Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Origins of oscillation criteria of operator differential equations in Hilbert space

It is shown that, for a nonempty ergodic family {T_t: t?J} of continuous linear operators on a topological vector space, the solution by iteration of the simultaneous linear functional equations x ? T_t(x) = h, t?J, is a special case of mean ergodic theory for a semigroup of linear operators.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

The hilbert series of the polynomial identities for the tensor square of the grassmann algebra

LetE be the Grassmann (or exterior) algebra of an infinite-dimensional vector space over a fieldK of characteristic 0. We compute the Hilbert series of the relatively free algebraF _m(varE?_K E)=K(x _l,...,x _m) /(T(E?_K E)∩K〈x _l,...,x _m〉) of the variety of algebrasvarE?_KE, whereT(E?_K E) is the set of all polynomial identities forE?_K EE from the free associative algebraK〈x ₁,x₂…〉.     

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

A pointwise abelian Ergodic theorem for Lp semigroups, 1 ⩽ p < ∞

Let (X, ∑, μ) be a σ-finite measure space and L_p(μ) = L_p(X, ∑, μ), 1 ? p ? ∞, the usual Banach spaces of complex-valued functions. Let {T_t: t ? 0} be a strongly continuous semigroup of positive L_p(μ) operators for some 1 ? p < ∞. Denote by R_λ the resolvent of {T_t}. We show that f?L_p(μ) implies λR_λf(x) → f(x) a.e. as λ → ∞.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=x_a, x'(a)=x'_a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies e_k=0(h²). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch² and e'(t)?C₁h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that e^R_k=0(h⁴). Thus our results are comparable to Runge-Kutta with half the function evaluations.     

Initially and eventually dominating nonoscillatory solutions in integro-differential difference equations

For the system T′(t) + p(t) T(t) + q(t) T(t ? τ) = ∝₀^tK(t ? μ) T(μ) dμ for t ? 0, T(t) = g(t) for t ∈ [?τ, 0], conditions have been obtained which ensure that a solution of this system is dominated by a nonoscillatory solution in the interval ¦τ, ∞).     

A direct numerical procedure for the Cauchy problem for the heat equation

For the Cauchy problem, u_t = u_xx, 0 < x < 1, 0 < t ? T, u(0, t) = f(t), 0 < t ? T, u_x(0, t) = g(t), 0 < t ? T, a direct numerical procedure involving the elementary solution of υ_t = υ_xx, 0 < x, 0 < t ? T, υ_x(0, t) = g(t), 0 < t ? T, υ(x, 0) = 0, 0 < x and a Taylor's series computed from f(t) ? υ(0, t) is studied. Continuous dependence better than any power of logarithmic is obtained. Some numerical results are presented.     

On oscillation of solutions of higher order forced functional differential inequalities

Oscillation criteria for the class of forced functional differential inequalities x(t){L_nx(t) + f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n even, and x(t){L_nx(t) ? f(t, x(t), x[g₁(t)],…, x[g_m(t)]) ? h(t)} ? 0, for n odd, are established.     

A combinatorial interpretation for two transformations of series that commute with compositional inversion

For a formal power series $\text{g(t) =}\text{1}\text{[1 + ∑}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{h}{}_{\mathrm{n}}\text{t}{}^{\mathrm{n}}\text{]}$ with nonnegative integer coefficients, the compositional inverse $\text{f}\text{(t) = t · f(t)}$ of $\text{g}\text{(t) = t · g(t)}$ is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of h_i colors. Since the compositional inverse of the Euler transformation of f(t) is the star transformation [[g(t)]^?1 ? 1]^?1 of g(t), [2], it follows that the Euler transformation of f(t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of h_i colors for i > 1, and h₁ ? 1 colors for i = 1.     

Numerical solutions with a priori error bounds for coupled self-adjoint time dependent partial differential systems

This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) u_t)_t = Q(t)u_xx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), u_t(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oR^r×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied.     

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

We introduce the notion of ?-unique bounded solution to the nonlinear differential equation x′ = f(x) ? h(t), where f: ? → ? is a continuous function and h(t) is an arbitrary continuous function bounded on ?. We derive necessary and sufficient conditions for the existence and ?-uniqueness of bounded solutions to this equation.     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)_k?1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)_k?1 of E(M), we consider the Rademacher averages k_∑εk⊗xk as elements of the noncommutative function space and study estimates for their norms ‖k_∑εk⊗xkE_‖ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖k_∑εk⊗xkE_‖ is equivalent to the infimum of over all yk, zk in E(M) such that xk=yk+zk for any k?1. Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖k_∑εk⊗xkE_‖ is equivalent to . We also study Rademacher averages _∑i,jεi⊗εj⊗xij for doubly indexed families (xij)_i,j of E(M).