performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

connes moscovici characteristic map is a lie algebra morphism let h be a hopf algebra with a modular pair in involution character 1 let a be a module algebra over h equipped with a non degenerated character invariant 1 trace tau we show that connes moscovici characteristic map varphi tau hc character 1 h rightarrow hc lambda a is a morphism of graded lie algebras we also have a morphism phi of batalin vilkovisky algebras from the cotorsion product of h text cotor h bbbk bbbk to the hochschild cohomology of a hh a a let k be both a hopf algebra and a symmetric frobenius algebra suppose that the square of its antipode is an inner automorphism by a group like element then this morphism of batalin vilkovisky algebras phi text cotor k vee mathbb f mathbb f cong text ext k mathbb f mathbb f hookrightarrow hh k k is injective

calculating discriminants by higher direct images the author uses the homological algebra to construct for any line bundle y on a nonsingular projective variety x the complex if s whose determinant is equal to the equation of the dual variety xv this generalizes the cayley koszul complexes defined by gelfand kapranov and zelevinski the formulas for the codimension and degree of xv in terms of complexes f y are given in the second part of the article the general technique is applied to classical discriminants and hyperdeterminants this paper consists of two parts in the first part 1 and 2 we give the method for calculating generalized discriminants considered by gelfand kapranov and zelevinski cf g k z the generalized discriminant is defined for any nonsingular projective variety x as the defining equation of the dual variety xv the method for calculating these equations introduced here uses the conormal construction and the technique of higher direct images for each line bundle y on x we define the canonical complex if s obtained by pushing down the twisted koszul resolutions associated to the canonical desingularisation of the dual variety xv the complex if y allows to calculate explicitly the degree of the discriminant and give the expression for it in terms of buchsbaum eisenbud multipliers associated to if y the general technique is described in 1 where all general formulas are proved section 2 deals with the case of g discriminants they occur when the nonsingular variety x is the g orbit of the highest weight vector in the irreducible representation of the reductive group g in the second part 3 and 4 we investigate the determinantal expressions of the discriminants they arise naturally in the context of the complexes f y if the complex 1f 2 has only two terms it reduces to a matrix in such cases the discriminant becomes the determinant of this matrix section 3 deals with the case of the discriminant of a homogeneous polynomial of degree m classically such formulas were known for the case of two variables sylvester and bezout formulas and three variables which is treated in the book of salmon s in 3 we give a list of such formulas in the case of two variables this list includes the sylvester expression 3 2 m arbitrary a 1 and bezout expression 3 2 m arbitrary a m 2 moreover these formulas appear as the extreme cases of the whole series of determinantal expressions of similar type section 3 3 deals with the case of three variables received by the editors march 27 1992 and in revised form october 5 1992 1991 mathematics subject classification primary 13d25 14n05 secondary 13d02 14m15 15a72 1994 american mathematical society 0002 9947 94 1 00 25 per page

resolutions of determinantal ideals the submaximal minors for many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix to put the problem more precisely suppose r is a commutative ring and x are variables with 1 i m and 1 j q n if we let r r x be the polynomial ring over r then we have the generic matrix x and we may form the ideal zp in r generated by the p x p minors of this matrix the problem then is to find an explicit free resolution of the ideal zp over the ring r it was proved by eagon and hochster lo that r i has a resolution of length m p l n p l but their proof consisted in showing that the ideal zp is perfect it did not provide a construction of the resolution in fact it is not known whether the betti numbers of the ideal zp depend on the characteristic of the ground ring r in 121 lascoux succeeded in giving an explicit resolution provided that the ground ring r contained the field q of rational numbers his construction rests heavily on the theory of schur functors and the fact that in characteristic zero the schur functors are the irreducible representations of the general linear group over the integers however the construction breaks down despite the fact that one can define the schur functors over an arbitrary commutative ring see 1 2 14 151 for various constructions of schur functors in analyzing the work of lascoux and its subsequent reworking by nielsen 131 some basic facts seemed to clamor for attention one was that within a resolution of r z there appeared to be two types of boundary maps one of degree 1 and one of degree p the maps of degree 1 were maps between sums of schur functors of fixed durfee square k see section 2 for definitions while the maps of degree p were from sums of schur functors of

deformation theory of representation of prop erad s i in this paper and its follow up we study the deformation theory of morphisms of properads and props thereby extending quillen s deformation theory for commutative rings to a non linear framework the associated chain complex is endowed with an l infinity algebra structure its maurer cartan elements correspond to deformed structures which allows us to give a geometric interpretation of these results to do so we endow the category of prop erad s with a model category structure we provide a complete study of models for prop erad s a new effective method to make minimal models explicit that extends the koszul duality theory is introduced and the associated notion is called homotopy koszul as a corollary we obtain the co homology theories of al gebras over a prop erad and of homotopy al gebras as well their underlying chain complex is endowed with an l infinity algebra structure in general and a lie algebra structure only in the koszul case in particular we make the deformation complex of morphisms from the properad of associative bialgebras explicit for any minimal model of this properad the boundary map of this chain complex is shown to be the one defined by gerstenhaber and schack as a corollary this paper provides a complete proof of the existence of an l infinity algebra structure on the gerstenhaber schack bicomplex associated to the deformations of associative bialgebras

characteristic free representation theory of the general linear group on etudie systematiquement les z formes des representations rationnelles du groupe lineaire generale

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

a koszul duality for props the notion of prop models the operations with multiple inputs and multiple outputs acting on some algebraic structures like the bialgebras or the lie bialgebras in this paper we generalize the koszul duality theory of associative algebras and operads to props

the odd origin of gerstenhaber brackets batalin vilkovisky operators and master equations using five basic principles we treat gerstenhaber lie brackets batalin vilkovisky bv operators and master equations appearing in mathematical and physical contexts in a unified way the different contexts for this are given by the different types of feynman graphs that underlie the particular situation two of the maxims we bring forth are 1 that extending to the non connected graphs gives a commutative multiplication forming a part of the bv structure and 2 that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs and is responsible for the bv operator being a differential our treatment results in uniform general theorems these allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature the more general point of view also allows us to disentangle the necessary from the circumstantial

associahedra cyclohedra and a topological solution to the a deligne conjecture abstract we give a topological solution to the a deligne conjecture using associahedra and cyclohedra to this end we construct three cw complexes whose cells are indexed by products of polytopes giving new explicit realizations of the polytopes in terms of different types of trees we are able to show that the cw complexes are cell models for the little discs the cellular chains of one complex in particular which is built out of associahedra and cyclohedra naturally act on the hochschild cochains of an a algebra yielding an explicit topological and minimal solution to the a deligne conjecture along the way we obtain new results about the cyclohedra such as new decompositions into products of cubes and simplices which can be used to realize them via a new iterated blow up construction

graph theoretic analysis of network input output processes the control of dynamical processes in networks is considered in the case where measurement and actuation capabilities are sparse and possibly remote specifically we study control of a canonical network dynamics when only one network component s state can be measured and only one in general different component can be actuated to do so we characterize the finite and infinite zeros of the resulting siso system in terms of the graph topology using these results we establish graph theoretic conditions under which there are zeros in the closed right half plane these conditions depend on the length strength and number of the paths from the component where the input is applied to the component where the measurements are made then we present the implications of these conditions on the controller design task focusing in stabilizations destabilization of network processes under static negative feedback

on time scale designs for networks we motivate the problem of designing a subset of the edge weights in a graph to shape the spectrum of an associated linear time invariant dynamics we address a canonical design problem of this form by applying time scale assignment methods and give graph theoretic characterisations of the designed dynamics

graph theoretic characterisations of zeros for the input output dynamics of complex network processes we examine linear network dynamics in which an input can be applied at only one network component and measurements can only be made at one in general different component the infinite zero and finite zero structures of these dynamics are characterised explicitly in terms of the network s graph matrix using the special coordinate basis for linear system these algebraic characterisations are then used to identify relationships between features of the network s graph topology and the state matrix of its finite zero dynamics

on operad structures of moduli spaces and string theory we construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy lie algebras gravity algebras and batalin vilkovisky algebras these algebras appeared recently in the context of string theory and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory

deformation theory of representations of prop erad s ii in this paper and its follow up merkulov and vallette j reine angew math we study the deformation theory of morphisms of properads and props thereby extending quillen s deformation theory for commutative rings to a non linear framework the associated chain complex is endowed with an l algebra structure its maurer cartan elements correspond to deformed structures which allows us to give a geometric interpretation of these results to do so we endow the category of prop erad s with a model category structure we provide a complete study of models for prop erad s a new effective method to make minimal models explicit that extends the koszul duality theory is introduced and the associated notion is called homotopy koszul as a corollary we obtain the co homology theories of al gebras over a prop erad and of homotopy al gebras as well their underlying chain complex is endowed with an l algebra structure in general and a lie algebra structure only in the koszul case in particular we make the deformation complex of morphisms from the properad of associative bialgebras explicit for any minimal model of this properad the boundary map of this chain complex is shown to be the one defined by gerstenhaber and schack as a corollary this paper provides a complete proof of the existence of an l algebra structure on the gerstenhaber schack bicomplex associated to the deformations of associative bialgebras

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

dominant eigenvalue minimization with trace preserving diagonal perturbation subset design problem abstract motivated by network resource allocation needs we study the problem of minimizing the dominant eigenvalue of an essentially nonnegative matrix with respect to a trace preserving or fixed trace diagonal perturbation in the case where only a subset of the diagonal entries can be perturbed graph theoretic characterizations of the optimal subset design are obtained in particular the design is connected to the structure of a reduced effective graph defined from the essentially nonnegative matrix also the change in the optimum is studied when additional diagonal entries are constrained to be undesignable from both an algebraic and graph theoretic perspective these results are developed in part using properties of the perron complement of nonnegative matrices and the concept of line sum symmetry some results apply to general essentially nonnegative matrices while others are specialized for sub classes e g diagonally symmetrizable or having a single node cut

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

modules and morita theorem for operads associative rings a b are called morita equivalent when the categories of left modules over them are equivalent we call two classical linear operads p q morita equivalent if the categories of algebras over them are equivalent we transport a part of morita theory to the operadic context by studying modules over operads as an application of this philosophy we consider an operadic version of the sheaf of linear differential operators on a super manifold m and give a comparison theorem between algebras over this sheaf on m and m red

topological conformal field theories and calabi yau categories abstract this is the first of two papers which construct a purely algebraic counterpart to the theory of gromov witten invariants at all genera these gromov witten type invariants depend on a calabi yau a category which plays the role of the target in ordinary gromov witten theory when we use an appropriate a version of the derived category of coherent sheaves on a calabi yau variety this constructs the b model at all genera when the fukaya category of a compact symplectic manifold x is used it is shown under certain assumptions that the usual gromov witten invariants are recovered the assumptions are that open closed gromov witten theory can be constructed for x and that the natural map from the hochschild homology of the fukaya category of x to the ordinary homology of x is an isomorphism

connection graph stability method for synchronized coupled chaotic systems this paper elucidates the relation between network dynamics and graph theory a new general method to determine global stability of total synchronization in networks with different topologies is proposed this method combines the lyapunov function approach with graph theoretical reasoning in this context the main step is to establish a bound on the total length of all paths passing through an edge on the network connection graph in particular the method is applied to the study of synchronization in rings of 2k nearest neighbor coupled oscillators a rigorous bound is given for the minimum coupling strength sufficient for global synchronization of all oscillators this bound is explicitly linked with the average path length of the coupling graph contrary to the master stability function approach developed by pecora and carroll the connection graph stability method leads to global stability of synchronization and it permits not only constant but also time dependent interaction coefficients in a companion paper blinking model and synchronization in small world networks with a time varying coupling see this issue this method is extended to the blinking model of small world networks where in addition to the fixed 2 k nearest neighbor interactions all the remaining links are rapidly switched on and off independently of each other

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

grassmannian and string theory the infinite dimensional grassmannian manifold contains moduli spaces of riemann surfaces of all genera this well known fact leads to a conjecture that non perturbative string theory can be formulated in terms of the grassmannian we present new facts supporting this hypothesis in particular it is shown that grassmannians can be considered as generalized moduli spaces this statement permits us to define corresponding string amplitudes at least formally one can conjecture that it is possible to explain the relation between non perturbative and perturbative string theory by means of localization theorems for equivariant cohomology this conjecture is based on the characterization of moduli spaces relevant to string theory as sets consisting of points with large stabilizers in certain groups acting on the grassmannian we describe an involution on the grassmannian that could be related to s duality in string theory