Rasa Kazakevičiūtė-Makovska
email: makowski@am.bi.ruhr-uni-bochum.de

of interacting membranes and strings

Keywords: membranes, strings, hybrid structures, constitutive equations, membrane-string attachments

The equations governing large deformations of hybrid structures are derived from basic principles of mechanics. The considered structures comprise membrane elements with strings arranged inside and possibly along boundaries of the entire structure. No special assumptions are imposed on the mechanical properties of membrane and string elements. The principle of virtual work is formulated under weak continuity hypotheses so that the derived theory takes into account general types of membrane-string attachments. The theory of mechanical constitutive equations for the considered structures is developed and the form of elastic constitutive equations is discussed. A number of specific aspects of string-membrane interactions are analyzed and possible applications of the developed theory are indicated.

Introduction

In continuum mechanics, perfectly flexible (true) membranes and strings are defined as material surfaces and curves, respectively, having no or only insignificant resistance to bending. These theoretical concepts are well-developed (see e.g. Libai and Simmonds [START_REF] Libai | The Nonlinear Theory of Elastic Shells[END_REF] for membranes and Antman [START_REF] Antman | Nonlinear Problems of Elasticity[END_REF] for strings) and they have long been used to model diverse phenomena ranging from mechanics of tension structures to fluid capillarity and biomembranes.

In a more recent work [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF], Atai and Steigmann presented the theory of perfectly flexible hybrid structures consisting of interacting strings (cables) and membranes. Assuming that strings and membranes as well as the interactions between them are perfectly elastic, they derived the complete set of equilibrium equations for two classes of membrane-string attachments (interactions) from the principle of stationary total potential energy. In the first one, the cable is fixed to the membrane at each boundary point and deforms with it as an embedded (material) string, while in the second one the cable and the membrane may slide relative to each other without separating.

Their theory can be successfully applied in modeling hybrid tension structures in civil engineering 1 Tel.: +49 234 3226037; fax: +49 234 3214154

A c c e p t e d m a n u s c r i p t

2 as well as in modeling various phenomena in physical and biological sciences. In such applications, however, it would be desirable to take into account the energy dissipation due to the inevitable irreversibility of the modeled processes and possible other types of attachments. This in turn requires that the a priori assumption of perfect elasticity must be rejected and the formulation of the theory cannot be based on variational formulations.

The static problem of a membrane supported by cables arranged along edges has been also considered by De Tommasi and his coworkers [START_REF] Tommasi | Statics of edge cables in tent structure[END_REF][START_REF] Tommasi | Statics of tent structures[END_REF]. Assuming that the cable is fixed to the membrane boundary (the first type of attachment studied in [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF]), they characterized external loads which can be carried by the structures at equilibrium. In particular, they have shown that in the absence of distributed loading only surfaces with negative Gaussian curvature are admitted as equilibrium configurations of such structures, the fact which have being used in the design of tensioned structures by Otto [START_REF] Otto | Tensile Structures[END_REF] (the proof of this theorem may also be based on the Mohr's circle construction applied to the surface curvature tensor, see e.g. Nutbourne [START_REF] Nutbourne | A circle diagram for local differential geometry[END_REF]). Under the additional assumption that the contact forces between cable and membrane are everywhere contained in the tangent plane to the membrane along the edge cable, it has been found in [START_REF] Tommasi | Statics of tent structures[END_REF] that the equilibrium of the cable and membrane requires that cable equilibrium configurations coincide with the characteristic lines associated with the differential equation of the membrane equilibrium.

In [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF][START_REF] Tommasi | Statics of edge cables in tent structure[END_REF][START_REF] Tommasi | Statics of tent structures[END_REF] cables arranged along the boundary of membrane structure have only been considered with the additional assumption of perfect elasticity of the membrane and cables. A more general approach to the statics of hybrid structures comprised of interacting membranes and strings has been outlined in the author's earlier works [START_REF] Kazakeviciute-Makovska | Statical deformations of flexible membranes coupled to strings[END_REF][START_REF] Kazakeviciute-Makovska | Modelling of fabric structures in civil engineering[END_REF]. The aim of this paper is to show that the equations governing large deformations of hybrid structures with membrane and string elements arranged in any way and of any material behavior can be derived directly from the basic principles of mechanics. The principle of virtual work is next formulated by a simple and rigorous procedure without too restrictive continuity hypotheses, thus allowing not only for both types of attachments considered in [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF] but also some more general kinds. Some as yet unstudied aspects of the problem of constitutive equations for hybrid structures with general form of string-membrane interactions are also examined. Finally, a number of specific aspects of string-membrane interactions are analyzed and some interesting applications of the developed theory are indicated.

The paper is organized as follows. The spatial configurations of hybrid structure comprised of membranes and strings and modeled as piecewise smooth surface is described in Section 2. In Section 3 the integral balance laws of forces and moments of forces are formulated for the hybrid structure subject to a general system of loading. In Section 4 the local equilibrium conditions are derived from the integral balance laws. Next, in Section 5, the principle of virtual work for the

Spatial configurations of hybrid structure

Classically, strings and membranes are modeled as one-and two-dimensional continua whose spatial configurations are represented by smooth curves and surfaces, respectively. Hybrid structures considered in this paper comprise membranes with strings arranged inside and possibly along boundaries of the entire structures. Their spatial configurations cannot be represented by smooth surfaces. Hence, it is assumed that a deformed (unknown) configuration of a hybrid structure may be represented as a piecewise smooth surface. Such a surface, denoted by M , is the union of a finite number of smooth surface elements k M . Edges of M , at which two distinct surface elements meet, form a network, denoted by N , consisting of all smooth curves i C

including their end points α c (Fig. 1). Mathematically, such a surface is defined by the following conditions (see e.g. Protter and Morrey [START_REF] Protter | A First Course in Real Analysis[END_REF]): , , , ,

D D * k k l M M M M k l (1)
and , for some , for some

* i i k l N C C M M k l . ( 2
)
Here

D k
M denotes the interior of the surface element k M and k M its boundary. The position vectors of points in \ M N and C will be denoted by y and y , respectively, and α y will denote the position vectors of isolated nodes α c .

Balance laws of forces and moments of forces

Classical theories of perfectly flexible membranes and strings are based on the assumption that they have resistance to stretching but not to bending, and hence they cannot support couples.

Consequently, the mechanical interactions between parts of the hybrid structure and between the hybrid structure and its environment are described by forces, but not by couples, consisting of (Fig.

2):

q -surface distributed force (defined at points \ \ y M N ) N -surface stress tensor (defined at points \ \ y M N ) q -line distributed force (defined at points y C N ) n -line stress vector (defined at points y C N ) Considering any part of the hybrid structure which in the deformed configuration is represented by Π M , the laws expressing the balance of forces and moments of forces take the form

\{ } { } \{ } \{ } 0 i i i α c Π α Π\C Π c C Π da dl dl q N ν q n f , ( 3
) \ \ { } { } \ \{ } \{ } 0 i i i i α α c Π α Π C Π c C Π da dl dl y q y Nν y q y n y f . ( 4
)
Here { } n n is the stress inside the string at the point i c . Moreover, ν is the unit normal vector to the boundary curve Π lying in the tangent plane.

In general, the surface distributed force q will encompass the force due to excess pressure on both sides of the membrane, the membrane body force due to gravity as well as the membrane inertia force. The same meaning is assigned to the line distributed force q .

The balance laws ( 3) and ( 4) are consistent with the general concepts of perfectly flexible membranes and strings (Libai and Simmonds [START_REF] Libai | The Nonlinear Theory of Elastic Shells[END_REF], Antman [START_REF] Antman | Nonlinear Problems of Elasticity[END_REF]). Such membranes and strings can only carry internal forces tangent to their deformed configurations. They have to be distinguished from thin shells and rods with finite bending stiffness but which are subject to a combination of external loads and boundary conditions which produce extensional and/or compressive forces with no or only negligible bending. In contrast, perfectly flexible membranes and strings considered in this paper cannot support compressive forces.

Equilibrium equations

Balance of forces

In the theory of smooth membranes, the integral balance laws may be replaced by partial differential equations (the local equilibrium equations) at interior points of the membrane whenever all fields are sufficiently smooth for the surface divergence theorem to be applicable (Gurtin and Murdoch [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]). The extension of that theorem to piecewise smooth surfaces and fields was presented in [START_REF] Kazakeviciute-Makovska | Statical deformations of flexible membranes coupled to strings[END_REF][START_REF] Kazakeviciute-Makovska | Modelling of fabric structures in civil engineering[END_REF] and the main results are given in Appendix (see also Forte and Vianello [START_REF] Forte | On surface stresses and edges forces[END_REF] in the context of piecewise smooth boundaries of three dimensional bodies).

Under the assumption that the surface stress tensor N is differentiable at all points \ \ y M N , the patchwork surface divergence theorem yields (see Appendix)

\{ } { } \{ } a b i Π c Π\C C Π dl div da dl Nν N N ν , (5)
( ) ( ) ( ) ) 1 ) ( ) ( ) ( 1 ( ) ( ) ( 2 1 1 α i n i s i c Π α i C Π s s dl n n n n n . ( 7
)
Here s denotes the derivative with respect to s and α n is the number of strings which meet at the nodes c c . Substituting ( 5) and ( 7) into (3), the balance law of forces for the hybrid structure takes the

form 1 ( ) ( ) ( ) ) ) 1 ( 1 1 ( ) ( ( 0 a b α n s iα α i Π\C C Π div da dl N q n q Nν n f . ( 8
)
Since the balance laws of mechanics are required to hold simultaneously for every part of the structure, the integrands of each integral and the last term in (8) must vanish separately by virtue of arbitrariness of Π . From this condition the following local equilibrium equations of forces are obtained:

0 divN q (9)
at points \ \ y M N , 0 a b s n q Nν [START_REF] Protter | A First Course in Real Analysis[END_REF] at points y C N and

1 1 0 α n i α i n f ( 11
)
at nodes α c N .

The most important result of the above derivation are the equilibrium equations ( 10) and [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF] for the string elements. The equilibrium equation ( 9) is well-known in the theory of smooth membranes here derived in the coordinate free notation (c.f. [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]). This equation written in the component form for any local parametrization of the membrane takes the form of three scalar equations (see e.g. Green and Zerna [START_REF] Green | Theoretical Elasticity[END_REF]). ( ) ( ) ( ) ( )

2 2 1 1 2 1
) ). ) Introducing ( 12) and ( 13) into ( 4) and rearranging the terms, the balance law of moments of forces is obtained in the form

\ 1 { ( ) ( )} { ( ) } ( ( ) ) . ( ) ( )} ) } ( ( ) ) . \ { ( 1 ( ( ) 1 ( 1 { ( { 0 a b α T T Π C n s s i α α α i C Π
Equation ( 16) expresses the balance of moments of forces at regular points of hybrid structures with two basic implications. For any direction with the unit tangent vector ν , the surface traction ν n Nν has no component along the normal to the membrane, and hence the stress tensors N must be of the form N N I , where N is the tangential tensor field on M . Moreover, by virtue of (16) the stress tensor N must be symmetric, i.e. N N T . Let us also note that ( )

N N div div I
, so that the surface stress tensor N in the local equilibrium equation of forces ( 9) may be replaced by the tangential stress tensor N . These results obtained here for hybrid structures are well-known in the theory of smooth membranes (see e.g. Gurtin and Murdoch [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]).

The condition of vanishing of the line integral in [START_REF] Truesdell | The Non-Linear Field Theories of Mechanics[END_REF] requires that

By implication, "the jump terms" disappear from [START_REF] Song | Analysis of tensioned membrane structure considering cable sliding[END_REF] and this equation reduces to the form wellknown in the theory of smooth strings (see e.g. Antman [START_REF] Antman | Nonlinear Problems of Elasticity[END_REF]):

0 s y n . ( 21
)
The condition [START_REF] Leewood | Design and analysis of fabric structures with the finite element method[END_REF] requires that the stress vector n in the string must be tangent to its deformed configuration, and hence it must be of the form N n e, where N is the tension and s e y denotes the unit vector tangent to C .

Finally, it follows ( 14) that at the nodes α c of the strings network the following conditions must be satisfied

1 ( ) ( 1 (
)

0 α n i α α i y n y f (22)
which, with the use of [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF], may be written as

1 1 ( ) ( ) ( 1 1 ( ) ( ) 1 ) ( ) 1 0 α α n n i α i i i y n y n . ( 23
)
Since α i y y for all strings having α c as a common node, the condition ( 23) is trivially satisfied.

Principle of virtual work

The equilibrium equations for the hybrid structure derived in the previous sections may be reformulated in the form of the virtual work principle. To this end, let w , w and α w be any spatial vector fields which are defined at points \ \ y M N , y C N and at nodes α c , respectively, and collectively denoted by ( , , ) ( , , w α w w w . As direct implication of the equilibrium equations ( 9), [START_REF] Protter | A First Course in Real Analysis[END_REF] and ( 11), we have

( ; ) 0 0 w G ( , ( 24
)
where for any part ( of the hybrid structure which in the deformed configuration occupies the domain Π M . Under the regularity assumptions of the previous section, the terms in (25) containing the surface divergence of N and the derivative of n can be transformed, using the patchwork surface divergence theorem and the corresponding theorem for line integrals (see Appendix):

1 ( ; ) ( ) • ( ) • ( ) • ) • ( )• ( )• 1 ( 1 1 ( )• ( ( w a b α n s iα α α i Π\C C Π div da dl N q w n q Nν w n f w G ( (25)
\{ } ( )• • • • { } ( )• \{ } • • • \{ } • • \{ } a b i Π\C Π\C C Π Π c div da D da dl dl N w N w Nν w Nν w (26) and 1 • ( • ) • • ( • ) ( ( • ) ), • ( • ) • • ( • ) • • ) ) 1 • ) ), ( • ) ( ( 1 ( • ) ( ( 1 • • α i s s s C Π CΠ CΠ n s i i c Π α i C Π
G G G ( ( ( (28)
Here

\ 1 1 ( ; ) • ( • • • ) ( ( • ) ( )• ) • ( • • • ) \ \ • ( • \ ( • 1 1 ( ( • ) ( )• ) 1 1 ( • ) ( )• ) 1 1 w a b a b α int s Π C C Π n n i i α α i i D da dl N w n w Nν w Nν w n w n w G ( (29) and \
\ { } ( ; ) • • • ( • ) ( • ) { } ( • ) ( • ) ( • \ \{ } \ • • • \ \{ } • • \{ } w i i ext i α α c Π α Π C Π c C Π da dl dl q w Nν w q w n w f w G ( . (30)
The equation (28) holds for every part ( of the hybrid structure, whenever the balance laws are satisfied. The converse theorem is also true, as it can be shown by reversing the whole procedure of derivation leading to (28). With w considered as the generalized virtual displacement, the integral identity (28) represents the principle of virtual work of internal and external forces defined by ( 29) and (30), respectively.

The principle (28) provides the weak form of the integral balance law (3) for the entire hybrid structure with any type of strings network and undergoing possibly non-smooth deformations. Moreover, this principle applies independently of the mechanical properties of membranes and strings which must be specified by constitutive equations.

Stress power and principle of irreversibility

According to the standard concepts of continuum mechanics, the constitutive equations must be consistent with the law of thermodynamics. Within purely mechanical theories of hybrid structures, the law of thermodynamics implies that the temporal increase in free energy of any part of the hybrid structure ( must be less than or equal to the power expended on ( . Letting ( ) (36)

F ( D L υ FF . ( 35
Here λ y is the string deformation vector being the one-dimensional counterpart of the surface deformation gradient F and the prime stands for the derivative with respect to the arc length parameter S in the reference configuration of the string. Similarly to (35), the spatial velocity vector l is given by

1 1 s λ l υ λ . ( 37
)
The stress power density resulting from the interactions of string and membrane elements can be obtained as follows

0 0 ˆ• • • • 0 0 • • • • 0 a b a b a b a b σ λ Sν υ Sν υ Nν υ Nν υ , ( 38
)
where 0 ν is the unit normal vector to the boundary curve Π in the reference configuration of the hybrid structure.

In consistency with the expression (33) for the stress power, the free energy of ( must be of the form \ ( ) ( )

Π C C Π ΨdA ψ φ dL dL Π C C \ Π \ Π C C \ \ ΨdA ψ φ d ( ) ( C Π ψ ( Π F . ( 39
)
Here Ψ and ψ are the free energy densities of the membrane and string elements, and φ is the free energy density resulting from the string-membrane interactions. All densities are measured in the reference configuration. Introducing (33) and (39) into (31), the energy im-balance requires that the following two conditions must be satisfied simultaneously

0 0 Δ σ Ψ , ( 40
) ˆ( ) 0 ˆ( ) 0 δ σ σ ψ φ (41)
with σ , σ and σ being given by (34), ( 36) and (38), respectively.

Elastic constitutive equations

According to the results of the previous section, the constitutive equations for hybrid structures must be consistent with the dissipation inequalities ( 40 The assumption leading to (42) is natural in the analysis of elastic structures because the stress power (38) is non-zero for discontinuous deformations only, in which case there is the energy dissipation.

For perfectly elastic (hyperelastic) hybrid structures the free energies Ψ and depend only on the surface deformation gradient F and the stretch vector λ , respectively, that is ( ), ( ). ( ), ( ).

Ψ Ψ ψ ψ F λ ( 43
)
In this special case, there is no energy dissipation and the local im-balance laws (40) and (42) take the form

1 ( ( ) ( ))• 0 ( ( ) ( ))• 0 1 ) 1 T Δ j Ψ F N F F F (44)
and

( ( ))• 0 ( ( ))• 0 δ ψ λ n λ λ . ( 45
)
The conditions ( 44) and (45) must hold for all F and λ , respectively, yielding the constitutive equations for perfectly elastic hybrid structures in the classical form:

1 ( ( )) , ( ) ( ( )) , ( ) 1 ( T j Ψ ψ F λ N F F n λ . ( 46
( ( )) ( ( )) 1 1 T λ j Ψ λ ψλ C N F C F n λ . ( ( )) 1 1
) 48
The equations (48) provide the most general form of the constitutive equations satisfying the principle of frame indifference. Specific forms of the elastic constitutive equations for elastomeric and biological membranes have been considered in [START_REF] Kazakeviciute-Makovska | Non-linear response functions for transversely isotropic elastic membranes[END_REF] and for anisotropic fabric membranes in [START_REF] Kazakeviciute-Makovska | Modelling of fabric structures in civil engineering[END_REF].

The use of the general constitutive equations (46) together with the additional assumption that loadings acting on the hybrid structure are conservative reduces the integral identity (28) to the principle of stationary total potential energy which has been used by Atai and Steigmann [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF] in their formulation of the theory of hybrid structures. However, the modeling of elastomeric and biological membranes requires that inelastic effects, such as stress softening, permanent set and hysteresis, are also properly accounted for. These aspects of the constitutive modeling have been considered in [START_REF] Kazakeviciute-Makovska | Damage-induced stress-softening effects in elastomeric and biological membranes[END_REF] for smooth membranes and the same approach may be adopted for hybrid structures.

Analysis and discussion

According to the results obtained in this paper, the problem of determination of static equilibrium shapes for hybrid structures reduces to the solution of the field equations ( 9) and [START_REF] Protter | A First Course in Real Analysis[END_REF] for appropriate constitutive equations and external boundary conditions. Obviously, the admissible shapes of hybrid structures depend on the nature of the membrane-string interactions. These interactions are described by the equilibrium equation ( 10) and the corresponding expression (38)

for the stress power, and both must be considered simultaneously in the analysis of possible types of membrane-string attachments.

For the following discussion, it will be convenient to rewrite the equilibrium equation [START_REF] Protter | A First Course in Real Analysis[END_REF] in form

( ) ( ) ) 0 s ν ν n q n n , ( 49
)
where ν n N ν and ν n N ν are the stresses which membrane elements M and M exert on the string C being the common boundary of both. Correspondingly, the stress power density per unit length of the deformed string (38) may be rewritten as

ˆ( )• ( • • ) ( )• ( • • ) ) • ( • • ) ν ν ν ν σ n n υ n υ n υ . ( 50
)
Here υ is the velocity of the string, υ and υ are the limits of the velocities for the adjacent Consequently, the stress vectors and the velocity vectors appearing in the expression (50) for the stress power may be represented by their components relative to the basis { , , } e m b so that

A c c e p t e d m a n u s c r i p t 3 hybrid

 3 structure is derived from the local equations representing the balance of forces. The principle of irreversibility for such structures is formulated in Section 6 and the constitutive equations for elastic behavior of hybrid structures are discussed in Section 7. Section 8 contains the analysis of a few special types of membrane-string attachments and their relevance in the design of tensioned fabric structures. Certain possible applications of the theory in the field of cell cleavage are pointed out.

Fig. 1 .A c c e p t e d m a n u s c r i p t 4 From

 14 Fig. 1. Spatial configuration of the hybrid structure

αfFig. 2 .A c c e p t e d m a n u s c r i p t 5 Moreover

 25 Fig. 2. Forces acting in the deformed configuration of hybrid structure

  ... 1, 2,... i n, is the set of points at which strings i C intersect the boundary curve Π and i i c

  stands for the surface divergence operator (seeGurtin and Murdoch [11]) in the deformed configuration of the hybrid structure and the jump a b Nν at any point y C is defined byNν N ν N ν a b . (6)Here, the orientation of the curve C which is a common boundary of two membrane elements M and M is assumed to coincide with the orientation of the boundary curve M of M . Moreover,(.) denote the limiting values of the respective quantity computed on both sides of the curve C .Let us further assume that the curve C is given in the parametric form ( ) s y = y , where s indicates the arc length parameter in the deformed configuration of the string. Then the application of the integration by parts to the terms i i c Π n in the balance law of forces (3) yields2 1 1

A c c e p t e d m a n u s c r i p t 7 4. 2

 72 Balance of moments of forcesUnder the same regularity assumptions, the application of the patchwork surface divergence theorem together with the use of some differential identities yields (see Appendix) inclusion operator in the deformed configuration of the hybrid structure, s D grad denotes the surface gradient operator (see Gurtin and Murdoch[START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]

  is satisfied if the integrands of each integral and the last term vanish separately by virtue of the arbitrariness of Π . The condition of vanishing of the surface integral in (14law of forces (9) holds, then the first term in (15) vanishes and equation (15) reduces to the form T T IN NI .

  if the balance law of forces[START_REF] Protter | A First Course in Real Analysis[END_REF] holds. Then the first term in[START_REF] Ishii | Form finding analysis in consideration of cutting patterns of membrane structures[END_REF] the balance of moments of all forces acting on the string, including the internal stresses due to interactions of the string with adjacent membrane elements. Because y y y at all points y C N , it follows from (19) that

A c c e p t e d m a n u s c r i p t 9

 9

  (26) and (27) into (25) and rearranging the terms, the expression ( ; ) w G ( may be written as the difference of two terms, so equation (24) takes the form (

)

  The line elements in the reference and deformed configuration are related by λ , ds λdS , where • • λ λ λ is the stretch of string. Using this relation, the stress power density for string elements is obtained in the form

  ) and (41). If the deformation of the entire structure is continuous across the strings, then υ υ at all points y C N . By implication, the stress power density (38) resulting from the interactions of string and membrane elements vanishes, ˆ0 0 σ , and the corresponding free energy must be set to zero. In effect, the energy im-balance (41) reduces to the

  denotes the right Cauchy-Green deformation tensor for membranes and • • λ λ λ is the stretch of the string. Then, by implication, the constitutive equations (46) for the

A c c e p t e d m a n u s c r i p t 13

 13

  membrane elements. The equation (49) and expression (50) apply to any string in the network to which two membrane elements are attached in one or another way to be discussed below. Here let us only note that (49) and (50) apply also to the case of the string attached to the boundary of the hybrid structure, in which case we simply set 0 that the stress vector n is necessarily tangent to the deformed axis of the string, κ is the curvature of C and m denotes the principle normal vector. With the standard definition of the binormal vector b e m , the vector equilibrium equation (49) may be expressed in component form as

  Figure(s)

 A c c e p t e d ma n u s c r i p t 10 denote the free energy of ( , this requirement takes the form of energy im-balance (see Truesdell and Noll [START_REF] Truesdell | The Non-Linear Field Theories of Mechanics[END_REF] in the context of three dimensional bodies) ( ) ( ; ) ( ) 0 int ( ; ) ( ) 0

where ( ; ) v int P ( is the mechanical power of internal forces, i.e. the stress power of the part ( of the hybrid structure.

In mechanics, the power is defined as the rate of working of all forces acting on the body under consideration. The mechanical power for any part ( of the hybrid structure follows directly from (25) by assuming that w is the vector field of real velocities ( , , ) ( , , ) v

The expression for the stress power int P can be obtained from (29) by taking w to be the generalized vector of real velocity fields v . Under the continuity condition α υ υ at the nodes α c , the stress power of the hybrid structure may be written relative to an arbitrarily chosen reference configuration in the form

where σ and σ are the stress power densities of membrane and string elements, respectively, and σ signifies the stress power density resulting from the interactions of string and membrane elements.

From now on, Π denotes a part of the hybrid structure in a fixed reference configuration and all densities are measured per unit area and unit length in that configuration.

The relation between area elements in the reference and deformed configurations of the smooth membrane parts is given by the classical formula da jdA , where j detF and F y y denotes the surface deformation gradient relative to the reference configuration. With the help of this relation, the stress power density for membrane elements may be expressed as

is the surface stress tensor in the reference configuration (see Gurtin and

Murdoch [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]) and the spatial velocity gradient L is given by 1 1

A c c e p t e d m a n u s c r i p t 14

Now different types of membrane-string attachments may be considered.

The simplest type of membrane-string connection is that described by Atai and Steigmann [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF] as stiff attachment. In this case, υ υ υ and the stress power (53) vanishes, i.e. ˆ0 0 σ for all possible deformations of the structure. Thus, the nature of the membrane-string connection is entirely determined by the equilibrium equations (52). In the case of the string arranged along a part of the external boundary of the structure and in the absence of distributed loads, the equations (52) reduce to the form

This very special case has been discussed in detail in De Tommasi et al. [START_REF] Tommasi | Statics of edge cables in tent structure[END_REF][START_REF] Tommasi | Statics of tent structures[END_REF].

The more general type of the membrane-string connection is that of the boundary string admitting sliding but without separation of the string and adjacent membrane elements. In this case,

so the stress power (53) simplifies to the form

For sliding without friction described by Atai and Steigmann [START_REF] Atai | Coupled deformations of elastic curves and surfaces[END_REF], the membrane elements exert no traction on the string so that

and the stress power vanishes, ˆ0 0 σ , as in the case of stiff attachment. Moreover, the first equilibrium equation in (52) reduces to the form

In the special case when there are no distributed forces along the string, • 0 0 q e and the equation (58) reduces to the form 0 0 s N , so that the tension is constant along the string. This is the second type of attachment considered in [START_REF] Tommasi | Statics of edge cables in tent structure[END_REF][START_REF] Tommasi | Statics of tent structures[END_REF] for the case of boundary string.

The problem of membrane-string interactions naturally arises in the design of tension fabric structures. However, in that case, the problem has been invariably approached by various numerical methods in which the membranes and strings are treated as separate structural elements connected only at isolated points (see e.g. the discussion in [START_REF] Ishii | Form finding analysis in consideration of cutting patterns of membrane structures[END_REF][START_REF] Ishida | Theoretical analysis for equilibrium state of membrane structures considering slide between cable and membrane element[END_REF][START_REF] Song | Analysis of tensioned membrane structure considering cable sliding[END_REF]). The general theory presented in this paper provides the theoretical basis for analytical studies of membrane tension structures with numerous special types of membrane-string attachments. Moreover, the principle of virtual work formulated in Section 5 may be used as the basis for the finite element formulations following the approach of [START_REF] Tabarrok | Nonlinear analysis of tension structures[END_REF][START_REF] Leewood | Design and analysis of fabric structures with the finite element method[END_REF] successfully applied in the design of smooth fabric structures. The membranes and hybrid structures with no flexural rigidity will become wrinkled as soon as one of the principal

A c c e p t e d m a n u s c r i p t

15 stresses is non-positive. This requires the application of special methods, such as the dynamic relaxation [START_REF] Haseganu | Analysis of partly wrinkled membranes by the method of dynamic relaxation[END_REF], in the design of these structures.

Perfectly flexible membranes and strings have also been used in modeling numerous phenomena in biomechanics (see e.g. [START_REF] Fung | Biomechanics: Mechanical Properties of Living Tissues[END_REF][START_REF] Lardner | Elastic models of cytokinesis[END_REF]). An interesting application of the theory involving the membrane-string interactions arises in modeling the cytokinesis process [START_REF] Kazakeviciute-Makovska | Statical deformations of flexible membranes coupled to strings[END_REF].

Appendix A. Patchwork surface divergence theorem

If M is a smooth surface whose boundary M is piecewise smooth, u is a smooth vector field on M and S is a smooth tensor field such that ( )

x E T M S at x M , then the following surface divergence theorem holds (Gurtin and Murodch [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF]):

Here ( ) x x T M ν is the outward unit normal to M , div and D are the surface divergence and surface gradient operators (see Section 4).

A vector (tensor) field u is said to be piecewise smooth on a surface M (not necessarily smooth) if there exists a partition

M of M such that for each i M the surface gradient Du exists at all points of i M D and Du is continuous on M M M . Consider now a piecewise smooth surface M consisting of two smooth surface elements M and M having a curve C as the common boundary. If u is a piecewise smooth vector field and S is a piecewise smooth tensor field, then the divergence theorem (A.1) can be applied in each surface element separately. Thus, one gets: where the jumps are defined as in (A.3) with the tensor product replaced by corresponding product.


Thomas Forst
email: thomasf@ikfe.de

Matthias M Weber

Mirjam Löbig

Ute Lehmann

Jürgen Müller

; Cloth Hohberg

Christiane Friedrich

Winfried Fuchs

Andreas Pfützner

Pioglitazone in Addition to Metformin Improves Erythrocyte Deformability in Patients with Type 2 Diabetes mellitus

Keywords: Pioglitazone, Diabetes mellitus Type 2, Erythrocyte Deformability, Adiponectin Clinical Science Immediate

The aim of this study was to compare the effect of pioglitazone or glimepiride on erythrocyte deformability in T2DM.

The study covered 23 metformin treated T2DM patients with an HbA1c above 6.5%. Patients were randomized to receive either 15 mg Pioglitazone (PIO) bid. or 1 mg Glimepiride (GLIM) bid. in combination with 850 mg Metformin bid. for 6 months. Blood samples were taken for the measurement of fasting glucose, HbA1c, fasting insulin, intact proinsulin, adiponectin, and heamatocrit (Hct). In addition, the erythrocyte elongation index (EI) was measured using laserdiffractoscopy.

Both treatments significantly improved HbA1c levels (PIO -0.9 ± 1.1%; GLIM -0.6 ± 0.4%; p<0.05 respectively) and end up in comparable HbA1c levels after 6 months (PIO 6.5 ± 1.2; GLIM 6.2 ± 0.4) Treatment with PIO reduced fasting insulin levels (-8.7 ± 15.8 mU/L; p=0.098), intact proinsulin levels (-11.8 ± 9.5 pmol/L; p<0.05), and Hct (-1.3 ± 2.3%; p=0.09), while adiponectin levels increased (8.2 ± 4.9 µg/mL; p<0.05). No significant change in these parameters could be observed during GLIM treatment. PIO improved the EI, resulting in a significant increase in EI at all physiological shear stress ranges (0.6 to 6.0 Pa; p<0.05).

The improvement in EI correlated with the increase in adiponectin levels (r=0.74; p<0.001), and inversely with intact proinsulin levels (r=-0.47; p<0.05). This is the first study showing an improvement in EI during treatment with pioglitazone, which was associated with an increase in adiponectin and a decrease in intact proinsulin levels.

2010-03-09 page 3 of 21

Objective

Patients with type 2 diabetes mellitus suffer from an increased incidence of micro-and macrovascular complications. It is widely accepted that rheological properties of erythrocytes are of considerable significance in many circulatory disorders like atherosclerosis, essential hypertension and diabetes mellitus [START_REF] Marcinkowska-Gapinska | Blood fluidity and thermography in patients with diabetes mellitus and coronary artery disease in comparison to healthy subjects[END_REF][START_REF] Koltai | The effect of blood glucose levels on hemorheological parameters, platelet activation and aggregation in oral glucose tolerance tests[END_REF][START_REF] Zilberman-Kravits | Increased red cell aggregation is correlated with HbA1C and lipid levels in type 1 but not type 2 diabetes[END_REF]. Several clinical studies document decreased erythrocyte deformability in diabetic patients [START_REF] Hach | C-peptide and its C-terminal fragments improve erythrocyte deformability in type 1 diabetes patients[END_REF][START_REF] Brown | Association of reduced red blood cell deformability and diabetic nephropathy[END_REF][START_REF] Garnier | Erythrocyte deformability in diabetes and erythrocyte membrane lipid composition[END_REF][START_REF] Shin | Progressive impairment of erythrocyte deformability as indicator of microangiopathy in type 2 diabetes mellitus[END_REF], and even in children with short duration of diabetes, impaired erythrocyte deformability was observed [START_REF] Linderkamp | Impaired deformability of erythrocytes and neutrophils in children with newly diagnosed insulin-dependent diabetes mellitus[END_REF]. Impaired tissue perfusion, as observed in diabetes mellitus, could be attributed to a reduction in erythrocyte deformability and an increase in blood viscosity [9;10]. It is crucial for the erythrocytes to pass through the capillaries in order to supply oxygen to the surrounding tissues, and decreased erythrocyte deformability contribute to an impaired oxygen and nutrition supply in diabetic microangiopathy [11;12]. Insulin resistance as observed in patients with diabetes mellitus type 2 is associated with endothelial dysfunction [13;14] and an increase in blood viscosity [15;16]. Pioglitazone was shown to improve insulin sensitivity [START_REF] Miyazaki | Improved glycemic control and enhanced insulin sensitivity in type 2 diabetic subjects treated with pioglitazone[END_REF], to improve endothelial function [START_REF] Maegawa | Short-term low-dosage pioglitazone treatment improves vascular dysfunction in patients with type 2 diabetes[END_REF][START_REF] Forst | Influence of glucose control and improvement of insulin resistance on microvascular blood flow and endothelial function in patients with diabetes mellitus type 2[END_REF][START_REF] Horio | Pioglitazone-induced insulin sensitization improves vascular endothelial function in nondiabetic patients with essential hypertension[END_REF][START_REF] Papathanassiou | Pioglitazone vs glimepiride: Differential effects on vascular endothelial function in patients with type 2 diabetes[END_REF] and to improve the overall cardiovascular risk profile in diabetic and in non-diabetic subjects [START_REF] Forst | Effect of simvastatin and/or pioglitazone on insulin resistance, insulin secretion, adiponectin, and proinsulin levels in nondiabetic patients at cardiovascular risk--the PIOSTAT Study[END_REF][START_REF] Forst | Pleiotrophic and anti-inflammatory effects of pioglitazone precede the metabolic activity in type 2 diabetic patients with coronary artery disease[END_REF][START_REF] Mazzone | Effect of pioglitazone compared with glimepiride on carotid intima-media thickness in type 2 diabetes: a randomized trial[END_REF][START_REF] Nissen | Comparison of pioglitazone vs glimepiride on progression of coronary atherosclerosis in patients with type 2 diabetes: the PERISCOPE randomized controlled trial[END_REF]. In contrast, there is increasing evidence, that antidiabetic treatment with sulfonylurea, especially in combination with metformin, might have a negative impact on the cardiovascular outcome in patients with diabetes mellitus type 2 [START_REF] Evans | Risk of mortality and adverse cardiovascular outcomes in type 2 diabetes: a comparison of patients treated with sulfonylureas and metformin[END_REF][START_REF] Pfutzner | IRIS II study: intact proinsulin is confirmed as a highly specific indicator for insulin resistance in a large cross-sectional study design[END_REF][START_REF] Rao | Is the combination of sulfonylureas and metformin associated with an increased risk of cardiovascular disease or all-cause mortality?: a meta-analysis of observational studies[END_REF]. The aim of this investigation was to compare the effect of adding pioglitazone or glimepiride on erythrocyte deformability in patients with type 2 diabetes mellitus already treated with metformin.

Methods and Results

Patients

This is a sub analysis of the patients from the PIOfix study, treated at the ikfe study site in Mainz, in whom erythrocyte deformability was measured immediately after venous blood draw. Patients had to be on a stable treatment with metformin monotherapy in a maximal individually tolerated dosage (850 -2000 mg) within the last 12 weeks and with an HbA1c ≥ 6.5 %. Subjects were excluded if they had been treated with insulin or other oral antidiabetic drugs with the exception of metformin within the last 12 weeks. All other concomitant treatments were kept stable during the study. Patients were randomised to pioglitazone 15 mg bid. or glimepiride 1 mg bid. in combination with 850 mg metformin bid.

Fasting blood samples were taken at baseline and after 24 weeks of treatment for the measurement of plasma glucose, HbA1c, insulin, intact proinsulin, adiponectin, haematocrit, and the assessment of erythrocyte deformability.

The study was conducted in accordance with the Declaration of Helsinki and was approved by the local ethical committee. All subjects gave written informed consent.

Measurement of Erythrocyte Deformability

Measurement of erythrocyte deformability was performed using laserdiffractoscopy from freshly obtained venous blood samples. Laserdiffractoscopy was performed using Rheodyn SSD shear stress diffractometer and corresponding Dextran FP-60 24 mPas (Myrenne GmbH, Roetgen, Germany). The method of laserdiffractoscopy has been described in detail previously [START_REF] Kunt | The effect of human proinsulin Cpeptide on erythrocyte deformability in patients with type 1 diabetes mellitus[END_REF]. In summary a He-Ne-Laser detects deformation of erythrocytes between two parallel glass discs, one of them rotating and executing defined shear stresses as per following formula: According to Baskurt et al. the maximal RBC elongation index (EI max ) and the shear stress required for half of this maximal deformation (SS 1/2 ) was calculated [START_REF] Baskurt | Parameterization of red blood cell elongation index--shear stress curves obtained by ektacytometry[END_REF].

Additional Laboratory Measurements

Blood samples for all other laboratory measurements were centrifuged and kept at -20°C until analysis. Plasma glucose concentrations were determined by the glucose dehydrogenase method (Super GL, RLT, Möhnesee-Delecke, Germany). Insulin and intact proinsulin was measured by a chemiluminescence assay (Invitron, Monmouth, UK), adiponectin was measured by an Sandwich-Assay (ELISA) (TECOmedical, Sissach, Switzerland).

HbA1c was determined by high-performance liquid chromatography (Menarini Diagnostics, Neuss, Germany).

Subjects sample size considerations and statistical analysis

No clinical information was available about the effect of pioglitazone or glimepiride on erythrocyte deformability in patients with diabetes mellitus type 2. Therefore, no confirmatory study size calculation was possible and the obtained results were interpreted in an exploratory sense.

For the erythrocyte deformability two-sided p-values for within-and between-group treatment differences were calculated for the absolute change from baseline, considering an ANCOVA model with the fixed effect factors for treatment group and centre and with the baseline value as covariate. For all other within and between group comparisons, two-sided p-values resulting from the one-and two-sample t-tests are reported. Pearson correlation coefficients are calculated to assess the relationship between different variables. P-values < 0.05 were considered to be statistically significant.

Results:

Twenty three patients with type 2 diabetes were included in the analysis. Eleven patients were randomised to the pioglitazone group and twelve patients were randomised to the glimepiride group. Treatment with pioglitazone increased erythrocyte deformability at all shear stress rates tested in our study protocol (figure 1A). In the physiological shear stress range of 0.6 to 6.0 Pa a significant improvement in erythrocyte deformability compared to baseline could be observed during treatment with pioglitazone. In contrast, glimepiride treatment showed a slight, albeit non-significant deterioration in erythrocyte deformability within our study (figure 1B). In the physiological shear stress range in between 0.6 Pa and 6.0 Pa, the change from baseline in the erythrocyte elongation index (EI) was significantly different in between the two treatment groups. During pioglitazone treatment, EI max decreased from 82.1± 8.5 to 76.9 ± 17.8 % (p<0.01), and the SS 1/2 decreased from 6.3 ± 1.4 to 5.2 ± 3.7 Pa (p < 0.001). No significant effect of glimepiride treatment on EI max (from 87.9 ± 19.9 to 86.3 ± 15.1 %) or SS 1/2 (from 6.4 ± 3.2 to 6.7 ± 2.5) could be observed in our study.

During pioglitazone treatment Hct slightly decreased from 42.5 ± 2.8 % to 41.1 ± 3.8 % (p=0.09), while a slight increase from 41.0 ± 2.4 % to 41.2 ± 2.6 % (n.s.) could be observed during treatment with glimepiride A significant correlation could be observed in between the increase in adiponectin plasma levels and the increase in the EI (r=0.74; p<0.001; figure 2) while an inverse relationship could be observed in between the EI and intact proinsulin plasma levels (r=-0.47; p<0.05; Pioglitazone is used for the treatment of insulin resistance in patients diabetes mellitus type 2.

Beyond improving blood glucose control, treatment with pioglitazone was shown to effect micro-and macrovascular blood flow, and to reduce the risk for cardiovascular complications in patients with type 2 diabetes [10;19;23;33-35]. Pioglitazone was shown to reduce inflammation [START_REF] Pfutzner | Pioneer study: PPARgamma activation results in overall improvement of clinical and metabolic markers associated with insulin resistance independent of long-term glucose control[END_REF], to improve endothelial function [START_REF] Papathanassiou | Pioglitazone vs glimepiride: Differential effects on vascular endothelial function in patients with type 2 diabetes[END_REF], to reduce the intima media layer of the carotid artery [24;36], and to decrease atherosclerotic plaques in the cardiovascular system [START_REF] Nissen | Comparison of pioglitazone vs glimepiride on progression of coronary atherosclerosis in patients with type 2 diabetes: the PERISCOPE randomized controlled trial[END_REF]. In contrast to glimepiride, pioglitazone improved erythrocyte deformability at all physiological shear stress ranges. The improvement in erythrocyte deformability was found to be associated with an increase in plasma adiponectin concentrations and a decrease in intact proinsulin concentrations.

Adiponectin, an adipokine released from the visceral adipose tissue, was found to increase insulin sensitivity, to improve endothelial function and to evolve several anti-inflammatory and anti-oxidative effects [37;38]. Decreased adiponectin plasma levels contribute to the development of insulin resistance and type 2 diabetes mellitus, and a causal relationship between low adiponectin levels and the development of cardiovascular disease has been suggested [39;40] . Treatment with pioglitazone was shown to reduce intima media thickness of the carotid artery, which was found to be associated with an increase in adiponectin levels [START_REF] Forst | Pharmacological PPARgamma stimulation in contrast to beta cell stimulation results in an improvement in adiponectin and proinsulin intact levels and reduces intima media thickness in patients with type 2 diabetes[END_REF]. Erythrocyte deformability is highly dependent on the fluidity of the erythrocyte membrane [42;43]. In hypertensive patients it was shown that the membrane fluidity of erythrocytes was inversely related to plasma adiponectin levels, indicating that hypoadiponectinemia might be associated with decreased membrane fluidity and decreased erythrocyte deformability [START_REF] Tsuda | Adiponectin and membrane fluidity of erythrocytes in normotensive and hypertensive men[END_REF]. It has been demonstrated that adiponectin stimulates the production of NO in endothelial cells [START_REF] Chen | Adiponectin stimulates production of nitric oxide in vascular endothelial cells[END_REF]. Many studies demonstrated an interrelation between NO and erythrocyte deformability [46;47]. In our study, treatment with pioglitazone in type 2 diabetic patients increased plasma adiponectin levels and improved erythrocyte deformability over a wide range of physiological shear stress rates. Therefore it seems conceivable, that treatment with pioglitazone in type 2 diabetic patients improve erythrocyte deformability by increasing plasma adiponectin levels and subsequent stimulation of the endothelial NO release.

In type 2 diabetic patients, an increased haematocrit effects endothelial function [START_REF] Froom | Blood viscosity and the risk of death from coronary heart disease[END_REF], and is assumed to contribute to an increase in the risk coronary vascular disease [START_REF] Natali | Haematocrit, type 2 diabetes, and endothelium-dependent vasodilatation of resistance vessels[END_REF]. In accordance with previous studies [50;51], our study confirmed a slight decrease in Hct during treatment with pioglitazone. Also, no association could be observed in between the improvement in erythrocyte flexibility and the reduction in Hct, interferences in between changes in Hct and erythrocyte deformability can not be excluded, due the limited study size.

In conclusion this is the first study to show an improvement in erythrocyte deformability during treatment with pioglitazone in patients with type 2 diabetes mellitus. In consideration with data from the literature, it could be assumed, that the increase in adiponectin levels caused by pioglitazone treatment might affect erythrocyte membrane fluidity and thereby improve erythrocyte deformability. Our results further confirm important effects of pioglitazone in the modification of the cardiovascular risk profile which can be observed beyond glucose control.

Study limitation

A potential limitation of our findings is that the data were obtained from a subgroup treated at one site within a large multi-centre study. Because erythrocyte deformability was a secondary endpoint and no confirmatory study size estimation for this parameter deemed possible, the findings should be interpreted in an exploratory sense. Even if no association was found in between erythrocyte deformability and HbA1c levels or lipid parameters in the study, insufficient statistical power do not allow to rule out such a relationship.

Clinical

  Science Immediate Publication. Published on 28 May 2010 as manuscript CS20100161 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100161 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. 2010-03-09 page 4 of 21

  stress, r = laser beam distance from rotation center, = viscosity, h = height of gap between discs, t = time) Adjusting for equipment-specific values (r = 25mm, h = 0,5mm und = 24mPas) the formula is condensed to (Pa) Pa * rpm (rpm = revolutions per minute) The applied shear stress range is electronically regulated and consists of 8 levels (0,3 Pa; 0,6 Pa; 1,2 Pa; 3 Pa; 6 Pa; 12 Pa; 30 Pa; 60 Pa). Clinical Science Immediate Publication. Published on 28 May 2010 as manuscript CS20100161 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100161 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. 2010-03-09 page 5 of 21 The erythrocyte deformability measurement detects scattered light intensities along orthogonal axes (A, B) of red blood cells within the laser diffraction light cone. The erythrocyte Elongation Index (EI) is calculated by the following equation: EI (%) = ((A-B)/(A+B))*100.

Clinical

  Science Immediate Publication. Published on 28 May 2010 as manuscript CS20100161 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100161 A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited 2010-03-09 page 9 of 21

Figure 1 AFigure 1 BFigure 2 :Figure 3 :

 1123 Figure 1 A. Erythrocyte Elongation Index (EI) at baseline and 6 months after treatment with pioglitazone (mean ± SD); □ baseline, ■ 6 months treatment with pioglitazone; * = p<0.05 baseline vs. 6 months)

Figure 1 A

 1 Figure 1 A:

Figure 1B :

 1B Figure 1B:

Figure 2

 2 Figure 2:

Figure 3 :

 3 Figure 3:

LDL (mg/dl) 105.9 ± 34.3 68.8 ± 30.5 * Triglycerids 200.1 ± 31.3 149.3 ± 50.3 *Table 1 :

 1 Clinical characteristics of the investigated groups (Mean ± SD; * = p<0.05) Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

			Clinical Science Immediate Publication. Published on 28 May 2010 as manuscript CS20100161 Clinical Science Immediate Publication. Published on 28 May 2010 as manuscript CS20100161
		Shear stress	0.30	0.60	1.20	3.00	6.00	12.00	30.00	60.00
		(PA)
		GLIM (EI; %)	-0.4±1.7	-0.5±1.1	-1.1±2.5	Pioglitazone -1.5±3.1 -1.4±2.9	-1.3±2.9	Glimepiride -1.3±3.5 -1.3±3.9
		n PIO (EI; %)	1.3±2.1	2.4±1.3* ,$	3.2±2.2* ,$	11 3.3±2.8* ,$	3.1±2.9*	2.7±2.8	12 2.5±2.6	2.7±2.6
	THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100161 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100161	Age (years) Male / Female Body Mass Index Duration of Diabetes (years) HbA1c (%) Syst. Blood Pressure (mmHg) Diast. Blood Pressure (mmHg) HDL (mg/dl) A c c e p t e d M a n u s c r i p t 56 ± 9 58 ± 12 9 / 2 7 / 5 33.5 ± 5.1 32.0 ± 3.6 6.1 ± 3.9 6.4 ± 4.8 7.5 ± 0.9 6.8 ± 0.3 * 128.6 ± 11.8 132.5 ± 16.9 82.0 ± 8.2 81.5 ± 9.7 43.4 ± 4.9 39.3 ± 5.6 Table 2: Change in erythrocyte elongation index (delta EI) from baseline (* p<0.05 vs. baseline; $ p<0.05 vs. GLIM) A c c e p t e d M a n u s c r i p t
		2010-03-09 2010-03-09							page 15 of 21 page 16 of 21
		© 2010 The Authors Journal compilation © 2010 Portland Press Limited

Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited 2010-03-09 page 17 of 21


B H Jasiulis
email: jasiulis@math.uni.wroc.pl

J K Misiewicz
email: j.misiewicz@wmie.uz.zgora.pl

On the connections between weakly stable and pseudo-isotropic distributions

Keywords: weakly stable distribution, symmetric stable distribution, α -symmetric distribution, scale mixture Mathematics Subject Classification: 60A10, 60B05, 60E05, 60E07, 60E10

) with the condition that for all random variables Q1, Q2 there exists a random variable Θ such that

where X, X , Q 1 , Q 2 , Θ are independent. Some of the weakly stable distributions turn out to be the extreme points for the class of pseudoisotropic distributions, where the distribution is pseudo-isotropic if all its one-dimensional projections are the same up to a scale parameter.

We show here that the scaling function for pseudo-isotropic distribution can define a generalized distribution iff it is an α-norm for some α > 0.

Introduction

The idea of generalized convolutions was studied by K. Urbanik in a series of papers, see [START_REF] Urbanik | Generalized convolutions[END_REF][START_REF] Urbanik | Generalized convolutions II[END_REF][START_REF] Urbanik | Generalized convolutions III[END_REF][START_REF] Urbanik | Generalized convolutions IV[END_REF][START_REF] Urbanik | Generalized convolutions V[END_REF]. This area turned out to be interesting and it has been developed extensively till now.

In seventies of the last century there appeared a series of papers written mainly by Kucharczak and Urbanik (see e.g. [START_REF] Kucharczak | Quasi-Stable functions[END_REF][START_REF] Urbanik | Generalized convolutions[END_REF]), where the authors introduced the idea of weakly stable distributions on [0, ∞). More precisely, a probability measure µ on [0, ∞) was called weakly stable if

∀ a, b > 0 ∃ λ on [0, ∞) T a µ * T b µ = µ • λ,

where (T a µ)(A) = µ(A/a), (µ • λ)(A) = µ(A/s)λ(ds) for every Borel set

A in [0, ∞). Currently measures µ for which this condition holds are rather called R + -weakly stable. They are extensively studied, see e.g. [START_REF] Kucharczak | Transformations preserving Weak Stability[END_REF][START_REF]kovich. On Symmetric Stochastic Convolutions[END_REF][START_REF] Vol'kovich | On infinitely decomposable measures in algebras with stochastic convolution. Stability Problems of Stochastic models[END_REF][START_REF] Vol'kovich | Multidimensional B-stable distributions and some generalized convolutions. Stability Problems of Stochastic models[END_REF].

Recently Misiewicz, Oleszkiewicz and Urbanik (see [START_REF] Misiewicz | Classes of measures closed under mixing and convolution. Weak stability[END_REF]) gave a definition and showed some basic properties of weakly stable distributions on R n , or on a separable Banach spaces E. The definition of weakly stable distributions is almost the same as the one given by Kucharczak and Urbanik, except for the fact that constants a, b can be any real numbers. The authors obtained in [START_REF] Misiewicz | Classes of measures closed under mixing and convolution. Weak stability[END_REF] the full characterization of weakly stable distributions with non-trivial discrete part. In [START_REF] Misiewicz | Weak stability and generalized weak convolution for random vectors and stochastic processes[END_REF] J.K. Misiewicz studied basic properties of the weak generalized convolution based on weakly stable distribution.

In this paper we present some new results on weak generalized convolutions and their connections with pseudo-isotropic distributions, i.e. multidimensional symmetric distributions having all one-dimensional margins the same up to a scale parameter. We show that pseudo-isotropy defines a weak generalized convolution iff the corresponding scale function is the p norm on R n . This result links the geometry of L p -spaces with functional equations, theory of pseudo-isotropic distributions and generalized convolutions.

Throughout this paper we denote by L(X) the distribution of the random vector X. If random vectors X and Y have the same distribution we will write X d = Y. By P(IE) we denote the set of all probability measures on a separable Banach space (or on a set) IE. We will use the simplified notation P(IR) = P, P([0, +∞)) = P + . For every a ∈ IR and every probability measure µ we define the rescaling operator T a : P(IE) → P(IE) as follows

T a µ(A) =        µ(A/a) for a = 0; δ 0 (A)
for a = 0, for every Borel set A ∈ IE. Equivalently T a µ is the distribution of the vector aX if µ is the distribution of the vector X. The scale mixture µ • λ of a measure µ ∈ P(IE) with respect to the measure λ ∈ P is defined by:

µ • λ(A) def = IR T s µ (A) λ(ds).
It is easy to see that µ • λ is the distribution of the random vector XΘ if µ = L(X), λ = L(Θ), X and Θ are independent. In the language of characteristic functions we obtain

µ • λ(t) = IR µ(ts)λ(ds).
Notice that for a symmetric random vector X independent of random vari-

(λ 1 • µ) * (λ 2 • µ) = λ • µ.
If µ is not symmetric then the measure λ is uniquely determined. This fact was proven in [START_REF] Misiewicz | Classes of measures closed under mixing and convolution. Weak stability[END_REF] for a weakly stable measure µ, and in [START_REF] Urbanik | Remarks on B-stable Probability Distributions[END_REF] for a R + -weakly stable measure µ. If the measure µ is symmetric then only the symmetrization of λ is uniquely determined (see [START_REF] Misiewicz | Classes of measures closed under mixing and convolution. Weak stability[END_REF], Remark 1). In this case we can always replace the measure λ by its symmetrization (

1 2 δ 1 + 1 2 δ -1 ) • λ.
For convenience in this paper we assume that for symmetric µ the measure λ is concentrated on [0, ∞) taking if necessary |λ| instead of λ.

Generalized weak convolution

Definition 2. Let µ ∈ P(IE) be a nontrivial weakly stable measure, and let λ 1 , λ 2 be probability measures on IR. If

(λ 1 • µ) * (λ 2 • µ) = λ • µ,
then the weak generalized convolution of the measures λ 1 , λ 2 with respect to the measure µ (notation λ 1 ⊕ µ λ 2 ) is defined as follows

λ 1 ⊕ µ λ 2 =        λ if µ is not symmetric; |λ| if µ is symmetric.
In the case of symmetric weakly stable distribution µ we take λ 1 ⊕ µ λ 2 = |λ| in order to get uniqueness of this measure. If Θ 1 , Θ 2 are random variables with distributions λ 1 , λ 2 respectively then the random variable with distribution λ 1 ⊕ µ λ 2 will be denoted as Θ 1 ⊕ µ Θ 2 . Thus we have

Θ 1 X + Θ 2 X d = Θ 1 ⊕ µ Θ 2 X, where X, X , X have distribution µ, Θ 1 , Θ 2 , X , X and Θ 1 ⊕ µ Θ 2 , X are
independent. Now it is easy to see that the following lemma holds.

Lemma 1. If the weakly stable measure µ ∈ P(IE) is not trivial then ⊕ µ is commutative and associative. Moreover for all λ 1 , λ 2 , λ ∈ P λ 1 ⊕ µ λ 2 is uniquely determined and the following conditions hold:

(i) λ ⊕ µ δ 0 = λ (λ ⊕ µ δ 0 = |λ| if µ is symmetric); (ii) T a λ 1 ⊕ µ λ 2 = T a λ 1 ⊕ µ T a λ 2 ; (iii) (pλ 1 + (1 -p)λ 2 )⊕ µ λ = p (λ 1 ⊕ µ λ)+(1-p) (λ 2 ⊕ µ λ) ∀ p ∈ [0, 1]; (iv) if λ n → λ then λ n ⊕ µ ν → λ ⊕ µ ν for all ν ∈ P.
In 1964 K. Urbanik introduced the definition of a generalized convolution as an commutative and associative binary operator on the space P 2 + of all probability measures on [0, ∞) taking values in P + such that the properties Notice that for a weakly stable measure µ we have that (T cn δ n 1 ) • µ = T c n (µ n ) thus if the condition (v) holds for the weak generalized convolution, B.H. JASIULIS 1 AND J.K. MISIEWICZ 2 then µ is in the domain of attraction of some stable measure, which we do not want to assume.

Urbanik calls the generalized convolution regular if there exists a nontrivial homomorphism h :

P + → R such that h(λ 1 λ 2 ) = h(λ 1 )h(λ 2 ) and h(pλ 1 + (1-p)λ 2 ) = ph(λ 1 )+(1-p)h(λ 2 ).
It is easy to see that the weak generalized convolution ⊕ µ is regular with h(δ x ) = µ(x).

pseudo-isotropic distributions

We give here only the definition and an elementary description of pseudoisotropic distributions in the scope required for this paper. More about pseudo-isotropic distributions one can find in [START_REF] Misiewicz | Pseudo-isotropic measures[END_REF], [START_REF] Misiewicz | Sub-stable and pseudo-isotropic processes[END_REF].

Definition 3. The random vector X = (X 1 , . . . , X n ) or its distribution µ on R n is pseudo-isotropic if there exists a function c : R n → [0, ∞) and a function f on R, f (-t) = f (t) such that for each ξ ∈ R n , and every t ∈ R

E exp {i < ξt, X >} = f (c(ξ)|t|) .
It was shown (see e.g. [START_REF] Misiewicz | Sub-stable and pseudo-isotropic processes[END_REF]) that c is a continuous quasi-norm on R n , i.e. it has all the properties of a norm except that the triangular inequality holds with some constant. Moreover, by putting ξ = (1, 0, . . . , 0) we see that

E exp {itX 1 } = f (c(1, 0, . . . , 0))|t|)
thus f (|t|) is, up to a scale parameter, equal the characteristic function of the variable X 1 . We can always assume that c(1, 0, . . . , 0) = 1. Then

∀ ξ ∈ R n < ξ, X >= n k=1 ξ k X k d = c(ξ)X 1 .
This property reminds the one in the definition of strictly stable variables, except that here we do not have independent and identically distributed components of the random vector X. Let

M(c, n) = {µ ∈ P(R n ) : µ(ξ) = f (c(ξ)), f ∈ Φ} ,
where Φ is the set of all real characteristic functions on R. Thus M(c, n)

is the set of all pseudo-isotropic distributions on R n with the fixed quasinorm c. It is evident that M(c, n) is closed under rescaling, convex linear combinations and weak limits since the set of characteristic functions of the form f (c(ξ)) with the fixed function c has these properties, thus M(c, n) is convex and weakly closed. In some cases this set contain only one element

δ 0 , for example M( • α , n) = {δ 0 } if α ∈ (2, ∞]
and n ≥ 3 (see [START_REF] Misiewicz | Sub-stable and pseudo-isotropic processes[END_REF]). In view of this fact we say that the quasi-norm c on R n is admissible if the set M(c, n) contains more than one element.

ψ a,b (c(ξ)) = ∞ 0 ϕ(c(ξ)t)λ(dt).
This means that ν = µ • λ and consequently

T a µ * T b µ = µ • λ.
Assume now that x ε µ(dx) < ∞ and let α = sup{p ≤ 2 :

x p µ(dx) < ∞}. Since µ is symmetric we see that µ belongs to the domain of attraction (see e.g. [?]) of some non-trivial symmetric α-stable measure γ α and

T n -1/α µ * n → γ α . ( * * )
This means, in particular, that γ α ∈ M(c, n) and there exists λ ∈ P such that γ α = µ • λ. The measure λ is not equal δ 0 since otherwise we would have γ α = δ 0 which is excluded. Now the equation ( * * ) together with the

trivial equality µ = µ • δ 1 implies finally that T n -1/α δ ⊕ µ n 1 → λ.

Pseudo-isotropic generalized convolutions

Let (X, Y ) be a pseudo-isotropic random vector with the scaling function (a quasi-norm) c : R 2 → [0, ∞), i.e. for every a, b ∈ R we have that aX

+ bY d = c(a, b)X.
We define a binary operation ⊕ c on the set P + by the following

λ 1 ⊕ c λ 2 = L (c(Θ 1 , Θ 2 )) ,
where

L(Θ 1 ) = λ 1 , L(Θ 2 ) = λ 2 . It is easy to see that if (Θ 1 , Θ 2 ) and (X, Y ) are independent then Θ 1 X + Θ 2 Y d = c(Θ 1 , Θ 2 )X.
∈ (0, ∞] such that c(s, t) =        (s α + t α ) 1/α for α < ∞; max{s, t} for α = ∞.
The assumption 2) in the Bohnenblust theorem was supposed to be the weakest possible substitute of the continuity of the function c. In our case the function c is continuous as the quasi-norm on R 2 , thus we can directly apply this result. Now, with this particular function c the conditions (ii) ÷ (iv) hold trivially. To see the opposite implication notice first that for α ∈ (0, 2] it is enough to take X, Y independent, identically distributed For α ∈ (2, ∞) we use the fact that every two dimensional Banach space embeds isometrically into some L 1 -space (see e.g. [START_REF] Misiewicz | Norm dependent positive definite functions and measures on vector spaces[END_REF]). Thus for each such α the function exp{-(s, t) α } is a characteristic function of some symmetric random vector, say (X, Y ). Obviously the vector (X, Y ) is pseudo-isotropic with the quasi-norm (s, t) α .

Remark. Notice that the considered generalized convolution ⊕ • p is regular with the homomorphism e -t p . If p > 2 then the generalized convolution

  (i)÷(iv) of the previous lemma hold with ⊕ µ replaced by and (v) there exists a sequence of positive numbers (c n ) such that T c n δ n 1 converges weakly to a measure ν = δ 0 (here λ n = λ • • • λ denotes the generalized convolution of n identical measures λ).

Proposition 1 .

 1 Let c : R n → [0, ∞) be an admissible quasi-norm. If there exists µ ∈ M(c, n) such that the set E = {T a µ : a ≥ 0} is the set of all extreme points of M(c, n), then µ is weakly stable. Moreover, if there existsε > 0 such that R n x ε µ(dx) < ∞then for the binary operation ⊕ µ the condition (v) from the Urbanik definition of generalized convolution holds. Proof. Let µ(ξ) = ϕ(c(ξ). Notice that the measure T a µ * T b µ has the characteristic function ϕ(|a|c(ξ))ϕ(|b|c(ξ)) = ψ a,b (c(ξ)) which is also some function dependent on c(ξ) only. Thus the corresponding measure ν, ν(ξ) = B.H. JASIULIS 1 AND J.K. MISIEWICZ 2 ψ a,b (c(ξ)) belongs to M(c, n). Since E is the set of all extreme points of M(c, n) then there exists a measure λ on [0, ∞) such that

  symmetric α-stable random variables. Then (X, Y ) is pseudo-isotropic with the quasi-norm c(s, t) = (s, t) α . For α = ∞ we shall take two independent symmetric standard Cauchy random variables Z 1 , Z 2 , and define(X, Y ) = (Z 1 + Z 2 , Z 1 -Z 2 ). Then we have aX + bY = (a + b)Z 1 + (a -b)Z 2 d = (|a + b| + |a -b|) Z 1 d = max{|a|, |b|}X.

  The question arises under which conditions the operation ⊕ c defines an generalized convolution? Theorem 1. Let c : R 2 → [0, ∞) be the quasi-norm defined by a pseudoisotropic random vector (X, Y ). Then the binary operation ⊕ c defines an generalized convolution on P + if and only if there exists α ∈ (0, ∞] such that c(t, s) = (t, s) α .Proof. Notice first that for constant random variables Θ 1 ≡ t, Θ 2 ≡ s we have c(Θ 1 , Θ 2 ) ≡ c(t, s). This means in particular that δ t ⊕ c δ s = δ c(t,s) . we shall check under what conditions the operation ⊕ c is such that the conditions (i) ÷ (v) hold for ⊕ µ and replaced by ⊕ c . The condition (i) states that δ 0 is the unit element for ⊕ c , i.e. δ 0 ⊕ c λ = λ for each λ ∈ P + . Applying this condition for the measure λ = δ s we would have that δ s = c(δ 0 , δ s ) = δ c(0,s) , which means that for every s > 0 we shall have c(0, s) = s. Since the operation ⊕ c is supposed to be commutative then B.H. JASIULIS 1 AND J.K. MISIEWICZ2

	4) c(0, 1) = 1;
	5) c(c(r, s), t) = c(r, c(s, t)) for all r, s, t ≥ 0
	then there exists α
	Now Now we came to the assumptions of the spectacular result of Bohnenblust
	(see Th. 4.1 in [1]), which states that if c is any function defined over the
	quadrant s, t ≥ 0 having the following properties:
	1) c(rs, rt) = rc(s, t) for all r, s, t ≥ 0;

we have also c(t, s) = c(s, t) for all s, t ≥ 0. Associativity of the operation ⊕ c (applied only to measures concentrated at one point) means that for all r, s, t ≥ 0 we have (δ r ⊕ c δ s ) ⊕ c δ t = δ r ⊕ c (δ s ⊕ c δ t ). This means that for the function c the following condition holds

∀ r, s, t ≥ 0 c(c(r, s), t) = c(r, c(s, t

)).

2) c(s, t) ≤ c(s 1 , t 1 ) for all 0 ≤ s ≤ s 1 , 0 ≤ t ≤ t 1 ;

3) c(s, t) = c(t, s) for all s, t ≥ 0;

Acknowledgement. The authors are indebted to the referee for his valuable remarks and comments.