classification of irreducible weight modules over w algebra w 2 2 in this paper we show that the support of an irreducible weight module over the w algebra w 2 2 which has an infinite dimensional weight space coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional as a corollary we obtain that every irreducible weight module over the w algebra w 2 2 having a nontrivial finite dimensional weight space is a harish chandra module and hence is either an irreducible highest or a lowest weight module or an irreducible module of the intermediate series

from jack polynomials to minimal model spectra in this note a deep connection between free field realizations of conformal field theories and symmetric polynomials is presented we give a brief introduction into the necessary prerequisites of both free field realizations and symmetric polynomials in particular jack symmetric polynomials then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods while these results on the representation theory of the minimal models are all known this note exploits the full power of jack polynomials to present significant simplifications of the original proofs in the literature

robust control in the gap a state space solution in the presence of a single input delay a solution is provided to the problem of sub optimal robust design in the gap for a system with a single input lag the problem is shown to reduce to two matrix algebraic riccati equations that solve an allied linear quadratic gaussian h sub 2 problem and a parameterized differential matrix riccati equation the analysis intertwines finite dimensional delay equations and distributed abstract evolution models the parameterized family of suboptimal compensators is presented in terms of a neutral equation model

on staggered indecomposable virasoro modules in this article certain indecomposable virasoro modules are studied specifically the virasoro mode l0 is assumed to be nondiagonalizable possessing jordan blocks of rank 2 moreover the module is further assumed to have a highest weight submodule the left module and that the quotient by this submodule yields another highest weight module the right module such modules which have been called staggered have appeared repeatedly in the logarithmic conformal field theory literature but their theory has not been explored in full generality here such a theory is developed for the virasoro algebra using rather elementary techniques the focus centers on two different but related questions typically encountered in practical studies how can one identify a given staggered module and how can one demonstrate the existence of a proposed staggered module given just the values of the highest weights of the left and right modules themselves subject to simple necessary conditions invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module the possible values of these invariants form a vector space of dimension 0 1 or 2 and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace possibly empty the number of invariants and affine restrictions is purely determined by the structures of the left and right modules moreover in order to facilitate applications the expressions for the invariants and restrictions are given by formulas as explicit as possible they generally rely on expressions for virasoro singular vectors finally the text is liberally peppered throughout with examples illustrating the general concepts these have been carefully chosen for their physical relevance or for the novel features they exhibit

the deformed lie algebra of differential operators of order at most one in this paper we study the algebraic properties of the deformed lie algebra l of differential operators of order at most one we determine all the derivations central extensions and automorphism group of l furthermore the reducibility criterion for the verma modules of the universal central extension of l is achieved through a study of the shapovalov form

singular dimensions of the n 2 superconformal algebras ii the twisted n 2 algebra we introduce a suitable adapted ordering for the twisted n 2 superconformal algebra i e with mixed boundary conditions for the fermionic fields we show that the ordering kernels for complete verma modules have two elements and the ordering kernels for g closed verma modules just one therefore spaces of singular vectors may be two dimensional for complete verma modules whilst for g closed verma modules they can only be one dimensional we give all singular vectors for the levels 1 and for both complete verma modules and g closed verma modules we also give explicit examples of degenerate cases with two dimensional singular vector spaces in complete verma modules general expressions are conjectured for the relevant terms of all primitive singular vectors i e for the coefficients with respect to the ordering kernel these expressions allow to identify all degenerate cases as well as all g closed singular vectors they also lead to the discovery of subsingular vectors for the twisted n 2 superconformal algebra explicit examples of these subsingular vectors are given for the levels 1 and finally the multiplication rules for singular vector operators are derived using the ordering kernel coefficients this sets the basis for the analysis of the twisted n 2 embedding diagrams

the standard h problem and the maximum principle the general linear case the classical relatively simple ideas of linear quadratic lq optimization are used in a time domain treatment of the standard h infty problem given the power of time domain analysis the problem can be solved in the general framework of linear time varying possibly infinite dimensional finite as well as infinite horizon systems under no structural restrictions such as block dimensions zero blocks in the d operator etc in the spirit of recent finite dimensional linear time invariant lti results the solution is given in terms of two coupled riccati equations it includes a criterion for suboptimality and a parametrization of all suboptimal compensators results pertinent to lti periodic and asymptotic systems are obtained as corollaries

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings in this paper we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping moreover strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions

on geometric and analytic constraints in the h infty fixed lag smoothing limiting factors on achievable performance in the continuous and discrete time fixed lag smoothing problems are shown to include geometric constraints inherited from the fixed interval linfin problem and analytic constraints affected by the preview interval thus the fixed lag problem is viewed as a constrained version of its fixed interval counterpart the analysis sheds light on the dependence of achievable performance on the preview length and in particular on the generic presence of performance saturation at a finite preview interval

robust controller design for delay systems in the gap metric a robust stabilization problem in the gap metric is discussed for a system with delays in control the authors derive a design procedure of robust controllers based on finite dimensional ricatti equations and it is shown that the resulting controller has a structure of observer based predictive action by employing completing the square argument a game theoretic interpretation is also provided on the trade off between the initial uncertainties and attenuating the disturbance caused by plant uncertainties

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

h spl infin deconvolution filtering prediction and smoothing a krein space polynomial approach this correspondence is concerned with the infinite horizon h spl infin deconvolution for discrete time invariant systems described by polynomial equations a simple frequency domain approach in krein space is developed for the deconvolution problem with the help of an arma innovation model solutions to the problems of deconvolution filtering prediction and fixed lag smoothing are obtained the estimators are calculated by performing one j spectral factorization

h sup spl infin control and estimation with preview part i matrix are solutions in continuous time preview control and fixed lag smoothing allow a noncausal component in the controller estimator time domain variational analysis is used in a reduction to an open loop differential game leading to a complete necessary and sufficient characterization of suboptimal values and an explicit state space design in terms of a parameterized nonstandard algebraic matrix riccati equation in a general continuous time linear system setting the solution offers insight into the appropriate structure of the associated hamiltonian where the state and co state are not the usual state of the original dynamic system and that of its adjoint rather the state and co state are selected to capture the respective lumped effects of initial data and future input selection in the allied game

extensions of thickened verma modules of the virasoro algebra abstract let g denote the virasoro lie algebra h its cartan subalgebra and s h the symmetric algebra on h in this paper we consider thickened verma modules m which are u g s h bimodules satisfying m s h c m where m is the usual verma module with highest weight h we determine ext 1 m m to be s h s h where is up to a c algebra automorphism of s h a product of irreducible factors of the determinant of the shapovalov matrix this result provides a conceptual explanation of the factorization of the shapovalov determinant and implies that the inverse of the shapovalov matrix has only simple poles

game theory approach to h sub spl infin optimal discrete time fixed point and fixed lag smoothing optimal h sub spl infin fixed point and fixed lag discrete time smoothing estimators are developed by applying a game theory approach a deterministic discrete time game is defined where the estimator plays against nature nature determines the system initial condition the driving input and the measurement noise whereas the estimator tries to find an estimate that brings a prescribed cost function that is based on the error of the estimation at a fixed time instant to a saddle point equilibrium the latter estimate yields the h sub spl infin optimal fixed point smoothing differently from the usual case in h sub spl infin optimal estimation and control the critical value of the scalar design parameter of the smoothing game is obtained in closed form explicitly in the terms of the corresponding h sub 2 solution unlike the h sub 2 case the recursive application of the h sub spl infin fixed point smoothing algorithm does not lead to fixed lag smoothing in the h sub spl infin norm sense the h sub spl infin fixed lag smoothing filter is derived by augmenting the state vector of the system with additional delayed states

squaring down lti systems a geometric approach in this paper the problem of reducing a given lti system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory first it will be shown how an lti system can be turned into a left invertible system thus preserving key system properties like stabilizability phase minimality right invertibility relative degree and infinite zero structure moreover the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane by duality the scheme of a right inverter will be derived straightforwardly moreover the squaring down problem will be addressed in fact when the left and right reduction procedures are applied together a system with an unequal number of inputs and outputs is turned into a square and invertible system furthermore as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems

representation theory of neveu schwarz and ramond algebras i verma modules abstract in this article we study the structure of verma modules of n 1 super virasoro algebras as applications we construct bernstein gel fand gel fand type resolutions this article is the detailed and expanded version of iohara and koga c r acad sci paris ser i 328 1999 381

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

brief paper minimax fir smoothers for deterministic continuous time state space models in order to design a smoother for a deterministic continuous time state space model a new performance criterion is proposed which is given by a ratio of the current estimation error to the weighted energy of the deterministic disturbance applied during the recent finite horizon among smoothers with the deadbeat property and finite impulse response fir structure a minimax fir smoother mfs is obtained to optimize the proposed performance criterion to begin with the functional optimization problem is formulated with respect to kernel functions of the mfs and then its solution is explicitly presented the mfs depends only on inputs and outputs on the finite recent horizon and is independent of any a priori state information the mfs is first represented in an integral form for simple representation and then a differential form is introduced for efficient numerical computation as in h iir smoothing and h 2 iir filtering it is shown that the proposed mfs for a deterministic system can be interpreted as the minimum variance unbiased fir smoother for a stochastic system

brief on the h fixed lag smoothing how to exploit the information preview the problem of the continuous time h fixed lag smoothing over the infinite horizon is studied the first solution to the problem is derived in terms of one algebraic riccati equation of the same dimension as in the filtering case and the mechanism by which the performance improvements with respect to the h filtering occur is clarified it is shown that the h smoother exploits the information preview in an h 2 manner

on the structure of verma modules over the w algebra w 2 2 in this paper we describe the structure of verma modules over the w algebra w 2 2 we show that either a verma module over w 2 2 is irreducible or its maximal submodule is cyclic

on hybrid projection methods for asymptotically quasi φ nonexpansive mappings the purpose of this paper is to consider the problem of approximating a common fixed point of two asymptotically quasi f nonexpansive mappings based on hybrid projection methods strong convergence theorems are established in a real banach space

irreducible virasoro modules from tensor products ii abstract in this paper we obtain a class of virasoro modules by taking tensor products of a finite number of irreducible virasoro modules of the form i a i defined in 19 with irreducible highest weight modules v h or with irreducible virasoro modules ind n defined in 24 i e the tensor product i 1 m i a i v where v is v h or ind n we obtain the necessary and sufficient conditions for such tensor product modules to be irreducible and determine the necessary and sufficient conditions for two of them to be isomorphic we compare the tensor products with other known irreducible non weight virasoro modules we also characterize these irreducible tensor products

moduli spaces of curves and representation theory we establish a canonical isomorphism between the second cohomology of the lie algebra of regular differential operators on x of degree 1 and the second singular cohomology of the moduli space hat f g 1 of quintuples c p z l wherec is a smooth genusg riemann surface p a point onc z a local parameter atp l a degreeg 1 line bundle onc and a class of local trivializations ofl atp which differ by a non zero factor the construction uses an interplay between various infinite dimensional manifolds based on the topological spaceh of germs of holomorphic functions in a neighborhood of 0 in x and related topological spaces the basic tool is a canonical map from hat f g 1 to the infinite dimensional grassmannian of subspaces ofh which is the orbit of the subspaceh of holomorphic functions on x vanishing at under the group auth as an application we give a lie algebraic proof of the mumford formula n 6n2 6n 1 1 where n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg whose fiber overc is the space of differentials of degreen onc

a spectral factorization approach for h spl infin estimation problems in discrete time this note deals with the h filtering prediction and smoothing problems for discrete time linear systems the j spectral factorization approach is pursued and for each of the three problems a computationally efficient procedure to derive a j spectral factor is presented the construction of filter smoother and predictor hinges on the stabilizing solution of the same algebraic riccati equation this fact allows for the avoidance of any fictitious augmentation of the state space dimensions

brief paper an efficient algorithm for discrete time h preview control an efficient numerical solution to the output feedback discrete time h preview control problem is presented the first step of the solution comes from embedding the preview control problem in a two degree of freedom generalised regulator framework in which the augmented system state includes a segment of the frequency weighted previewed signal an efficient method for solving the resulting inflated discrete algebraic riccati equations dares is then found it is demonstrated that one need only solve an estimation riccati equation with dimension that of the plant and a control riccati equation with dimension that of the plant and reference weighting taken together the reduced control riccati equation is the product of a recursive reduction enabled by the zero eigenvalues in the associated closed loop matrix an added feature of the presented algorithm is that no large matrix computations are required and no large matrices need be stored only the edges of the full information riccati equation solution are required when synthesising the controller

lq control for ito type stochastic systems with input delays this paper considers the linear quadratic lq control problem for the ito type stochastic system with input delays due to simultaneous appearances of diffusion terms dependent on the state and the control as well as delays in the dynamic system the problem is very involved and remains to be solved we not only provide the solvable condition of the problem but also the explicit expression of the causal and adapted controller for a kind of lq problems all of these are based on a stochastic riccati equation the key technique is to pursue the explicit cost value of the lq problem by fbsde and derive the analytical controller via the interplay between the original problem and its equivalent abstract description

a modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in banach spaces in this paper we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi asymptotically nonexpansive mappings strong convergence theorems are established in a uniformly smooth and strictly convex banach space which also enjoys the kadec klee property our results improve and extend the corresponding results announced by qin et al 2010 and many authors

geometry of the analytic loop group abstract we introduce and study a notion of analytic loop group with a riemann hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity u g with non trivial central charge we introduce a poisson structure and study properties of its poisson dual group we prove that the hopf poisson structure is isomorphic to the semi classical limit of the center of u g it is a geometric realization of the center then the symplectic leaves and corresponding equivalence classes of central characters of u g are parameterized by certain g bundles on an elliptic curve

convergence theorems for mixed equilibrium problems variational inequality problem and uniformly quasi ϕ asymptotically nonexpansive mappings abstract we introduce a modified block hybrid projection algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi asymptotically nonexpansive mappings the set of the variational inequality for an inverse strongly monotone operator the set of solutions of the mixed equilibrium problems then we obtain strong convergence theorems for the sequences generated by this process in a 2 uniformly convex and uniformly smooth banach space our results extend and improve ones from several earlier works

linear quadratic regulation and stabilization of discrete time systems with delay and multiplicative noise this paper is concerned with the long standing problems of linear quadratic regulation lqr control and stabilization for a class of discrete time stochastic systems involving multiplicative noises and input delay these fundamental problems have attracted resurgent interests due to development of networked control systems an explicit analytical expression is given for the optimal lqr controller more specifically the optimal lqr controller is shown to be a linear function of the conditional expectation of the state with the feedback gain based on a riccati zxl difference equation it is also shown that the system is stabilizable in the mean square sense if and only if an algebraic riccati zxl equation has a particular solution these results are based on a new technical tool which establishes a non homogeneous relationship between the state and the costate of this class of systems and the introduction of a new lyapunov function for the finite horizon optimal control design

convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces the purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi f nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of banach spaces our results improve and extend the corresponding results announced by many others

strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces abstract in this paper we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a banach space then we study the strong and weak convergence of the sequences

singular dimensions of the n 2 superconformal algebras i verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1 following a method developed for the virasoro algebra by kent we introduce the concept of adapted orderings on superconformal algebras we prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space we apply this method to the topological n 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological verma modules 0 1 2 or 3 depending on the type of verma module and the type of singular vector as a consequence we prove the conjecture of gato rivera and rosado on the possible existing types of topological singular vectors 4 in chiral verma modules and 29 in complete verma modules interestingly we have found two dimensional spaces of singular vectors at level 1 finally by using the topological twists and the spectral flows we also obtain the maximal dimensions of the singular vector spaces for the neveu schwarz n 2 algebra 0 1 or 2 and for the ramond n 2 algebra 0 1 2 or 3

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

the mumford form and the polyakov measure in string theory an explicit formula is derived for the mumford form on the moduli space of algebraic curves according to the belavin knizhnik theorem this gives a formula for the polyakov bosonic string measure

classification of harish chandra modules over some lie algebras related to the virasoro algebra abstract in this paper we provide a uniform method to thoroughly classify all harish chandra modules over some lie algebras related to the virasoro algebras we first classify such modules over the lie algebra w s for s 0 1 2 with this result and method we can also do such works for some lie algebras related to the virasoro algebra including the several kinds of schrodinger virasoro lie algebras which are open up to now

h sub infinity fixed lag smoothing filter for scalar systems the solution of the h sub infinity optimal linear fixed lag smoothing problem is considered using a polynomial matrix description for the discrete system the smoother is obtained from the solution of a linear equation and a spectral factorization calculation the pole zero properties of the optimal smoother are obvious in the polynomial representation and insights into the measurement noise rejection properties of the smoother are obtained allowance is made for both dynamic cost weighting and colored measurement noise the polynomial form of the smoothing filter may be incorporated into a self tuning algorithm and a simple adaptive smoother is discussed

on designing h_ infty fault detection filter for linear discrete time varying systems this technical note deals with the problem of finite horizon h fault detection for linear discrete time varying systems using an observer based fault detection filter fdf with residual feedback as a residual generator a new problem formulation of fault detection is first presented then the design of finite horizon h fdf is converted into a minimum problem of certain quadratic form through building a relationship with krein space projection and applying innovation analysis a sufficient and necessary condition for the minimum is derived and a solution to the h fdf is obtained by recursively computing riccai recursions numerical examples are given to show the effectiveness of the proposed method

classification of finite irreducible conformal modules over a class of lie conformal algebras of block type abstract we classify finite irreducible conformal modules over a class of infinite lie conformal algebras b p of block type where p is a nonzero complex number in particular we obtain that a finite irreducible conformal module over b p may be a nontrivial extension of a finite conformal module over vir if p 1 where vir is a virasoro conformal subalgebra of b p as a byproduct we also obtain the classification of finite irreducible conformal modules over a series of finite lie conformal algebras b n for n 1

when does the h sup spl infin fixed lag smoothing performance saturate for a finite smoothing lag a notable difference between the h sup 2 and h sup spl infin smoothing is that the achievable performance in the latter problem might saturate as the function of the smoothing lag in the sense that there might exist a finite smoothing lag for which the achievable performance level is the same as for the infinite smoothing lag in this note necessary and sufficient conditions under which such a saturation occurs are derived in particular it is shown that the h sup spl infin performance saturates only if the h sup spl infin norm of the optimal error system is achieved at the infinite frequency i e if the worst case disturbance for the infinite smoothing lag case can be arbitrarily fast and thus in a sense unpredictable

necessary and sufficient condition for two player stackelberg strategy this technical note revisits the open loop stackelberg strategy for a two player game by introducing a new costate which captures the future information of the control input we present a necessary and sufficient condition for the existence and uniqueness of the two player game the optimal strategy is designed in terms of three decoupled and symmetric riccati equations which improves the existing results greatly on computation

a hybrid iteration scheme for equilibrium problems variational inequality problems and common fixed point problems in banach spaces abstract in this paper we introduce an iterative process which converges strongly to a common element of a set of common fixed points of finite family of closed relatively quasi nonexpansive mappings the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an inverse strongly monotone mapping in banach spaces

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

geometric realization of conformal field theory on riemann surfaces conformal field theory on a family of riemann surfaces is formulated we derive equations of motion of vacua which are parametrized by moduli of riemann surfaces and show that these vacua are characterized uniquely by these equations our theory has a deep connection with sato s theory of kp equations

strong convergence of shrinking projection methods for quasi φ nonexpansive mappings and equilibrium problems the purpose of this paper is to consider the convergence of a shrinking projection method for a finite family of quasi f nonexpansive mappings and an equilibrium problem strong convergence theorems are established in a uniformly smooth and strictly convex banach space which also enjoys the kadec klee property

the adapted ordering method for lie algebras and superalgebras and their generalizations in 1998 the adapted ordering method was developed for the representation theory of the superconformal algebras in two dimensions it allows us to determine maximal dimensions for a given type of space of singular vectors to identify all singular vectors by only a few coefficients to spot subsingular vectors and to set the basis for constructing embedding diagrams in this paper we present the adapted ordering method for general lie algebras and superalgebras and their generalizations provided they can be triangulated we also review briefly the results obtained for the virasoro algebra and for the n 2 and ramond n 1 superconformal algebras

stabilization of discrete time systems with multiplicative noise and multiple delays in the control variable this paper focuses on the stabilization problem of discrete time systems with multiplicative noise and multiple delays in the control variable first a transformation is constructed to convert the original system into a delay free system with its input matrix containing future noise next the finite horizon linear quadratic regulation problem for the delay free system subject to a constraint on the control is investigated we provide the necessary and sufficient condition for the problem having a unique solution and the optimal feedback control using riccati type difference equations finally we show that the original system can be stabilized in the mean square sense if and only if the set of solutions to the riccati type difference equations is convergent this further yields the fact that the system under consideration can be stabilized if and only if the algebraic riccati type equations have a unique solution such that a specific matrix is positive definite

strong convergence of a proximal type algorithm in a banach space in this paper we study strong convergence of the proximal point algorithm it is known that the proximal point algorithm converges weakly to a solution of a maximal monotone operator but it fails to converge strongly then in math program 87 2000 pp 189 202 solodov and svaiter introduced the new proximal type algorithm to generate a strongly convergent sequence and established a convergence property for it in hilbert spaces our purpose is to extend solodov and svaiter s result to more general banach spaces using this we consider the problem of finding a minimizer of a convex function

h sup spl infin control and estimation with preview part ii fixed size are solutions in discrete time h sup spl infin preview control and fixed lag smoothing problems are solved in general discrete time linear systems via a reduction to equivalent open loop differential games to prevent high order riccati equations found in some solutions the state of the hamilton jacobi system resides in a quotient space of an auxiliary extended state space system the dimension of that auxiliary space is equal to the state space dimension of the original system ignoring the delay

control for ito stochastic systems with input delay this paper examines the long standing problem of linear quadratic regulation and stabilization for ito stochastic systems with input delay this problem remains a primary challenge because the separation principle does not hold for ito stochastic systems this paper presents a complete solution to the problem 1 the sufficient and necessary solvability condition of the optimal control and the analytical controller are given based on the modified riccati differential equation defined herein 2 the sufficient and necessary stabilization condition in mean square sense is explored we show that the ito stochastic system with input delay is stabilized if and only if the modified algebraic riccati equation developed in this paper has a unique positive definite solution the essential obstacle encountered in this paper concerns a delayed forward backward stochastic differential equation d fbsde which is mathematical challenging

on the structure of verma modules over virasoro and neveu schwarz algebras in the paper we present a different proof of the theorem of b l feigin and d b fuchs about the structure of verma modules over virasoro algebra we state some new results about the structure of verma modules over neveu schwarz the proof has two advantages first it is simpler in the most interesting cases for example in the so called minimal models second it can be generalized for neveu schwarz algebra for some class of verma modules

on simplicity of vacuum modules we find necessary and sufficient conditions of irreducibility of vacuum modules over affine lie algebras and superalgebras from this we derive conditions of simplicity of minimal w algebras moreover in the case of the virasoro and neveu schwarz algebras we obtain explicit formulas for the vacuum determinants

an iterative method for total quasi φ asymptotically nonexpansive semi groups and generalized mixed equilibrium problems in banach spaces in this paper we introduce an iterative scheme by the modified halpern mann type for total quasi asymptotically nonexpansive semi groups and prove the strong convergence theorem using our new iterative process which converges strongly to a common element of the set of common fixed points for total quasi asymptotically nonexpansive semi groups and the set of solutions of generalized mixed equilibrium problem in a uniformly smooth and strictly convex banach space with the kadec klee property using the properties of the generalized f projection operator our results extend and improve the corresponding recent results in the literature

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra