L q theory for a generalized Stokes System

Regularity properties of solutions to the stationary generalized Stokes system are studied. The extra stress tensor is assumed to have a growth given by some N-function, which includes the situation of p-growth. We show results about differentiability of weak solutions. As a consequence we obtain the gradient L ^q estimates for the problem. These estimates are applied to the stationary generalized Navier Stokes equations.     

Analysis of a parabolic cross-diffusion population model without self-diffusion

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H¹ estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler-Galerkin approximation, discrete entropy estimates, and L¹ weak compactness arguments. Furthermore, employing the entropy-entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.     

Global existence for systems of parabolic conservation laws in several space variables

We prove the global existence of solutions of the Cauchy problem for certain systems of conservation laws with artificial viscosity terms added. The system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix, and the initial data is assumed to be close to a constant in L² ∩ L^∞. In particular, our result applies to the equations of compressible fluid flow in two and three space variables.     

On the stability of sheet invariant sets of two-dimensional periodic systems

In the paper small C ¹-perturbations of differential equations are considered. The concepts of a weakly hyperbolic set K and a sheet ? for a system of ordinary differential equation are introduced. Lipschitz property is not assumed to hold. It is shown that if the perturbation is small enough, then there is a continuous mapping h: ? → ? ^Y , where ? ^Y is a sheet of the perturbed system.     

Construction and Strong Feller Property of Distorted Elliptic Diffusion with Reflecting Boundary

Using Dirichlet form techniques we prove existence of a diffusion process with singular drift on an open domain Ω???? ^d , d?∈??, d?≥?2, and generalized reflection at the boundary. The boundary is assumed to be C ²-smooth except for a sufficiently small set. We prove an elliptic regularity result which gives L ^p -strong Feller property for the semigroup and resolvent for p?≥?2 with $p > \frac{d}{2}$ . This result allows us to construct the process pointwisely except for an explicitly known set. For starting points outside this known set the process solves the corresponding martingale problem. The results are applied to prove the existence of the stochastic dynamics of a finite interacting particle system and for the Ginzburg-Landau interface model with a hard wall.     

Stability of compromise solution in multicriteria decision-making problems

Stability is a desirable attribute of the nondominated solution in the decision-making problem with multiple noncommensurable objectives. It is assumed that the decision space is given by a system of inequalitiesG(x)≤b, b ∈ R ^N . In this paper, sufficient conditions are determined for a compromise solution to be stable with respect to the changes ofb.     

Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method

This paper deals with the control policy of a removable and unreliable server for an M/M/1/K queueing system, where the removable server operates an F-policy. The so-called F-policy means that when the number of customers in the system reaches its capacity K (i.e. the system becomes full), the system will not accept any incoming customers until the queue length decreases to a certain threshold value F. At that time, the server initiates an exponential startup time with parameter γ and starts allowing customers entering the system. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. A matrix analytical method is applied to derive the steady-state probabilities through which various system performance measures can be obtained. A cost model is constructed to determine the optimal values, say (F^∗, μ^∗, γ^∗), that yield the minimum cost. Finally, we use the two methods, namely, the direct search method and the Newton-Quasi method to find the global minimum (F^∗, μ^∗, γ^∗). Numerical results are also provided under optimal operating conditions.     

Matrice de connexion d'un système aux q-différences confluant vers un système différentiel et matrices de monodromie

We built in [6] a version with elliptic coefficients of the Birkhoff connection matrix and showed its role in the classification of regular singular q-difference systems. We give here a geometrical interpretation: when q tends to 1, P tends to a locally constant matrix P such that the (finitely many) values (a)⁻¹ (b) are the monodromy matrices of the limiting differential system (assumed to be non-resonant at 0 and ∞) at the singularities on C^*.     

Oscillation of solutions of second-order nonlinear differential equations of Euler type

We consider the nonlinear Euler differential equation t²x^″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t²x^″+a(t)g(x)=0.     

The space clcv(? n ) with the Hausdorff-Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is 1, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space clcv(? ⁿ ) of nonempty closed convex subsets of ? ⁿ with the Hausdorff-Bebutov metric.     

On the closure in W1q of the attainable subspace of linear time lag systems

This paper presents the necessary and sufficient condition for the closedness in the topology of W₁^q(T ? 1, T; Rⁿ) of the set of all complete final states x_{[T ? 1,T]} of a linear retarded system, which can be attained by L^p controls. The criterion is fully computable. The analyticity of the matrix coefficients of the system equation is assumed. A number of related results are presented, including a criterion for the closedness of the set of all trajectories of a nondelayed linear system with measurable coefficients.     

Pullback attractors for a semilinear heat equation on time-varying domains

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains Ot in RN are obtained from a bounded base domain O by a C²-diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

On Formal Maps Between Generic Submanifolds in Complex Space

Let H:(M,p)→(M ^′,p ^′) be a formal mapping between two germs of real-analytic generic submanifolds in ? ^N with nonvanishing Jacobian. Assuming M to be minimal at p and M ^′ holomorphically nondegenerate at p ^′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ^′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.     

Noncommutative topological dynamics and compact actions on C1-algebras

The classical notions of topological transitivity and minimality of a topological dynamical system are extended and analyzed in the context of C¹-dynamical systems. These notions are compared with other notions naturally arising in noncommutative ergodic theory. As an application, a C¹-algebra version of a theorem of Araki, Haag, Kastler, and Takesaki (Comm. Math. Phys.53 (1977), 97–134) about the correspondence between a compact automorphism group (here assumed to be abelian) and its fixed-point subalgebra is proved in the presence of a commuting topologically transitive action. A variation of this theorem in the setting of standard W¹-inclusions is also presented.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

Limit distributions of the χ 2 statistic of K. Pearson in a sequence of independent trials

We study the χ ² statistic of K. Pearson in a sequence of independent and, generally, inhomogeneous trials with a fixed number of outcomes. It is assumed that the probabilities of occurrence of outcomes of the trials satisfy certain conditions. This problem statement embraces familiar results for the χ ² statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the χ ² statistic. For the χ ² statistic centered and normalized in a suitable way, we find limit distributions (the normal one, the distribution of the sum of the squares of normal random variables and, in particular, the χ ² distribution). Conditions for the convergence to the corresponding limit distributions are given.     

Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

In this paper, we consider the second order Hamiltonian system $\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= ?d ²/dt ² ?A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.     

Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound ??(B(x,r))??Cr ^d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.