Inclusion Theorems Research Articles

Double summability factor theorems and applications

Weobtain sufficient conditions for the series ∑∑ amn , which is absolutely summable of order k by a double triangular matrix method A , to be such that ∑∑ amnλmn is absolutely summable of order k by a double triangular matrix B . As corollaries we obtain a number of inclusion theorems. Mathematics subject classification (2000): 40G99, 40G05, 40D15.

On a summability factor theorem

Let 1 < k s < ∞ . In the present paper we obtain sufficient conditions for a series ∑ an , which is absolutely summable of order k by a weighted mean method to be such that ∑ anλn is absolutely summable of order s by a triangular matrix. As corollary of this result we obtain an inclusion theorem. Mathematics subject classification (2000): 40F05, 40D25, 40G99.

Polynomial approach for design and robust analysis of Lateral missile control

A polynomial approach is presented for the design and analysis of a sideslip velocity missile autopilot. The controller is designed using polynomial eigenstructure assignment (PEA). The missile model is described by linear parameter varying (LPV) matrices. The design renders to a closed-loop system independent of the choice of equilibria. Thus, if the operating points are in the vicinity of the equilibria, then only one linear model will describe closed-loop dynamics, regardless of the rate of change of the operating points. Parametric stability margins for uncertainty in the controller parameters and aerodynamic derivatives are analysed using a finite version of the Nyquist Theorem. The Finite Nyquist Theorem (FNT) exploits the polynomial framework to assess robustness for a parametric uncertain system with the Finite Inclusion Theorem (FIT). The design and analysis approach is applied to a single-input single-output (SISO) tail-controlled missile in the cruciform fin configuration. Simulations show good tracking of sideslip velocity with closed-loop system, fast responses and good parametric robustness against six uncertain parameters.

Reynold's Transport Theorem for Differential Inclusions

The well-known Reynold's Transport Theorem deals with the integral over a time-dependent set (that is evolving along a smooth vector field) and specifies its semiderivative with respect to time. Here reachable sets of differential inclusions are considered instead. Dispensing with any assumptions about the regularity of the compact initial set, we give sufficient conditions on the differential inclusion for the absolute continuity (of the integral) with respect to time and its weak derivative is formulated as a Hausdorff integral over the topological boundary.

Collective fixed points and a system of coincidence theorems in product [formula omitted]-spaces and applications

By using a continuous selection theorem, some new collective fixed point theorems and coincidence theorems for the families of set-valued mappings defined on the product space of noncompact F C -spaces are proved under very weak assumptions. As applications, some nonempty intersection theorems, inclusion theorems and existence theorems of solutions for systems of inequalities are established in the product space of noncompact F C -spaces. These results generalize many known results from the recent literature.

A Unified Approach to Dynamic Contact Problems in Viscoelasticity

In this paper we consider mathematical models describing dynamic viscoelastic contact problems with the Kelvin–Voigt constitutive law and subdifferential boundary conditions. We treat evolution hemivariational inequalities which are weak formulations of contact problems. We establish the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations. These results are consequences of an existence theorem for second order evolution inclusions. In a particular case we deliver sufficient conditions under which the solution to a hemivariational inequality is unique. Finally, applications to several unilateral and bilateral problems in contact mechanics are provided.

Interconnected Systems with Uncertain Couplings: Explicit Formulae formu-Values, Spectral Value Sets, and Stability Radii

In this paper we study the variation of the spectrum of block-diagonal systems under perturbations of compatible block structure with fixed zero blocks at arbitrarily prescribed locations ("Gershgorin-type perturbations"). We derive explicit and computable formulae for the associated mu-values. The results are then applied to characterize spectral value sets and stability radii for such perturbed systems. By specializing our results to the scalar diagonal case, the classical eigenvalue inclusion theorems of Gershgorin, Brauer, and Brualdi are obtained as corollaries. Moreover it follows that the inclusion regions of Brauer and Brualdi are optimal for the corresponding perturbation structures.

Inclusion theorems of convolution operators associated with normalized hypergeometric functions

For the normalized Gaussian hypergeometric function zF ( a , b ; c ; z ) given by F ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a , n ) ( b , n ) ( c , n ) ( 1 , n ) z n , | z | < 1 , the author aims at finding conditions on a , b and c such that the convolution operator zF ( a , b ; c ; z ) * f ( z ) satisfies some inclusion results in certain class of analytic functions and in particular when f ∈ P γ ( β ) , where P γ ( β ) = f ∈ A : Re e i ϕ ( 1 - γ ) f ( z ) z + γ f ′ ( z ) - β > 0 , ϕ ∈ R , z ∈ Δ . A convolution result for the class P γ ( β ) is also given. Analogous results for the confluent hypergeometric functions are also discussed.

Lacunary statistical convergence of multiple sequences

Quite recently, Mursaleen and Edely [M. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. (in press)], defined the statistical analogue for double sequences x = { x k , l } as follows: A real double sequence x = { x k , l } is said to be P-statistically convergent to L provided that for each ϵ > 0 P − lim m , n 1 m n { numbers of ( j , k ) : j < m and k < n , | x j , k − L | ≥ ϵ } . In this paper we introduce and study lacunary statistical convergence for double sequences and we shall also present some inclusion theorems.

Existence results for neutral functional differential inclusions in Banach algebras

In this paper an existence theorem for the first-order functional differential inclusions in Banach algebras is proved under the mixed generalized Lipschitz and Carathéodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.

Dual-Stiffness Sensor for Damage Detection,Localization, and Prognostics.

The concept of a dual stiffness sensor that measures material degradation at the micromechanics scale and introduces damage variables to reflect average material degradation at a macromechanics scale is developed. The idea of a mechanical multifunctional dual stiffness sensor for in situ real-time stiffness and energy density measurements is motivated by Eshelby’s equivalent inclusion theorem. A complete analytical derivation is presented and the use of a dual stiffness/energy sensor to diagnose damage and to determine the remaining life of structures is evaluated. A numerical verification using finite element analysis is presented, and the results indicate that the sensor can be used for diagnostics and prognostics of structures. A distributed array of sensors is used to determine the onset and location of damage. Experimental results presented for uniaxial experimental tests under monotonic quasi-static loading conditions indicate that the concept can easily be implemented. For a sensor pair of known geometry and material properties, the Eshelby tensors are independent of the host: A methodology was thus developed to detect material properties of different hosts and was used to predict accurately elastic material properties of diverse materials.

A note on properties and computations of matrix pseudospectra

Recently, Shen et al. has published a paper concerning the properties and computations of matrix pseudospectra. However, some of the main results seem to be incomplete. As a complement of the paper, in this note we first make several examples to show the incompleteness of that results, then we give the modified results through giving more appropriate definitions replacing the one given in [Y. Shen, J. Zhao, H. Fan, Properties and computations of matrix pseudospectra, Applied Mathematics and Computation 161 (2005) 385–393]. An interesting inclusion theorem for pseudospectra is given, which provides preferable bounds for standard pseudospectra. Some other properties are also explored.

Robust Analysis for Missile Lateral Acceleration Control Using Finite Inclusion Theorem.

In this paper, polynomial eigenstructure assignment design approach is applied to the lateral acceleration control for a tactical missile model. The tail-controlled missile in the cruciform n conguration is modelled as a second-order quasi-linear parameter-varying system. Polynomial eigenstructure assignment is seen as an approach to perform dynamic inversion where the controller has a chosen structure. Although developed primarily for LTI systems, the design approach uses here the explicit parametrisation of the LPV system and thus lead to a LPV controller. Keeping the polynomial framework, robustness to parametric uncertainties is then assessed based on a nite version of the Nyquist Theorem, the Finite Nyquist Theorem and its extension to polynomial families, the Finite Inclusion Theorem. Simplifying the value set using the Mapping Theorem, the multi-linear polynomial family is captured and its D-stable robustness assessed. Finally, simulations show good tracking of lateral acceleration with fast response time and lead to discussion on the robust properties of the closed-loop system.

SPECTRAL MAPPING THEOREMS FOR HOLOMORPHIC FUNCTIONAL CALCULI

A spectral inclusion theorem and a spectral mapping theorem are proved for the functional calculus for sectorial operators. Most proofs are generic, so that similar results can be obtained for other functional calculi. Applications are a new proof for the spectral mapping theorem for fractional powers and the identity for any injective sectorial operator.

Corrigendum: Corrections to ‘Inclusion theorems for non-linear difference equations with applications’

Corrigendum: Corrections to ‘Inclusion theorems for non-linear difference equations with applications’

On Some Euler Sequence Spaces of Nonabsolute Type

In the present paper, we introduce Euler sequence spaces e 0 r and e of nonabsolute type that are BK-spaces including the spaces c 0 and c and prove that the spaces e 0 r and e are linearly isomorphic to the spaces c 0 and c, respectively. Furthermore, some inclusion theorems are presented. Moreover, the α-, β-, γ- and continuous duals of the spaces e 0 r and e are computed and their bases are constructed. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes $$\left( {e_c^r :\ell _p } \right)$$ and $$\left( {e_c^r :c} \right)$$ are established, and characterizations of some other classes of infinite matrices are also derived by means of a given basic lemma, where 1 ≤ p ≤ ∞.

General inclusion theorems for absolute summability of order k ≥ 1

We establish a general inclusion theorem for absolute summability of order k 1, involving two lower triangular matrices. As corollaries we obtain a number of other inclusion theorems.

On boundary value problems for second-order discrete inclusions

We prove some existence theorems regarding solutions to boundary value problems for systems of second-order discrete inclusions. For a certain class of right-hand sides, we present some lemmas showing that all solutions to discrete second-order inclusions satisfy an a priori bound. Then we apply these a priori bounds, in conjunction with an appropriate fixed point theorem for inclusions, to obtain the existence of solutions. The theory is highlighted with several examples.

General inclusion relations for absolute summability

We obtain sufficient conditions for the series an , which is absolutely summable of order k by a triangular matrix method A ,1 < k s < ∞, to be such that an is absolutely summable of order s by a triangular matrix B. As corollaries, we obtain a number of inclusion theorems.

Locating real eigenvalues of a spectral problem in fluid‐solid type structures

Exploiting minmax characterizations for nonlinear and nonoverdamped eigenvalue problems, we prove the existence of a countable set of eigenvalues converging to ∞ and inclusion theorems for a rational spectral problem governing mechanical vibrations of a tube bundle immersed in an incompressible viscous fluid. The paper demonstrates that the variational characterization of eigenvalues is a powerful tool for studying nonoverdamped eigenproblems, and that the appropriate enumeration of the eigenvalues is of predominant importance, whereas the natural ordering of the eigenvalues may yield false conclusions.