boson realization of sp 4 r ii the generating kernel formulation in a previous paper of this series the matrix elements were discussed with respect to boson states of an operator k2 required for the boson realization of the sp 4 r lie algebra in the present paper it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp 4 r coherent states the results have relevance for the determination of the matrix elements of the generators of the sp 4 r lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group and are in principle generalizable to symplectic algebras of higher dimensions

subduction coefficients of brauer algebras and racah coefficients of o n and sp 2 m ii racah coefficients irreducible representations of brauer algebras are discussed in the non standard basis a method for evaluating subduction coefficients sdcs i e the transformation coefficients between standard and non standard bases of brauer algebras is outlined non trivial sdcs of for are derived racah coefficients of o n and sp 2m can be derived from subduction coefficients of brauer algebras by using the schur weyl duality relation between and o n or sp 2m

discrete adjoint approach for nonlinear unsteady aeroelastic design optimization presented is a discrete adjoint approach for computing nonlinear unsteady aeroelastic geometric design sensitivities which can subsequently be used for nonlinear unsteady aeroelastic geometric des

explicit evaluation of coupling coefficients for the most degenerate representations of so n explicit results for particular coupling coefficients for the most degenerate representations of so n are given the isoscalar factors which allow a recursive calculation of arbitrary coupling coefficients from those of so 3 are derived 6j and 9j symbols for most degenerate representations are also briefly discussed

more on coupling coefficients for the most degenerate representations of so n we present explicit closed form expressions for the general group theoretical factor appearing in the topology of a high temperature expansion of so n symmetric lattice models this object which is closely related to 6j symbols for the most degenerate representation of so n is discussed in detail

internal labelling problem an algorithmic procedure combining the decomposition of casimir operators induced by the embedding of a subalgebra into a semisimple lie algebra with the properties of commutators of subgroup scalars an analytical algorithm for the computation of missing label operators with the commutativity requirement is proposed two new criteria for subgroups scalars to commute are given the algorithm is completed with a recursive method to construct orthonormal bases of states as examples to illustrate the procedure four labelling problems are explicitly studied

coupling coefficients of so n and integrals involving jacobi and gegenbauer polynomials the expressions for the coupling coefficients 3j symbols for most degenerate symmetric representations of orthogonal groups so n in a canonical basis with so n restricted to so n 1 and different semicanonical or tree bases with so n restricted to so n so n n n n are considered respectively in context of integrals involving triplets of the gegenbauer and the jacobi polynomials since the directly derived triple hypergeometric series do not reveal the apparent triangle conditions of the 3j symbols they are rearranged using their relation with semistretched isofactors of the second kind for the complementary chain sp 4 su 2 su 2 and analogy with the stretched 9j coefficients of su 2 into formulae with more rich limits for summation intervals and obvious triangle conditions the isofactors of class one representations of orthogonal groups or class two representations of unitary groups and of course the related integrals involving triplets of the gegenbauer and the jacobi polynomials turn into double sums in the cases of canonical so n so n 1 or u n u n 1 and semicanonical so n so n 2 so 2 chains as well as into the 4f3 1 series under more specific conditions some ambiguities of the phase choice of the complementary group approach are adjusted as well as problems with an alternating sign parameter of so 2 representations in the so 3 so 2 and so n so n 2 so 2 chains

multiple hypergeometric functions and 9 j coefficients it is well known that the 9 j recoupling coefficient appearing in the quantum theory of angular momentum has 72 symmetries however the triple sum series expression for the 9 j coefficient exhibits none of these symmetries here a stretched 9 j coefficient for which a closed form single term expression exists is considered and the type of summation theorems the triple sum series reduces is investigated for any of the 72 symmetries apart from well known single summation theorems for hypergeometric functions this analysis gives rise to new summation theorems for double and triple hypergeometric functions

missing label operators in the reduction o p o p 2 o 2 i consider the missing label problem for basis vectors of an o p representation corresponding to a group reduction chain with links o p o p 2 o 2 a chain with these links is required if the basis vectors are to be of definite weight i obtain two different sets of missing label operators which together with the casimir operators of group and subgroups from a complete set of labeling operators whose eigenvectors provide a canonical basis in the o p representation space the problem is solved for both the even and odd dimensional orthogonal groups

a class of 6 j symbols for so 2l 1 in terms of rotation matrices for so 3 it is shown that a 6 j symbol for so 2l 1 in which four primitive spinor representations 1 21 2 1 2 appear is directly related to an so 3 rotation matrix possessing a rank of l 1 2 and characterised by the euler angles 0 1 2 pi 0 an interpretation is given in terms of a rotation operator exp 1 2i pi ty acting in the combined spin and quasispin space of an atomic l shell whose states mod t mt are defined in a quasiparticle scheme in which four coupled spinors 1 21 2 1 2 are used

the control of the boundary value problem for linear impulsive integro differential systems recently there has been a lot of activity in the investigation of differential equations with solutions which undertake discontinuities or jumps at w x some definite instants 1 4 at the same time in the theory of impulsive differential equations there is a number of problems which are connected with the results of the theory of integral and integro differential equations w x and which are not well investigated yet 3 5 8 the purpose of this paper is to consider a problem of finding conditions for controllability of the boundary value problem for a linear impulsive system of integro differential equations the problem of the control of the linear impulse system was w x considered in 9 moreover the existence results and the integral representation of solutions of the second kind volterra integro summary equation and integro differential equations are obtained with impulse actions at fixed moments of time necessary and sufficient conditions of the solvability of the boundary value problem for linear impulsive systems of integro differential equations are defined q 1999 academic press

the continuous adjoint method for the design of hydraulic turbomachines abstract this article presents the development and application of the continuous adjoint method for designing optimizing the shape of hydraulic turbomachines the reynolds averaged flow equations are solved in the rotating reference frame and the terms arising from the differentiation of the coriolis and centripetal forces are taken into account in the formulation of the adjoint equations the objective functions presented in this article can be used for achieving a the optimal collaboration of the runner impeller with the draft tube by controlling the meridional and circumferential velocity profiles at the exit of the runner b the operation at the desired hydraulic head and or c the cavitation suppression all of them are used to improve an existing francis runner it is important to note that the objective function related to cavitation is by definition non differentiable and a way to effectively handle it is proposed the continuous adjoint method is presented in its most general form and could readily be adapted to other objective functions in the field of hydraulic turbomachines

application of poisson maps on coadjoint orbits of sp 6 r group to many body dynamics the canonical transformation approach generated by the semisimple subgroup gcm 3 sp 6 r is applied to reduction of the lie poisson bracket on coadjoint orbits of the sp 6 r group and the poisson coalgebra spo 6 is determined investigating the construction of the n particle phase induced by this reduction we identify the poisson coalgebra spo 6 as the algebra of quadratic o k k n 1 invariant forms on symplectic 6 k 12 dimensional phase space t o k 3 o k k 3 the general classification scheme of poisson orbits for spo 6 is found and applied to the classification of coadjoint orbits of the sp 6 r group occurring in the decomposition of n particle phase spaces we show that the spo k k 4 6 poisson action on some class of surfaces determined by casimir invariants is not transitive the poisson maps for all classes of orbits spo 4 and spo 6 are found the quantum unitary irreducible representations of spo 4 are obtained

adjoint based sensitivity formulation for fully coupled unsteady aeroelasticity problems a method to compute sensitivities for use in gradient based shape optimization applied to unsteady aeroelasticity problems is presented the method takes into account the coupling between all three fundamental aspects of computational aeroelasticity namely unsteady flow equations time varying structural response to aerodynamic loads and dynamic meshes that accommodate geometric deformations the devised method provides discretely exact sensitivities of time integrated and non time integrated objective functions with respect to design variables that control the shape of geometry the algorithm is formulated in a general manner and can be readily extended to coupled multidisciplinary problems involving any number of disciplines the algorithm is applied to a simple two dimensional flutter model to demonstrate the proof of concept

6j symbols for symmetric representations of so n as the double series the corrected triple sum expression of alisauskas 1987 j phys a math gen 20 35 for the recoupling racah coefficients 6j symbols of the symmetric most degenerate representations of the orthogonal groups so n previously derived from the fourfold sum expression of alisauskas also related to the result of hormes and junker 1999 j phys a math gen 32 4249 is rearranged into three new different double sum expressions related to the hypergeometric kampe de feriet type series and a new triple sum expression with preferable summation condition the regge type symmetry of special 6j symbols of the orthogonal groups so n in terms of special kampe de feriet f1 41 3 series is revealed the recoupling coefficients for antisymmetric representations of symplectic group sp 2n are derived using their relation with the recoupling coefficients of the formal orthogonal group so 2n

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

almost periodic solutions for abstract impulsive differential equations this paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in banach space the investigations are carried out by means of the fractional powers of operators we construct an example to illustrate the feasibility of our results

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

inhomogeneous boson realization of indecomposable representations of lie algebras by making use of the differential realization of lie algebras in the space of inhomogeneous polynomials of a certain number of variables the corresponding inhomogeneous boson realization of lie algebras is given a new kind of indecomposable representations of lie algebras are studied on the universal enveloping algebra of heisenberg weyl algebra its subspaces and its quotient spaces the finite dimensional representations are naturally obtained on the subspaces of fock space as an example the indecomposable and irreducible representations of the lie algebra su 2 are discussed in detail

explicit canonical tensor operators and orthonormal coupling coefficients of su 3 the canonical unit su 3 tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators with the renormalization factors expressed in terms of the denominator functions of biedenharn gustafson lohe louck and milne the matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of asherova and smirnov the self conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique the corresponding extreme isoscalar factors of the clebsch gordan wigner coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of su 3 with its considerable simplication for the boundary values of parameters the general isofactors are also expanded in the different ways in terms of their boundary values the new classes of the generaliz

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

discrete adjoint based approach for optimization problems on three dimensional unstructured meshes a comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented by linearizing each procedure in the entire optimization problem transposing each linearization and reversing the sequential order of operations the adjoint of the complete optimization problem including flow equations and mesh motion equations is constructed in a modular and verifiable fashion this construction is also shown to produce minimal memory overheads and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis these techniques are implemented in a three dimensional unstructured multigrid navierstokes solver and demonstrated on a transonic drag reduction problem for a wing body configuration

on the existence of solutions for nonlinear first order implicit impulsive integro differential equations abstract by using the monch fixed point theorem the purpose of this paper is to obtain the new existence theorems of solutions for a new class of initial value problems of nonlinear first order implicit impulsive integro differential equations in banach spaces under some weaker conditions

some 6 j symbols for symplectic and orthogonal groups by cerkaski s method the method devised by cerkaski 1987 has been used for the group sp 2n to find some 6 j symbols for which a single multiplicity index is required this approach possesses some advantages over that based on racah s techniques for so 3 the formulae for the 6 j symbols are comparatively simple and they automatically exhibit the symmetries of jucys 1973 elementary means are used to show that the eigenvalues of casimir s operator c2 for sp 2n are identical to those of c2 for o 2d if the authors take d n and replace the irreps of sp 2n by their transposed counterparts for o 2d where d is integral or half integral this enables the formulae for sp 2n to be used to generate expressions for some 6 j symbols for the orthogonal groups o 2d and so 2d illustrations are provided by so 2l 1 and o 4

a set of commuting missing label operators for so 5 so 3 the two missing label problem for basis vectors of an so 5 irreducible representation reduced according to the principal so 3 subalgebra is considered a pair of commuting hermitian operators which are scalars with respect to the so 3 subalgebra are explicitly constructed within the so 5 enveloping algebra one label generating operator is of fourth order and the other of sixth order in the so 5 basis elements

algebraic expressions for some generalised 6 j symbols for so n and g2 a class of generalised 6 j symbols which enter naturally when a second pair of modes epsilon tau 2 is added to the octahedral jahn teller system gamma 8 epsilon tau 2 has been evaluated for the group so 5 the method involves the fictitious boson configurations dwd and can be readily generalised to lwl the resulting 6 j symbols for so 2l 1 can be used to produce others by means of generalisations of the equations satisfied by the ordinary 6 j symbols for so 3 extensions to so 2l 2 and g2 are described as well as the weight space symmetries of jucys 1973 the relation between 6 j symbols and isoscalar factors is illustrated for so 7 contains implies g2

on the evaluation of the u 3 content of the matrix elements of one and two body operators an expression for the u 3 content of the matrix elements of one and two body operators in elliott s basis is obtained three alternative ways of evaluating this content with increasing performance in computing time are presented all of them allow an exact representation of that content in terms of integers avoiding rounding errors in the computer codes the role of dual bases in dealing with nonorthogonal bases is also clarified

internal labelling operators and contractions of lie algebras we analyse under which conditions the missing label problem associated with a reduction chain of simple lie algebras can be completely solved by means of an inonu wigner contraction naturally related to the embedding this provides a new interpretation of the missing label operators in terms of the casimir operators of the contracted algebra and shows that the available labelling operators are not completely equivalent further the procedure is used to obtain upper bounds for the number of invariants of affine lie algebras arising as contractions of semi simple algebras

another proof of the triple sum formula for wigner 9j symbols we give a simple proof of the alisauskas jucys triple sum formula for wigner 9j symbols

some coupling and recoupling coefficients for symmetric representations of son a new expression of son contains implies son 1 isoscalar factors reduced wigner coefficients for coupling l1 l2 to l1l2 is obtained special classes of these factors including those used together with special isofactors of sun contains implies son in on scalar basis functions depending on the microscopic collective variables of a nucleus are shown as being proportional to the 6j coefficients of su2 with standard or quartervalued parameters group chain reflection symmetry of isofactors is discovered which allows the author to interchange certain reflected parameters of irreducible representations of the group and its subgroup recoupling coefficients 6l coefficients for symmetric representations of son n or 5 are expanded in terms of 6j coefficients of su2 with standard or quartervalued parameters new expressions for special clebsch gordan and 6j coefficients of su2 so3 are presented

collectivity and geometry iii the three dimensional case in the sp 6 sp 2 o 3 chain for closed shells as was indicated in previous papers general hamiltonians for systems of n particles in three dimensional space can be formulated in the enveloping algebra of the symplectic group sp 6n this group admits among others the subgroup sp 6 o n and as has been noticed by many authors collective hamiltonians can be formulated in the enveloping algebra of sp 6 so that their eigenstates can be characterized by a definite irreducible representation irrep of this group the mathematical problem is then to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of this group as well as of appropriate subgroups in the present series of papers the subgroups chosen where sp 2 o 3 as the casimir operator of sp 2 when n is formally related to the bohr mottelson vibrational hamiltonian bmvh while o 3 gives the angular momentum of the state we give an algorithm for determining these states that closely parallels the procedure followed for the bmvh programs are being developed to convert our algorithm into a computational tool for determining collective excitations in nuclei

on a class of 6j coefficients with one multiplicity index for groups sp 2n so 2n and so 2n 1 an important class of 6j symbols for the groups sp 2n so 2n and so 2n 1 with one nontrivial multiplicity index is investigated an appropriate choice of a basis in the multiplicity space is made and the so called canonical form for 6j symbols is obtained their expressions depending on the roots of an nth order equation and explicit expressions for some simple class of representations are obtained

a pair of commuting scalars for g 2 su 2 su 2 g 2 su 2 su 2 is a two missing labels problem and therefore in order to give a complete and orthogonal specification of states of irreducible representations of g 2 in an su 2 su 2 basis one needs to find a pair of commuting hermitian operators which are scalar with respect to the su 2 su 2 subalgebra a theorem due to peccia and sharp states that there are apart from the lie algebra invariants twice as many functionally independent scalars as missing labels here two commuting su 2 su 2 scalars are obtained both of sixth order in the g 2 basis elements they are in fact combinations of five scalars of different tensorial types indicating that the functionally independent ones are in general insufficient to provide the lowest order commuting scalars an expression for the sixth order invariant of g 2 is also obtained

collectivity and geometry ii the two dimensional case in a previous paper of this series we showed that nuclear collective behavior can be related to the symplectic geometry of the many nucleon system when we introduce in it appropriate constraints we showed in that paper that a full discussion of collective behavior in the many body system requires in a space of d dimensions the basis for the irreducible representations of the sp 2d group in the chain sp 2d sp 2 o d as well as the matrix elements of the generators of sp 2d in this basis in the present paper we implement this program fully for d 2 both because of the mathematical interest of the chain sp 4 sp 2 o 2 and as a stepping stone to the chain sp 6 sp 2 o 3 which is central to the discussion of collective motions

exponential stability of linear delay impulsive differential equations for an impulsive delay differential equation with bounded delay and bounded coefficients the following result is established if each solution is bounded on 0 together with its derivative for each bounded right hand side then the equation is exponentially stable an explicit stability theorem is presented

algebraic expressions for some so n racah coefficients in this paper the algebraic expressions for so n racah coefficients involving the coupling of three irreducible representations 1 0 over dot m 1 m 2 m 3 0 over dot are given by using the building up process their properties are discussed in detail c 1998 american institute of physics s0022 2488 98 01811 8

negative dimensional groups in quantum physics the author shows that an explicit definition of negative dimensional classical groups in terms of their action on grassmann representation spaces together with weyl s character formula for the grassmann tensorial representations of such groups leads to a framework in which some surprising negative dimensional properties of group theoretic invariants arise naturally as an application he shows that the spectra of the grassmann harmonic oscillator and of grassmann angular momentum are related to their bosonic counterparts by a simple analytic continuation in the dimension

commutativity of missing label operators in terms of berezin brackets we obtain a criterion on the commutativity of polynomials in the enveloping algebra of a lie algebra in terms of an involution condition with respect to the berezin bracket as an application it is shown that the commutativity requirement of missing label operators for reduction chains in the missing label problem can be solved analytically

a class of generalised 9 j symbols for sp 2n the symbolic method of kramers 1930 is used to obtain an expression for a 9 j symbol in terms of multiple products of spinor invariants this technique is generalised from the unitary compact symplectic group sp 2 to sp 2n and a generating function is found for a class of multiplicity free 9 sigma symbols where sigma denotes an irreducible representation of sp 2n of the form sigma 1 sigma 2 sigma n for which sigma i 0 i 1 schwinger s generating function for a 9 j symbol is recovered by setting n 1 a specialisation to 6 sigma symbols is made by setting one of the nine irreducible representations equal to the scalar 00 0 and the method is checked by working out a sample 6 sigma symbol previously obtained by an aufbau approach

collectivity and geometry i general approach in the last decade an extensive literature appeared in which the microscopic collective behavior of nuclei was associated with definite irreducible representations irreps of either the o n or sp 6 groups where n a 1 and a is the number of nucleons it became clear that the two approaches are equivalent as problems with 3n degrees of freedom are characterized by a definite irrep of the group sp 6n and for its subgroup sp 6 o n the irrep of o n determines that of sp 6 and vice versa thus one can consider that collective effects appear when one introduces the constraint that in the many body system the states are restricted to a definite irrep of o n and thus also of sp 6 and the hamiltonians are in the enveloping algebra of sp 6 rather than in that of sp 6n once sp 6 becomes the paramount group of collective motions the problem is to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of its subgroups what subgroups to choose rowe and rosensteel

transformation formulas for double hypergeometric series related to 9 j coefficients and their basic analogs in a recent paper alisauskas deduced different triple sum expressions for the 9 j coefficient of su 2 and suq 2 for a singly stretched 9 j coefficient these reduce to different double sum series using these distinct series we deduce a set of new transformation formulas for double hypergeometric series of kampe de feriet type and their basic analogs these transformation formulas are valid for rather general parameters of the series although a common feature is that all the series appearing here are terminating it is also shown that the transformation formulas deduced here generate a group of transformation formulas thus yielding an invariance group or symmetry group of particular double series

asymptotic behavior of linear impulsive integro differential equations asymptotic equilibria of linear integro differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro differential equations are established a new gronwall bellman type lemma for integro differential inequalities is proved an example is given to demonstrate the validity of one of the results

control and optimal response problems for quasilinear impulsive integrodifferential equations abstract in various real world applications there is a necessity given to steer processes in time more and more it becomes acknowledged in science and engineering that these processes exhibit discontinuities our paper on theory of control especially optimal control and on theory of dynamical systems gives a contribution to this natural or technical fact one of the central results of our paper is the pontryagin maximum principle l s pontryagin v g boltyanskii r v gamkrelidze e f mishchenko the mathematical theory of optimal processes interscience publishers john wiley new york 1962 which is considered in sufficient form for the linear case of impulsive differential equations the problem of controllability of boundary value problems for quasilinear impulsive system of integrodifferential equations is investigated the control consists of a piecewise continuous function part as well as impulses which act at a variable time by studying the optimal control of response we give a first inclusion of an objective function by this pioneering contribution we invite to future research in the wide field of optimal control with impulses and in modern challenging applications

collectivity and geometry iv sp 6 sp 2 o 3 basis states for open shells using generating function methods branching rules for sp 6 sp 2 o 3 are derived the branching rules suggest an integrity basis or set of elementary permissible diagrams in terms of which the subgroup basis states are defined they correspond to vibrational or bohr mottelson type states in the nuclear symplectic model

analytical expressions for the matrix elements of the non compact symplectic algebra analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp 3 r it is shown that the expressions are exact for representations of the type sigma 1 sigma 2 sigma 3 and for all states of an arbitrary representation that are multiplicity free with respect to the u 3 subalgebra furthermore they are remarkably accurate in general

discrete adjoint approach for modeling unsteady aerodynamic design sensitivities a discrete adjoint approach is presented for computing steady and unsteady aerodynamic design sensitivities for compressible viscous flows about airfoil configurations the nominal flow solver method is based on a harmonic balance solution technique which is capable of modeling both steady and nonlinear periodic unsteady flows the computer code for the discrete adjoint solver which is derived from the nominal harmonic balance solver has been generated with the aid of the advanced automatic differentiation software tool known as taf transformation of algorithms in fortran

unsteady aerostructure coupled adjoint method for flutter suppression an unsteady aerostructure coupled adjoint method is presented for computing shape sensitivities of unsteady flows with fluid structure interactions the nominal flow solver is an unsteady euler solver based on a stationary cartesian grid using transpiration boundary conditions with hilbert transformation a novel damping related cost function is defined for which the shape sensitivity can be computed by the new unsteady aerostructure coupled adjoint method the capability of the new method is demonstrated by a successful airfoil shape optimization for flutter suppression