Pseudo real closed fields,pseudo p-adically closed fields and NTP2

The main result of this paper is a positive answer to the Conjecture 5.1 of [14] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then $Th(M)$ is NTP₂ if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then $Th(M)$ is NTP₂. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then $Th(M)$ is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking.     

Groups definable in separably closed fields

We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field is definably isomorphic to the group of -rational points of an algebraic group defined over .

     

Existentially closed ordered difference fields and rings

We describe classes of existentially closed ordered difference fields and rings. We show an Ax‐Kochen type result for a class of valued ordered difference fields (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Special transformations in algebraically closed valued fields

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan (2006) [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV F_S(0,0), that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (V F,Γ)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.     

Arbitrary decompositions into open and closed trails

The problem of arbitrary decomposition of a graph G into closed trails i.e. a decomposition into closed trails of prescribed lengths summing up to the size of the graph G was first considered in the case of the complete graph G=K_n (for odd n) in connection with vertex-distinguishing coloring of the union of cycles.Next, the same problem was investigated for other families of graphs.In this paper we consider a more general problem: arbitrary decomposition of a graph into open and closed trails. Our results are based on and generalize known results on decomposition of a graph into closed trails. Our results also generalize some results concerning decomposition of a graph into open trails. We here emphasize that the known results on the closed case are basic ingredients for the proof of the open and closed case.     

Note on generalizing theorems in algebraically closed fields

The generalization properties of algebraically closed fields of characteristic and of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that admits finite term bases, and admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some , ( 1's) is provable in steps, then is provable. Received: February 1, 1996     

Critical point equation and closed conformal vector fields

In this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non-trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.     

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

We use energy estimates to study the long time stability of multidimensional planar viscous shocks for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength are allowed, subject to an appropriate Evans function condition. The main tools are a conjugation argument that allows us to replace the eigenvalue equation by a problem in which the dependence of the coefficients is removed and degenerate Kreiss-type symmetrizers designed to cope with the vanishing of the Evans function for zero frequency.

     

Conjugate points and shocks in nonlinear optimal control

We investigate characteristics of the Hamilton-Jacobi-Bellman
equation arising in nonlinear optimal control and their relationship with weak and strong local minima. This leads to an extension of the Jacobi conjugate points theory to the Bolza control problem. Necessary and sufficient optimality conditions for weak and strong local minima are stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.

     

Slow Solutions of a Differential Inclusion and Vector Optimization

The present work develops a new approach for studying the dynamic evolution of a vector optimization problem. We introduce a convenient differential inclusion that rules the dynamics of the optimization problem. Actually we consider a sort of ‘gradient system’ defined by vector valued functions. The main tool used is a completely new adaptation to the vector problem of the notion of pseudogradient, which is a well-known concept in the modern critical point theory. Finally we study a special class of solutions of the above quoted differential inclusion: the slow solutions.     

Lie group transformations for self‐similar shocks in a gas with dust particles

In this paper, Lie group of transformation method is used to investigate the self‐similar solutions for the system of partial differential equations describing one‐dimensional unsteady plane flow of an inviscid gas with large number of small dust particles. The forms of the drag force D and the heat transfer rate Q experienced by the particle not in equilibrium with the gas have been derived for which the system of equations admits self‐similar solutions. A particular solution to the problem in one case have been found out and is used to study the effect of the dust particles on the similarity exponent. Copyright © 2009 John Wiley & Sons, Ltd.     

Categorical anatomy of closed graph and open mapping theorems

Recent development of the theory of general topological vector spaces (without local convexity, hence without duality theory), as in [1], has exposed more of the essential (distinctly topological) nature of a good deal of the locally convex theory, resulting in a remarkably high survival rate of theorems, with (it is claimed) greater simplicity and elegance of proofs. This is especially true of the closed graph theorem. Associated concepts, such as barrelledness and the various kinds of completeness, can be described and related in a manner which demands extension to uniform spaces and beyond. In the setting of categories and functors, we believe these ideas acquire an elegance and unity which considerably widens their scope while shedding light on their homeground in functional analysis. The aim of this paper, which takes a much broader perspective than Pelletier [10], is to distill some of this categorical essence, and to point out new problems, both topological and categorical, which present themselves along the way.     

Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics

The formation of a current sheet in a weakly collisional plasma can be modelled as a finite-time singularity solution of magnetohydrodynamic equations. We use an exact self-similar solution to confirm and generalise a previous finding that, in sharp contrast to two-dimensional solutions in standard MHD, a finite-time collapse to a current sheet can occur in Hall MHD. We derive a criterion for the finite-time singularity in terms of initial conditions, and we use an intermediate asymptotic solution for the evolution of an axial magnetic field to obtain a general expression for the singularity formation time. We illustrate the analytical results by numerical solutions.     

On the equations for p-adic closed and open strings

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems. The text was submitted by the author in English.     

一个有快慢进展的TB模型的全局稳定性分析

建立了一个有快慢进展、接种和治疗的TB模型,定义了模型的基本再生数R0,通过构造Lyapunov函数来研究解的渐近性态.证明了当R01时,无病平衡点是全局渐近稳定的;也证明了当R0>1时,惟一的地方病平衡点是全局渐近稳定的.     

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach themselves to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We use large-system asymptotics to characterize system performance, and augment this with simulations, which suggest that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.     

Compton scattering in intense magnetic fields

Compton scattering in intense magnetic fields in the general frame of reference is studied with the help of the QED perturbation theory in the incoming interaction picture. A general expression for the cross section is derived which reduces naturally to the one in the electron-rest frame of reference. This expression can be approximately simplified for the scattering of a high-energy electron with a low-frequency photon. Based on this simplified expreaaion, spectrum functions, as well as power spectra of scattered photons with high energies resulting from the inverse Compton scattering are calculated which manifest clearly a feature of resonances. Project supported by the National Natural Science Foundation of China (Grant No. 19573008) and the Science Research Division of Shanghai Jiaotong University.     

A short note on groups in separably closed valued fields

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.