The geometric algebraCℓ 3 as a model for a projective plane

We show that the geometric algebraCℓ ₃ can be used as a model for the real projective plane, in the sense that the axioms defining the plane and their duals can be proved as theorems. However, it seems that there is some difficulty in using a geometric algebra to model a projective space over a noncommutative division ring.     

Towards a de Bruijn–Erdős Theorem in the L_1-Metric

A well-known theorem of de Bruijn and Erd?s states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness. In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$ - or $y$ -coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines. If points of $X$ are allowed to share their coordinates, then either there is a line that contains all $n$ points, or $X$ induces at least $n/37$ distinct lines.     

关于广义(α,β)-度量的若干Ricci曲率性质

本文研究了广义(α,β)-度量的Ricci曲率和Ricci曲率张量.首先,在一定条件下,本文给出了强Einstein广义(α,β)-度量的一个等价刻画.进一步,得到了广义(α,β)-度量是Ricci-齐次Finsler度量的一个充分必要条件.     

Soliton Solutions for a Singular Schrödinger Equation with Any Growth Exponents

This paper is concerned with a kind of quasilinear Schrödinger equation with combined nonlinearities, a convex term with any growth and a singular term, in a bounded smooth domain. Multiplicity results are obtained by critical point theory together with truncation arguments and the method of upper and lower solutions.     

Ricci Curvature and Fundamental Group

By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor (1968) are generalized.     

Can Soliton Attractors Exist in Realistic 3+1-D Conservative Systems?

High-energy physicists already know that stable attractors (solitons) can exist in 3+1-dimensional conservative Lagrangian systems, so long as the definition of an attractor is based on weak notions of stability and the fields admit topological charge. This paper explores the possibility of attractors in Lagrangian field theories without topological charge, using a new, stronger concept of stability—Convective quantized Asymptotic Orbital Stability (ChAOS) . Under certain conditions, ChAOS is related to additive Liapunov stability or energetic stability. Russian physicists have argued that such stability tends to require topological charge; however, this paper describes systems which avoid those arguments, and suggests how numerical examples might be constructed. Solitons have been proposed to explain the existence and nature of elementary particles within the Feynman version of quantum theory; Section 6cites this literature, as well as new possibilities for alternative versions with testable nuclear implications.     

Kähler metrics with cone singularities along a divisor of bounded Ricci curvature

Let D be a smooth divisor in a compact complex manifold X and let \(\beta \in (0,1)\). We use the liner theory developed by Donaldson (Essays in Mathematics and Its Applications, Springer, Berlin, pp 49–79, 2012) to show that in any positive co-homology class on X there is a Kähler metric with cone angle \(2\pi \beta \) along D which has bounded Ricci curvature. We use this result together with the Aubin–Yau continuity method to give an alternative proof of a well-known existence theorem for Kähler–Einstein metrics with cone singularities.     

Multiplicity mod 2 as a Metric Invariant

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of ℝ ⁿ . In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S ⁿ is preserved by the antipodal map. The study of such subsets of S ⁿ enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2 under some extra assumptions on the tangent cone. We also prove a real version of a theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic bi-Lipschitz homeomorphisms.     

连续环群的Ricci形式

本文给出了连续环群的Ricci形式是闭的充分必要条件．     

射影Ricci平坦的Kropina度量

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.     

Job challenge as a motivator in a principal–agent setting

We analyze the tradeoff between monetary and non-monetary incentives in a principal–agent framework where the principal chooses job-challenge (job-design) to provide non-monetary incentives. We capture the construct of job-challenge as a motivator based on Atkinson (1958). In: Atkinson, J.W. (Ed.), Motires in Fantasy, Action and Society, Van Nostrand, Princeton, NJ. Specifically, more challenging jobs reduce the probability of success but increase the marginal productivity of the agent's effort. This provides the agent with a sense of accomplishment from which the agent derives utility. We show that there are three effects of job-challenge when we consider information asymmetries. The substitution effect arises because non-monetary incentives can be used to substitute for monetary incentives. The risk effect arises because job challenge can be used to impose some of the risk needed to mitigate the impact of hidden-action, rather than having to impose risk through monetary incentives which could be costlier to the principal. The informativeness effect arises because the final outcome is more informative on the hidden-action of the agent and thus less risk needs to be imposed through the monetary incentives than in a model that does not feature job-challenge. The cost to the principal of designing challenging jobs arises from the decreased expected output.     

Soliton,kink and antikink solutions of a 2-component of the Degasperis–Procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis–Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained.     

Schrödinger Equation as a Self-Consistent Field

It is well known that the Schrödinger equation can be reduced to the Hamilton–Jacobi equation in Bohmian mechanics. Corresponding new equations of the Vlasov and Lamb types are derived, and their stationary solutions are investigated.     

Ricci流与超Ricci流上的Li-Yau-Hamilton Harnack不等式 献给余家荣教授100华诞

本文对赋予依赖时间变化的加权紧致与完备Riemann流形上的时变Witten Laplace算子的热方程的正解证明Li-Yau-Hamilton型微分Harnack不等式和Harnack不等式.特别地,本文对赋予Ricci流或倒向Ricci流的紧致与完备Riemann流形上的Laplace-Beltrami算子的热方程的正解证明Li-Yau-Hamilton型微分Harnack不等式和Harnack不等式.     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

The Isoperimetric Inequality in Steady Ricci Solitons

The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^n-1is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric,the author utilize the result of Guan-Li-Wang to get his conclusion. For the sake of the soliton structure, the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally s...     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.