Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EX^t < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [?^?1 EX^t]^1?t. We show that there exist positive ?₁ ≥ ?₂ such that for all 0 <?≤?₁ the function g(t) = [?^-1EX^t]^1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?₂ the function log g(t) is nonincreasing in [0, c].     

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, ∞).     

Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

We assume that Ω^t is a domain in ?³, arbitrarily (but continuously) varying for 0?t?T. We impose no conditions on smoothness or shape of Ω^t. We prove the global in time existence of a weak solution of the Navier–Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q_{[0, T)} := {( x , t);0?t?T, x ∈Ω^t}. The solution satisfies the energy‐type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd.     

On the problem of determining the parameters of a layered elastic medium and an impulse source

Considering the linear system of elasticity equations describing the wave propagation in the half-space ? ₊ ³ = {x ∈ ?³ | x ₃ > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x ₃. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x ₃ ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.     

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.     

Optimal control of longitudinal vibrations of composite rods with the same wave propagation time in each part

We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave propagation time through each part of the rod is the same and T is a multiple of that time.     

On the order of convergence of a semigroup to the identity operator

We describe classes of vectors f from a Hilbert space H for which the quantity ‖T(t)f?f‖, where T(t)=e ^?tA , t≥0, and A is a self-adjoint nonnegative operator in H, has a certain order of convergence to zero as t→+0.     

On elliptic modular foliations

In this article we consider the three parameter family of elliptic curves E^t: y² − 4(x − t₁ )³ + t² (x − t₁) + t₃ = 0, t ∈ ?³, and study the modular holomorphic foliation ?ω, in ?³ whose leaves are constant locus of the integration of a l-form ω over topological cycles of E_t. Using the Gauss—Manin connection of the family E^t, we show that ?ω is an algebraic foliation. In the case , we prove that a transcendent leaf of ?ω contains at most one point with algebraic coordinates and the leaves of ?ω corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family E^t, we find a uniformization of ?ω in T, where T ⊂ ?³ is the locus of parameters t for which E_t is smooth. We find also a real first integral of ?ω. restricted to T and show that ?ω is given by the Ramanujan relations between the Eisenstein series.     

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.     

A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

For positive constants a > b > 0, let P _T (t) denote the lattice point discrepancy of the body tT _a,b , where t is a large real parameter and T = T _a,b is bounded by the surface $$ \partial \tau _{a,b} :\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {(a + b\cos \alpha )\cos \beta } \\ {(a + b\cos \alpha )\sin \beta } \\ {b\sin \alpha } \\ \end{array} } \right), 0 \leqq \alpha ,\beta < 2\pi . $$ In a previous paper [12] it has been proved that $$ P_\tau (t) = \mathcal{F}_{a,b} (t)t^{3/2} + \Delta _\tau (t), $$ where F _a,b (t) is an explicit continuous periodic function, and the remainder satisfies the (“pointwise”) estimate Δ _T (t) ? t ^11/8+? . Here it will be shown that this error term is only ? t ^1+? in mean-square, i.e., that $$ \int\limits_0^T {(\Delta _\tau (t))^2 dt} \ll T^{3 + \varepsilon } $$ for any ? > 0.     

Padé-type approximants with orthogonal generating polynomials

The interpolation of the function x → 1/(1 ? xt) generating the series f(t) = ∑^∞_{i = 0}c_itⁱ at the zeros of an orthogonal polynomial with respect to a distribution d α satisfying some conditions will give us a process for accelerating the convergence of f_n(t) = ∑ⁿ_{i = 0}c_itⁱ. Then, we shall see that the polynomial of best approximation of x → 1/(1 ? xt) over some interval or its development in Chebyshev polynomials T_n or U_n are only particular cases of the main theorem.At last, we shall show that all these processes accelerate linear combinations with positive coefficients of totally monotonic and oscillating sequences.     

Harmonic analysis and asymptotic behavior of solutions to the abstract cauchy problem

LetC _ub ( $\mathbb{J}$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $\mathbb{J}$ 2 ε {?⁺, ?} with values in a Banach spaceX. Let ? be a class fromC _ub( $\mathbb{J}$ ,X). We introduce a spectrumsp?(φ) of a functionφ εC _ub (?,X) with respect to ?. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ? to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ ε ?, whereA is the generator of aC ₀-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩i? is countable, all bounded uniformly continuous mild solutions on ?⁺ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ?⁺ in the cases (i)T(t) is a uniformly exponentially stableC ₀-semigroup andφ εC _ub(?,X); (ii)T(t) is a uniformly bounded analyticC ₀-semigroup,φ εC _ub (?,X) andσ(A) ∩i sp(φ) = Ø. Under the condition (i) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), then the solutions belong to ?. In case (ii) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), andT(t) is almost periodic, then the solutions belong to ?. The existence of mild solutions on ? to (*) is also discussed.     

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

Optimal policies for inventory models with some specified markets and finite time horizon

The paper presents the deterministic finite time horizon inventory lot size model, without backlogs and with no lead time, for a single commodity, with some specified markets. The specified markets are represented by the family b(t)=kt^r of demand functions where k>0, r>?2 are known parameters and t stands for time, 0<t₀?t?T. The strict positivity of t₀. compared to the restrictive condition t₀=0 which has been already solved, is crucial and implies entirely different analytical techniques. An important special case is the affine function (r = 1) partly treated already by Donaldson [3]. The problem is to find the optimal schedule of replenishments, i.e., the number and timings of orders.The problem is completely resolved (compared to a recent heuristic by Silver [8]) and the solution is given in a closed form and is proven to be unique. Numerical examples are provided.     

Parameter-dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial-value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = T^ρ A(t)T^1?ρ + T^ρ B(t)T^1?ρ R(t) + R(t)T^ρC(t) T^1?ρ + R(t)T^ρD(t)T^1?ρ R(t), R(0)=R₀, where 0 ? ρ ? 1 and where the functions R and A through D take on values in the cone of non-negative bounded linear operators on L¹ (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.     

Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces

Let D be nonempty open convex subset of a real Banach space E. Let be a continuous pseudocontractive mapping satisfying the weakly inward condition and let be fixed. Then for each t∈(0,1) there exists satisfying yt∈tTyt+(1−t)u. If, in addition, E is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if {yt} remains bounded as t→1; in this case, {yt} converges strongly to a fixed point of T as t→¹⁻. Moreover, an explicit iteration process which converges strongly to a fixed point of T is constructed in the case that T is also Lipschitzian.     

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.