vector norms as lyapunov functions for linear systems a unified theory of quadratic and piecewise linear lyapunov functions for continuous and discrete time linear systems is presented the key to this work is the description of these lyapunov functions by vector norms the main results are sufficient and necessary conditions for a vector norm to be a lyapunov function as well as a method based on these conditions of constructing such lyapunov functions

constrained optimal control of hybrid systems with a linear performance index we consider the constrained finite and infinite time optimal control problem for the class of discrete time linear hybrid systems when a linear performance index is used the finite and infinite time optimal solution is a piecewise affine state feedback control law in this paper we present algorithms that compute the optimal solution to both problems in a computationally efficient manner and with guaranteed convergence and error bounds both algorithms combine a dynamic programming exploration strategy with multiparametric linear programming and basic polyhedral manipulation

further results on infinity norms as lyapunov functions for linear systems this note continues on the results proposed in two previous papers for computing vector 1 or infin norm based lyapunov functions for linear discrete time systems by presenting a finitely terminating algorithm for the construction of lyapunov functions of that class the algorithm utilizes the solution of a finite sequence of feasibility linear programs lps with very few constraints or equivalently very simple algebraic tests in contrast to solving the original problem where a bilinear matrix equation with rank and norm constraint needs to be solved

extensions of pade discretization for linear systems with polyhedral lyapunov functions for generalized jordan structures recently we showed that certain types of polyhedral lyapunov functions for linear time invariant systems are preserved by diagonal pade approximations under the assumption that the continuous time system matrix ac has distinct eigenvalues in this technical note we show that this result also holds true in the case that ac has non trivial jordan blocks

exact lagrangian submanifolds in simply connected cotangent bundles we consider exact lagrangian submanifolds in cotangent bundles under certain additional restrictions triviality of the fundamental group of the cotangent bundle and of the maslov class and second stiefel whitney class of the lagrangian submanifold we prove such submanifolds are floer cohomologically indistinguishable from the zero section this implies strong restrictions on their topology an essentially equivalent result was recently proved independently by nadler 16 using a different approach

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

on the bottleneck tree alignment problems given a set w of k sequences and a tree t with k leaves labeled with a unique sequence in w a label tree is used to assign a sequence label to each internal node of the tree t the cost of a tree edge is defined as the distance such as the hamming distance or the levenshtein edit distance between the sequence labels of a pair of nodes in the edge the bottleneck edge of a label tree is the edge with the maximum cost in the label tree the bottleneck tree alignment problem is concerned with the determination of a label tree with minimum cost of the bottleneck edge a lifted label tree is a label tree in which the sequence label of each internal node is equal to the sequence label of some child of the node the bottleneck lifted tree alignment problem involves the minimization of cost of the bottleneck edge among all possible lifted label trees of the tree t this paper shows that when the distance function is a metric the bottleneck tree alignment problem is np complete even when the tree structure resembles a binary tree for special cases an exact algorithm is used to solve the bottleneck lifted tree alignment problem in polynomial time a 2 h 1 approximation algorithm is used to solve the bottleneck tree alignment problem where h is the height of the tree t it is observed that the bound is existentially tight finally this paper shows that any lifted label tree is an optimal solution to the bottleneck tree alignment problem if the distance function is an ultrametric

model predictive control based on linear programming the explicit solution we study model predictive control mpc schemes for discrete time linear time invariant systems with constraints on inputs and states that can be formulated using a linear program lp in particular we focus our attention on performance criteria based on a mixed 1 norm namely 1 norm with respect to time and norm with respect to space first we provide a method to compute the terminal weight so that closed loop stability is achieved we then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it the piecewise affine form allows to eliminate online lp as the computation associated with mpc becomes a simple function evaluation besides practical advantages the availability of the explicit structure of the mpc controller provides an insight into the type of control action in different regions of the state space and highlights possible conditions of degeneracies of the lp such as multiple optima

a cotangent fibre generates the fukaya category abstract we prove that the algebra of chains on the based loop space recovers the derived wrapped fukaya category of the cotangent bundle of a closed smooth oriented manifold the main new idea is the proof that a cotangent fibre generates the fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by cieliebak and latschev

geometric composition in quilted floer theory abstract we prove that floer cohomology of cyclic lagrangian correspondences is invariant under transverse and embedded composition under a general set of assumptions

new approach to the stability of chemical reaction networks piecewise linear in rates lyapunov functions piecewise linear in rates pwlr lyapunov functions are introduced for a class of chemical reaction networks crns in addition to their simple structure these functions are robust with respect to arbitrary monotone reaction rates of which mass action is a special case the existence of such functions ensures the convergence of trajectories towards equilibria and can be used to establish their asymptotic stability with respect to the corresponding stoichiometric compatibility class we give the definition of these lyapunov functions prove their basic properties and provide algorithms for constructing them examples are provided relationship with consensus dynamics are discussed and future directions are elaborated

floer cohomology and geometric composition of lagrangian correspondences abstract we prove an isomorphism of floer cohomologies under geometric composition of lagrangian correspondences in exact and monotone settings

fusion rings and geometry the algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper a formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings in general and for current algebra theories in particular it is shown that fusion rings lead to a natural orthogonal polynomial structure the rings are expressed through generators and relations the relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties in general the fusion algebras are mapped to affine varieties which are the locus of the relations the connection with modular transformations is investigated in this picture it is explained how chiral algebras arising in n 2 superconformal field theory can be derived from fusion rings in particular it is argued that theories of the type su n k su n 1 are the n 2 counterparts of grassmann manifolds and that there is a natural identification of the chiral fields with schubert varieties which is a graded algebra isomorphism

on infinity norms as lyapunov functions for linear systems the note presents a construction of the lyapunov function defined by l sub spl infin sup n vector norm for both continuous and discrete time linear systems in contrast to the previous results no restrictions on the positions or multiplicity of system matrix eigenvalues are required

pade discretization for linear systems with polyhedral lyapunov functions this technical note has been motivated by the need to assess the preservation of polyhedral lyapunov functions for stable continuous time linear systems under numerical discretization of the transition matrix this problem arises when discretizing linear systems in such a manner as to preserve a certain type of stability of the discrete time approximation our main contribution is to show that a continuous time system and its pade discretization of any order and sampling always share at least one common piecewise linear polyhedral lyapunov function

lyapunov function construction by linear programming in this paper the lyapunov function given by the infinity norm is considered a new proof of the theorem of molchanov and pyatintskii 1986 is given the presented proof enables us to give an explicit calculation of the matrix which was only demonstrated to exist in the original paper combining the obtained result with the idea of scaling faces of a polytope gives the method of fitting the shape of level surfaces of the lyapunov function given by the infinity norm to the specified demands

positively invariant polyhedral sets of discrete time linear systems the problem of the existence of positively invariant polyhedral sets for linear discrete time dynamical systems is studied in the first part of the paper necessary and sufficient conditions for a

homological mirror symmetry for calabi yau hypersurfaces in projective space we prove homological mirror symmetry for a smooth d dimensional calabi yau hypersurface in projective space for any d 3 for example d 3 is the quintic threefold the main techniques involved in the proof are the construction of an immersed lagrangian sphere in the d dimensional pair of pants the introduction of the relative fukaya category and an understanding of its grading structure a description of the behaviour of this category with respect to branched covers via an orbifold fukaya cate gory a morse bott model for the relative fukaya category that allows one to make explicit computations and the introduction of certain graded categories of matrix factorizations mirror to the relative fukaya category

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

nearby lagrangians with vanishing maslov class are homotopy equivalent we prove that the inclusion of every closed exact lagrangian with vanishing maslov class in a cotangent bundle is a homotopy equivalence we start by adapting an idea of fukaya seidel smith to prove that such a lagrangian is equivalent to the zero section in the fukaya category with integral coefficients we then study an extension of the fukaya category in which lagrangians equipped with local systems of arbitrary dimension are admitted as objects and prove that this extension is generated in the appropriate sense by local systems over a cotangent fibre whenever the cotangent bundle is simply connected this generation statement is used to prove that every closed exact lagrangian of vanishing maslov index is simply connected finally we borrow ideas from coarse geometry to develop a fukaya category associated to the universal cover allowing us to prove the result in the general case

a novel unified approach to invariance conditions for a linear dynamical system in this paper we propose a novel simple and unified approach to explore sufficient and necessary conditions i e invariance conditions under which four classic families of convex sets namely polyhedra polyhedral cones ellipsoids and lorenz cones are invariant sets for a linear discrete or continuous dynamical system for discrete dynamical systems we use the theorems of alternatives i e farkas lemma and s lemma to obtain simple and general proofs to derive invariance conditions this novel method establishes a solid connection between optimization theory and dynamical system also using the s lemma allows us to extend invariance conditions to any set represented by a quadratic inequality such sets include nonconvex and unbounded sets for continuous dynamical systems we use the forward or backward euler method to obtain the corresponding discrete dynamical systems while preserves invariance this enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems finally some numerical examples are presented to illustrate these invariance conditions

complete modules and torsion modules suppose that r is a ring and that a is a chain complex over r inside the derived category of differential graded a modules there are naturally defined subcategories of a torsion objects and of a complete objects under a finiteness condition on a we develop a morita theory for these subcat egories find conceptual interpretations for some associated algebraic functors and in appropriate commutative situations identify the associated functors as local homology or local cohomology some of the results are suprising even in the case r z and a z p 1 introduction let r be a ring and r mod the derived category of chain complexes of left modules see section 1 2 we choose a fixed complex a which is perfect in other words isomorphic in mod to a complex of finite length in which the entries are finitely generated projective modules we de clare another complex n to be a trivial if hom a a0 0 where hom denotes the chain complex of morphisms in r mod going further we say that x is a torsion if hom x at 0 for all a trivial n9 and that x is a complete if homr n9x 0 for all a trivial n we then study the category ators of a torsion complexes and the category acomp of a complete complexes both of these are triangulated full subcategories of r mod it turns out that these categories are equivalent to one another and also equivalent to the derived category of dif ferential graded modules over the endomorphism complex of a we construct approximation functors celu r mod at0rs and a r mod acomp for an object m of r mod9 the complex celu m is a kind of a cellular approxi mation to m in the sense that it is the best approximation to m which can be cobbled together from a and its suspensions the complex ma is the bousfield localization of m with respect to a homology theory on r mod derived from a we provide algebraic formulas for the functors and find that the functors are related in interesting ways one of which involves an arithmetic square we also show that if cciia r has certain finite dimensionality properties then an object m of r mod is a torsion or a complete if and only if the homology groups 77 m individually satisfy appropriate torsion or completeness conditions suppose now that r is a commutative ring that 7 c r is a finitely generated ideal that a r i9 and that k is the associated koszul complex the complex a

construction of invariant polytopic sets with specified complexity the problem of construction of controlled invariant polytopic sets with specified complexity for linear systems subject to linear state and control constraints is investigated first geometric conditions for the enlargement of a polytopic set by adding a new vertex in order to produce a polytopic set of specified complexity are established next conditions for such an enlargement of controlled invariant sets to preserve the controlled invariance property are presented the established theoretical results are used to develop methods for the construction of admissible controlled invariant sets with specified complexity two numerical examples show how these results can be used for the computation of monotonic sequences of admissible controlled invariant sets of specified complexity

stability of artificial neural networks with impulses sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a hopfield type neural network with lipschitzian activation functions without assuming their boundedness monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time both the continuous time and their corresponding discrete time networks are considered the sufficient conditions of the discrete time network do not restrict the step size appearing in the discretization process and these conditions approach as the step size tends to zero those of the conditions of the continuous time networks the sufficient conditions are in terms of the parameters of the network only and are easy to verify also when the impulsive jumps are absent the results reduce to those of the non impulsive systems

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

a new algorithm for computing similarity between rna structures abstract the primary structure of a ribonucleic acid rna molecule is a sequence of nucleotides bases over the four letter alphabet a c g u the secondary or tertiary structure of an rna is a set of base pairs nucleotide pairs which form bonds between a u and c g for secondary structures these bonds have been traditionally assumed to be one to one and non crossing we consider the edit distance between two rna structures this is a notion of similarity introduced in proceedings of the tenth symposium on combinatorial pattern matching lecture notes in computer science vol 1645 springer berlin 1999 p 281 between two rna molecule structures taking into account the primary the secondary and the tertiary structures in general this problem is np hard for tertiary structures in this paper we consider this notion under some constraints we present an algorithm and then show how to use this algorithm for practical applications

generating the fukaya categories of hamiltonian g manifolds abstract let g be a compact lie group and let k be a field of characteristic p geq 0 such that h g has no p torsion if p 0 we show that a free lagrangian orbit of a hamiltonian g action on a compact monotone symplectic manifold x split generates an idempotent summand of the monotone fukaya category mathcal f x k if and only if it represents a nonzero object of that summand slightly more general results are also provided our result is based on an explicit understanding of the wrapped fukaya category mathcal w t g k through koszul twisted complexes involving the zero section and a cotangent fibre and on a functor d b mathcal w t g k to d b mathcal f x times x k canonically associated to the hamiltonian g action on x we explore several examples which can be studied in a uniform manner including toric fano varieties and certain grassmannians

simple lp type criteria for positively invariant polyhedral sets let s be a convex polyhedral set in the state space of a linear dynamical system a new proof of an algebraic condition for the set s to be positively invariant is given by the dual set of s the condition obtained in this correspondence can be examined by solving a set of standard lp type formulas it is also shown that the conventional algebraic conditions have redundancies in the lp type formulas discrete time systems as well as continuous time systems are considered

set valued observer design for a class of uncertain linear systems with persistent disturbance and measurement noise in this paper a class of linear systems affected by parameter variations additive noise and persistent disturbances is considered the problem of designing a set valued state observer which estimates a region containing the real state for each time instant is investigated the techniques for designing the observer are based on positive invariant set theory by constructing a set induced lyapunov function it is shown that the estimation error converges exponentially to a given compact set with an assigned rate of convergence

lagrangian fibers of gelfand cetlin systems abstract a gelfand cetlin system is a completely integrable system defined on a partial flag manifold whose image is a rational convex polytope called a gelfand cetlin polytope motivated by the study of nishinou nohara ueda 24 on the floer theory of gelfand cetlin systems we provide a detailed description of topology of gelfand cetlin fibers in particular we prove that any fiber over an interior point of an r dimensional face of the gelfand cetlin polytope is an isotropic submanifold and is diffeomorphic to t r n for some smooth manifold n and t r s 1 r we also prove that such n s are exactly the vanishing cycles shrinking to points in the associated toric variety via the toric degeneration we also devise an algorithm of reading off lagrangian fibers from the combinatorics of the ladder diagram

simple fast algorithms for the editing distance between trees and related problems ordered labeled trees are trees in which the left to right order among siblings is significant the distance between two ordered trees is considered to be the weighted number of edit operations insert delete and modify to transform one tree to another the problem of approximate tree matching is also considered specifically algorithms are designed to answer the following kinds of questions 1 what is the distance between two trees 2 what is the minimum distance between t 1 and t 2 when zero or more subtrees can be removed from t 2 3 let the pruning of a tree at node n mean removing all the descendants of node n the analogous question for prunings as for subtrees is answered a dynamic programming algorithm is presented to solve the three questions in sequential time o t 1 times t 2 times min textit depth t 1 textit leaves t 1 times min textit depth t 2 textit leaves t 2 and space o t 1 times t 2 compared with o t 1 times t 2 t

brief on absolute stability analysis by polyhedral lyapunov functions this paper addresses the problem of computing a polyhedral lyapunov function plf for a linear differential inclusion ldi a numerically efficient algorithm is presented to compute a plf for an ldi it is also pointed out that for the case of a polyhedron defined by vertices rather than by faces one can state algebraic stability conditions analogous to those given by molchanov and pyatnitskiy 1986

design of stabilizing switching control laws for discrete and continuous time linear systems using piecewise linear lyapunov functions in this paper the stability of switched linear systems is investigated using piecewise linear lyapunov functions in particular we identify classes of switching sequences that result in stable trajectories given a switched linear system we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system in the proposed approach we assume that each individual subsystem is stable and admits a piecewise linear lyapunov function based on these lyapunov functions we compose global lyapunov functions that guarantee stability of the switched linear system a large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space the approach is applied to both discrete time and continuous time switched linear systems

a compact hybrid feature vector for an accurate secondary structure prediction amino acid propensity score is one of the earliest successful methods used in protein secondary structure prediction however the score performs poorly on small sized datasets and low identity protein sequences based on current in silico method secondary structure can be predicted from local folds or local protein structure in biology the evolution of secondary structure produces local protein structure with different lengths to precisely predict secondary structures we propose a derivative feature vector dps that utilizes the optimal length of the local protein structure dps is the unification of amino acid propensity score and dihedral angle score this new feature vector is further normalized to level the edges prediction is performed by support vector machines svm over the dps feature vectors with class labels generated by secondary structure assignment method ssam and secondary structure prediction method sspm all experiments are carried out on rs126 sequences the results from this proposed method also highlight the overall accuracy of our method compared to other state of the art methods the performance of our method was acceptable specifically in dealing with low number and low identity sequences

a novel document similarity measure based on earth mover s distance in this paper we propose a novel measure based on the earth mover s distance emd to evaluate document similarity by allowing many to many matching between subtopics first each document is decomposed into a set of subtopics and then the emd is employed to evaluate the similarity between two sets of subtopics for two documents by solving the transportation problem the proposed measure is an improvement of the previous om based measure which allows only one to one matching between subtopics experiments have been performed on the tdt3 dataset to evaluate existing similarity measures and the results show that the emd based measure outperforms the optimal matching om based measure and all other measures in addition to the texttiling algorithm the sentence clustering algorithm is adopted for document decomposition and the experimental results show that the proposed emd based measure does not rely on the document decomposition algorithm and thus it is more robust than the om based measure

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method