k-Dirac Operator and Parabolic Geometries

In this paper we construct particular differential operators which are invariant with respect to the canonical action of the principal group of a particular type of parabolic geometry. These operators form sequences which are related to the minimal resolutions of the $k$ -Dirac operator studied in Clifford analysis.     

Affine Biharmonic Curves in 3-Dimensional Homogeneous Geometries

Every 3-dimensional Riemannian manifold with 4-dimensional isometry group admits a normal almost contact structure compatible to the metric. In this paper we study affine biharmonic curves in model spaces of Thurston geometry except Sol.     

On Meet-Complements in Cohn Geometries

Within the frame of projective lattice geometry, the present paper investigates classes of meet-complements in Cohn geometries and especially in Ore and Bezout geometries. The algebraic background of these geometries is given by torsion free modules over domains — in particular Ore and Bezout domains. ¹     

On Scattered Convex Geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.     

On Characterization of Absolute Geometries

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einführung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if ${\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}$ . Denoting the foot of a on the line ${\overline{c, d}}$ with ${a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}$ , the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if ${a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}$ or ${a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}$ respectively. Let LS be the set of all L-S quadrangles and LS _r , LS _h or LS _e the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS =  LS _r or LS =  LS _h or LS =  LS _e . To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” ${o:=\overline{a,c}\cap\overline{b,d}}$ , and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.     

On Parabolic Cylinder Functions

In this paper certain properties of the parabolic cylinder functionsas formulated by Wells & Spence (1945) are discussed. Theparticular features studied are certain orthogonality propertiesand the character of the cigenvalues when the parabolic cylinderfunctions are considered as arising from essentially two-parametereigenvalue problems.     

On the Order Dimension of Convex Geometries

We study the order dimension of the lattice of closed sets for a convex geometry. We show that the lattice of closed subsets of the planar point set of Erd?s and Szekeres from 1961, which is a set of 2 ^n???2 points and contains no vertex set of a convex n-gon, has order dimension n???1 and any larger set of points has order dimension at least n.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

On Levels in Arrangements of Curves

Abstract. Analyzing the worst-case complexity of the k -level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O( nk^ 1-1/(9· 2 ^s-3 ) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s . Previously, nontrivial results were known only for the case s=1 and s=2 . We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O( nk ^7/9 log ^2/3 k) . The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.     

On Rectifiable Curves in the Plane

This paper gives a formula of integral for the rectifiable curves in the plane by the Hausdorff fractional derivation and integral. Received August 10, 1999, Revised December 27, 1999, Accepted January 14, 2000     

On 4-Dimensional Mapping Tori and Product Geometries

The paper gives simple necessary and sufficient conditions fora closed 4-manifold to be homotopy equivalent to the mappingtorus of a self homotopy equivalence of a PD₃-complex. Thisis a homotopy analogue of the Stallings and Farrell fibrationtheorems available in other dimensions. The paper also considers4-manifolds which admit a geometry of Euclidean factor typeand complex surfaces which fibre over S¹.     

论平面上的可求长曲线

本文利用Hausdorff分数型导数与积分理论给出平面上可求长曲线的积分表 达式.     

On Levels in Arrangements of Curves

   Abstract. Analyzing the worst-case complexity of the k -level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O( nk^ 1-1/(9· 2 ^s-3 ) ) ) for curves that are graphs of polynomial functions of an arbitrary fixed degree s . Previously, nontrivial results were known only for the case s=1 and s=2 . We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O( nk ^7/9 log ^2/3 k) . The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.     

On a Parabolic Sine-Gordon Model

We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

On Maximally Inflected Hyperbolic Curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert’s method we show that for any integers \(d\) and \(r\) such that \(4\le r \le 2d^2-2d\) , there is a non-singular hyperbolic curve of degree \(2d\) in \({\mathbb R}^2\) with exactly \(r\) line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree \(6\) .     

On Projective Invariants of Curves

It is shown that for the action of the projective group on curves in the plane leaving a point fixed, the differential invariants depending on several points are derivable from the well known 4-point invariants.