Perturbation Theorem Research Articles

Multiplicative Perturbations of ConvolutedC-Cosine Functions and ConvolutedC-Semigroups

We obtain the multiplicative perturbation theorems for convolutedC-cosine functions (resp., convolutedC-semigroups) andn-times integratedC-cosine functions (resp.,n-times integratedC-semigroups) forn∈ℕ. Moreover, we obtain some new results for perturbations onC-cosine functions (resp.,C-semigroups). Some examples are presented.

On Perturbation of ConvolutedC-Regularized Operator Families

Of concern are two classes of convolutedC-regularized operator families: convolutedC-cosine operator families and convolutedC-semigroups. We obtain new and general multiplicative and additive perturbation theorems for these convolutedC-regularized operator families. Two examples are given to illustrate our abstract results.

Fredholm perturbations and seminorms related to upper semi-Fredholm perturbations

In this paper, we investigate the stability in the set of Fredholm perturbations under composition with bounded operators. Moreover, we introduce the concept of a measure of non-Fredholm perturbations, which allows us to give a general approach to the question of obtaining perturbation theorems for semi-Fredholm operators. Finally, we prove some localization results about the Wolf, the Schechter and the Browder essential spectrum of bounded operators on a Banach space X.

Schrödinger type evolution equations with monotone nonlinearity of non-power type

Existence of unique strong solutions is established forSchrödinger type evolution equations withmonotone nonlinearity.The proof is based on a perturbation theorem for$m$-accretive operators in a complex Hilbert space.

Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces

We prove a Miyadera–Voigt type perturbation theorem for strong Feller semigroups. Using this result, we prove well-posedness of the semilinear stochastic equation on a separable Banach space E, assuming that F is bounded and measurable and that the associated linear equation, i.e. the equation with , is well-posed and its transition semigroup is strongly Feller and satisfies an appropriate gradient estimate. We also study existence and uniqueness of invariant measures for the associated transition semigroup.

Perturbations of invariant subspaces of unreduced Hessenberg matrices

Non-structured perturbation of invariant subspaces of unreduced, i.e. nonderogatory Hessenberg matrices is considered. Some perturbation results for the generalized eigenvectors and the characteristic polynomial of unreduced upper Hessenberg matrices are given. Two theorems are on the perturbation of invariant subspaces which are somewhat similar to the sinΘ theorems of Davis and Kahan apart from the residual, which we do not have here. Dense perturbations of unreduced Hessenberg matrices are also considered. Finally, we prove an invariant subspace perturbation theorem for nonderogatory matrices.

Output feedback nonlinear control for electro-hydraulic systems

In this paper we present an output feedback nonlinear control for position tracking of electro-hydraulic systems (EHSs). Although previous nonlinear control methods improved the position tracking performance of EHS, all of the methods require full state feedback. However, due to cost and space limitations, it is not always possible to measure the full state of the EHS. The proposed method consists of a high gain observer and a passivity-based controller. The high gain observer is designed to estimate the full state, and the passivity-based control is implemented for position tracking. In order to design the passivity-based controller with the high gain observer, a defined Lyapunov condition guarantee that the origin of the tacking error dynamics is exponentially stable by selecting the controller gain. The stability of the closed-loop is studied using the singular perturbation theorem. The performance of the proposed method is validated through simulations and experiments.

Jost solution and the spectral properties of the matrix-valued difference operators

In this paper, we find polynomial type Jost solution of the selfadjoint matrix-valued difference equation of second order. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem we prove that, the selfadjoint operator L generated by the matrix-valued difference expression of second order has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of L and prove that it has a finite number of simple real eigenvalues.

From exact stochastic to mean-field ODE models: a new approach to prove convergence results

In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

Thermalization in Nature and on a Quantum Computer

In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.

The rotation of eigenspaces of perturbed matrix pairs

We present new sinΘ theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. We assess the sharpness of the new estimates in terms of effectivity quotients (the quotient of the measure of the perturbation and the estimator). Known relative sinΘ theorems for eigenspaces of positive definite Hermitian matrices are included as special cases in our approach. Our experiments indicate that relative sinΘ theorems are asymptotically sharp when the rotation is measured in the appropriate matrix dependent scalar product but not always in the ordinary Euclidean scalar product.

Hitting time of quantum walks with perturbation

The hitting time is the required minimum time for a Markov chain-based walk (classical or quantum) to reach a target state in the state space. We investigate the effect of the perturbation on the hitting time of a quantum walk. We obtain an upper bound for the perturbed quantum walk hitting time by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on the definition of quantum hitting time given in MNRS algorithm, we further compute the delayed perturbed hitting time (DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the upper bound for DPQHT is actually greater than the difference between the square root of the upper bound for a perturbed random walk and the square root of the lower bound for a random walk.

ON PERTURBATION OF LOCAL INTEGRATED COSINE FUNCTIONS

In this paper, we apply the contraction mapping theorem to obtain some bounded and unbounded perturbation theorems concerning local integrated cosine functions. Some growth conditions of perturbations of local integrated cosine functions are also established.

An Augmented Lagrangian Algorithm for Solving SemiinfiniteProgramming

We present a smooth augmented Lagrangian algorithm for semiinfinite programming (SIP). For this algorithm, we establish a perturbation theorem under mild conditions. As a corollary of the perturbation theorem, we obtain the global convergence result, that is, any accumulation point of the sequence generated by the algorithm is the solution of SIP. We get this global convergence result without any boundedness condition or coercive condition. Another corollary of the perturbation theorem shows that the perturbation function at zero point is lower semi‐continuous if and only if the algorithm forces the sequence of objective function convergence to the optimal value of SIP. Finally, numerical results are given.

On Stable Perturbations of the Generalized Drazin Inverses of Closed Linear Operators in Banach Spaces

We investigate the stable perturbation of the generalized Drazin inverses of closed linear operators in Banach spaces and obtain some new characterizations for the generalized Drazin inverses to have prescribed range and null space. As special cases of our results, we recover the perturbation theorems of Wei and Wang, Castro and Koliha, Rakocevic and Wei, Castro and Koliha and Wei.

Research on an Anti-Perturbation Kalman Filter Algorithm

Abstract. An improved Kalman filter algorithm is proposed by two kinds of anti-perturbation method which is derived according to the perturbation theorem of inverse matrix. Furthermore, direction-correcting has been merged into the algorithm by using multiple hypothesis testing theory which can detect the current direction of a target. Finally, Both quantitative and qualitative analysis are given in detail. The measurements and experiments based on indoor positioning demonstrate that the improved algorithm(named IKF) has great performance.

Application of the transfer matrix for tuning the CSNS-DTL

In the construction of a drift tube LINAC (DTL), many factors caused during the fabrication and assembly of the structure cells cause the electric field distribution not as the same as the design curve. To solve this problem, the traditional way is to solve the equation of Slater's perturbation theorem to obtain the electrical field variation due to local frequency shift. However, that is very difficult under complicated conditions. Since the field perturbation equation is similar to the particle's transverse motion equation, which can be simply solved by using the transfer matrix method, we thus propose to apply a transfer matrix method in tuning the DTL. We demonstrate the availability and advantages of this method with 3D microwave code simulation and the LabVIEW calculation program. After two iterations, the initial error of the electric field of 19.5% has been improved greatly down to 1.3%–4.5%. This indicates that the transfer matrix method is very useful and convenient for the simplification of tuning procedures.

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

The $\varepsilon$-pseudospectral abscissa and radius of an $n\times n$ matrix are, respectively, the maximal real part and the maximal modulus of points in its $\varepsilon$-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for $\varepsilon$ sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rank-one perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with $\varepsilon$, and may be less than 10 in many practical cases of interest.

Erratum to: The scalar curvature flow on S n —perturbation theorem revisited

Erratum to: The scalar curvature flow on S n —perturbation theorem revisited

Virial theorem and hypervirial theorem in a spherical geometry

The virial theorem in the one- and two-dimensional spherical geometry are presented in both classical and quantum mechanics. Choosing a special class of hypervirial operators, the quantum hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann–Feynman theorem, these relations can be used to formulate a perturbation theorem without wavefunctions, corresponding to the hypervirial-Hellmann–Feynman theorem perturbation theorem of Euclidean geometry. The one-dimensional harmonic oscillator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method.