We consider a stochastic ranking process, which is a mathematical model of the ranking in the web page of online bookstores or posting web pages. We give a scaling limit of tagged particle dynamics. In this limit the scaled tagged particles jumps to the top of the list when its own Poisson clock rings and moves deterministically along a curve otherwise. This curve is characteristic curve of a system of quasi linear PDE, which is mentioned in [11, 14]. We also give a scaling limit of multi-tagged particle dynamics, in which the motion of the particles are independent.

In this paper we consider "parabolically" scaled centered tagged particle dynamics for a stochastic ranking process (regarded as a particle system), which is driven according to an algorithm for self-organizing linear list of a finite number of items. We let the number of items to infinity and show that the scaled tagged particle weakly converges to a "diffusion" processes with occasional jumps, in which the particle jumps to 0 when its own Poisson clock rings and behaves as a "diffusion" process otherwise. The "diffusion" is decomposed into a sum of independent continuous Markov Gaussian processes with a random covariance. Intuitively, each component process is constructed by infinitely many particles having the same intensity behind the tagged particle. This random covariance depends only on its own last Poisson time. In multi-tagged particle system, "hyperbolically" scaled tagged particles decompose [0,1] interval into L + 1 layers, where L is a number of tagged particles. Intuitively, infinitely many particles in each layer construct a "diffusion" processes, which is interpreted as a shrunk version of that in the single tagged particle case. Each "parabolically" scaled centered tagged particle holds in common these "diffusion" processes if the corresponding layer is behind the corresponding "hyperbolically" scaled tagged particle.

Let $\mathcal{T}$ (Δ) be the universal Teichmüller space, viewed as the set of all Teichmüller equivalent classes [f] of quasiconformal mappings f of Δ onto itself. The notion of completing triangles was introduced by F. P. Gardiner. Three points [f], [g] and [h] are called to form a completing triangle if each pair of them has a unique geodesic segment joining them. Otherwise, they form a non-completing triangle. In this paper, we construct two Strebel points [f] and [g] such that [f], [g] and [id] form a non-completing triangle. A sufficient condition for points [f], [g] and [id] to form a completing triangle is also given.

In this paper, we study a generalized functional Φ_F related to the conformality of maps between Riemannian manifolds. We derive the first variation formula and the second variation formula of Φ_F, then we study the stability of F-stationary map from or into the standard sphere. We also introduce the F-stress energy tensor associated to Φ_F which is naturally linked to conservation law.

On the basis of A. L. Carey, D. Crowley, M. K. Murray's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Dixmier-Douady class.

The distance estimator method is well-known as an iterative method which estimates the Euclidean distance between a given point and Julia set. Although it brings a remarkable effect in drawing Julia set, it seems to be not known about how accurate this method is. In the present paper, we give explicit estimates on this method.

In this paper, we mainly investigate complete or compact spacelike hypersurfaces with constant mean curvature or constant scalar curvature in Lorentzian manifolds L₁ⁿ⁺¹. We give a new estimate of the Laplacian ΔS of the squared length S of the second fundamental form of such spacelike hypersurfaces. Finally, we give partial generalizations of some Conjectures in Lorentzian manifolds L₁ⁿ⁺¹.

For a family of Ornstein-Uhlenbeck processes having an infinitesimal size of noise, we prove precise asymptotics for large deviations of an integral form over the continuous path space. The main ingredients in the proof are an exponential change of measures and subtle Taylor expansions with suitable estimates. An application of these precise integral large deviations to partial differential equations is included.

We establish how a two-dimensional local field can be described as a locally convex space once an embedding of a local field into it has been fixed. We study the resulting spaces from a functional analytic point of view: in particular we study bounded, c-compact and compactoid submodules of two-dimensional local fields.

Nevanlinna showed that two distinct non-constant meromorphic functions on C must be linked by a Möbius transformation if they have the same inverse images counted with multiplicities for four distinct values. Later on, Gundersen generalized the result of Nevanlinna to the case where two meromorphic functions share two values ignoring multiplicity and share other two values with counting multiplicities. In this paper, we will extend the results of Nevanlinna-Gundersen to the case of two holomorphic mappings into Pⁿ(C) sharing (n + 1) hyperplanes ignoring multiplicity and other (n + 1) hyperplanes with multiplicities counted to level 2 or (n + 1).

For a compact Riemann surface X of genus g, we will construct a holomorphic section of the line bundle π₁^*K_X^{g(g+1)(g+2)/6} $\otimes$ π₂^*K_X^{g(g+1)(g+2)/6} over X × X whose zero set consists exactly of the points (P,Q) with the cardinalities of the Weierstrass gap sets G(P,Q) greater than the minimal value (g² + 3g)/2.