Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.     

Refinement of Vahlen's Theorem for Minkowski Bases of Three-Dimensional Lattices

In the paper, the analog of Vahlen's theorem for Minkowski bases of three-dimensional lattices is sharpened.     

Unit disk of the smallest self-perimeter in a Minkowski plane

On a Minkowski plane with nonsymmetric metric, we find all unit disks with self-perimeter 6, the smallest possible value.     

Curves in the Minkowski plane and their contact with pseudo-circles

We study the caustic, evolute, Minkowski symmetry set and parallels of a smooth and regular curve in the Minkowski plane.     

Large triangles contained in the unit disk of a Minkowski plane

We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are ${\frac{3}{2}}$ . We also prove that C permits to inscribe a triangle whose sides are of lengths at least ${\frac{3}{2}}$ in the positive orientation, or that they are of lengths at least ${\frac{3}{2}}$ in the negative orientation. The ratio ${\frac{3}{2}}$ in both the theorems is best possible.     

A random covering of the plane by convex figures

Translated from Issledovaniya po Prikladnoi Matematike, No. 6, pp. 10–13, 1979.     

Continuity of the solution to the even logarithmic Minkowski problem in the plane

This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R~2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.     

Representing orders on the plane by translating convex figures

Given a finite collection of disjoint, convex figures on the plane, is it possible to assign to each a single direction of motion so that this collection of figures may be separated, through an arbitrary large distance, by translating each figure one at a time, along its assigned direction? We present a computational model for this separability problem based on the theory of ordered sets.     

A characterization of the semi-isometries of a Minkowski plane over a field K

Under certain hypotheses on the field K,W.Bens has proved that a map of the plane K², which preserves a single Lorents-Minkowski — distance is semilinear and injective [5],[6]. We shall generalise this result for every field K with char K ? {2,3,5 } and for K = GF(5^m), m> 1.     

Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C~2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C~2-smooth curves on surfaces. We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

How the elliptic integrals K and E arise from circles and points in the Minkowski plane

Various self-circumferences of unit circles placed on a Minkowskian plane are expressed in terms of elliptic integrals of the first and second kinds. Landen's transformation is interpreted in this geometric context, and the self-circumferences are estimated asymptotically in the case of circles whose boundaries come very close to the origin.     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

拟半连续格和交半连续格

作为半连续格的推广, 引入了拟半连续格的概念. 讨论了拟半连续格的基本性质. 在拟半连续格上得到了类似于拟连续偏序集的一些主要结果. 同时探讨了半连续格、拟半连续格、交半连续格、交连续格、强连续格几种不同结构之间的关系. 最后,讨论了半连续函数空间仍是半连续格的条件.