jordan all derivable points in the algebra of all upper triangular matrices abstract let tm n be the algebra of all n n upper triangular matrices we say that l tm n is a jordan derivable mapping at g if st ts s t s t t s t s for any s t tm n with st g an element g tm n is called a jordan all derivable point of tm n if every jordan derivable linear mapping at g is a derivation in this paper we show that every element in tm n is a jordan all derivable point

on bilinear maps determined by rank one matrices with some applications abstract let m n be the algebra of all n n matrix over a field f a a rank one matrix in m n in this article it is shown that if a bilinear map from m n m n to m n satisfies the condition that u v i a whenever u v a then there exists a linear map from m n to m n such that x y x y x y m n if is further assumed to be symmetric then there exists a matrix b such that x y tr xy b for all x y m n applying the main result we prove that if a linear map on m n is desirable at a rank one matrix then it is a derivation and if an invertible linear map on m n is automorphisable at a rank one matrix then it is an automorphism in other words each rank one matrix in m n is an all desirable point and an all automorphisable point respectively

global output feedback stabilisation of a class of stochastic systems with unknown growth rate this paper is concerned with the global output feedback stabilisation problem for a class of stochastic systems with unknown growth rate the output of the system under consideration contains unkno

all derivable points in nest algebras abstract suppose that a is an operator algebra on a hilbert space h an element v in a is called an all derivable point of a for the strong operator topology if every strong operator topology continuous derivable mapping at v is a derivation let n be a complete nest on a complex and separable hilbert space h suppose that m belongs to n with 0 m h and write m for m or m our main result is for any alg n with p m p m if m is invertible in alg n m then is an all derivable point in alg n for the strong operator topology

multiplicative mappings at some points on matrix algebras let mn be the algebra of all n n matrices and let mn mn be a linear mapping we say that is a multiplicative mapping at g if st s t for any s t mn with st g fix g mn we say that g is an all multiplicative point if every multiplicative linear bijection at g with in in is a multiplicative mapping in mn where in is the unit matrix in mn we mainly show in this paper the following two results 1 if g mn with detg 0 then g is an all multiplicative point in mn 2 if is an multiplicative mapping at in then there exists an invertible matrix p mn such that either s psp 1 for any s mn or t pttrp 1 for any t mn

comments on design of optimal dead beat controllers it is shown that the system considered in the numerical example of the above paper belongs to a class of systems with n states and r inputs for which the set of all minimum time dead beat mtdb controllers is parameterized by an arbitrary matrix of order lambda 1 r n x n lambda r where is the greatest integer less than the ratio n r since the example treats the case of n 3 r 2 the mdtb controller is parameterized by a single scalar element

characterization of generalized jordan left derivations on real nest algebras abstract let a be a real nest algebra of b h where h is a real and separable hilbert space we show that the following conditions are equivalent for a weak topology continuous linear map a b h 1 is a left preserving kernel into range mapping i e t ker t ran t for any t a 2 is a generalized left inner derivations i e t t a bt for some a b b h 3 is a generalized jordan left derivations i e t 2 t t t t t i t for any t a 4 is a left 1 preserving kernel into range mapping i e t ker t ran t for any rank one operator t a

dead beat control of linear discrete time systems the paper considers the problem of determining the various classes of state feed back dead beat controllers for linear multivariable discrete time systems unlike earlier work attention is not restricted to the special case of minimum time dead beat controllers a general dead beat control problem is posed and solved the solution is based principally on an approach of closed loop eigenstructure assignment the minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix the concept of the greatest minimum of free parameters is introduced finally the method is illustrated by a numerical example

nonlinear mappings on upper triangular matrices derivable at zero point abstract let t n f be the set of all n n upper triangular matrices over a field f recently by using a technique from graph theory l wang in 8 proved that a bijective mapping on t n f with f a finite field preserves zero product i e x y 0 whenever x y 0 if and only if can be decomposed into the product of an inner automorphism an automorphism of t n f induced by an automorphism of f and a regular automorphism of the zero divisor graph of t n f algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra this viewpoint encourages us to determine nonlinear mappings on t n f which are derivable at zero point i e x y x y 0 whenever x y 0 it is proved that a mapping on t n f is derivable at zero point if and only if is the sum of an inner derivation an additive derivation of t n f induced by an additive derivation of f and a regular mapping on t n f

minimum gain minimum time deadbeat controllers abstract this paper considers the design of minimum gain minimum time deadbeat controllers mgmtdc for linear discrete time systems a new direct method for constructing the mgmtdc of minimum frobenius norm is developed compared with many existing techniques the proposed method gives analytic expressions for the constructed mgmtdc requires less computational effort and does not require that the transition matrix be nonsingular

local derivations on certain csl algebras in this paper it is shown that every norm continuous linear local derivation from an arbitrary csl algebra whose lattice is generated by finitely many independent nests into any ultraweakly closed subalgebra which contains the algebra is an inner derivation and that every norm continuous linear local derivation from an arbitrary csl algebra whose lattice is completely distributive into any ultraweakly closed subalgebra which contains the algebra is a derivation

adaptive stabilization of a class of nonlinear systems with nonparametric uncertainty we consider adaptive stabilization for a class of nonlinear second order systems interpreting the system states as position and velocity the system is assumed to have unknown nonparametric position dependent damping and stiffness coefficients lyapunov methods are used to prove global convergence of the adaptive controller furthermore the controller is shown to be able to reject constant disturbances and to asymptotically track constant commands for illustration the controller is used to stabilize the van der pol limit cycle the duffing oscillator with multiple equilibria and several other examples

local derivations on operator algebras abstract when attempting to find sufficient conditions for a linear mapping to be a derivation an obvious candidate is the concept of a local derivation local derivations on operator algebras have been investigated in recent papers of kadison j algebra130 1990 494 509 and larson and sourour proc symp pure math 51 1990 187 194 a local derivation is a norm continuous linear map from an operator algebra a into an a bimodule m which agrees with some derivation at each point in the algebra we show that if a is the direct limit of finite dimensional csl algebras via extendable embeddings e g a triangular af algebra then a local derivation on a must be a derivation further we show that for many finite dimensional operator algebras any inner local derivation must be an inner derivation

maps determined by action on identity product elements abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 6 invertible for a given element z m n r a map on m n r is called preserving z product if x y z whenever xy z a map on m n r is called derivable at the given point z if x y x y z whenever xy z using elementary matrix technique we show that if an invertible linear map on m n r preserves identity product then it is a jordan automorphism and a linear map on m n r is derivable at the identity matrix if and only if it is an inner derivation

derivable maps and derivational points abstract for an algebra a and an a bimodule m let l a m be the set of all linear maps from a to m a map l a m is called derivable at c a if a b a b c for all a b a with ab c we call an element c a a derivational point of l a m if l a m the condition is derivable at c implies is a derivation we characterize derivable maps by means of peirce decompositions and determine derivational points for some general bimodules as a special case we see that for a nest algebra a on a hilbert space h every 0 c a is a derivational point of l a b h

generalized derivable mappings at zero point on some reflexive operator algebras abstract let a be a subalgebra with the unit operator i in b h we say that a linear mapping from a into b h is a generalized derivable mapping at zero point if st s t s t s i t for any s t a with st 0 in this paper we show the following main result every norm continuous generalized derivable mapping at zero point on finite csl algebras is a generalized derivation

parameterization of the class of deadbeat controllers this technical note deals with the problem of parameterizing the class of deadbeat controllers for a given system through the minimum of parameters it is shown that if n and m are dimensions of the state vector and control vector respectively and the i s are the controllability indexes then almost the whole class of deadbeat controllers is explicitly parameterizable through mn mu 1 3 mu 2 2m 1 mu m parameters

all derivable points in matrix algebras abstract let m n be the algebra of all n n matrices we say that an element g m n is an all derivable point in m n if every derivable linear mapping at g i e st s t s t for any s t m n with st g is a derivation we mainly show in this paper that a matrix g is an all derivable point in m n if and only if g 0

characterizations of derivations on triangular rings additive maps derivable at idempotents abstract let t be a triangular ring an element z t is said to be a full derivable point of t if every additive map from t into itself derivable at z i e a b a b z for every a b t with ab z is in fact a derivation in this paper under some mild conditions on triangular ring t we show that some idempotent elements of t are full derivable points as an application we get that for any nontrivial nest n in a factor von neumann algebra r every nonzero idempotent element q satisfying pq q qp p for some projection p n is a full derivable point of the nest subalgebra alg n of r

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

time optimal output regulators for linear multivariate discrete time systems part 2 all classes of right invertible systems in part 1 of this paper hassan and amin 1987 a simple algebraic solution for the time optimal output regulator problem was presented this solution which consists of a state feedback control law has been obtained for all classes of right invertible decouplable systems s a b c e the results of part 1 are here extended to all classes of right invertible systems s a b c e a set of optimal output deadbeat indices called the optimal set is defined and related to the observability indices of the optimal closed loop system the time optimal output regulator problem for a right invertible non decouplable system s a b c e is resolved by transforming s into a decouplable system sc a b cc ec having the optimal output deadbeat index of s first an algorithm is presented to construct iteratively in a well defined optimal sense a unimodular left compensator l z and a compensated decouplable system sc a b cc ec from the state space parameters of s then a family of optimal state fee

eigenstructure assignment via linear state feedback control in fahmy and o reilly 1982 a b a parametric solution for the linear state feedback eigenstructure assignment problem is developed this approach is here modified in such a way as to accommodate the case where the set of closed loop eigenvalues and the set of open loop eigenvalues have elements in common it is particularly shown that each common eigenvalue introduces a number of partially free design parameter vectors in addition to the original free ones the paper concludes with illustrative numerical examples

jordan higher all derivable points in triangular algebras abstract let t be a triangular algebra we say that d d n n n l t is a jordan higher derivable mapping at g if d n st ts i j n d i s d j t d i t d j s for any s t t with st g an element g t is called a jordan higher all derivable point of t if every jordan higher derivable linear mapping d d n n n at g is a higher derivation in this paper under some mild conditions on t we prove that some elements of t are jordan higher all derivable points this extends some results in 6 to the case of jordan higher derivations

lyapunov stable adaptive stabilization of non linear time varying systems with matched uncertainty this paper considers direct adaptive stabilization of multi input non linear time varying systems with full state feedback and state and time dependent uncertainty the novel result of this paper is a single parameter adaptive controller that yields a lyapunov stable closed loop system we demonstrate the controller on a non linear spring mass damper a three degree of freedom mathieu equation and a 4th order non linear time varying system

time optimal output regulators for linear multivariable discrete time systems part 1 right invertible decouplable systems corsetti and houpis 1985 presented a simple algebraic solution for the timeoptimal output regulator problem this solution has been obtained for a special class of right invertible decouplable systems s a b c satisfying the condition that cb is of full rank in this paper a general method is presented to extend the results of corsetti and houpis to all classes of right invertible decouplable systems s a b c e it is shown that a simple state feedback control law drives each output of s a b c e in a minimum number of steps equal to the order of the associated infinite zero this state feedback represents the optimal solution for the considered time optimal output regulator problem a class of optimal solutions parametrized by a free parameter matrix k is obtained for non square systems the properties of the resulting optimal closed loop system is given and a numerical example is worked out to illustrate the generality and feasibility of the proposed method

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

all derivable points of operator algebras abstract let a be an operator subalgebra in b h where h is a hilbert space we say that an element z a is an all derivable point of a for the norm topology strongly operator topology etc if every norm topology strongly operator topology etc continuous derivable linear mapping at z i e st s t s t for any s t a with st z is a derivation in this paper we show that every invertible operator in the nest algebra alg n is an all derivable point of the nest algebra for the strongly operator topology we also prove that every nonzero element of the algebra of all 2 2 upper triangular matrixes is an all derivable point of the algebra

idempotent elements determined matrix algebras abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 2 invertible v be an r module it is shown in this article that if a symmetric bilinear map from m n r m n r to v satisfies the condition that u u e u whenever u 2 u then there exists a linear map f from m n r to v such that x y f x y x y m n r applying the main result we prove that an invertible linear transformation on m n r preserves idempotent matrices if and only if it is a jordan automorphism and a linear transformation on m n r is a jordan derivation if and only if it is jordan derivable at all idempotent points