gaudin models with irregular singularities abstract we introduce a class of quantum integrable systems generalizing the gaudin model the corresponding algebras of quantum hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine kac moody algebra at the critical level extending the construction of higher gaudin hamiltonians from b feigin et al 1994 17 to the case of non highest weight representations of affine algebras we show that these algebras are isomorphic to algebras of functions on the spaces of opers on p 1 with regular as well as irregular singularities at finitely many points we construct eigenvectors of these hamiltonians using wakimoto modules of critical level and show that their spectra on finite dimensional representations are given by opers with trivial monodromy we also comment on the connection between the generalized gaudin models and the geometric langlands correspondence with ramification

optimal preconditioners for systems defined by functions of toeplitz matrices abstract we propose several circulant preconditioners for systems defined by some functions g of toeplitz matrices a n in this paper we are interested in solving g a n x b by the preconditioned conjugate method or the preconditioned minimal residual method namely in the cases when g z are the functions e z sin z and cos z numerical results are given to show the effectiveness of the proposed preconditioners

solution of toeplitz normal equations by sine transform based preconditioning abstract the normal equations constructed by a toeplitz matrix are studied in order to find a suitable preconditioner related to the discrete sine transform new results are given about the structure of the product of two toeplitz matrices which allow the cgn method to achieve a superlinear rate of convergence this preconditioner outperforms the circulant one for the iterative solution of toeplitz least squares problems such strategy can also be applied to nonsymmetric linear systems a block generalization is discussed

a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

design of networked control systems with packet dropouts this note is concerned with stability and controller design of networked control systems ncss with packet dropouts new ncs models are provided considering both single and multiple packet transmissions both sensor to controller s c and controller to actuator c a packet dropouts are modeled and their history behavior is described by different independent markov chains in term of the given models sufficient conditions for stochastic stability are derived in the form of linear matrix inequalities lmis and corresponding control laws are given numerical examples illustrate the effectiveness of the results

quantization of the shift of argument subalgebras in type a abstract given a simple lie algebra g and an element g the corresponding shift of argument subalgebra of s g is poisson commutative in the case where is regular this subalgebra is known to admit a quantization that is it can be lifted to a commutative subalgebra of u g we show that if g is of type a then this property extends to arbitrary thus proving a conjecture of feigin frenkel and toledano laredo the proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

recursive based pcg methods for toeplitz systems with nonnegative generating functions in this paper we consider the solutions of symmetric positive definite but ill conditioned toeplitz systems an x b here we propose to solve the system by the recursive based preconditioned conjugate gradient method the idea is to use the inverse of am the principal submatrix of an with the gohberg semencul formula as a preconditioner for an the inverse of am can be generated recursively by using the formula until m is small enough the construction of the preconditioners requires only the entries of an and does not require the explicit knowledge of the generating function f of an we show that if f is a nonnegative bounded and piecewise continuous even function with a finite number of zeros of even order the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers hence the conjugate gradient method when applied to solving the preconditioned system converges very quickly numerical results are included to illustrate the effectiveness of our approach

toeplitz determinants with perturbations in the corners abstract the paper is devoted to exact and asymptotic formulas for the determinants of toeplitz matrices with perturbations by blocks of fixed size in the four corners if the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity then standard perturbation theory yields asymptotic expressions for the perturbed determinants this premise is not satisfied for matrices generated by so called fisher hartwig symbols in that case we establish formulas for pure single fisher hartwig singularities and for hermitian matrices induced by general fisher hartwig symbols

the extension of the concept of the generating function to a class of preconditioned toeplitz matrices abstract it is well known that the generating function f l 1 of a class of hermitian toeplitz matrices a n f describes very precisely the spectrum of each matrix of the class recently this theory has been extended in various senses in three directions the non hermitian case the block case and the case of preconditioned toeplitz matrices here by using some results of the first and the second extension we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices moreover we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate gradient methods previously defined in the literature finally we perform some numerical experiments which confirm the theoretical idea

augmented lagrangian approach to design of structured optimal state feedback gains we consider the design of optimal state feedback gains subject to structural constraints on the distributed controllers these constraints are in the form of sparsity requirements for the feedback matrix implying that each controller has access to information from only a limited number of subsystems the minimizer of this constrained optimal control problem is sought using the augmented lagrangian method notably this approach does not require a stabilizing structured gain to initialize the optimization algorithm motivated by the structure of the necessary conditions for optimality of the augmented lagrangian we develop an alternating descent method to determine the structured optimal gain we also utilize the sensitivity interpretation of the lagrange multiplier to identify favorable communication architectures for structured optimal design examples are provided to illustrate the effectiveness of the developed method

essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices

superlinear convergence for pcg using band plus algebra preconditioners for toeplitz systems the paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned toeplitz systems t n f x b where the generating function f is known a priori real valued nonnegative and has isolated roots of even order the preconditioner that we propose is a product of a band toeplitz matrix and matrices that belong to a certain trigonometric algebra the basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand alone preconditioner as a result we obtain a flexible preconditioner which can be applied to the system t n f x b infusing superlinear convergence to the pcg method the important feature of the proposed technique is that it can be extended to cover the 2d case i e ill conditioned block toeplitz matrices with toeplitz blocks we perform many numerical experiments whose results confirm the theoretical analysis and effectiveness of the proposed strategy

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

opers with irregular singularity and spectra of the shift of argument subalgebra joint work with boris feigin and edward frenkel math qa 0712 1183 let g be a semisimple complex lie algebra and u g its universal enveloping algebra the algebra u g bears a natural filtration by the degree with respect to the generators the associated graded algebra is the symmetric algebra s g c g by the poincare birkhoff witt theorem the commutator on u g defines the poisson bracket on s g let zs g s g be the center of s g with respect to the poisson bracket and let g g be a regular semisimple element due to the result of mischenko and fomenko 1978 the algebra a s g generated by the elements where zs g is commutative with respect to the poisson bracket and has maximal possible transcendence degree equal to 12 dim g rk g if is a regular element contained in the cartan subalgebra h g then h a and if f is the principal nilpotent element then the subalgebra af contains the principal nilpotent subalgebra zg f it was recently shown that the shift of argument subalgebras can be quantized

a characterization of convex problems in decentralized control ast we consider the problem of constructing optimal decentralized controllers we formulate this problem as one of minimizing the closed loop norm of a feedback system subject to constraints on the controller structure we define the notion of quadratic invariance of a constraint set with respect to a system and show that if the constraint set has this property then the constrained minimum norm problem may be solved via convex programming we also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback these results are developed in a very general framework and are shown to hold in both continuous and discrete time for both stable and unstable systems and for any norm this notion unifies many previous results identifying specific tractable decentralized control problems and delineates the largest known class of convex problems in decentralized control as an example we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate we also show that symmetric synthesis is included in this classification and provide a test for sparsity constraints to be quadratically invariant and thus amenable to convex synthesis

korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices abstract in this paper we are concerned with the approximation of toeplitz matrices generated by continuous 2 periodic functions f i r with i for this purpose we define a class of matrix algebras related to suitable choices of gram functions where we look for good preconditioners in particular we construct these preconditioners through linear operators and linear positive operators lpos approximating in some sense the function f then by making use of some matrix versions 39 42 41 of the korovkin and weierstrass theorems we analyze the convergence features of old and new preconditioners finally among the given results are adapted in order to deal with l1 generating functions and the related preconditioners are compared with the ones devised by using band toeplitz matrices 6 36

a unifying approach to abstract matrix algebra preconditioning in earlier papers tyrtyshnikov 42 and the first author 14 considered the analysis of clustering properties of the spectra of specific toeplitz preconditioned matrices obtained by means of the best known matrix algebras here we generalize this technique to a generic banach algebra of matrices by devising general preconditioners related to convergent approximation processes 36 finally as case study we focus our attention on the tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric toeplitz systems can be constructed in this class

global low rank enhancement of decentralized control for large scale systems this note proposes a new control strategy which is computationally attractive for systems of large dimensions the main idea is to supplement decentralized feedback with a global additive term which is computed as a product of two low rank matrices this feature is of critical importance for systems that cannot be adequately stabilized using standard decentralized control the low rank matrices can be efficiently obtained using linear matrix inequalities and the resulting control is suitable for implementation in a multiprocessor environment simulations on a platoon of vehicles demonstrate that such a control can significantly improve the robustness of the closed loop system with respect to uncertain nonlinearities

stability of model based networked control systems with time varying transmission times in model based networked control systems mb ncss an explicit model of the plant is used to produce an estimate of the plant state behavior between transmission times in this paper the stability of mb ncss is studied when the controller actuator is updated with the sensor information at nonconstant time intervals networked control systems with transmission times that are varying either within a time interval or are driven by a stochastic process with identically independently distributed and markov chain driven transmission times are studied sufficient conditions for lyapunov stability are derived for stochastically modeled transmission times almost sure stability and mean square sufficient conditions for stability are introduced

how to choose the best iterative strategy for symmetric toeplitz systems in the last decade many efficient iterative solvers for n x n symmetric also hermitian or generic toeplitz systems have been devised r h chan and g strang siam j sci stat comput 10 1989 pp 104 119 t f chan siam j sci stat comput 9 1988 pp 766 771 d bini and f di benedetto proceedings 2nd spaa conference 1990 pp 220 223 d bini and p favati siam j matrix anal appl 14 1993 pp 500 507 t huckle siam j matrix anal appl 13 1992 pp 767 777 f di benedetto g fiorentino and s serra comput math appl 26 1993 pp 35 45 s serra siam j matrix anal appl 17 1996 pp 1007 1019 g fiorentino and s serra calcolo 28 1991 pp 283 305 s serra siam j matrix anal appl 20 1998 pp 31 44 in all these methods it is assumed that the considered systems have the form an f x b where the symbol f the generating function is an l1 function and the entries of an f along the kth diagonal coincide with the kth fourier coefficient of f the proof of convergence and sometimes even the definition of these solvers are derived by using some symbolic or analytic properties of f while this function is often unknown in real applications in this paper by means of the knowledge of the coefficients of an f only we propose and discuss an algorithm to economically determine the minimal features of f allowing us to choose and define the best iterative strategy as a consequence of the analysis of the proposed procedure we find some results on the structure of quasi algebra of the set a n f f in l 1 finally we exhibit several numerical experiments which confirm the effectiveness of the proposed idea

on algebras of toeplitz plus hankel matrices we characterize a wide class of maximal algebras of toeplitz plus hankel matrices by exploiting properties of displacement operators

analysis of preconditioning techniques for ill conditioned toeplitz matrices several preconditioning techniques for solving toeplitz systems are known in literature but their convergence features are completely understood only in the well conditioned case we study the application of tau circulant and hartley preconditioners to ill conditioned toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system

superlinear pcg methods for symmetric toeplitz systems in this paper we deal with the solution by means of preconditioned conjugate gradient pcg methods of n x n symmetric toeplitz systems a n f x b with nonnegative generating function f here the function f is assumed to be continuous and strictly positive or is assumed to have isolated zeros of even order in the first case we use as preconditioner the natural and the optimal approximation of a n f proposed by bini and di benedetto and we prove that the related pcg method has a superlinear rate of convergence and a total arithmetic cost of o n log n ops under the second hypothesis we cannot guarantee that the natural matrix is positive definite while for the optimal we show that in the ill conditioned case this can be really a bad choice consequently we define a new matrix for preconditioning the given system then by applying the sherman morrison woodbury inversion formula to the preconditioned system we introduce a small constant number of subsidiary systems which can be solved again by means of the previous pcg method finally we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis

matrix algebras in optimal preconditioning the theory and the practice of optimal preconditioning in solving a linear system by iterative processes is founded on some theoretical facts understandable in terms of a class v of spaces of matrices including diagonal algebras and group matrix algebras the v structure lets us extend some known crucial results of preconditioning theory and obtain some useful information on the computability and on the efficiency of new preconditioners three preconditioners not yet considered in literature belonging to three corresponding algebras of v are analyzed in detail some experimental results are included

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

stabilization of networked control systems with sparse observer controller networks in this technical note we provide a set of stability conditions for linear time invariant networked control systems with arbitrary topology using a lyapunov direct approach we then use these stability conditions to provide a novel low complexity algorithm for the design of a sparse observer based control network we employ distributed observers by employing the output of other nodes to improve the stability of each observer dynamics to avoid unbounded growth of controller and observer gains we impose bounds on their norms the effects of relaxation of these bounds is discussed when trying to find the complete decentralization conditions

feigin frenkel center in types b c and d for each simple lie algebra mathfrak g consider the corresponding affine vertex algebra v mathrm crit mathfrak g at the critical level the center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of feigin and frenkel about two decades ago however only recently simple formulas for the generators of the center were found for the lie algebras of type a following talalaev s discovery of explicit higher gaudin hamiltonians we give explicit formulas for generators of the centers of the affine vertex algebras v mathrm crit mathfrak g associated with the simple lie algebras mathfrak g of types b c and d the construction relies on the schur weyl duality involving the brauer algebra and the generators are expressed as weighted traces over tensor spaces and equivalently as traces over the spaces of singular vectors for the action of the lie algebra mathfrak sl 2 in the context of howe duality this leads to explicit constructions of commutative subalgebras of the universal enveloping algebras mathrm u mathfrak g t and mathrm u mathfrak g and to higher order hamiltonians in the gaudin model associated with each lie algebra mathfrak g we also introduce analogues of the bethe subalgebras of the yangians mathrm y mathfrak g and show that their graded images coincide with the respective commutative subalgebras of mathrm u mathfrak g t

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

a korovkin based approximation of multilevel toeplitz matrices with rectangular unstructured blocks via multilevel trigonometric matrix spaces in this paper we are concerned with the approximation of multilevel toeplitz matrices generated by multivariate rectangular matrix valued continuous also l2 functions f i p rightarrow cal c s times t with i pi pi we first define a class cal a t of multilevel trigonometric matrix spaces with unstructured s times t blocks second we introduce a multi sequence of linear approximation operators cal p n x n n 1 n 2 ldots n p acting on complex matrices of dimensions n 1 cdots n p s times n 1 cdots n p t n j in cal n and giving the minimizer of the distance of x from a fixed representative of cal a t with regard to the frobenius norm then by stating and proving new versions of the korovkin theorem for f ranging in the banach space bf c i p cal c s times t cdot infty and by linking these theorems to the operators cal p n we prove the goodness of the proposed approximation in terms of the strong and weak equidistribution of the eigenvalues singular values of a n f and cal p n a n f the discussed results provide a very uniform tool for treating the preconditioning problem with applications to a wide variety of structures as analyzed in the twin paper s serra capizzano calcolo in press

bethe subalgebras in twisted yangians we study analogues of the yangian of the lie algebra mathfrak g mathfrak l n for the other classical lie algebras mathfrak s mathfrak o n and mathfrak s mathfrak p n we call them twisted yangians they are coideal subalgebras in the yangian of mathfrak g mathfrak l n and admit homomorphisms onto the universal enveloping algebras u mathfrak s mathfrak o n and u mathfrak s mathfrak p n respectively in every twisted yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical lie algebra the images in u mathfrak s mathfrak o n and u mathfrak s mathfrak p n of these subalgebras are also maximal commutative

center of the quantum affine vertex algebra in type a abstract we consider the quantum vertex algebra associated with the double yangian in type a as defined by etingof and kazhdan we show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level

toeplitz matrices and determinants with fisher hartwig symbols abstract this paper is concerned with toeplitz matrices generated by symbols of the form a t r 1 r t t r 2 r t t r r b t t 1 where t 1 t r are pairwise distinct points on the unit circle b is sufficiently smooth b t 0 t 1 and ind b 0 t t r r is defined as exp i r arg t t r with arg t t r and r r are complex numbers satisfying re r 1 2 re r 1 2 weighted hardy spaces l 2 1 and l 2 2 are defined such that the toeplitz operator t a l 2 1 l 2 2 is bounded and invertible and that the finite section method is applicable to t a considered as acting from l 2 1 onto l 2 2 these results are then applied to prove a conjecture of fisher and hartwig adv in chem phys 15 1968 333 353 which asserts that the determinants of the n n sections of t a are asymptotically equal to g b n n q e a n where q r 2 r 2 and g b e a are certain nonzero constants under quite general conditions on the smoothness of b these constants are completely identified

matrix algebra preconditioners for multilevel toeplitz systems do not insure optimal convergence rate in the last decades several matrix algebra optimal and superlinear preconditioners those assuring a strong clustering at the unity have been proposed for the solution of polynomially ill conditioned toeplitz linear systems the corresponding generalizations for multilevel structures are neither optimal nor superlinear see e g contemp math 281 2001 193 concerning the notion of superlinearity it has been recently shown that the proper clustering cannot be obtained in general see linear algebra appl 343 344 2002 303 siam j matrix anal appl 22 1 1999 431 math comput 72 2003 1305 in this paper by exploiting a proof technique previously proposed by the authors see contemp math 323 2003 313 we prove that the spectral equivalence and the essential spectral equivalence up to a constant number of diverging eigenvalues are impossible too in conclusion optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel multigrid techniques

emergence of a singularity for toeplitz determinants and painleve v we obtain asymptotic expansions for toeplitz determinants corresponding to a family of symbols depending on a parameter t for t positive the symbols are regular so that the determinants obey szeg s strong limit theorem if t 0 the symbol possesses a fisher hartwig singularity letting t 0 we analyze the emergence of a fisher hartwig singularity and a transition between the two different types of asymptotic behavior for toeplitz determinants this transition is described by a special painleve v transcendent a particular case of our result complements the classical description of wu mccoy tracy and barouch of the behavior of a 2 spin correlation function for a large distance between spins in the two dimensional ising model as the phase transition occurs

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results

segal sugawara vectors for the lie algebra of type g2 explicit formulas for segal sugawara vectors associated with the simple lie algebra mathfrak g of type g 2 are found by using computer assisted calculations this leads to a direct proof of the feigin frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level as an application we give an explicit solution of vinberg s quantization problem by providing formulas for generators of maximal commutative subalgebras of u mathfrak g we also calculate the eigenvalues of the hamiltonians on the bethe vectors in the gaudin model associated with mathfrak g

komleva type expansions and asymptotics for linear operators abstract in this paper we prove asymptotic komleva type expansions valid for sequences of linear operators t n approximating the identity in supremum norm over the space of the continuous functions in particular under suitable mild conditions on the sequence t n we obtain rational expansions for t n f that are of special interest in a numerical analysis context as special cases of these results we find asymptotic expansions for exponential type and de la vallee poussin polynomial operators the case of the cesaro sums is discussed in connection with the komleva theory but the main asymptotic results are proved by using other tools coming from a context of structured linear algebra some numerical applications of the theoretical part are then discussed