NS Condition of Admissibility for the Linear Estimator of Normal Mean with Unknown Variance

Suppose Y - N（β, σ^2 In）, where β ∈ R^n and σ^2 〉 0 are unknown. We study the admissibility of linear estimators of mean vector under a quadratic loss function. A necessary and sufficient condition of the admissible linear estimator is given.     

Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

Under simple conditions, we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem. For the resonance case at infinity, we do not need to assume any more conditions to ensure the boundedness of the （PS） sequence of the corresponding functional.     

Existence of Multiple Positive Solutions of Inhomogeneous Semi–Linear Elliptic Problems Involving Critical Exponents ∗

We consider the obstacle problem for the degenerate Monge-Ampére equation. We prove the existence of the greatest viscosity sub-solution u(x) below a given obstacle φ(x), and its C ^{1, 1}-regularity which is optimal. Then the solution satisfies the concave uniformly elliptic equation if it doesn't touch the obstacle. We use the author's previous work to show the C ^{1, α}-regularity of the free boundary, ?{u(x) = φ(x)}. Finally, we discuss the stability of this free boundary.     

Memetic Algorithms for Multiple Interference Cancellations of Linear Array Based on Phase-Amplitude Perturbations

A novel method based on the memetic algorithm for the design of multiple interference cancellations of a linear array antenna by phase-amplitude perturbations is proposed. The adaptive array antenna is capable of sensing the presence of interference sources and suppressing the interferences in the interfering directions. This technique can increase the signal-to-interference ratio. The memetic algorithm is applied to find the weighting vector which makes the pattern nulling optimization of the proposed adaptive antenna. This technique is also able to do the cancellation of multiple interferences for different incident directions.     

Families of Auslander–Reiten components for simply connected differential graded algebras

Peter Jørgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form ${{\mathbb {Z}}A_\infty}Peter J?rgensen introduced the Auslander–Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form \mathbb ZA_￥{{\mathbb {Z}}A_\infty} and that the Auslander–Reiten quiver of a d-dimensional sphere consists of d − 1 such components. We show that this is essentially the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter family for all n.     

Multiple Solutions for Systems of Sturm–Liouville Boundary Value Problems

In this paper, the authors discuss the existence of multiple solutions to a class of second-order Sturm–Liouville boundary value systems. Their proofs are based on variational methods and critical point theory.     

Normal Form for Families of Hamiltonian Systems

We consider perturbations of integrable Hamiltonian systems in the neighborhood of normally parabolic invariant tori. Using the techniques of KAM-theory we prove that there exists a canonical transformation that puts the Hamiltonian in normal form up to a remainder of weighted order 2d ＋ 1. And some dynamical consequences are obtained.     

Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

Denote by γ the Gauss measure on ℝ ⁿ and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} to L ¹γ. This result is in sharp contrast both with the fact that (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H ¹γ to L ¹γ, where H ¹γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case (rI-D)^iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space \mathfrakh¹\mathbbRⁿ{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L ¹ℝ ⁿ . We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L ² μ, and with a kernel satisfying certain analytic assumptions, is bounded from H ¹ μ to L ¹ μ if and only if it is bounded from \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} to L ¹ μ. Here H ¹ μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered.     

Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems

In this paper, we study the generalized Douglas–Rachford algorithm and its cyclic variants which include many projection-type methods such as the classical Douglas–Rachford algorithm and the alternating projection algorithm. Specifically, we establish several local linear convergence results for the algorithm in solving feasibility problems with finitely many closed possibly nonconvex sets under different assumptions. Our findings not only relax some regularity conditions but also improve linear convergence rates in the literature. In the presence of convexity, the linear convergence is global.     

Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.     

Crank–Nicolson Scheme for Abstract Linear Systems

This paper studies the Crank–Nicolson discretization scheme for abstract differential equations on a general Banach space. We show that a time-varying discretization of a bounded analytic C₀-semigroup leads to a bounded discrete-time system. On Hilbert spaces, this result can be extended to all bounded C₀-semigroups for which the inverse generator generates a bounded C₀-semigroup. The presentation is based on C₀-semigroup theory and uses a functional analysis approach.     

Criteria of stability for a Class of Linear Systems

In this paper,employing the geometric criteria of stabilty we obtain some stable results of linear differential equations with impulsive effects.     

Linear convergence of Frank–Wolfe for rank-one matrix recovery without strong convexity

Mathematical Programming - We consider convex optimization problems which are widely used as convex relaxations for low-rank matrix recovery problems. In particular, in several important problems,...     

Input–Output Identifiability of Continuous-Time Linear Systems

In time-domain identification of linear systems the aim is to estimate the impulse response or transfer function of a linear system to within a given tolerance using a finite number of noisy observations of the output. Whether this is possible depends on the model set, that is, a given set to which the system is assumed to belong a priori. We give necessary and sufficient conditions on the model set to ensure that such identification is possible in the continuous-time case.     

Analysis of a Linear Sampling Method for Identification of Obstacles

Some difficulties are pointed out in the methods for identification of obstacles based on the numerical verification of tile inclusion of a function in the range of an operator. Numerical examples are given to illustrate theoretical conclusions. Alternative methods of identification of obstacles are mentioned： the Support Function Method （SFM） and the Modified Rayleigh Conjecture （MRC） method.