Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H ¹(Ω) × L ²(?Ω) with boundary data from L ²(?Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H ¹(Ω) into L ²(Ω) or L ²(?Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.     

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.     

Propagating reaction fronts in an ionic autocatalytic system: The effects of electric field intensity and ionic strength

An ionic chemical system which has two components, A⁺ and B⁺, reacting with a quadratic autocatalytic rate law and two further, nonreacting components C⁻ and D⁺ is considered. The system is shown to be characterized by the dimensionless parameters δ_B, I and d₀ representing the ratio of the diffusion coefficients of B⁺ and A⁺, the (constant) applied electric current and the ionic strength of the system (the initial concentration of D⁺), respectively. The travelling wave equations for this system are discussed first, and parameter ranges identified where forward-propagating kinetic waves can exist. These are then seen, from numerical integrations of the original initial-value problem, to be the long time structures, when this is the case. When this is not the case, qualitatively different behaviour is observed depending on whether δ_b > 1 or δ_B <1, the form of which is dependent on the size and direction of the applied electric field as well as on the ionic strength. This is described in detail and shown to agree with previous studies for d₀ = 0 and do large.     

Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems

We obtain a finite-dimensional Perron effect of change of values λ ₁ ≤ … ≤ λ _n < 0 of all arbitrarily specified negative characteristic exponents of the n-dimensional system of linear approximation with infinitely differentiable bounded coefficients to arbitrarily specified, arranged in ascending order, values β _k ≥ λ _k , k = 1, …, n, of characteristic exponents of all nontrivial solutions of an n-dimensional nonlinear differential system with an infinitely differentiable perturbation of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. Each value β _k is realized by all nontrivial solutions of the perturbed system issuing from the difference R ^k |R ^k?1 of embedded subspaces R ¹ ? R ² ? … ? R ⁿ .     

Some new researches on the general connectivity index of benzenoid systems

Given H a benzenoid system, we find an expression of the general connectivity index, denoted by ?? _?? (H), in terms of inlet features and internal vertex features of H. As a consequence, the extremal n-benzenoid systems with maximal general connectivity index ?? _?? are completely characterized. Moreover, by constructing a combinatorial bijection, we prove that the linear chain L _n is the unique extremal n-benzenoid system with minimal general connectivity index ?? _?? if and only if ??>?? ^?, where ?? ^? is the positive root of the equation 2×6 ^x ?9 ^x =0.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R ¹ ? R ² ? ... ? R ⁿ .     

Local controllability and non-controllability for a 1D wave equation with bilinear control

We consider a linear wave equation, on the interval (0,1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.

•

For every T>2, this system is locally controllable in H³×H², in time T, with controls in L²((0,T),R).

•

For T=2, this system is locally controllable up to codimension one in H³×H², in time T, with controls in L²((0,T),R): the reachable set is (locally) a non-flat submanifold of H³×H² with codimension one.

•

For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L²((0,T),R), is contained in a non-flat submanifold of H³×H², with infinite codimension.

The proof of these results relies on the inverse mapping theorem and second order expansions.     

On the exponential dichotomy of linear difference equations

We consider a system of linear difference equationsx ⁿ⁺¹ =A (n)xⁿ in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln εZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k εZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.     

On the multiplicative completion of centered systems

In their consideration of the completeness phenomenon in L²(a,b), Boas and Pollard have shown that a cofinite subset of a complete system can be transformed into a complete system by multiplying each member of the system by a fixed, bounded measurable function. Following a trail blazed by Braun and extensively developed by Kazarian, one can show that some systems, complete in L^p(E), can be transformed, à la Boas and Pollard, in such a manner that the new system is, in fact, a quasibasis for L^p(E). Here it is shown that certain families of centered functions can be transformed in this manner.     

Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms

A class of cubic Hamiltonion system with the higher-order perturbed term of degree n=5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C¹₉⊃2[C²₃⊃2C²₂] (let C_m^k denote a nest of limit cycles which encloses m singular points, and the symbol `⊂' is used to show the enclosing relations between limit cycles, while the sign `+' is used to divide limit cycles enclosing different critical points. Denote simply C_m^k+C_m^k=2C_m^k, etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem.     

On the stability of invariant sets of leaves of three-dimensional periodic systems

The small C ¹ perturbations of differential equations are studied. The concepts of a weakly hyperbolic set K and a leaf ? are introduced for a system of ordinary differential equations. The Lipschitz condition is not supposed. It is shown that, if the perturbation is small enough, then there exists a continuous mapping h: ? → ? ^Y , where ? ^Y is a leaf of the perturbed system.     

Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter ε, τ=o(ε), ε→0. We establish the following general framework: If there exists an a priori L^∞ bound that is uniformly with respect to ε for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L^∞ bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in L^p to the equilibrium solutions of these systems is established, provided that the relaxation time τ tends to zero faster than ε.     

Large time behavior of entropy solutions to some hyperbolic system with dissipative structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t →∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.     

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.     

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.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Minimum Covering for Hexagon Triple Systems

A hexagon triple is a graph consisting of three triangles of the form (a, x, b), (b, y, c), and (c,z,a), where a, b, c, x, y, z are distinct. The triangle (a, b, c) is called the inside triangle and the triangles (a, x, b), (b,y,c), and (c, z, a) are called outside triangles. A 3k-fold hexagon triple system of order n is a pair (X, H), where H is an edge-disjoint collection of hexagon triples which partitions the edge set of 3kK ⁿ with vertex set X. Note that the outside triangles form a 3k-fold triple system. If the 3k-fold hexagon triple system (X, H) has the additional property that the inside triangles form a k-fold triple system, then (X, H) is said to be perfect. A covering of 3kK ⁿ with hexagon triples is a triple (X, H, P) such that: 1.3kK ⁿ has vertex set X. 2.P is a subset of E(λ K ⁿ ) with vertex set X for some λ, and 3.H is an edge disjoint partition of E(3kK ⁿ )∪ P with hexagon triples. If P is as small as possible (X, H, P) is called a minimum covering of 3kK ⁿ with hexagon triples. If the inside triangles of the hexagon triples in H form a minimum covering of kK ⁿ with triangles, the covering is said to be perfect. A complete solution for the problem of constructing perfect 3k-fold hexagon triple system and perfect maximum packing of 3kK ⁿ with hexagon triples was given recently by the authors [2]. In this work, we give a complete solution of the problem of constructing perfect minimum covering of 3kK ⁿ with hexagon triples.     

The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system

The blow-up solutions of the Cauchy problem for the Davey-Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo-Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey-Stewartson system is developed in R³. Thirdly, the mass concentration is established for all the blow-up solutions of the system in R². Finally, the existence of the minimal blow-up solutions in R² is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t→T (blow-up time) is in detail investigated in terms of the ground state.