on constructing constrained control lyapunov functions for linear systems this works considers the problem of control of linear systems with input constraints the notion of control lyapunov functions clfs is first generalized and relaxed to define a constrained control lyapunov function cclf and a constructive procedure for cclf is subsequently presented

on the relation between the positive inertia index and negative inertia index of weighted graphs abstract let g w be an edge weighted graph and let a g w be its adjacency matrix the positive inertia index respectively the negative inertia index of g w denoted by p g w resp p g w is defined to be the number of positive eigenvalues resp negative eigenvalues of a g w clearly p g w p g w is the rank of g w whereas p g w p g w is defined as the signature of g w in this paper some upper and lower bounds on p g w p g w are determined for the weighted chordal graphs resp bipartite graphs complete graphs cactus graphs join graphs bisplit graphs power graphs

an upper bound of the nullity of a graph in terms of order and maximum degree abstract let g be a finite undirected graph without loops and multiple edges by and n we respectively denote the nullity the maximum vertex degree and the order of g in 10 it was proved that n 2 n 1 when g is a tree this result was generalized to a bipartite graph by 25 for a reduced bipartite graph g the above inequality was improved to n 2 2 ln 2 by 27 however the problem of bounding the nullity of an arbitrary graph g in terms of n and is left open for more than ten years in this article we aim to solve such a left problem we prove that 1 n for an arbitrary graph g with order n and maximum degree and the equality holds if and only if g is the disjoint union of some copies of k

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

brief an analysis and design method for linear systems subject to actuator saturation and disturbance we present a method for estimating the domain of attraction of the origin for a system under a saturated linear feedback a simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant this condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis moreover the condition can be expressed as linear matrix inequalities lmis in terms of all the varying parameters and hence can easily be used for controller synthesis this condition is then extended to determine the invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements the effectiveness of the developed methods is illustrated with examples

a numerically robust state space approach to stable predictive control strategies recent work with predictive control strategies has shown that the stable gpc sgpc approach has significant computational and numerical advantages however sgpc is cast in the transfer function framework which limits its application here we develop a means of extending the results to state space models and show that improved numerical conditioning can be obtained for many stable predictive control strategies

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

on the nullity of a graph with cut points let g be a simple graph of order n and a g be its adjacency matrix the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in the spectrum of a g denote by ck and lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in ck respectively in 1998 sciriha i sciriha on singular line graphs of trees congr numer 135 1998 73 91 show that the order of every tree whose line graph is singular is even then gutman and sciriha i gutman i sciriha on the nullity of line graphs of trees discrete math 232 2001 35 45 show that the nullity set of l0 is 0 1 in this paper we investigate the nullity of graphs with cut points and deduce some concise formulas then we generalize scirihas result showing that the order of every graph g is even if such a graph g satisfies that g ck and l g k 1 and the nullity set of lk is 0 1 k k 1 for any given k where l g denotes the line graph of the graph g

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

lyapunov based model predictive control of stochastic nonlinear systems in this work we design a lyapunov based model predictive controller lmpc for nonlinear systems subject to stochastic uncertainty the lmpc design provides an explicitly characterized region from where stability can be probabilistically obtained the key idea is to use stochastic lyapunov based feedback controllers with well characterized stabilization in probability to design constraints in the lmpc that allow the inheritance of the stability properties by the lmpc the application of the proposed lmpc method is illustrated using a nonlinear chemical process system example

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

free braided differential calculus braided binomial theorem and the braided exponential map braided differential operators i are obtained by differentiating the addition law on the braided covector spaces introduced previously such as the braided addition law on the quantum plane these are affiliated to a yang baxter matrix r the quantum eigenfunctions expr x v of the i braided plane waves are introduced in the free case where the position components xi are totally noncommuting a braided r binomial theorem and a braided taylor theorem expr a f x f a x are proven these various results precisely generalize to a generic r matrix and hence to n dimensions the well known properties of the usual one dimensional q differential and q exponential as a related application it is shown that the q heisenberg algebra px qxp 1 is a braided semidirect product c x c p of the braided line acting on itself a braided weyl algebra and similarly for its generalization to an arbitrary r matrix

characterizing the mixed graphs with exactly one positive eigenvalue and its application to mixed graphs determined by their h spectra abstract a mixed graph is obtained by orienting some edges of a simple graph the positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its hermitian adjacency matrix including multiplicities in this paper we study the positive inertia indices of mixed graphs our main results provide a characterization of mixed graphs with positive inertia index 1 as an application some classes of mixed graphs determined by their h spectra are fixed

characterization of graphs whose signature equals the number of odd cycles abstract let g be a simple graph with vertex set v g and edge set e g the signature s g of g is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix a g in 20 it was proved that c 1 g s g c 1 g where c 1 g denotes the number of odd cycles in g a problem arises naturally what graphs have signature attaining the upper bound c 1 g resp the lower bound c 1 g in this paper we focus our attention on this problem characterizing graphs g whose signature equals c 1 g resp c 1 g

nakayama automorphism and rigidity of dual reflection group coactions abstract we study homological properties and rigidity of group coactions on artin schelter regular algebras

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

signature of power graphs abstract the signature s g of a graph g is defined as the difference between its positive inertia index and the negative inertia index in 2013 h ma et al 2013 8 conjectured that c 3 g s g c 5 g for an arbitrary simple graph g where c i g denotes the number of cycles in g with length i modulo 4 in 2014 l wang et al 10 proved that c 3 t k s t k c 5 t k for any tree t and for any k 2 in this paper we prove that c 3 g k s g k c 5 g k for any simple graph g and for any k 2 thus extend the main result of 10 to more general cases

relations between the inertia indices of a mixed graph and those of its underlying graph abstract given a graph g the mixed graph d g is obtained from g by orienting some of its edges where g is called the underlying graph of d g let p d g n d g resp p g n g be the positive inertia index and negative inertia index of d g resp g in this paper we first establish the inequalities d g p d g p g d g and d g n d g n g d g where d g is the dimension of cycle space of g furthermore all the corresponding extremal graphs are characterized

positive and negative inertia index of a graph abstract in this paper the positive and negative inertia index of trees unicyclic graphs and bicyclic graphs are discussed the methods of calculating them are obtained and an inequality about the difference between positive and negative inertia index is proved moreover a conjecture is proposed

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

skew divided difference operators in the nichols algebra associated to a finite coxeter group abstract let w s be a finite coxeter system with root system r and with set of positive roots r for r v w w we denote by w and w v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of w generalizing the work of liu 15 we prove that w v can be written as a polynomial with nonnegative coefficients in where r in fact we prove the stronger and analogous statement in the nichols woronowicz algebra model for schubert calculus on w after bazlov 4 we draw consequences of this theorem on saturated chains in the bruhat order and partially treat the question when w v can be written as a monomial in where r in an appendix we study related combinatorics on shuffle elements and bruhat intervals of length two

the signature of line graphs and power trees abstract let g be a graph and let a g be the adjacency matrix of g the signature s g of g is the difference between the positive inertia index and the negative inertia index of a g ma et al 2013 10 conjectured that c 3 g s g c 5 g where c 3 g and c 5 g respectively denote the number of cycles in g which have length 4 k 3 and 4 k 5 for some integers k 0 and proved the conjecture holds for trees unicyclic or bicyclic graphs it is known that s g 0 if g is bipartite and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existing results in this paper we show that the conjecture holds for line graphs and power trees

the k theory of the flag variety and the fomin kirillov quadratic algebra abstract we propose a new approach to the multiplication of schubert classes in the k theory of the flag variety this extends the work of fomin and kirillov in the cohomology case and is based on the quadratic algebra defined by them more precisely we define k theoretic versions of the dunkl elements considered by fomin and kirillov show that they commute and use them to describe the structure constants of the k theory of the flag variety with respect to its basis of schubert classes

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

constrained stabilization of continuous time linear systems in this paper we consider the stabilization problem for linear continuous time systems under state and control constraints we show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain continuous feedback stabilizing control and we show how to use existing numeric procedures for discrete time systems to solve the continuous time problem we propose a new discontinuous stabilizing control law for scalar input systems which has the advantage of being successfully applicable to systems with quantized control

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

global stabilization of linear systems via bounded controls abstract this paper is concerned with global stabilization of linear systems by bounded controls two types of nonlinear control laws with one made of cascade connections of saturation functions and the other one consisting of nested connections of saturation functions are firstly proposed for multiple integrators system the possible ranges for the controllers parameters are extended with respect to existing results simultaneously the so called state dependent saturation functions are introduced into these two types of control laws to replace the standard saturation functions which can improve the transient performance of the closed loop system significantly the results are then extended to a class of linear systems that can be globally stabilized by bounded controls an illustrative example is presented to show the effectiveness and advantages of the proposed approaches

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

signed graphs with cut points whose positive inertia indexes are two abstract a signed graph g consists of an underlying graph g and a sign function which assigns each edge uv of g a sign u v either positive or negative the adjacency matrix of g is defined as a g a u v with a u v u v a u v where a u v 1 if u v v g are adjacent and a u v 0 otherwise the positive inertia index of g written as p g is defined to be the number of positive eigenvalues of a g recently yu et al 2016 12 characterized the signed graphs g with pendant vertices such that p g 2 in this paper we extend the above work to a more general case characterizing the signed graphs g with cut points whose positive inertia index is 2

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

braided doubles and rational cherednik algebras we introduce and study a large class of algebras with triangular decomposition which we call braided doubles braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational cherednik algebras we classify braided doubles in terms of quasi yetter drinfeld qyd modules over hopf algebras which turn out to be a generalisation of the ordinary yetter drinfeld modules to each braiding a solution to the braid equation we associate a qyd module and the corresponding braided heisenberg double this is a quantum deformation of the weyl algebra where the role of polynomial algebras is played by nichols woronowicz algebras our main result is that any rational cherednik algebra canonically embeds in the braided heisenberg double attached to the corresponding complex reflection group

on semiglobal stabilizability of antistable systems by saturated linear feedback it was recently established that a second order antistable linear system can be semiglobally stabilized on its null controllable region by saturated linear feedback and a higher order linear system with two or more antistable poles can be semiglobally stabilized on its null controllable region by more general bounded feedback laws we will show in this note that a system with three real valued antistable poles cannot be semiglobally stabilized on its null controllable region by the simple saturated linear feedback

on the stabilizing property of siorhc it is shown that the stabilizing property of siorhc stabilizing i o receding horizon control holds for general stabilizable discrete time linear plants irrespective of the condition that the plant has no poles at the origin

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

stability margins of nonlinear receding horizon control via inverse optimality using the nonlinear analog of the fake riccati equation developed for linear systems we derive an inverse optimality result for several receding horizon control schemes this inverse optimality result unifies stability proofs and shows that receding horizon control possesses the stability margins of optimal control laws

duality for generalised differentials on quantum groups abstract we study generalised differential structures 1 d on an algebra a where a a 1 given by a b a d b need not be surjective the finite set case corresponds to quivers with embedded digraphs the hopf algebra left covariant case to pairs 1 where 1 is a right module and a right module map and the hopf algebra bicovariant case corresponds to morphisms a 1 in the category of right crossed or drinfeld radford yetter modules over a when a u g the generalised left covariant differential structures are classified by cocycles z 1 g 1 we then introduce and study the dual notion of a codifferential structure 1 i on a coalgebra and for hopf algebras the self dual notion of a strongly bicovariant differential graded algebra d augmented by a codifferential i of degree 1 here is a graded super hopf algebra extending the hopf algebra 0 a and where applicable the dual super hopf algebra gives the same structure on the dual hopf algebra accordingly group 1 cocycles correspond precisely to codifferential structures on algebraic groups and function algebras among general constructions we show that first order data 1 on a hopf algebra a extends canonically to a strongly bicovariant differential graded algebra via the braided super shuffle algebra the theory is also applied to quantum groups with 1 c q g dually paired to 1 u q g

heuristically enhanced feedback control of constrained discrete time linear systems recent advances in computer technology have spurred new interest in the use of feedback controllers based upon on line minimization for the control of constrained linear systems still the use of computers in the feedback loop has been hampered by the fact that the amount of time available for computation in most sampled data systems is not enough to achieve a complete solution using conventional algorithms several ad hoc techniques have been proposed but their applicability is restricted by the lack of supporting theory in this paper we present a theoretical framework to analyze the stability of the closed loop system resulting from the use of on line optimization in the feedback loop using these results we show that a suboptimal algorithm based upon the use of heuristic search techniques yields asymptotically stable systems provided that enough computation power is available to solve at each sampling interval an optimization problem considerably simpler than the original the controller presented in this paper is valuable for situations where the customary approaches of using pontryagin s minimum principle or storing a family of extremal curves are not applicable due to limitations in the computational resources available

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

robust model predictive control of stable linear systems a new robust model predictive control mpc algorithm is presented for stable constrained linear plants that is a direct generalization of the nominally stabilizing regulator presented by rawlings and muske model uncertainty is parametrized by a list of possible plants robust stability is achieved through the addition of constraints that prevent the sequence of optimal controller costs from increasing for the true plant asymptotic stability is demonstrated through a lyapunov argument simulation experiments demonstrate the performance of the algorithm for two example processes

an explicit description of null controllable regions of linear systems with saturating actuators we give simple exact descriptions of the null controllable regions for general linear systems with saturating actuators the description is in terms of a set of extremal trajectories of the anti stable subsystem for lower order systems or systems with only real eigenvalues this description is further simplified to result in explicit formulae for the boundaries of the null controllable regions

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

stabilization of exponentially unstable linear systems with saturating actuators we study the problem of stabilizing exponentially unstable linear systems with saturating actuators the study begins with planar systems with both poles exponentially unstable for such a system we show that the boundary of the domain of attraction under a saturated stabilizing linear state feedback is the unique stable limit cycle of its time reversed system a saturated linear state feedback is designed that results in a closed loop system having a domain of attraction that is arbitrarily close to the null controllable region this design is then utilized to construct state feedback laws for higher order systems with two exponentially unstable poles

lie theory and coverings of finite groups we introduce the notion of an inverse property ip quandlec which we propose as the right notion of lie algebra in the category of sets to any ip quandle we construct an associated group gc for a class of ip quandles which we call locally skew and when gc is nite we show that the noncommutative de rham cohomology h 1 gc is trivial aside from a single generator that has no classical analogue if we start with a group g then any subsetc gnfeg which is ad stable and inversion stable naturally has the structure of an ip quandle ifc also generates g then we show that gc g with central kernel in analogy with the similar result for the simply connected covering group of a lie group we prove that this covering map gc g is an isomorphism for all nite crystallographic reection groups w withc the set of reections and that c is locally skew precisely in the simply laced case this implies that h 1 w k when w is simply laced proving in particular a conjecture for sn in 11 we obtain similar results for the dihedral groups d6m we also consider c zp 1 zp 1 as a locally skew ip quandle lie algebra of sl2 z and show that gc b3 the braid group on 3 strands the map b3 sl2 z which therefore arises naturally as a covering map in our theory coincides with the restriction of the usual universal covering map sl2 r sl2 r to the inverse image of sl2 z

the positive and the negative inertia index of line graphs of trees abstract the positive resp negative inertia index of g denoted by p g resp n g is defined to be the number of positive resp negative eigenvalues of its adjacency matrix the nullity of g denoted by g is defined to be the multiplicity of the eigenvalue zero in the adjacency spectrum of g in 2001 gutman and sciriha 11 proved that the nullity of the line graph of a tree is either 0 or 1 in this paper we consider the positive and the negative inertia index for the line graph l t of a tree t showing that e t 1 2 p l t e t 1 where e t denotes the number of internal non pendant edges contained in t the extremal trees for which p l t attains the upper bound and the lower bound are respectively characterized it is shown that l t is nonsingular if p l t attains the upper bound and l t is singular if p l t attains the lower bound

fibered quadratic hopf algebras related to schubert calculus abstract we introduce and study certain quadratic hopf algebras related to schubert calculus of the flag manifold

nichols woronowicz algebra model for schubert calculus on coxeter groups abstract we realise the cohomology ring of a flag manifold more generally the coinvariant algebra of an arbitrary finite coxeter group w as a commutative subalgebra of a certain nichols woronowicz algebra in the yetter drinfeld category over w this gives a braided hopf algebra version of the corresponding schubert calculus the nilcoxeter algebra and its action on the coinvariant algebra by divided difference operators are also realised in the nichols woronowicz algebra we discuss the relationship between fomin kirillov quadratic algebras kirillov maeno bracket algebras and our construction