Approximation of SDE solutions using local asymptotic expansions

We develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

Analysis of two components parallel repairable degenerate system with vacation

<abstract><p>This article studies a parallel repairable degradation system with two similar components and a repairman who can take a single vacation. Suppose that the system consists of two components that cannot be repaired "as good as new" after failures; when the repairman has a single vacation, the fault component of system may not be repaired immediately, namely, if a component fails and the repairman is on vacation, the repair of the component will be delayed, if a component fails and the repairman is on duty, the fault component can be repaired immediately. Under these assumptions, a replacement policy $ N $ based on the failed times of component 1 is studied. The explicit expression of the system average cost rate $ C(N) $ and the optimal replacement policy $ N^{\ast} $ by minimizing the $ C(N) $ are obtained, which means the two components of the system will be replaced at the same time if the failures of component 1 reach $ N^{\ast} $. To show the advantage of a parallel system, a replacement policy $ N $ of the cold standby system consisting of the two similar components is also considered. The numerical results of both systems are given by the numerical analysis. The optimal replacement policy $ N^* $ for both systems are obtained. Finally, the comparison of numerical results shows the advantages of the parallel system.</p></abstract>

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper, stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied, and we use distribution delay to simulate the delay in nutrient conversion. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. First, we state that this model has a unique global positive solution for any initial value, which is helpful to explore its stochastic properties. Furthermore, we prove the stochastic ultimate boundness of the solution of system. Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established, which means the microorganism will be extinct. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.

Null controllability of a coupled degenerate system with the first and zero order terms by a single control

This paper concerns the null controllability of a system of m linear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subset ω of Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system.

A magnetic white dwarf with five H α components

ABSTRACT G183−35 is an unusual white dwarf that shows an H α line split into five components, instead of the usual three components seen in strongly magnetic white dwarfs. Potential explanations for the unusual set of lines include a double degenerate system containing two magnetic white dwarfs and/or rotational modulation of a complex magnetic field structure. Here, we present time-resolved spectroscopy of G183−35 obtained at the Gemini Observatory. These data reveal two sets of absorption lines that appear and disappear over a period of about 4 h. We also detect low-level (0.2 per cent) variability in optical photometry at the same period. We demonstrate that the spectroscopic and photometric variability can be explained by the presence of spots on the surface of the white dwarf and a change in the average field strength from about 4.6 to 6.2 MG. The observed variability is clearly due to G183−35’s relatively short spin period. However, rotational modulation of a complex magnetic field by itself cannot explain the changes seen in the central H α component. An additional source of variability in the line profiles, most likely due to a chemically inhomogeneous surface composition, is also needed. We propose further observations of similar objects to test this scenario.

Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

The Regularity of Solution for Weakly Coupled System Derived by Microwave Heating Model

In this paper, we study the regularity of the weak solution of the coupled system derived from the microwave heating model with frequency variable. We first show that the weak solution E of the system is Hölder continuous near the boundary of S = ∂ Ω . The main idea of the proof is based on the estimation of linear degenerate system in Campanato space. Then we show that the solution u of the heat conduction equation is Hölder continuous with exponent α 2 . Finally, under the appropriate conditions we show that the coupled system with microwave heating has a weak solution. Moreover the regularity of the weak solution is studied.

2-factor Newton method for solving the constrained optimization problem with the singular Kuhn—Tucker system

A new method for solving the inequality constrained optimization problem is proposed for the case when the system of necessary optimality conditions of Kuhn—Tucker is degenerate. This situation occurs for example in the case when strict complementarity conditions fails in solution point. The reduction of the inequalities con- strained optimization problem to the equalities constrained problem is substantiated and the use of a new 2-fac- tor Newton method for the effective solution of the obtained degenerate system of optimality conditions is shown.

Między stronami mocy. Przyczynek do analizy ustrojowej uniwersów fantastycznych

BETWEEN TWO SIDES OF POWER: A CONTRIBUTION TO CONSTITUTIONAL ANALYSIS OF SCIENCE-FICTION UNIVERSAIn popular parlance the Star Wars universe often serves as an example of a binary division between good Rebels and evil Empire. However, a detailed legal analysis of the turbulent political and legal history of the good old Republic and its transformation into the Empire casts doubts on this popular opinion.The Republic seems to be a degenerate system, based on exploitation of the weak, slavery and dominance of the military order the Jedi, exercising power without any democratic control. Surprisingly, the transformation of the Republic into the Empire was formally admissible, and backed up by republican constitutional principles. Moreover, it has been purported here that the political system of the galaxy had very strong feudal relics and allowed both for vendettas and the right to rebel against the goverment. The Rebellion was in fact a counterrevolutionary movement whose main goal was to re-establish the ancient regime and anihilate the last two representatives of the schismatic Jedi sect the Siths, while the Empire was trying to establish a ruthless, but effective system of government. Thus, what we have here is not a battle of Good against Evil but simply a civil war between conservative terrorists and authoritarian reformers. Surprisingly, a short-lived victory of the Rebellion leads to a social and economic crisis, while the restoration of the Empire by the New Order guarantees stability of the economic and political system. Moreover, imperial feuds and vendettas impact only the major players, while the commoners are not directly affected.