## Abstract

A.E. Green F.R.S. and P.M. Naghdi developed two theories of thermoelasticity, called type II and type III, which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time *t*=0 and at a later time *t*=*T*. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time.

## 1. Introduction

Green & Naghdi (1991, 1992, 1993) develop an analysis in a rational way to produce a fully consistent theory of thermoelasticity which we believe is capable of incorporating thermal pulse transmission in a very logical manner. We believe that only extensive mathematical and physical analyses of the developments of Green and Naghdi will reveal the usefulness of the theory, and it is to this goal that the present paper is addressed.

In 1991, Green & Naghdi developed a theory for describing the behaviour of a continuous body which relies on an entropy balance law rather than an entropy inequality. Their thermodynamics introduces a quantity, *T*, which is the ‘empirical’ temperature, and a term(1.1)which they refer to as the thermal displacement variable. Note that ** x** is the spatial coordinate in the reference configuration of the body. Green & Naghdi (1993) show how classical thermoelasticity may be incorporated within their theory. However, they also develop, in detail, a theory they call thermoelasticity of type II, which allows for heat transmission at finite speed and for which there is no energy dissipation. The linearized equations for this theory are derived by Green & Naghdi (1993) in their equations (3.15) and (3.16) for an isotropic material. They establish uniqueness for their type II theory for

*an isotropic material*. In fact, this theory has attracted considerable further interest. Extensive reviews of hyperbolic thermoelasticity, and of the Green–Naghdi theories in particular, are given by Chandrasekharaiah (1998) and Hetnarski & Ignaczak (1999), who show that such theories, which admit the possibility of heat travelling as a wave of finite speed, are increasingly important in the modern world. Of particular interest to this paper are that Chandrasekharaiah (1996) proves uniqueness, Iesan (1998) produces continuous dependence results, Nappa (1998) establishes spatial decay theorems and Quintanilla (2002) studies the question of existence. Decay in one space dimension is studied in Quintanilla & Racke (2003), while Zhang & Zuazua (2003) establish very interesting decay results in two and three dimensions. Further results of structural stability and decay type are given by Quintanilla (2001

*a*,

*b*, 2003). Quintanilla & Straughan (2000) use logarithmic convexity and Lagrange identity arguments to yield uniqueness and growth without requiring sign definiteness of the elasticities, while Quintanilla & Straughan (2004) derive a fully nonlinear acceleration wave analysis. It is pertinent to note that Green & Naghdi (1993) write, ‘This type of theory, … thermoelasticity type II, since it involves no dissipation of energy is perhaps a more natural candidate for its identification as thermoelasticity than the usual theory’. In addition, Green & Naghdi (1991) observe. ‘This suggests that a full thermoelasticity theory—along with the usual mechanical aspects—should more logically include the present type of heat flow [type II] instead of the heat flow by conduction [classical theory, type I]’, where the words in brackets have been added for clarity. Since theories that allow heat transport as a wave may exhibit the phenomenon of blow-up in finite time (e.g. Quintanilla & Straughan 2002), it is vital to study the mathematical behaviour of the Green–Naghdi theories.

Green & Naghdi (1992) also develop another thermoelasticity theory based on their 1991 work which they call type III thermoelasticity. This again allows for heat transport at finite speed, but there *is* dissipation in this theory. The linearized isotropic equations are given in Green and Naghdi (1992) as equations (3.19) and (3.20). Green & Naghdi (1992) study the behaviour of one-dimensional waves in their theory. The approach of Green and Naghdi, which is based on an entropy balance law, is developed in generality for various theories of continuous media in Green & Naghdi (1995*a*--*c*).

*A priori* bounds and questions of structural stability have been the focus of attention in many recent articles in thermoelasticity, both for the classical theory and for type II and type III theories. We mention that Ames & Payne (1991, 1994, 1995) establish a series of results on continuous dependence for the backward in time problem, for a unilateral problem, and for the initial-time geometry problem, respectively. Rionero & Chirita (1987) show how one can use a weighted Lagrange identity argument to establish uniqueness and continuous dependence results for classical thermoelasticity on unbounded spatial regions without requiring the solution to decay at infinity.

In this paper, we continue our investigation into the study of thermoelasticity of type II and type III by analysing a completely new, but important, class of problems within these theories.

Recently, a new class of non-standard problems has been shown to be relevant to many applied mathematical situations. This is where the data are not given at time *t*=0, but instead as a linear combination at times *t*=0 and *t*=*T* (see Payne & Schaefer 2002; Payne *et al*. in press; Ames *et al*. 2004*a*,*b* and references therein). Such problems were originally introduced as a means of stabilizing solutions to the improperly posed problem when the data is given at *t*=*T* and one wishes to compute the solution backward in time (cf. Ames *et al*. 1998; Ames & Payne 1999 and references therein). Accurate solution of such problems is essential to many real applied mathematical situations, such as image reconstruction from noisy data (see e.g. Ames & Straughan 1997, ch. 8). Ames *et al*. (2004*a*) study a non-standard problem for the parabolic-like abstract equation *u*_{t}+*Au*=*f*. Ames *et al*. (2004*b*) investigate a non-standard problem for the classical (parabolic) diffusion equation with the spatial domain being an infinite cylinder. While both papers of Ames *et al*. (2004*a*,*b*) are dealing with non-standard problems, neither has any bearing on the work herein which deals with equations that are second order in time. Payne & Schaefer (2002) primarily study a non-standard problem for the abstract equation *u*_{tt}+*Au*=*F*. However, they briefly investigate a similar non-standard problem for the equation *u*_{tt}+*au*_{t}+*Au*=0, for *a*>0, a constant. This equation does contain dissipation but the order is lower than that contained in the equations of thermoelasticity studied here. The techniques of Payne & Schaefer (2002) for their equation with dissipation do not evidently apply directly to thermoelasticity, and so we need to employ a different avenue. Payne *et al*. (2004) study a non-standard problem for the Maxwell–Cattaneo equations. To do this, they reduce their problem to an investigation of the single equation , for constants *τ*, *ξ*_{1}, *ξ*_{2}>0. They derive very interesting bounds, but we stress that the systems of equations studied here are very different. Our equations are coupled systems of four‐second order in time equations and require different techniques from those of Payne *et al*. (2004). In thermoelasticity of type II, there is no dissipation, but the interaction terms are different from what one finds in the single abstract equation of Payne & Schaefer (2002). Also, the method we employ is different from both Payne & Schaefer (2002) and Payne *et al*. (2004). We have to work with the temperature equation as it stands, but we find it necessary to first differentiate the momentum equation before we commence our analysis.

The object of this paper is to obtain solution estimates in appropriate measures of the solution to thermoelasticity of type II or type III, given data as a linear combination at *t*=0 and *t*=*T*. While Payne and his co-workers have successfully studied such problems in wave-like equations, we believe that this is the first such study in the important area of thermoelasticity, especially in the Green–Naghdi theories of type II and type III. We obtain solution estimates for the displacement, temperature, and strain, under a variety of conditions on the coupling constants. These estimates lead to continuous dependence on appropriate terms on the model, and to uniqueness under the stated conditions. We also show that such problems do not always possess a unique solution, and thereby delimit the class of coupling constants for which the problems are physically useful. In addition to establishing *a priori* solution bounds for a solution to thermoelasticity of type II or III, we show that the time region of Payne *et al*. (2004) is sharp by demonstrating that there is non-uniqueness outside of this region. This result is extended to type III thermoelasticity. We also consider non-homogeneous boundary conditions in thermoelasticity. This aspect is not investigated in Payne & Schaefer (2002) nor in Payne *et al*. (2004).

## 2. Equations and notation

The relevant equations of anisotropic inhomogeneous thermoelasticity of type II for a body with a centre of symmetry are (cf. Quintanilla 2002):

(2.1)where *u*_{i} and *θ* are the displacement vector and temperature field. Standard indicial notation is employed throughout, and a superposed dot denotes ∂/∂*t*. The quantities *ρ*, *f*_{i} and *r* are density, body force and heat supply, respectively, while *c*>0 is a constant. The tensor *a*_{ij}(** x**) is a coupling tensor.

The appropriate equations for type III theory may be derived from Green & Naghdi (1992), and for a centrosymmetric body are

(2.2)where the notation is as above. We give information on the tensors *k*_{ij} and *b*_{ij} as we deal with the respective theories, although *b*_{ij} and *k*_{ij} are always symmetric.

Let *Ω* be a bounded domain in with boundary *Γ* smooth enough to allow applications of the divergence theorem. Denote by (,) and ‖.‖ the inner product and norm on *L*^{2}(*Ω*). Either set of equations (2.1) or (2.2) hold on *Ω*×(0, *T* ) for some time *T*(≤∞), and we consider boundary conditions of the form:(2.3)

Non-homogeneous boundary conditions are considered in §5.

The other conditions imposed on the problem are that (** x** is suppressed and only

*t*values given)(2.4)(2.5)Here,

*α*and

*β*are given real numbers with

*g*

_{i}(

**),**

*x**h*

_{i}(

**),**

*x**α*

_{0}(

**) and**

*x**α*

_{1}(

**) prescribed functions.**

*x*The elastic coefficients are symmetric and positive definite in the sense that(2.6)for all second‐order tensors *ξ*_{ij}. Also, the thermal conductivity *k*_{ij} is positive definite with *b*_{ij} non-negative, i.e.(2.7)for all *ξ*. In addition, the interaction coefficients *a*_{ij}(** x**) remain bounded together with their derivatives, so(2.8)for some positive constant

*a*

_{1}. The functions

*ρ*,

*c*,

*f*

_{i}and

*r*depend on

**but not**

*x**t*, with

*ρ*and

*c*>0.

We denote by *A*(*u*, *v*) and *K*(*ϕ*, *ψ*) the bilinear forms(2.9)and further introduce the operator notation(2.10)

It is sometimes natural to work with the thermal displacement variable (cf. Green & Naghdi 1992, 1993), and the use in Quintanilla & Straughan (2000, 2004). However, the temperature field *θ* is perhaps a variable for which we have a more physical feeling, and so, while we could employ *ζ*, we choose to work directly with the function *θ*. This necessitates that we here work with a higher derivative ‘energy-like’ function and so differentiate equations (2.1)_{1} or (2.2)_{1} to derive the governing equation(2.11)

Since we have introduced an extra derivative, we require a condition supplementary to equations (2.4) and (2.5). To derive this, we evaluate equations (2.1)_{1} or (2.2)_{1} at *t*=*T* and at *t*=0 and employ equations (2.4) and (2.5) to calculate(2.12)where *F*_{i} is defined by(2.13)Sometimes it is convenient to express in terms of as(2.14)

## 3. Energy bounds when |*α*|, |*β*|>1

Payne & Schaefer (2002) divide their analysis of the equation *u*_{tt}+*Au*=*F* with conditions *αu*(0)+*u*(*T*)=*g* and *βu*_{t}(0)+*u*_{t}(*T*)=*h* into the cases |*α*|, |*β*|>1, and |*α*|, |*β*|<1, and further show that |*α*|<1, |*β*|>1 or |*α*|>1, |*β*|<1 does not define a well posed problem. For type II thermoelasticity we broadly find results in agreement with this. For type III thermoelasticity, however, the situation for |*α*|, |*β*|<1 is very different, as we show in §4.

For now, we proceed with |*α*|, |*β*|>1 and deal with type III thermoelasticity. However, the analysis holds for type II thermoelasticity since *b*_{ij} satisfies equation (2.7). We stress that, for simplicity, we consider conditions (2.4) and (2.5), where the same *α* and *β* are employed in these auxiliary conditions. The general non-standard problem in thermoelasticity of type II or III would replace *α* with *γ*(>0), and *β* with *δ*(>0), in equation (2.5), where *α*, *β*, *γ*, *δ* may all be different. This is very different from the Payne & Schaefer (2002) case. The analysis is technically more complicated though, and so we restrict attention to the simpler equations (2.4) and (2.5).

We multiply equation (2.11) by and integrate over *Ω*, using the boundary conditions (2.3), to find(3.1)Similarly, by multiplying equation (2.2)_{2} by and integrating, we derive(3.2)

Now, define *E*_{1}(*t*) by(3.3)Upon the addition of equations (3.1) and (3.2), we obtain(3.4)By integration of equation (3.4) from 0 to *t*, we thus find, for any 0≤*t*≤*T*,(3.5)where, in the last two terms, the dependence of *θ* on **x** is suppressed and only the *t*-dependence is highlighted. In particular, from equation (3.5), we discover(3.6)We now substitute conditions (2.4), (2.5) and (2.12) to remove terms evaluated at *T* from inequality (3.6). This leads to(3.7)In deriving inequality (3.7), we have used the arithmetic–geometric mean inequality, with *γ*_{1}>0 arbitrary, on the term which arises. We then used Poincaré's inequality *λ*_{1}‖*f*‖^{2}≤‖∇*f*‖^{2} for a function *f* zero on *Γ*, and then employed inequality (2.7).

The next step is to use the arithmetic–geometric mean equality on each of the underlined terms on the left of inequality (3.7) for the arbitrary constants *γ*_{2}, …, *γ*_{5}>0. For example,

In this way, we find, from inequality (3.7)(3.8)

Finally, since |*α*|>1, |*β*|>1, we select *γ*_{1}, *γ*_{2}, *γ*_{3}, *γ*_{4}, *γ*_{5}, small enough thatThus, the coefficients of the terms on the left of inequality (3.8) are positive. In this way, we determine computable constants *k*_{1}, …, *k*_{6}, such that(3.9)Note that *A*_{0} contains only data and so is a known function. We now return to equation (3.5) and see that(3.10)We use the arithmetic–geometric mean inequality for arbitrary *γ*_{6}, *γ*_{7}>0 to derivewith a similar bound for . Now pick *γ*_{6}<*λ*_{1}*k*_{0}, and then, from inequality (3.10), we determine constants *k*_{7} and *k*_{8}, such that(3.11)

Upon use of the estimate (3.9) in inequality (3.11), we thus see there are computable constants *c*_{1}, …, *c*_{6}, such that(3.12)Since the quantity *B*_{0} contains only known data this is an *a priori* bound for the function *E*_{1}(*t*) for 0≤*t*≤*T*.

Of course, equation (3.12) yields an estimate for ‖*θ*(*t*)‖^{2} and ‖∇*θ*(*t*)‖^{2} by use of inequality (2.7) and Poincaré's inequality. However, because of the structure of *E*_{1} we still need to derive an estimate for ‖** u**(

*t*)‖ and ‖∇

**(**

*u**t*)‖. In order to achieve such an estimate, we multiply equation (2.2)

_{1}by and integrate over

*Ω*to see that(3.13)where

*U*(

*t*) is given by(3.14)Thus, for any 0≤

*t*≤

*T*,(3.15)In particular, evaluating this equation at

*t*=

*T*and using conditions (2.4), (2.5), (2.8) and the arithmetic–geometric mean inequality, we may find(3.16)

The terms on the left are expanded and the arithmetic–geometric mean inequality is employed on the and *u*(0)*g* terms for arbitrary positive constants *δ*_{1} and *δ*_{2}. We further use the arithmetic–geometric mean inequality as follows, for *δ*_{3}>0 at our disposal,where condition (2.6) and Poincaré's inequality have been employed. With the aid of this inequality, we thus determine, from inequality (3.16),(3.17)

The last two expressions arise from the last term in inequality (3.7) and are bounded by a piece of *E*_{1}(*T*). Thus, by choosing *β*^{2}>1+*δ*_{1}, , we determine computable constants *d*_{1}, …, *d*_{4}, such that(3.18)where *C*_{0} is now a data term.

We next return to equation (3.15) and estimatefor constants *ζ*_{1}, *ζ*_{2}>0 at our disposal. We choose *ζ*_{1} so small that *ζ*_{1}<*λ*_{1}*a*_{0} and bound the *A*(*u*(0), *u*(0)) term by *U*(0), then bound the last two terms by *E*_{1}(*t*), to arrive at(3.19)

Upon utilising the estimates (3.12) and (3.18) in inequality (3.19), we finally derive constants , such that(3.20)

The term *D*_{0} involves only data and thus we have an *a priori* bound for *U*(*t*) which in turn yields an *a priori* bound for either ‖**u**(*t*)‖^{2} or ‖∇**u**(*t*)‖^{2}.

**Remark 3.1.** *In the conditions* *(2.4) and (2.5)**, we have used the same values of α and β for both u _{i}, θ and *, .

*We could carry through the analysis of this section with values of α, β in*

*(2.4)*

*but different values, say γ, δ, in*

*(2.5)*.

*The same remark applies to the analysis of the next section.*

**Remark 3.2.** *We can derive similar estimates to those of this section also for the classical equations of thermoelasticity. When |α|<1, |β|<1, the situation in classical thermoelasticity is not unlike that of type III and is discussed in the next section.*

## 4. Energy bounds when |*α*|, |*β*|<1

When |*α*|, |*β*|<1, we are able to progress only with type II thermoelasticity. The reason for this is clarified below. For the moment, however, we restrict our attention to type II theory.

We commence by observing that equations (2.1) are invariant under time reversal, *t*→−*t*. Furthermore, we rewrite the ‘initial conditions’ (2.4) and (2.5) to put the coefficients *α*, *β*, etc., in front of the *T* terms, e.g. . We effectively have the problem of §3 with 0 and *T* reversed. In §3 we bounded *E*_{1}(*T*) by *E*_{1}(0) and then used |*α*|>1, |*β*|>1 to bound *E*_{1}(0) by data. We here interchange the roles of 0 and *T* and bound *E*_{1}(0) by *E*_{1}(*T*), and then use the fact that 1/|*α*|>1, 1/|*β*|>1 to bound *E*_{1}(*T*) by data. In this way we find a bound for *E*_{1}(*t*) in terms of data for all 0≤*t*≤*T*.

To handle *U*, we write

We then eliminate the quantities at *t*=0 in favour of those at *t*=*T*. In this way, we derive a bound of form *U*(*T*)≤data. This in turn leads to a bound *U*(0)≤data. Then from equation (3.15), we may derive an estimate of the form *U*(*t*)≤data, 0≤*t*≤*T*.

When |*α*|<1, |*β*|<1 in type III theory, we have not been able to find a suitable bound. Since Payne & Schaefer (2002) and Payne *et al*. (2004) did find bounds for a wave equation with dissipation, it is worth examining why things break down.

Payne & Schaefer (2002) establish an *a priori* bound for a solution to the equationfor *A*, a densely defined symmetric linear operator, and *a*>0 constant. They require the restriction |*α*|, |*β*|<e^{−aT}. If one inspects their (clever) proof carefully, then it hinges on being able to bound the dissipation function by , where, in their case, , the norm and inner product being on an appropriate function space. For type III thermoelasticity, if we work with *E*_{1}(*t*) as in §3, then we encounter a dissipation of the form(4.1)whereas the *θ* part in *E*_{1} essentially involves and ‖∇*θ*‖^{2}. Thus, we cannot bound the dissipation term in equation (4.1) by since the function in (4.1) is effectively . The analogue of the Payne & Schaefer (2002) proof, in going from eqn (6.4) to inequality (6.5) of their paper, breaks down.

In fact, we do not expect to find a bound for *E*_{1}(*t*) for |*α*|, |*β*|<e^{−aT} in type III thermoelasticity (*a* in this case is a constant related to the first eigenfunction in the membrane problem, and has nothing to do with the *a* in Payne & Schaefer 2002). We explain below the reason why, in that we show there is non-uniqueness in this range. To see this, we begin by analysing a problem studied in Payne *et al*. (2004), namely,(4.2)

This problem is not the same as type III thermoelasticity. However, it is useful as an illustration because type III thermoelasticity contains higher order dissipation. A similar non-uniqueness result to that given for the simple model is provided below for the full type III thermoelasticity theory. Payne *et al*. (2004) obtain an exponential decay bound for an energy function, and for the *L*^{2}(*Ω*) norm of *θ* provided |*α*|>e^{−aT}, where *a* is defined in equation (2.6) of their paper. We now show that for *α*<0 this restriction cannot be exceeded; this is very different from the Payne & Schaefer (2002) situation. In fact, if we let *θ*=*f*(*t*)ϕ_{n}, where *ϕ*_{n} is the *n*th eigenfunction of the membrane problemwe then find that gives a non-zero solution to equation (4.2), where . It is easy to check *θ* satisfies the ‘initial conditions’ of equation (4.2) for *g*=*h*=0. Thus, since *θ*≡0 is one solution, we have non-uniqueness. This requires the coefficients to satisfy and *α* to have the value For *n*=1, we have the Payne *et al*. (2004) limit, . Thus, for there is non-uniqueness. We stress again that the solution estimate obtained by Payne *et al*. (2004) holds only for Our non-uniqueness result is new and completes the picture by showing that one cannot extend the Payne *et al*. (2004) bounds beyond their time-interval.

Thus, if the dissipation is too strong (as it is in type III thermoelasticity), we do not expect to find bounds for *E*_{1}(*t*) and *U*(*t*) for |*α*|<1, |*β*|<1. To provide evidence in support of this remark, we now demonstrate the analogous non-uniqueness result to that above for the equations of type III thermoelasticity. In one space dimension, we may write the equations of thermoelasticity of type III aswhere for simplicity we take *ρ*, , *a*_{1}, *c*, *k*, *b* to be constants. Suppose these equations are defined on {*x*∈(0, *π*)}×{*t*∈(0, *T*)} with *u*=0, *θ*_{x}=0 at *x*=0, *π*, and

Then, if for any , the constants satisfy the restriction and are solutions to the problem. Since *u*≡0, *θ*≡0 is also a solution we have non-uniqueness, with . Thus, for , we can construct an infinite number of solutions for appropriate coefficients.

**Remark 4.1.** *The above non-uniqueness behaviour has a similar analogue in classical thermoelasticity. In some sense, this shows type III thermoelasticity behaves more like classical thermoelasticity than type II, a fact also observed through the propagation of nonlinear acceleration waves by* *Quintanilla & Straughan (2004)*.

**Remark 4.2.** *Payne & Schaefer (2002)* *also show |α|>1, |β|<1 or |α|<1, |β|>1 lead to non-uniqueness for the abstract wave equation. We believe that this aspect carries over to the thermoelastic models studied here.*

## 5. Non-homogeneous boundary conditions

The analysis in §§3 and 4 has required *u*_{i}=0 and *θ*=0 on the spatial boundary, *Γ*. In practice, one frequently requires inhomogeneous boundary conditions of the form(5.1)

Thus, let *u*_{i}, *θ* be a solution to equations (2.1) or (2.2) with conditions (2.4) and (2.5), but with the boundary conditions (2.3) replaced by boundary conditions (5.1). To derive estimates for suitable norms of *u*_{i} and *θ* in the inhomogeneous problem we introduce functions *v*_{i}, *ψ*, which solve the system(5.2)in *Ω*×(0, *T*], with boundary conditions(5.3)

The functions *v*_{i}, *ψ* satisfy *standard* initial conditions(5.4)

If we consider type II thermoelasticity, then *b*_{ij}≡0 in system (5.2). Since this is the standard boundary-initial-value problem for type II or type III thermoelasticity, then existence of a solution is known (Quintanilla 2002). Thus, ‖**u**(*t*)‖, ‖*ψ*(*t*)‖ are known in various norms for 0≤*t*≤*T*.

Introduce the difference functions *w*_{i}=*u*_{i}−*v*_{i}, *ϕ*=θ−*ψ*. Then, *w*_{i}, *ϕ* solves the problemin *Ω*×(0, *T*) with *w*_{i}=ϕ=0 on *Γ*, and the non-standard conditionsTo obtain an estimate for *u*_{i} we let ‖*u*‖ be a suitable norm for *u*_{i}, e.g. in *L*^{2}(*Ω*) or then from the triangle inequalityThe quantity ‖* v*(

*t*)‖ is known and ‖

*(*

**w***t*)‖ may be found by the estimates in §§3 and 4 in terms of the data functions

*g*

_{i},

*h*

_{i},

*α*

_{0},

*α*

_{1},

*k*

_{i}, , ℓ, and the functions

*v*

_{i}(

*T*) and

*ψ*(

*T*). Since these solve a standard boundary-initial-value problem, they are known and so we can find bounds for ‖

*(*

**u***t*)‖. Likewise, ‖

*θ*(

*t*)‖ and may be estimated in terms of data. Of course, the boundary data and

*θ*

^{B}are involved in the bounds through the functions

*(*

**v***t*),

*(*

**v***T*), etc.

## Acknowledgments

This work was supported by the project ‘Aspects of stability in thermomechanics’ BFM 2003-00309 of the Spanish Ministry of Science and Technology.

## Footnotes

- Received March 1, 2004.
- Accepted July 13, 2004.

- © 2005 The Royal Society