Volume 18, Issue 1

In [9] S. Yoshiara determines possible automorphism group of doubly transitive dimensional dual hyperovals. He shows that a doubly transitive dual hyperoval D is either isomorphic to the Mathieu dual hyperoval or the dual hyperoval is defined over 𝔽 2 , and if the hyperoval has rank n , the automorphism group has the form E ⋅ S , with an elementary abelian group E of order 2 n and S a subgroup of GL( n ,2) acting transitively on the nontrivial elements of E . Moreover Yoshiara describes the possible candidates for S . In this paper we assume that S is non-solvable and show that then the dimensional dual hyperoval is a bilinear quotient of a Hyubrechts dual hyperoval.

In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion ), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.

We define generalized null 2-type submanifolds in the m -dimensional Euclidean space 𝔼 m . Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the condition Δ H = f H + gC for some smooth functions f , g and a constant vector C in 𝔼 m , where Δ and H denote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼 3 and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, completeness or compactness, as well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.

Given six points on a conic, Pascal’s theorem gives rise to a configuration called the hexagrammum mysticum . It contains 20 Steiner points and 20 Cayley–Salmon lines. By a classical theorem due to von Staudt, the Steiner points fall into 10 conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. We show that the C-S lines are pairwise conjugate precisely when the original sextuple is tri-symmetric . The variety of tri-symmetric sextuples turns out to be arithmetically Cohen–Macaulay of codimension two. We determine its SL 2 -equivariant minimal resolution.

We investigate Steiner’s Porism in finite Miquelian Möbius planes constructed over the pair of finite fields GF( q ) and GF( q 2 ), for an odd prime power q . Properties of common tangent circles for two given concentric circles are discussed and with that, a finite version of Steiner’s Porism for concentric circles is stated and proved. We formulate conditions on the length of a Steiner chain by using the quadratic residue theorem in GF( q ). These results are then generalized to an arbitrary pair of non-intersecting circles by introducing the notion of capacitance, which turns out to be invariant under Möbius transformations. Finally, the results are compared with the situation in the classical Euclidean plane.

We prove a correspondence theorem for singular tropical surfaces in ℝ 3 , which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin’s lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in ℝ 3 . As application we show that there are pencils of real surfaces of degree d in ℙ 3 containing at least (3/2) d 3 + O ( d 2 ) singular surfaces, which is asymptotically comparable to the number 4( d − 1) 3 of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces [12].

In this paper we study an extension of Yamabe solitons for inequalities. We show that a Riemannian complete non-compact shrinking Yamabe soliton ( M , g , V , λ ) has finite fundamental group, provided that the scalar curvature is strictly bounded above by λ . Furthermore, an instance of illustrating the sharpness of this inequality is given. We also mention that the fundamental group of the sphere bundle SM is finite.

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body K by a circumscribed polytope P with a given number of facets. These bounds are of particular interest if K is elongated. To measure the elongation of the convex set, its isoperimetric ratio V j ( K ) 1/ j V i ( K ) −1/ i is used.

Let q be an odd prime power. We prove that a planar function f from 𝔽 q to itself can be written as an affine Dembowski–Ostrom polynomial if and only if the projective plane derived from f is a commutative semifield plane.

We study smooth quartic surfaces in ℙ 3 which admit a group of projective automorphisms isomorphic to the icosahedron group.