The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

The initial and boundary value problem for the degenerate parabolic equation v_t = Δ(?(v)) + F(v) in the cylinder $\text{Ω × ¦0, ∞), Ω ? R}{}^{\mathrm{n}}$ bounded, for a certain class of point functions ? satisfying ?′(v) ? 0 (e.g., $\text{?(v) = ¦v¦}{}^{\mathrm{m}}\text{sign}\text{v}$) is considered. In the case that F(v) sign $\text{v ? C(1 + ¦?(v)¦}{}^{\mathrm{\alpha }}\text{), α < 1}$, the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that $\text{F(v)}\text{?(v)}$ is nondecreasing a condition is given on the initial state v(x, 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

Point processes of exits by bivariate Gaussian processes and extremal theory for the χ2-process and its concomitants

Let ζ(t), η(t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that ζ(t) and η(t) are independent for all t, and consider the movements of a particle with time-varying coordinates (ζ(t), η(t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R³ with its points located on the cylinder {(t, u cos θ, u sin θ); t ≥ 0, 0 ≤ θ < 2π}. It is shown that if r(t) log t → 0 as t → ∞, the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a χ²-process, χ²(t) = ζ²(t) + η²(t), P{sup_0≤t≤Tχ²(t) ≤ u²} → e^?τ if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u ×}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞. Furthermore, it is shown that the points in R³ generated by the local ?-maxima of χ²(t) converges to a Poisson process in R³ with intensity measure (in cylindrical polar coordinates) (2πr²)^?1dtdθdr. As a consequence one obtains the asymptotic extremal distribution for any function g(ζ(t), η(t)) which is “almost quadratic” in the sense that $\text{g}{}^1\text{(r}\text{cos}\text{θ, r}\text{sin}\text{θ) =}\text{1}\text{2}\text{(r}{}^2\text{? g(r}\text{cos}\text{θ, r}\text{sin}\text{θ))}$ has a limit $\text{g}{}^1\text{(θ)}$ as r → ∞. Then if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

A convergent two-time method for periodic differential equations

Two timing, an ad hoc method for studying periodic evolution equations, can be given a rigorous justification when the problem is in standard form, u = ?f(t, u). First solve $\text{dw}\text{dσ}\text{= ?(I ? M) f(σ, w)}\text{for}\text{w(σ, v)}$, where M is the mean value operator and v is any initial value. Then w(σ, v) is periodic in σ but does not satisfy the original equation. Now, force a solution u(t), using nonlinear variation of constants, in the form w(σ, v(τ)), where σ = t is the fast time and τ = ?t is the slow time. With the resulting differential equation for v, one reads off from its nonconstant solutions thè approximate transient behavior of u(t) for times of order ?^?1. On the other hand, the equilibrium points (constant solutions) v₀ correspond to steady state (periodic solutions) of the original system. Interesting applications, such as to one-dimensional wave equations with cubic damping, can be given.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

Uniqueness and continuous dependence results for solutions of singular differential equations

Solutions of Cauchy problems for the singular equations $\text{u}{}_{\mathrm{tt}}\text{+ (}\text{Ψ(t)}\text{t}\text{) u}{}_{\mathrm{t}}\text{= Mu}$ (in a Hilbert space setting) and $\text{u}{}_{\mathrm{t}}\text{+ Δu +}\text{∑}\text{m}\text{i=1}\text{((}\text{k}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)(}\text{?}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{)) + g(t)u=0}\text{in}\text{ω × |0,T)}$, ω={(x₁,…,x_M)ε$\text{R}$^m: 0 < x_i < c_i for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced.     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

L p -decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the L ^p (2 ? p ? +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (?(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w ₀(x), z ₀(x)) lies in (H ³ × H ²) (?) and |v ₊ ? v _?| + ∥w ₀∥₃ + ∥z ₀∥₂ is sufficiently small. Furthermore, the L ^p (2 ? p ? +∞) convergence rates of the solutions are also obtained.     

On the null-controllability of nonlinear delay systems with restrained controls

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{) + B(t) u(t) + f(t, x}{}_{\mathrm{t}}\text{, u(t))}$ when the values of the control functions u lie in an m-dimensional unit cube C^m of E^m. Conditions are placed on f which guarantee that if the uncontrolled system $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{)}$ is uniformly asymptotically stable and if the linear control system x(t) = L(t, x_t) + B(t) u(t) is proper, then (1) is null-controllable.