General Linear Operator Research Articles

Solution to dynamic matrix linear systems by operator exponential functions

In this paper, we present a solution of general dynamic matrix linear systems based on operator theory. For general linear operators defined in matrix form the concept of operator exponential functions is first proposed, and some properties are then established. With the proposed operator exponential functions as a tool, the solution to general matrix linear systems is derived. Furthermore, by using the obtained results on dynamic matrix systems, the solution of a class of general linear matrix equations is expressed in the form of infinite integration.

Approximation of Conjugate Functions by General Linear Operators of Their Fourier Series at the Lebesgue Points

Abstract The pointwise estimates of the deviations r T͂n,A,Bf (·) - f͂͂ (·) and T͂n,A,Bf (·) - f͂͂ (·,ε) in terms of moduli of continuity ω̃f and r ω̃f are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal [3, Theorem 1, p. 437].

Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition - Application to fractional variational problems

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. The main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ being the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].

Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition – Application to fractional variational problems

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. The main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ being the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].

Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions

In this note we present some uniqueness and comparison results for a class of problem of the form(0.1)−Lu=H(x,u,∇u)+h(x),u∈H01(Ω)∩L∞(Ω), where Ω⊂RN, N≥2 is a bounded domain, L is a general elliptic second order linear operator with bounded coefficients and H is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp.

A general analytical approximation for nonlinear vibrations analysis of continuous systems using renormalization group method

We propose a general procedure based on renormalization group method (RGM) and Green’s function method to determine the analytical approximation of nonlinear continuous systems with general linear and nonlinear operators. The nonlinear operator has a general form (not necessarily quadratic and cubic form) and can be function of displacement, velocity and acceleration or combination of thereof. In addition, the procedure can handle all types of resonances and there is no restriction on the number of modes which are considered in the analysis. Furthermore, for a specified form of the linear operator (e.g. wave operator), the nonlinear and non-homogeneous boundary conditions can be considered in the analysis. Although, the analysis is general, but it is assumed that the boundary data is small and therefore the boundary layer is not observed. To show the effectiveness of the procedure, some examples are presented.

Variational-Based Reduced-Order Model in Dynamic Substructuring of Coupled Structures Through a Dissipative Physical Interface: Recent Advances

This paper deals with a variational-based reduced-order model in dynamic substructuring of two coupled structures through a physical dissipative flexible interface. We consider the linear elastodynamic of a dissipative structure composed of two main dissipative substructures perfectly connected through interfaces by a linking substructure. The linking substructure is flexible and is modeled in the context of the general linear viscoelasticity theory, yielding damping and stiffness operators depending on the frequency, while the two main dissipative substructures are modeled in the context of linear elasticity with an additional classical viscous damping modeling which is assumed to be independent of the frequency. We present recent advances adapted to such a situation, which is positioned with respect to an appropriate review that we carry out on the different methods used in dynamic substructuring. It consists in constructing a reduced-order model using the free-interface elastic modes of the two main substructures and, for the linking substructure, an appropriate frequency-independent elastostatic lifting operator and the frequency-dependent fixed-interface vector basis.

Matrix transformations between certain sequence spaces over the non-Newtonian complex field.

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the field ℂ* and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets over ℂ* to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

Value Distribution and Linear Operators

AbstractNevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

Array-representation integration factor method for high-dimensional systems

High order spatial derivatives and stiff reactions often introduce severe temporal stability constraints on the time step in numerical methods. Implicit integration method (IIF) method, which treats diffusion exactly and reaction implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reactions and diffusions. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. Motivated by a compact representation for IIF (cIIF) for Laplacian operators in two and three dimensions, we introduce an array-representation technique for efficient handling of exponential matrices from a general linear differential operator that may include cross-derivatives and non-constant diffusion coefficients. In this approach, exponentials are only needed for matrices of small size that depend only on the order of derivatives and number of discretization points, independent of the size of spatial dimensions. This method is particularly advantageous for high-dimensional systems, and it can be easily incorporated with IIF to preserve the excellent stability of IIF. Implementation and direct simulations of the array-representation compact IIF (AcIIF) on systems, such as Fokker–Planck equations in three and four dimensions and chemical master equations, in addition to reaction–diffusion equations, show efficiency, accuracy, and robustness of the new method. Such array-presentation based on methods may have broad applications for simulating other complex systems involving high-dimensional data.

Some new versions of Hahn-Banach theorem

In 1997, L.-G. Huang and X. Zhang introduced the cone metric space, which is the generalization of the metric space. Further, some authors introduced the cone normed space. In this paper, we introduce the conceptions of bounded linear operator from a normed space to a cone normed space and bounded linear operator between two cone normed spaces. A general Hahn-Banach type linear operator extension theorem from a linear space to a partial ordering linear space, with the order is induced by a minimal cone, is proven. Two preserved-norm bounded linear operator extension theorems, one is from a normed space to a cone normed space and another is between two cone normed spaces, and some corollaries are given.

Minimization and parameter estimation for seminorm regularization models with I-divergence constraints

In this paper, we analyze the minimization of seminorms ‖L · ‖ on under the constraint of a bounded I-divergence D(b, H · ) for rather general linear operators H and L. The I-divergence is also known as Kullback–Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data but also in the case of multiplicative Gamma noise. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. A central part of this paper consists in proving relations between the parameters of I-divergence constrained and penalized problems. To solve the I-divergence constrained problem, we consider various first-order primal–dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least-squares problem which can be solved based on Morozov’s discrepancy principle by a Newton method. We prove that these algorithms produce not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which converges to a regularization parameter so that the corresponding penalized problem has the same solution. Furthermore, we derive a rule for automatically setting the constraint parameter for data corrupted by multiplicative Gamma noise. The performance of the various algorithms is finally demonstrated for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm

This paper introduces a generalization and automation of the Wiener Hermite expansion with perturbation (WHEP) technique to solve a class of stochastic nonlinear partial differential equations with a perturbed nonlinearity. The automated algorithm generates the deterministic resultant linear equations according to the application of a general linear differential operator and the input parameters. Sample output with different nonlinearities, orders and corrections are presented. The resultant equations are solved numerically and the ensemble average and variance are computed and compared with previous research work. Higher order solutions with higher corrections are computed to show the importance of the generalization of the WHEP technique. The current work extends the use of WHEP for solving stochastic nonlinear differential equations.

On Fitzpatrick functions of monotone linear operators

Bauschke, Borwein and Wang have shown in [H.H. Bauschke, J.M. Borwein, X. Wang, Fitzpatrick functions and continuous linear monotone operators, Siam J. Optimization, 18 (3) (2007), 789–809] that if FT(⋅,⋅) denotes the Fitzpatrick function of a continuous linear monotone operator T on a separable real Hilbert space H, then FT(x,u)=2qT+∗((u+T∗x)/2), where qA∗ denotes the Fenchel conjugate of the function qA:H→R sending x to 2−1〈x,Ax〉 for an arbitrary continuous positive symmetric operator A∈B(H). Here, T+≔(T+T∗)/2≥0 and A† denotes the Moore–Penrose type inverse of a positive symmetric operator A. The main result of the present paper is a sharpening of the result, achieved by showing that dom(qA∗)=ran(A1/2) and that FT(x,u)=4−1‖(T+1/2)†(u+T∗x)‖2 on dom(qA∗), where dom(f) is the set on which an extended real-valued function f is finite. We will also find a formula for the Fitzpatrick function of a general n-cyclic monotone linear operator in terms of a corresponding ordinary monotone linear operator.

Blind source separation of images based upon fractional autocorrelation

Blind source separation (BSS) is a process in which mixed signals are separated into their original sources. Both the sources as well as the mixing coefficients are unknown but a priori information about statistical behavior and about the mixing model might be available. We here suggest a generalization of our previous research that showed a new BSS algorithm based on general cross correlation linear operators applied on the sources that are to be separated. In that approach in cases of negligible cross-correlation between the source signals, a very good separation could be obtained. Here we propose to use the fractional Fourier transform in order to reduce the correlation between the source signals and to further enhance the obtained separation performance. We present reduced dependence on the cross-correlation between the source images, resulting in better separation of the mixed sources.

Asymptotic properties of iterates of certain positive linear operators

In this paper we prove Korovkin type theorem for iterates of general positive linear operators T:C[0,1]→C[0,1] and derive quantitative estimates in terms of modulus of smoothness. In particular, we show that under some natural conditions the iterates Tm:C[0,1]→C[0,1] converges strongly to a fixed point of the original operator T. The results can be applied to several well-known operators; we present here the q-MKZ operators, the q-Stancu operators, the genuine q-Bernstein–Durrmeyer operators and the Cesaro operators.

Some inclusion relationships and integral-preserving properties of certain subclasses of p-valent meromorphic functions associated with a family of linear operator

Abstract By making use of a general linear operator we introduce several new subclasses of multvalent meromorphic functions and investigate various inclusion relationships. Several interesting integral-preversing properties are also discussed.

Theory and application of general linear image processing

Image modeling in terms of physical/mathematical formation and the human visual system plays an important role in developing new image processing techniques. Parameters such as scale, orientation, texture, morphology, and color must be taken into consideration in order to solve complicated image processing problems efficiently. Although the classical linear image processing (CLIP) approach has played a central role in image processing, it is not necessarily the best and even the right choice. General linear image processing (GLIP) approaches based on different image modeling techniques have been studied to overcome some of the problems associated with CLIP. Indeed, using general linear algebra, it is possible to develop entirely new general linear operators and transforms. It is even possible to define entirely new general linear operations (addition, scalar multiplication, convolution, etc.) in order to describe how images are combined, amplified, transformed, analyzed, compared, and measured.

Approximation of integrable functions by general linear operators of their fourier series

The pointwise estimates of the deviation Tn,A,Bf(·)−f(·) in terms of moduli of continuity −w·f and w·f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.: J. Math. Anal. Appl., 220, 434–450 (1998) Theorem 1, p. 437].