On a class of weak Landsberg metrics

In this paper, we discuss a class of Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We characterize weak Landsberg metrics in this class and show that there exist weak Landsberg metrics which are not Landsberg metrics in dimension greater than two.     

All regular Landsberg metrics are Berwald

This article negatively answers the long standing problem concerning the existence of non-Berwald Landsberg metrics.     

On almost regular automorphisms

Among other things, we prove that a polycyclic group admitting an automorphism of order two with finitely many fixed elements is abelian-by-finite. An example shows that this result cannot be extended to finitely generated soluble groups.     

On the class of generalized Landsberg manifolds

In 2000, Bejancu–Farran introduced the class of generalized Landsberg manifolds which contains the class of Landsberg manifolds. In this paper, we prove three global results for generalized Landsberg manifolds. First, we show that every compact generalized Landsberg manifold is a Landsberg manifold. Then we prove that every complete generalized Landsberg manifold with relatively isotropic Landsberg curvature reduces to a Landsberg manifold. Finally, we show that every generalized Landsberg manifold with vanishing Douglas curvature satisfies \(\mathbf{H}=0\).     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

On a class of almost hypoelliptic polynomials

The paper gives criteria for almost hypoelliptic polynomials P of two and three variables, providing that P(ξ) → ∞ as |ξ| → +∞.     

On almost regular automorphisms of prime order

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 206–208, September–October, 1992.     

On a new class of Finsler metrics

In this paper, the geometric meaning of (α,β)-norms is made clear. On this basis, a new class of Finsler metrics called general (α,β)-metrics are introduced, which are defined by a Riemannian metric and a 1-form. These metrics not only generalize (α,β)-metrics naturally, but also include some metrics structured by R. Bryant. The spray coefficients formula of some kinds of general (α,β)-metrics is given and the projective flatness is also discussed.     

On a class of almost unbiased ratio estimators

Summary Murthy and Nanjamma [4] studied the problem of construction of almost unbiased ratio estimators for any sampling design using the technique of interpenetrating subsamples. Subsequently, Rao [7], [8] has given a general method of constructing unbiased ratio estimators by considering linear combinations of the two simple estimators based on the ratio of means and the mean of ratios. However, it is difficult to choose an optimum weight (Rao [9]) which minimizes the variance of the combined estimator since the weights are random in certain cases. In this note, we consider a different method of combining these estimators and obtain a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derive an optimum in this class. The case of simple random sampling where a similar class of almost unbiased ratio estimators can be developed is briefly discussed. The results are illustrated by means of simple numerical examples.     

On a class of projectively flat Finsler metrics

In this paper, we classify locally projectively flat general $(\alpha ,\beta )$-metrics $F=\alpha \varphi (b^2,\frac{\beta }{\alpha })$ on an $n(\ge 3)$-dimensional manifold if α is of constant sectional curvature and ${\varphi }_1\ne 0$. Furthermore, we find equations to characterize this class of metrics with constant flag curvature and determine their local structures.     

On a class of weakly almost periodic mappings

Communicated by Karl H. Hofmann     

On a class of weakly Einstein Finsler metrics

In this paper we study a special class of Finsler metrics—m-Kropina metrics which are defined by a Riemannian metric and a 1-form. We prove that a weakly Einstein m-Kropina metric must be Einsteinian. Further, we characterize Einstein m-Kropina metrics in very simple conditions under a suitable deformation, and obtain the local structures of m-Kropina metrics which are of constant flag curvature and locally projectively flat with constant flag curvature respectively.     

On countable connected locally connected almost regular Urysohn spaces

We construct connected, locally connected, almost regular, countable, Urysohn spaces. This answers a problem of G.X. Ritter. We show that there are 2^c such non-homeomorphic spaces. We also show that there are 2^c non-homeomorphic spaces which are further rigid. We discuss the group of homeomorphisms of such spaces.The following question was raised by G.X. Ritter: Does there exist a countable connected locally connected Urysohn space which is almost regular? We answer this question in the affirmative and in fact, show that not only are there as many as 2^c such spaces but that there are just as many rigid spaces with the same properties. Furthermore we show that every countable Urysohn space is a subspace of such a space. We also prove that every countable group is isomorphic to the group of autohomeomorphisms of some connected locally connected almost regular Urysohn space. Examples are given of groups of order c which can be represented in this manner.     

On the class of almost L-weakly and almost M-weakly compact operators

In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.     

On a subclass of the class of generalized Douglas-Weyl metrics

We study a class of Finsler metrics whose Douglas curvature is constant along any Finslerian geodesics. This class of Finsler metrics is a subclass of the class of generalized Douglas-Weyl metrics and contains the class of Douglas metrics as a special case. We find a condition under which this class of Finsler metrics reduces to the class of Landsberg metrics. Then we show this class of metrics contains the class of R-quadratic metrics.     

On smooth solutions of a class of almost hypoelliptic equations

For a class of non-degenerate (regular) almost hypoelliptic equations, which are partially hypoelliptic with respect to certain variables, infinitely differentiable in a specific strip solutions are extracted.     

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b ₁ = k?λ?1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k ? 4b ₁ + 3 (and µ(u, w) + µ(u, z) = 2k ? 4b ₁ + 4). If k = 3b ₁ + γ with γ ≥ ?2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b ₁ = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.     

On a class of Riemannian metrics arising from Finsler structures

On the slit tangent bundle of Finsler manifolds, we introduce a class of metrics and study the relation between Levi-Civita connection, Vaisman connection, vertical foliation, and Reinhart spaces. We show that the Levi-Civita and the Vaisman connections induce the same connections in the structural bundle if and only if the base manifold is Landsbergian. Moreover every foliated Reinhart manifold reduces to a Riemannian manifold.