Explicit Representation Formula Research Articles

On the Drazin inverse of the linear combinations of two idempotents in a complex Banach algebra

Let A be a complex Banach algebra with unit and let p,q be two idempotents in A. For α,β∈C\\{0}, the authors obtain an explicit representation formula for the Drazin inverse of αp+βq and give the upper bound of the corresponding Drazin index by using the presented method. As a consequence, the main previously published results on Drazin inverse of αp+βq are corollaries of our theorem. Finally, the authors apply the theorem to an example which cannot be solved by previously known results.

Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations

For autonomous linear integral equations, we establish an explicit asymptotic representation formula of solutions, developing the spectral analysis for the generator of the solution semigroup as well as the one for the formal adjoint operator associated with the generator.

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every \({\varepsilon > 0}\), we consider the Green’s matrix \({G_{\varepsilon}(x, y)}\) of the Stokes equations describing the motion of incompressible fluids in a bounded domain \({\Omega_{\varepsilon} \subset \mathbb{R}^d}\), which is a family of perturbation of domains from \({\Omega\equiv \Omega_0}\) with the smooth boundary \({\partial\Omega}\). Assuming the volume preserving property, that is, \({\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}\) for all \({\varepsilon > 0}\), we give an explicit representation formula for \({\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}\) in terms of the boundary integral on \({\partial \Omega}\) of \({G_0(x, y)}\). Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.

Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated to SL(2) acting on a manifold of dimension 4. This is further reduced to the simplest Lie system: the Riccati equation. Lie reduction permits explicit representation formulas for various initial value problems. Additionally, a concise (hyperbolic) Weierstrass-type representation formula is derived. Finally, a number of open problems are framed.

A sup × inf-type inequality for conformal metrics on Riemann surfaces with conical singularities

A sup × inf-type inequality is proved for the regular part of conformal factors for Riemannian metrics on surfaces with conical singularities of positive order α>0. The proof is based on the explicit representation formula for solutions of the singular Liouville equation and generalizes to the singular case an argument by I. Shafrir.

L p -Estimates for a Linear Problem Arising in the Study of the Motion of an Isolated Liquid Mass

We prove L p -estimates for solutions of a linear problem arising as a result of linearization of a free boundary problem for a viscous capillary drop. The proof is rather elementary; it is based on the explicit representation formula for the solution of a model problem in the half-space and on the theorem on L p -multipliers in the Fourier integrals. Bibliography: 20 titles.

A Novel Approach to Calculation of Reproducing Kernel on Infinite Interval and Applications to Boundary Value Problems

A new analytical method for the computation of reproducing kernel is proposed and tested on some examples. The expression of reproducing kernel on infinite interval is obtained concisely in polynomial form for the first time. Furthermore, as a particular effective application of this method, we give an explicit representation formula for calculation of reproducing kernel in reproducing kernel space with boundary value conditions.

A REPRESENTATION FORMULA FOR THE p-ENERGY OF METRIC SPACE-VALUED SOBOLEV MAPS

We give an explicit representation formula for the p-energy of Sobolev maps with values in a metric space as defined by Korevaar and Schoen [Comm. Anal. Geom.1(3–4) (1993) 561–659]. The formula is written in terms of the Lipschitz compositions introduced by Ambrosio [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)3(17) (1990) 439–478], thus further relating the two different definitions considered in the literature.

Anisotropic elastic moduli reconstruction in transversely isotropic model using MRE

Magnetic resonance elastography (MRE) is an elastic tissue property imaging modality in which the phase-contrast based MRI imaging technique is used to measure internal displacement induced by a harmonically oscillating mechanical vibration. MRE has made rapid technological progress in the past decade and has now reached the stage of clinical use. Most of the research outcomes are based on the assumption of isotropy. Since soft tissues like skeletal muscles show anisotropic behavior, the MRE technique should be extended to anisotropic elastic property imaging. This paper considers reconstruction in a transversely isotropic model, which is the simplest case of anisotropy, and develops a new non-iterative reconstruction method for visualizing the elastic moduli distribution. This new method is based on an explicit representation formula using the Newtonian potential of measured displacement. Hence, the proposed method does not require iterations since it directly recovers the anisotropic elastic moduli. We perform numerical simulations in order to demonstrate the feasibility of the proposed method in recovering a two-dimensional anisotropic tensor.

Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations

Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer’s rule) for the minimum norm least squares solutions of quaternion matrix equations AX=B,XA=B and AXB=D.

Sticky particle dynamics with interactions

We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler–Poisson system in the attractive regime our approach yields an explicit representation formula for the solutions.

Flux intensity functions for the Laplacian at axisymmetric edges

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of . Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley & Sons, Ltd.

Flux intensity functions for the Laplacian at polyhedral edges

We present explicit representation formulas for the coefficients of the singularities associated with mixed boundary value problems for the Poisson equation in two-dimensional domains with corners and three-dimensional domains with straight edges including cracks. We rely on partial Fourier analysis of the boundary value problem in the vicinity of the singularities to derive an asymptotic expansion of the solution. Hence, the edge flux intensity functions are expressed in terms of Fourier series and we give integral expressions from which the associated Fourier coefficients can be computed independently and in parallel. Our method does not require the knowledge of the dual singular solutions of the adjoint problems. Moreover, the constructive nature of the formulas provides in a straight forward way a strategy for the construction of efficient numerical methods for the accurate computation of the stress intensity factors and the edge flux intensity functions. Also, the formulas can be used to construct postprocessing algorithms for the standard finite element approximation of the solution of the boundary value problem to improve the accuracy. Examples that illustrate the efficiency and robustness of the formulas are also given.

How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”

Navier–Cauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how—by means of the Principle of Virtual Powers—it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green–Saint-Venant tensor and all its N − 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.

Symmetric determinantal representation of polynomials

We give an elementary proof, only using linear algebra, of a result due to Helton, Mccullough and Vinnikov, which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil. We give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2.

A variational problem for couples of functions and multifunctions with interaction between leaves

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions. Mathematics Subject Classification. 49Q20, 54C60.

Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay

For linear Volterra difference equations with infinite delay, we obtain an explicit asymptotic representation formula of solutions by means of the formal adjoint equation associated with a certain bilinear form.

Smoothness and Function Spaces Generated by Homogeneous Multipliers

Differential operators generated by homogeneous functionsψof an arbitrary real orders>0(ψ-derivatives) and related spaces ofs-smooth periodic functions ofdvariables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and Triebel-Lizorkin spaces. Explicit representation formulas forψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.

Sharp Observability Estimates for Heat Equations

The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in Fattorini and Russell (Arch Rational Mech Anal 43:272–292, 1971) to the multi-dimensional setting and improves those available in the literature, namely those by Miller (J Differ Equ 204(1):202–226, 2004; SIAM J Control Optim 45(2):762–772, 2006; Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 17(4):351–366, 2006) and Tenenbaum and Tucsnak (J Differ Equ 243(1):70–100, 2007). Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.

On Generalisation of Frongello Linking Algorithm

New explicit representation formulae for the well-known Frongello Linking Algorithm are presented and two new families of Linking Algorithms are created based on simple algebraic analysis of these formulae (which will be known as cF and gF Linking Algorithms). Although these new algorithms cF and gF can be considered as the generalisations of Frongello, they stand alone as a novel approach to Arithmetic Attribution Linking and do not have the well-known weaknesses of the original and Modified Frongello. The corresponding analysis of cF and gF Linking Algorithms shows the usefulness of a new approach to Performance Measurement Linking Algorithms. As the most powerful and universal of the two Algorithms presented here, gF should be considered as an ultimate possible generalisation of Frongello. In turn, cF serves as a “halfway thought” version of gF, its construction is much simpler but power of outcome is almost the same. Unlike the standard and Modified Frongello Algorithms, gF is also independent of the month order, symmetric and homogeneous. The simplicity of cF and gF Linking Algorithms makes them easy to use and hence more appropriate for practitioners without advanced degrees in mathematics.