convergence of directional methods under mild differentiability and applications abstract a semilocal convergence analysis for directional methods under mild differentiability conditions is provided in this study using our idea of recurrent functions we provide sufficient convergence conditions as well as the corresponding errors bounds the results are extended to hold in a hilbert space setting and a favorable comparison is provided with earlier works 6 7 8 9 10 11 20 numerical examples are also provided in this study

a family of quasi newton methods for nonlinear equations with direct secant updates of matrix factorizations a new family of least change secant methods for solving systems of nonlinear algebraic equations is introduced at each iteration of the methods of this family the matrix b k is represented as a product of the form a k 1 r k where the matrices a k and r k have a determined structure and a k 1 r k 1 are chosen by the use of a least change secant rule the first and second methods of broyden math comp 19 1965 pp 577 593 math comp 25 1971 pp 285 294 and the methods of schubert math comp 24 1970 pp 27 30 johnson and austria siam j numer anal 20 1983 pp 315 325 and chadee tech report sol 85 8 dept of operations research stanford university stanford ca 1985 are members of this family local and superlinear convergence results are proved and particular members of the family that seem to be of practical usefulness are introduced

integrable xyz spin chain with boundaries we consider a general class of boundary terms of the open xyz spin 1 2 chain compatible with integrability we have obtained the general elliptic solution of a k matrix obeying the boundary yang baxter equation using the r matrix of the eight vertex model and derived the associated integrable spin chain hamiltonian

robustness analysis a case study the stability test of polynomials whose coefficients depend multilinearly on interval parameters is considered the authors describe and compare four brute force solution approaches these are eigenvalue calculation zero exclusion from a specified value set algebraic tests of real and complex hurwitz roots and the parameter space method they are applied to a simple example with two parameters and third order polynomial an interesting feature of the example is that it can have an isolated unstable point the example may be useful as a benchmark for future approaches to the multilinear problem all four methods are shown to be feasible for the simple example but they require effort

directional newton methods in n variables directional newton methods for functions f of n variables are shown to converge under standard assumptions to a solution of f x 0 the rate of convergence is quadratic for near gradient directions and directions along components of the gradient of f with maximal modulus these methods are applied to solving systems of equations without reversion of the jacobian matrix

fixed point optimization algorithms for distributed optimization in networked systems this paper considers a networked system with a finite number of users and deals with the problem of minimizing the sum of all users objective functions over the intersection of all users constraint sets onto which the projection cannot be easily implemented the main objective of this paper is to devise distributed optimization algorithms which enable each user to find the solution of the problem without using other users objective functions and constraint sets to reach this goal we first introduce easily implementable nonexpansive mappings of which the intersection of the fixed point sets is equal to the constraint set in the problem we formulate the problem as a convex minimization problem over the intersection of the fixed point sets of the nonexpansive mappings we then present an iterative algorithm based on the conventional incremental subgradient methods which use the projection for solving the problem the algorithm can be implemented by using nonexpansive mappings other than the project

robust wideband beamforming by the hybrid steepest descent method this paper uses the hybrid steepest descent method hsdm to design robust smart antennas several design criteria as well as robustness are mathematically described by a finite collection of closed convex sets in a real euclidean space desirable beamformers are defined as points of the generalized convex feasible set which is well defined even in the case of inconsistent design criteria a quadratic cost function is formed by the correlations of the incoming data and the hsdm constructs a point sequence that strongly converges to the unique minimizer of the cost function over the generalized convex feasible set numerical examples validate the proposed design

the general uq sl 2 invariant xxz integrable quantum spin chain the boundary conditions are constructed for the general xxz quantum integrable spin chain such that the resulting system is invariant under the quantum algebra uq sl 2 action the result generalizes that due to pasquier and saleur 1990 obtained for spin 1 2 and spin 1

extensions of the small mu test for robust stability the notion of a topologically arcwise connected set is used to develop a general graph topology based robustness test the test provides a switching criterion among different inequality tests and possibly among different nominal models when it is applied to extend the small mu test the result reduces conservatism and allows a more natural application of the test

hecke algebraic approach to the reflection equation for spin chains we use the structural similarity of certain coxeter artin systems to the yang baxter and reflection equations to convert representations of these systems into new solutions of the reflection equation we construct certain bethe ansatz states for these solutions using a parametrization suggested by abstract representation theory

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

analysis of two chebyshev like third order methods free from second derivatives for solving systems of nonlinear equations in this paper two chebyshev like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations the methods can be obtained by having different approximations to the second derivatives present in the chebyshev method we study the local and third order convergence of the methods using the point of attraction theory the computational aspects of the methods are also studied using some numerical experiments including an application to the chandrasekhar integral equations in radiative transfer

algorithms of common solutions for variational inclusions mixed equilibrium problems and fixed point problems in this paper we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real hilbert space furthermore we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

fusion procedure for the belavin model with open boundary conditions we have recently used unitarity and cross unitarity properties of an symmetric r matrix to construct the transfer matrix t u for an n site open spin chain here we give a fusion procedure for such a chain and we prove that the fused transfer matrix is commutative with the original transfer matrix t u

stability and performance robustness analysis with highly structured uncertainties necessary and sufficient test the author presents an extension of a technique of analyzing the stability of linear time invariant interconnected systems with uncertainties each element of the frequency response matrix is assumed to lie inside a polygon at each frequency it is shown that the exterior boundary of the image of the cartesian product of these polygons under the map det i m delta is contained in the boundary of the image of the cartesian product of the boundaries of these polygons under the same map thus the image can be constructed from the images of the edges the computational technique of r r e de gaston and m g safonov 1988 can then be applied to perform a nonconservative robustness analysis of the feedback system

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on the fusion of face and vertex models some new solutions of the yang baxter or star triangle relations are found using the fusion procedure for belavin s 1981 zn zn symmetric model the relation between the fusion procedure of the vertex and face models is also constructed

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

some new properties of the structured singular value j c doyle et al 1982 have shown that a necessary and sufficient condition for robust stability or robust performance in the h sup infinity frame work may be formulated as a bound on the structured singular value mu of a specific matrix m which includes information on the system model the controller the model uncertainty and the performance specifications often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices t which are of direct interest to the engineer e g sensitivity or complementary sensitivity the present paper shows how to derive bounds on sigma t from bounds on mu m

interval expansion method for nonlinear equation in several variables an interval expansion method is presented to find the solution of nonlinear equation in several variables with guaranteed accuracy in this method the existence theorem for the solution of nonlinear equation is obtained by using two newly defined operators a new algorithm for solving nonlinear equation in several variables is given furthermore we prove that the new method is global convergent under mild conditions

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

acceleration method combining broadcast and incremental distributed optimization algorithms this paper considers a networked system consisting of an operator which manages the system and a finite number of subnetworks with all users and studies the problem of minimizing the sum of the operator s and all users objective functions over the intersection of the operator s and all users constraint sets when users in each subnetwork can communicate with each other they can implement an incremental subgradient method that uses the transmitted information from their neighbor users since the operator can communicate with users in the subnetworks it can implement a broadcast distributed algorithm that uses all available information in the subnetworks we present an iterative method combining broadcast and incremental distributed optimization algorithms our method has faster convergence and a wider range of application compared with conventional distributed algorithms we also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topol

on the convergence of a damped secant method with modified right hand side vector we present a convergence analysis for a damped secant method with modified right hand side vector in order to approximate a locally unique solution of a nonlinear equation in a banach spaces setting in the special case when the method is defined on r i our method provides computable error estimates based on the initial data such estimates were not given in relevant studies such as herceg et al 1996 krejic 2002 numerical examples further validating the theoretical results are also presented in this study

approximate methods for convex minimization problems with series parallel structure abstract consider a problem of minimizing a separable strictly convex monotone and differentiable function on a convex polyhedron generated by a system of m linear inequalities the problem has a series parallel structure with the variables divided serially into n disjoint subsets whose elements are considered in parallel this special structure is exploited in two algorithms proposed here for the approximate solution of the problem the first algorithm solves at most min m n 1 subproblems each subproblem has exactly one equality constraint and at most n variables the second algorithm solves a dynamically generated sequence of subproblems each subproblem has at most n 1 equality constraints where is the total number of variables to solve these subproblems both algorithms use the authors projected newton bracketing method for linearly constrained convex minimization in conjunction with the steepest descent method we report the results of numerical experiments for both algorithms

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

a viscosity method with no spectral radius requirements for the split common fixed point problem abstract this paper is concerned with an algorithmic solution to the split common fixed point problem in hilbert spaces our method can be regarded as a variant of the viscosity approximation method under very classical assumptions we establish a strong convergence theorem with regard to involved operators belonging to the wide class of quasi nonexpansive operators in contrast with other related processes our algorithm does not require any estimate of some spectral radius the technique of analysis developed in this work is new and can be applied to many other fixed point iterations numerical experiments are also performed with regard to an inverse heat problem

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

on the convergence of a damped newton like method with modified right hand side vector we present a convergence analysis for a damped newton like method with modified right hand side vector in order to approximate a locally unique solution of a nonlinear equation in a banach spaces setting in the special case when the method is defined on r m our method provides computable error estimates based on the initial data such estimates were not given in relevant studies such as numerical examples further validating the theoretical results are also presented in this study

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

constant solutions of reflection equations and quantum groups to the yang baxter equation an additional relation can be added this is the reflection equation that appears in various places with or without spectral parameter e g in factorizable scattering on a half line integrable lattice models with nonperiodic boundary conditions noncommutative differential geometry on quantum groups etc two forms of spectral parameter independent reflection equations are studied chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations for a variety of known solutions of the yang baxter equation the constant solutions of the reflection equations are given various quadratic algebras defined by the reflection equations are also given explicitly

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations abstract in this work we take advantage of the powerful quadratic programming theory to obtain optimal solutions of scheduling problems we apply a methodology that starts in contrast to more classical approaches by formulating three unrelated parallel machine scheduling problems as 0 1 quadratic programs under linear constraints by construction these quadratic programs are non convex therefore before submitting them to a branch and bound procedure we reformulate them in such a way that we can ensure convexity and a high quality continuous lower bound experimental results show that this methodology is interesting by obtaining the best results in literature for two of the three studied scheduling problems

projected subgradient techniques and viscosity methods for optimization with variational inequality constraints in this paper we propose an easily implementable algorithm in hilbert spaces for solving some classical monotone variational inequality problem over the set of solutions of mixed variational inequalities the proposed method combines two strategies projected subgradient techniques and viscosity type approximations the involved stepsizes are controlled and a strong convergence theorem is established under very classical assumptions our algorithm can be applied for instance to some mathematical programs with complementarity constraints

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

on the stability bounds of singularly perturbed systems the stability bound problem of linear time invariant singularly perturbed systems is considered a set of new stability conditions based on the frequency domain representation is derived the stability criteria can be easily verified by computing certain singular values within finite frequency intervals illustrative examples show that the proposed criteria actually induce a less conservative epsilon bound than the existing criteria and for certain cases an infinite epsilon bound may be obtained

a uniformly convergent finite difference scheme for a singularly perturbed semilinear equation boundary value problems for singularly perturbed semilinear elliptic equations are considered special piecewise uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter numerical methods composed of standard monotone finite difference operators and these piecewise uniform meshes are shown theoretically to be uniformly with respect to the singular perturbation parameter convergent numerical results are also presented which indicate that in practice the method is first order accurate

direct secant updates of matrix factorizations this paper presents a new context for using the sparse broyden update method to solve systems of nonlinear equations the setting for this work is that a newton like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory jacobian matrix approximation whenever required the total cost of obtaining each jacobian matrix or the cost of factoring it to solve for the newton step is assumed to be sufficiently high to make it attractive to keep the same jacobian approximation for several steps this paper suggests the extremely convenient and apparently effective technique of applying the sparse broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required the strategy is shown to be locally and q superlinearly convergent and some encouraging numerical results are presented

proximal point algorithms for nonsmooth convex optimization with fixed point constraints abstract the problem of minimizing the sum of nonsmooth convex objective functions defined on a real hilbert space over the intersection of fixed point sets of nonexpansive mappings onto which the projections cannot be efficiently computed is considered the use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem with the focus on fixed point approximation techniques two algorithms are devised for solving the problem one blends an incremental subgradient method which is a useful algorithm for nonsmooth convex optimization with a halpern type fixed point iteration algorithm the other is based on an incremental subgradient method and the krasnosel ski mann fixed point algorithm it is shown that any weak sequential cluster point of the sequence generated by the halpern type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the krasnosel ski mann type algorithm which also belongs to the solution set numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence

convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi nonexpansive mappings this paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set it is assumed that each user s constraint set can be expressed as a fixed point set of a certain quasi nonexpansive mapping this enables us to consider the case in which the projection onto the constraint set cannot be computed efficiently this paper proposes two methods for solving the problem of minimizing the sum of their nondifferentiable convex objective functions over the intersection of their fixed point sets of quasi nonexpansive mappings in a real hilbert space one method is a parallel subgradient method that can be implemented under the assumption that each user can communicate with other users the other is an incremental subgradient method that can be implemented under the assumption that each user can communicate with its neighbors investigation of the two methods convergence properties for a constant step size reveals that with a small constant step size they approximate a solution to the problem consideration of the case in which the step size sequence is diminishing demonstrates that the sequence generated by each of the two methods strongly converges to the solution to the problem under certain assumptions convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods efficiency this paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods

hard constrained inconsistent signal feasibility problems we consider the problem of synthesizing feasible signals in a hilbert space in the presence of inconsistent convex constraints some of which must imperatively be satisfied this problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones the resulting convex optimization problem is analyzed and numerical solution schemes are presented along with convergence results the proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems an application to signal synthesis is demonstrated

fusion procedure for open chains the authors have generalized sklyanin s approach of constructing open integrable quantum spin chains to the case of pt invariant r matrices they formulate a fusion procedure for such chains in particular they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated applications of these results include constructing open integrable higher spin chains as well as obtaining functional equations for transfer matrix eigenvalues which may be solved by an analytical bethe ansatz

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

form factors and complete spectrum of xxx antiperiodic higher spin chains by quantum separation of variables the antiperiodic transfer matrices associated to higher spin representations of the rational 6 vertex yang baxter algebra are analyzed by generalizing the approach introduced recently in the framework of sklyanin s quantum separation of variables sov for cyclic representations spin 1 2 highest weight representations and also for spin 1 2 representations of the 6 vertex reflection algebra such sov approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on bethe ansatz and baxter q operator in particular we both prove the completeness of the sov characterization of the transfer matrix spectrum and its simplicity then the derived characterization of local operators by sklyanin s quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products

robustness with simultaneous pole and zero movement across the jω axis in this paper we present some new tests for assessing the robustness of the stability property of a feedback design the tests offer a wider scope of application than those previously described in the literature the main result is a robustness test which can be used to assess closed loop stability when the nominal plant model and perturbed plant model representing uncertainty have different numbers of poles and zeros in the right half plane this test also indicates over which frequencies good sensitivity and robust stability are compatible objectives and over which frequencies a compromise has to be made contrary to popular belief if there are no uncertain zeros which can move across the imaginary axis then good sensitivity and robust stability are compatible objectives over the same frequencies

integrability and fusion algebra for quantum mappings we apply the fusion procedure to a quantum yang baxter algebra associated with time discrete integrable systems notably integrable quantum mappings we present a general construction of higher order quantum invariants for these systems as an important class of examples we present the yang baxter structure of the gel fand dikii mapping hierarchy that we have introduced in previous papers together with the corresponding explicit commuting family of quantum invariants

vi constrained hemivariational inequalities distributed algorithms and power control in ad hoc networks we consider centralized and distributed algorithms for the numerical solution of a hemivariational inequality hvi where the feasible set is given by the intersection of a closed convex set with the solution set of a lower level monotone variational inequality vi the algorithms consist of a main loop wherein a sequence of one level strongly monotone hvis are solved that involve the penalization of the non vi constraint and a combination of proximal and tikhonov regularization to handle the lower level vi constraints minimization problems possibly with nonconvex objective functions over implicitly defined vi constraints are discussed in detail the methods developed in the paper are then used to successfully solve a new power control problem in ad hoc networks

an m matrix and majorant approach to robust stability and performance analysis for systems with structured uncertainty uncertain multiinput multioutput systems described in the frequency domain are discussed the theory of nonnegative matrices and m matrices is used along with majorant bounding techniques to develop robust stability and performance results for two types of uncertainty the first type is uncertainty with norm bounded subblocks the second type is uncertainty that has subblocks with known patterns but unknown gains this latter uncertainty representation allows the elements of a given uncertainty block to be correlated for uncertainties of this type a recursive analysis methodology is developed which yields increasingly nonconservative results throughout this paper the performance analysis determines norm bounds on the deviations of the outputs from their nominal values the results are illustrated by a numerical example

robust stability bounds for a class of discrete time regulators with computation delays abstract robust stability of discrete time regulators which utilize state predictors to compensate computation delays is considered novel expressions for the return difference matrices and the complementary sensitivity matrices at the input and the output of the regulator are found to obtain simple bounds for unstructured perturbations robust stability for perturbations of the system matrix and or the gain matrix is also considered under certain restrictions on the nominal system simple bounds for the perturbations are obtained directly from the characteristic equation it is shown that as far as the effect of the computation delays is concerned these bounds are explicitly related to those for the unstructured perturbations

on spin systems related to the temperley lieb algebra the spectrum of the transfer matrices constructed from the spectral parameter dependent temperley lieb r matrix is found using functional relations identical to those of the spin 1 2 xxz magnet

directional secant method for nonlinear equations we present a directional secant method a secant variant of the directional newton method for solving a single nonlinear equation in several variables under suitable assumptions we prove the convergence and the quadratic convergence speed of this new method numerical examples show that the directional secant method is feasible and efficient and has better numerical behaviour than the directional newton method

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

newton like method with modification of the right hand side vector this paper proposes a new newton like method which defines new iterates using a linear system with the same coefficient matrix in each iterate while the correction is performed on the right hand side vector of the newton system in this way a method is obtained which is less costly than the newton method and faster than the fixed newton method local convergence is proved for nonsingular systems the influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented relevant numerical examples are used to demonstrate the advantages of the proposed method

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi

plant and controller reduction problems for closed loop performance model reduction problems which consider preserving closed loop performance h sub 2 h sub infinity and mu in the presence of reduction error are developed these are formulated as weighted multiplicative error problems for plant reduction and weighted additive error problems for controller reduction with the weight function incorporating explicitly such control information as the desired sensitivity operator bound the setpoint disturbance spectrum and the plant uncertainties these problems are efficiently solved using the frequency weighted balanced realization technique the benefits of these reduction problems are illustrated with examples taken from the control of a binary distillation column