

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Abstract: Abstract In this paper, we study oscillatory properties of neutral differential equations. Moreover, we discuss some examples that show the effectiveness and the feasibility of the main results. PubDate: 2021-09-21 DOI: 10.1007/s00574-021-00276-3

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Abstract: Abstract For a vertex v of a connected graph G, the status of v is defined as the sum of the distances from v to all other vertices in G. The minimum status of G is the minimum of status of all vertices of G. We give the largest values for the minimum status of series-reduced trees with fixed parameters such as maximum degree, number of pendant vertices, diameter, matching number and domination number, and characterize the unique extremal trees. PubDate: 2021-09-20 DOI: 10.1007/s00574-021-00278-1

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Abstract: We fix a mistake in the argument leading to the proof that the family of foliations introduced in the paper does not have an algebraic solution apart from the line at infinity PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00219-4

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Abstract: Abstract In this note we show that the exceptional algebraic set for an infinite discrete group in \(PSL(3,{{\mathbb {C}}})\) should be a finite union of: complex lines, copies of the Veronese curve or copies of the cubic \(xy^2-z^3\) . PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00218-5

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Abstract: Abstract This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00220-x

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Abstract: Abstract In this work, we consider a quasi-homogeneous, corank 1, finitely determined map germ f from \((\mathbb {C}^2,0)\) to \((\mathbb {C}^3,0)\) . We consider the invariants m(f(D(f))) and J, where m(f(D(f))) denotes the multiplicity of the image of the double point curve D(f) of f and J denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of \(f(\mathbb {C}^2)\) . We present formulas to calculate m(f(D(f))) and J in terms of the weights and degrees of f. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00225-6

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Abstract: Abstract We prove that unicritical polynomials \(f(z)=z^d+c\) which are semihyperbolic, i.e., for which the critical point 0 is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric \(\rho (z) dz \) , where \(\rho (z) = 1+{{\,\mathrm{dist}\,}}(z,P(f))^{-1+1/d}\) has singularities on the postcritical set P(f). We also show that this metric is Hölder equivalent to the usual Euclidean metric. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00228-3

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Abstract: Abstract We give an upper bound on the largest eigenvalue of the signless Laplacian matrix of a Hamiltonian graph. This bound is applied to obtain sufficient spectral conditions for the non-existence of Hamiltonian cycles. Under certain additional assumptions we provide a polynomial time decisive spectral criterion for the Hamiltonicity of a given graph with sufficiently large minimum vertex degree. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00211-y

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Abstract: Abstract In this paper, we classify the Einstein hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) . We use the characterization of the hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) whose tangent component of the unit vector field spanning the factor \(\mathbb {R}\) is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) must have constant sectional curvature. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00216-7

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Abstract: Abstract In this paper, we investigated influence of \({\mathcal {M}}_p\) -supplemented p-subgroups contained in a normal subgroup on structure of finite groups. We obtained some new criteria for p-supersolvability of finite groups, decided all possible non-Abelian pd-chief factors under our assumption and gave a Frobenius type theorem. Some earlier results on \({\mathcal {M}}_p\) -supplemented p-subgroups appear as particular cases. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00223-8

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Abstract: Abstract We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas’s work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of orthogonal projections of the surface to 3-space. Cross-ratio invariants at \(P_3(c)\) -points are used to recover two moduli in the 4-jet of a projective parametrization of the surface and identify the stable configurations of the asymptotic curves of the surface. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00221-w

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Abstract: Abstract Let C be a smooth non-rational projective curve over the complex field \(\mathbb {C}\) . If A is an abelian subvariety of the Jacobian J(C), we consider the Abel-Prym map \(\varphi _A : C \rightarrow A\) defined as the composition of the Abel map of C with the norm map of A. The goal of this work is to investigate the degree of the map \(\varphi _A\) in the case where A is one of the components of an isotypical decomposition of J(C). In this case we obtain a lower bound for \(\deg (\varphi _A)\) and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in a few examples. In particular, these examples show that both bounds are sharp. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00226-5

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Abstract: Abstract This paper deals with foliations by curves [s] of degree \( r \ge 2 \) on \( \mathbb {P}^2 \) with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional \( Y \subset S \) if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every non-degenerate foliation in degrees \( 2 \le r \le 5 \) does have a minimal subscheme and that the set of non-degenerate foliations of degree \( r \ge 6 \) that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree \( r \ge 7 \) that do not have a minimal subscheme at all. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00214-9

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Abstract: Abstract We consider a new class of singularities called mixed maps from Oka’s class. In this new setting we prove the existence of Milnor–Hamm fibration on the tube and sphere. Moreover, we discuss the problem of existence of a Milnor vector field for this class. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00229-2

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Abstract: Abstract We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into \(\mathbb {R}^4\) ) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00217-6

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Abstract: Abstract We consider the stable ruled surface \(S_1\) over an elliptic curve. There is a unique foliation on \(S_1\) transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00224-7

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Abstract: Abstract We define the path coalgebra and Gabriel quiver constructions as functors between the category of k-quivers and the category of pointed k-coalgebras, for k a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel k-quiver functor is left adjoint to the corresponding path coalgebra functor. We dualize, obtaining adjoint pairs of functors (contravariant and covariant) for pseudocompact algebras. Using these tools we describe precisely to what extent presentations of coalgebras and pseudocompact algebras in terms of path objects are unique, giving an application to homogeneous algebras. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00227-4

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Abstract: Abstract We establish q-analogues for three summation formulae about the \(\lambda \) -extended Catalan numbers. PubDate: 2021-09-01 DOI: 10.1007/s00574-020-00210-z