system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

a fast fourier transform on multipoles fftm algorithm for solving helmholtz equation in acoustics analysis this article presents a fast algorithm for the efficient solution of the helmholtz equation the method is based on the translation theory of the multipole expansions here the speedup comes from the convolution nature of the translation operators which can be evaluated rapidly using fast fourier transform algorithms also the computations of the translation operators are accelerated by using the recursive formulas developed recently by gumerov and duraiswami siam j sci comput 25 1344 1381 2003 it is demonstrated that the algorithm can produce good accuracy with a relatively low order of expansion efficiency analyses of the algorithm reveal that it has computational complexities of o na where a ranges from 1 05 to 1 24 however this method requires substantially more memory to store the translation operators as compared to the fast multipole method hence despite its simplicity in implementation this memory requirement issue may limit the application of this algorithm to solving very lar

efficient fmm accelerated vortex methods in three dimensions via the lamb helmholtz decomposition vortex methods are used to efficiently simulate incompressible flows using lagrangian techniques use of the fmm fast multipole method allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods both equations require field evaluation of constrained divergence free vector valued quantities velocity vorticity and cross terms from these these are usually evaluated by performing several fmm accelerated sums of scalar harmonic functions we present a formulation of vortex methods based on the lamb helmholtz decomposition of the velocity in terms of two scalar potentials in its original form this decomposition is not invariant with respect to translation violating a key requirement for the fmm one of the key contributions of this paper is a theory for translation for this representation the translation theory is developed by introducing conversion operators which enable the representation to be restored in an arbitrary reference frame using this form efficient vortex element computations can be made which need evaluation of just two scalar harmonic fmm sums for evaluating the velocity and vorticity evolution terms details of the decomposition translation and conversion formulae and sample numerical results are presented

evaluation modules for the q tetrahedron algebra let f denote an algebraically closed field and fix a nonzero q f that is not a root of unity we consider the q tetrahedron algebra q over f it is known that each finite dimensional irreducible q module of type 1 is a tensor product of evaluation modules this paper contains a comprehensive description of the evaluation modules for q this description includes the following topics given an evaluation module v for q we display 24 bases for v that we find attractive for each basis we give the matrices that represent the q generators we give the transition matrices between certain pairs of bases among the 24 it is known that the cyclic group z 4 acts on q as a group of automorphisms we describe what happens when v is twisted via an element of z 4 we discuss how evaluation modules for q are related to leonard pairs of q racah type

billiard arrays and finite dimensional irreducible uq sl2 modules abstract we introduce the notion of a billiard array this is an equilateral triangular array of one dimensional subspaces of a vector space v subject to several conditions that specify which sums are direct we show that the billiard arrays on v are in bijection with the 3 tuples of totally opposite flags on v we classify the billiard arrays up to isomorphism we use billiard arrays to describe the finite dimensional irreducible modules for the quantum algebra u q sl 2 and the lie algebra sl 2

the equitable presentation for the quantum group uq g associated with a symmetrizable kac moody algebra g abstract we consider the quantum group u q g associated with a symmetrizable kac moody algebra g we display a presentation for u q g that we find attractive we call this the equitable presentation for g sl 2 the equitable presentation has generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1

computation of galerkin double surface integrals in the 3 d boundary element method the galerkin boundary element method bem also known as the method of moments is a powerful method for solving the laplace equation in three dimensions there are advantages to galerkin formulations for integral equations as they treat problems associated with kernel singularity and lead to symmetric and better conditioned matrices however the galerkin method requires the computation of double surface integral over pairs of triangles there are many semianalytical methods to treat these integrals which all have some issues and are discussed in this paper novel methods inspired by the treatment of these kernels in the fast multipole method are presented for computing all the integrals that arise in the galerkin formulation to any accuracy integrals involving completely geometrically separated triangles are nonsingular and are computed using a technique based on spherical harmonics and multipole expansions and translations which require the integration of polynomial functions over the triangles integrals involving cases where the triangles have common vertices edges or are coincident are treated via scaling and symmetry arguments combined with automatic recursive geometric decomposition of the integrals the methods are validated and example results are presented

high performance blas formulation of the multipole to local operator in the fast multipole method the multipole to local m2l operator is the most time consuming part of the far field computation in the fast multipole method for laplace equation its natural expression though commonly used does not respect a sharp error bound we here first prove the correctness of a second expression we then propose a matrix formulation implemented with basic linear algebra subprograms blas routines in order to speed up its computation for these two expressions we also introduce special data storages in memory to gain greater computational efficiency this blas scheme is finally compared for uniform distributions to other m2l improvements such as block fft fft with polynomial scaling rotations and plane wave expansions when considering runtime extra memory storage numerical stability and common precisions for laplace equation the blas version appears as the best one

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

acoustic scattering by arbitrary distributions of disjoint homogeneous cylinders or spheres a t matrix formulation is presented to compute acoustic scattering from arbitrary disjoint distributions of cylinders or spheres each with arbitrary uniform acoustic properties the generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers by employing field expansions based on orthogonal harmonic functions continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal analytic expressions to eliminate the need for integral equations the effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant numerical results establish the validity and efficiency of the outer iteration procedure for nested objects two and three dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two dimensional approximations to three dimensional scattering of elevation focused beams

robust stability analysis for feedback interconnections of time varying linear systems feedback interconnections of causal linear systems are studied in a continuous time setting the developments include a linear time varying ltv generalization of vinnicombe s nu gap metric and an integral quadratic constraint based robust l2 stability theorem for uncertain feedback interconnections of potentially open loop unstable systems these main results are established in terms of toeplitz wiener hopf and hankel operators and the fredholm index for a class of causal linear systems with the following attributes i a system graph i e subspace of l2 input output pairs admits normalized strong right i e image and left i e kernel representations and ii the corresponding hankel operators are compact these properties are first verified for stabilizable and detectable ltv state space models to initially motivate the abstract formulation and subsequently verified for frequency domain multiplication by constantly proper callier desoer transfer functions in analysis that confirms consistency of the developments with the time invariant theory to conclude the aforementioned robust stability theorem is applied in an illustrative example concerning the feedback interconnection of distributed parameter systems over a network with time varying gains less

calculation of acoustical scattering from a cluster of scatterers the problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed the problem is formulated by using the t matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm mlfma and the fast multipole method fast fourier transform fmmfft method and the efficiency of the two methods is compared it was observed that in general the mlfma performs better than the fmmfft algorithm however when the scatterers are distributed uniformly on a rectangular grid the fmmfft algorithm performs as good as the mlfma the accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres

fast multipole method for the biharmonic equation in three dimensions the evaluation of sums matrix vector products of the solutions of the three dimensional biharmonic equation can be accelerated using the fast multipole method while memory requirements can also be significantly reduced we develop a complete translation theory for these equations it is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering the translation of a pair of elementary solutions of the laplace equations the extension of the theory to the case of polyharmonic equations in r 3 is also discussed an efficient way of performing the fmm for biharmonic equations using the solution of a complex valued fmm for the laplace equation is presented compared to previous methods presented for the biharmonic equation our method appears more efficient the theory is implemented and numerical tests presented that demonstrate the performance of the method for varying problem sizes and accuracy requirements in our implementation the fmm for the biharmonic equation is faster than direct matrix vector product for a matrix size of 550 for a relative l 2 accuracy e 2 10 4 and n 3550 for e 2 10 1 2

stability analysis with dissipation inequalities and integral quadratic constraints this technical note considers the stability of a feedback connection of a known linear time invariant system and a perturbation the input output behavior of the perturbation is described by an integral quadratic constraint iqc iqc stability theorems can be formulated in the frequency domain or with a time domain dissipation inequality the two approaches are connected by a non unique factorization of the frequency domain iqc multiplier the factorization must satisfy two properties for the dissipation inequality to be valid first the factorization must ensure the time domain iqc holds for all finite times second the factorization must ensure that a related matrix inequality when feasible has a positive semidefinite solution this technical note shows that a class of frequency domain iqc multipliers has a factorization satisfying these two properties thus the dissipation inequality test with an appropriate factorization can be used with no additional conservatism

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

the tetrahedron algebra the onsager algebra and the sl2 loop algebra let k denote a field with characteristic 0 and let t denote an indeterminate we give a presentation for the three point loop algebra sl2 k t t 1 t 1 1 via generators and relations this presentation displays s4 symmetry using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras each of which is isomorphic to the onsager algebra

linear transformations that are tridiagonal with respect to the three decompositions for an lr triple abstract recently paul terwilliger introduced the notion of a lowering raising or lr triple and classified the lr triples an lr triple is defined as follows fix an integer d 0 a field f and a vector space v over f with dimension d 1 by a decomposition of v we mean a sequence v i i 0 d of 1 dimensional subspaces of v whose sum is v for a linear transformation a from v to v we say a lowers v i i 0 d whenever a v i v i 1 for 0 i d where v 1 0 we say a raises v i i 0 d whenever a v i v i 1 for 0 i d where v d 1 0 an ordered pair of linear transformations a b from v to v is called lr whenever there exists a decomposition v i i 0 d of v that is lowered by a and raised by b in this case the decomposition v i i 0 d is uniquely determined by a b we call it the a b decomposition of v consider a 3 tuple of linear transformations a b c from v to v such that any two of a b c form an lr pair on v such a 3 tuple is called an lr triple on v let be nonzero scalars in f the triple a b c is an lr triple on v said to be associated to a b c let v i i 0 d be a decomposition of v and let x be a linear transformation from v to v we say x is tridiagonal with respect to v i i 0 d whenever x v i v i 1 v i v i 1 for 0 i d let x be the vector space over f consisting of the linear transformations from v to v that are tridiagonal with respect to the a b and b c and c a decompositions of v there is a special class of lr triples called q weyl type in the present paper we find a basis of x for each lr triple that is not associated to an lr triple of q weyl type

the equitable basis for mathfrak sl _2 this article contains an investigation of the equitable basis for the lie algebra mathfrak sl 2 denoting this basis by x y z we have x y 2x 2y quad y z 2y 2z quad z x 2z 2x we determine the group of automorphisms g generated by exp ad x exp ad y exp ad z where x y z is the basis for mathfrak sl 2 dual to x y z with respect to the trace form u v tr uv and study the relationship of g to the isometries of the lattices l mathbb z x oplus mathbb z y oplus mathbb z z and l mathbb z x oplus mathbb z y oplus mathbb z z the matrix of the trace form is a cartan matrix of hyperbolic type and we identify the equitable basis with a set of simple roots of the corresponding kac moody lie algebra mathfrak g so that l is the root lattice and frac 1 2 l is the weight lattice of mathfrak g the orbit g x of x coincides with the set of real roots of mathfrak g we determine the isotropic roots of mathfrak g and show that each isotropic root has multiplicity 1 we describe the finite dimensional mathfrak sl 2 modules from the point of view of the equitable basis in the final section we establish a connection between the weyl group orbit of the fundamental weights of mathfrak g and pythagorean triples

computation of scattering from clusters of spheres using the fast multipole method a t matrix based method of solution of the multiple scattering problem was presented by the authors j acoust soc am 112 2688 2701 2002 this method can be applied to the computation of relatively small problems since the number of operations required grows with the number of spheres n as o n3 and with the sixth power of the wave number the use of iterative techniques accelerated using the fast multipole method fmm can accelerate this solution as presented by koc and chew j acoust soc am 103 721 734 1998 originally in this study we present a method that combines preconditioned krylov subspace iterative techniques fmm accelerated matrix vector products a novel fmm based preconditioner and fast translation techniques that enable us to achieve an overall algorithm in which the cost of the matrix vector multiplication grows with n as o n logn and with the third power of the wave number we discuss the convergence of the iterative techniques selection of the truncation number errors in the solution and other issues the results of the solution of test problems obtained with the method for n approximately 10 2 10 4 for different wave numbers are presented

recursions for the computation of multipole translation and rotation coefficients for the 3 d helmholtz equation we develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the helmholtz equation in spherical coordinates these expressions are based on the derivation of recurrence relations some of which to our knowledge are presented here for the first time the symmetry and other properties of the coefficients are also examined and based on these efficient procedures for calculating them are presented our expressions are direct and do not use the clebsch gordan coefficients or the wigner 3 j symbols although we compare our results with methods that use these to prove their accuracy for evaluating an nt term truncation of the translated series involving o n t 2 multipoles our expressions require o n t 3 evaluations compared to previous exact expressions that require o n t 5 operations

very strictly passive controller synthesis with affine parameter dependence this paper considers linear parameter varying lpv control within a passive systems framework a very strictly passive vsp controller is presented in state space form where the state space matrices have an affine dependence on plant parameters the resulting controller is a vsp lpv controller guaranteed to stabilize nonlinear passive systems via the passivity theorem an boldsymbol mathcal h infty inspired control synthesis method is presented along with an experimental comparison between the proposed vsp lpv controller and a self scheduled lpv controller experimental results demonstrate effective joint angle and rate tracking of a two link flexible joint manipulator when the proposed vsp lpv controller is used

dual polar graphs the quantum algebra uq sl2 and leonard systems of dual q krawtchouk type abstract in this paper we consider how the following three objects are related i the dual polar graphs ii the quantum algebra u q sl 2 iii the leonard systems of dual q krawtchouk type for convenience we first describe how ii and iii are related for a given leonard system of dual q krawtchouk type we obtain two u q sl 2 module structures on its underlying vector space we now describe how i and iii are related let denote a dual polar graph fix a vertex x of and let t t x denote the corresponding subconstituent algebra by definition t is generated by the adjacency matrix a of and a certain diagonal matrix a a x called the dual adjacency matrix that corresponds to x by construction the algebra t is semisimple we show that for each irreducible t module w the restrictions of a and a to w induce a leonard system of dual q krawtchouk type we now describe how i and ii are related we obtain two u q sl 2 module structures on the standard module of we describe how these two u q sl 2 module structures are related each of these u q sl 2 module structures induces a c algebra homomorphism u q sl 2 t we show that in each case t is generated by the image together with the center of t using the combinatorics of we obtain a generating set l f r k of t along with some attractive relations satisfied by these generators

taylor expansion based fast multipole method for 3 d helmholtz equations in layered media abstract in this paper we develop a taylor expansion te based fast multipole method fmm for low frequency 3d helmholtz green s function in layered media two forms of taylor expansions with either non symmetric or symmetric derivatives of layered media green s functions are used for the implementations of the proposed te fmm in the implementation with non symmetric derivatives an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives meanwhile the implementation based on symmetric derivatives is more robust and pre computed tables for the high order derivatives in translation operators are used numerical tests in layered media have validated the accuracy and o n complexity of the proposed algorithms

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

a method to compute periodic sums in a number of problems in computational physics a finite sum of kernel functions centered at n particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries even though the finite sum can be efficiently computed via fast summation algorithms such as the fast multipole method fmm the periodized extension is usually treated via a different algorithm ewald summation accelerated via the fast fourier transform fft a different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper the method splits the periodized sum in to two parts the first comprising the contribution of all points outside a large sphere enclosing the box and some of its neighbors is approximated inside the box by a collection of kernel functions sources placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions the second part comprising the part inside the sphere and including the box and its immediate neighborhood is treated via available summation algorithms the coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential imposed on a circumspherical surface for the box while the method is presented in general details are worked out for the case of evaluating electrostatic potentials and forces results show that when used with the fmm the periodized sum can be computed to any specified accuracy at a cost that is twice that of the free space fmm with the same accuracy several technical details and efficient algorithms for auxiliary computations are provided as are numerical comparisons

lowering raising triples and uq sl2 abstract we introduce the notion of a lowering raising or lr triple of linear transformations on a nonzero finite dimensional vector space we show how to normalize an lr triple and classify up to isomorphism the normalized lr triples we describe the lr triples using various maps such as the reflectors the inverters the unipotent maps and the rotators we relate the lr triples to the equitable presentation of the quantum algebra u q sl 2 and lie algebra sl 2

multiple scattering by multiple spheres a new proof of the lloyd berry formula for the effective wavenumber we provide the first classical derivation of the lloyd berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers our approach clarifies the assumptions under which the lloyd berry formula is valid more precisely we derive an expression for the effective wavenumber which assumes the validity of lax s quasi crystalline approximation but makes no further assumptions about scatterer size and then we show that the lloyd berry formula is obtained in the limit as the scatterer size tends to zero

efficient recursive generation of the scalar spherical multipole translation matrix new efficient recursive procedures for generating the translation matrix of the scalar spherical multipole field are described they are based on a new set of recurrence relations that result when the angular momentum operator is applied to the spherical multipole field their efficiency and accuracy are compared analytically and through a computer experiment with those of the brute force method and an existing recursive procedure

a scalar potential formulation and translation theory for the time harmonic maxwell equations we develop a computational method based on the debye scalar potential representation which efficiently reduces the solution of maxwell s equations to the solution of two scalar helmholtz equations one of the key contributions of this paper is a theory for the translation of maxwell solutions using such a representation since the scalar potential form is not invariant with respect to translations the translation theory is developed by introducing conversion operators which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame advantages of this representation include the fact that only two helmholtz equations need be solved and moreover the divergence free constraints are satisfied automatically by construction truncation error bounds are also presented the availability of a translation theory and error bounds for this representation can find application in methods such as the fast multipole method for illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the t matrix method the resulting system was solved using an iterative method based on gmres the results of the computations agree well with previous computational and experimental results

calculation of the addition coefficients in electromagnetic multisphere scattering theory one of the most intractable problems in electromagnetic multisphere scattering theory is the formulation and evaluation of vector addition coefficients introduced by the addition theorems for vector spherical harmonics this paper presents an efficient approach for the calculation of both scalar and vector translational addition coefficients which is based on fast evaluation of the gaunt coefficients the paper also rederives the analytical expressions for the vector translational addition coefficients and discusses the strengths and limitations of other formulations and numerical techniques found in the literature numerical results from the formulation derived in this paper agree with those of a previously published recursion scheme that completely avoids the use of the gaunt coefficients but the method of direct calculation proposed here reduces the computing time by a factor of 4 6

an lr pair that can be extended to an lr triple abstract fix an integer d 0 a field f and a vector space v over f with dimension d 1 by a decomposition of v we mean a sequence v i i 0 d of 1 dimensional f subspaces of v such that v i 0 d v i direct sum consider f linear transformations a b from v to v then a b is called an lr pair whenever there exists a decomposition v i i 0 d of v such that a v i v i 1 and b v i v i 1 for 0 i d where v 1 0 and v d 1 0 by an lr triple we mean a 3 tuple a b c of f linear transformations from v to v such that any two of them form an lr pair in the present paper we consider how an lr pair a b can be extended to an lr triple a b c

finite dimensional irreducible uq sl2 modules from the equitable point of view abstract we consider the quantum algebra u q sl 2 with q not a root of unity we describe the finite dimensional irreducible u q sl 2 modules from the point of view of the equitable presentation

a matrix version of the fast multipole method we present a matrix interpretation of the three dimensional fast multipole method fmm the fmm is for efficient computation of gravitational electrostatic potentials and fields it has found various applications and inspired the design of many efficient algorithms the one dimensional fmm is well interpreted in terms of matrix computations the three dimensional matrix version reveals the underlying matrix structures and computational techniques used in fmm it also provides a unified view of algorithm variants as well as existing and emerging implementations of the fmm

computation of the head related transfer function via the fast multipole accelerated boundary element method and its spherical harmonic representation the head related transfer function hrtf is computed using the fast multipole accelerated boundary element method for efficiency the hrtf is computed using the reciprocity principle by placing a source at the ear and computing its field analysis is presented to modify the boundary value problem accordingly to compute the hrtf corresponding to different ranges via a single computation a compact and accurate representation of the hrtf termed the spherical spectrum is developed computations are reduced to a two stage process the computation of the spherical spectrum and a subsequent evaluation of the hrtf this representation allows easy interpolation and range extrapolation of hrtfs hrtf computations are performed for the range of audible frequencies up to 20khz for several models including a sphere human head models the neumann ku 100 fritz and the knowles kemar kemar manikins and head and torso model the kemar manikin comparisons between the different cases are provided comparis

robust synthesis for linear parameter varying systems using integral quadratic constraints a robust synthesis algorithm is developed for a class of uncertain linear parameter varying lpv systems the uncertain system is described as an interconnection of a nominal lpv system and a block structured uncertainty the nominal part is a gridded lpv system with state matrices that are arbitrary functions of the parameter the input output behavior of the uncertainty is described by integral quadratic constraints iqcs the robust synthesis problem leads to a non convex optimization the proposed algorithm is a coordinate wise descent similar to the well known dk iteration for synthesis it alternates between an lpv synthesis step and an iqc analysis step both steps can be efficiently solved as semidefinite programs the derivation of the synthesis algorithm is less obvious for lpv systems as compared to its lti counterpart due to the lack of a valid frequency response interpretation the main contribution is the construction of the iterative synthesis algorithm using time domain dissipation inequalities and a scaled system analogous to that appearing in synthesis it is shown that the proposed algorithm ensures that the robust performance is non increasing at each iteration step the effectiveness of the proposed method is demonstrated on a simple numerical example

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem the numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximate but fast methods such as the fast multipole method however since these methods are only approximate it is important to have an estimate for the error introduced in such calculations an analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained the error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed the total error introduced in a multilevel implementation is also investigated numerically

a mesh free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators a multiple scattering approach is presented to compute the solution of the helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere the solution is represented in terms of partial wave expansions and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces this approach yields high order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable numerical results are presented to demonstrate the accuracy and efficiency of the method

generalized robust gain scheduled pid controller design for affine lpv systems with polytopic uncertainty in the paper a generalized guaranteed cost output feedback robust gain scheduled pid controller synthesis is presented for affine linear parameter varying systems under polytopic model uncertainty the controller synthesis is generalized in a sense that it covers robust robust gain scheduled and robust switched with arbitrary switching algorithm pid controller design the proposed centralized decentralized controller method is based on bellman lyapunov equation guaranteed cost and parameter dependent quadratic stability the proposed sufficient robust stability and performance conditions are derived in the form of bilinear matrix inequalities bmi which can efficiently be solved or further linearized as the main result the suggested performance and stability conditions without any restriction on the controller structure are convex functions of the scheduling and uncertainty parameters hence there is no need for applying multi convexity or other relaxation techniques and consequently the proposed solution delivers a less conservative design method the viability of the novel design technique is demonstrated and evaluated through numerical examples

upper triangular matrices and billiard arrays abstract fix a nonnegative integer d a field f and a vector space v over f with dimension d 1 let t denote an invertible upper triangular matrix in mat d 1 f using t we construct three flags on v we find a necessary and sufficient condition on t for these three flags to be totally opposite in this case we use these three totally opposite flags to construct a billiard array b on v it is known that b is determined up to isomorphism by a certain triangular array of scalar parameters called the b values we compute these b values in terms of the entries of t we describe the set of isomorphism classes of billiard arrays in terms of upper triangular matrices

analysis of radiative scattering for multiple sphere configurations an analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented the analysis builds upon the previously developed superposition solution in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble the addition theorems for vector spherical harmonics which transform harmonics from one coordinate system into another are rederived and simple recurrence relations for the addition coefficients are developed the relations allow for a very efficient implementation of the order of scattering solution technique for determining the scattered field coefficients for each sphere prediction of the radiative absorption and scattering characteristics of small particles is important to researchers in a number of fields e g atmospheric modelling analysis of radiative transfer from flames and development of nonintrusive laser based optical diagnostic methods computation of the radiative characteristics of spherical particles from lorenz mie theory is practically a trivial matter due to the existence of efficient computer codes bohren huffman 1983 however it is not unusual to encounter situations in which the individual particles while spherical in shape are so close together that the isolated sphere assumption inherent in lorenz mie theory is questionable a common example is soot formed in combustion processes electron micrographs of the individual soot particles reveal them to be agglomerates of a large number of primary spherical particles dobbins

a new fast multipole boundary element method for two dimensional acoustic problems abstract a new fast multipole boundary element method bem is presented in this paper for solving large scale two dimensional 2d acoustic problems based on the improved burton miller formulation this algorithm has several important improvements the fast multipole bem employs the improved burton miller formulation and successfully overcomes the non uniqueness difficulty associated with the conventional bem for exterior acoustic problems the improved burton miller formulation contains only weakly singular integrals and avoids the numerical difficulties associated to the evaluation of the hypersingular integral it leads to the numerical implementations more efficient and straightforward furthermore the fast multipole method fmm and the approximate inverse preconditioned generalized minimum residual method gmres iterative solver are adopted to greatly improve the overall computational efficiency the numerical examples with neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole bem for solving large scale 2d acoustic problems in a wide range of frequencies

a fast multipole method for the three dimensional stokes equations many problems in stokes flow and linear elasticity require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions in the fluid mechanics setting these are typically the stokeslet the kernel of the single layer potential or the stresslet the kernel of the double layer potential in this paper we present a simple and efficient method for the rapid evaluation of such fields using a decomposition into a small number of coulombic n body problems following an approach similar to that of fu and rodin y fu g j rodin fast solution methods for three dimensional stokesian many particle problems commun numer meth en 16 2000 145 149 while any fast summation algorithm for coulombic interactions can be employed we present numerical results from a scheme based on the most modern version of the fast multipole method h cheng l greengard v rokhlin a fast adaptive multipole algorithm in three dimensions j comput phys 155 1999 468 498 this approach should be of value in both the solution of boundary integral equations and multiparticle dynamics

the classification of leonard triples of qracah type abstract let k denote an algebraically closed field let v denote a vector space over k with finite positive dimension by a leonard triple on v we mean an ordered triple of linear transformations in end v such that for each of these transformations there exists a basis of v with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal there is a family of leonard triples said to have qracah type this is the most general type of leonard triple we classify the leonard triples of qracah type up to isomorphism we show that any leonard triple of qracah type satisfies the z 3 symmetric askey wilson relations